Spatial Economics for Granular Settings*
Jonathan I. Dingel
Chicago Booth, NBER, and CEPR

Felix Tintelnot
University of Chicago, NBER, and CEPR

20 September 2023

Abstract
We examine the application of quantitative spatial models to the growing body of fine spatial
data used to study local economic outcomes. In granular settings in which people choose from alarge set of potential residence-workplace pairs, observed outcomes in part reflect idiosyncratic
choices. Using both Monte Carlo simulations and event studies of neighborhood employment
booms, we demonstrate that calibration procedures that equate observed shares and modeled
probabilities perform very poorly in these high-dimensional settings. Parsimonious specificationsof spatial linkages deliver better counterfactual predictions. To quantify the uncertainty aboutcounterfactual outcomes induced by the idiosyncratic component of individuals’ decisions, we
introduce a quantitative spatial model with a finite number of individuals. Applying thismodel to Amazon’s proposed second headquarters in New York City reveals that its predictedconsequences for most neighborhoods vary substantially across realizations of the individual
idiosyncrasies.

Keywords: commuting, exact hat algebra, overfitting, quantitative spatial economics
JEL codes: C25, F16, R1, R13, R23*

We thank four anonymous referees, Rodrigo Adão, Gabriel Ahlfeldt, Treb Allen, Stephane Bonhomme, Kirill
Borusyak, Victor Couture, Teresa Fort, Lars Hansen, Elhanan Helpman, Thomas Holmes, John Huizinga, Erik Hurst,
Kyle Mangum, Yuhei Miyauchi, Eduardo Morales, Stephen Redding, Jordan Rosenthal-Kay, Esteban Rossi-Hansberg,
Chris Severen, Sebastian Sotelo, Daniel Sturm, Oren Ziv, and numerous conference and seminar participants forhelpful feedback. We are grateful to Junbiao Chen, Levi Crews, Reigner Kane, Xianglong Kong, Daniil Iurchenko,
John Ruf, Ye Sun, Linghui Wu, Shijian Yang, and Mingjie Zhu for excellent research assistance. Dingel thanks
the James S. Kemper Foundation Faculty Research Fund at the University of Chicago Booth School of Business.
This work was completed in part with resources provided by the University of Chicago Research Computing Center.
We gratefully acknowledge support from the National Science Foundation under grant number 2018609. jdingel@chicagobooth.edu and tintelnot@uchicago.edu

1Introduction

Economists increasingly use quantitative spatial models to evaluate urban policies such as infrastructure investments and land-use planning. The growing availability of economic data observed atfine spatial scales offers tremendous potential for new insights. However, to inform policymakers,
these models need to produce credible counterfactual predictions when applied to granular settings.
Quantitative spatial models feature links between pairs of locations that shape the geography of
economic outcomes (Redding and Rossi-Hansberg, 2017; Proost and Thisse, 2019). In many of thesemodels, individuals choose a residence and a workplace based on the characteristics of locations,the links between them, and an idiosyncratic component. A setting with granular geographic units
has many locations and therefore a great number of links between pairs of locations. Two concerns
arise as the number of spatial links becomes large relative to the number of decision makers. First,
if one uses a high-dimensional parameterization of spatial linkages, such as one parameter per pairof locations, the modeler runs the risk of fitting the model to the idiosyncratic components ofdecisions rather than capturing spatial linkages. Second, counterfactual changes in local economic
outcomes are uncertain because they depend on the idiosyncratic preferences of a finite number ofindividuals.
In this paper, we examine the application of quantitative spatial models to granular settings.
In doing so, we make three main contributions. First, we show that the calibration procedure mostoften used to compute counterfactual outcomes, which perfectly matches the observed baseline
outcomes, does poorly in predicting changes in outcomes in granular settings. We show how to use
simulations that require only the inputs typically used in this procedure to diagnose the problem.
We confirm the poor predictive performance using event studies of neighborhood employment
booms in New York City. Second, we demonstrate that both a parsimonious parameterization
using only a transit-time covariate and parameterizations using matrix approximations to capture unobserved spatial linkages deliver better counterfactual predictions in this high-dimensionalsetting. Third, we develop a spatial model with a finite number of individuals that we use toquantify the variability of counterfactual outcomes induced by individual idiosyncrasies. Applying
this model to Amazon’s proposed second headquarters in New York City shows that the predicted
counterfactual changes for most census tracts vary considerably with the idiosyncratic componentof preferences.
Section 2 reviews the computation of counterfactual outcomes in quantitative spatial models. In the conventional model with a continuum of individuals, counterfactual changes can becharacterized by a system of equations containing counterfactual endogenous outcomes relative tobaseline endogenous outcomes, counterfactual exogenous parameters relative to baseline exogenous
parameters, constant elasticities, and baseline equilibrium shares (a formulation known as “exact
hat algebra”). Researchers have used two approaches that differ greatly in their parameterization of
the spatial links between residences and workplaces that determine the baseline equilibrium shares.
The covariates-based approach makes the cost of commuting a function of pair-specific covariates,
such as transit time. The calibrated-shares procedure calibrates a cost parameter for each pair of1

locations to match the observed commuting shares. The contrast in baseline equilibrium sharesproduces contrasting predictions about counterfactual outcomes.
Calibrating pair-specific commuting costs to match observed commuting shares offers advantages and disadvantages. The number of commuters between a pair of locations in a finite sample
depends on three components: spatial links correlated with included covariates, orthogonal spatial
links, and individual-specific idiosyncrasies. The potential advantage of calibrating pair-specificcosts to match observed shares is that this can capture spatial links not predicted by covariates
like transit times, such as many Columbia University employees living in nearby university-owned
residences. The potential disadvantage is that parameterizing the model’s spatial links by matching
all observed variation means overfitting the model: the calibrated parameters conflate the spatial
links common to all individuals and individual-specific idiosyncrasies in the observed data.
Section 3 examines the application of quantitative spatial models to granular empirical settings.
The commuting matrices used in urban economics often have so many more pairs of locations thancommuters that many of the cells must contain none. Small flows constitute a substantial shareof total commuting: more than 40% of New York City commuters have a residence-workplacepairing populated by five or fewer individuals. Such settings can pose small-sample problems forthe calibrated-shares procedure, which equates observed shares and underlying probabilities.
We use Monte Carlo simulations to examine the finite-sample behavior of the calibrated-sharesprocedure. The data-generating process is the parsimonious covariates-based parameterizationof New York City in 2010, which uses the same covariate employed to estimate the commuting
elasticity for the calibrated-shares procedure. These simulations show that small-sample bias canseriously impair the calibrated-shares procedure when the model is otherwise correctly specified.
While the procedure works well as the number of individuals becomes arbitrarily large, it producesmuch worse predictions in simulations using the actual number of individuals in New York City.
Next, we examine how well the incumbent approaches predict changes in commuting flows using
event studies. Commuting flows are the key spatial linkage between neighborhoods in many modelsof cities. We study events in which there is a large, discrete increase in employment in a single census
tract, often arising from the arrival or expansion of a large employer. We increase the productivity
parameters for these tracts to match their increases in total employment and compare the predictedchanges in bilateral commuting flows to those observed in the data. When examining 83 tract-level
employment booms in New York City, we find that the covariates-based approach—–using transit
times as the only bilateral covariate—–predicts the observed changes in commuting flows better
than the calibrated-shares procedure in 80 of these events. While predictive performance varies
across events, regressing the observed change in commuters on the covariates-based approach’s
prediction typically yields an estimated slope of one, whereas the median estimated slope for thecalibrated-shares procedure is negative. In other words, the counterfactual predictions from thecalibrated-shares procedure are negatively correlated with the observed changes in commuting in
more than half of the 83 events.
We explore additional approaches that offer the advantage of a more flexible specification while2

guarding against the disadvantage of overfitting. One specification augments the covariates-based
approach to capture unobserved spatial linkages that have an interactive fixed effects structure(Bai, 2009; Chen, Fernández-Val, and Weidner, 2021). This more flexible specification modestly
improves predictive performance. An alternative approach computes baseline shares used in theexact hat algebra from rank-restricted matrix approximations without using covariates.

This

simple, computationally cheap transformation delivers substantially better predictions than using
the observed shares, almost matching the performance of the covariates-based approach.1 Both theinteractive-fixed-effects specification and the data-smoothing procedure allow one to incorporate
unobserved components when applying continuum models to granular settings.
In Section 4, we depart from treating the data as a finite sample from the continuum model andintroduce a quantitative spatial model with a finite number of individuals. We use this model toquantify the variability of counterfactual outcomes induced by individual idiosyncrasies. Where thecontinuum model would yield a single equilibrium allocation, our framework produces a distributionof economic outcomes. Notably, outcomes with strictly positive probabilities may not arise in aparticular realization of individual idiosyncrasies.
Our model retains the general-equilibrium approach employed in prior quantitative spatial models while introducing a finite number of individuals. The key modeling challenge is that individual
decisions affect equilibrium wages and rents, but it is computationally infeasible to enumerate all
the possible labor allocations and resulting prices. To overcome this combinatorial challenge, we
assume that individuals make choices based on wage and rent beliefs that are the equilibrium
prices from the continuum model with the same economic primitives. This simplification of agents’beliefs to solve an otherwise intractable decision problem is similar to assumptions used in industrialorganization (Weintraub, Benkard, and Van Roy, 2008) and macroeconomics (Krusell and Smith,1998). In our application to New York City, decisions based on these beliefs are optimal for the
overwhelming majority of individuals. At realized wages and rents, which differ from the beliefs
about these prices, 96% of individuals would not change their residence-workplace choice if given
the opportunity.
Our model with finite individuals can be estimated using the same data used to estimate thecontinuum model. When individuals’ price beliefs are the continuum-case equilibrium prices, theestimated parameters of this model are identical to those of the covariates-based continuum model.
Given the same parameter values, the mean equilibrium labor allocation is equal to the equilibrium
labor allocation of the continuum model. By contrast, equilibrium prices solve a system of non-linear
equations, so their mean values are not necessarily equal to the continuum-case prices. Crucially,
the model with a finite number of individuals characterizes the dispersion in quantities and pricesthat arises from idiosyncrasies.
1

Smoothing the observed shares using rank-restricted matrix approximations before employing them in exact
hat algebra is reminiscent of, yet distinct from, practices such as applying the Hodrick and Prescott (1997) filter
before calibrating macroeconomic models, grouping firms into classes using k-means clustering before estimating amodel of workers and firms (Bonhomme, Lamadon, and Manresa, 2019), and smoothing observations before inverting
conditional choice probabilities in dynamic models (e.g., Kalouptsidi, 2014; Hsiao, 2022).

3

In Section 5, we contrast methods for evaluating the local economic effects of Amazon’s proposed—
but later abandoned—HQ2 in New York City. The changes in the residents and land rents predictedby the covariates-based approach differ considerably from the predictions of the calibrated-sharesprocedure. We find considerable uncertainty generated by individual idiosyncrasies for outcomesat the neighborhood level and even for larger groupings of tracts. The variation in counterfactual
outcomes due to individual idiosyncrasies in the model with a finite number of individuals is larger
than that due to parameter uncertainty in the covariates-based continuum model. The sizable
uncertainty regarding the local effects of a headline-grabbing potential employment boom suggeststhat counterfactual predictions would be highly uncertain in many granular settings.
Our paper introduces a quantitative spatial model suitable for applications using small spatial
units, such as census tracts. Social scientists study neighborhoods to understand housing markets,
intergenerational mobility, racial segregation, and many other phenomena. Spatially precise satellite imagery (Donaldson and Storeygard, 2016) and phone movement data (Couture et al., 2022)
allow even finer investigations where there are few decision makers, as anticipated by Holmes andSieg (2015, p.106). Granular data are valuable in part because a growing body of evidence shows
highly localized agglomeration economies (Rosenthal and Strange, 2020).2 We provide tools for
applying quantitative spatial models to these granular settings.
Our event studies of neighborhood employment booms contribute to a small literature assessing
the predictive power of quantitative spatial models. These studies compare model-predicted changes
to observed outcomes using plausibly exogenous shocks: Ahlfeldt et al. (2015) use Berlin’s divisionand reunification, Monte, Redding, and Rossi-Hansberg (2018) use million-dollar-plant openings,
and Adão, Arkolakis, and Esposito (2022) use the “China shock” of Autor, Dorn, and Hanson(2013). Kreindler and Miyauchi (2023) use a gravity model and phone-derived commuting flows to
predict workplace wages in cities that lack conventional data coverage. More broadly, economists
have evaluated the predictions of quantitative trade models about changes in trade policy (Kehoe,
2005; Kehoe, Pujolàs, and Rossbach, 2017; Adão, Costinot, and Donaldson, 2023). We explore thepredictive power of quantitative spatial models in granular settings.
Finally, our framework relates to prior research about the roles of individual idiosyncrasies
in economic outcomes and parameter estimation. A growing literature examines the importanceof firm-specific shocks for aggregate fluctuations in macroeconomics (Gabaix, 2011; Carvalho andGrassi, 2019) and international economics (di Giovanni, Levchenko, and Mejean, 2014; Gaubert
and Itskhoki, 2021). Our paper joins a line of research that examines how individual idiosyncrasies
affect economists’ inferences about the economy. Ellison and Glaeser (1997) and Schoefer andZiv (2022) address the role of individual manufacturing plants when computing the geographic
concentration of industries and geographic variation in productivity, respectively. Mogstad et al.
(2020) raise concerns about inferring ranks of small geographic units. The finite numbers of firms
2

Arzaghi and Henderson (2008) estimate that productivity gains from interactions among advertising firms occur
primarily within 500 meters. Ahlfeldt et al. (2015) estimate production and residential externalities that decay bymore than half within two minutes of travel time. Rossi-Hansberg, Sarte, and Owens (2010) estimate that externalities
from non-market interactions decline by half approximately every 1,000 feet.

4and shipments have also been studied as one explanation for zeros in international trade (Eaton,
Kortum, and Sotelo, 2013; Armenter and Koren, 2014), and the finite number of consumers is one
explanation for zeros in product-level sales data (Quan and Williams, 2018; Hortaçsu et al., 2023;
Gandhi, Lu, and Shi, 2023). Similar to our framework, Panigrahi (2022) models a finite set offirms that form links based on beliefs from a limiting economy with a continuum of firms. Our
paper makes two contributions to these various strands of the literature. First, we demonstrate
the importance of avoiding overfitting when using continuum models to predict counterfactual
outcomes. Second, we introduce a spatial model with a finite number of individuals that quantifies
the uncertainty about counterfactual outcomes induced by individual idiosyncrasies.

2

Computing counterfactual outcomes in continuum models

This section describes the computation of counterfactual outcomes in quantitative spatial models.
We use a baseline model featuring a continuum of individuals making residential and workplacelocation choices, as in Ahlfeldt et al. (2015) and Redding and Rossi-Hansberg (2017).

2.1

Primitives

We consider a closed economy populated by a measure of L individuals who each supply one unitof labor. Discrete locations are indexed by k or n. Each location has a fixed quantity of land,
Tk , owned by immobile landlords who consume only goods.3 Each location has the technology to
produce a differentiated good (i.e., the Armington assumption). These goods are freely traded.
Individuals have Cobb-Douglas preferences over goods and land, devoting α of their expenditure to the latter. They have constant elasticity of substitution (CES) preferences over the setof differentiated goods, with elasticity of substitution σ > 1. Individuals (indexed by i) haveidiosyncratic preferences for pairs of residential and workplace locations, such that i’s indirect
utility from residing in k and working in n isi
Ukn
= ϵ ln

|



wn



rkα P 1−α δkn
{z
}

i
+νkn
,

(1)

≡Ukn

where wn denotes the wage in location n, rk denotes the land rent in k, P denotes the common CES
price index for goods, δkn denotes the commuting cost between k and n, Ukn denotes the mean
i is the idiosyncratic preference of individual i for residing in k andutility of choice kn, and νknworking in n. The commuting elasticity ϵ governs the importance of mean utility relative to thei is drawn from the standard Gumbel distribution.4
idiosyncratic preference. We assume that νkn

Commuting costs have two components: time (δ̄kn ) and disutility (λkn ), so δkn = δ̄kn × λkn .
3This simplifying assumption follows Monte, Redding, and Rossi-Hansberg (2018).
The cumulative distribution function of the Gumbel (type-1 extreme
value) distribution with a location parameteriiof zero and scale parameter of one is F (νkn
) = exp − exp −νkn.
45

Time spent commuting is not spent working, so individuals residing in k and working in n earn
only wn /δ̄kn because they only spend 1/δ̄kn of their time working.
Production of each location’s differentiated good is linear in labor. Goods output in locationn is qn = An Ln , where An is that location’s productivity and Ln is the quantity of labor supplied
by workers working in n. The goods market is perfectly competitive, so the price of location n’shP
i1/(1−σ)1−σ.
output is wn /An for all consumers. Thus, the CES price index is P =
(w
/A
)
n
n
n

2.2

Equilibrium

Let ℓkn denote the measure of people residing in k and working in n. Using the Gumbel distribution
assumption, one can integrate over the idiosyncratic preferences to obtain the fraction of people
residing in k and working in n:wnϵ (rkα δkn )−ϵ
ℓkn
=P
−ϵ .
ϵ
α
L
′ ′ w ′ r ′ δk ′ n′
k ,n

n

(2)

k

Goods market clearing equates each location’s output to the quantity demanded. Labor suppliedPP
in location n is Ln =k ℓkn /δ̄kn . Each individualk ℓkn /δ̄kn , and thus output there is An
devotes 1 − α of their expenditure to differentiated goods and α of their expenditure to land, whileimmobile landlords spend all of their income on differentiated goods, such that total expenditureP
on differentiated goods equals aggregate income. Aggregate income is Y ≡k,n ykn , whereykn ≡ wn ℓkn /δ̄kn is labor income earned in workplace n by workers residing in k. The demand foreach differentiated good stemming from CES preferences means that equating quantity suppliedand quantity demanded requires
An

X ℓkn
k

δ̄kn

=

(wn /An )−σ
Y
P 1−σ

∀n.

(3)

Note that goods market clearing implies labor market clearing.
Similarly, land market clearing equates the fixed land endowment Tk to the quantity demanded
by individuals, who devote a constant fraction α of their expenditure to land:
Tk =

α X
ykn
rk n

∀k.

(4)

Equilibrium is a set of wages {wn }, rents {rk },and labor{ℓkn } such that equations (2),  allocation

1+ϵ
σ+ϵ

(3), and (4) hold. Appendix C.1 shows that if

αϵ
1+αϵ

≤ 12 , this equilibrium exists and is

unique by application of Theorem 1 of Allen, Arkolakis, and Li (2023).

2.3

Procedures for counterfactual predictions

In this model, counterfactual changes can be characterized by a system of equations containing
counterfactual endogenous outcomes relative to baseline endogenous outcomes, counterfactual ex6

ogenous parameters relative to baseline exogenous parameters, constant elasticities, and baseline
equilibrium shares. This is known as “exact hat algebra” in the trade literature (Dekle, Eaton, andKortum, 2008; Costinot and Rodrı́guez-Clare, 2014). Denote the counterfactual value of a variable′

x by x′ and denote the counterfactual-baseline ratio of a variable x > 0 by x̂ ≡ xx (and imposex̂ = 0 if x = 0). As described in Appendix C.2, tedious manipulation of equations (2), (3), and (4)yields the following system of equations characterizing the counterfactual equilibrium:
! 1
1−σ
1
ykn
P
ŵn = Ânŷkn
P̂ Ŷ σ−1′k ′ yk nkXyknr̂k = T̂k−1
ŷkn P′n ykn′n
−εŵnε r̂kα δ̄ˆkn λ̂kn
ℓ̂kn = P−εℓ k ′ n′
ˆ
ε
α δ̄
′
′
′
′
ŵ
r̂
λ̂
k′ ,n′ n′k′ k n k n
L
X

(5)
(6)

if ℓkn > 0.

(7)This system defines the relative endogenous outcomes ŵn , r̂k , and ℓ̂kn in terms of relative exogenous
parameters Â , T̂ , δ̄ˆ , and λ̂ , elasticities σ, α, and ϵ, and baseline shares ℓkn and ykn .
n

k

kn

kn

L

Y

There are many ways to estimate or calibrate the model’s parameters to deliver the baseline
shares. At one end of the spectrum, one can parsimoniously parameterize the time and disutility
components of commuting costs as functions of only observed characteristics.5 To compute counkn
terfactual equilibria, one would plug the fitted model’s values of ℓL
and yYkn into equations (5)–(7).

At the other end of the spectrum, one can calibrate millions of commuting-cost parameters sothat the model’s baseline equilibrium matches the observed shares. In this case, one would plugkn
and yYkn into equations (5)–(7) to compute counterfactual outcomes. The latterthe observed ℓL

calibrated-shares procedure has been widely used in spatial economics recently, often in granularsettings with as many residence-workplace pairs as decision makers.6 Between these two extremes
lie a host of strategies to determine the baseline equilibrium shares, some of which we discuss indetail below.
We emphasize the distinction between using the comparative statics defined by equations (5)–(7)to compute counterfactual outcomes and fitting the model’s parameters. Because equations (5)–(7)show that computing counterfactual outcomes only requires knowing the model’s parameters upkn
to the point where the model delivers the shares ℓL
and yYkn , others have used the phrase “exact

hat algebra” to refer to both rewriting the equations in hats and calibrating combinations of model5

A prominent example of this covariates-based approach is Ahlfeldt et al. (2015). In this approach, equation (2)is generically not satisfied by the observed values of ℓkn and the parameterization of δkn . One justification for themodel not fitting the observed data would be that the observed set of commuters is a sample from the population, as
in Ahlfeldt et al. (2015). Similarly, when using a sample describing domestic trade transactions, Holmes and Stevens(2014) employ a covariates-based approach for trade costs.
6
The calibrated-shares procedure has decades-deep roots in computable general equilibrium models of international
trade (Rutherford, 1995). Recent studies in spatial economics employing this technique include Allen, Arkolakis, andLi (2016); Perez-Cervantes (2016); Waddell and Sarte (2016); Monte, Redding, and Rossi-Hansberg (2018); Heblich,
Redding, and Sturm (2020), Krebs and Pflüger (2019); Severen (2021); and Owens, Rossi-Hansberg, and Sarte (2020).

7

parameters to rationalize observed shares. In fact, the system of equations defines counterfactual
outcomes regardless of how one estimates or calibrates the parameters of the baseline equilibrium.
The key question is how to fit the model’s parameters to data.

3

Counterfactual analysis in granular settings

We assess the predictive performance of quantitative spatial models in granular empirical settings.
In Section 3.1, we introduce data on tract-to-tract commuting flows in New York City. Thisempirical setting, like many studied in previous research, has a large number of location pairs
relative to the number of decision makers. After estimating the commuting elasticity and otherparameters in Section 3.2, we present a Monte Carlo exercise in Section 3.3 that reveals that thecalibrated-shares procedure performs poorly in these settings because it overfits. We further assess
predictive performance using event studies of neighborhood employment booms in New York City
in Section 3.4. There is a stark contrast between the two incumbent approaches in predicting
the change in the number of people from each residential census tract who work in the booming
tract. While the approach using commuting costs based on transit times predicts quite well, thecalibrated-shares procedure’s predicted changes are in fact negatively correlated with the observedchanges in the majority of events. We also propose and assess alternative specifications that cancapture unobserved components of spatial linkages while guarding against overfitting.

3.1

Commuting flows in granular settings

Urban economists often study empirical settings in which the fine spatial resolution of the geographic units employed makes the number of decision makers modest relative to the number ofpairs of locations. Owens, Rossi-Hansberg, and Sarte (2020) study 1.3 million people commuting
between 1.3 million pairs of tracts in the Detroit urban area. Severen (2021) studies the Los
Angeles metropolitan area, which has 6.7 million commuters and more than 6 million pairs oftracts. Tsivanidis (2023) models the behavior of Bogotá’s 8 million residents commuting between
almost 7 million pairs of tracts. Zárate (2023) examines Mexico City, which has 9 million people
commuting between 13 million pairs of tracts. At an even finer spatial resolution, Ahlfeldt et al.
(2015) model the behavior of about 3 million Berliners choosing among about 254 million pairsof city blocks. In this section, we examine tract-to-tract commuting flows in New York City.
Appendix D.1 reports comparable summary statistics for Detroit and Minneapolis-St. Paul.
We use data on commuting between census tracts in New York City taken from the Longitudinal
Employer-Household Dynamics, Origin-Destination Employment Statistics (LODES).7 Tracts aredefined by the US Census Bureau to be relatively uniform in population size, such that the
typical tract has 4,000 residents. One implication of granularity is that some data providers
7
Davis et al. (2019) use the 2010 wave to compute the joint distribution of residences and workplaces for New
York City. Owens, Rossi-Hansberg, and Sarte (2020) use the 2014 wave of these data for the Detroit urban area toestimate a gravity model of commuting.

8perturb observations to protect confidentiality. For example, the published LODES tract-level
workplace employment counts are infused with noise. Moreover, the published LODES commuting
matrices report synthetically generated residence-workplace pairs.8 These confidentiality-protecting
procedures are both a symptom of the granular setting and an additional cause for caution wheninterpreting the value of any individual observation.
Small flows constitute a large share of tract-to-tract commuting in New York City. New York
City has about 2.5 million resident-employees and 4.6 million tract pairs. Thus, its commuting
matrix is almost necessarily sparse, as the average cell contains about one-half of a commuter. In
fact, 85% of tract pairs have zero commuters between them, so the average positive cell contains
about three people. As shown in Figure 1, more than half of the tract pairs with a positive numberof commuters have only one commuter. Crucially, Figure 1 shows that small flows constitute a
substantial share of total commuting: 41.1% of New York City commuters have five or fewer commuters in their cell of the commuting matrix. Furthermore, the commuting flows are impersistentover time and exhibit asymmetric zeros (see Appendices D.3 and D.4). Introducing multiple worker
types would exacerbate these patterns by reducing the number of decision makers per choice foreach type.9
Even when studying larger geographic units, there may be few decision makers per pair oflocations, particularly when the data describe only a sample of respondents. For example, whilesome US counties have millions of residents, Appendix D.2 documents that more than half ofthe positive commuting counts between pairs of counties represent the behavior of five or fewer
survey respondents. More broadly, small-sample bias may complicate the analysis of other spatial
phenomena, such as migration. Appendix D.5 shows that state-to-state migration flows reported
in American Community Survey (ACS) data also represent a small number of respondents, arefrequently zero, and are impersistent across time (see Foschi et al. (2023) for related findings).

3.2

Estimation

We estimate the continuum model using the tract-to-tract commuting data for New York Cityin 2010. Following the literature, we set α = 0.24 and σ = 4 prior to estimating the remaining
parameters.10
We assume that the time component of commuting costs, δ̄kn , is observed, while the disutility
component, λkn , is unobserved. We construct the observed part of the commuting costs by assuming
8

Graham, Kutzbach, and McKenzie (2014): “For each job in a workplace cell, LODES draws from a Dirichlet
multinomial posterior distribution of possible residential locations. . . . The prior adds uncertainty, so that evencommutes with few or no observed flows may appear to have a job. Conversely, even when there are commuters froman origin in the likelihood, that residence may not be drawn and thus would not appear in LODES.”
9
For example, individuals in each of the four educational attainment categories reported in the LODES data livein all but one or two residential tracts and are employed in more than 99% of workplace tracts, so each type would
have a choice set with more than 4.5 million elements.
10
Davis and Ortalo-Magne (2011) suggest α = 0.24. Monte, Redding, and Rossi-Hansberg (2018) use σ = 4 when
examining goods trade between US counties, citing estimates for international trade between countries. Absent an
estimate of tract-level labor demand elasticities, we also report results obtained when using σ = 1.1 and σ = ∞.
None of the main conclusions of this paper are sensitive to the particular value for σ.

