Spatial economics for granular settings

Jonathan I. Dingel (Chicago Booth)
Felix Tintelnot (UChicago)
October 2023

Local economic shocks at fine spatial resolution
Local economic shocks. . .
 Workplace employment/productivity
e.g., Amazon’s proposed HQ2 in New York City
 Land/housing/residential amenities
e.g., Detroit’s neighborhood revitalization projects
 Transportation costs
e.g., Berlin’s new U5 connection of underground lines. . . at fine spatial resolution (Rosenthal and Strange, 2020)
 Arzaghi and Henderson (2008): productivity gains from interactions within 500 meters
 Rossi-Hansberg, Sarte and Owens (2010): housing externalities halve every 1,000 feet
 Ahlfeldt et al. (2015): production & residential externalities halve within 1-2 minutes
2 / 38

Quantitative spatial models in granular settings
 Spatial linkages (commuting, trade, local externalities, etc) govern theincidence of local economic shocks Want “an empirically relevant quantitative model to perform generalequilibrium counterfactual policy exercises” (Redding and Rossi-Hansberg, 2017)
 Continuum of agents → realized shares = model probabilities

Literature Consider a granular setting. Two concerns arise when the number of spatial
links is large relative to the number of decision makers:
1. Risk of overfitting the model to the idiosyncratic components of individual
decisions
2. Counterfactual outcomes may be sensitive to the idiosyncratic components
of individual decisions
3 / 38

Quantitative spatial models in granular settings
 Spatial linkages (commuting, trade, local externalities, etc) govern theincidence of local economic shocks Want “an empirically relevant quantitative model to perform generalequilibrium counterfactual policy exercises” (Redding and Rossi-Hansberg, 2017)
 Continuum of agents → realized shares = model probabilities

Literature Consider a granular setting. Two concerns arise when the number of spatial
links is large relative to the number of decision makers:
1. Risk of overfitting the model to the idiosyncratic components of individual
decisions
2. Counterfactual outcomes may be sensitive to the idiosyncratic components
of individual decisions
3 / 38

Spatial economics for granular settings: Roadmap
Computing counterfactuals in continuum models
Counterfactual analysis in granular empirical settings
Applying continuum model to NYC 2010
Monte Carlo: Calibrated-shares procedure overfits data
Event studies: Neighborhood employment booms
A spatial model with a finite number of individuals
Application to Amazon’s HQ2
4 / 38

Computing counterfactual
outcomes in continuum models

Continuum model: Economic environment
 Each location has productivity A and land endowment T Measure L individuals w/ one unit of labor and hired by competitive firms
producing freely traded goods differentiated by location of production
 Individuals have Cobb-Douglas preferences over goods (1 − α) and land (α)
 Commuting costs: δkn = δ̄kn × λkn|{z} |{z}timedisutility Individuals have idiosyncratic tastes for pairs of residential and workplacelocations, such that i’s utility from living in k and working in n is


wn
i
i
i iid
Ukn = ϵ ln α 1−α
+ νkn
νkn
∼ T1EV(1)
rk P
δkn
5 / 38

Continuum model: Equilibrium
Given economic primitives (α, ϵ, σ,L,{An },{Tk },{δkn }), an equilibrium is a setof wages {wn }, rents {rk }, and labor allocation {ℓkn } such thatℓknwϵ (rα δkn )−ϵ= P n ϵk α−ϵLk′ ,n′ wn′ (rk′ δk′ n′ )labor allocation (gravity):
goods markets:Ankland markets:(wn /An )−σ
Y
P 1−σ
δ̄kn
α X ℓkn
Tk =wn
rk n δ̄kn| {z }

X ℓkn

=

∀k, n

(2)

∀n

(3)

∀k, n

(4)

ykn

1+ϵ
σ+ϵ



αϵ
1+αϵ



≤ 12 =⇒ unique equilibrium (Allen, Arkolakis and Li, 2023)
6 / 38

Continuum model: Counterfactual outcomes′

Define x̂ ≡ xx . Counterfactual equilibrium system can be expressed as

1
! 1
! 1−σ
X  ŵn′ 1−σ X ykn′ 1−σ1yknŶ σ−1ŵn = Ânŷkn PYÂn′k ′ yk ′ n
k
k
n′
X
ykn
r̂k = T̂k −1ŷkn Pn′ ykn′n
1, if ℓkn = 0




−ε

ε
αˆ
ŵ
r̂
δ̄
λ̂
ℓ̂kn =nk kn kn
if ℓkn > 0

−ε


P
ℓ k ′ n′
 k′ ,n′ ŵnε ′ r̂kα′ δ̄ˆk′ n′ λ̂k′ n′
L

X

(5)
(6)

(7)

“Exact hat algebra”: Compute ŵn , r̂k , and ℓ̂kn given elasticities σ, α, and ϵ, baselineknshares ℓLand yYkn , and relative exogenous parameters Ân , T̂k , δ̄ˆkn and λ̂kn .
7 / 38

