THE

Vol. 137

2022

Issue 3

THE SLOPE OF THE PHILLIPS CURVE: EVIDENCE FROM
U.S. STATES∗JONATHON HAZELLJUAN HERREÑOEMI NAKAMURA
JÓN STEINSSON
We estimate the slope of the Phillips curve in the cross section of U.S. statesusing newly constructed state-level price indices for nontradeable goods back to
1978. Our estimates indicate that the slope of the Phillips curve is small andwas small even during the early 1980s. We estimate only a modest decline in theslope of the Phillips curve since the 1980s. We use a multiregion model to infer
the slope of the aggregate Phillips curve from our regional estimates. Applying
our estimates to recent unemployment dynamics yields essentially no missing
disinflation or missing reinflation over the past few business cycles. Our results
imply that the sharp drop in core inflation in the early 1980s was mostly due toshifting expectations about long-run monetary policy as opposed to a steep Phillips
curve, and the greater stability of inflation between 1990 and 2020 is mostly due
to long-run inflation expectations becoming more firmly anchored. JEL Codes:
E30.
∗ We thank Tomas Breach, Massimiliano Cologgi, and Victoria de Quadrosfor excellent research assistance. We thank John Molino, our project coordinatorat the BLS, for all his help throughout the project. We thank Hassan Afrouzi,
Regis Barnichon, Olivier Blanchard, Chi-Young Choi, Jordi Galı́, Yuriy Gorodnichenko, Pierre-Olivier Gourinchas, Greg Kaplan, Chen Lian, Silvana Tenreyro,
John Williams, and seminar participants at various institutions for valuable comments and discussions. We thank the National Science Foundation, the Alfred P.
Sloan Foundation, the Smith Richardson Foundation, the Institute for New Economic Thinking, and the Julis-Rabinowitz Center for Public Policy and Financefor financial support.


C The Author(s) 2022. Published by Oxford University Press on behalf of the President

and Fellows of Harvard College. All rights reserved. For Permissions, please email: journals.permissions@oup.com
The Quarterly Journal of Economics (2022), 1299–1344. https://doi.org/10.1093/qje/qjac010.
Advance Access publication on February 16, 2022.

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I. INTRODUCTION

According to this formulation, inflation π t is determined by threefactors: expected inflation Et π t+1 , the output gap—measured here
as the difference between unemployment ut and the natural rateof unemployment unt—and cost-push shocks ν t . The slope of thePhillips curve κ represents the sensitivity of inflation to the output
gap (i.e., to an increase in demand).
The episode in U.S. economic history that has perhaps moststrongly influenced the profession’s thinking regarding the slopeof the Phillips curve is the Volcker disinflation. In the early 1980s,
Paul Volcker’s Federal Reserve sharply tightened monetary policy. Unemployment rose sharply and inflation fell sharply. The
conventional interpretation of this episode is that it provides evidence for a relatively steep Phillips curve.
One way to formalize this conventional interpretation is to
assume that inflation expectations are adaptive: βEt π t+1 = π t−1
in equation (1). This yields the accelerationist Phillips curve:(2)πt = −κ ut − unt + νt .
Stock and Watson (2019) estimate κ in this equation and refer to it as the “Phillips correlation.” They measure π t by theannual change in 12-month core personal consumption expenditures (PCE) inflation, and ut − unt by the Congressional Budget
Office (CBO) unemployment gap, both at a quarterly frequency.
Figure I reproduces this analysis. It suggests that the slope of thePhillips curve was steep prior to and during the Volcker disinflation (0.67 for the period 1960–1983) but has flattened considerably
since then (to only 0.03 for the period 2000–2019q1).1
The insensitivity of inflation to changes in unemployment
between 1990 and 2020 led many economists to suggest that thePhillips curve had disappeared—or was “hibernating.” During
the Great Recession, unemployment rose to levels comparable
to those during the Volcker disinflation, yet inflation fell by
1. See also Ball and Mazumder (2011), Kiley (2015b), and Blanchard (2016).

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The Phillips curve is a formal statement of the common intuition that if demand is high in a booming economy, this will
provoke workers to seek higher wages and firms to raise prices. A
well-known formulation is the New Keynesian Phillips curve:(1)
πt = β Et πt+1 − κ ut − unt + νt .

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The black solid line is a regression line for 2000–2019. The dark gray broken
line is a regression for 1984–1999. The light gray dash-dot line is a regression line
for 1960–1983. The year-over-year change in inflation is the four-quarter change
in the (backward-looking) four-quarter moving average of headline PCE inflation.
The unemployment gap is the four-quarter (backward-looking) moving average ofthe gap between the unemployment rate and the natural rate of unemployment.
Authors’ calculations. The figure replicates figure 1 from Stock and Watson (2019).

much less. The “missing disinflation” during and after the GreatRecession then gave way to “missing reinflation” in the late 2010sas unemployment fell to levels not seen in 50 years but inflation
inched up only slightly. A similar debate raged in the late 1990s,
when unemployment was also very low without this leading tomuch of a rise in inflation. Some have argued that the apparent
flattening of the Phillips curve signals an important flaw in theKeynesian model.
There is an alternative interpretation of these facts that emphasizes the anchoring of long-term inflation expectations in the
United States (Bernanke 2007; Mishkin 2007). Figure II plots
long-term inflation expectations from the Survey of Professional
Forecasters. During the 1980s, long-term inflation expectations
fluctuated a great deal. In particular, they fell rapidly over the

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FIGURE I
Stock and Watson’s Changing Phillips Correlation

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THE QUARTERLY JOURNAL OF ECONOMICSPCE Core Inflation and Long-Term Inflation Expectations
The gray line plots 10-year-ahead inflation expectations for the Consumer Price
Index (CPI). From 1990 onward, these come from the Survey of Professional Forecasters. For the 1980s, these come from Blue Chip and are available on the Research and Data site of the Federal Reserve Bank of Philadelphia. The black line
plots 12-month core CPI inflation using the Bureau of Labor Statistics’ researchseries. This research series uses current methods to calculate inflation back intime.

period of the Volcker disinflation. In sharp contrast, since 1998,
long-term inflation expectations have been extremely stable.
An alternative to the standard narrative of the Volcker disinflation is that the decline in inflation was driven not by a steep
Phillips curve but by shifts in beliefs about the long-run monetary
regime in the United States that caused the rapid fall in long-run
inflation expectations we observe in Figure II. To see how this canbe the case, it is useful to solve equation (1) forward and assume
for simplicity that unemployment follows an AR(1) process. This
yields(3)πt = −ψ ũt + Et πt+∞ + ωt ,where ũt denotes the deviation of unemployment from its
long-run expected value, Et π t+∞ represents long-term inflation