9

0

0

.2

.2

.4

.4

.6

.6

.8

.8

1

1

Figure 1: Number of commuters between pairs of tracts in New York City

0

10

20

30

40

Number of individuals
Fraction of tract pairs

Cumulative share of commuters

Notes: This figure describes tract pairs in New York City by the number of individuals who reside in the origin
tract and work in the destination tract in 2010 LODES data. The sample is restricted to the 15% of tract pairs thathave a strictly positive number of commuters. The blue histogram depicts the fraction of tract pairs reporting eachnumber of commuters, for each value from 1 to 39 and values of 40 or greater. The black increasing step function
depicts the cumulative share of commuters by the number of commuters between the pair of tracts. New York Cityhas 2,160 residential census tracts.

that each worker has H hours that are spent either working or commuting. We compute δ̄kn =H
11
H−tkn −tnk , where H = 9 hours and tkn is the transit time from k to n according to Google Maps.

Assumptions about the unobserved disutility component, λkn , vary widely across methods, as
described below.
One way to estimate the continuum model using the covariates-based approach is to interpret
observed data on residence-workplace choices as a finite sample from the continuum model. The
covariates-based specification assumes there is no unobserved disutility component of commuting
costs, λkn = 1 ∀k, n. In that case, given observed values of commuting costs δkn = δ̄kn , theremaining parameters can be estimated by maximum likelihood. Interpreting the right side ofequation (2) as a probability and denoting the number of individuals who chose the kn pair by ℓkn ,the log likelihood function is
#
"
−ϵ

 X
wnϵ rkα δ̄kn
i
i
′ ′
.
L≡
ℓkn ln P(Ukn > Uk′ n′ ∀k n ̸= kn) =ℓkn ln Pϵ r α δ̄ ′ ′ −ϵ
w
′
′
′
′
k
n
k ,n
n
k
k,n
k,n
X

(8)

This is the canonical conditional-logit likelihood of McFadden (1974) applied to location choices,
11

Given H hours, 1/δ̄kn is the share of that time spent working if the individual resides in k and works in n. The
semi-elasticity of commuting flows with respect to transit time tkn = tkn + tnk is −ϵ/ (H − tkn ). Using H = 8 or
H = 10 yields very little change in the model fit relative to the H = 9 results reported in Table 1. For New York
City, we use Google Maps public-transit times collected by Davis et al. (2019). We impute missing observations for
fewer than 4.1% of tract pairs in New York City by predicting transit times using physical distance.

10

as in McFadden (1978). The k- and n-specific terms are captured by residence and workplace fixed
effects, respectively. The maximization of this likelihood function is numerically equivalent to a
Poisson pseudo maximum likelihood (PPML) estimator that is available for a variety of software
packages (Guimarães, Figueirdo, and Woodward, 2003).12
The calibrated-shares procedure, which calibrates λkn , also requires an estimate of the commuting elasticity ϵ. One might make two distinct assumptions about the unobserved component

λkn . If E λ−ϵkn |rk , wn , δ̄kn = 1, the constant elasticity function (2) can be estimated using the
PPML estimator (Silva and Tenreyro, 2006). This yields the same estimate of the commuting
elasticity ϵ as maximizing equation (8). If one takes the logarithm of each side of equation (2),
assumes E ln λkn |rk , wn , δ̄kn = 0, and restricts the estimation sample to observations for whichℓkn is strictly positive, then the commuting elasticity can be estimated by ordinary least squares(OLS). As emphasized by Silva and Tenreyro (2006), these two estimators can yield very different
parameter estimates.
We apply both the PPML and OLS estimators to New York City in 2010. Our estimateof the commuting cost elasticity is presented in column 1 of Table 1. The estimate of ϵ ≈ 8,
which is comparable to the value of 6.8 estimated by Ahlfeldt et al. (2015) for commuting within
Berlin, implies that idiosyncratic preferences are modestly dispersed. Estimating the commuting
elasticity using OLS yields a much lower elasticity, largely because of the well-understood selection
bias associated with omitting four-fifths of observations from the estimation sample.13 In what
follows, we use the ϵ estimate reported in column 1 for both the covariates-based approach and thecalibrated-shares procedure.
Table 1: Commuting elasticity estimates
PPML/MLE

OLS

Commuting cost

-7.986(0.307)-2.307(0.0516)

Model fit (R2 or pseudo-R2 )Location pairs
Commuters

0.662
4,628,878
2,488,905

0.561
690,6732,488,905

Notes: All specifications include residence fixed effects and workplace fixed effects. The “PPML/MLE” column
presents the results from maximum likelihood estimation of equation (8). The “OLS” column presents the results of
estimating the log version of equation (2) by ordinary least squares, omitting observations in which ℓkn = 0. The
model-fit statistic is the pseudo-R2 for MLE and R2 for OLS. We report the PPML standard errors (clustered by k
and by n), which are larger than the logit MLE standard errors associated with maximizing equation (8).

kn
and yYkn toGiven the elasticities σ and ϵ, one only needs the model-implied baseline shares ℓL
12

See Sotelo (2019) for a discussion of the relationship between the multinomial and PPML estimators in thecontext of gravity models. In practice, we use the Stata package ppmlhdfe of Correia, Guimarães, and Zylkin (2020).
13
Applying the PPML estimator to the OLS sample of strictly positive flows yields an estimate of ϵ = 4.4. Thus
about two-thirds of the discrepancy between the OLS and MLE estimates reflects the omission of the zero flows. This
selection bias also affects the fixed effects (see Appendix D.6).

11

compute counterfactual outcomes using equations (5)–(7). The calibrated-shares procedure useskn
the observed shares ℓL
and yYkn .14 For the covariates-based approach, the model parameters {Tk }and {An } can be computed from the estimated fixed effects and the market-clearing conditions.
In particular, the residence fixed effect is proportional to rk−αϵ , and the workplace fixed effect is
proportional to wnϵ . Given α, σ, ϵ, {δkn }, {rk }, and {wn }, equations (2), (3), and (4) can be solvedkn
to obtain {Tk } and {An }, and one can compute the model-implied baseline shares ℓL
and yYkn .

The calibrated-shares procedure is more data demanding than our covariates-based parameterization. Let I denote the number of individuals and N the number of locations in the economy. The
covariates-based approach estimates two parameters (An and Tn ) for each of the N locations,whereas the calibrated-shares procedure introduces an unobserved component for each pair oflocations and therefore calibrates more than N 2 parameters to rationalize the observed commuting
flows. For the case of New York City, I/N exceeds one thousand, while I/N 2 is about one-half. In
this setting, we have limited concerns about overfitting the residence and workplace fixed effects
and a much greater concern about overfitting the pair-specific commuting costs.

3.3

Monte Carlo: The calibrated-shares procedure in a granular setting

We use Monte Carlo simulations to assess how well the calibrated-shares procedure for computing
counterfactual outcomes performs in a granular setting. In anticipation of examining New York City
neighborhood employment booms in the next section, the data-generating process is the estimatedcovariates-based model of New York City in 2010. We imagine the researcher observes equilibrium
wages and a finite-sample labor allocation drawn from a multinomial distribution defined byprobabilities given by the right side of equation (2). The researcher then uses the calibrated-sharesprocedure to predict the consequences of a “counterfactual” productivity increase in one workplace
tract. The “true” consequences of this productivity change are given by the counterfactual labor
allocation and prices in the covariates-based continuum model. If the observed sample had an
infinite number of individuals, the realized labor allocation would equal the true shares, and thecalibrated-shares procedure would perfectly predict the changes caused by the productivity increase.
In short, our simulations examine the finite-sample behavior of the calibrated-shares procedureusing the actual number of individuals in New York City (I ≈ 2.5 million).
We contrast the true and predicted changes in the number of commuters from each residential
origin to the workplace with increased productivity. As detailed in Appendix A.1, the modeled
productivity increases for both the covariates-based model and the calibrated-shares procedure areengineered so that the total employment increases in the “treated” tracts are perfectly predicted.
We assess the predictive power by contrasting true and predicted changes in bilateral commuting
flows and rents. We do so by computing the mean squared error (MSE) and by regressing the14
The potential advantage of the calibrated-shares procedure is capturing links that covariates miss. The largest
commuting flow in the 2010 LODES data for New York City is 827 commuters who reside between 110th and 114thStreets in Morningside Heights and work at adjacent Columbia University. The model that uses transit times predicts
only 70 of the 827 observed commuters, failing to capture the effect of the university’s dual role as employer andlandlord.

12

Figure 2: Calibrated-shares procedure overfits in Monte Carlo simulations
B. Ratio of models’ prediction errors
.5
.4
.3

0

0

.1

50

10

.2

Fraction20

Density
100
150

30

200

40

250

A. Regression of true on predicted changes

.2

.4

.6

.8

1
0

0

Covariates−based: slope
Calibrated−shares: slope

Covariates−based: intercept
Calibrated−shares: intercept

0

.002
.004
.006
Covariates−based MSE / Calibrated−shares MSE.008

Notes: This figure depicts the regression coefficients and mean squared errors from 100 simulations in which
I = 2, 488, 905, the data-generating process is the estimated covariates-based model of New York City in 2010,
and the counterfactual change is a 9% increase in productivity in one workplace tract. See Appendix A.1
for a detailed description of the simulation procedure. “Covariates-based” means estimating the covariatesbased parameterization described in Section 3.2 using the simulated data. “Calibrated-shares” means using theobserved (simulated) shares in the exact hat algebra defined by equations (5)–(7). In Panel A, the densities forthe covariates-based model are on the left vertical axis and the densities for the calibrated-shares procedure areon the right vertical axis.

observed changes on the predicted changes. With an unbiased forecast, the regression of observedon predicted changes would yield a slope coefficient of one and an intercept of zero. Of course,
the estimated coefficients in a single simulation will also reflect finite-sample noise. Simulating thisprocess 100 times yields a distribution of regression coefficients and MSEs, as shown in Figure 2.
Before assessing the calibrated-shares procedure’s predictive performance, we note that thecovariates-based model is well-behaved in the finite sample. The covariates-based model is correctly
specified because λkn = 1 ∀k, n in the data-generating process. It delivers accurate predictions
when applied to a finite sample: its slope and intercept coefficients are centered on one and zero,
respectively. It also accurately predicts the changes in rents (see Appendix Figure A.1).
The calibrated-shares procedure makes worse predictions. While it perfectly fits the commuting
shares from the finite sample, it produces less accurate predictions about the changes in commutersand rents caused by the productivity increase. This illustrates an overfitting problem: a more
flexible parameterization improves in-sample fit but worsens out-of-sample performance (Hastie,
Tibshirani, and Friedman 2009, p. 221; Belloni, Chernozhukov, and Hansen 2014, p. 30). Regressing the true change in commuters on the change predicted by the calibrated-shares procedure yields
a slope of only 0.78 in the median simulation in Figure 2. Its forecast error is larger than that of thecovariates-based model, which has a mean squared error that is 1% of the calibrated-shares forecast
error in the median simulation. Its predictions about rents are similarly inferior: the regression
slopes for rents are centered on 0.20, and the covariates-based model’s mean squared error for rents13

is 0.005% of the calibrated-shares forecast error (see Appendix Figure A.1).
Table 2: Calibrated-shares procedure’s finite-sample performance
A. Regressand is continuum change in commuters

I

2.5

125

250

2560

Calibrated-shares: slope
0.782 0.876 0.948 0.974 0.986 0.995
Calibrated-shares: intercept 0.269 0.153 0.064 0.032 0.017 0.007
Calibrated-shares: MSE
0.225 0.113 0.045 0.023 0.011 0.005
B. Regressand is finite-sample change in commuters
I
2.5
5
12.5
25
50
125

0.997
0.004
0.002

1.000
0.000
0.000

250

2560

Calibrated-shares: slope
Calibrated-shares: intercept
Calibrated-shares: MSE

0.982
0.022
0.169

0.998
0.002
0.017

-0.4081.724
17.022

5

0.194
0.982
8.486

12.5

0.669
0.404
3.400

25

0.835
0.202
1.699

50

0.913
0.106
0.851

0.968
0.040
0.340

Notes: This table reports the average values of the slope coefficient, intercept coefficient, and mean squared
error from 100 simulations as we vary I, the number of individuals in the simulated economy. The column
titles denote the number of individuals in millions. The “2.5” million case corresponds to I = 2, 488, 905,
which is the number of individuals who reside and work in New York City in the 2010 LODES data. In PanelA, the regressand is the change in commuters with a continuum of individuals. This change does not varyacross simulations. In Panel B, the regressand is the simulation-specific change in commuters from a realization
drawn from the model using pre-shock parameter values to a realization drawn from the model using post-shock
parameter values.

Since the only element of the data-generating process at odds with the assumptions of thecalibrated-shares procedure is the finite number of individuals, these simulation results demonstratethat finite-sample bias can severely limit that procedure’s predictive power. The procedure suffersfrom an overfitting problem when the number of individuals is small relative to the number ofparameters. As shown in Panel A of Table 2, the calibrated-shares procedure performs better as
we increase the number of individuals. Only when the number of individuals is one to two orders ofmagnitude larger do the calibrated-shares procedure’s predictions become closely correlated withoutcomes. Its rent predictions become closely correlated with outcomes only when the number ofindividuals is two to three orders of magnitude larger (Appendix Table A.1).
In any empirical application, the researcher does not observe changes in outcomes for a continuum of individuals. The observed changes in outcomes come from finite-sample draws fromthe pre- and post-shock data-generating processes. We simulate this setting by drawing two labor
allocation realizations, the first from the estimated model of New York City in 2010 and thesecond from that model after the productivity increase in one workplace tract. Because the two
realizations are independent draws, commuter counts are mean reverting: absent a shock, thechanges are negatively correlated with the initial counts. This does not change the slope coefficientfor the covariates-based model’s predictions (see Appendix Figure A.2), but it severely degrades thepredictive performance of the calibrated-shares procedure. For the smallest number of individuals
in Panel B of Table 2, the calibrated-shares procedure’s predicted changes are negatively correlatedwith the simulated changes. As explained in Appendix C.8, this particular symptom of overfitting
14

is a case of Galton’s fallacy.15
In the simulations described thus far, the covariates-based model is correctly specified. The
calibrated-shares procedure’s potential advantage over the covariates-based specification is that it
can capture spatial links not predicted by covariates. In Appendix A.3, we conduct simulations inwhich the covariates-based model is misspecified because λkn ̸= 1 in the data-generating process.
Tables A.2 and A.3 show that, for a given number of individuals, the calibrated-shares procedure’s
performance improves relative to the covariates-based model as the share of variation in δknorthogonal to the included covariates increases. It achieves this by calibrating λkn for each pairof locations, which is highly data demanding. As the number of individuals grows, the overfitting
problem diminishes and the advantage of a more flexible parameterization is realized. For the actual
number of individuals in this empirical setting, the covariates-based specification outperforms thecalibrated-shares procedure even when there is substantial misspecification due to variation in theunobserved component of commuting costs. Given these findings, we have reason to expect thatthe calibrated-shares procedure will perform poorly in granular empirical settings.

3.4

Event studies: Predicting commuting responses to a local shock

We now examine how various approaches are able to predict changes in commuting flows usingneighborhood employment booms in New York City. As in the Monte Carlo simulations, weinvestigate changes in commuting patterns to workplace tracts that had large increases in employment. In particular, we conduct “event studies” in which we examine the commuting flows
to the 83 workplace tracts in New York City that had a two-year increase in total employment
from 2010 to 2012 of at least 400 employees and at least 12.5% from a 2010 level of at least 400
employees. We focus on local employment booms because these large changes are likely driven
by workplace-specific shocks, such as new office openings or an expansion by a large employer,rather than resident-workplace-specific shocks. For example, in 2011, Tiffany & Co. moved its
corporate headquarters to 260,000 square feet of office space at 200 Fifth Avenue. In late 2010,
Google acquired a building of nearly 3 million square feet at 111 Eighth Avenue. These locations
are two of the 83 workplace tracts we examine.16
To assess the predictive performance of a model parameterization, we use its baseline sharesfor New York City in 2010 and equations (5)–(7) to compute the increase in productivity required
to match the observed 2010–2012 change in employment for the “treated” tracts.17 Since different
procedures fit the 2010 data differently, these productivity increases could vary, but in practice they15

One can use the model of Section 4 to take finite-sample draws from a pre- and post-shock data-generatingiprocess without inducing mean reversion. One does so by fixing the idiosyncratic preferences {νkn
} to be the same in
the two draws. A Monte Carlo exercise using this assumption shows that the calibrated-shares procedure still suffersfrom an overfitting problem and predicts outcomes worse than the covariates-based specification (see Appendix A.4).
16
Figure B.1 depicts the employment changes for these two tracts.
17
Because the 83 employment booms simultaneously occurred from 2010 to 2012, we compute the vector of 83
productivity increases that matches the observed changes in employment in these 83 workplace tracts. The results
obtained when computing 83 separate productivity increases, which neglects the simultaneity, are virtually identicalto those in Figure 3.

15

are very similar. Since the productivity increases are defined so that both procedures match theobserved increase in employment, we examine their predictions for the change in bilateral commuter
counts.
3.4.1

Covariates-based model outperforms calibrated-shares procedure
Figure 3: Comparison of models’ predictive performance across 83 events.15

1

Fraction.1

1.5

Covariates−based: slope
Covariates−based: intercept
Calibrated−shares: slope
Calibrated−shares: intercept

0

0

.05

.5

Density

B. Ratio of models’ prediction errors.2

2

A. Regression of observed on predicted changes

−3

−2

−1

0

1

2

3

.1

2

C. No extensive margin: Regression coefficients

D. No extensive margin: Ratio of prediction errors.2

1

Fraction
.1
.15

1.5

No extensive margin: slope
No extensive margin: intercept
Calibrated−shares: slope
Calibrated−shares: intercept

0

0

.05

.5

Density.2 .3 .4 .5 .6 .7 .8 .9 1 1.1 1.2
Covariates−based MSE / Calibrated−shares MSE

−3

−2

−1

0

1

2

3

.5

.6
.7
.8
.9
1
1.1
1.2
No extensive margin MSE / Calibrated−shares MSE

Notes: “Calibrated-shares” refers to using the observed shares in the exact hat algebra defined by equations (5)–(7). “Covariates-based” in Panels A and B refers to the parameterization described in Section 3.2. “No extensive
margin” in Panels C and D is a hybrid parameterization in which δkn = δ̄kn if ℓkn > 0 in the 2010 data andδkn = ∞ if ℓkn = 0 in the 2010 data. In Panel A, 3 slope and 1 intercept coefficients are not depicted for thecalibrated-shares procedure. In Panel C, 6 slope coefficients are not depicted for the no extensive margin model.
3 slope and 1 intercept coefficients are not depicted for the calibrated-shares procedure. Vertical lines depict themedians of the slope coefficient distributions.

Figure 3 contrasts the changes in commuter flows predicted by the covariates-based model andthe calibrated-shares procedure for all 83 employment booms from 2010 to 2012. For each booming
tract and each counterfactual procedure, we regress the observed change in the number of residents
16

from each residential tract who work in the treated workplace tract on the predicted change. An
unbiased prediction procedure should yield a slope of one and an intercept of zero. Panel A of
Figure 3 depicts the distribution of these coefficients for both procedures. For our covariatesbased model, the slope coefficients are roughly centered on one (median of 0.99), and the intercept
coefficients are roughly centered on zero (median of 0.01).
The calibrated-shares procedure does not perform as well. Across the 83 events, the medianslope coefficient is -0.28, and the median intercept coefficient is 0.60. That is, the calibrated-sharesprocedure’s predictions are negatively correlated with observed outcomes in more than half of theevents. As a result, the covariates-based approach typically has a lower forecast error. Panel B
of Figure 3 contrasts the two models’ MSEs for each event. The covariates-based approach has alower MSE than the calibrated-shares procedure in 80 of the 83 events.
Looking at in-sample fit for the 2010 commuting flows would suggest a very different contrast
between the two approaches: the χ2 test statistic for in-sample fit for the calibrated-shares procedure is literally zero, while the estimated covariates-based model would be rejected by a χ2 test(see Appendix B.6). This warns against evaluating model specifications by in-sample fit. The
covariates-based specification, while not perfectly describing the observed outcomes in 2010, ismuch more informative about how commuting flows change with local employment booms. The
calibrated-shares procedure predicts changes in commuting flows very poorly.
The covariate-based approach’s superior predictive power is not due solely to the presenceof zeros in the commuting matrix. To illustrate this, we produce predictions using a hybrid
parameterization in which residence-workplace pairs that have zero commuters in the 2010 data areassigned infinite commuting costs rather than the covariates-based values. Treating the zeros thisway substantially impairs the model’s predictive performance, as shown in Figure 3C. Nonetheless,this no-extensive-margin parameterization of the covariates-based approach produces predictions
that have lower MSEs than the predictions of the calibrated-shares procedure, as shown in Panel Dof Figure 3.
3.4.2

Temporal aggregation and geographic aggregation

One may try to smooth out idiosyncratic elements of the data before fitting the model. For example,
one might pool multiple years of data. In many empirical settings, particularly historical contexts,
consecutive years of data are not available. In the case of LODES, annual data are available since
2002. We average commuter and wage observations for 2008–2010 before applying the covariatesbased approach and the calibrated-shares procedure.
Pooling three years of data yields a modest improvement for the calibrated-shares procedure(see Figure B.2). Its predictions are now positively correlated with observed outcomes for 66% of
the events. Nonetheless, the covariates-based approach typically forecasts the changes in commuter
counts much better. The covariates-based approach’s slope coefficients are closer to one (median of
0.98 vs. 0.16), and its intercept coefficients are closer to zero (0.03 vs. 0.34). The covariates-based
approach applied to the pooled data has a lower MSE than the pooled calibrated-shares procedure17

in 73 of the 83 events.
Using larger spatial units is another potential way to address the overfitting problem. We canaggregate the 2,160 residential tracts into 195 Neighborhood Tabulation Areas (NTAs) defined by
the New York City Department of City Planning. We consider both aggregating the tract-level
predictions up to NTA-level predictions and defining locations in the model to be NTAs.
When we aggregate the predicted changes in number of commuters to the booming workplace
tracts from residential tracts up to residential NTAs, the covariates-based model still outperforms
the calibrated-shares procedure (see Figure B.2). When aggregating predictions, the covariatesbased model has a lower MSE than the calibrated-shares procedure in 55 of the 83 events.
Estimating the model using NTAs reduces the number of locations by an order of magnitude andthus the number of location pairs by two orders of magnitude. 35 NTAs had 2010–2012 employment
booms. As described in Appendix B.3, the covariates-based approach and the calibrated-sharesprocedure perform similarly well when they are applied to NTAs. The covariates-based model has
a lower MSE than the calibrated-shares procedure in 18 of the 35 events. In line with our Monte
Carlo results, the calibrated-shares procedure is viable when locations are defined as NTAs suchthat there are more than 50 individuals per pair of locations.
Using larger spatial units avoids the overfitting problem that arises when the calibrated-sharesprocedure is applied to granular settings. But doing so leaves one unequipped to address the many
research questions that concern small spatial units such as neighborhoods. Research does not needto avoid granular settings: the covariates-based approach performs well when applied to small
spatial units.
3.4.3

Exact hat algebra using shares from an approximated matrix

Temporal and geographic aggregation are simply two of many statistical tools that researchers
could use to reduce idiosyncratic noise in granular settings. We now consider a noise-reduction
strategy that can be applied to a single year of data from a granular setting (without geographic
aggregation). The idea is to use the shares from an approximation of the commuting matrix inthe exact hat algebra, which will reduce overfitting to the extent that the approximation reduces
idiosyncratic noise.
In particular, we use a rank-restricted singular value decomposition (SVD) of the observed
commuting matrix. The commuting matrix L = {ℓkn } of dimension K × N can be decomposed
into three separate matrices U, Σ, and V such that L = UΣVT . This is an SVD of the commuting
matrix where U is a K × K orthonormal matrix (the left singular vectors matrix), Σ is a nonnegative K × N diagonal matrix with diagonal elements representing the singular values sorted
in descending order, and V is an N × N orthonormal matrix (the right singular vectors matrix).
The rank-r-restricted SVD retains only the r largest singular values and sets the remaining singularvalues to zero. We replace negative elements of this rank-restricted matrix by zeros and rescale allvalues so that the total number of individuals in the resulting approximation is equal to that in18

the observed commuting matrix.18
This approximation requires the researcher to choose the rank r. The rank-1 approximation
would underfit the data because it is merely the product of a k component and an n component,
lacking a bilateral kn component.

Using a full-rank matrix would mean using the observed

commuting matrix. The statistics literature suggests rules of thumb for selecting the rank, suchas visually identifying a point in a “scree plot” in which adjacent singular values differ little orselecting all SVD layers that explain more than a given share of the total variation (Cohen, 2021,
p.496). Alternatively, one could also use a Monte Carlo analysis to pick the optimal rank. We
discuss these approaches in Appendix B.4 and use them in our application to New York City. More
importantly, it is computationally trivial to vary the rank in this approximation, making it easy to
conduct sensitivity analyses.
When studying neighborhood employment booms in New York City, using such an approximation of the commuting matrix yields substantially improved predictive performance over calibrating
the model to the observed shares. We use an approximation derived from a rank-16 SVD, selectedby a 0.5% threshold for the share of singular values. A Monte Carlo simulation using the covariatesbased model as the data-generating process would have selected a rank of 18 (see Appendix B.4.2).
Comparing the observed and rank-approximated commuting matrices visualized in Appendix Figure B.4 suggests that these approximations capture much of the relevant observed variation.19
Plugging this approximation of the shares into equations (5)–(7) yields counterfactual predictions
that are much closer to the predictions from the covariates-based model estimated on the full data.
As shown in Figure 4, the median regression slope is 0.81, and the mean squared error is similar tothat of the covariates-based approach. This performance is not very sensitive to the precise choiceof rank used in the approximation. Appendix Table B.1 reports the slope and MSE for a range ofranks. The MSE for the median event is similar for ranks 5 through 20.
Using a noise-reducing approximation of the matrix of spatial outcomes in exact hat algebra
may be fruitful in other contexts. The approximation guards against overfitting the spatial linkages
between pairs of locations even in the absence of a high-quality covariate that predicts bilateralflows. Moreover, the computational costs of our suggested approximation are very low, as any
modern linear algebra library allows one to compute the SVD of million-element matrices very
quickly.
3.4.4

Interactive fixed effects

One can leverage the explanatory power of observed covariates while allowing for more flexibility
by extending the covariates-based approach to have an explicitly modeled unobserved component
18

This replacement means the approximated matrix may not be rank r. A non-negative rank-r approximation,
which is costlier to compute, delivers very similar counterfactual predictions. See Appendix B.4.3.
19
One feature of the observed matrix that these low-rank approximations do not capture is the large counts on
the diagonal of the matrix: the proclivity for individuals to reside and work in the same tract. Appendix Figure B.4
shows that both an approximation derived from a rank-200 SVD and the covariates-based model do capture thispattern.

19

Figure 4: Predictive performance across 83 events using low-rank approximation
B. Ratio of models’ prediction errors.3

1

Fraction.2

1.5

Covariates−based: slope
Covariates−based: intercept
Low−rank SVD−based: slope
Low−rank SVD−based: intercept

0

0

.1

.5

Density.4

2

A. Regression of observed on predicted changes

−3

−2

−1

0

1

2

3

.7
.8
.9
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7
Covariates−based MSE / Low−rank SVD−based MSE

Notes: This figure depicts the predictive performance of using an approximation of the commuting
matrix in exact hat algebra. The approximation is derived from a rank-16 singular value decompositionof the observed commuting matrix, as described in Appendix B.4. The covariates-based model is thesame as in Figure 3. Panel A depicts the distributions of slope and intercept coefficients, and the verticallines denote the median slope coefficients.

of commuting costs. As detailed in Appendix B.5, we estimate an interactive-fixed-effects (IFE)
specification that assumes the disutility component of commuting costs has an R-dimensional factor
structure, λkn = exp(ψk′ γn ) (Bai, 2009; Chen, Fernández-Val, and Weidner, 2021). For intermediate
values of R, this specification is more flexible than the covariates-based model (R = 0) without goingto the extreme of one parameter per pair of locations (the full-rank case). As with the covariatesbased model, one can use the equilibrium shares of the estimated IFE model with equations (5)–(7)to compute counterfactual outcomes.
A rank-1 IFE specification offers a modest improvement over the covariates-based model in theevent studies. As an encompassing specification, the IFE necessarily outperforms the covariatesbased model (CBM) in terms of in-sample fit (see pseudo-R2 in Appendix Table B.4 and χ2 test in
Appendix B.6). The event studies, summarized in Figure 5, examine its out-of-sample predictive
performance. In Panel A, the IFE slope coefficient for the median event is 0.85, which is farther
from 1.0 than the CBM. While not centered on 1.0, its distribution of slope coefficients is narrowersuch that the IFE slope coefficient is closer to 1.0 in 42 of the 83 events. The IFE’s forecast error issmaller than that of the CBM in the typical event: Panel B shows that its MSE is lower in 47 of the
83 events. Thus, the more flexible interactive-fixed-effects specification offers a modest improvementin predictive performance. The performance of the rank-2 and rank-3 IFE specifications, reported
in Appendix Figure B.6, is very similar.