Continuum model: Fitting the model to data
 In general, we distinguish defining a system of equations to solve forcounterfactual equilibria from fitting a model’s parameters
 Exact hat algebra concerns comparative statics, not calibration
 “Calibrated shares” (often used interchangeably with “exact hat algebra”)
 Dominant approach to counterfactual analysis in quantitative spatial models
uses observed shares in equations (5)-(7) Implicitly calibrates parameters so model exactly delivers the observed shares(e.g., ℓkn = 0 =⇒ δkn = ∞) Covariates-based approach (e.g., Ahlfeldt et al. 2015)
 Parameterize δkn as function of observed covariates
 Use fitted model’s values of the baseline shares in equations (5)-(7) As we will discuss, many alternatives lie between these two approaches
8 / 38

Counterfactual analysis in
granular empirical settings

Commuting flows in granular settings

.8

.8

.6

.6

.4
.2
0

 Gravity model predicts 2014 value
better than 2013 value for bottom 95%
of tract pairs.4

 44% of NYC tract pairs with positive
flow in 2013 were zeros in 2014.2

 41.1% of commuters in cell with ≤ 50 85% of tract pairs have zerocommuters between them

1

1

NYC has 2.5 million resident-employees and 4.6 million tract pairs.

0

10

20

30

40

Number of individuals
Fraction of tract pairs

Cumulative share of commuters

Source: Longitudinal Employer-Household Dynamics, Origin Destination Employment Statistics. LODES
employment counts are noise-infused and LODES flows are synthetically generated.
Detroit

MSP

Counties

Impersistent: Detroit

Imp: counties

Asymmetric zeros

State-to-state migration

9 / 38

Contrasting parameterizations of commuting costs
 Pick α = 0.24, σ = 4, L = number of employed individuals
 Seek values of {δkn }, ϵ, {Tk }, {An }δkn = δ̄kn × λkn
|{z}
|{z}
observedunobserved

H Compute {δ̄kn } from Google Maps transit times: δ̄kn = H−tkn−tnk

1. Covariates-based approach:
Assume λkn = 1 ∀k, n
2. Calibrated-shares procedure:
Assume structural error λkn appropriately orthogonal
10 / 38

Estimating the commuting elasticity
Covariates-based: Logit log likelihood function(McFadden, 1974, 1978; Guimarães, Figueiredo and Woodward,
2003)"
ln L =XXknℓkn ln P

wnϵ rkα δ̄kn

ϵk′ ,n′ wn′

−ϵrkα′ δ̄k′ n′#−ϵ11 / 38

Estimating the commuting elasticity

Covariates-based: Logit log likelihood function
"
#
−ϵ
XX
wnϵ rkα δ̄kn
ln L =ℓkn ln P
−ϵϵαnk′ ,n′ wn′ rk′ δ̄k′ n′kCalibrated shares: Commuting gravity eqn
−ϵwnϵ rkα δ̄kn λkn
ℓkn
=P
−ϵ
ϵ
α
L
k′ ,n′ wn′ rk′ δ̄k′ n′ λk′ n′E λ−ϵkn |· = 1 → estimate ϵ by PPMLE (Silva andTenreyro, 2006)
11 / 38

Estimating the commuting elasticity for NYC in 2010

Covariates-based: Logit log likelihood function
#
"
−ϵ
XX
wnϵ rkα δ̄kn
ln L =ℓkn ln P
−ϵϵαnk′ ,n′ wn′ rk′ δ̄k′ n′kCalibrated shares: Commuting gravity eqn
−ϵwnϵ rkα δ̄kn λkn
ℓkn
=P
−ϵ
ϵ
α ′ ′
L
′ n′
w
r
λ
δ̄
′
′
′
′
k
n
k
n
k
k ,n

NYC (2010)
PPML/MLE
Commuting cost

-7.986(0.307)

Model fit (pseudo-R2 )Location pairs
Commuters

0.662
4,628,878
2,488,905

Notes: Specification includes residence fixed
effects and workplace fixed effects.

11 / 38

Estimating the commuting elasticity for NYC in 2010
Covariates-based: Logit log likelihood function
"
#
−ϵ
XX
wnϵ rkα δ̄kn
ln L =ℓkn ln Pϵα ′ ′ −ϵ
w
r
δ̄
′
′
′
′
k
n
n
n
k
k ,n
k
Calibrated shares: Commuting gravity eqn
−ϵwnϵ rkα δ̄kn λkn
ℓkn
=P
−ϵ
ϵ
α
L
k′ ,n′ wn′ rk′ δ̄k′ n′ λk′ n′

NYC (2010)
PPML/MLE
Commuting cost

-7.986(0.307)

Model fit (pseudo-R2 )Location pairs
Commuters

0.662
4,628,878
2,488,905

Notes: Specification includes residence fixed
effects and workplace fixed effects.