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FIGURE II

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expectations, and the parameter ψ is proportional to κ in equation (1). (Section II presents a more detailed derivation.) This formulation of the Phillips curve makes clear that changes in beliefs
about the long-run monetary regime feed strongly into current
inflation: the coefficient on Et π t+∞ in equation (3) is one. Furthermore, in the presence of substantial variation in Et π t+∞ , therelationship between π t and ũt may be essentially uninformative
about the slope of the Phillips curve (ψ and κ). In particular, ifchanges in Et π t+∞ comove negatively with ũt (as they would during an imperfectly credible shift in the long-run inflation target)
the Phillips curve would appear to be steeper than it actually was.
Sargent (1982) emphasizes that hyperinflations tend to end
quickly—much too quickly to be explained by even a very large
value of κ in the Phillips curve. In these episodes, it is clear thatthe primary cause of the abrupt fall in inflation is an abrupt fallin Et π t+∞ associated with an abrupt change in the policy regime.
Volcker’s monetary policy constituted a sharp regime shift thatwas imperfectly credible at the outset but became gradually morecredible as time passed (Erceg and Levin 2003; Goodfriend andKing 2005; Bianchi and Ilut 2017). This regime shift led to a large
and sustained decline in long-term inflation expectations over the
1980s but also a transitory rise in unemployment. Perhaps it was
this large change in inflation expectations that was the primarycause of the rapid fall in inflation over this period rather thanhigh unemployment working through a steep Phillips curve.
This discussion highlights an important identification problem researchers face when they seek to estimate the slope of thePhillips curve: inflation expectations may covary with the output
gap. Standard methods for estimating the Phillips curve aim toaddress this issue by controlling for inflation expectations Et π t+1
when estimating equation (1). A challenge with this approach isthat estimates are quite sensitive to details of the specification.
Mavroeidis, Plagborg-Møller, and Stock (2014) show that reasonable variation in the choice of data series, the specification, andthe time period used yield a wide range of estimates for κ roughly
centered on a value of zero (i.e., they are equally likely to havethe “right” as the “wrong” sign). Mavroeidis, Plagborg-Møller, andStock (2014, 124) point to a weak-instruments problem in driving
these results: there simply isn’t enough variation in the aggregate data to separately identify the coefficients on unemployment
and expected inflation. They conclude: “the literature has reacheda limit on how much can be learned about the New KeynesianPhillips curve from aggregate macroeconomic time series. New

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identification approaches and new datasets are needed to reachan empirical consensus.”
In addition to the identification problem discussed above,
researchers seeking to estimate the slope of the Phillips curve
face the classic simultaneity problem of distinguishing demand
shocks from supply shocks. Supply shocks (unt and ν t ) yield positive comovement of inflation and unemployment (stagflation). If
the variation used to identify the slope of the Phillips curve iscontaminated by such shocks, the estimated slope will be biased
toward zero and may even have the “wrong” sign. Fitzgerald andNicolini (2014) and McLeay and Tenreyro (2019) point out thata central bank conducting optimal monetary policy will seek tooffset aggregate demand shocks. If the central bank is successful,
the remaining variation in inflation will be only due to supply
shocks, a worstcase scenario for the simultaneity problem.
Can cross-sectional data help overcome these problems? Several recent papers have argued that they can. Fitzgerald andNicolini (2014) and McLeay and Tenreyro (2019) show that using regional data helps overcome the simultaneity problem of
distinguishing demand and supply shocks: central banks cannot
offset regional demand shocks using a single national interest
rate. These papers as well as Kiley (2015a), Babb and Detmeister (2017), Hooper, Mishkin, and Sufi (2019), and Fitzgerald et al.
(2020) make use of city-level inflation data produced by the Bureau of Labor Statistic (BLS) to estimate regional Phillips curves.
Beraja, Hurst, and Ospina (2019) use regional wage data to estimate wage Phillips curves.
We contribute to this regional Phillips curve literature in several ways. First, we show formally how estimating the Phillips
curve using regional data provides a solution to the problem ofshifting values of Et π t+∞ confounding the estimation of the slopeof the Phillips curve. We derive a regional Phillips curve in a simple benchmark multiregion model of a monetary union. The modelclarifies the interpretation of the slope of regional Phillips curves
relative to that of the aggregate Phillips curve. We also use themodel to show that changes in the long-run monetary regime areabsorbed by time fixed effects when the regional Phillips curveis estimated using a panel data specification. The intuition isthat such long-run regime changes are common to all regions andtherefore cancel out across regions in the monetary union.
Using our cross-section specification, we estimate a modest
flattening of the Phillips curve when we split our sample in
1990: the Phillips curve in the post-1990 sample is flatter by aTHE SLOPE OF THE PHILLIPS CURVE

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2. Carvalho et al. (2021) reach a similar conclusion using very different methods. They propose a model for long-run inflation expectations and show how theirmodel generates the result that the Volcker disinflation was driven by shifting
long-run inflation expectation and also that long-run inflation expectations become anchored in the 1990s onward.

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factor of two. This contrasts sharply with empirical specifications
that make use of time series variation: a specification without
time fixed effects yields a 50–100 times steeper Phillips curve forthe pre-1990 sample. We interpret this as evidence that shifting
long-run inflation expectations seriously confound estimates of
the Phillips curve based on time series variation in the pre-1990
sample.
Our cross-sectional estimates indicate that the slope of thePhillips curve is small and was small even during the 1980s.
Combining our estimate of the slope of the Phillips curve withan estimate of the persistence of fluctuations in unemployment,
we find that a 1 percentage point increase in unemployment reduces inflation by about 0.34 percentage points, that is, ψ = 0.34
in equation (3). This implies that only a modest fraction of thelarge changes in inflation in the early 1980s can be accounted forby the direct effect of increasing unemployment working throughthe slope of the Phillips curve. In contrast, movements in long-run
inflation expectations were large over this period, as is evident
from Figure II. In particular, long-run inflation expectations fellby about 4 percentage points from 1981 to 1986, accounting forabout two-thirds of the fall in core inflation during this period.
We conclude that a majority of the rapid decline in core inflation
during the Volcker disinflation arose from a rapid decline of longterm inflation expectations, associated with a rapidly changing
monetary regime.2
Our estimates of the slope of the Phillips curve imply essentially no missing disinflation during the Great Recession or
missing reinflation in the late 2010s or late 1990s. In other words,
our cross-sectional estimates are consistent with the magnitudeof movements in aggregate inflation after 1990. We conclude thatthe stability of inflation since 1990 is due to long-run inflation
expectations becoming more firmly anchored. These conclusions
echo those of Jorgensen and Lansing (2019).
Our analysis uses new state-level consumer price indicesfor the United States that we have constructed back to the1970s. Prior to our work, state-level price indices based on
BLS micro-price data have not existed. The BLS has publishedcity-level inflation series for a group of relatively large cities.