20

Figure 5: Predictive performance of the rank-1 IFE specification across 83 events
B. Ratio of models’ prediction errors.3

1

Fraction.2

1.5

Covariates−based: slope
Covariates−based: intercept
Rank 1 IFE: slope
Rank 1 IFE: intercept

0

0

.1

.5

Density.4

2

A. Regression of observed on predicted changes

−3

−2

−1

0

1

2

3

.7 .75 .8 .85 .9 .95 1 1.05 1.1 1.15 1.2 1.25 1.3
Covariates−based MSE / Rank 1 IFE MSE

Notes: This figure summarizes the predictive performance of the covariates-based model and the rank-1
IFE specification. Panel A depicts the distributions of slope coefficients, and the vertical lines denote themedian slope coefficients.

3.4.5

Sensitivity analysis

One difference between the approaches described above is in their use of estimated or observed
wages. The covariates-based model and interactive-fixed-effects specification estimate wages using
workplace fixed effects. By contrast, observed wages are used in the calibrated-shares procedureand in the exact hat algebra with the SVD-approximated shares. We have verified that using
alternative wage values—such as equal wages across locations—does little to alter the predictionsof the calibrated-shares procedure in the event studies.
We find similar event-study results when employing different assumptions about preferences
and production. Thus far, we have assumed idiosyncratic preferences are spatially independent, asin prior research. Making residential locations in the same NTA closer substitutes in a nested-logit
demand system has little effect on the predictions produced by either method (see Appendix B.7).
Similarly, the changes in commuters predicted by both methods are not sensitive to the elasticity
of substitution between goods σ (see Appendix B.8) nor to introducing local increasing returns at
workplaces (see Appendix B.9).

3.5

Counterfactuals in continuum models: Takeaways

The predictive performance of quantitative spatial models applied to granular empirical settings
is highly sensitive to the baseline equilibrium shares used in the exact hat algebra. The incumbent methods—the covariates-based approach and calibrated-shares procedure—produce sharply
contrasting predictions for bilateral commuting flows because they parameterize spatial linkages
very differently.

Using shares from a covariates-based model predicts changes in commuting

flows remarkably well. Using observed shares to calibrate millions of parameters suffers from an

21overfitting problem. Aggregating data across a few years does not remedy this problem. Using
shares produced from a low-rank approximation of the commuting matrix alleviates the overfitting
problem and produces predictions comparable to the covariates-based approach. An interactivefixed-effects specification that extends the covariates-based approach offers modest improvements
in predictive performance at greater computational cost.
In which spatial settings is the calibrated-shares procedure likely to suffer from an overfitting
problem? Generally, the answer depends on the number of individuals observed, the number of location pairs, the true choice probabilities, and the counterfactual shock. To examine the calibratedshares procedure’s finite-sample performance in a specific setting of interest, we recommend using acovariates-based model as the data-generating process in Monte Carlo simulations, as in Section 3.3.
The inputs necessary for such simulations are the same as those used in the calibrated-sharesprocedure, assuming the researcher has used a covariate to estimate the commuting elasticity. In
our simulations of a local employment boom in New York City, observing a sample containing atleast 50 individuals per pair of locations would have been adequate to apply the calibrated-sharesprocedure.
More broadly, our results caution against using observed shares from large matrices in exact
hat algebra. For example, Antràs and Chor (2022, p.333) discuss the relevance of our findings forcalibrating trade models with thousands of parameters using detailed trade matrices. Using sharesproduced by data-smoothing procedures, such as low-rank approximations, or using more parsimonious models is likely to deliver better counterfactual predictions in high-dimensional settings.

4

A spatial model with a finite number of individuals

In many quantitative spatial models, individuals choose a residence and a workplace based on
characteristics of locations, the links between them, and an idiosyncratic component. In a modelwith a continuum of individuals, the idiosyncratic components are integrated out, so that particularrealizations of these idiosyncrasies play no role in economic outcomes.
We introduce a spatial model with a finite number of individuals in which equilibrium outcomes
depend on individual idiosyncrasies. Its outcomes differ from the continuum model in two ways.
First, because quantitative spatial models are not linear, the equilibrium prices in a model with acontinuum of individuals need not equal the average equilibrium prices from a model with finite
individuals. Second, counterfactual changes in quantities and prices depend on which individuals
change their decisions in response to the exogenous shock, which depends on their vectors ofunobserved idiosyncratic preferences. There is therefore uncertainty about realized counterfactual
outcomes, as the distribution of idiosyncrasies induces a distribution of counterfactual changes.
The key challenge in modeling a finite number of people is that individuals’ decisions affectwages and rents. This raises two issues. First, do individuals internalize the effects of their own
choices on local labor supplies and land demands? Second, are individuals able to enumerate theprices induced by every possible combination of others’ choices? In the interest of tractability,22

we assume that individuals have common point-mass beliefs about wages and rents. Therefore,
individuals act as price takers, as in Gabaix (2011). In our application, we assume that, given themodel parameters, individuals have beliefs about wages and rents that are the equilibrium prices of
the conventional continuum model. We label these beliefs “continuum-case rational expectations”
because they would be rational if there were a continuum of individuals.20 This model with a finite
number of individuals is intentionally similar to the conventional continuum model: they coincide
as the number of individuals becomes arbitrarily large and can be estimated similarly.

4.1

Setup

Our model with a finite number of individuals features the same utility function, technology, andcommuting costs as in Section 2.21 The set of primitives Υ ≡ {L, {An }, {Tk }, {δ̄kn }, {λkn }, α, ϵ,σ} is the same as in the continuum model. It differs from the continuum model in two important
respects. First, we consider an economy with an aggregate labor endowment L embodied in I
individuals each supplying LI units of labor. Second, we introduce a timing assumption: individuals
first choose their residence-workplace pairs, and then all markets clear given individuals’ residential
and workplace locations. This timing assumption would be of no consequence if there were acontinuum of individuals.
i by ν I . As before, theseWe denote the set of all idiosyncratic residence-workplace draws νkn

idiosyncratic draws from the standard Gumbel distribution are independent across individuals
and location pairs. The superscript I indicates that the set of draws grows with the number ofindividuals.
We make the following assumptions about information and expectations. All workers know
the economic primitives Υ and have (common) expectations about the equilibrium variables rk andwn . We assume that the price expectations are point-mass beliefs, such that each individual assigns
100% probability to a single vector of wages and a single vector of land prices. Denote these beliefi }.
vectors by {w̃n } and {r̃k }. Worker i knows her idiosyncratic preferences {νknDecisions are made and markets clear in the following order. Based on beliefs {w̃n } and {r̃k },each worker chooses the residential location and the work location that maximize expected utilityiŨkn= ϵ ln

|

!
w̃n
i
+νkn
,
P̃ 1−α r̃kα δkn
{z
}

(9)

≡Ũkn

20

Similar simplifications to agents’ beliefs have been made in industrial organization (e.g., oblivious equilibriumin Weintraub, Benkard, and Van Roy 2008) and macroeconomics (e.g., Krusell and Smith 1998). Specifically, in an
oblivious equilibrium, a finite number of firms optimize assuming that the industry state is constant over time andequals its long-run expected value, even though the industry state is not truly constant in an environment with afinite number of firms and idiosyncratic shocks. In the heterogeneous agents model of Krusell and Smith (1998),
agents condition their choices on moments from the distribution of state variables (rather than tracking the entire
distribution).
21
Model extensions that introduce trade costs, residential amenities, local increasing returns, and production
employing land are presented in Appendix C.7.

23

where P̃ =hP

1−σn (w̃n /An )cannot relocate.22

i1/(1−σ). After these decisions are made, workers are immobile andAs detailed below, realized equilibrium land prices rk and wages wn are thosethat clear goods, labor, and land markets given individuals’ residential and workplace locations.
The assumption that individuals have point-mass beliefs about wages, {w̃n }, and rents, {r̃k },considerably simplifies the analysis. Otherwise, individuals would need to compute the prices
associated with all possible residence-workplace allocations and then solve for the fixed-pointprobabilities that each of these occurs. The set of feasible allocations is large. For example, an
economy with only 10 individuals and 16 residence-workplace pairs would have more than 3 million
possible allocations.23 For empirically relevant magnitudes, computing the number of possible
allocations leads to such large numbers that they cause overflow problems in standard software.
While our approach is general in the sense that it allows for any kind of point-mass beliefs about
wages and rents, in our analysis below we assume that the expected prices are the equilibrium wages
and rents of the continuum model with the same economic primitives Υ. The combination of these
beliefs and our timing assumption is more palatable to the extent that individuals would not change
their decisions if given the opportunity to re-optimize after observing realized prices. We computethe magnitude of “ex post regrets” in Section 4.4.

4.2

Equilibrium

For expositional clarity, we distinguish between a trade equilibrium, which clears markets taking
individuals’ locations as given, and a commuting equilibrium with finitely many individuals, in which
individuals choose locations based on beliefs about the trade equilibrium that will result.
The trade equilibrium clears markets given individuals’ location choices. Given a labor allocation {ℓkn }, goods market clearing is identical to the continuum model and given by equation (3).
Similarly, land market clearing is identical to the continuum model and given by equation (4).
Definition 4.1. Trade equilibrium. Given the labor allocation {ℓkn } and economic primitives Υ,
a trade equilibrium is a set of wages {wn } and land prices {rk } such that equations (3) and (4) aresatisfied.
Remark. The trade equilibrium is unique. As shown in Appendix C.5, given the labor allocation{ℓkn }, there is a unique set of relative wages satisfying equation (3). Equation (4) can be rewrittenP
as rk = Tαk n δ̄ℓkn wn , so there is a unique set of land prices associated with that wage vector.
kn

22
The assumption that individuals make irreversible decisions is common in static spatial models. For example,
in the open-city model of Ahlfeldt et al. (2015), individuals choose to live in Berlin based on expected utility, whichis equal to the reservation level of utility in the wider economy. Individuals who choose Berlin and realize utility
below the city-wide average cannot leave. Redding and Rossi-Hansberg (2017), Heblich, Redding, and Sturm (2020),
Owens, Rossi-Hansberg, and Sarte (2020), and Brinkman and Lin (2022) make the same irreversibility assumption.
In dynamic location choice models (e.g., Diamond, McQuade, and Qian 2018), moving costs cause agents to notnecessarily change locations in response to shocks.
23
With I individuals and N 2 residence-workplace pairs, the set of possible allocations (the support of the

2
2
−1)!
−1
multinomial distribution) contains I+N= (I+Nelements. For I = 10 and N = 4, this is about 3.27 × 106 .
N 2 −1(N 2 −1)!I!

24

Individuals’ choices of location are made on the basis of their beliefs about the subsequent
trade equilibrium. Given belief vectors {w̃n } and {r̃k } and the idiosyncratic workplace-residence
utility draws ν I , individuals choose the residence-workplace pair that maximizes their utility inequation (9).
Definition 4.2. Commuting equilibrium with finitely many individuals.

Given a number ofindividuals I, economic primitives Υ, idiosyncratic residence-workplace draws ν I , and a set ofpoint-mass beliefs ({w̃n }, {r̃k }), a commuting equilibrium with finitely many individuals is a labor
allocation {ℓkn }, wages {wn }, and land prices {rk } such thatP
i > Ũ ′ ′ + ν i′ ′ ℓkn = LI Ii=1 1{Ũkn + νkn
kn
k′ n′ ∀(k , n ) ̸= (k, n)}; and wages {wn } and land prices {rk } are a trade equilibrium, per Definition 4.1, given the labor

allocation {ℓkn }.
We now define a set of price beliefs that are the equilibrium prices of the continuum model from
Section 2 with the same parameters for economic primitives.
Definition 4.3. Rational expectations for the continuum case. Given economic primitives Υ, {w̃n }and {r̃k } are “continuum-case rational expectations” if {wn } = {w̃n } and {rk } = {r̃k } constitutean equilibrium of the continuum model defined in Section 2.2 for the same economic primitives Υ.
Distinguishing between the number of individuals I and the aggregate labor supply L allows us
to apply the law of large numbers to locational decisions as I → ∞ without changing aggregate labor
supply. As Appendix C.6 shows, taking I → ∞ causes the labor allocation in the limiting case to be
the same as in the continuum model in equation (2). Note that given a set of idiosyncratic residenceworkplace draws ν I and a set of point-mass beliefs ({w̃n }, {r̃k }), the commuting equilibrium withfinitely many individuals of Definition 4.2 is unique. Provided that the parametric condition for
uniqueness in the continuum model in Section 2.2 is satisfied, the wage and rent beliefs that satisfy
Definition 4.3 will also be unique.

4.3

Estimation of economic primitives

One can estimate a covariates-based specification of the model with a finite number of individuals
in the same way we estimated its continuum-model counterpart in Section 3.2. We take α, σ, δ̄kn
as given, impose λkn = 1 ∀kn, and assume that ν i are drawn independently across individuals andlocation pairs from a standard Gumbel distribution.24 This distributional assumption implies thatthe log likelihood function is
#
"
−ϵ
h
i X
w̃nϵ r̃kα δ̄kn
i
i
′ ′
L≡
ℓkn ln P(Ũkn > Ũk′ n′ ∀k n ̸= kn) =ℓkn ln P.
ϵ r̃ α δ̄ ′ ′ −ϵ
w̃
′
′
′
′
k
n
k
,n
n
k
k,n
k,n
X

24

(10)

The assumption that λkn = 1 can be relaxed by allowing the unobserved disutility component of commuting costs
to have a factor structure, λkn = exp(ψk′ γn ), as in Section 3.4.4 and Appendix B.5. If individuals have continuum-caserational expectations, the economic primitives of this specification of the model with a finite number of individuals
match those of the interactive-fixed-effects specification of the continuum model.

25This matches equation (8), except these residence and workplace fixed effects depend on rent
and wage beliefs rather than the realized equilibrium rents and wages. The maximum-likelihood
estimate of the commuting elasticity ϵ is identical to that in the covariates-based continuum model(in column 1 of Table 1). The remaining economic primitives {Tk } and {An } can be inferredfrom the maximum-likelihood estimates using assumptions about the rent and wage beliefs. Given
values of the commuting elasticity ϵ and the land expenditure share α, individuals’ beliefs about
land prices and wages are transformations of the estimated fixed effects. In particular, the residence
fixed effect is proportional to r̃k−αϵ , and the workplace fixed effect is proportional to w̃nϵ . If {r̃k } and{w̃n } are continuum-case rational expectations, plugging these estimated beliefs into equation (2)yields the continuum-case labor allocation. Given α, σ, {r̃k }, {w̃n }, {δkn } and that continuum-caselabor allocation, equations (3) and (4) can be solved to obtain {Tk } and {An }. The estimated landand productivity parameters are identical to the estimates from the covariates-based continuum
model.

4.4

Ex post regret

The realized equilibrium rents and wages differ generically from the point-mass beliefs about rents
and wages that govern individuals’ choices of residences and workplaces. Because equilibrium prices
solve a non-linear system of equations, the average realized equilibrium prices may differ from thepoint-mass beliefs. As described in Appendix E.1, these differences are small at our baseline
parameter values: the 95th -percentile tract’s absolute percentage point deviation of mean realized
price from the continuum-case rational expectation is 0.72% for wages and 0.03% for rents.25
Realized equilibrium prices vary with individual idiosyncrasies. Figure E.2 presents histogramsof the standard deviation of wages and rents across 100,000 simulations of our estimated model of
New York City divided by the continuum-case rational expectations of these prices. Across tracts,
the median level of this measure of dispersion is 0.016 for wages and is 0.032 for rents. While thisdispersion may be considered sizable, these differences between expectations and realized prices
do not necessarily imply that individuals would move if we relaxed the irreversibility assumption.
Individuals’ decisions depend on their idiosyncratic preferences, so many are inframarginal.
Using our estimated model of New York City, we investigate how often individuals would choosea different residence-workplace pair at the realized equilibrium prices. We define the magnitude
of “ex post regret” as the increase in income an individual would require as compensation to notchange their choice given the realized equilibrium rents and wages. In particular, at realized prices{rk } and {wn }, for individual i who chose residence-workplace pair kn, ex post regret χi is definedsuch that χi solves


max
ϵ
ln
′ ′
k ,n

wn′
P 1−α rkα′ δk′ n′+ νki ′ n′


=





(1 + χi )wn
i
ϵ ln
+
ν
kn ,
P 1−α rkα δkn

25These differences are larger when the labor demand elasticity σ is lower: 3.61% for wages and 0.05% for rents
when σ = 1.1.

26

where the left side is the individual’s maximum utility at realized prices, the right side is theindividual’s utility from kn with their income multiplied by (1 + χi ), and by definition kn was thei . Individual i’s regret is zero if kn is their optimal choice at realized prices.
choice maximizing Ũk,n

In this setting, we find that ex post regret is quantitatively modest. In simulations reported
in detail in Appendix E.3, 96% of individuals would not want to change their residence-workplacechoice. For the 4% who would want to switch, the median ex post regret χi is equal to 0.7%.
Given the large return to making the model tractable and computationally feasible, we judge themagnitude of ex post regret when using continuum-case rational expectations to be quite modest.

4.5

Contrasts with the continuum model

The model with a finite number of individuals has a fundamental contrast with the continuum
model: individual idiosyncrasies affect equilibrium outcomes. Given economic primitives Υ, thismodel produces a distribution of outcomes associated with the distribution of idiosyncratic preference shocks ν I , whereas the continuum model delivers deterministic outcomes. Such a contrast
has been emphasized by studies investigating the sensitivity of aggregate outcomes to firm-levelproductivity shocks (e.g., Gabaix 2011; di Giovanni, Levchenko, and Mejean 2014; Eaton, Kortum,and Sotelo 2013; Gaubert and Itskhoki 2021).
Given the same economic primitives Υ, the expected change in the equilibrium labor allocationof the model with a finite number of individuals is equal to the change in the equilibrium labor
allocation of the continuum model. This is because the multinomial distribution of residenceworkplace outcomes {ℓkn } defined by the probabilities in equation (10) has expected values equal
to the quantities in equation (2). By contrast, the model’s equilibrium prices solve a system of nonlinear equations, so their expected values are not necessarily equal to the continuum-case prices.
For our baseline parameter values, these differences are small.
We focus on the implications of these idiosyncratic components for predicting counterfactual
outcomes. We compute the change in outcomes associated with a change in economic primitives
from Υ to Υ′ for a given realization of idiosyncratic preferences ν I .26 The set of individuals
who change their decisions in response to the change in economic primitives depends on theparticular realized vectors of idiosyncratic preferences. Simulating realizations of ν I therefore yields
a distribution of counterfactual changes. The dispersion in this distribution represents uncertainty
about counterfactual predictions stemming from individual idiosyncrasies.

5

Application to Amazon HQ2 in Long Island City

We now examine the predictions of quantitative spatial models for the economic consequencesof Amazon’s aborted second headquarters (HQ2) in Long Island City, a controversial and widely
discussed proposal. In November 2018, the company announced it would hire more than 25,000
26This is akin to the computation of counterfactual changes holding a realization of firms’ idiosyncratic unit costs
fixed in Eaton, Kortum, and Sotelo (2013, Section 6.1).

27

employees in 4 million square feet of office space.27

However, Amazon scrapped the projectin February 2019 after facing a fierce backlash from local politicians and community members
concerned about corporate subsidies and gentrification.
This large counterfactual change to a single workplace allows us to illustrate how choices ofmethods for counterfactual analyses affect the predictions of quantitative spatial models. Section 5.1
shows how the geographic incidence of a productivity shock on local rents is governed by the parameterization of spatial linkages reflected by the model’s baseline equilibrium shares. Section 5.2 uses
the model with a finite number of individuals to quantify the uncertainty about these counterfactual
outcomes. Individual idiosyncrasies make neighborhood-level counterfactual changes considerably
uncertain, even when examining a productivity shock shifting 1% of employment to a single tract.

5.1

Geographic incidence of Amazon HQ2

We analyze the consequences of a productivity increase in the Long Island City tract that would
cause employment to rise by 25,000 workers from the 2010 baseline levels. Figure 6A depicts
the distribution of residences among workers employed in the treated Long Island City tract
in 2010. To contrast the predictions of the continuum model when using the covariates-basedmodel and the calibrated-shares procedure, we find the productivity increases ÂCBM
and ÂCSPn∗n∗that raise employment by 25,000 in the Amazon workplace tract n∗ , holding all other economic
primitives fixed. Appendix F.1 reports additional results from using alternative values of the labordemand elasticity σ, from using a nested-logit demand system, and from the SVD-based and IFE
specifications of the continuum model.
The counterfactual changes in residential populations are sensitive to the method used: while
the covariates-based approach predicts small changes in all tracts, the calibrated-shares procedure
predicts more varied and less spatially correlated changes. The covariates-based model’s predictedchanges in the number of residents in each tract, shown in Figure 6B, are modest in size and closely
related to transit times to Long Island City. The calibrated-shares procedure’s predicted changes,
shown in Figure 6C, are very large for some residential tracts. The calibrated-shares procedure
predicts increases in residents in neighborhoods scattered across Queens and Staten Island, but very
little increase in residents in Manhattan neighborhoods with short commutes to Long Island City.
The calibrated-shares procedure suggests that 49% of the workers in Long Island City would residein Queens, while the covariates-based model predicts only 33%. The calibrated-shares procedure’s
predictions are closely tied to the initial numbers of residents working in the treated tract: thespatial patterns in Figure 6A and Figure 6C are remarkably similar.
The incidence of Amazon HQ2 on residential rents predicted by the two methods differs consider27

Amazon had requested proposals for HQ2 locations from cities and states across North America. In November
2018, it announced two winners, saying that it would hire more than 25,000 employees each in Long Island City in New
York and Arlington in Virginia (Amazon, 2018). More details can be found in the Memorandum of Understanding
between New York State, New York City and Amazon. Berkes and Gaetani (2023) introduce a model that features
productivity spillovers and skill heterogeneity to analyze Amazon HQ2. Following Berkes and Gaetani (2023), the“treated” tract in our counterfactual scenario is 36081000700.

28

Figure 6: Amazon HQ2 counterfactual change in residents
A. Residents working in treated tract

B. Covariates-based predictions

C. Calibrated-shares predictions

Notes: Panel A depicts the number of residents in each tract who work in the treatment tract in the LODES 2010
data. The legend of Panel A reports the percentiles corresponding to the integer number of residents, excluding thetreatment tract. Panels B and C depict changes in the number of residents predicted by the covariates-based modeland calibrated-shares procedure, respectively. The legend percentile cutoffs in Panels B and C correspond to thosein Panel A. The covariates-based predictions for residents describe both the continuum model and the model withfinite individuals because the change in the former equals the expected change in the latter.

ably. The greater variation in changes in the number of residents predicted by the calibrated-sharesprocedure is accompanied by greater variation in the predicted changes in real rents, r̂k /P̂ . The
calibrated-shares procedure predicts rent increases of 1.6% to 10.4% for the top decile of tracts(Figure 7C). It suggests that in 12 census tracts, real rents would increase by at least 5%. By
contrast, the covariates-based model predicts more modest increases of 0.8% to 1.3% for the top
decile of affected tracts (Figures 7A and 7B).28 The geographic incidence differs notably. Figure 7
shows that the calibrated-shares procedure predicts much larger rent increases in neighborhoods far
from Amazon HQ2, such as Staten Island.29 By contrast, the political opposition to Amazon HQ2
based on concerns about gentrification largely concerned housing prices in nearby constituencies(Goodman, 2019).
Unlike the changes in residents and rents, the changes in workers, wages, and welfare predictedby the two methods are quite similar. Appendix Figures F.1 and F.2 show the changes in workers
and wages. These predictions are similar because the geographic pattern of employment changes in
response to a productivity shock is closely tied to the initial number of workers employed in eachtract, which the two methods agree upon. The citywide welfare consequence of the productivity
shock, derived in Appendix C.4, is also very similar across methods. This occurs because the welfare28
The covariates-based continuum model and model with a finite number of individuals predict very similar rentchanges (compare Panels A and B).
29
Appendix Figure F.3 plots rent changes against distance to the treated tract for both methods.

29

Figure 7: Predicted changes in rents
A. Covariates-based: Continuum

B. Covariates-based: Finite

C. Calibrated shares

Notes: These maps depict percent changes in rents, (r̂k /P̂ − 1) × 100. Panel A depicts the change in thecovariates-based continuum model. Panel B depicts the mean change across 100 simulations of the modelwith finite individuals. Panel C depicts each tract’s predicted rent change using the calibrated-sharesprocedure.

effect of this single-tract employment boom is largely invariant to spatial linkages: computing
counterfactual changes using a parameterization with no spatial linkages, δkn = 1 ∀kn, also yields
a very similar welfare consequence.30

5.2

Uncertainty about counterfactual predictions induced by idiosyncrasies

Counterfactual predictions come with various forms of uncertainty. For example, to capture uncertainty about the parameters used to produce counterfactual predictions, researchers can bootstrap
the counterfactual predictions of a continuum model by sampling from the data or the distributionof estimated parameters. In addition to parameter uncertainty, when the number of individuals is
finite, there is a distribution of equilibrium outcomes associated with the distribution of individuals’
idiosyncratic preferences. That is, counterfactual predictions about a change in economic primitives(from Υ to Υ′ ) are subject to uncertainty stemming from individual idiosyncrasies.
The magnitude of this uncertainty relative to the predicted counterfactual changes depends
on the empirical setting and the counterfactual shock. We consider the creation of 25,000 jobs in
one tract to be a large shock, as Amazon’s HQ2 choice was a high-profile corporate decision that
30
This finding does not imply that the two methods will always yield similar welfare conclusions. For counterfactual
scenarios where spatial linkages matter for citywide welfare consequences, they may differ. For example, the citywide
welfare benefit of faster commuting between two tracts that had zero commuters in the 2010 data would bezero under the calibrated-shares procedure but positive when using the baseline shares from the covariates-based
parameterization.