Covariates-based approach: Solve for {Tk } and {An } using fixed effects (∝ rk−αϵ andwnϵ ) and equations (2), (3), and (4)Calibrated-shares procedure: Use estimated ϵInteractive fixed effects

11 / 38

Monte Carlo: Applying each procedure to finite data
 DGP is estimated covariates-based model for NYC in 2010
 Simulated “event”: ↑ productivity of 200 Fifth Ave tract by 9%
 Apply calibrated-shares procedure and covariates-based approach(Increase An to match total employment increase in simulated data) Does the procedure predict the change in the number of commuters fromeach residential tract working in the “treated” tract?
 Regress “true” changes on predicted changes (2160 obs per simulation)

Ideally, want slope = 1 and intercept = 0
 Compute forecast errors (MSE for “true” vs predicted changes)
12 / 38

Monte Carlo: Calibrated-shares procedure overfits

.5
.4
.3
Fraction

0

0

.1

50

10

.2

20

Density
100
150

30

200

40

250

Apply each procedure to simulated “2010” data. 100 simulations w/ I = 2, 488, 905
Changes in commuter counts (ℓ′kn̄ − ℓkn̄ )

.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

I

2.5

5

12.5

25

50

125

250

2560

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

0.782
0.269
0.225

0.876
0.153
0.113

0.948
0.064
0.045

0.974
0.032
0.023

0.986
0.017
0.011

0.995
0.007
0.005

0.997
0.004
0.002

1.000
0.000
0.000

.008

Changes in rentsΛ

13 / 38

Monte Carlo: Calibrated-shares procedure overfits.25.2
Fraction
.15

0

0

.05

1

1

.1

2

Density

2

3

3

4

Apply each procedure to simulated “2010” data. 100 simulations w/ I = 2, 488, 905
Changes in commuter counts (ℓ′kn̄ − ℓkn̄ ) via finite-sample draws from pre- and post- DGPs

0

.5

1

1.5

2
0

−.5

Covariates−based: slope
Covariates−based: intercept

Calibrated−shares: slope
Calibrated−shares: intercept

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

.8

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

2.5

5

12.5

25

50

125

250

2560

-0.4081.724
17.022

0.194
0.982
8.486

0.669
0.404
3.400

0.835
0.202
1.699

0.913
0.106
0.851

0.968
0.040
0.340

0.982
0.022
0.169

0.998
0.002
0.017

.88

14 / 38

Using tract-level events to evaluate model performance
Kehoe (2005): “it is the responsibility of modelers to demonstrate that their models arecapable of predicting observed changes, at least ex post”

How well do models predict changes in commuting flows?
 Look at 83 tract-level employment booms (≥+12.5%) in NYC in 2010–2012
 We raise productivities in tracts to match observed changes in total
employment
 Does the model predict changes in bilateral commuting flows to that
destination? (n.b. total employment change need not be exogenous)
 Regress observed changes on predicted changes
 Contrast forecast errors (MSEs)
15 / 38

Comparison of models’ predictive performance across 83 events

2

Covariates-based model much better at predicting change in number ofcommuters from each residential tract to booming workplace tract.2.15
Fraction
.1
.05

1
0

0

.5

Density

1.5

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

.1

−3

−2

−1

0

1

2

3

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

No extensive margin

16 / 38

Comparisons with alternative approaches

We compare the performance of the covariates-based specification to otherparameterization methods, including
 using pooled pre-event data for 2008-2010,
 aggregating counterfactual predictions to the Neighborhood Tabulation Area
(NTA) level, using a low-rank approximation of the commuting matrix computed withsingular value decomposition (SVD), and using fitted values from an enriched covariates-based model includinginteractive fixed effects.
17 / 38

.4
0

.1

.5

1

Fraction
.2
.3

1.5

Covariates−based: slope
Covariates−based: intercept
Calibrated−shares: slope
Calibrated−shares: intercept0

Density.52Comparisons over 83 events with pooled data

−3

−2

−1

0

1

2

3

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

18 / 38.2

2Comparisons over 83 events at NTA level.15
Fraction
.1
.05

1

0

0

.5

Density

1.5

Covariates−based
Calibrated−shares

−1

0

1

Slope

2

3

0

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

3.5

Estimation at NTA level

19 / 38

Calibrated “fitted” shares: SVD
For this exercise, we replace the commuting matrix with a low-rankapproximation.
 For the commuting matrix L = [ℓkn ] and fixed rank r, solve

min L̃ − L

s.t. rank(L̃) ≤ r

L̃ SVD factors L as U SV ′ , where U and V are orthonormal and S is diagonal
and non-negative, with L’s singular values as entries.
 By keeping the largest r values in S and setting the rest to zero, we obtain
the optimal rank r approximation, per the Eckart-Young theorem.