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3. For example, Nishizaki and Watanabe (2000) find evidence of Phillips curve
flattening in their baseline specification with no time fixed effects, but this evidencechanges dramatically when time fixed effects are added.
4. Much of the expenditure weight for housing derives from owner-occupied
housing. However, rents are used to measure inflation for all shelter, due to thedifficulty of backing out the user cost of housing from actual house prices in atheoretically appealing way. The expenditure weight of the CPI less food andDownloaded from https://academic.oup.com/qje/article/137/3/1299/6529257 by guest on 13 July 2022

But it has refrained from reporting inflation indices for smaller
metropolitan areas (and for states). Our new state-level priceindices use all the available underlying micro data gathered by
the BLS. We also construct state-level price indices for nontradeables and tradeables. We focus our analysis on the behavior of
the prices of nontradeable goods. This is important. For prices set
at the national level—as is more likely for tradeables—the slopeof the regional Phillips curve will be zero no matter how large theslope of the aggregate Phillips curve is.
A notable conclusion of the recent regional Phillips curve literature has been that the estimated slope of the regional Phillips
curve has tended to be steeper than the slope estimated for theaggregate Phillips curve. The theoretical framework we develop
helps explain why this is the case. We show that panel data estimates of the regional Phillips curve by prior researchers are
estimates of ψ in equation (3) as opposed to estimates of κ inequation (1). This means that they are not directly comparable tomuch of the aggregate literature. We discuss how researchers canconvert estimates of ψ to κ and explain what other statistics thisconversion depends on (primarily the degree of persistence of theunemployment variation used to estimate ψ). Our analysis highlights the importance of the exact specification used in estimating
regional Phillips curves.3
The regional setting, along with our new inflation indices,
allow us to leverage new forms of variation in estimating thePhillips curve. We develop a new tradeable-demand spillovers instrument building on insights from Nguyen (2014). This instrument is based on the idea that supply shocks in tradeable sectors will differentially affect demand in nontradeable sectors inregions that are differentially exposed to the shocked tradeable
sectors: for example, an oil boom will increase demand for restaurant meals in Texas. In carrying out our regional analysis, we arecareful to account for the fact that roughly 42% of the expenditure
weight in core inflation is on the shelter component of housing services, which are measured by rents.4 We estimate the slope of theregional Phillips curve for rents and show that it is substantiallyTHE SLOPE OF THE PHILLIPS CURVE

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II. THE POWER AND PROBLEM OF LONG-RUN INFLATIONEXPECTATIONS
To appreciate the value of using regional variation to estimate
the slope of the Phillips curve, it is useful to understand the centralenergy is 77.7%, and 32.3 percentage points out of this expenditure weight arerents.

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steeper than the regional Phillips curve for nontradeables excluding housing. We use the combination of these estimates topredict the behavior of aggregate core inflation, which includes
rents, and show that these predictions match the greater aggregate cyclicality of core inflation than core inflation excluding housing, a fact emphasized by Stock and Watson (2019). We conclude
from this that the behavior of rent prices play an important rolein determining the slope of the regional and aggregate Phillips
curves.
In addition to the papers already mentioned, our work builds
on the vast empirical and theoretical literature on the Phillips
curve. The literature originates with Phillips (1958) and Samuelson and Solow (1960). Friedman (1968) and Phelps (1967) emphasized the importance of including an inflation expectations termin the Phillips curve. Gordon (1982) emphasized the importance of
supply shocks. Important early papers that estimate the New Keynesian Phillips curve include Roberts (1995), Fuhrer and Moore(1995), Galı́ and Gertler (1999), and Sbordone (2002), but see also
works cited in Mavroeidis, Plagborg-Møller, and Stock (2014). Important recent publications estimating the Phillips curve include
Ball and Mazumder (2011, 2019), Coibion and Gorodnichenko(2015), Stock and Watson (2019), Barnichon and Mesters (2020),
and Del Negro et al. (2020). Our article is also related to a recent
literature that assesses the missing disinflation during the GreatRecession (see Christiano, Eichenbaum, and Trabandt 2015; Del
Negro, Giannoni, and Schorfheide 2015; Gilchrist et al. 2017;
Crump et al. 2019).
The article proceeds as follows. Section II derives equation (3)and explains the problem of regime change in estimating thePhillips curve. Section III describes our main framework for interpreting the regional Phillips curve. Section IV describes our
new state-level inflation indices. Section V presents our empiricalresults. Section VI concludes.

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role of long-run inflation expectations in determining aggregate
inflation. To this end, we solve equation (1) forward to get(4)πt = −κ Et∞

β j ut+ j + ωt ,

njwhere ωt ≡ Et ∞j=0 β (κut+ j + νt+ j ). This equation illustrates how
inflation at time t is determined by the path of unemployment out
into the infinite future.5 We can furthermore decompose the variation in future unemployment ut+j into a transitory and permanent component. Define the transitory component of variation in
unemployment to be ũt = ut − Et ut+∞ , where Et ut+∞ is the permanent component of the variation in unemployment. Using theseconcepts, we can rewrite equation (4) as(5)πt = −κ Et∞

β j ũt+ j −j=0κEt ut+∞ + ωt .
1−βAssuming that shocks to unt and ν t are transitory, equation (1) imκ
plies that Et πt+∞ = − 1−β
Et ut+∞ . We can then rewrite equation (5)as(6)πt = −κ Et∞

β j ũt+ j + Et πt+∞ + ωt .

j=0

5. While the most popular microfoundation of the New Keynesian Phillips
curve—and the one we develop in Section III—is based on the price rigidity assumptions in Calvo (1983), this equation or something very similar arises fromseveral other microfoundations. Roberts (1995) shows that the same Phillips curvearises from Rotemberg’s (1982) quadratic costs of price adjustment model and Taylor’s (1979, 1980) model of staggered contracts (the timing of the output gap termis slightly different in the Taylor model). Furthermore, Gertler and Leahy (2008)
develop the same Phillips curve as a linear approximation of a model with Ss
foundations. In the case of the Rotemberg model in continuous time, the derivation does not rely on a linear approximation around a zero-inflation steady state.
Models based on information frictions yield Phillips curves that are not forward
looking. These models, however, typically assume no price rigidity. Incorporating
price rigidity into these models would make their Phillips curves forward looking
as well. Sbordone (2002), Galı́, Gertler, and Lopez-Salido (2005), and Rudd andWhelan (2005) develop approaches to estimating the Phillips curve on aggregate
data using versions of equation (4).