30

elicited numerous proposals from local policymakers. Are the counterfactual changes in outcomes
across different parts of New York City large relative to the uncertainty driven by the idiosyncraticcomponents of individuals’ location decisions?
The predicted tract-level changes in residents and rents are small relative to the idiosyncratic
variation in this setting. We summarize uncertainty by simulating the counterfactual change fromΥ to Υ′ for 100 realizations of idiosyncratic preferences ν I and construct a 90% confidence interval
by reporting the 5th and 95th percentiles of the distributions of changes. Figure 8A plots the 90%
confidence interval for each tract’s predicted change in residents against its mean predicted change.
Every tract’s 90% confidence interval for the change in residents has a positive upper bound and anegative lower bound. In this sense, none of the predicted changes in residents are distinguishable
from zero. Figure 8C plots the 90% confidence interval for each tract’s predicted rent change against
its predicted change. For about 15% of residential tracts, the predicted changes in real rents are
distinguishable from zero.
Since the predicted changes in the number of workers for some tracts are larger, many of them
are meaningfully large relative to the idiosyncratic variation. As the number of workers varies acrosstracts much more than the number of residents, so too do the widths of the confidence intervals.
Figure 8B depicts these confidence intervals for the changes in the number of workers. For hundredsof workplace tracts, the 90% confidence interval includes both negative and positive values.31 For
tracts predicted to lose many workers, the predicted change is sufficiently large relative to theuncertainty that even the 95th percentile of changes is negative. For the vast majority of workplace
tracts, the uncertainty over real wages is sufficiently small that the 5th -percentile change for thereal wage is greater than zero (see Figure 8D).32
The uncertainty stemming from individual idiosyncrasies in the model with finitely many
individuals is larger than the variation in counterfactual predictions associated with parameter
uncertainty in the continuum model in this setting. To compute the latter, we sample with replacement 2.5 million individuals from the observed commuting matrix 100 times. For each of these
100 samples, we estimate the covariates-based continuum model and compute its counterfactual
predictions. Figure 8 shows that the variation in counterfactual changes induced by parameter
uncertainty is much smaller than the variation induced by different realizations of idiosyncrasies.
We also investigate the uncertainty stemming from time-varying fundamentals by re-doing thecounterfactual analysis using 2008 data (instead of 2010 data). When using the covariates-based
parameterization, the counterfactual predictions are not very sensitive to the choice of baseline
year: all the real rent changes and more than 95% of the real wage changes produced by the 2008
estimated model are inside the 90% confidence intervals of the predictions from the 2010 estimated31

While workplaces are generically substitutes for one another, the Amazon expansion alters the pattern of
residential rents, which can cause some individuals to choose a new residence-workplace pair that includes a nonAmazon workplace. An (idiosyncratic) increase in labor supply can even reduce real wages in such workplaces, as
shown in Figure 8D.
32
If labor demand were perfectly elastic σ = ∞, equilibrium wages would be independent of residual labor supply
and would exhibit no uncertainty: wn ∝ An .

31

Figure 8: Variation in counterfactual changes: Idiosyncrasies vs. parameter uncertainty
B. Changes in workers−20

Change in residents
−10
0
10
20

Change in workers−500 −400 −300 −200 −100

030

A. Changes in residents

−4

−2
0
2
Mean change in the number of residents
Idiosyncrasies 95th percentile
Parameter uncertainty 95th pctile

4

−500

Idiosyncrasies 5th percentile
Parameter uncertainty 5th pctile

−400
−300
−200
−100
Mean change in the number of workers
Idiosyncracies 95th percentile
Parameter uncertainty 95th pctile

Idiosyncracies 5th percentile
Parameter uncertainty 5th pctile

D. Changes in wages

−2

−20

0

Percent change
24Percent change
−10
0
10

6

20

C. Changes in rents

0

0

.5
1
Mean percent change in rent
Idiosyncrasies 95th percentile
Parameter uncertainty 95th pctile

1.5

.2

Idiosyncrasies 5th percentile
Parameter uncertainty 5th pctile.4.6
Mean percent change in wage

Idiosyncrasies 95th percentile
Parameter uncertainty 95th pctile.81

Idiosyncrasies 5th percentile
Parameter uncertainty 5th pctile

Notes: These plots depict variation in counterfactual changes induced by individual idiosyncrasies in the modelwith a finite number of individuals alongside variation attributable to parameter uncertainty about {An }, {Tk },and ϵ in the covariates-based continuum model. We summarize the former by plotting the 5th and 95th percentilesof predicted changes in quantities and prices across 100 simulations of the model with a finite number ofindividuals. In Panels A, B, C, and D, the horizontal axes correspond to the values depicted in Figures 6B,
F.1B, 7B, and F.2B, respectively. Panels B and D exclude the Amazon HQ2 workplace tract. In Panel A, the
90% confidence interval for the change in residents includes zero for all tracts. In Panel B, one outlier with an
employment decline of 1,483 is not depicted, and the 90% confidence interval for the change in employment
includes zero for 413 of the 2142 non-Amazon workplaces. The 90% confidence interval for the change in
employment for the Amazon workplace tract is 24,777 to 25,349. In Panel C, real rent changes are (r̂k /P̂ −1)×100,
and 322 out of 2160 origin tracts have a positive change in rents at the 5th percentile. In Panel D, the real wagechanges are (ŵn /P̂ − 1) × 100, and 1950 out of 2143 destination tracts have a positive change in wages at the 5th
percentile. We quantify parameter uncertainty by producing 100 counterfactual predictions by sampling withreplacement 2,488,905 individuals from the observed commuting matrix 100 times. For each of these 100 samples,
we estimate the covariates-based continuum model and compute its counterfactual predictions. The plots depict
the 5th and 95th percentiles of predicted changes in quantities and prices across these 100 bootstrap samples.
The variation attributable to parameter uncertainty is considerably smaller than the variation attributable toindividual idiosyncrasies: the bootstrap 5th and 95th percentiles are almost indistinguishable.

32model depicted in Figure 8.33
As we move from tract-level outcomes to examining larger geographic units, the uncertainty
about counterfactual outcomes induced by individual idiosyncrasies diminishes but remains substantial. If we aggregate tract-level changes into 20 bins based on distance to the Amazon HQ2 tract,
the mean predicted changes in residents and rents decline with distance to the Amazon HQ2 tract.
For all but the first and twentieth ventiles, the 90% confidence interval for the change in residents
includes zero (see Table F.1). For every ventile, the 90% confidence interval for the rent change
includes the citywide change. The uncertainty about the change in real aggregate output stemmingfrom individual idiosyncrasies is negligible.34 When we apply the model with a finite number ofindividuals to 195 Neighborhood Tabulation Areas rather than tracts, the 90% confidence interval
for the change in residents includes zero for 194 of the 195 NTAs. The confidence intervals for thepredicted change in real rent include the citywide change for 132 NTAs (see Appendix Figure F.7).
The uncertainty about counterfactual tract-level changes in residents and rents is similar whenresidential locations are more substitutable and when labor demand is more elastic. Uncertainty issimilar when we employ a nested-logit specification in which residential locations in the same NTA
are closer substitutes (see Appendix Figure F.8). When labor demand is perfectly elastic (σ = ∞),wages are invariant to individual idiosyncrasies. Uncertainty about rent changes is similar to theuncertainty in the baseline case (σ = 4), though the mean rent increase is smaller so more tracts
have confidence intervals that include zero (see Appendix Figure F.9).
The predicted consequences of the most closely followed corporate headquarters decision inrecent memory for local economic outcomes are subject to considerable uncertainty stemming fromindividual idiosyncrasies. This notable uncertainty would be missed by the conventional continuum
approach. Given the magnitude of uncertainty stemming from individual idiosyncrasies that we
document in this case, we believe this uncertainty is likely relevant in many empirical applications
that consider the geographic distribution of economic outcomes.
When should a researcher quantify the uncertainty about counterfactual predictions attributable
to individual idiosyncrasies? Our results suggest the scale of the predicted changes is key: individual idiosyncrasies induce little uncertainty about aggregate outcomes. When studying spatiallyfine outcomes, the answer depends on the empirical setting, such as the size and nature of thecounterfactual shock and the expected number of individuals choosing a pair of locations. To
assess whether they can rule out uncertainty attributable to individual idiosyncrasies, researchers
can repeatedly draw finite counterfactual realizations from the continuum model to compute a
distribution of differences between finite-sample outcomes at counterfactual parameters Υ′ and the
expected outcome at baseline parameters Υ. This distribution of differences is more dispersed thanthe distribution of counterfactual changes from the model with a finite number of individuals thatfixes the vector of idiosyncratic preferences ν I .35 If the dispersion of the former is small, then the33
By contrast, the calibrated-shares procedure’s predictions are much more sensitive to the use of 2008 versus 2010
data. The correlation between the real rent changes shown in Figure 7C and those obtained using the 2008 data is
only 0.23 (0.73 for real wages).
34
The 90% confidence interval for the increase in aggregate output is 0.41% to 0.43%.
35
This distribution of differences is more dispersed than the distribution of counterfactual changes because an33

uncertainty stemming from individual idiosyncrasies is small. For example, if the distribution ofcontinuum-model differences does not include zero, then the confidence interval from the model withfinite individuals excludes zero. When uncertainty attributable to individual idiosyncrasies cannot
be ruled out, computing and reporting this uncertainty will produce more informative predictions
about local economic outcomes.

6

Conclusion

Economists increasingly use spatially fine data and quantitative spatial models to compute counterfactual general-equilibrium outcomes.

The smaller the number of individuals behind eacheconomic outcome reported in these granular settings, the less compelling the conventional modelingassumption that there is a continuum of individuals. These applied general-equilibrium models
need to reliably predict economic changes if policymakers are to use them to inform their decisions(Kehoe, 2005; Bryan, Glaeser, and Tsivanidis, 2020; Adão, Arkolakis, and Esposito, 2022).
We document that some conventional methods perform poorly when applied to granular settings.
Calibrating pair-specific parameters when there are many pairs of locations relative to the numberof decision makers fits the model to the idiosyncratic component of individual decisions. We
recommend that researchers use Monte Carlo simulations to verify the finite-sample behavior of
their procedures for producing counterfactual predictions in their empirical setting. We show how
to use low-rank matrices to capture unobserved spatial linkages without overfitting when computing
counterfactual outcomes using exact hat algebra.
Counterfactual outcomes for particular locations vary with individual idiosyncrasies. We introduce a model with a finite number of individuals in which equilibrium outcomes depend in parton the idiosyncratic component of individuals’ choices. This estimated model yields a distributionof equilibrium outcomes when computing counterfactual scenarios. Evaluating the counterfactual
consequences of Amazon’s proposed second headquarters in Long Island City, we find that there isconsiderable uncertainty about most of the predicted changes in local outcomes. Since all empiricalsettings feature a finite number of individuals, the continuum assumption has been made in theinterest of modeling convenience, not realism. The model with a finite number of individuals is
tractable and features a covariates-based specification that can be estimated using the same data
as its continuum counterpart. Thus, it can be applied in the same settings in which economists
have thus far assumed a continuum.

idiosyncratically high outcome at counterfactual parameters Υ′ would also be higher at baseline parameters Υ given
the same idiosyncratic preferences ν I . This contrast in dispersion can be seen by contrasting Figure 9 in Dingel
and Tintelnot (2021), which depicted such differences, to Figure 8 above. Sampling from the continuum model
is computationally cheaper than computing counterfactual outcomes holding idiosyncratic preferences fixed in themodel with a finite number of individuals.

34

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40

Appendix
A

Monte Carlo simulations

We conduct a Monte Carlo simulation to examine the finite-data performance of the covariatesbased approach and the calibrated-shares procedure. In these simulations, the data-generatingprocess is our estimated covariates-based model of New York City in 2010. We impose a counterfactual 9% increase in productivity (Ân = 1.09) for the “treated” tract containing 200 Fifth
Avenue, which generates an increase in employment matching the observed change from 2010 to2012.36 There are no changes in the productivity of other workplace tracts (Ân = 1), no changes
in land endowments (T̂ = 1 ∀k), and no changes in commuting costs (δ̄ˆ = 1 ∀k, n). In thekknlimiting case, as I → ∞, the calibrated-shares procedure would perfectly describe the changes
in commuting flows associated with this productivity increase. Thus, any predictive failure whendrawing finite-sample realizations from this data-generating process are due to problems stemmingfrom the finite number of individuals.

A.1

Monte Carlo exercise: Continuum changes

In the continuum model with parameter values estimated in Section 3.2, the “true” change inoutcomes is given by combining the relative changes in exogenous parameters and the baseline
shares using equations (5)–(7).
We compute predicted changes using the covariates-based model and the calibrated-sharesprocedures in 100 simulations of finite-sample data. In each of the 100 simulations, we implement
the following steps:
1. To generate “observed” baseline data, draw a finite-sample realization of the labor allocationfrom the parameterized continuum model.
2. Estimate ϵ and other parameters of the covariates-based model using the “observed” realization. Compute baseline shares ℓkn and ykn for the covariates-based model using theseparameter estimates. Compute baseline shares ℓkn and ykn = ℓkn wn /δ̄kn for the calibratedshares procedure using the finite-sample realization of ℓkn drawn in the previous step and the“true” values of wn and δ̄kn used in the data-generating process.
3. Using equations (5)–(7), compute the increase in productivity required to match the truechange in employment for the “treated” tract for both the covariates-based approach and thecalibrated-shares procedure.
4. Compute the predicted change in commuter counts (ℓkn (A′ ) − ℓkn (A)) and rents (r̂k /P̂ − 1)
using both the covariates-based approach and the calibrated-shares procedure.
36

In 2011, Tiffany & Co. moved its corporate headquarters to 260,000 square feet of office space at 200 Fifth
Avenue.

Appendix - 1

5. Regress the true change in commuters destined for the treated tract on the changes predictedby the covariates-based approach. Regress the true change in commuters destined for thetreated tract on the changes predicted by the calibrated-shares procedure. Regress the truechange in real rents on the changes predicted by the covariates-based approach and calibratedshares procedure, respectively. Compute the mean squared error of each set of predictions.
The results of these 100 simulations are reported in Figure 2, Panel A of Table 2, Figure A.1,
and Table A.1. Figure A.1 shows that the calibrated-shares procedure predicts changes in rents
poorly with the number of individuals in the data. Table A.1 shows that the slope coefficient
becomes close to one when the number of individuals is two or three orders of magnitude larger.
Figure A.1: Monte Carlo: Calibrated-shares procedure predict rents poorly
A. Regression of true on predicted changes.4
Fraction
.2

0

0

.1

10

100

Density

20

200

.3

30

300

B. Ratio of models’ prediction errors

.2

.4

.6

.8

1
0

0

Covariates−based: slope
Calibrated−shares: slope

Covariates−based: intercept
Calibrated−shares: intercept

0.0001 .0002 .0003 .0004 .0005 .0006 .0007 .0008 .0009
Covariates−based MSE / Calibrated−shares MSE.001

Notes: This figure describes predicted rent changes from the same simulations that produced the predicted commuterchanges in Figure 2 and Panel A of Table 2. This figure depicts the regression coefficients and mean squared errors
from 100 simulations in which I = 2, 488, 905, the data-generating process is the estimated covariates-based model of
New York City in 2010, and the counterfactual change is an 9% increase in productivity in one workplace tract. The
predicted change in real rents is compared to the continuum-model change in real rents.

Table A.1: Calibrated-shares procedure’s finite-sample performance predicting rent changes
I
Calibrated-shares: slope
Calibrated-shares: intercept
Calibrated-shares: MSE

2.5

5

12.5

25

50

125

250

2560

0.192
0.808
417.560

0.311
0.689
225.459

0.537
0.464
85.3280.696
0.304
43.4690.820
0.180
21.960

0.918
0.082
9.332

0.958
0.042
4.884

0.996
0.004
1.049

Notes: This table describes predicted rent changes from the same simulations that produced the predicted commuterchanges in Panel A of Table 2.

A.2

Monte Carlo exercise: Finite-sample changes

In any empirical application, the researcher does not observe changes in outcomes for a continuumof individuals. The observed changes in outcomes come from finite-sample draws from the pre- andAppendix - 2

post-shock data-generating processes. We simulate this setting by drawing two labor-allocation
realizations for each simulation, the first from the estimated model of New York City in 2010 andthe second from that model with 9% higher productivity for the treated workplace tract. For eachsimulation, we execute six steps. The first four are identical to those described in the previous
section, Appendix A.1. The last two are
5. Generate “observed” post-shock data by drawing a finite-sample realization of the labor
allocation from the continuum model at the counterfactual productivity vector, which has
9% higher productivity for the treated workplace tract. Compute “observed” changes incommuter counts by subtracting the step-1 baseline realization of commuter counts fromthese realized post-shock commuter counts.
6. Regress the “observed” changes in commuter counts destined for the treated workplace tract
on the changes predicted by the calibrated-shares procedure. Regress the “observed” changes
in commuter counts destined for the treated workplace tract on the changes predicted by thecovariates-based approach.
The results of these simulations are reported in Panel B of Table 2 and Figure A.2. The
additional finite-sample noise raises the MSEs for both methods relative to the simulations describedin Section A.1.
Figure A.2: Calibrated-shares procedure overfits in finite-sample simulations.25

B. Ratio of models’ prediction errors.2
Fraction
.15

0

0

.05

1

1

.1

2

Density

2

3

3

4

A. Regression of finite-sample changes on predictedchanges

0

.5

1

1.5

2
0

−.5

Covariates−based: slope
Covariates−based: intercept

Calibrated−shares: slope
Calibrated−shares: intercept

.8

.82
.84
.86
Covariates−based MSE / Calibrated−shares MSE.88

Notes: This figure depicts the regression coefficients and MSEs from 100 simulations in which I = 2, 488, 905,
the data-generating process is a finite-sample realization from the estimated covariates-based model of New
York City in 2010, and a finite-sample realization from that model with a 9% increase in productivity in
one workplace tract. “Covariates-based” means applying the parameterization described in Section 3.2 to thesimulated data. “Calibrated-shares” means using the observed (simulated) shares in the exact hat algebra
defined by equations (5)–(7).

A.3

Monte Carlo simulations with unobserved disutility, λkn ̸= 1

Next, we consider a data-generating process in which commuting costs have both an observed andunobserved component in order to examine the relative predictive performance of the covariatesAppendix - 3

based model and calibrated-shares procedure in granular settings. The commuting costs are now
determined byln δkn = ln δ̄kn + ln λkniidln λkn ∼ N 0, Λ × Var(ln δ̄kn ) ,Λ ∈ R+ .

The parameter Λ governs the variance of the unobserved component of commuting costs, statedrelative to the variance of the observed component. The simulations shown in Table 2 are the case inwhich Λ = 0, so that λkn = 1 ∀k, n. When Λ > 0, the covariates-based model omits an unobserved
component of commuting costs that the calibrated-shares procedure may, in principle, capture.
Since ln δ̄kn and ln λkn are orthogonal by assumption, the covariates-based model’s predictions will
be unbiased, but its forecast errors will increase with the magnitude of unobserved variation Λ. For
Λ = 0.1, 0.25, 0.5, and 1.0, we compute 100 simulations using 100 realizations of the λkn matrix.
Tables A.2 and A.3 present the results of these simulations for continuum-model changes andfinite-sample changes, respectively. As established in the Λ = 0 case, which appears in Table 2, thecalibrated-shares procedure is highly data demanding. For all values of Λ reported in Tables A.2
and A.3, the calibrated-shares procedure’s predictions become unbiased (slope near one) only as thenumber of individuals becomes orders of magnitude larger than the true value. Since the calibratedshares procedure can capture the unobserved component of commuting costs, its mean forecast
error becomes smaller than that of the covariates-based model when the unobserved variation is
sufficiently large. For the number of individuals in the empirical setting (I = 2, 488, 905), thisoccurs only when the unobserved component of commuting costs varies at least a quarter as muchas the observed component (Λ ≥ 0.25) when we evaluate continuum-model changes and more thanhalf as much as the observed component (Λ > 0.5) when we evaluate finite-sample changes.

A.4

Monte Carlo simulations: Finite-sample changes with fixed ν IThis appendix revisits the Monte Carlo investigation of Section 3.3 using a data-generating process
in which the economic primitives change from Υ to Υ′ but the set of idiosyncratic preferences
ν I is fixed. This leverages the model introduced in Section 4 as the data-generating process.
Contrasted with the finite-sample outcomes with iid-ν idiosyncrasies, these finite-sample simulations
are substantially closer to the continuum-change outcomes.
Figure A.3 is the fixed-ν analogue of Figure 2, depicting the distributions of the regression
coefficients and relative MSE for the covariates-based model and calibrated-shares procedure. The
covariates-based model outperforms the calibrated-shares procedure. The regression coefficientresults are very similar to Figure 2. The covariates-based model remains an unbiased predictor,
while the calibrated-shares procedure has a median slope of 0.79. The ratio of mean squared errors
is closer to one in these simulations because the noisy finite-sample outcomes mean that both
approaches have larger forecast errors than in the continuum-outcomes case. The MSE ratio in the
median simulation is about 84%.
Fixing the idiosyncratic preferences reduces the excess churn in simulations with iid ν draws:
Appendix - 4

Table A.2: Monte Carlo simulations with λkn ̸= 1: Regressand is continuum change
Slope (mean)
Covariates-based Calibrated-shares

MSE (mean)
Covariates-based Calibrated-shares

Λ

I

0
0
0
0
0
0
0
0

2.5
5
12.5
25
50
125
250
2560

0.99850.99920.99950.99980.99991.00001.00001.00000.78170.87590.94790.97370.9864
0.9946
0.99710.99970.00140.00070.00030.00010.00010.00000.00000.00000.2252
0.11300.04520.02270.01120.00450.00230.0002

0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1

2.5
5
12.5
25
50
125
250
2560

1.00051.00131.00201.0022
1.0022
1.00231.00231.00230.79010.88180.94800.97480.98670.9951
0.99710.99970.03710.03640.03600.03590.03580.03580.03580.03580.22540.11360.04480.02260.01130.00450.00230.0002

0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25

2.5
5
12.5
25
50
125
250
2560

1.00331.00441.00451.00471.00491.00491.00491.00490.82270.9021
0.9581
0.97880.98950.99580.99790.99970.23280.23210.23180.23160.23150.23150.23150.23150.22640.11270.04520.02260.01130.00450.00230.0002

0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5

2.5
5
12.5
25
50
125
250
2560

1.00481.00561.0062
1.0063
1.00651.00661.00661.00660.8907
0.9416
0.97640.9876
0.99320.99760.99900.99991.05131.05041.05011.04981.04971.04971.04971.04970.22630.11220.04500.02260.01130.00450.00220.0002

1
1
1
1
1
1
1
1

2.5
5
12.5
25
50
125
250
2560

0.9954
0.99650.99720.99690.9971
0.9971
0.99720.99720.96880.9837
0.99330.99650.99820.99940.99950.99986.3762
6.37496.37456.37506.37486.37476.37466.37460.2176
0.10920.0441
0.02180.01090.00440.00220.0002

Notes: This table reports the mean outcomes across 100 simulated event studies for both the covariates-basedmodel and the calibrated-shares procedure. The data-generating process is the covariates-based model estimatedon data for New York City in 2010 augmented by an unobserved component of commuting costs of magnitude Λ(see text). The regressand is the continuum change in commuters. The value of I is stated in millions of individuals.
The value labeled “2.5” million is in fact I = 2, 488, 905. The Λ = 0 results also appear in Panel A of Table 2.

Appendix - 5

Table A.3: Monte Carlo simulations with λkn ̸= 1: Regressand is finite-sample change
Slope (mean)
Covariates-based Calibrated-shares

MSE (mean)
Covariates-based Calibrated-shares

Λ

I

0
0
0
0
0
0
0
0

2.5
5
12.5
25
50
125
250
2560

0.9796
0.9745
1.0073
1.00771.00071.00260.99980.9999-0.4075
0.19430.66950.83530.91350.9676
0.9821
0.998214.3836
7.17222.87371.43510.71870.28740.14280.014017.02228.48593.39981.69950.85060.34030.16920.0166

0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1

2.5
5
12.5
25
50
125
250
2560

1.0013
1.00670.98361.00131.0012
1.00271.00041.0024-0.34390.2516
0.66570.83540.91490.96780.98130.998314.3945
7.25562.90031.47480.75480.32400.17950.049616.98688.55993.39231.70150.85190.34010.16990.0166

0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25

2.5
5
12.5
25
50
125
250
2560

1.0056
0.99600.99781.0032
1.0072
1.00801.00481.0049-0.1295
0.38210.73340.86420.9343
0.97600.98770.998514.75227.4577
3.10751.6716
0.94790.52000.37820.245717.1357
8.53073.40101.70500.84780.34000.17040.0166

0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5

2.5
5
12.5
25
50
125
250
2560

1.01540.98951.0002
1.00801.00261.00501.00851.00690.32140.62220.84710.9246
0.9561
0.98480.99360.999215.4545
8.19273.91982.48791.7592
1.33981.19291.062917.01978.4782
3.3937
1.69240.85590.33970.16940.0165

1
1
1
1
1
1
1
1

2.5
5
12.5
25
50
125
250
2560

0.98970.97881.00130.99730.99790.99980.99830.99870.80440.89380.95780.97880.9881
0.99820.99671.000420.7317
13.6669
9.18487.84617.0405
6.69496.50386.392916.7998
8.5801
3.39621.70510.84290.34000.16850.0165

Notes: This table reports the mean outcomes across 100 simulated event studies for both the covariates-basedmodel and the calibrated-shares procedure. The data-generating process is the covariates-based model estimatedon data for New York City in 2010 augmented by an unobserved component of commuting costs of magnitude Λ(see text). The regressand is the simulation-specific change in commuters from a realization drawn from the modelusing pre-shock parameter values to a realization drawn from the model using post-shock parameter values. The
value of I is stated in millions of individuals. The value labeled “2.5” million is in fact I = 2, 488, 905. The Λ = 0
results also appear in Panel B of Table 2.
Appendix - 6

Figure A.3: Calibrated-shares procedure performs poorly in fixed-ν I simulations
B. Ratio of models’ prediction errors

Fraction
.2

0

0

.1

2

4

Density
5

.3

6

.4

10

8

A. Regression of observed on predicted changes

0

.5

1

1.5
0

−.5

Covariates−based: slope
Covariates−based: intercept

Calibrated−shares: slope
Calibrated−shares: intercept

.7

.75
.8
.85
Covariates−based MSE / Calibrated−shares MSE.9Notes: This figure depicts the regression coefficients and MSEs from 100 simulations in which I = 2, 488, 905,
the data-generating process is the estimated covariates-based model of New York City in 2010 with fixed
idiosyncratic preferences ν I , and the counterfactual change is a 9% increase in productivity in one workplace
tract.

across 100 simulations, the standard deviation of the total employment increase in the “treatedtract” is about 180 in the iid-ν case and 51 in the fixed-ν case. Eliminating excess churn reduces
the forecast error for both approaches, but their relative predictive performance is similar to theiid case. Across 100 simulations, the covariates-based model’s average MSEs with iid and fixed
idiosyncratic shocks are 14.38 and 1.23, respectively. The calibrated-shares procedure’s average
MSEs are 17.02 and 1.45, respectively.

B

Event studies

This appendix presents additional details regarding the event studies examined in Section 3.4.

B.1

Two employment booms

Figure B.1 depicts the time series of total employment in two New York City census tracts that
contain 200 Fifth Avenue and 111 Eighth Avenue. Between 2010 and 2012, Tiffany & Co. andGoogle both substantially increased their employment in these two tracts. These tracts are two of
the 83 examined in the event studies in Section 3.4.

B.2

Event studies using temporal and geographic aggregation

Figure B.2 present the results of pooling multiple years of data and aggregating the predictedchanges in number of commuters to the booming workplace tracts from residential tracts up to
residential NTAs.

Appendix - 7

Figure B.1: Employment in two of the event-study tracts
Tract containing 111 Eighth Avenue

Employment (000s)
4
6
8

10

12

Employment (000s)
12 14 16 18 20 22

Tract containing 200 Fifth Avenue

2

3

4

5

6

7

8

9 10 11
Year (2000s)

12

13

14

15

16

17

2

3

4

5

6

7

8

9 10 11
Year (2000s)

12

13

14

15

16

17

Notes: This figure depicts the number of primary jobs held by New York City residents in tracts
36061005800 and 36061008300 in the LODES data.

B.3

Estimating the model using larger spatial units

One may try to avoid the calibrated-shares procedure’s overfitting problem by analyzing larger
spatial units. Studying New York City’s 195 Neighborhood Tabulation Areas (NTAs) instead oftracts reduces the number of locations by an order of magnitude and the number of location pairsby two orders. To do so, we aggregate tract-to-tract commuter counts up to NTA-to-NTA flows,
compute NTA-to-NTA commuting costs as the average of tract-to-tract commuting costs δ̄kn , andcompute NTA-level wages using employment-weighted averages of tract-level wages. We use boththe covariates-based model and the calibrated-shares procedure to predict changes in NTA-to-NTA
commuting flows for the 35 NTAs that had 2010-2012 employment growth of at least 12.5% and at
least 2,000 employees in 2010.
When analyzing these substantially larger spatial units, the covariates-based approach and thecalibrated-shares procedure perform similarly. Figure B.3 shows that the covariates-based model’s
median slope coefficient is closer to one. The covariates-based model has a lower MSE than thecalibrated-shares procedure in 18 of the 35 events.