 We replace all negative values with zeros and rescale so thatPP
ℓkn = L̃kn , and use the observed wages from 2010.
20 / 38

.3.1

1

Fraction.2

1.5

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

0

0

.5

Density.4

2

Calibrated “fitted” shares: SVD, rank 16−3Choosing rank

−2

−1

Alternative ranks

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

Matrix visualizations

21 / 38

Calibrated “fitted” shares: Interactive fixed effects

Interactive fixed effects is a generalization of the covariates-based model andrepresents a midpoint between the covariates-based andcalibrated-observed-shares.
 For the covariates-based specification, we assumed that there were no
unobserved commuting costs λkn = 1 ∀kn.
 Now let λkn = exp(ψk′ γn ), with ψk and γn both R × 1 vectors.
R is the rank of the implied factor structure Estimate ψ and γ, residence FEs, workplace FEs, and commuting elasticity ϵby maximum likelihood.
22 / 38

.3
0

.5

.1

1

Fraction.2

1.5

Covariates−based: slope
Covariates−based: intercept
Rank 1 IFE: slope
Rank 1 IFE: intercept0

Density.4

2

Calibrated “fitted” shares: Rank 1 interactive fixed effects

−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

23 / 38

Counterfactuals in continuum models: Takeaways

We examined varied strategies for estimating/calibrating the model of baseline
shares Calibrating millions of parameters using millions of observed shares has
severe overfitting problem
 Time aggregation (pooling 3 years) is insufficient Parsimonious transit-time parameterization performs well in event studies
 SVD (of ranks 6 to 16) performs similarly More flexible interactive-fixed-effect specification offers modest improvement
24 / 38

A spatial model with a finite
number of individuals

A spatial model with a finite number of individuals
Goal: examine the sensitivity of counterfactual outcomes to the idiosyncraticcomponent of individual decisions
In the limit (I → ∞), the equilibrium of our model with an integer number ofindividuals is (almost surely) the equilibrium of the continuum model
Modeling concerns raised by the integer number of individuals:
 Individuals must have beliefs about equilibrium wages and land prices!
2(I + N 2 − 1)!
I +N −1=
I = 10, N = 4 =⇒ 3.27 × 106
22(N − 1)!I!
N −1
 There will be a distribution of equilibria for each set of parameters Υ
25 / 38

Model: Economic environment

 Each location has productivity A and land endowment T
 I individuals are endowed with L/I units of labor and hired by competitive
firms producing freely traded goods differentiated by location of production
 Individuals have Cobb-Douglas preferences over goods and land
 Individuals have idiosyncratic tastes for residence-workplace pairs Workers know primitives Υ ≡ (L, {An }, {Tk }, {δ̄kn }, {λkn }, α, ϵ, σ) andhave (common) point-mass beliefs r̃k and w̃n about land prices and wagesi Worker i knows own idiosyncratic preferences {νkn
} but not the full set ofI
idiosyncratic residence-workplace draws ν

26 / 38

Model: Economic environment

 Each location has productivity A and land endowment T
 I individuals are endowed with L/I units of labor and hired by competitive
firms producing freely traded goods differentiated by location of production
 Individuals have Cobb-Douglas preferences over goods and land
 Individuals have idiosyncratic tastes for residence-workplace pairs Workers know primitives Υ ≡ (L, {An }, {Tk }, {δ̄kn }, {λkn }, α, ϵ, σ) andhave (common) point-mass beliefs r̃k and w̃n about land prices and wagesi Worker i knows own idiosyncratic preferences {νkn
} but not the full set ofI
idiosyncratic residence-workplace draws ν

26 / 38

Timing: Individuals choose labor allocation, then markets clear
1. Workers choose the kn pair that maximizesiŨkn= ϵ lnw̃nP̃ 1−α r̃kα δkn

!
i
+ νkn

given point-mass beliefs r̃k and w̃n
2. After choosing kn based on their beliefs, workers are immobile and cannot
relocate
3. Given the labor allocation {ℓkn } and economic primitives Υ, a tradeequilibrium is a set of wages {wn } and land prices {rk } that clears all
markets.
27 / 38

Commuting equilibrium with a finite number of individuals

Given primitives Υ, idiosyncratic residence-workplace draws ν I , and point-mass
beliefs {w̃n }, {r̃k }, a commuting equilibrium with a finite number ofindividuals, I, is defined as a labor allocation {ℓkn }, wages {wn }, and land
prices {rk } such thatPi
 ℓkn = LI Ii=1 1{Ũkn(ν I ) > Ũki ′ n′ (ν I ) ∀(k ′ , n′ ) ̸= (k, n)}; and wages {wn } and land prices {rk } are a trade equilibrium given the labor
allocation {ℓkn }.

28 / 38

Convergence to the continuum model equilibrium Definition: Given primitives Υ ≡ (L, {An }, {Tk }, {δ̄kn }, {λkn }, α, ϵ, σ), w̃
and r̃ are “continuum-case rational expectations” if w̃ and r̃ constitutea trade equilibrium for the labor allocation {ℓkn } given by equation (2).
 Result: As I → ∞, if individuals’ point-mass beliefs are continuum-caserational expectations, then the equilibrium quantities and prices of themodel with a finite number of individuals coincide (almost surely) with thoseof the continuum model.