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j=0

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Finally, let’s assume for simplicity that ũt follows an AR(1) process with autocorrelation coefficient equal to ρ u . In this casej
Et ũt+ j = ρu ũt and we can rewrite equation (6) as(7)πt = −ψ ũt + Et πt+∞ + ωt ,6. Equations (6) and (7) remain valid in the case where the coefficient on Et π t+1
in equation (1) is equal to one rather than β. In this case, the long-run Phillips
curve is vertical, and long-run unemployment and inflation are unrelated. Thatis, inflation still satisfies πt = −κ Et ∞j=0 ũt+ j + ωt + Et πt+∞ , but Et ut+∞ is now
independent of Et π t+∞ . The forward sum is bounded even though β = 1, becauseũt has zero unconditional mean.

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κ
where ψ = 1−βρ.
uThis way of writing the Phillips curve highlights the importance of long-run inflation expectations in determining inflation
at the aggregate level. Long-run inflation expectations Et π t+∞appear with a coefficient of one in equation (7). In other words,
current inflation moves one for one with changes in long-run inflation expectations. These long-run expectations are determinedby the private sector’s beliefs about the long-run monetary regime
being followed by the central bank (the long-run inflation target).
Variation in beliefs about the long-run monetary regime therefore
have very large effects on current inflation.6
Equation (7) implies that inflation can vary dramaticallywithout any variation in ũt if there is substantial variation in
long-run inflation expectations. In this case, the relationship between inflation and ũt may be entirely uninformative about theslope of the Phillips curve. Worse still, variation in long-run inflation expectations may be correlated with variation in ũt . For example, it seems very plausible that Volcker’s willingness to allow
unemployment to rise to very high values in the early 1980s—andthe fact that he was not forced to resign—signaled to the public that he was serious about bringing down inflation (and had
the backing of the president to do this). Such a correlation will
impart an upward bias on estimates of the slope of the Phillips
curve unless variation in inflation expectations can be controlledfor. But in practice, controlling for inflation expectations is hardbecause of weak instruments (Mavroeidis, Plagborg-Møller, andStock 2014) and because direct measures of inflation expectations
may be imperfect. So, a rapid drop in inflation expectations may
masquerade as a steep Phillips curve.
Why has the Phillips curve appeared to flatten over the pastfew decades? Figure II shows that since roughly 1998, long-term

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inflation expectations have been firmly anchored at close to 2%.
This has led to a collapse of the covariance between Et π t+∞ andunemployment and therefore eliminated any bias associated withpoorly proxied variation in inflation expectations. A fall in thisbias will appear from the perspective of the (misspecified) accelerationist Phillips curve (such as the one we discuss in the
introduction) as a flatter curve.
One piece of corroborating evidence for this view is the close
relationship between π t and Et π t+1 in the data. Recall that thestandard formulation of the New Keynesian Phillips curve—equation (1)—implies that it is the gap between π t and βEt π t+1 —let’scall this the inflation gap—that must be explained by demandpressure (the κut term) or supply shocks (κunt + νt ). Figure III
plots SPF forecasts of inflation over the next year along with four
different measures of current inflation. The difference between
the two series is approximately equal to the inflation gap π t −βEt π t+1 .
The measure of current inflation plotted in the top left panelof Figure III is the 12-month change in the overall CPI. This conventional way of comparing current inflation and inflation expectations over the next year suggests that these series are closely
related, but that there is nevertheless substantial variation in
the gap between them (the inflation gap). Moving to the top rightpanel, we measure current inflation by the 12-month change incore CPI inflation, excluding food and energy. The inflation gap
measured this way is quite a bit smaller. Evidently, commodities
account for a large part of the inflation gap for the overall CPI.
However, a substantial inflation gap remains in the early 1980s.
The measure of current inflation plotted in the bottom left
panel of Figure III is the 12-month change in the core PCE. The
advantage of this series is that it makes use of current measurement methods, retroactively applied. In this case, the inflation
gap is very small. A similar message emerges in the bottom
right panel using the 12-month change in the core CPI researchseries published by the BLS. This series also uses consistent,
modern methods to calculate inflation back in time. A particularly
important measurement change for our purposes occurred in
1983, when the BLS switched to using rent inflation as a proxy foroverall housing inflation, including for owner-occupied housing(rental equivalence). Before that time, housing services inflationin the CPI was constructed from a weighted average of changes
in house prices and mortgage costs (i.e., interest rates). This

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Each panel shows the comparison of the one-year-ahead forecast of the GDP
deflator coming from the Survey of Professional of Forecasters and a measure of
inflation. The top left panel uses the published headline CPI. The top right panel
excludes food and energy by plotting the published measure of the core CPI. The
bottom panels correct for changes in the methodology of inflation measurement.
The bottom left panel uses PCE inflation, which has maintained a stable methodology, and the bottom right panel uses the Constant Methodology Research Series
for core CPI published by the Bureau of Labor Statistics. We use forecasts of the
GDP deflator because forecasts for the CPI are not available before 1980.

earlier approach essentially baked in a strong relationship between Volcker’s actions to curb the Great Inflation and measured
CPI inflation, since interest rates (and house prices) fed directly
into the CPI.7
7. These choices are consequential because the housing component of the CPI
has a weight of roughly one-third in the overall CPI. Online Appendix B.2 presents
our attempt to replicate the pre-1983 BLS housing methodology on more modern

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FIGURE III
π t versus Et π t+1

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III. A MODEL OF THE REGIONAL PHILLIPS CURVE
We develop a two-region, New Keynesian, open-economy
model featuring tradeable and nontradeable sectors. We derivea regional Phillips curve in this model and show how it relates
to the aggregate Phillips curve. The model demonstrates a chief
benefit of regional data: time and state fixed effects “differenceout” changes in long-run inflation expectations. The model also
illustrates the importance of using nontradeable inflation whenestimating the slope of the Phillips curve using regional data.
III.A. Model Setup
Our model consists of two regions that belong to a monetaryand fiscal union. We refer to the regions as Home (H) and Foreign (F). The population of the entire economy is normalized toone. The population of the home region is denoted by ζ . Labor
is immobile across regions. In each region, there is a single labor market. Household preferences, market structure, and firm
behavior take the same form in both regions. Below we describe
the economy of the home region. All prices in the economy are denominated in “dollars,” a digital currency issued by the federal
government.9 Throughout, we adopt the following conventions
data. The main conclusion from this is that this methodology would have led tomuch more variable (and cyclical) inflation over the past few decades.
8. We discuss this in more detail in Online Appendix B.1.
9. In other words, we are considering an economy in the cashless limit (Woodford 1998, 2003).

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The overall message that emerges from Figure III is that the
inflation gap for core inflation measured using modern methods
is tiny throughout our sample period. Importantly, this includes
the period of the Volcker disinflation. This is suggestive evidence
that the slope of the Phillips curve was small throughout our sample period: unemployment varied a great deal in the early 1980s
and again in the Great Recession without much variation in the
inflation gap. However, the panels in Figure III illustrate wellthat this conclusion is sensitive to the details of how inflation ismeasured.8 It is also sensitive to whether the expectations data
used come from the SPF or from the Michigan Survey of Consumers as Coibion and Gorodnichenko (2015) emphasize, and isalso sensitive to the exact timing of the variables.