B.4

Noise reduction via low-rank approximation

B.4.1

The approximation procedure

One method for guarding against idiosyncratic noise in the observed commuting matrix is to use
values of ℓkn derived from a low-rank approximation of the matrix rather than the observed values
when plugging ℓkn and ykn = ℓkn wn /δ̄kn into equations (5)–(7). That is, for some particular rankr, replace the commuting matrix L = [ℓkn ] withL̃ = argmin

M∈RK×N∥L − M∥2F ,subject torank(M) ≤ r,

(B.1)

where ∥ · ∥F is the Frobenius matrix norm. The solution to this problem can be computed fromthe singular value decomposition (SVD) of the original matrix L. In particular, the optimal
approximation L̃ is given by the following algorithm (Hastie, Tibshirani, and Wainwright, 2015):
Appendix - 8

Figure B.2: Models’ predictive performance with temporal and geographic aggregation
B. Temporal aggregation: Prediction errors.4

1

Fraction
.2
.3

1.5

Covariates−based: slope
Covariates−based: intercept
Calibrated−shares: slope
Calibrated−shares: intercept

0

0

.1

.5

Density.5

2

A. Temporal aggregation: Regressions

−3

−2

−1

0

1

2

3

D. Geographic aggregation: Prediction errors.2

2

C. Geographic aggregation: Regressions.05

1

Fraction
.1

1.5

.15

Covariates−based
Calibrated−shares

0

0

.5

Density.1 .2 .3 .4 .5 .6 .7 .8 .9 1 1.1 1.2 1.3 1.4
Covariates−based MSE / Calibrated−shares MSE

0
−1

0

1

2

3

.5
1
1.5
2
2.5
3
Covariates−based MSE / Calibrated−shares MSE

3.5

Notes: This figure contrasts the covariates-based model and calibrated-shares procedure’s predictionsfor 83 tract-level employment booms when employing temporal aggregation or geographic aggregation.
Panels A and B depict results when using baseline data for 2008-2010 instead of only 2010. Panels C
and D depict predictive performance after aggregating the tract-level predictions underlying Panels A and
B of Figure 3 up to 195 Neighborhood Tabulation Areas. We sum the predicted change in commuters tothe booming workplace tract across all residential tracts in an NTA. In panel A, One slope coefficient is
not depicted for the calibrated-shares procedure. In Panel C, only the slope coefficients are depicted.
Vertical lines depict the medians of the slope coefficient distributions.

 Compute the SVD of L = UΣVT , where U is a K × K orthonormal matrix, Σ is a non-negative K ×N diagonal matrix with diagonal elements representing the singular values sorted
in descending order, and VT is an N × N orthonormal matrix.
 Keep only the first (largest) r singular values in Σ, to obtain L̃ := U

!
Σr 0
0

0

V T , where Σris the upper left r × r block in Σ.
However, this approximation is imperfect for our purposes on two accounts: first, while theoriginal matrix L is non-negative (as it consists of counts), there is no guarantee that the rank-r
approximation is also non-negative. We therefore set all negative entries in L̃ to 0. Additionally,L̃ must be rescaled so that the sum across its entries matches the original; that is, we obtain L̃SVD
rAppendix - 9

Figure B.3: Models’ predictive performance across 35 NTA-level employment booms
B. Ratio of models’ prediction errors.15

1

Fraction.1

1.5

Covariates−based: slope
Covariates−based: intercept
Calibrated−shares: slope
Calibrated−shares: intercept

0

0

.05

.5

Density.2

2

A. Regression of observed on predicted changes

−3

−2

−1

0

1

2

3

.4

.6 .8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6
Covariates−based MSE / Calibrated−shares MSE

Notes: This figure depicts predictive performance when fitting both approaches using data aggregated up
to the 195 Neighborhood Tabulation Areas defined by the New York City Department of City Planning. In
panel A, 10 intercept coefficients are not depicted for the covariates-based model. 5 intercept coefficientsare not depicted for the calibrated-shares procedure. Panel A depicts the distributions of slope coefficients,
and the vertical lines denote the median slope coefficients.

by replacing
negative entries in the rank-r SVD approximation with zeros, then rescaling such thatP  SVD P= k,n ℓkn .
k,n L̃rkn

Figure B.4 visualizes the approximated matrices derived from three choices of rank. alongside
the observed 2010 data and the fitted values from the covariates-based and rank-1 interactivefixed-effects specifications. (The IFE procedure is described in detail in Appendix B.5.) The
SVD-approximated commuting matrices preserve many features of the observed commuting matrix
even under fairly stringent rank restrictions. In particular, workplaces with a large number ofcommuters are preserved, as are clusters of tracts with comparatively high or low commuting
shares. Some features, however, are discarded by the approximation. Most notably, the higher
values on the diagonal of the matrix (which corresponds with commuters who choose to live andwork in the same tract) are missing from low-rank approximations (e.g., rank 5 and rank 16) but arecontained in higher-rank approximations (e.g., rank 200). This is because the diagonal of a matrix
is a fundamentally “high-rank” phenomenon and cannot in general be replicated by a low-rankapproximation.
Table B.3 summarizes the sparsity generated by approximations derived from various ranks,
including both the SVD-based procedure described above and the non-negative factorization procedure described in Section B.4.3.
B.4.2

Selecting the rank

When applying any rank-restricted algorithm, the rank of the approximation is a key choice.
Since we are using low-rank approximation to reduce idiosyncratic noise, the choice of the rank
parameter r balances the trade-off between the signal and noise in observed commuting flows. We
Appendix - 10

Figure B.4: Approximations of the commuting matrix
A. Observed commuting matrix

B. Fitted values, CBM

C. Rank-5 SVD

D. Rank-16 SVD

E. Rank-200 SVD

F. Fitted values, IFE rank 1

Notes: This figure depicts the observed 2010 commuting matrix, the fitted covariates-based model (CBM) of ℓkn ,approximations derived from rank-5, rank-16, and rank-200 singular value decompositions, and a rank-1 interactivefixed-effects specification. Rows and columns are ordered by FIPS codes, so that all but eight adjacent entries are in
the same county.

present multiple procedures from which a researcher can select to choose the rank used; some arecomputationally and conceptually simple (drawing from the established norms in machine learning),
while a Monte Carlo simulation is more complex and tailored to a particular context. We describe
each procedure in turn and present results, including their suggested rank.
Selecting the rank by simple methods

After computing the SVD of the commuting matrix,

we obtain an ordered list of singular values σ1 , σ2 , . . . , σN with σi ≥ σj for i < j. These are depicted
in a scree plot in Figure B.5, which normalizes the singular values by dividing each by the sumσof all singular values (i.e., P i′σi ) and restricts attention to the 25 largest singular values. Thisi

normalization allows us to interpret each singular value as representing a fraction of the matrix’s
total variation (Cohen, 2021, p.499).
Figure B.5 allows us to evaluate two distinct rank criteria. The first is an informal procedure
Appendix - 11

Figure B.5: Scree plot (ordered singular values) for NYC 2010 commuting matrix

Notes: This figure shows the magnitude of successive singular values for the NYC 2010 commuting matrix. The
dashed line indicates the threshold of 0.5% of the sum of the singular values.

known as the “elbow” method, which is a method of visual inspection that attempts to identify
the singular values that are substantially larger than the rest. The second, the variance criterion,
is numerically founded but nonetheless requires a cutoff chosen by the researcher (Cohen, 2021,
p.496). For the variance criterion, we compare the magnitude of each singular value with 0.5%Pof the sum of singular values; that is, we identify the set {σi | σi ≥ 0.005 j σj }. This cutoff isshown with a dashed horizontal line in the scree plot. The two methods offer substantially different
answers in our case, with the elbow method suggesting a lower rank of 4 and the 0.5% variancecriterion suggesting a rank of 16. We select the more generous of these values and use rank 16 as
our preferred specification in the main text.
Selecting the rank using Monte Carlo simulations Another method for selecting the rank
is dependent on the research setting permitting Monte Carlo simulations. One can select therank based on ranks’ performance across Monte Carlo simulations (using regression coefficientsand MSEs summarizing commuting-flow predictions). While substantially more computationallycomplex than the simple methods above, this procedure yields results that are more closely tied tothe particular structure of the empirical setting.
Starting from the steps outlined in Appendix A.1 across 100 Monte Carlo draws from thecontinuum model, we follow steps 1 through 5 with two exceptions. First, we consider only changes
in commuter flows, not rents; and second, we substitute the SVD-approximated commuting countsℓ̃SVD
kn for the “observed” counts, ℓkn , which are drawn from the data-generating process. Note that
this requires us to approximate the commuting matrix separately for each of the Monte Carlo draws.
As in Appendix A.1, we estimate the commuting elasticity ϵ for each of the Monte Carlo draws.
The inputs into this commuting elasticity estimation are the “observed” values as in the covariatesbased model, not the SVD-approximated commuting matrix. Finally, we raise productivity in thesame tract (containing 200 Fifth Avenue) as in the Appendix A.1 simulations.
Appendix - 12

Table B.1 reports the performance across ranks in these Monte Carlo simulations. Rank 18
performs best in these simulations based on MSEs and is nearly best based on regression coefficients.
Rank 16, which is suggested by the variance criteria shown in Figure B.5, performs very similarly.
The lower panel of Table B.1 presents the event study performance (for the same 83 employment
booms as in Section 3.4.3) for approximations derived from a wide variety of ranks. The rank-16
results are shown in Figure 4.
B.4.3

Non-negative matrix factorization

While the previously outlined SVD procedure is one means of implementing noise reduction,
truncating the commuting shares to be non-negative means that the resulting approxmiation is
not necessarily low-rank. Another approach is to use an algorithm that imposes the non-negativity
constraint. That is, we find a solution to the problem
L̃NNMF = argminM∈RK×N∥L − M∥2F

subject torank(M) ≤ r, and [M]kn ≥ 0.

(B.2)

Solving this problem is more difficult and, unlike in the unconstrained case in which thesolution is uniquely determined by SVD, there may be multiple (local) optima.

We use theGreedy Coordinate Descent algorithm from Hsieh and Dhillon (2011) with a fixed initialization as
implemented in the Julia package NMF. This procedure produces low-rank approximated matricesthat deliver very similar performance as the computationally simpler SVD procedure describedabove. The results for event studies are shown in Table B.2.
B.4.4

Results

Table B.1 reports the event study and Monte Carlo performance of the SVD-approximated matrices.
Table B.2 reports the event study performance of the non-negative matrix factorization (NNMF)
approximations across ranks. As the non-negative matrix factorization appears to provide no
meaningful improvement over the simpler SVD approximation procedure, we do not present any
further analysis via Monte Carlo simulations for that procedure. Finally, Table B.3 reports the
proportion of the approximated commuting matrix’s entries that are zero across ranks for both
approximation methods. While we used δ̄kn to calculate ykn = wn ℓkn /δ̄kn in line with the rest of
the paper, we have verified that the SVD approximation predictions are nearly identical when using
ykn = wn ℓkn .

B.5

Interactive fixed effects

B.5.1

Modeling unobserved commuting costs

This section extends the covariates-based model to incorporate unobserved disutility components ofcommuting costs. We assume that λkn has a low-rank factor structure. Specifically, we parameterize

Appendix - 13

Table B.1: Summary of SVD Monte Carlo and event study results by rank
1

2

3

4

5

6

8

Rank
12
14

10

15

16

18

20

50

100

500 1000 1500 2143

Monte Carlo performance
Slope 1.03 1.05 1.05 1.04 1.02 1.01 1.00 1.00 1.00 1.00 1.00 1.00 1.00 .99 .91 .78 .61 .78 .78 .78
Int. -.039 -.060 -.057 -.049 -.021 -.014 -.002 -.005 -.005 -.004 -.003 -.001 .002 .011 .110 .265 .268 .268 .269 .269
MSE .1583 .0462 .0461 .0460 .0370 .0357 .0331 .0320 .0321 .0316 .0313 .0309 .0305 .0305 .0893 .2214 .5637 .2252 .2252 .2252
Event study performance
Slope .73 .70 .71 .80 .83 .86 .85 .83 .83 .82 .82 .81 .80 .79 .62 .32 -.43 -.47 -.47 -.46
Int.
.06 .14 .14 .09 .08 .08 .10 .13 .13 .14 .14 .15 .16 .17 .30 .51 .80 .82 .82 .82
MSE 10.53 10.38 10.37 10.29 10.27 10.26 10.27 10.30 10.32 10.32 10.38 10.40 10.45 10.48 10.96 11.71 12.94 13.23 13.35 13.39

Notes: This table reports the predictive performance, measured by the average values of the regression slope andintercept coefficients and MSE, using SVD approximations of various ranks across two cases: a) 100 Monte Carlosimulations, with each drawn from the parameterized continuum model with a 9% productivity shock to the tractcontaining 200 Fifth Avenue, where the performance is evaluated relative to the continuum model’s predictions;and b) the 83 event studies examined in Section 3.4.

Table B.2: Summary of NNMF event study results by rank
1

2

3

4

5

Rank
8
10

6

12

14

16

18

20

15

50

Event study performance
Slope .73 .70 .71 .78 .80 .81 .87 .87 .86 .86 .87 .83 .82 .82 .70
Int.
.06 .14 .14 .11 .09 .10 .09 .10 .10 .10 .10 .13 .14 .14 .24
MSE 10.53 10.38 10.37 10.28 10.27 10.28 10.25 10.29 10.24 10.26 10.26 10.39 10.36 10.41 10.85
Notes: This table reports the predictive performance, as measured by the average values of the regression slope
and intercept coefficients and MSE, across the 83 event studies examined in Section 3.4 using various ranks of
NNMF low-rank approximated commuting matrices.

Table B.3: Percentage (%) of approximated commuting matrix entries equal to zero
1

2

3 4

5

6

8

10

Rank
14 15

12

16 18 20 50 100 500 1000 1500 2143

SVD
0 1.9 2 6 11 12 15 15 16 17 17 17 17 18 22 26 3741
4285
NNMF 0 .4 .1 3 4.1 3.9 3.6 1.9 1.6 1.5 1.4 1.4 1 1.1 .5
Notes: This table reports the proportion of entries in the approximated commuting matrices, across ranks, thatare equal to zero. These values are shown for both the SVD and the NNMF procedures, though we omit high-rankapproximations via NNMF because of computational burden. The SVD procedure results in many more zeros,
particularly for larger ranks, than the NNMF, largely due to the generation of negative values by SVD prior to ourtruncating them to zero.

the overall commuting costs δkn as
δkn = δ̄kn × exp(ψk′ γn ),| {z }=λIFEkn

where ψk and γn are R × 1 vectors that give rise to an R-dimensional factor structure. The log
likelihood function then becomes"ln LIFE =Xk,nℓkn ln P

−ϵwnϵ rkα δ̄kn exp(ψk′ γn )ϵk′ ,n′ wn′

#

−ϵ .
rkα′ δ̄k′ n′ exp(ψk′ ′ γn′ )Appendix - 14

(B.3)

In practice, to implement the estimation of the parameters, we follow Chen, Fernández-Val,
and Weidner (2021), who estimate interactive fixed effects in a Poisson model. Here, multiple
practical difficulties arise. Most critically, the likelihood is no longer convex; in particular, there is
the possibility of multiple local optima. To guard against this, we use multiple randomized starting
points and select the parameter estimates that produce the highest likelihood. Another difficultyis that the computational cost of this procedure rises rapidly with the factor dimension R. We
consider R ≤ 3. Note that in all specifications, additive fixed effects are already included to obtain
estimates for wnϵ and rkα ; that is, R gives the dimension of the “purely” interactive fixed effects.37
In the main text, we show the estimates from the model with R = 1 for two reasons. First, we
take advantage of the relative computational ease of the R = 1 case to estimate from a large numberof starting points and find a consistent convergence pattern that suggests our estimate is the true
optimum. Second, the eigenvalue ratio test suggested by Chen, Fernández-Val, and Weidner (2021)
indicates that R = 1 is the optimal factor structure among the set of options {1, 2}. Possible choicesof R span the range from the covariates-based specification (when R = 0) to the calibrated-sharesprocedure (the full-rank case).38 The event study results for rank R = {2, 3} are very similar to
those for rank R = 1 and are shown below.
B.5.2

Estimation results

Table B.4 presents the estimates from maximizing the log likelihood function in equation (B.3)
using the algorithm outlined by Chen, Fernández-Val, and Weidner (2021). We show results for
R = 0, 1, 2, 3, where R = 0 is identical to the covariates-based model and the OLS column is
included for the sake of comparison. The magnitude of the commuting elasticity ϵ diminisheswith the rank of the factor structure. As expected, the model fit metrics increase with the factor
structure rank, as higher ranks correspond to more free parameters to attain better (in-sample) fit.
B.5.3

Event study results for rank R = {2, 3}

In Figure B.6, we display the event study performance for IFE factor structure ranks 2 and 3. The
extreme similarity in performance across the ranks suggests that rank 1 is sufficient to capture mostof the expected gains from using IFE over the CBM.
37

This differs from the notation in Chen, Fernández-Val, and Weidner (2021), who include the additive fixed effects
in the factor dimension R. Our purely interactive factor dimension is, in their notation, R2 .
38
Consider the case of a 2 × 2 commuting matrix A. This matrix can be decomposed into two 2 × 2 matrices (e.g.,
AI). However, there are many such decompositions, illustrating a failure of identification when fitting the commuting
matrix with full-rank interactive fixed effects. Notice also that simply limiting the dimensions of the decomposition is
not sufficient. In the 2 × 2 case, the only matrices A that can be written as the product of a 2 × 1 with a 1 × 2 vector
are those with rank 1. As our commuting matrix is full rank, this suggests why a very high rank (and accompanying
identification issues) would be needed to reproduce the calibrated-shares procedure using only high-rank interactive
fixed effects.

Appendix - 15

Table B.4: Commuting elasticity estimates from IFE specification
OLS
Commuting cost(pseudo-)R2-2.307
0.561

Location pairs
Commuters

690,6732,488,905

R=0

MLE
R=1 R=2

R=3-7.9840.662-7.1760.685-6.3590.701-6.652
0.694

4,628,880
2,488,905

Notes: All specifications include residence fixed effects and workplace fixed effects. The columns under“MLE” present the results from the maximum likelihood estimation described in the text, with R denoting theinteractive fixed effects factor structure rank. The “OLS” column presents the results of estimating the log
version of equation (2) by ordinary least squares, omitting observations where ℓkn = 0. The model-fit statistic
is pseudo-R2 for MLE and R2 for OLS.

Figure B.6: Event study performance: varying factor structure rank, IFE

2.4

A. Rank 2.3
Fraction
.2
0

.5

.1

1

Density

1.5

Covariates−based: slope
Covariates−based: intercept
Rank 2 IFE: slope
Rank 2 IFE: intercept.7 .75 .8 .85 .9 .95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35
Covariates−based MSE / Rank 2 IFE MSE

0
−3

−2

−1

0

1

2

3

2

.3

B. Rank 3.2
Fraction
.1

1
0

0

.5

Density

1.5

Covariates−based: slope
Covariates−based: intercept
Rank 3 IFE: slope
Rank 3 IFE: intercept

−3

−2

−1

0

1

2

3

.7 .75 .8 .85 .9 .95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35
Covariates−based MSE / Rank 3 IFE MSE

Notes: This figure summarizes the predictive performance of the covariates-based model and the ranks 2 and3 IFE specifications, analogous to Figure 5. Panel A depicts the distributions of slope coefficients, and thevertical lines denote the median slope coefficients.

Appendix - 16

B.6

Tests of in-sample fit

In this appendix section, we assess the in-sample fit of different model parameterizations. The χ2
test contrasts the expected and observed frequencies of outcomes and compares this difference to a
critical value from the test-statistic distribution. Since in our application the expected frequenciesof outcomes are small, the test statistic may not follow the asymptotic χ2 distribution. We therefore
construct the test statistic’s distribution using a parametric bootstrap.
P (ℓobs −E[ℓ ])2
The test statistic is χ2 = kn kn E[ℓknkn, where ℓobs
kn is the observed number of individuals]
who reside in k and work in n in the data and E[ℓkn ] is the expected number of commuters (given
by ℓkn in equation (2)). We construct the model-implied distribution of this test statistic from aparametric bootstrap in the following way:
1. We calculate the test statistic using
χ2 simulated =X (ℓsim − E[ℓkn ])2kn

kn

E[ℓkn ]

,

iid

ℓsim
kn ∼ Multinomial(I, {pkn })′ ′i > Uiwhere pkn ≡ Pr(Ukn
k′ n′ ∀(k , n ) ̸= (k, n)) is the model-implied probability of choosing

kn defined in equation (8) and I = 2, 488, 905 is the total number of commuters.
2. We repeat the first step 1,000 times to obtain the parametric-bootstrap-based distributionfˆ χ2 simulated .
The χ2 test statistic for the covariates-based model is extremely large (calculated as 8.445e299,
though numerical error may be sizable). This very large value of the test statistic is driven by outlier
observations for which the model’s expected number of commuters is extremely small (i.e., nearly
equal to zero). By comparison, the test statistics for the interactive-fixed-effects specifications aremuch smaller: 1.005e7, 1.268e7, and 8.968e6 for ranks R = 1, 2, and 3, respectively. Nonetheless,
all these test statistics exceed the 99th percentiles of the distributions of the χ2 test statistics shownin Figure B.7A.
By contrast, the calibrated-shares procedure has perfect in-sample fit by construction and a χ2
test statistic of zero. This fit is too good to be true: the test statistic of zero for the observed data
is lower than all simulated values in the distribution generated by the parametric bootstrap.
We also assess the in-sample fit produced by the rank-restricted singular value decomposition(SVD) introduced in Section 3.4.3. One issue for this specification is that the rank-restricted
SVD assigns zeros to some residence-workplace pairs that have positive counts in the data. A
zero-probability event occurring rejects the parameterized model. We calculate the mean squared
error (MSE) as an alternative test statistic that sidesteps the divide-by-zero issue. Specifically, we
1 P
obs
2
calculate MSE = K×N
kn (ℓkn − E[ℓkn ]) , where K is number of residence tracts and N is thenumber of workplace tracts. The parameterized bootstrapped distributions of MSE test statistics
are shown in Figure B.7B. The approximated commuting matrix derived from a rank-110 SVD
would lie within the 99-percent confidence interval, whereas higher-rank approximations exhibit an
in-sample fit that is too good to be true.
Appendix - 17

Figure B.7: Distributions of χ2 and MSE test statistics
A. Distributions of χ2 test statistics

0

0

50

5.000e−06

100.00001150.000015200.00002

B. Distributions of MSE test statistics

4.4e+06

4.6e+06

4.8e+06
CBM
IFE R25.0e+06

5.2e+06

0.53

0.54

IFE R1IFE R3

SVD R16
SVD R200

0.54

0.55

SVD R100
SVD R500

P (ℓobs −E[ℓ ])2
Notes: The test statistic in the left panel is χ2 = kn knE[ℓknkn, where ℓobs
kn is the number of commuters from k]
to n and E[ℓkn ] is the expected value of commuters from k to n (given by ℓkn in the continuum model’s equation (2)).
We depict the distributions between their 0.5th and 99.5th percentiles. The test statistics using the commuter counts
in the 2010 LODES data are 8.445e299, 1.005e7, 1.268e7, and 8.968e6 for thePCBM, rank-1 IFE, rank-2 IFE, and
2
obs
obs
1
rank-3 IFE, respectively. The test statistic in the right panel is MSE = K×N
kn (ℓkn − E[ℓkn ]) , where ℓkn is thenumber of commuters from k to n, E[ℓkn ] is the expected value of commuters, K is number of residence tracts, and N
is the number of workplace tracts. We depict the distributions between their 0.5th and 99.5th percentiles. The test
statistics using the commuter counts in the 2010 LODES data are 1.467, 0.5725, 0.3581, and 0.1799 for the rank-16,
rank-100, rank-200, and rank-500 singular value decompositions, respectively.

B.7

Spatially correlated idiosyncratic preferences

This appendix section relaxes the assumption that idiosyncratic preferences are independent acrossgranular spatial units. In particular, preferences for residence-workplace pairs have a nested-logit
structure, such that individual i’s indirect utility of residing in k and working in n isi
= ϵ̊ ln
Ůkn
|



wn



rkα P 1−α δkn
{z
}

i
,
+ν̊kn

(B.4)

≡Ůkn

where Ůkn denotes the mean utility of choice kn and the parameter ϵ̊ governs the importance ofmean utility relative to idiosyncratic preferences in the nested-logit specification. We allow thei to be correlated across residences within Neighborhood Tabulation
idiosyncratic preferences ν̊kn

Areas (NTAs).
that the vector of idiosyncratic preferences has a cumulative distribution
 We assume
P

ζ
P
i /ζ
−ν̊kn
F (ν̊ i ) = exp − z, where Bz denotes the set of residence-workplace kn(k,n)∈Bz epairs with the residence k in NTA z, and ζ governs the correlation of idiosyncratic preferencesamong alternatives in the set Bz . Whereas draws of idiosyncratic preferences between residenceworkplace pairs with residences in distinct NTAs remain uncorrelated, the within-residence-NTA
correlation of idiosyncratic preferences is 1 − ζ 2 . Given equation (B.4) and this distribution ofAppendix - 18

idiosyncratic preferences, the fraction of people residing in k and working in n is
hP
iζ

ϵ̊/ζ
α δ ′ ′ −ϵ̊/ζϵ̊/ζ α
−ϵ̊/ζ
w
r
′
′
′
′
(k ,n )∈Bz nk kn
wn (rk δkn )×
Pkn = PhPiζ ,ϵ̊/ζ α

P
−ϵ̊/ζ
ϵ̊/ζ
α δ ′ ′ −ϵ̊/ζwr(k′ ,n′ )∈Bz wn′ (rk′ δk′ n′ )z′(k′ ,n′ )∈B ′ n′k′ k n|{zz}=Pkn|Bz

|

{z

}

=PBz

where PBz denotes the probability of residing in NTA z and Pkn|Bz denotes the probability of
selecting residence-workplace pair kn among residents of NTA z.
Under the assumption of nested-logit preferences and no unobserved disutility (δkn = δ̄kn ∀kn),
the counterfactual-baseline ratio of commuters is
"
!ϵ̊/ζ

ŵn
α
r̂ δ̄ˆ

ℓ̂kn =

P(k′ ,n′ )∈Bz

×

k kn

ŵn′
r̂α δ̄ˆ ′ ′z ′ PBz′

× Pk′ n′ |Bzk′ k n"PP(k′ ,n′ )∈Bz′

#ζ−1

ϵ̊/ζ

ŵn′
r̂α δ̄ˆ ′ ′#ζ if ℓkn > 0.

ϵ̊/ζ× Pk′ n′ |Bz′

k′ k nThis specification yields the following log likelihood function:ln L ≡

Xℓkn ln Pkn =k,nXϵ̊/ζ α−ϵ̊/ζ × wn (rk δkn )ℓkn ln hPϵ̊/ζ
(k′ ,n′ )∈Bz wn′

z′


.