29 / 38

Estimating the finite model

Likelihood (McFadden, 1974, 1978; Guimarães, Figueiredo and Woodward, 2003)
#
"
−ϵ
XX
w̃nϵ r̃kα δ̄kn
ln L =ℓkn ln P
α ′ ′ −ϵ
ϵ
r̃
δ̄
w̃
′
′
′
′
k
n
n
n
k
k ,n
k
 Solve for {Tk } and {An } using fixed effects (∝ r̃k−αϵ and w̃nϵ ) undercontinuum-case rational expectations
 This estimation procedure yields same ϵ, {Tk }, and {An } as thecovariates-based continuum model
30 / 38

Ex post regret is small

 Individuals make residence-workplace choices based on wage and rent beliefs
 The realized equilibrium wages and rents will differ

Price dispersion

 Calculate ex post regret χi at realized prices for i who chose kn :


 




wn′
(1 + χi )wn
i
i
max
ϵ ln
+ νk′ n′ = ϵ ln
+ νkn
k′ ,n′
P 1−α rkα′ δk′ n′
P 1−α rkα δkn
 Quantitatively modest: 96% would not want to switch

Switchers Conditional on wanting to switch, median ex-post regret χi is 0.7%.
Price dispersion in ex post regret simulations

31 / 38

Comparison with continuum model Idiosyncratic residence-workplace draws ν I → distributions of equilibrium
quantities and prices (for given primitives Υ) Mean equilibrium outcomes:
 Mean commuter counts coincide with those from the continuum model
ℓkn
L

h
i
i
i
′
′
= E Pr(Ukn> Uk′ n′ ∀(k , n ) ̸= (k, n)) Land prices and wages are solved from a non-linear system of equations

 Variance of equilbrium outcomes due to idiosyncrasies
 Confidence interval for residents, workers, wages, and prices

 In counterfactual exercises: Change from Υ to Υ′ for given ν I
32 / 38

Comparison with continuum model Idiosyncratic residence-workplace draws ν I → distributions of equilibrium
quantities and prices (for given primitives Υ) Mean equilibrium outcomes:
 Mean commuter counts coincide with those from the continuum model
ℓkn
L

h
i
i
i
′
′
= E Pr(Ukn> Uk′ n′ ∀(k , n ) ̸= (k, n)) Land prices and wages are solved from a non-linear system of equations

 Variance of equilbrium outcomes due to idiosyncrasies
 Confidence interval for residents, workers, wages, and prices

 In counterfactual exercises: Change from Υ to Υ′ for given ν I
32 / 38

Comparison with continuum model Idiosyncratic residence-workplace draws ν I → distributions of equilibrium
quantities and prices (for given primitives Υ) Mean equilibrium outcomes:
 Mean commuter counts coincide with those from the continuum model
ℓkn
L

h
i
i
i
′
′
= E Pr(Ukn> Uk′ n′ ∀(k , n ) ̸= (k, n)) Land prices and wages are solved from a non-linear system of equations

 Variance of equilbrium outcomes due to idiosyncrasies
 Confidence interval for residents, workers, wages, and prices

 In counterfactual exercises: Change from Υ to Υ′ for given ν I
32 / 38

Application to Amazon’s HQ2

Counterfactual: Amazon HQ2 in Long Island City

 Amazon’s 2017 RFP for HQ2 with 50,000 employees elicited 238 proposals
 NYC proposed four possible sites (and controversial tax breaks)
 Split siting announced in 2018 would have put 25,000 employees in Long
Island City
 Quantitative questions: What would happen to NYC neighborhoods withthis local employment boom? Are these changes large relative to uncertainty
stemming from idiosyncratic component of indivduals’ decisions?

33 / 38

Contrasting predictions for changes in residents
Calibrated-shares predictions are spatially idiosyncratic

Covariates-based model

Calibrated-shares procedure
34 / 38

Contrasting predictions for changes in residents
Calibrated-shares predictions are tightly tied to initial residents

Covariates-based model

Calibrated-shares procedure

Workers

Residents working at AHQ2
tract

34 / 38

Contrasting predictions for changes in rents

Covariates-based model
Wages

Calibrated-shares procedure
35 / 38

Sizable uncertainty about predicted changes from idiosyncrasies

Changes in workers−20

Change in residents
−10
0
10
20

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

030Changes in residents

−4

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

4

Idiosyncrasies 5th percentile
Parameter uncertainty 5th pctile

−500

−400
−300
−200
−100
Mean change in the number of workers
Idiosyncracies 95th percentile
Parameter uncertainty 95th pctile0

Idiosyncracies 5th percentile
Parameter uncertainty 5th pctile

36 / 38

Sizable uncertainty about predicted changes from idiosyncrasies
Changes in wages

−2

−20

0

Percent change
24Percent change
−10
0
10

6

20

Changes in rents

0.5
1
Mean percent change in rent
Idiosyncrasies 95th percentile
Parameter uncertainty 95th pctile