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unless otherwise stated. Lowercase variables are the logs of uppercase variables. Hatted variables denote the percentage deviationof a variable from its steady-state value. Steady-state values arerecorded without time subscripts.

E0

∞


β t u(C Ht , NHt ),t=0

where CHt is per capita consumption of a composite consumption good, NHt is per capita employment, and β is the household’s
subjective discount factor. We follow Greenwood, Hercowitz, andHuffman (1988) in assuming that the function u(CHt , NHt ) takes
the form(8)u(C Ht , NHt ) =1−σ −1
1+ϕ −1NHt
C Ht − χ 1+ϕ−1
1 − σ −1,

where ϕ is the household’s Frisch elasticity of labor supply, σdetermines the household’s elasticity of intertemporal substitution, and χ governs the intensity of the household’s disutility of labor. We refer to this preference specification as GHH
preferences.
The composite consumption good CHt is a constant elasticity ofTand nontradeables
substitution (CES) index over tradeables C Ht
N
C Ht given by
 1
1
N η−1
T η−1
C Ht = φ Nη C Ht η + φTη C Ht η

η
η−1

,

where η is the elasticity of substitution between tradeables andnontradeables and φ T and φ N are the household’s steady-state
expenditure shares on tradeable and nontradeable goods, respecN
T
and C Htare themselves composite goods described furtively. C Ht
ther below. Nontradeable goods are only consumed in the region
where they are produced. In contrast, the market for tradeable
goods is completely integrated across regions. Hence, home andforeign households may face different prices for nontradeables but
face the same prices for tradeable goods. The expenditure share

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1. Households. The representative household in the homeregion seeks to maximize the utility function

1314

THE QUARTERLY JOURNAL OF ECONOMICS

on tradeable and nontradeable goods must sum to 1, that is, φ N +φ T = 1.
Nis given by
The composite nontradeable good C Ht
1
0

θ −1
N
C Ht
(z) θ dz

θ
θ −1

,

N
where C Ht(z) denotes consumption of variety z of nontradeable
goods in the home region. The home price of this nontradeable
N(z). The parameter θ > 1 denotes the elasticity ofvariety is PHt
substitution between different nontradeable varieties.
T
is a CES aggregate overHome tradeable consumption C Ht
tradeable goods produced in the home and foreign regions given
by


η−1
η−1
1
1
T
H η TH η
F η TF ηC Ht= τ HtC Ht+ τ Ht
C Ht

(9)

η
η−1

,

TH
TF
where C Ht
and C Htare home consumption of composite tradeable goods produced in the home and foreign regions, respectively. We assume (for simplicity) that the elasticity of substitution between home-produced and foreign-produced tradeables
is η (the same as the elasticity of substitution between tradeables
and nontradeables). Demand for home-produced and foreignHandproduced tradeables is subject to shocks denoted by τ Ht
F
HF
τ Ht , respectively. We normalize τ Ht + τ Ht = 1. For simplicity, we
do not allow for home bias in tradeable consumption. Thus, we
set τ HH = τ FH = ζ, that is, the share of spending on goods fromthe home region in each region is equal to the size of the homeregion.
The home and foreign composite tradeable goods are CES
indices given by

TH
=
C Ht

1
0

θ −1
TH
C Ht
(z) θ dz

θ
θ −1

and

TF
C Ht
=

1
0

θ −1
TF
C Ht
(z) θ dz

θ
θ −1

,

TH
TF
(z) and C Ht(z) are home consumption of varieties ofwhere C Ht
tradeable goods produced in the home and foreign region, respectively. The prices of these home-produced and foreign-producedTT(z) and PFt(z), respectively.
tradeable good varieties are PHt

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N
=
C Ht

1315

THE SLOPE OF THE PHILLIPS CURVEHouseholds maximize utility subject to a sequence of budget
constraints:
N
N
TH T
TF T
C Ht
PHt
+ C HtPHt + C Ht
PFt + Et [MHt,t+1 BH,t+1 ]where BHt is a random variable denoting payoffs of the statecontingent portfolio held by households in period t; MHt,t+1 is the
one-period-ahead stochastic discount factor of the home represenN
T
T
, PHt
, and PFt
are price indices that give
tative household; PHt
N
TH
TF
the minimum cost of purchasing a unit of C Ht
, C Ht, and C Ht
, respectively; WHt is the nominal wage received by workers in regionT
H; and  N
Ht and  Ht are the profits of nontradeable and tradeable firms in the home region. There is a complete set of financial
markets across the two regions. To rule out Ponzi schemes, we
assume that household debt cannot exceed the present value of
future income in any state.
We present the first-order necessary conditions for household
optimization in Online Appendix A.1. As we noted, the problemof the foreign household is analogous. We therefore refrain fromdescribing it in detail here. For simplicity, we do not allow for
tradeable-demand shocks to foreign tradeable consumption as we
do for home tradeable consumption.
2. Firms. There is a continuum of firms in the tradeable andnontradeable sectors. Firms are indexed by z and firm z specializes in the production of differentiated good z. Labor is the only
variable factor of production used by firms.
We begin by discussing the nontradeable sector. The output of
N(z). The production
good z in the nontradeable sector is denoted Y Ht
function of firm z in this sector isNNN(z) = ZHt
NHt
(z),
Y Ht

(10)

N
N
(z) is the amount of labor demanded by firm z and ZHt
where NHt
is a productivity shock.
Firm z in the nontradeable sector maximizes its value:

Et

∞

j=0

N
N
N
MHt,t+ j PH,t+j (z)Y H,t+ j (z) − W H,t+ j NH,t+ j (z)Downloaded from https://academic.oup.com/qje/article/137/3/1299/6529257 by guest on 13 July 2022

T
 BHt + WHt NHt +  N
Ht +  Ht ,

1316

THE QUARTERLY JOURNAL OF ECONOMICS

given demand for its good, which isNN(z) = ζ C Ht
Y Ht

N
(z)
PHt

−θ
.

N
PHt

T
T
T
(z) = ZHt
NHt
(z),
Y Ht
T
(z) is the amount of labor demanded by the firm prowhere NHt
T
is a productivity shock.
ducing good z and ZHt
Firm z in the tradeable sector maximizes its value:

Et

∞


T
T
T
MHt,t+ j PH,t+j (z)Y H,t+ j (z) − W H,t+ j NH,t+ j (z)

j=0given demand for its good. Demand in the tradeable sector comesfrom both the home and foreign regions. Firm z’s demand is thus
given by
T
(z) =
Y Ht



TH
TH
ζ C Ht+ (1 − ζ ) C Ft



T
(z)
PHt
T
PHt

−θ
.