−ϵ̊/ζ iζ
rkα′ δk′ n′

P hP

k,n

−ϵ̊/ζ iζ−1α′′rk′ δk nϵ̊/ζ
(k′ ,n′ )∈Bz′ wn′

We sequentially estimate the parameters of the nested-logit model (Train, 2009). In the firststep, we maximize the log likelihood with respect to ϵ̊/ζ, residence fixed effects, workplace fixed
effects, and nest-specific fixed effects. Given the fixed effects, the estimate of ϵ̊/ζ will be equal tothe estimate of ϵ in the logit case. Note that the residence fixed effects are unique only up to a−αϵ̊/ζ

nest-specific normalization. We normalize the residence fixed effects relative to rk

for some

residence k in each NTA z. We index the baseline residence within a NTA as k̄(z). Once we impose
−αϵ̊
the normalization, we have residence fixed effects r rk
. We can therefore write
k̄(z)



−αϵ̊

rk
rk̄(z)

Pkn|Bz =P(k′ ,n′ )∈Bz



rk ′
rk̄(z)−ϵ̊wnϵ̊ δ̄kn

−αϵ̊

1/ζ

wnϵ̊ ′ δ̄k−ϵ̊′ n′
1/ζ =rk

−αϵ̊rk̄(z)Iz−ϵ̊
wnϵ̊ δ̄kn

1/ζ
,

and

PBz

=P

−αϵ̊ ζ
rk̄(z)
Iz−αϵ̊ ζ
z ′ rk̄(z ′ ) Iz ′where Iz is the inclusive value of nest z. In the second step, we compute the inclusive values using
the estimates from the first step. Having the estimates of the inclusive values, Iˆz , we maximize the

Appendix - 19,

following log likelihood function with respect to rk̄(z) :ln L̃ ≡

X


X


z

(k,n)∈Bzℓkn  ln PBz =

X


X


z

(k,n)∈Bzℓkn  × ln  P−αϵ̊ ˆζrk̄(z)Izz′−αϵ̊ ˆζrk̄(z′ ) Iz ′Note that we cannot simultaneously identify rk̄(z) and parameter ζ because the number ofparameters exceeds the number of alternatives. We assume that the value of parameter ζ is known,in order to estimate rk̄(z) up to a proportionality coefficient. We compute the equilibrium prices
and the remaining economic primitives {Tk } and {An } from the estimated fixed effects and themarket-clearing conditions.
We estimate the model for two values of parameter ζ: 0.25 and 0.75. We estimate each stage ofthe model by using the Poisson pseudo maximum likelihood estimator described in Section 3.2.Whenζ is 0.75, the elasticity parameter ϵ̊ is 5.99 (= 7.99 × ζ). The implied productivities are unchanged,
and the implied land endowments have a correlation of 0.99 with the land endowments in thebaseline model. When ζ is 0.25, the elasticity parameter ϵ̊ is 2.0. The implied productivities areunchanged, and the implied land endowments have a correlation of 0.61 with the land endowmentsin the baseline model.
As long as the nested-logit model and the logit-model agree on the unconditional choice probabilities (i.e., they imply the same Pk′ n′ ), we note that in the nested-logit model there is greater
within-nest substitution in response to a change in commuting costs than in the simple logit model.
This can be seen in the second of the following three elasticities, noting that Pk′ n′ |Bz > Pk′ n′ :−ϵ(1 − Pk′ n′ |Bz ) − ϵ̊(Pk′ n′ |Bz − Pk′ n′ )∂ ln Pkn= ϵ ζPk′ n′ + (1 − ζ)Pk′ n′ |Bz′′∂ ln δk nϵ̊P ′ ′kn

if k ′ = k, n′ = n and (k, n) ∈ Bz
if k ′ ̸= k, (k ′ , n′ ) ∈ Bz and (k, n) ∈ Bz
if (k ′ , n′ ) ∈/ Bz and (k, n) ∈ Bz

We use the model to compare the predictive performance of the covariates-based model andthe calibrated-shares procedure in 83 employment booms that are detailed in Section 3.4. Figure B.8 contrasts the predictions of the two procedures for each value of the parameter ζ. As thefigure shows, the results are robust to the alternative assumptions about preferences. Even whenidiosyncratic preferences are highly correlated within residence NTAs, the covariates-based model
considerably outperforms the calibrated-shares procedure.

B.8

Alternative labor demand elasticities

We compare the predictive performance of the covariates-based model and the calibrated-sharesprocedure for varying values of σ, the local labor demand elasticity. For both σ = 1.1 and σ = ∞,the covariates-based model has a lower mean squared error than the calibrated-shares procedurein 80 of the 83 events.39 The covariates-based model outperforms the calibrated-shares procedure
39

Note that the specification with σ = 1.1 does not satisfy the sufficient condition for uniqueness.

Appendix - 20

Figure B.8: Comparison of methods’ predictive performance with nested-logit ν i
B. ζ = 0.75: Ratio of prediction errors.15

1

Fraction.1

1.5

Covariates−based: slope
Covariates−based: intercept
Calibrated−shares: slope
Calibrated−shares: intercept

0

0

.05

.5

Density.22

A. ζ = 0.75: Regression coefficients

−3

−2

−1

0

1

2

3

.1

.15

1

Fraction.1

1.5

Covariates−based: slope
Covariates−based: intercept
Calibrated−shares: slope
Calibrated−shares: intercept

0

0

.05

.5

DensityD. ζ = 0.25: Ratio of prediction errors.22

C. ζ = 0.25: Regression coefficients.2 .3 .4 .5 .6 .7 .8 .9 1 1.1 1.2
Covariates−based MSE / Calibrated−shares MSE

−3

−2

−1

0

1

2

3

.1

.2 .3 .4 .5 .6 .7 .8 .9 1 1.1 1.2
Covariates−based MSE / Calibrated−shares MSE

Notes: The pairs of panels in this figure depict results analogous to those in Figure 3 for the nested-logit model
introduced in this appendix with nesting parameter ζ = 0.75 (Panels A and B) and ζ = 0.25 (Panels C and D).
The results for ζ = 0.75 and ζ = 0.25 are not numerically identical, but they typically coincide to two decimal
places and are visually indistinguishable.

regardless of the value of σ because their predictions are not very sensitive to σ. Varying σ does
not meaningfully change the slope, intercept, or MSEs. As an example, Figure B.9 shows that theslope coefficients obtained when using σ = 1.1 or σ = ∞ are very similar to those for σ = 4.0 in
the calibrated-shares procedure.

B.9

Local increasing returns to scaleThis appendix section details the process for computing counterfactual predictions with local
increasing returns and shows the results of such an exercise. In short, the predictions from boththe covariates-based model and the calibrated-shares procedure are not sensitive to local increasing
returns over the range of relevant values.
Per Appendix C.7.4, we assume that the increasing returns are external to the firm and that
Appendix - 21

Figure B.9: Event-study slope coefficients by local labor demand elasticity σ
B. Slope coefficients for σ = ∞ versus σ = 4.0

−4

−3

−2

−1

0

1

1
0
−1
−2
−3
−4

Slope coefficient for CSP predictions when sigma = inf

1
0
−1
−2
−3
−4

Slope coefficient for CSP predictions when sigma = 1.1

A. Slope coefficients for σ = 1.1 versus σ = 4.0−4−3

Slope coefficient for calibrated−shares predictions when sigma = 4.0
Slope

−2

−1

0

1

Slope coefficient for calibrated−shares predictions when sigma = 4.0

45 Degree LineSlope

45 Degree Line

Notes: The slope coefficients come from regressions of the observed change in commuters on the predictedchange in commuters for each of the 83 tract-level employment booms.

the production function in location n is given by qn = Ān Lη+1
n . This implies the goods market
clearing condition is
!1+η
X

Ān

ℓkn /δ̄kn=

wn−σ ĀσnkPk ℓkn /δ̄kn
1−σ
P

ησ
Y.

This can be rearranged to solve for wn :σ−1σ

wn = Ān! ση−1−ησXℓkn /δ̄kn

σ−1

1

P σ Y σ.

k

To express the counterfactual outcomes via exact hat algebra, take ratios to obtain:1η+1ŵn = Āˆn

X
k

ykn
ŷkn Pk ′ yk ′ n!1−σ(σ−1)(η+1)

11P̂ η+1 Ŷ (σ−1)(η+1) .

This expression for changes in wages generalizes equation (5) to the case of local increasing returns.
Equations (6) and (7) are unchanged. The change in the CES price index isXP̂ = kŵnĀˆ L̂η!1−σ P

n n

 1
1−σ
y
k kn 
.
Y

In the event studies of 83 local employment booms, the covariates-based model outperforms thecalibrated-shares procedure regardless of the value of η; varying η does not meaningfully change theslope, intercept, or MSEs. The differences in predicted commuting flows typically differ only on theorder of 10−2 relative to the baseline, even when using values of η larger than the sufficient condition
for the equilibrium uniqueness. As an example, Figure B.10 shows that the slope coefficients
Appendix - 22

Figure B.10: Event-study slope coefficients by local increasing returns η
B. Slope coefficients for η = 0.1 versus η = 0

−2

−1

0

1

−2

−1

0

1
−3

−3

Slope coef. for CSP predictions when η = 0.1

−4

−4

1
0
−1
−2
−3
−4

Slope coef. for CSP predictions when η = 0.0028

A. Slope coefficients for η = 0.0028 versus η = 0−4

Slope coef. for CSP predictions when η = 0
Slope

−3

−2

−1

0

1

Slope coef. for CSP predictions when η = 0

45 Degree LineSlope

45 Degree Line

Notes: The slope coefficients come from regressions of the observed change in commuters on the predictedchange in commuters for each of the 83 tract-level employment booms. The baseline case, η = 0, is themodel without increasing returns.

obtained when using η = .0028 or η = 0.1 are very similar to those for η = 0 in the calibratedshares procedure.

Appendix - 23

C

Theory: Proofs and extensions

Section C.1 provides a condition on parameter values sufficient for the existence and uniquenessof the continuum model’s equilibrium. Section C.2 details the exact hat algebra for computing
counterfactual outcomes. Section C.3 provides further details on the calibrated-shares procedure.
Section C.5 shows that the trade equilibrium of the model with a finite number of individuals is
unique. Section C.6 shows that the continuum model is a limiting case of the model with a finite
number of individuals. The following subsection presents extensions of the bare-bones model in
Section 4 that introduce trade costs, the use of land in production, residential amenities, and local
increasing returns.

C.1

Existence and uniqueness of continuum model’s equilibrium

We derive a set of parameter values sufficient for the existence and uniqueness of the continuum
model’s equilibrium. As shown in Theorem 1 of Allen, Arkolakis, and Li (2023), the spectral radiusof the elasticity matrix characterizes the existence and uniqueness properties of the system.
The equilibrium of the continuum model can be written asX
rk1+αϵ Tk=
Dkn wn1+ϵαnX−ϵLn wn =
Dkn rk−αϵ

(C.1)
(C.2)

k
σ
σ−1
A1−σ
n Ln wn = P

XLn′ wn′ ,

(C.3)

n′

where, for notational convenience, we defineΘ≡XX

wnϵ (rkα δkn )−ϵn−1 −(1+ϵ)
Dkn ≡ LΘ δkn.
kEquations (C.1)–(C.3) are 3×N equations in 3×N unknowns {rn , wn , Ln }n=1,2,...,N given valuesfor Θ and P . Although Θ and P are endogenous, they do not vary across locations. Accordingly,
they fall under the class of endogenous scalars discussed in Remark 4 of Allen, Arkolakis, and Li(2023), which leave the conclusion of their Theorem 1 Part ii.b unchanged.
Following Remark 5 of Allen, Arkolakis, and Li (2023), we construct the intermediate matrices
Γ and B:
1 + αϵ 0 0


Γ= 0
−ϵ 1
0
σ 1



0

1+ϵ 0




B = −αϵ

0


0 .

0

1

1

Appendix - 24

Then we can construct the final matrix A defined by (A)hh′ ≡


0

1+ϵ 0



1
1+αϵ


BΓ−1 = −αϵ00  0

0

1

1

0
1
− σ+ϵ
σ
σ+ϵ

0

0


BΓ−1 hh′ :



1 ,
σ+ϵ 
ϵ
σ+ϵ



0

 αϵ
=  1+αϵ

A

0

1+ϵ
σ+ϵ

1+ϵ
σ+ϵ

0


0 

σ−1
σ+ϵ



1+ϵ
σ+ϵ

where we’ve assumed σ > 1.
We want to show that the spectral radius, the absolute value of the maximum eigenvalue, of A
(denoted ρ(A)) equals one. The reasoning in Remark 6 in Appendix B.1.5 of Allen, Arkolakis, andLi (2023) demonstrates that ρ(A) ≥ 1. Thus, it remains to show that ρ(A) ≤ 1. To do so, we will
use Lemma 1 in Appendix B.1.6 from Allen, Arkolakis, and Li (2023). Consider the matrix


λ

 αϵ
λI − A = − 1+αϵ
0

1+ϵ
− σ+ϵ

1+ϵ
− σ+ϵ



λ

0

− σ−1
σ+ϵ

1+ϵ
λ − σ+ϵ.

Define the function f (λ) ≡ det(λI − A), and denote its kth derivative by fk (λ):
 


 



1+ϵ
1+ϵ
αϵ
1+ϵ
1+ϵ
αϵ
σ−1
f (λ) = λ − λ
−
λ−
−
σ+ϵ
σ+ϵ
1 + αϵσ+ϵσ+ϵ
1 + αϵ
σ+ϵ

 


1+ϵ
1+ϵ
αϵ
f1 (λ) = 3λ2 − 2λ
−
σ+ϵ
σ+ϵ
1 + αϵ2(1 + ϵ)f2 (λ) = 6λ −
.
σ+ϵ
3

2



Lemma 1 of Allen, Arkolakis, and Li (2023) states that ρ(A) ≤ 1 if and only if fk (1) ≥ 0 for
k = 0, 1, 2.


 



1+ϵ
1+ϵ
αϵ
1+ϵ
αϵ
2+ϵ−σ
−
+
σ+ϵ
σ+ϵ
1 + αϵσ+ϵ1 + αϵ
σ+ϵ

 


1+ϵ
1+ϵ
αϵ
f1 (1) = 3 − 2
−
σ+ϵ
σ+ϵ
1 + αϵ2(1 + ϵ)f2 (1) = 6 −.
σ+ϵf (1) = 1 −Both f1 (1) ≥ 0 and f2 (1) ≥ 0 for any σ > 1 and α, ϵ > 0. Inequality f (1) ≥ 0 can be restated as


1+ϵ
σ+ϵ



αϵ
1 + αϵ



1
≤ .
2

(C.4)

Note that condition (C.4) is satisfied by our baseline parameter values (α = 0.24, σ = 4, ϵ = 7.986).
Under condition (C.4), ρ(A) ≤ 1, which means that ρ(A) = 1. Thus, by Theorem 1 Part ii.b of
Allen, Arkolakis, and Li (2023), if condition (C.4) is true, the equilibrium exists and is unique up
to scale.

Appendix - 25

C.2

Exact hat algebraThis section shows how to express a counterfactual equilibrium of the model in terms of counterfactual endogenous outcomes relative to baseline endogenous outcomes, counterfactual exogenous
parameters relative to baseline exogenous parameters, constant elasticities, and baseline equilibrium
shares. This method has been dubbed “exact hat algebra” in the international trade literature.
One can solve for the counterfactual equilibrium variables associated with combinations ofcounterfactual-to-baseline ratios of productivities Ân , land endowments T̂k , and commuting costs
λ̂ and δ̄ˆ .
kn

kn

The derivation of ŵn starts from equation (3):
An

X ℓkn
δ̄kn

k

=

(wn /An )−σ
Y
P 1−σ

∀n.

Rearranging terms to isolate wn and then taking ratios yields
σ−1σŵn = Ân

 −1

ℓ′kn
σ
k δ̄ ′
 P kn ℓknk δ̄kn

P

|

{z

}

−1
≡L̂nσ

!1
 σ−1 P′σP′ σk′ ,n′ yk′ n′P.
P
k′ ,n′ yk′ n′| {z } |
{z
}


≡P̂

σ−1
σ

1

≡Ŷ σ

Next, we rewrite this in terms of ℓ̂ˆkn and baseline shares:
δ̄kn

 −1

P

σ−1

ŵn = Ân σ σℓ̂kn ℓknk δ̄ˆ δ̄kn 
P knℓkn k δ̄knσ−1

1P̂ σ Ŷ σ
−1

σ−1
σ

= Ân

!σ
ℓkn
δ̄kn
P ℓ
δ̄ˆkn k δ̄kn
kn

X ℓ̂knkσ−1

1P̂ σ Ŷ σ−1σ
Multiply both sides by ŵn σ , simplify, and take both sides to the power σ−1
to obtain

σ−1
σ

ŵn

σ−1
σ

= Ân

ŵn = Ân

!− 1
σ
σ−1
1
ykn
ŷkn PP̂ σ Ŷ σk ′ yk ′ n
k
! 1
1−σ
X
1
ykn
ŷkn PP̂ Ŷ σ−1 ,k′ yk′ nXkwhich is equation (5).

Appendix - 26

(C.5)

To derive P̂ , rearrange terms and employ goods market clearing: ′ 1−σ
P̂

1−σ

=

X
n

wn
A′n

P  wn′ 1−σ =n′An′X  ŵn 1−σ
Ân

n



wn
An

1−σ

P  wn′ 1−σ =n′An′X  ŵn 1−σ X ykn
n

Ân

k

Y

.

This impliesP̂ =X  ŵn 1−σ X ykn
Ân

n

k

! 1

1−σ

.

Y

This shows that relative goods price index P̂ can be written in terms of relative endogenous wages,
relative exogenous productivities, and baseline (earnings) shares.
The ratio of counterfactual nominal output Y ′ to baseline nominal output Y is
P
P
′
X
Y′
ykn
k,n yknk,n ŷkn ykn
Ŷ =
=P
= P
=
ŷkn
Y
Y
k,n yknk,n yknk,nWith the changes in output, changes in prices, and changes in wages, equation (5) is expressed
entirely in terms of hats and shares.
Next, we derive r̂k . Dividing the counterfactual version of equation (4) by the baseline version
yieldsy′T̂k = r̂k−1 Pn kn .
n ykn
PRearranging this expression yieldsy′r̂k = T̂k−1 Pn kn = T̂k−1
n ykn
P

P
X
ykn
n ŷkn ykn
P= T̂k−1ŷkn P,yn kn
n′ ykn′nwhich is equation (6).
Finally, taking the ratio of counterfactual ℓ′kn to baseline ℓkn using equation (2) yields, after

Appendix - 27

considerable manipulation, equation (7):′Pℓ̂kn =

P

= P

′

′

′

wnϵ (rkα δ̄kn λkn )−ϵ
′ϵ
′α ′
′
−ϵ
k′ ,n′ wn′ (rk′ δ̄k′ n′ λk′ n′ )−ϵϵ (r α δ̄wn
k kn λkn )ϵα−ϵk′ ,n′ wn′ (rk′ δ̄k′ n′ λk′ n′ )ŵnϵ (r̂kα δ̄ˆkn λ̂kn )−ϵ
′
′ϵ
′α ′
−ϵ
′ wn′ (rk′ δ̄k′ n′ λk′ n′ )
ϵ
α
−ϵ
w
(r
δ̄
λ
)
′
′
′
′
′
′
k ,n
n′ k ′ k n k n
′

Pk ,n

=

ŵnϵ (r̂kα δ̄ˆkn λ̂kn )−ϵ′′′′wϵ (rα δ̄ ′ ′ λ ′ ′ )−ϵwnϵ′ (rkα′ δ̄k′ n′ λk′ n′ )−ϵP n′ ϵk′ kαn k n−ϵk′ ,n′ wϵ ′ (rα′ δ̄k′ n′ λk′ n′ )−ϵk′ ,n′ wn′ (rk′ δ̄k′ n′ λk′ n′ )
n
k

P
=P

ŵnϵ (r̂kα δ̄ˆkn λ̂kn )−ϵ.
ϵαˆ−ϵ ℓk′ n′′ ŵ (r̂ δ̄ ′ ′ λ̂ ′ ′ )

k′ ,n

n′

k′ k nkn

LTogether, equations (5), (6), and (7) deliver ŵn , r̂k , and ℓ̂kn given the elasticities σ and ϵ,commuting costs δ̄kn , baseline equilibrium values ℓkn and wn , and counterfactual-to-baseline ratiosÂ , T̂ , λ̂ , and δ̄ˆ .
n

k

C.3

kn

kn

Calibrated-shares procedure

The calibrated-shares procedure uses the observed ℓkn and ykn = wn ℓkn /δ̄kn in equations (5)–(7).
This implicitly calibrates combinations of model parameters. For example, this procedure implicitly
rationalizes zero-commuters observations with infinite commuting costs, ℓkn = 0 ⇐⇒ δkn = ∞.
−ϵ=This procedure cannot characterize cases in which ℓkn = 0 and ℓ′kn ̸= 0 because the object δ̂kn

ϵ
δkn
′
is not sensibly defined if δkn = ∞ and δkn < ∞.
δ′knThis procedure does not identify the parameters An , Tk , and δkn . Given the elasticities σ andϵ and the baseline equilibrium shares ℓkn and ykn , equations (2) (3), and (4) are insufficient toseparately identify Tk and δkn . One would also need to observe land prices rk .
When implementing this procedure in Sections 3.4 and 5.1, we compute tract-level workplace
wages using LODES and ZIP Business Patterns data following Owens, Rossi-Hansberg, and Sarte(2020).

C.4

Welfare changes in the continuum model

Recall from equation (1) that the indirect utility of an individual i residing in k and working in nisi
Ukn
= ϵ logwn
rkα P 1−α δkn

Appendix - 28



i
+ νkn
.

Following equation (3.10) in Train (2009), one can show that the expected utility Ū of each (exante identical) worker, each of whom selects her utility-maximizing residence-workplace pair, is


X
 i


−ϵ
i
 + c ∀k, n,
Ū = Eν i Ukn |Ukn
≥ Uki ′ n′ , ∀k ′ , n′ = log 
wnϵ ′ rkα′ P 1−α δk′ n′(C.6)

k′ ,n′

where c is a constant. Note that c is unknown since the level of utility cannot be measured. DenoteCkn = wn / rkα P 1−α δ̄kn as the real consumption for individuals residing in k and working in n.
We can rewrite equation (C.6) in terms of real consumption:
Ū = log Xϵ + c,
Ckn
λ−ϵ
kn

k,n

where we use the decomposition of commuting costs δkn into time δ̄kn and disutility λkn withδkn = δ̄kn λkn .
Now, consider a counterfactual equilibrium. Recall that the counterfactual-to-baseline ratio of′xa variablex > 0 is denoted by x̂ ≡ x . The expected utility in the counterfactual equilibrium isP−ϵ + c. We define 1 + ψ as the equivalent variation in consumption′ ϵ ′Ū ′ = log
k,n (Ckn ) (λkn )such that



X
X
′ ϵ ′
 + c = ϵ log(1 + ψ) + Ū .
[(1 + ψ)Ckn ]ϵ λ−ϵ) (λkn )−ϵ  + c = log Ū ′ = log  (Ckn
kn


k,n

k,n

In other words, the equivalent variation in consumption is defined such that multiplying the realconsumption of individuals in all residence-workplace pairs by 1 + ψ in the baseline equilibrium(holding prices and other allocations fixed) would generate the same welfare outcome as thecounterfactual equilibrium with new prices and allocations. Re-arranging the equation above yields
an expression for 1 + ψ in terms of welfare difference Ū ′ − Ū :

1 + ψ = exp



1 ′
Ū − Ū ,ϵwhich is our measure of equivalent variation.
In practice, 1 + ψ can be calculated from changes in wages (ŵn ), rents (r̂k ), and price index(P̂ ), given exogenous changes in productivities (Ân ), land endowments (T̂k ), and commuting costs(δ̂ = δ̄ˆ λ̂ ), as well as initial wages (w ) and commuting flows (ℓ ). One can show that
kn

n

kn kn

kn


1

−ϵ ϵ
X ℓkn
 ,
1+ψ =
ŵϵ r̂α P̂ 1−α δ̂kn
L n k

(C.7)

k,n

where ŵn , r̂k , and P̂ can be solved from equations (5), (6), and (7) (with detailed derivations
Appendix - 29

described in Appendix C.2).

C.5

The trade equilibrium is unique

Recall that the trade equilibrium takes the labor allocation as given. Consequently, we just needto solve for wage and land rent vectors that satisfy equations (3) and (4) given arbitrary labor
allocation ℓkn . Given a labor allocation and vector of wages, equation (4) determines a uniquevector of land rents, so we can focus on solving for a unique vector of relative wages. RecallPP
1−σ 1/(1−σ)
Ln ≡ k δ̄ℓkn is total labor supply in location n. Substituting Ln and P =n (wn /An )kninto equation (3) yields(wn /An )−σY
1−σn′ (wn′ /An′ )An Ln = P∀n.

Note that this equation is homogeneous of degree zero in the wage vector. Selecting w1 = 1 as thenumeraire and writing the expression in terms of relative wages yields

wn =

An
A1

 σ−1 
σ

Ln
L1

−1/σ
.

This defines a unique vector of relative wages. Plugging those wages into equation (4) yields aunique vector of land rents.

C.6

Continuum model as limiting case of the model with a finite number ofindividuals

We derive the result that the equilibrium of the model with a finite number of individuals coincides
with the equilibrium of the continuum model as the number of individuals becomes infinite, I → ∞.
Note that aggregate labor supply L is fixed, as each individual supplies L/I units of labor. The
key step is to show that equation (2) holds as I → ∞. Conditional on the labor allocation {ℓkn },the number of individuals plays no role in the trade equilibrium, which will coincide with that of acontinuum model.
Definition 4.2 says that the share of labor supplied by residents of k who work in n isIoℓkn
1X n i=1 Ũkn (ν I ) > Ũki ′ n′ (ν I ), ∀(k ′ , n′ ) ̸= (k, n) .
L
I
i=1

Since the idiosyncratic preferences are drawn from an independentand identically distributed ran- on
1 PIi
dom variable, the law of large numbers implies that I i=1 1 Ũkn (ν I ) > Ũki ′ n′ (ν I ), ∀(k ′ , n′ ) ̸= (k, n)i >converges to the mean of a binary random variable that is equal to one with probability P(Ũkn

Ũki ′ n′ , ∀(k ′ , n′ ) ̸= (k, n)). Thus,
 n
o


i
i
E 1 Ũkn> Ũki ′ n′ , ∀(k ′ , n′ ) ̸= (k, n)= P Ũkn> Ũki ′ n′ , ∀(k ′ , n′ ) ̸= (k, n) .

Appendix - 30

Using the probability in equation (10),
I
o
 n
o
1X n iilim
1 Ũkn (ν I ) > Ũki ′ n′ (ν I ), ∀(k ′ , n′ ) ̸= (k, n) = E 1 Ũkn
> Ũki ′ n′ , ∀(k ′ , n′ ) ̸= (k, n)
I→∞ I
i=1

w̃nϵ (r̃kα δkn )−ϵ.
ϵα−ϵk′ ,n′ w̃n′ (r̃k′ δk′ n′ )=Pw̃ϵ (r̃α δ)−ϵkn→ P ′ ′nw̃ϵk(r̃kn. Since the continuum-case rational expectations
As a result, as I → ∞, ℓLαδ′ ′ )−ϵk ,nn′k′ k nare the wages and land rents that clear markets for this labor allocation, the market-clearing pricesof the model with a finite number of individuals converge to those values.

C.7

Model extensions

C.7.1

Trade costs

Relative to the model in Section 4, we now assume that goods trade is subject to iceberg trade
costs: delivering a unit of the location-n variety to location k requires producing τnk ≥ 1 units in n.
Individuals consume differentiated goods at their residences, and the price of location n’s output
for consumers in location k is τnk wn /An . Thus, the price index in location k is rkα Pk1−α , where thehP
i1/(1−σ)1−σlocal CES price index for goods is Pk =
(τ
w
/A
)
.
n
n nk nIn this environment, the equivalent of equation (9) is such that, based on the beliefs {w̃n } and{r̃k }, each worker chooses the residential location and the work location that maximize expectedutility,w̃ni= ϵ lnŨknr̃kα P̃k1−α δkn{z}|where P̃k =hP

1−σn (τnk w̃n /An )i1/(1−σ)

!
i
,
+νkn

(C.8)

≡Ũkn

.

With trade costs, the goods-market-clearing condition, which is the equivalent of equation (3),equates quantity supplied and quantity demanded:
An

X ℓkn
k

δkn

−σ

= (wn /An )"X  τnk 1−σ
Pk
k

!#
X

ykn′

.