1.5

Idiosyncrasies 5th percentile
Parameter uncertainty 5th pctile

.2

.4
.6
Mean percent change in wage

Idiosyncrasies 95th percentile
Parameter uncertainty 95th pctile.81

Idiosyncrasies 5th percentile
Parameter uncertainty 5th pctile

37 / 38

Conclusions

Conclusions

 Finer spatial data are exciting but not a free lunch We need to evaluate the performance of applied GE models
 Monte Carlo and event studies: Calibrated-shares procedure performs poorly
in granular empirical settings Parsimonious covariates-based specification predicts quite well
 New tools: use fitted shares (e.g., low-rank matrix approximations) rather
than observed shares in exact hat algebra Uncertainty about counterfactual predictions induced by individual
idiosyncrasies can be sizable
38 / 38

Thank youPredicting the incidence of local economic shocks
Workplace employment: “The new 15-year lease agreement with property owner L&L
Holding Co. will allow Tiffany to unite employees at the company’s three headquarters
locations under one roof. Formerly known as the International Toy Center, the approximately
800,000-square-foot building at 200 Fifth recently emerged from a massive makeover at thehands of L&L.” (CP Executive, 30 Apr 2010 )Residential amenities: “A recent report by the Urban Institute warns of ‘green
gentrification,’ where public investment in green spaces – like the 606 trail –can raise property
values, attract development and wealthier residents, and price existing residents out of thearea.” (Chicago Reporter, 30 Jan 2020)

Transportation costs: “After completion, the U5 gap closure will give the major residential
areas in the east of Berlin a direct connection to the historic city centre, the government
district and the central station. . . Once the U5 gap has been closed, 20 percent of private
vehicle traffic is expected to shift to the new U5.” (projekt-u5.de)BackMany applications infer infinite costs from zeros
 Heblich, Redding and Sturm (2020): “For all pairs of boroughs with zerocommuting flows, our model implies prohibitive commuting costs, and we makethis assumption to ensure that the model is consistent with the observed data.”

 Monte, Redding and Rossi-Hansberg (2018): “model implies prohibitive
commuting costs for pairs with zero commuting flows” and “the model implies
prohibitive trade costs for pairs with zero trade”

 Severen (2021): “most pairs that are ever zero (in either 1990 or 2000) arealways zero. Always zero pairs do not contribute any variation to models withpair fixed effects”
BackGravity-based estimates better predict 2014 commuter counts# 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

ACS state-to-state migration matrix is s.t. sampling noise
2002
2001

0

1

2

3

4

5

6+

0
1
2
3
4
5
6+

0.59 0.16 0.13
0.40 0.17 0.17
0.32 0.16 0.21
0.23 0.16 0.21
0.26 0.13 0.15
0.13 0.09 0.12
0.03 0.03 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

 The ACS 2001 dataset has 644,427 prime-age individuals out of the total sample of
1,192,206 individuals.
 Of 73,101 individuals who moved residences, 80.6% migrated within their states and19.4% (14,215) moved between-states.
 Migration flows are winsorized at 6 and values are given as a percentage of 2001
migration flows, so rows sum to one.
BackSome counties migration flows are s.t. sampling noise

0

0

.2

200

.4

400

.6

600

.8

800

1

 35 million cross-county commuters between 79,188 pairs of counties within
120km (MRR 2018) in 2006–2010 American Community Survey
 Skewed: For the bottom 90% of pairs, the mean value is only 40 commuters 45% of county pairs within 120 km have zero commuters between them.

0

100

200

300

400

Number of individuals
Fraction of county pairs

Cumulative share of commuters

Some counties migration flows are s.t. sampling noise

.8

800

.6
.2
0

commuters report a number of commuters that is0 34% of county pairs with a positive number of

200

.4

(≤ 100 commuters)

600

commuters represent ∼ 5 or fewer respondents400

 55% of county pairs with a positive number of1

 35 million cross-county commuters between 79,188 pairs of counties within
120km (MRR 2018) in 2006–2010 American Community Survey
 Skewed: For the bottom 90% of pairs, the mean value is only 40 commuters 45% of county pairs within 120 km have zero commuters between them.
 ACS is a 1-in-20 representative sample

0less than the Census-reported margin of error.
Back

100

200

300

400

Number of individuals
Fraction of county pairs

Cumulative share of commuters

Zeros are asymmetric, but daily commutes are roundtrip journeys
When ℓkn > 0, we often observe ℓnk = 0:
 US counties: ℓnk = 0 for 22% of county pairs with ℓkn > 0.
 Detroit tracts: ℓnk = 0 for 66% of tract pairs with ℓkn > 0.
 Brazilian municipios: ℓnk = 0 for 49% of municipio pairs with ℓkn > 0.