The tradeable goods firms also have an opportunity to change
their price with probability 1 − α each period and must otherwise
keep their prices fixed.
We present the first-order necessary conditions for firm optimization in Online Appendix A.2. The problems of foreign firms
are analogous to those of home firms.
3. Government Policy and Equilibrium. The federal government operates a common monetary policy for the two regions. This
policy takes the form of the following interest rate ruler̂tn = ϕπ (πt − π̄t ) − ϕu (ût − ūt ) + εrt ,where, as elsewhere in the article, hatted variables denotedeviations from a zero-inflation steady state and lowercase

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Firm z can set its price freely with probability 1 − α as in Calvo(1983). With probability α the firm must keep its price unchanged.
Analogously to the nontradeable sector, the output of firm z
Tin the tradeable sector is denoted Y Ht(z). Its production functionisTHE SLOPE OF THE PHILLIPS CURVE

1317

III.B. Regional and Aggregate Phillips Curves
Taking a log-linear approximation of the model presented in
Section III.A around a zero-inflation steady state with balanced
trade yields the following regional Phillips curve for the inflationof nontradeable goods:
(11)

N
N
N
N
= β Et π H,t+1− κ ûHt − λ p̂Ht
+ ν Ht,π Ht

10. Prior work that allows for a time-varying inflation target includes Stockand Watson (2007), Ireland (2007), and Cogley and Sbordone (2008).

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variables are the logs of uppercase variables. Economy-wide inflation π t is a population-weighted average of inflation in the tworegions: π t ≡ ζ π Ht + (1 − ζ )π Ft , where π Ht = pHt − pH,t−1 isconsumer price inflation in the home region and π Ft is defined
analogously for the foreign region. In our model, we define unemployment in the home region simply as uHt = 1 − NHt . We
define foreign unemployment analogously. This implies that to afirst order ûHt = −n̂Ht and ûFt = −n̂Ft . Economy-wide unemployment is a population-weighted average of unemployment in thetwo regions, so ût = ζ ûHt + (1 − ζ )ûFt .
Importantly, we allow the monetary authority to have a timevarying inflation target π̄t .10 Since the long-run Phillips curve in
our model is not vertical, variation in long-run inflation yields
variation in long-run unemployment. We assume that the monetary authority targets an unemployment rate that is consistentπ̄t. We assume
with its long-run inflation target, that is, ūt = (1−β)κ
that ϕ π and ϕ u obey the Taylor principle, ensuring that the economy has a unique locally bounded equilibrium. εrt is a transitory
monetary shock, which we assume follows an exogenous AR(1)process.
For simplicity, the government levies no taxes, engages in no
spending, and issues no debt. In other words, there is no fiscal
policy. The digital currency issued by the government is in zeronet supply. The government’s monetary policy has no fiscal implications. An equilibrium in this economy is an allocation thatsatisfies household optimization, firm optimization, the government’s interest rate rule, and market clearing. We focus on theunique locally bounded equilibrium of the model. Implicitly werule out equilibria in which the inflation rate rises without boundusing the trigger strategy argument presented in Obstfeld andRogoff (1983).

1318

THE QUARTERLY JOURNAL OF ECONOMICS

and aggregate Phillips curve for overall inflation:(12)πt = β Et πt+1 − κ ût + νt ,NPHt
− 1 is the percentage deviation of the home relative price ofPHtN
nontradeables from its steady-state value of one, ν Ht
is a nontradeable home supply shock, ν t is a corresponding aggregate supply
shock, and the parameter κ = λϕ −1 , where λ = (1−α)(α1−αβ ) . We pro-vide a detailed derivation of these equations in Online Appendix A.
Equations (11) and (12) yield an important result: the slopesof the regional Phillips curve for nontradeables and the aggregate Phillips curve are the same in our model. These slopes areboth equal to κ. This result holds for the nontradeable regional
Phillips curve but does not carry over to the regional Phillips
curve for overall consumer price inflation—which includes bothtradeable and nontradeable inflation in the region. As we show in
Online Appendix A.8, the slope of the regional Phillips curve foroverall consumer price inflation is smaller by a factor equal to theexpenditure share on nontradeable goods.
Intuitively, the difference in the slope between the nontradeable and overall regional Phillips curves arises because all regions
share the tradeable goods and these goods are priced nationally.
The tradeable goods therefore do not contribute to differences in
inflation across regions, which means that the regional CPI is
made up partly of goods whose regional prices are insensitive toregional variation in unemployment. This makes the regional CPI
less sensitive to regional unemployment than the aggregate CPI
is to aggregate unemployment.
Our result that the slope of the nontradeable regional Phillips
curve is equal to the slope of the aggregate Phillips curve leads
us to focus our cross-sectional empirical work on inflation for
nontradeable goods. Earlier research that has estimated regional
Phillips curves has done so for overall consumer price inflation at
the regional level (e.g., Fitzgerald and Nicolini 2014; McLeay andTenreyro 2019). Our model suggests that results from such analysis are less directly informative about the slope of the aggregate
Phillips curve.
Our assumption that households have GHH preferences helps
simplify the derivation of the regional and aggregate Phillips
curves in our model—equations (11) and (12). GHH preferences

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N
N
N
N
where π Ht
= pHt
− pH,t−1
is home nontradeable inflation, p̂Ht=

THE SLOPE OF THE PHILLIPS CURVE

1319

III.C. Estimating the Slope of the Phillips Curve with Regional
Data
Next we solve the regional Phillips curve—equation (11)—forward to obtain
(13)

N
= −Et
π Ht

∞




N
N
N
β j κ ũH,t+ j + λ p̂H,t+
j + Et πt+∞ + ω Ht ,

j=0


N
j N
= Et ∞
where ũHt = uHt − Et uH,t+∞ and ω Ht
j=0 β ν H,t+ j .
A major benefit of estimating the slope of the Phillips curveusing regional data from a monetary union is that variationNterm in equain long-run inflation expectations—the Et πt+∞tion (13)—is constant across regions. This implies that variation

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imply that wealth effects on labor supply are zero, which eliminates the dependence of marginal costs on consumption. The absence of a consumption term in the Phillips curve plays a role inthe derivation of our result that the nontradeable regional Phillips
curve and the aggregate Phillips curve have the same slope. We
discuss this point at greater length in Online Appendix A.9. The
form of the Phillips curve in our model does not depend on the
structure of financial markets. We have assumed complete financial markets across regions, but the Phillips curve is the same ina model with incomplete markets across regions.
An important difference between equations (11) and (12) isNin
the presence of the relative price of nontradeables term λ p̂Htequation (11). This term implies that inflation in the nontradeables sector will be lower the higher the relative price of nontradeables. Conceptually, this term is very important. It pushes relative
prices toward parity in the long run. Also, it implies that even if
prices in the economy are very flexible—κ is very large—a local
boom will not result in unbounded inflation of home nontradeable
prices because demand for these goods is affected by their prices
relative to other prices in the economy. The mechanical reason
this term appears is that the inflation rate for nontradeable goods
is driven by variation in the real wage deflated by nontradeable
prices. Labor supply in the home region, however, is a function of
the real wage deflated by the home CPI. The real marginal-costvariable in the home nontradeable Phillips curve therefore gives
rise to an unemployment term and a relative price of nontradeables term.