(C.9)

n′

Equation (4), which clears the land market, is unchanged by the introduction of trade costs.
The definition of a trade equilibrium is akin to Definition 4.1, with equations (C.9) and (4)serving as the relevant market-clearing conditions. The definition of a commuting equilibrium withfinitely many individuals is akin to Definition 4.2 with {τkn } added to the list of economic primitives
i being defined in equation (C.8).
and Ũkn

Appendix - 31

C.7.2

Production uses land

Relative to the model in Section 4, we now assume that production of goods uses a Cobb-Douglas
combination of labor and land inputs. In particular, the quantity of output in workplace locationn is
qn = AnLn
1−β

1−β 

TnP
β

β
,

where Ln is labor and TnP is land used in production. Therefore, the unit cost of production inlocation n iscn =1 1−β β
w
rn
An n


1−σ 1/(1−σ) .
n cnPand the CES price index is P =Labor supplied in location n is given by Ln =ℓkn
k δkn .

P

From the first-order condition, landdemanded for use in production in location n can be expressed as
TnP =

β
1−β



wn
rn

X
k

ℓkn
.
δ̄kn

Substituting this expression into the production technology, output in location n is
1
qn =
An
1−β



wn
rn

β X
k

ℓkn
.
δ̄kn

As in Section 4, each individual devotes 1−α of their expenditure to differentiated goods and αof their expenditure to land, while immobile landlords spend all of their income on differentiatedgoods, such that total expenditure on differentiated goods equals aggregate income. With land usedin production, however, aggregate income now includes the income that accrues to each factor. Total
P
income that accrues to labor is n wn Ln . Total income that accrues to land used in production
equalsX

rn′ TnP′ =n′X ℓk′ n′
β X
β X
wn ′
=
wn′ Ln′ .
1−β ′
1−β ′
δ̄k′ n′′nkn

The demand for each differentiated good stemming from CES preferences means that demand is
proportional to aggregate income. Goods market clearing requires that quantity demanded equals
quantity supplied in each location n:
1
An
1−β



wn
rn

β X
k

ℓkn
c−σ
n
= 1−σ
P
δ̄kn

"

1
1−β

X

wn′

n′

X ℓk′ n′k′

δ̄k′ n′#.

Writing the left-hand side in terms of the unit cost and simplifying yields
wn

"
#
X
X ℓk′ n′
c1−σ
n
= 1−σ
wn′
.
P
δ̄kn
δ̄k′ n′
′
′

X ℓkn
k

n

Appendix - 32k

(C.10)

Residents spend a fraction α of their income on land, so their demand for land, denoted TnR , isgiven byTnR = αX ℓnn′ wn′δ̄nn′ rnk=

α X
ynn′ .
rn ′nClearing the land market means equating the fixed land supply Tn to the sum of the quantities ofland demanded by producers and residents:
Tn = TnP + TnR =

1
rn

"

#
X ℓnn′
β X ℓknwn′ .
wn + α
1−β
δ̄kn
δ̄nn′
′
k

(C.11)n

The definition of a trade equilibrium is akin to Definition 4.1, with equations (C.10) and (C.11)
serving as the relevant market-clearing conditions. The definition of a commuting equilibrium withfinitely many individuals is akin to Definition 4.2 with the Cobb-Douglas production parameter β
added to the list of economic primitives.
C.7.3

Residential amenities

Relative to the model in Section 4, we now assume that each location is endowed with a residential
amenity Bk . Individual i’s utility from residing in k and working in n is nowi= ϵ lnŨkn

Bk w̃nαr̃k P 1−α δkn



i
.
+ νkn

(C.12)

Note that local amenities can depend on local residential density as in Allen and Arkolakis (2014):
P
P
Bk = B̄k ( n ℓkn )−θ , where B̄k is the fundamental amenities in location k,
n ℓkn is the total
number of individuals residing in k, and θ ≥ 0 governs the strength of the congestion force. In thespecial case when θ = 0, local amenities are exogenous.
The definition of a trade equilibrium in Definition 4.1 remains unchanged. The definition of acommuting equilibrium with finitely many individuals is akin to Definition 4.2 with the fundamental
i being defined inresidential amenities {B̄k } and θ added to the list of economic primitives and Ũkn

equation (C.12).
C.7.4

Local increasing returns

Relative to the model in Section 4, we now assume that production exhibits local external economies
of scale. In particular, in each location n the linear production technology is qn = An Ln , where Ln
is the labor supply of workers working in location n and An ≡ Ān Lηn . Thus, the CES price indexhP
i1/(1−σ)η 1−σis P =
w
/(
Ā
L
)
.
n
n
n
n
1+η
P
Since output in location n is Ān, equating quantity supplied and quantity
k ℓkn /δ̄kn
demanded requires
!1+η
Ān

X

ℓkn /δ̄kn=

wn −σ ĀσnkAppendix - 33Pk ℓkn /δ̄kn
1−σ
P

ησ
Y.

(C.13)

The remaining results are unchanged after replacing equation (3) with equation (C.13). The
definition of a trade equilibrium is akin to Definition 4.1, with equations (C.13) and (4) serving as
the relevant market-clearing conditions. The definition of a commuting equilibrium with finitely
many individuals is akin to Definition 4.2 with {Ān } and η added to the list of economic primitives.
Local increasing returns enter the sufficient conditions for the existence and uniqueness of theequilibrium in the continuum model and thereby the uniqueness of the continuum-case rational
1
expectations. We assume that η < σ−1. We define ψ = 1 + η − ησ. Following the same steps as inAppendix C.1, the relevant matrix whose spectral radius we need to evaluate is


0

 αϵ
A =  1+αϵ
0

ψ(1+ϵ)
σ+ψϵ

0
|σ−ψ|
σ+ψϵ

1+ϵ
σ+ψϵ




0 .

1+ϵ
σ+ψϵ

As in Appendix C.1, ρ(A) ≥ 1 and ρ(A) ≤ 1 if and only if fk (1) ≥ 0 for k = 0, 1, 2.



 



1+ϵ
αϵ
1+ϵ
1+ϵ
αϵ
|σ − ψ|1+ϵ−ψ
1−−
σ + ψϵσ + ψϵ1 + αϵσ + ψϵσ + ψϵ1 + αϵσ + ψϵ





1+ϵ
1+ϵ
αϵ
f1 (1) = 3 − 2−ψ
σ + ψϵσ + ψϵ1 + αϵ
1+ϵf2 (1) = 6 − 2.
σ + ψϵf (1) = 1 −Notice that f2 (1) > f1 (1) under the parametric assumptions. We proceed by numerically finding
the values of η that satisfy f (1) ≥ 0 and f1 (1) ≥ 0 given values of α, σ, and ϵ. Under the baseline
parameter values (α = 0.24, σ = 4, ϵ = 7.986), f (1) ≥ 0 and f1 (1) ≥ 0 are satisfied when η ≤ 0.0029.
The equilibrium exists and is unique up to scale when agglomeration forces are modest.

C.8

Galton’s fallacy for Poisson counts

When a process is serially uncorrelated or mean reverting, changes will be negatively correlatedwith prior levels. Consider an AR(1) process such thatyt = (1 − ρ̃)µ̃ + ρ̃yt−1 + ϵt ,where µ̃ is the unconditional mean of yt , ρ̃ ∈ [0, 1) is its serial correlation, and ϵt is iid white noisewith mean zero and variance σϵ2 . If ρ̃ = 0, yt is independently and identical distributed.

Appendix - 34

This process can be rewritten by iterating and then taking the difference:yt = (1 − ρ̃)µ̃ + ρ̃yt−1 + ϵt
= µ̃ +∞Xρ̃i ϵt−ii=0yt+1 − yt = ϵt+1 + (ρ̃ − 1)∞Xρ̃i ϵt−i .

i=0

As a result, the initial level and subsequent change are negatively correlated: Cov (yt , yt+1 − yt ) =−1 2
1+ρ̃ σϵ < 0.

We now consider commuter counts, which, unlike the AR(1) process above, must be integers.
Consider a Poisson process with an AR(1)-style mean:
yt+1 ∼ Poisson ((1 − ρ)µ + ρyt )y0 ∼ Poisson (µ) ,where ρ ∈ [0, 1) is the serial correlation coefficient and ρ = 0 is the iid case. We define the initialcondition to define the unconditional mean.
The following identities are helpful in showing the relationship between a count and its subsequent change:
E[yt+1 ] = µ,

 E yt2 = µ2 +µ,1 − ρ2E[yt+1 yt ] = µ2 +

ρµ.
1 − ρ2Using these, we can show that the covariance is negative:
Cov (yt , yt+1 − yt ) = Cov (yt , yt+1 ) − Var (yt ) =

ρ
µ
ρ−1
−1
µ−
=
µ=
µ < 0.
2
2
2
1−ρ
1−ρ
1−ρ
1+ρ

This matches the expression for the continuous-valued distributions above.

Appendix - 35

D

Descriptive statistics and data sources

Section D.1 presents descriptive statistics for Detroit and Minneapolis-St. Paul analogous to thosefor New York City presented in the main text. Section D.2 examines commuting flows between
counties. Section D.3 shows that commuting flows are often impersistent. Section D.4 shows thatzero commuting flows are often asymmetric. Section D.5 shows that small and zero flows are alarge proportion of cells in the migration matrix and that these flows are impersistent over time.
Section D.6 shows that estimates of workplace fixed effects are biased by dropping observations
with zero commuters. Section D.7 describes the data sources we employ.

D.1

Tract-level commuting flows for Detroit and Minneapolis-St. PaulThis section presents summary statistics for tract-to-tract commuting counts in the Detroit urban
area and the Minneapolis-St. Paul metropolitan area in 2014 analogous to those reported for New
York City in Section 3.1. We chose these examples because Owens, Rossi-Hansberg, and Sarte(2020) study the Detroit urban area and Minnesota is the only state that reports employment by
establishment rather than firm in the LODES data (Graham, Kutzbach, and McKenzie, 2014).
The Detroit urban area has about 1.3 million resident-employees and 1.4 million tract pairs,
so the average cell in its commuting matrix is near one. Among the tract pairs, 74% have zerocommuters between them. In Detroit, 42.6% of commuters have five or fewer commuters in theircell of the commuting matrix.
The Minneapolis-St. Paul metropolitan area has about 1.5 million resident-employees and 0.6
million tract pairs, so the average cell in its commuting matrix has less than three commuters.
Among the tract pairs, 61% have zero commuters between them. In the Twin Cities, 24.3% ofcommuters have five or fewer commuters in their cell of the commuting matrix.
The spatial concentration of employment contributes to the sparsity of these commuting matrices. The median tract in Detroit has 465 employees working in it. Since Detroit has 1,166
residential tracts, at least 60% of locations must have zero residents commuting to this workplace.

D.2

County-level commuting flows

Even when studying larger geographic units, many pairs of locations may have small flows. Consider
US counties, which vary greatly in population size. In the 2006–2010 American Community Survey(ACS) data, there are 136 million commuters (with commutes less than 120 kilometers).40 Of
those 136 million, 101 million live and work in the same county, so there are 35 million crosscounty commuters between 79,188 pairs of counties. Thus, the average off-diagonal element of thecounty-to-county commuting matrix has 445 commuters. However, the distribution of commuters
is extremely uneven. The top 10 county pairs account for more than 2 million commuters alone.
For the bottom 90% of off-diagonal observations, the mean value is only 40 commuters. Almost40

We follow Monte, Redding, and Rossi-Hansberg (2018) by restricting attention to county pairs that are less than
120 kilometers apart.

Appendix - 36

0

10

20

30

40

1

1

.8

.8

.6

.6

.4
.2
0

.2
0

.4

.2
0

0

.2

.4

.4

.6

.6

.8

.8

1

1

Figure D.1: Number of commuters between pairs of tracts in Detroit and the Twin Cities

010

Fraction of tract pairs

20

30

40

50

60

70

Number of individuals

Number of individuals
Cumulative share of commuters

Fraction of tract pairs

Cumulative share of commuters

Notes: These histograms report the number of tract pairs in the Detroit urban area (left panel) and theMinneapolis-St. Paul metropolitan area (right panel) by the number of individuals who reside in the origin tract
and work in the destination tract in 2014 LODES data. These histograms restrict the samples to pairs of tracts
with a strictly positive number of commuters. Observations larger than the 99th percentile are winsorized (40commuters for Detroit urban area and 70 commuters for Minneapolis-St. Paul metropolitan area). Detroit has
1,166 residential census tracts. The Twin Cities has 789 residential census tracts.

half of the county pairs report zero commuters. Among those reporting positive values, Figure D.2
shows that this distribution is skewed, so that many thousands of county pairs have small numbersof commuters. Unlike census tracts, however, these small flows constitute a small share of the total
number of commuters. That is, US counties with tens of thousands of residents do not exhibit thepatterns we described for granular settings, but many county pairs have small commuting flows
and show similar patterns.
In practice, the uncertainty accompanying these small flows is severely compounded by thefact that the ACS is a 1-in-20 sample of the population. In three different five-year waves of the
ACS, nearly half of the county pairs within 120 kilometers of each other are reported to have zerocommuters, as shown in Table D.1. More than half of the non-zero county pairs report fewer than
100 commuters, therefore representing the behavior of five or fewer respondents. As a consequence,as shown in the third column of Table D.1, for more than one-third of the county pairs with positive
commuting flows, the Census-reported margin of error exceeds the reported number of commuters.
Commuting data from other countries show similar features of granular settings. For example,
one-quarter of Germany’s county (Kreisfreie Städte and Landkreise) pairs within 120 kilometers
have fewer than 10 commuters (Krebs and Pflüger, 2019). In Brazil’s 2010 Censo Demográfico,which reports estimates based on a 10% sample of the population, there are 81 million commuters(with commutes less than 60 kilometers).41 There are 7 million cross-municipality commuters
between 131,620 pairs of municipalities. Thus, the average off-diagonal element of the municipalityto-municipality commuting matrix has only 56 commuters.

About three-quarters of the cells inthis commuting matrix are empty.
41

Dingel, Miscio, and Davis (2021) use these data to construct metropolitan areas based on commuting flows.

Appendix - 37

0

0

.2

200

.4

400

.6

600

.8

800

1

Figure D.2: Number of commuters between US counties

0

100

200

300

400

Number of individuals
Fraction of county pairs

Cumulative share of commuters

Notes: This histogram depicts the number of county pairs by the estimated number of commuters in the 2006–2010
ACS. The sample is restricted to pairs of distinct counties within 120 kilometers, the smallest 90% of estimatedcommuter counts among such counties, and only those pairs reporting a strictly positive number of commuters.

D.3

Commuting counts are often impersistent

The integer values appearing in commuting matrices are not very persistent. A residence-workplacepair may have three commuters one year and none the next. Another pair of locations may double
its number of commuters, from one to two. While the conventional continuum approach interpretsthese changes as substantial economic shifts, the finite-sample perspective is that these changes
are not very informative if they are merely a symptom of small counts. We document substantial
churn in commuting counts from year to year, suggesting that there is considerable finite-sample
noise in addition to signal in these commuting counts.
Table D.2, which presents the transition matrix for pairs of tracts in the Detroit urban area and
New York City between the years 2013 and 2014, demonstrates considerable impersistence. In the
Table D.1: Zeros in US county-to-county commuting matrix
DatasetACS 2006-2010
ACS 2009-2013
ACS 2011-2015

Zero Pairs

Positive Pairs

MOE >X (%)

36,40335,54735,002

42,78543,64144,09637
36
35

Notes: This table reports the number of county pairs with zero commuters andnon-zero commuters for three editions of the ACS. The sample is restricted to pairsof counties within 120 kilometers of each other. The final column reports the share ofcounty pairs for which the Census-reported 90% margin of error exceeds the (strictly
positive) reported number of commuters.

Appendix - 38

Detroit urban area, for pairs with one to four commuters in 2013, the percentages appearing on the
diagonal of the transition matrix are quite low. A pair of tracts with one commuter in 2013 wasalmost three times as likely to have zero commuters in 2014 than to have one commuter. A pair withfour commuters in 2013 was more likely to appear in any other column in 2014 than to report fourcommuters again. The 86% of pairs that had zero commuters in both years may appear to suggest
persistence, but this is primarily a symptom of the fact that in both years about three-quarters ofobservations are zero.42 More than 130,000 pairs of tracts that had zero commuters in 2013 had atleast one commuter in 2014. At the same time, 39% of Detroit tract pairs with positive flow in 2013
were zeros in 2014 . Thus, while zeros are pervasive in this commuting matrix, they are not very
persistent. The results for New York City in Table D.2 are very similar to those for Detroit. The
commuter counts are so impersistent that, for many tract pairs, a gravity-based estimate predicts
a tract pair’s commuter count in 2014 better than its observed count in 2013 does.
Table D.2: The impersistence of commuting counts for tract pairs in Detroit and NYC(a) Detroit(b) NYC

2013

0

1

2014
2

3

4

5+

2013

0

1

2014
2

3

4

5+

0
1
2
3
4
5+

0.86
0.60
0.37
0.23
0.15
0.04

0.10
0.22
0.25
0.22
0.17
0.06

0.02
0.10
0.16
0.18
0.17
0.07

0.01
0.04
0.09
0.13
0.14
0.08

0.00
0.02
0.06
0.08
0.11
0.08

0.00
0.02
0.08
0.16
0.26
0.68

0
1
2
3
4
5+

0.91
0.65
0.39
0.24
0.15
0.03

0.07
0.20
0.25
0.22
0.17
0.05

0.01
0.08
0.15
0.17
0.17
0.06

0.00
0.04
0.09
0.12
0.14
0.07

0.00
0.02
0.05
0.08
0.11
0.07

0.00
0.02
0.07
0.16
0.27
0.71

Notes: This table describes pairs of tracts in the Detroit urban area (left panel) and New York City (right panel)by reported number of commuters in the 2013 and 2014 LODES. It is a transition matrix, in which each cell lists theshare of tract pairs in that row (number of commuters in 2013) that belong to that column (number of commutersin 2014). Each row sums to 100%, modulo rounding.

We note considerable churn even when using larger geographic units. Table D.3 shows that 22%of the county pairs reporting zero commuters in the 2006–2010 ACS reported a positive number ofcommuters in the following five-year interval. Conversely, about 18% of pairs reporting a positive
number of commuters had zero commuters in the following five-year interval. For pairs of counties
with a strictly positive number of commuters smaller than 111 in 2006–2011, the diagonal elementsof the transition matrix are less than half. For example, a pair of counties reported to have 71–90
commuters in 2006–2011 has only a 14% probability of appearing in the same bin in the following
five-year interval.
To the extent that the observed impersistence of commuting counts is a small-sample problem,
these findings caution against procedures that infer structural parameters from the relative magni42
If p ∈ [0, 1] of the pairs were randomly independently assigned zero in each period, then p2 of those pairs would
lie in the upper left cell of the transition matrix. Thus, even if zeros were randomly independently assigned to threequarters of the tract pairs in each period, nine-sixteenths of the pairs would be zero in both periods. That would not
be evidence of persistence.

Appendix - 39

Table D.3: The impersistence of commuting counts for pairs of US counties

2011−2015

2006−2010

Initial Share(%) 0
0
1−30
31−50
51−70
71−90
91−110111−500
501−1,500>1,500

45.88
19.64
5.47
3.42
2.50
1.86
12.02
5.01
4.07

0.78
0.35
0.16
0.08
0.05
0.02
0.00
0.00
0.00

1−30

31−50

51−70

71−90

91−110 111−500 501−1,500 >1,500

0.18
0.46
0.36
0.26
0.16
0.11
0.02
0.00
0.00

0.02
0.10
0.19
0.18
0.15
0.12
0.03
0.00
0.00

0.01
0.05
0.12
0.15
0.16
0.15
0.04
0.00
0.00

0.00
0.02
0.07
0.10
0.14
0.12
0.05
0.00
0.00

0.00
0.01
0.04
0.08
0.12
0.13
0.05
0.00
0.00

0.00
0.02
0.07
0.15
0.23
0.35
0.74
0.13
0.00

0.00
0.00
0.00
0.00
0.00
0.00
0.07
0.81
0.06

0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.06
0.94

Notes: This table presents a transition matrix for pairs of counties within 120 kilometers of each other byreported number of commuters in two editions of the ACS. The first column reports the percentage of county
pairs that were in each row in the 2006–2010 data. The remaining columns report the share of county pairs
within the row that appeared in the corresponding bin in the 2011–2015 data. By definition, the shaded cells
in each row sum to one, modulo rounding. The bin boundaries are arbitrary, since Figure D.2 shows no obvious
bunching. We found similar impersistence when using alternative boundaries.

tudes of these counts. The difference between one commuter and two commuters (or one respondentand two respondents in a finite sample) is little evidence that the latter outcome was twice asprobable. Similarly, procedures that rationalize observations with zero commuters by imposing
infinite commuting costs so that these are zero-probability events rule out potential margins ofadjustment based on zeros that appear to be largely transitory.
Tract-pair-level commuter counts are so impersistent that a gravity model estimated using anobserved bilateral characteristic like transit time or distance can sometimes predict future commuter
counts better than the observed value. Table D.4 shows that, for tract pairs with fewer than ten
commuters reported, a gravity-based estimate predicts the following year’s value better than its
current value does. The fitted values from a gravity model estimated using 2013 data have a higher
R2 for predicting observed 2014 values than the observed 2013 values in both Detroit and New
York City. Using the observed 2013 values yields better predictions only for the tract pairs withthe largest 2% of commuter counts.

D.4

Zeros are often asymmetric

The zeros in commuting matrices are often asymmetric. Denoting the number of commuters living
in residence k and working in workplace n by ℓkn , an observed zero is asymmetric when ℓnk = 0 andℓkn > 0. For US counties, ℓnk = 0 for 22% of county pairs with ℓkn > 0.

In Detroit, ℓnk = 0 for

66% of tract pairs with ℓkn > 0. For Brazilian municipalities, ℓnk = 0 for 49% of municipio pairs
with ℓkn > 0. These asymmetric flows are not explained by asymmetric numbers of total workers
or residents: in Detroit, ℓnk = 0 for 47% of tract pairs for which ℓkn > 0 and total employment inAppendix - 40

Table D.4: Gravity-based estimates predict 2014 values better than 2013 values do# of commuters

Share

Gravity: time2013 values

Gravity: distance2013 values≤5≤ 10
Panel B: NYC

0.960
0.983

0.384
0.494

0.308
0.473

0.367
0.465

0.307
0.472

≤5
≤ 10

0.978
0.990

0.362
0.474

0.306
0.475

0.373
0.477

0.306
0.473

Panel A: Detroit

Notes: Each row reports results for all tract pairs in Detroit (upper panel) or New York City (lower panel)with fewer than 5 or 10 commuters in the 2013 LODES data. The “Share” column reports the share of tract
pairs covered by the row. The “Gravity” columns report the R2 obtained by regressing the 2014 number ofcommuters on the number of commuters predicted by a gravity model estimated using 2013 data. The “Gravity:
time” column estimates the gravity equation as described in Section 3.2. The “Gravity: distance” column usesln δkn = ln distancekn rather than commuting costs defined in Section 3.2. The “2013 values” columns report theR2 obtained by regressing the 2014 number of commuters on the 2013 number of commuters. All estimated slope
coefficients are positive. The two “2013 values” columns differ because the regression sample is within observations
where predictions from the gravity model are available. In the distance specification, k = n observations are
dropped. For NYC, the commute is infeasible for two observations.

k and n differs by 10% or less.
The fact that commuting matrices’ zeros are often asymmetric poses a puzzle for calibration
procedures that rationalize zero-commuter observations by infinite commuting costs. This interpretation of zeros implies severely asymmetric commuting costs, even though daily commutes areround-trip journeys. If we believe that commuting from n to k is impossible because we observeℓnk = 0, how do the individuals who live in k and work in n commute home at the end of theday? The mechanisms generating prohibitive commuting costs would have to exhibit within-day
variation. But the most plausible source of intraday variation in commuting costs — congestion
caused by a large number of commuters — cannot explain residence-workplace pairs that have nocommuters.
In many empirical settings, asymmetric zeros may simply reflect small flows rather than evidenceof very asymmetric commuting costs. When most pairs of locations have zero commuters, andthe modal pair with a positive number reflects the decision of only one commuter or one survey
respondent, the difference between zero and one isn’t particularly informative.

D.5

State-to-state migration flows in ACS data

Using ACS data from 2001 and 2002, we show that small and zero flows are a large proportion ofcells in the migration matrix and that these flows are impersistent over time. The ACS 2001 dataset
has 644,427 prime-age individuals out of the total sample of 1,192,206 individuals. Of these, 73,101
individuals moved residences. Of the 73,101 individuals who moved, 80.6% migrated within theirstates. We observe a total of 14,215 between-state movers out of the 644,427 prime-age individuals

Appendix - 41

in the 2001 ACS.

0.1

Density.2.3.4

Figure D.3: Histogram of migration flows between US states, 2001

0

10

20
Number of individuals

30

40

Given the 14,125 between-state movers in 2001 across 2,550 state pairs, the average cell in the
migration matrix has 5.6 movers. About one-third are zero. The histogram of the positive stateto-state migration counts in Figure D.3 exhibits features similar to the histogram of tract-to-tractcommuting counts: a large number of very small flows and a long right tail. About 73% of cells
have fewer migrants than the average value of 5.6. Values of 1 to 5 each account for about 10% of
cells in the migration matrix.
Table D.5: Transition matrix for state migration flows between 2001 and 2002
2002
2001

0

1

2

3

4

5

6+

0
1
2
3
4
5
6+

0.59
0.40
0.32
0.23
0.26
0.13
0.03

0.16
0.17
0.16
0.16
0.13
0.09
0.03

0.13
0.17
0.21
0.21
0.15
0.12
0.06

0.05
0.10
0.09
0.09
0.10
0.14
0.06

0.03
0.06
0.07
0.08
0.06
0.14
0.06

0.01
0.05
0.07
0.09
0.06
0.06
0.07

0.02
0.06
0.09
0.14
0.25
0.31
0.69

Appendix - 42

The 2001–2002 transition matrix in Table D.5 shows considerable impersistence in migration
flows. The particular value of a small state-to-state flow is not stable from year to year, similar to
the transition matrix for commuting counts. In Table D.5, migration flows are winsorized at 6 andvalues are given as a percentage of 2001 migration flows, so the rows sum to one. The diagonal
elements, for which the number of migrants is the same across the two years, are quite small. A
significant portion of the positive migration flows in 2001 are zero in 2002, as shown in the firstcolumn of the table.
Foschi et al. (2023) also report that the ACS migration data exhibit finite-sample problems andsuggest using Internal Revenue Service migration counts instead.

D.6

OLS estimation bias
Figure D.4: Destination fixed effects from tract-to-tract gravity regressions

OLS (omit zeros) destination FE
−1
0
1
2
3
4

NYC (2010)−50
MLE destination FE

5

Notes: This plot depicts the destination fixed effects estimated in Table 1’s column 1 (horizontal axis) andcolumn 2 (vertical axis). See the notes to Table 1 for details.

Figure D.4 shows that estimates of workplace fixed effects are biased by dropping observations
with zero commuters. When only using positive-commuter observations, the residence and workplace fixed effects characterize the average number of commuters for a residence-workplace pair,conditional on the number of commuters being strictly greater than zero. These conditional averages
are necessarily greater than the unconditional averages, and the difference is larger for workplacelocations with fewer workers.43 Figure D.4 contrasts the OLS (vertical axis) and maximum likelihood (horizontal axis) estimates of the workplace fixed effects. The difference is stark: the range
43
Census tracts are defined so that the number of residents is similar across tracts, while there is tremendous
heterogeneity in total employment. Thus, the selection bias is evident in the workplace fixed effects. When estimating
the analogous gravity regression for county-to-county commuting flows, we find that the selection bias manifests inboth the origin and destination fixed effects.

Appendix - 43

of the OLS estimates is half that of the maximum likelihood estimates because of considerable
truncation from below. In essence, the popular practice of omitting zero-commuter observations
attributes low-employment destinations’ lower employment counts to infinite commuting costs, notlower wage beliefs (lower productivity). By contrast, our maximum likelihood estimator infers thatthese destinations are less attractive workplaces from the fact that many origins have zero residents
working in these destinations.