If infinite commuting costs rationalize ℓnk = 0, how do you go from k to n inmorning and return from n to k in evening?
 Commuting costs must switch between finite and infinite within each day Zero commuters cannot make congestion a source of intra-day variation
Back

.8

.8

.6

.6

.4

.4

.2
0

 74% of tract pairs have zerocommuters between them.2

 42.6% of commuters in cellwith ≤ 50

 The number of tract pairs andnumber of commuters are bothabout 1.3 million.

1

1

Statistics for Detroit0 ℓnk = 0 for 66% of tract pairs
with ℓkn > 0.
Back

10

20

30

Number of individuals
Fraction of tract pairs

Cumulative share of commuters

40

.8

.8

.6

.6

.4

.4

 LODES data for Minnesota
is reported by establishment(rather than firm)

1

1

Statistics for Minneapolis-St Paul00

 Zeros are asymmetric:ℓnk = 0 for 54% of tract
pairs with ℓkn > 0.

.2.2

 Zeros are pervasive: 61%

0

10

20

30

Fraction of tract pairs

Back

40

50

60

Number of individuals
Cumulative share of commuters

70Commuter counts are impersistent: Detroit
First symptom of finite noise: Little mass on transition
matrix’s diagonal
 39% of Detroit tract pairs
with positive flow in 2013
were zeros in 2014
 Gravity model predicts 2014
value better than 2013 value
for bottom 95% of tract
pairs
Back

2013

0

2014
2

3

4

5+

0
1
2
3
4
5+

0.86 0.10 0.02
0.60 0.22 0.10
0.37 0.25 0.16
0.23 0.22 0.18
0.15 0.17 0.17
0.04 0.06 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

1

Commuter counts are impersistent: 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

Back

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

Fixed effect estimates are biased by dropping zeros
NYC (2010)
OLS (omit zeros) destination FE
−1
0
1
2
3
4

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

Detroit (2014)

−6

−4

−2
0
MLE destination FE

2

4

−5

0
MLE destination FE

5

E (ℓkn |ℓkn > 0) ≥ E (ℓkn ) → popular procedure attributes lower employment
counts to infinite commuting costs, not lower wages/productivity
BackInteractive fixed effects specification

We can parameterize the unobserved commuting costs λkn as exp(ψk′ γn ), whereψk and γn are R × 1 vectors.
 The dimensions of ψ and γ determine the rank of the implied factor
structure.
 As R increases, the computational difficulty of estimating the resulting
model increases rapidly.

 For the covariates-based model estimation with only origin and destination
fixed effects, we write R = 0.
Back

Estimation: Interactive fixed effects

Commuting elasticity estimates for NYC 2010, varying Rϵ̂pseudo-R2
Location pairs
Commuters

R=0-7.98420.662

R=1-7.1762
0.684

R=2 R=3-6.6521 -6.3586
0.694
0.701
4,628,880
2,488,905

R=4-5.7359
0.706

Back

20

Density

30

0

0

10

10

20

Density

30

40

40

50

Price dispersion across finite-model equilibria

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 thefinite model estimated on 2010 data. Left panel depicts dispersion in tracts’ rents, which have a median valueof 0.032 (p5 = 0.021, p95 = 0.051). Right panel depicts dispersion in tracts’ wages, which have a medianvalue of 0.016 (p5 = 0.004, p95 = 0.054).
BackEx post regret in the finite model
Unconditional
distribution
p96
p97
p98

p99

Conditional
distribution
Mean Median

0.01500.01430.01500.01520.01440.01500.01500.01500.01540.01480.01060.01020.01060.01060.01030.01070.01050.01060.01090.01060.00730.00710.00720.00730.00710.00730.00720.00730.00740.00720.0083 0.0149 0.0106

0.0072s

Share withregret

1
2
3
4
5
6
7
8
9
10

0.04420.04330.04460.0446
0.04370.04440.04470.04450.04520.04440.0000 0.0011 0.0042
0.0000 0.0009 0.0039
0.0000 0.0012 0.0043
0.0000 0.0012 0.0043
0.0000 0.0010 0.0040
0.0000 0.0012 0.0042
0.0000 0.0013 0.0043
0.0000 0.0012 0.0043
0.0000 0.0014 0.0045
0.0000 0.0011 0.0042

0.00820.00780.00830.00840.00790.00830.00830.00840.00860.0082mean

0.04440.0000p95

0.00120.0042

Notes: The table reports the share of individuals with ex post regret and the utility gains of their desired switches in simulations of ourestimated finite model. The first column 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 utility gain based on full sample (I = 2, 488, 905). Columns under “Conditional distribution” report the distribution
of utility gain among those who would want to switch. The “mean” row reports the mean value across ten simulations.
BackPrice dispersion across finite-model equilibria

Simulation count

mean

p5

p10

p25

p50

75

p90

p95

Wage
100,00010

0.021 0.004 0.006 0.010 0.016 0.025 0.039 0.054
0.023 0.004 0.005 0.009 0.015 0.025 0.039 0.057

Rent
100,000
100.035 0.021 0.024 0.027 0.032 0.038 0.045 0.051
0.034 0.016 0.019 0.024 0.031 0.038 0.049 0.058