1320

THE QUARTERLY JOURNAL OF ECONOMICS(14)

πitN = −Et

∞




N
N
β j κui,t+ j + λ p̂i,t+j + αi + γt + ω̃it ,j=0

where i denotes region, α i denotes a set of region fixed effects,Nand γ t denotes a set of time fixed effects. Variation in Et πt+∞in equation (13) that is common across regions will be absorbedby the time fixed effect.11 Constant differences across regions inNwill be absorbed by the state fixed effects. To the extent
Et πt+∞
N
across regions (e.g.,that there is remaining variation in Et πt+∞
due to changing trends), it will be a part of the error term ω̃itN .
It is useful to relate equation (14) to the empirical specifications used in the recent regional Phillips curve literature. If weNfollow AR(1) processes with autocorassume that both uHt and p̂Ht
relation coefficients equal to ρ u and ρ pN , respectively, equation (14)
11. The time fixed effects also absorb time variation in the long-run expected
unemployment Et ut+∞ . We have therefore replaced ũi,t+ j in equation (13) with
ui,t+j in equation (14). This equation remains valid if β = 1. The forward sum inthe equation is still bounded, because uit has zero mean conditional on time fixed
effects.

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in long-run inflation expectations will be absorbed by time fixed effects in a panel specification. Intuitively, while short-run inflationN) will differ across regions due to differences
expectations (Et πt+1
in their economic circumstances, long-run inflation expectationsN) are independent of the current business cycle. They are(Et πt+∞determined by beliefs about the long-run monetary regime. In amonetary union like the United States, these beliefs will vary
uniformly across regions. This means that these expectations aredifferenced out in a panel regression with time fixed effects.
The result that long-run inflation expectations are constantacross regions (and sectors) in our model relies on productivity
and other drivers of real costs having a common trend in the longrun. If productivity growth (say) differs across regions even in thelong run, this will lead to persistent differences in nontradeableinflation (a Balassa-Samuelson effect). However, if this difference
is constant over time, it will be absorbed by region fixed effects ina panel specification.
These observations imply that that we can adopt an empiricalN
term in equation (13) withspecification that replaces the Et πt+∞time and region fixed effects:

THE SLOPE OF THE PHILLIPS CURVE

1321

simplifies to(15)

πitN = −ψuit − δ p̂itN + αi + γt + ω̃itN ,12. This same type of lack of comparability arises in some cases for different
estimates based on aggregate data. Some researchers use longer-term inflation
expectations, rather than one-period-ahead inflation expectations, to proxy forN
Et πt+1
when estimating the Phillips curve using aggregate data. Our analysis
shows, however, that when researchers use data on long-term inflation expectations, they (perhaps inadvertently) end up estimating ψ, not κ. To compare such
estimates with those based on a specification that controls for one-period-ahead
expectations, one must translate between the two, for example, by using the forκ
mula ψ = 1−βρor a version of this formula appropriate for (say) 10-years-aheaduinflation expectations.

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κ
λwhere ψ = 1−βρ
and δ = 1−βρ
. This equation is similar to the emu
pN
pirical specification used by much of the recent regional Phillips
curve literature. Comparing equations (14) and (15), we see thatan important difference between them is that the slope coefficient
is not the same. The slope coefficient in equation (14) is κ (whichis the same as the slope coefficient in equations (11) and (12)),κ. Sincewhereas the slope coefficient in equation (15) is ψ = 1−βρuunemployment is quite persistent, ψ is likely to be substantiallylarger than κ. Note that the AR(1) assumption we use to derive
equation (15) is not used in our estimation of κ.
A curious feature of the recent regional Phillips curve literature is that it has tended to yield larger estimates of the slopeof the Phillips curve than more traditional estimation strategies
based on aggregate data (Fitzgerald and Nicolini 2014; Babb and
Detmeister 2017; Hooper, Mishkin, and Sufi 2019; McLeay andTenreyro 2019). Comparing equations (14) and (15) provides a simple explanation for this discrepancy. The regional Phillips curve
literature has been estimating ψ in equation (15), while the more
traditional literature using aggregate variation has typically been
estimating κ. Since ψ κ, it is not surprising that the slope of thePhillips curve estimated in the regional literature has seemedlarge relative to traditional estimates.12
The difference between κ and ψ arises due to the different
ways equations (11) and (15) capture the effects of expected future unemployment on current inflation. In equation (11), theeffects of expected future unemployment on current inflation are
captured by the inflation expectations term Et π t+1 and the coefficient on current unemployment κ only reflects the effect of current

1322

THE QUARTERLY JOURNAL OF ECONOMICS

13. McLeay and Tenreyro (2019) control for inflation expectations at the census
region level when they estimate the regional Phillips curve. The variation acrossregions in these inflation expectations data is quite minimal. It may be the casethat the variation in this variable is quite attenuated relative to actual variation
in inflation expectations across the metropolitan statistical areas that form theregional units in their analysis.

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unemployment on current inflation. In contrast, the slope coefficient in equation (15) captures both the effect of current unemployment and the effect of expected future unemployment into theindefinite future on current inflation—that is, the fact that highunemployment today forecasts high unemployment in future periods.13
An advantage of estimating specifications such as equations (14) and (15) rather than equation (11) is that identifying theslope coefficient is less sensitive to the exact timing of changes in
inflation relative to inflation expectations. In Figure III, we showthat the difference between inflation and inflation expectations isquite sensitive to the exact measure of inflation.
We have so far manipulated the Phillips curve under the standard assumption of full-information rational expectations. However, the arguments we make above—solving the Phillips curve
forward—rely only on the weaker assumption that the law of iterated expectations holds. We elaborate on this point in Online
Appendix A.10, drawing on results from Adam and Padula (2011)
and Coibion, Gorodnichenko, and Kamdar (2018).
To derive a tractable empirical specification for the regional
Phillips curve in which the coefficient on unemployment is thesame as in the aggregate Phillips curve, we have made a numberof strong assumptions (perfect labor mobility within a region, no
labor mobility across regions, GHH preferences, production linear in labor, etc.). In the world, these assumptions are unlikely tohold exactly, and the empirical specification we estimate is thus
unlikely to yield exactly the slope of the aggregate Phillips curve.
Deriving an exact analytical mapping is nonetheless useful because it highlights in a transparent way the importance of certain
forces (e.g., inflation expectations). In more general models where
no exact analytical mapping between the slope of the regional andaggregate Phillips curves exists, our regional slope estimates canbe used as empirical targets in a moment-matching exercise. Even
aside from the simple case we analyze where the aggregate andregional slopes are equal, these moments are likely to provide

THE SLOPE OF THE PHILLIPS CURVE

1323

valuable information about the slope of the aggregate Phillips
curve (Nakamura and Steinsson 2018; Andrews, Gentzkow, andShapiro 2020).