D.7

Data sources

This subsection describes the data sources we employ.
Longitudinal Employer-Household Dynamics, Origin-Destination Employment Statistics (LODES). The block-level origin-destination data for New York City, Detroit, and MinneapolisSt. Paul and the block-level workplace area characteristics data for New York City are obtained
from LODES version 7.4, published by the US Census Bureau.44 Following Owens, Rossi-Hansberg,
and Sarte (2020), we use the primary job counts.
The origin-destination data for NYC, Detroit, and Minneapolis-St. Paul are in the following
files:
Geography

filepath within https://lehd.ces.census.gov/data/lodes/LODES7/

NYC

from ny/od/ny od main JT01 2002.csv.gz
to ny/od/ny od main JT01 2017.csv.gz

Detroitfrom mi/od/mi od main JT01 2009.csv.gz
to mi/od/mi od main JT01 2014.csv.gz

Minneapolis-St. Paul

from mn/od/mn od main JT01 2010.csv.gz
to mn/od/mn od main JT01 2014.csv.gz
from mn/od/mn od aux JT01 2010.csv.gz
to mn/od/mn od aux JT01 2014.csv.gz
from wi/od/wi od main JT01 2010.csv.gz
to wi/od/wi od main JT01 2014.csv.gz
from wi/od/wi od aux JT01 2010.csv.gz
to wi/od/wi od aux JT01 2014.csv.gz

The Minneapolis-St. Paul core-based statistical area includes counties in both Minnesota andWisconsin. The LODES aux files describe jobs in which the workplace is in the state and theresidence is outside the state.
The workplace area characteristic data for NYC are downloaded from https://lehd.ces.
census.gov/data/lodes/LODES7/ny/wac/, from ny wac S000 JT01 2002.csv.gz tony wac S000 JT01 2017.csv.gz for primary jobs.
In these files, the relevant variables are
44

See the official documentation for details of the data structure.

Appendix - 44

 w geocode 15-digit workplace census block code. We aggregate it into 11-digit FIPS census

tract code as the workplace identifier.
 h geocode 15-digit residence census block code. We aggregate it into 11-digit FIPS census

tract code as the residence identifier.
 S000 Total number of jobs.
 SE01, SE02, SE03 Number of jobs with earnings $1250/month or less, $1,251/month to$3,333/month, and greater than $3,333/month. We use the three variables and the maximum

wage obtained from ZIP Code Business Statistics to calculate the annual average payroll per
employee in the workplace tract.
Five-Year American Community Survey Commuting Flows and Employment Data. We
summarize 2006-2010 county-level commuting statistics using data from the American Community
Survey. It can be downloaded from Table 1. Residence County to Workplace County Flows for theUnited States and Puerto Rico Sorted by Residence Geography: 2006-2010.
 StateID h, StateID w 2-digit FIPS state code for residence or workplace.
 CountyID h, CountyID w 3-digit FIPS county code for residence or workplace.
 E The tract-to-tract number of commuters.

NYC tract-to-tract transit time. We use data from Davis et al. (2019) describing pairs of New
York City tracts in terms of the travel times by public transport. It can be downloaded from https://github.com/jdingel/DavisDingelMonrasMorales/raw/master/initialdata/input/tractpairs.
dta.zip.
 geoid11 orig Origin 11-digit FIPS geographic identifier.
 geoid11 dest Destination 11-digit FIPS geographic identifier.
 traveltime public Travel time by public transportation in minutes from Google Maps. We

use it to compute the commuting cost δ̄kn .
Detroit tract-to-tract transit time. Our data on tract-to-tract transit time in Detroit comefrom Kij GoogleTime.xlsx and Tract Classification.xlsx in the Owens, Rossi-Hansberg, andSarte (2020) replication files. Note that the benchmark year in our analysis is 2014. Our analysis
covers the 297 census tracts in Detroit, as well as the surrounding adjacent metro area (Wayne
County, Oakland County, and Macomb County), which includes 866 additional tracts.
 duration minutes Tract-to-tact travel time in minutes. We reshape the data and obtain

the IDs for workplace and residence tracts. We then use the travel time to compute thecommuting cost δ̄kn .
Appendix - 45 work ID Workplace identifier.
 tract str 11-digit FIPS geographic identifier that maps workplace and residence IDs tocensus tract code.
US census tract-level geographical information. The distance information for Detroit and
NYC are retrieved from the US census tract-level geographical information.
 geoid 11-digit FIPS census tract ID.
 intptlat, intptlong Current latitude and longitude of the interior point, used to calculate

geodesic distance between the centroids.
ZIP Code Business Statistics. We apply the method in Owens, Rossi-Hansberg, and Sarte(2020) to calculate a weighted wage from ZIP Code Business Patterns. The yearly employed data arefrom https://www2.census.gov/: econ2006/CB/sector00/CB0800CZ1.zip to econ2010/CB/sector00/CB0600CZ
for data between 2008 and 2010, and from
econ2011/CB/sector00/CB1100CZ11.zip to econ2013/CB/sector00/CB1100CZ11.zip for data
between 2011 and 2013.
 zipcode 5-digit zip code.
 emp f Flag for the range of the number of employees.
 emp Number of employees, replaced with the midpoint of the associated bins indicated by

emp f if censored.
 payqtr1 f, payann f Flags for first-quarter payroll and annual payroll, used for dropping

observations if classified as missing data.
 payann Annual payroll ($1,000).

ZIP-tract crosswork. To aggregate the zip-level weighted wage to the tract level, we join the ZIPdata with the HUD-USPS Zip-to-Tract crosswalk. The annual ZIP-to-tract crosswalks are available
from https://www.huduser.gov/portal/datasets/usps/:
ZIP TRACT 122010.xlsx to ZIP TRACT 122013.xlsx.
 ZIP 5-digit zip code.
 TRACT 11-digit FIPS census tract code.
 BUS RATIO The ratio of business addresses in the ZIP-tract intersection to the total number ofbusiness addresses in the entire ZIP. It is used as the proportion to reweight the employment
and annual payroll data. We sum by census tract to arrive at the tract-level employment andannual payroll for each census tract. Then, we divide payroll by employment to obtain the
tract-level average payroll per employee and its maximum value.
Appendix - 46

NYC Neighborhood Tabulation Areas (NTA)-tract crosswalk. To aggregate the NYC
census tract level data to the Neighborhood Tabulation Areas defined by the New York City
Department of City Planning, we use the 2010 census tract to 2010 NTA equivalency.
 NeighborhoodTabulationAreaNT 4-digit NTA code.
 CensusBureauFIPSCountyC 3-digit FIPS census county code.
 CensusTract 9-digit FIPS census tract code.

E

Finite model: Price dispersion and “ex post regret”

E.1

Contrast with prices from continuum model

Equilibrium prices in the model with a finite number of individuals solve a system of non-linear
equations, so their expected values are not necessarily equal to the continuum-case prices. The
P ℓknlabor supply to workplace n is Ln =k δ̄ . This is a random variable whose realized valuekndepends on the realization of ν I . Labor demand in workplace n, when there are local increasing
returns, can be written as a rearrangement of equation (C.13):σ−1σwn = Ān

η−
Ln

(1+η)
σ



1/σ

Y

.

P 1−σ

For brevity, define the inverse labor demand elasticity η̃ ≡ η − (1+η)σ . Choose the numeraire so that
Y
= 1.
P 1−σ

The equilibrium wage is given by the intersection of the labor demand curve and the randomly
realized quantity of labor supplied Ln . The equilibrium wage in workplace n as a function of thisrandom variable Ln is

σ−1

wn (Ln ) = Lη̃n Ān σ .
The equilibrium wage in the continuum model is wn (E [Ln ]), while the mean equilibrium wage
in the model with a finite number of individuals is E [wn (Ln )]. These values coincide when demandis perfectly elastic (η̃ = 0) or when increasing returns are so large that wages are linear in quantity
supplied (η̃ = 1), but otherwise wn () is not a linear operator and thus these values differ by Jensen’sP
inequality. Similarly, rearranging equation (4) shows that equilibrium rents, rk = Tαk n ykn , are a
non-linear function of realized labor quantities and wages.
Using fn (Ln ) to denote the probability mass function of supplied labor, the difference between
the continuum model’s wage and the mean wage in the model with a finite number of individualsisE [w(Ln )]−1=w(E [Ln ])Z ∞ "0Ln
E[Ln ]

η̃

#− 1 f (Ln )dLn .

As noted above, this difference is zero when labor demand is perfectly elastic (η = 0 & σ = ∞).

Appendix - 47

To quantify how the difference in wages varies with the labor demand elasticity, we compute
100,000 simulations of equilibrium outcomes in the model with a finite number of individuals for
100,000 realizations of ν I when σ = 4 and when σ = 1.1. We compute the average wage in eachtract across these 100,000 equilibria, excluding the rare instances in which a tract’s equilibrium
wage is undefined because its realized labor supply Ln is zero. The average realizations of wages
and rents across 100,000 simulations of the model with a finite number of individuals are very close
to the continuum-case rational expectation of these prices at our baseline parameter values. The
gap between the two model’s prices is larger when the labor demand elasticity is lower. Figure E.1
shows the two distributions of the wage difference across the 2,143 workplace tracts for the twolabor demand elasticities. The median tract’s absolute percentage point deviation of mean realized
price from the continuum-case rational expectation is 0.06% for wages and 0.01% for rents. Whenσ = 1.1, these differences are 0.32% for wages and 0.03% for rents. The values of the 95th -percentile
differences are reported in Section 4.4.

02

Density
4

6

8

Figure E.1: Gap between mean wage and continuum-case expectation of wage

0

1

2
3
4
Percent deviation from continuum real wages
σ = 1.15

σ = 4.0

Notes: This figure depicts the absolute percentage-point deviation of the mean realized wage from the continuumcase rational expectation of the wage across the 2,143 workplace tracts. We compute the mean wage using 100,000
simulations of the model with a finite number of individuals. This mean excludes the rare instances in whicha tract’s equilibrium wage is undefined because its realized labor supply Ln is zero. Deviations larger than five
percentage points are winsorized to five.

E.2

Dispersion of real wages and rents relative to beliefs

Figure E.2 plots the dispersion of prices for each tract in New York City for the estimated modelwith a finite number of individuals. We define price dispersion as the standard deviation of the pricein the commuting equilibrium with finitely many individuals across 100,000 simulations divided by
the continuum-case rational expectation of that price.

Appendix - 48

Figure E.2: Price dispersion across equilibria
B. Wage20

Density

30

0

0

10

10

20

Density

30

40

40

50

A. Rent

0

.1

.2
Rent

.3

.4

0

.05

.1

.15

Wage

Notes: The plots depict the dispersion of prices (rk /P or wn /P ) for each tract in New York City using the modelwith a finite number of individuals estimated on 2010 data. We define price dispersion as the standard deviationof the price in the commuting equilibrium with finitely many individuals across 100,000 simulations divided by thecontinuum-case rational expectation of that price. Panel A depicts the dispersion in tracts’ rents, which have amedian value of 0.032 (p5 = 0.021, p95 = 0.051). Panel B depicts the dispersion in tracts’ wages, which have amedian value of 0.016 (p5 = 0.004, p95 = 0.054).

E.3

Ex post regret

We compute the share of individuals with ex post regret and their magnitudes as follows. As
discussed in the main text, we define “ex post regret” as the income gain an individual would
accept to not switch to their best choice under realized wages and rents. For individual i who chose
residence-workplace pair kn, at realized prices r and w ex post regret, χi is defined implicitly as




max
ϵ ln
′ ′
k ,n

wn′
1−α
P
rkα′ δk′ n′+ νki ′ n′







(1 + χi )wni= ϵ ln+ νkn ,
P 1−α rkα δkn

i . Denote k and n as the optimal choices ofwhere by definition kn was the choice maximizing Ũk,nk ′ and n′ for individual i. Given kn, we can solve for the ex post regret χi based on the following
equation:
ϵ ln

wn
1−α
P
rkα δkn!
i+ νkn= ϵ ln(1 + χi )wn
P 1−α rkα δkn



i
+ νkn
,

which implies that
 


1 i
ν
−α −1
ϵ kn
wn rk δkn eχi =  1 i
 − 1.
−1e( ϵ νkn ) wn rk−α δkn(E.1)

We simulate 2.5 million individuals to compute this statistic. Each of these individuals uses
the point-mass beliefs from the continuum model. The realized wages and rents differ because of
idiosyncratic preference shocks that do not wash out in the aggregate.
Table E.1 reports the distribution of ex post regrets.
Appendix - 49

Table E.1: Distribution of ex post regrets

p95

Unconditional
distribution
p96
p97
p98

p99

Conditional
distribution
Mean Median

0.04420.04330.04460.0446
0.04370.04440.04470.04450.04520.04440.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00110.00090.00120.00120.00100.00120.00130.00120.00140.00110.00420.00390.00430.00430.00400.00420.00430.00430.00450.00420.00820.00780.00830.00840.00790.00830.00830.00840.00860.00820.01500.01430.01500.01520.01440.01500.01500.01500.01540.01480.01060.01020.01060.01060.01030.01070.01050.01060.01090.01060.00730.00710.00720.00730.00710.00730.00720.00730.00740.00720.04440.00000.00120.00420.00830.01490.01060.0072s

Share withregret

1
2
3
4
5
6
7
8
9
10
mean

Notes: The table reports the share of individuals with ex post regret and the distribution of ex post regrets
over their desired switches in simulations of our estimated model with a finite number of individuals. The firstcolumn identifies the simulation s. The second column reports the fraction of individuals who have ex post regret
and therefore would prefer a different choice given realized prices. Columns under “Unconditional distribution”
report the distribution of ex post regret based on the full sample (I = 2, 488, 905). Columns under “Conditional
distribution” report the distribution of ex post regret among those who would want to switch. The “Mean” rowreports the mean value across the 10 simulations.

Appendix - 50

Table E.2 compares the price dispersion in the 100,000 simulations depicted in Figure E.2
and the 10 simulations depicted in Table E.1. The simulations in Figure E.2 each require one
draw from a multinomial distribution with 4.6 million outcomes whose probabilities appear inequation (10). Generating the labor allocation for each simulation by this method requires lessthan one second of computing. The simulations in Table E.1 involve more than 11 trillion draws
from the type-1 extreme value distribution. Generating the labor allocation for each simulation by
this method requires about 100 hours of computing. Hence, the latter simulations are much morecomputationally expensive. It is heartening that the distributions of price dispersion summarizedin Table E.2 are very similar.
Table E.2: Price dispersion across simulation methods
Simulation count

mean

p5

p10

p25

p50

75

p90

p95

Wage
100,000
10

0.021
0.023

0.004
0.004

0.006
0.005

0.010
0.009

0.016
0.015

0.025
0.025

0.039
0.039

0.054
0.057

Rent
100,000
10

0.035
0.034

0.021
0.016

0.024
0.019

0.027
0.024

0.032
0.031

0.038
0.038

0.045
0.049

0.051
0.058

Notes: This table compares the price (rk /P or wn /P ) dispersion generated by the simulations
used in Section E.2 (100,000 simulations) and Section E.3 (10 simulations). As described in
the text, each tract’s price dispersion is the standard deviation of the price in the commuting
equilibrium with finitely many individuals across simulations divided by the continuum-caserational expectation of that price. The first column reports the average price dispersion acrossall tracts. The remaining columns report the level of price dispersion for tracts at selected
percentiles.

Appendix - 51

F

Amazon HQ2 counterfactual

F.1

Geographic incidence of Amazon HQ2Figure F.1: Amazon HQ2 counterfactual change in workers
A. Employment in 2010

B. Covariates-based predictions

C. Calibrated-shares predictions

Notes: Panel A depicts the number of workers employed in each tract in 2010. Panels B and C depict decreases
in the number of workers predicted by the covariates-based model and calibrated-shares procedure, respectively.
As total population is fixed, the number of workers decreases in all tracts except the Amazon HQ2 location. The
covariates-based predictions for workers describe both the continuum model and the model with finite individuals
because the change in the former equals the expected change in the latter.

Appendix - 52

Figure F.2: Predicted changes in wages
A. Covariates-based: Continuum

B. Covariates-based: Finite

C. Calibrated shares

Notes: These maps depict percent changes in wages, (ŵn /P̂ −1)×100. Panel A depicts the change in thecovariates-based continuum model. Panel B depicts the mean change across 100 simulations of the modelwith finite individuals. Panel C depicts each tract’s predicted wage change using the calibrated-sharesprocedure.

−.50Change in Rents (%)
.5
1

1.5

Figure F.3: Rent changes by distance to Amazon HQ20

12
Kilometers to AHQ2 (log scale)
95% CI
95% CI

3

4

Covariates−based model
Calibrated−shares procedure

Notes: This figure plots the changes in rents predicted by the covariates-based continuum model and thecalibrated-sharesprocedureas functions of (log) distance to the Amazon HQ2 tract. The percent change
in rents is

r̂k
−1
P̂

× 100.

Appendix - 53

As alternatives to σ = 4, we consider the values σ = 1.1 and σ = ∞. The predicted changes
in commuters to the Amazon HQ2 tract from both the covariates-based model and the calibratedshares procedure are invariant to the labor demand elasticity σ: regressing predictions for one valueof σ on predictions from another yields a slope of one and an intercept of zero to the fifth or sixth
decimal place. The predicted changes in real rents by residential tract from both the covariatesbased model and the calibrated-shares procedure are almost perfectly correlated (R2 ≥ 0.997),
but their scale changes with σ: a lower labor demand elasticity implies a higher average real rent
increase (see Figure F.4 contrasting σ = 4 and σ = ∞). The predicted changes in wages by
workplace tract necessarily vary with the labor demand elasticity: a lower labor demand elasticity
implies a higher average wage increase and range of wage changes. When σ = ∞, there is no spatial
variation in wage changes.
Figure F.4: Comparison of predictions of the change in real rents for σ = 4.0 and σ = ∞B. CSP comparison of real rents for σ = 4.0 and σ =∞0Predicted percentage change in rent (CBM), σ= inf
−.5
0
.5
1

Predicted percentage change in rent (CSP), σ= inf
2
4
6
8
10

A. CBM comparison of real rents for σ = 4.0 and
σ=∞

0

.5
1
1.5
Predicted percentage change in rent (CBM), σ = 4.00R−squared = 0.9974; Slope = 0.9514; Intercept = −0.3739; MSE = 0.1622

2
4
6
8
10
Predicted percentage change in rent (CSP), σ = 4.0

R−squared = 0.9995; Slope = 0.9877; Intercept = −0.4116; MSE = 0.1777

Notes: These plots compare the percent change in real rents, (r̂/P̂ −1)×100, predicted by the covariatesbased model (left panel) and calibrated-shares procedure (right panel) for the labor demand elasticitiesof σ = 4.0 and σ = ∞. In both panels, the predicted real rent changes associated with different values of
σ are almost perfectly correlated and only differ by a constant.

Appendix - 54

0

20
40
60
Predicted change in commuters (CBM)80

R−squared = 0.4843; Slope = 0.7032; Intercept = 3.4354; MSE = 39.6199

Predicted change in commuters (NL, ζ = 0.25)
20
40
60
80
0

0

0

Predicted change in commuters (IFE1)
20
40
60

Predicted change in commuters (SVD16)
20
40

60

80

Figure F.5: Predicted changes in commuters to the AHQ2 tract from SVD, IFE, and nested-logit
specifications

0

20
40
60
Predicted change in commuters (CBM)80

R−squared = 0.8360; Slope = 0.7824; Intercept = 2.5185; MSE = 10.7927

0

20
40
60
Predicted change in commuters (CBM)80

R−squared = 1.0000; Slope = 0.9998; Intercept = 0.0026; MSE = 0.0000

Notes: This figure compares the predicted changes in commuters to the Amazon HQ2 tract from theapproximated matrix derived from a rank-16 singular value decomposition (left panel), from the rank-1
interactive-fixed-effects specification (center panel), and from the nested-logit specification with ζ = 0.25
(right panel) to the predictions from the covariates-based specification.

Figure F.6: Predicted changes in residents from SVD, IFE, and nested-logit specifications
A. Rank-16 SVD

B. Rank-1 IFE

C. Nested logit

Notes: These maps depict the changes in residents predicted by the approximated matrix derived from a rank-16
singular value decomposition (left panel), the rank-1 interactive-fixed-effects specification (center panel), and thenested-logit specification with ζ = 0.25 (right panel). They should be compared to the predictions depicted in
Figure 6B.

F.2

Uncertainty about counterfactual predictions due to idiosyncrasies

Appendix - 55

Table F.1: Changes in residents and rents by distance ventile
Change in residents% change in rents

VentileDistance

Mean

p5

p95

Mean

p5

p95

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

[0.0, 3.2]
[3.2, 4.2]
[4.2, 5.3]
[5.3, 6.2]
[6.2, 7.1]
[7.1, 8.0]
[8.0, 9.1]
[9.1, 10.0][10.0, 10.8][10.8, 11.5][11.5, 12.1][12.1, 12.8][12.8, 13.5][13.5, 14.2][14.2, 15.0][15.0, 15.9][15.9, 16.7][16.7, 18.2][18.2, 20.0][20.0, 36.5]

1.0
0.6
0.5
0.5
0.3
0.2
0.2
0.0
-0.0
-0.0
-0.1
-0.2
-0.2
-0.2
-0.2
-0.2
-0.3
-0.3
-0.3
-1.1

0.1
-0.5
-0.4
-0.4
-0.6
-0.6
-0.7
-0.6
-0.7
-0.7
-0.8
-1.0
-0.9
-1.0
-0.8
-1.0
-1.0
-0.9
-1.1
-2.0

1.9
1.6
1.3
1.3
1.3
1.2
1.0
0.8
0.8
0.8
0.6
0.7
0.6
0.4
0.3
0.4
0.3
0.3
0.5
-0.2

0.8
0.7
0.7
0.7
0.7
0.7
0.6
0.5
0.6
0.6
0.5
0.6
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.4

0.4
0.6
0.5
0.6
0.6
0.4
0.5
0.4
0.4
0.4
0.4
-0.0
0.4
-0.7
0.4
0.4
0.3
0.3
0.4
-0.0

1.4
0.8
0.9
0.8
0.8
0.8
0.7
0.7
0.8
0.7
0.7
1.2
0.7
2.0
0.7
0.7
0.7
0.7
0.6
0.8

Notes: This table summarizes the distribution of average tract-level outcomes by ventile of distance to the Amazon HQ2 tract. Each ventile contains
about 108 residential tracts. The third and sixth columns report the meanvalue of the ventile’s average outcome across 100 simulations. The fourth andseventh columns report the 5th percentile of the 100 simulations, and the fifth
and eighth columns report the 95th percentile
 of the 100 simulations. The
percent change in real rents is expressed as r̂P̂k − 1 × 100. Shorter-distanceventiles tend to have larger increases in residents and rents in the mean
simulation. For all but the first and twentieth ventiles, the 90% confidence
interval for the change in residents across 100 simulations includes zero. The
rent increase in the first ventile in the mean simulation is about double thatin the 20th ventile. The 90% confidence interval of every ventile includes
0.6%, the citywide average rent increase.

Appendix - 56

Table F.2: Changes in employment and wages by distance ventile

Change in employment% change in wages

VentileDistance

Mean

p5

p95

Mean

p5

p95

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

[0.0, 3.2]
[3.2, 4.2]
[4.2, 5.3]
[5.3, 6.2]
[6.2, 7.1]
[7.1, 8.0]
[8.0, 9.0]
[9.0, 10.0][10.0, 10.7][10.7, 11.5][11.5, 12.1][12.1, 12.7][12.7, 13.5][13.5, 14.2][14.2, 15.0][15.0, 15.8][15.8, 16.7][16.7, 18.2][18.2, 19.9][19.9, 36.5]

-49.7
-39.3
-14.6
-26.4
-33.5
-7.2
-5.9
-4.3
-6.0
-4.3
-4.4
-4.6
-4.1
-5.0
-3.3
-4.6
-3.4
-5.6
-3.2
-5.0

-51.1
-40.5
-15.4
-27.2
-34.5
-7.7
-6.3
-4.6
-6.6
-4.7
-4.8
-5.0
-4.5
-5.4
-3.7
-5.0
-3.7
-6.0
-3.5
-5.5

-48.5
-38.2
-13.7
-25.5
-32.2
-6.7
-5.4
-3.8
-5.6
-3.9
-4.0
-4.1
-3.8
-4.6
-3.0
-4.2
-3.1
-5.2
-2.9
-4.6

0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4

0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.3
0.4
0.3
0.4
0.3

0.4
0.4
0.5
0.5
0.4
0.4
0.4
0.5
0.5
0.5
0.4
0.5
0.5
0.4
0.4
0.5
0.5
0.4
0.5
0.4

Notes: This table summarizes the distribution of average tract-level outcome
by ventile of distance to the Amazon HQ2 tract. Each ventile contains about
107 workplace tracts. The third and sixth columns report the mean value of theventile’s average outcome across 100 simulations. The fourth and seventh columns
report the 5th percentile of the 100 simulations, and the fifth and eighth columns
report the 95th percentile of the 100 simulations. The percent change in real wages
ŵnis expressed as P̂ − 1 × 100.

Appendix - 57

Figure F.7: Uncertainty about counterfactual changes induced by idiosyncrasies (NTAs)
B. Changes in workers

−100

−800

−50

Change in workers
−600
−400
−200

Change in residents
0
50
100

0

150

A. Changes in residents

−50

0
50
Mean change in the number of residents
95th percentile

100

−800

5th percentile

−600
−400
−200
Mean change in the number of workers
95th percentile

5th percentile

D. Changes in wages

.1

−5

.2

Percent change.3.4Percent change
0
5

.5

.6

10

C. Changes in rents

0

0

.5

1
1.5
Mean percent change in rent

95th percentile

2.3

5th percentile.35.4
Mean percent change in wage
95th percentile.45

5th percentile

Notes: These plots depict variation in counterfactual changes induced by individual idiosyncrasies in the NTAlevel model with a finite number of individuals. The panel depicts the 5th and 95th percentiles of predicted changes
in quantities and prices across 100 simulations of the NTA-level model with a finite number of individuals.

Appendix - 58

Figure F.8: Uncertainty about counterfactual changes induced by idiosyncrasies (nested logit)
B. Changes in workers

−20

−500

Change in residents
−10
0
10

Change in workers−400 −300 −200 −100

020

A. Changes in residents

−4

−2
0
2
Mean change in the number of residents
95th percentile

4

−500

−400
−300
−200
−100
Mean change in the number of workers

5th percentile

95th percentile

5th percentile

D. Changes in wages

−10

0

−5

Percent change
24Percent change
0
5

10

6

C. Changes in rents

0

0

.5
1
Mean percent change in rent
95th percentile

1.5

0

5th percentile

.2

.4
.6
Mean percent change in wage
95th percentile.81

5th percentile

Notes: These plots depict variation in counterfactual changes induced by individual idiosyncrasies in the nestedlogit specification (with parameter ζ = 0.25) of the model with a finite number of individuals. The panel depicts
the 5th and 95th percentiles of predicted changes in quantities and prices across 100 simulations of the modelwith a finite number of individuals.

Appendix - 59

Figure F.9: Uncertainty about counterfactual changes induced by individual idiosyncrasies (σ = ∞)C. Changes in rents

−4

−2
0
2
Mean change in the number of residents
95th percentile

5th percentile

4

Percent change
0
10
−10
−20

−20

−500

Change in residents
−10
0
10
20

Change in workers−400 −300 −200 −100

020

B. Changes in workers

30

A. Changes in residents

−500

−400
−300
−200
−100
Mean change in the number of workers
95th percentile

5th percentile

0

−.5

0
.5
Mean percent change in rent
95th percentile

5th percentile

Notes: The plots depict the 5th and 95th percentiles of predicted changes in quantities and prices across 100
simulations of the model with a finite number of individuals in which σ = ∞. In Panel A, the 90% confidence interval
for the change in residents includes zero for all tracts. In Panel B, the 90% confidence interval for the change in
employment includes zero for 505 of the 2142 non-Amazon workplaces. Panel B excludes the Amazon HQ2 workplace
tract. The 90% confidence interval for the change in employment for the Amazon workplace tract is 24,777 to 25,349.
In Panel C, real rent changes are (r̂k /P̂ − 1) × 100, and 0 out of 2160 origin tracts have a positive change in rents at
the 5th percentile.

Appendix - 60

1