Notes: This table compares the price (rk /P or wn /P ) dispersion generated by the simulations of the finite model (100,000 simulations) and the ex post regret calculations (10simulations).
Back

Monte Carlo: Calibrated-shares procedure overfits

.5
.4
.3

0

0

.1

50

10

.2

Fraction20

Density
100
150

30

200

40

250

Apply each procedure to simulated “2010” data. 100 simulations w/ I = 2, 488, 905
Changes in rents (r̂k /P̂ )

.2

.4

.6

.8

1
0

0

Covariates−based: slope
Calibrated−shares: slope

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

Back

Covariates−based: intercept
Calibrated−shares: intercept

0

.002
.004
.006
Covariates−based MSE / 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

ℓ̂kn and r̂k.008

Monte Carlo: Calibrated-shares procedure overfits

Back

Λ

I

0
0
0
0
0
0
0
0
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
1
1
1
1
1
1
1
1

2.5
5
12.5
25
50
125
250
2560
2.5
5
12.5
25
50
125
250
2560
2.5
5
12.5
25
50
125
250
2560
2.5
5
12.5
25
50
125
250
2560
2.5
5
12.5
25
50
125
250
2560

Covariates-based

Calibrated-shares

Covariates-based

Calibrated-shares

0.99850.99920.99950.99980.99991.00001.00001.00001.00051.00131.00201.0022
1.0022
1.00231.00231.00231.00331.00441.00451.00471.00491.00491.00491.00491.00481.00561.0062
1.0063
1.00651.00661.00661.0066
0.9954
0.99650.99720.99690.9971
0.9971
0.99720.99720.78170.87590.94790.97370.9864
0.9946
0.99710.99970.79010.88180.94800.97480.98670.9951
0.99710.99970.82270.9021
0.9581
0.97880.98950.99580.99790.99970.8907
0.9416
0.97640.9876
0.99320.99760.99900.99990.96880.9837
0.99330.99650.99820.99940.99950.99980.00140.00070.00030.00010.00010.00000.00000.00000.03710.03640.03600.03590.03580.03580.03580.03580.23280.23210.23180.23160.23150.23150.23150.23151.05131.05041.05011.04981.04971.04971.04971.04976.3762
6.37496.37456.37506.37486.37476.37466.37460.2252
0.11300.04520.02270.01120.00450.00230.00020.22540.11360.04480.02260.01130.00450.00230.00020.22640.11270.04520.02260.01130.00450.00230.00020.22630.11220.04500.02260.01130.00450.00220.00020.2176
0.10920.0441
0.02180.01090.00440.00220.0002

Relationship between ℓ̂kn and r̂k
The compensating variation Ψ is
Ψ=

1
ℓ̂kn

−ϵ

, ∀k, n.
ŵnϵ r̂kα P̂ 1−α δ̄ˆkn λ̂kn

Taking logarithms on both sides yields the log-log linear relationship betweenchanges in commuter counts (ℓ̂kn̄ ) and changes in rents (r̂k )log(ℓ̂kn̄ ) = −αϵ log(r̂k ) + C,
where C = ϵ log(ŵn ) + (1 − α) log(P̂ ) − ϵ log(δ̄ˆkn ) − ϵ log(λ̂kn ) − log(Ψ).
BackMonte Carlo DGP with unobserved λkn

Λ

I

0
0.1
0.25
0.5
1

2.5
2.5
2.5
2.5
2.5

Slope (mean)
Covariates-based Calibrated-shares
0.97961.00131.00561.01540.9897MSE (mean)
Covariates-based Calibrated-shares-0.4075-0.3439-0.1295
0.32140.8044Λ is the variance of λkn relative to variance of δkn
Back

14.3836
14.3945
14.752215.4545
20.731717.022216.9868
17.1357
17.0197
16.7998

Employment increases in the anchor-tenant 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

Notes: This figure depicts the number of primary jobs in tracts 36061005800 and 36061008300 in
the LODES data.
Back

17

2

Comparisons over 83 events with no extensive margin.2
Fraction
.1
.15
.05

1
0

0

.5

Density

1.5

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

.5

−3

Back

−2

−1

0

1

2

3

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

.15
.05

1

Fraction.1

1.5

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

0

0

.5

Density.22Comparisons over 35 events with NTA-level model

.4

−3

Back

−2

−1

0

1

2

3

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

Scree plot for NYC 2010 commuting matrix
Explanatory share of ordered singular values:

BackPrediction performance with alternative SVD ranks

1

2

3

4

5

6

8

10

12

Rank
14

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
BackVisualizations of commuting matrices

2010 LODES
Back

Covariates-based

SVD rank 16

IFE rank 1

Contrasting predictions for changes in land prices

Covariates-based model
Back

Calibrated-shares procedurePredictions for changes in workers

Number of workers
Back

Covariates-based model

Calibrated-shares procedure

Contrasting predictions for changes in wages

Covariates-based model
Back

Calibrated-shares procedure