The BLS does not publish state-level price indices. Prior work
has used metropolitan-level BLS price indices and cost of living
estimates from the American Chamber of Commerce Realtors
Association (ACCRA) to construct state-level price indices (seeDel Negro 1998; Nakamura and Steinsson 2014). An important
drawback of this approach is that the BLS imputes missing datausing data from other regions. Recent work has used scanner pricedata to construct state-level price indices (Beraja, Hurst, and Ospina 2019). An important drawback of scanner data is the shortsample period available.
We construct new state-level price indices for the UnitedStates based on the micro-price data the BLS collects for the purpose of constructing the CPI. Our sample period is 1978 to 2018(with a 26-month gap in 1986–1988 due to missing micro-data).
The micro-data that we base our price indices on are available
in the CPI Research Database at the BLS. The data for 1978–
1987 were constructed by Nakamura et al. (2018). The micro-pricedata in the CPI Research Database cover thousands of individual
goods and services, constituting about 70% of consumer expenditures. They are collected by BLS employees who visit outlets torecord prices. The database does not include the rent prices used
to construct the shelter component of the CPI. For this reason,
we analyze the behavior of rents separately. Prices are sampled
in 87 geographical areas across the United States. In New York,
Chicago, and Los Angeles, all prices are collected at a monthly
frequency. In other locations, food and energy prices are collected
monthly and the prices of other items are collected bimonthly. The
CPI Research Database is described in more detail in Nakamura
and Steinsson (2008) and Klenow and Kryvtsov (2008).
IV.A. State-Level Price Index Construction
Our methodology for constructing price indices is a simplified
version of the procedure used by the BLS to construct the CPI.
One key difference versus the BLS procedure, and a key reason
we do not simply use the BLS’s own price index software, is that

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IV. DATA AND CONSTRUCTION OF STATE-LEVEL PRICE INDICES

1324

THE QUARTERLY JOURNAL OF ECONOMICS

(16)

ri,t =

Pi,t
Pi,t−τ

 1τ

,where ri,t denotes the price relative, Pi,t denote the effective price,and τ denotes the number of months since the last time a price wascollected for this product. Several details are important. First, it isimportant to use the effective price rather than the raw collected
price. The difference between the collected and effective prices isthat the latter adjusts for changes in the number and size of theitems being priced (e.g., a 2L bottle of Diet Coke versus a two-packof 2L bottles of Diet Coke).
Second, we define a product not only by its characteristics(e.g., 2L bottle of Diet Coke), but also by the location in which it issold. To be precise, in the CPI Research Database, each product is
indexed by outlet, quote, and version. The quote is a very narrowly
described product, and the version is the exact specification of theitem that the price collector identifies in the store. We hold all
three of these parameters—outlet, quote, and version—fixed inconstructing a product’s price relative.
Third, we must decide what to do when prices are missing.
Missing prices occur when the product is unavailable because of atemporary stockout, or as a consequence of the bimonthly pricing
schedule used by the BLS for most products in most cities. Our
procedure is to divide the price change evenly among the periods
between successive price observations by taking the τ th root of
the price change and applying this price relative to all τ periods.
This implies that ri,t = ... = ri,t−τ +1 , where again τ is the numberof periods between successive price changes. There are several
other important details of our index construction procedure that
we describe in Online Appendix B.3.
We aggregate the price relatives in several steps. First, we
compute an unweighted geometric average of the price relatives within each entry level item (ELI) product category andstate. ELIs are relatively narrow product categories such as

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we do not impute missing price observations using inflation rates
calculated for other sectors or regions. We describe our procedure
below.
We start by calculating price relatives for individual products.
These are the fundamental building blocks of a matched-model
price index. For product i at time t, the formula we use to calculate
the price relative isTHE SLOPE OF THE PHILLIPS CURVE

1325

14. See the appendix to Nakamura and Steinsson (2008) for a list of the ELIs
used in the construction of the CPI.
15. Here we follow the BLS in using consumption weights. Rubbo (2020)
argues that production networks imply that product-level inflation should beweighted by sales shares.
16. We find that there is much more variablility across states in nontradeableinflation than tradeable inflation. For nontradeables, the first principal componentof state-level inflation captures only about 37% of the variance in the underlying
state-level series. In contract, for tradeables, the first principal component captures about 71% of the variance in the underlying state-level series. This patternis consistent with our argument in Section III.B that many tradeable goods are
priced nationally and do not respond to regional marginal costs.

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“full service meals and snacks” (restaurants) and “motorcycles”
defined by the BLS for the purpose of calculating the CPI.14 We
then calculate sectoral state-level price indices by computing aweighted geometric average of the ELI-state indices across the
ELIs in that state and sector. We use national weights from theConsumer Expenditure Survey (CEX) for 1998 to perform thisaggregation.15
Our empirical analysis focuses on nontradeables, but we also
construct state-level price indices for tradeables—which we simply define as the complement of nontradeables—and overall statelevel price indices. We construct a price index for nontradeables
based on our own categorization of BLS’s ELI product categories.
In doing this, we try to be conservative in our definition of what
constitutes a nontradeable good, since including tradeable goods
could lead to attenuation of the slope of the Phillips curve if tradeable goods are priced nationally. In contrast, the main downsideof excluding some nontradeable goods is less precise estimates.
The goods we classify as nontradeables account for roughly 44%
of nonhousing consumer expenditures. Importantly, our index of
nontradeables does not include housing services or transportation goods (mainly airline tickets).16 We estimate regional Phillips
curves for housing services separately in Section V using different
data. Online Appendix B.4 provides a detailed list of which ELI
categories we classify as nontradeable.
Our method for calculating state-level price indices aims toapproximate the nonshelter price index published by the BLS.
Online Appendix Figure C.1 illustrates our ability to match theofficial BLS data by comparing the evolution of 12-month inflation
at the aggregate level using our methodology with official CPI
inflation excluding housing. The figure shows that we are able to1326

THE QUARTERLY JOURNAL OF ECONOMICS

approximate the official BLS data very closely. This is true even forthe pre-1988 period when we rely on the micro-data recovered byNakamura et al. (2018), which likely have greater measurement
error.17

The measure of unemployment that we use as our measureof labor market slack in the Phillips curve is the quarterly, seasonally adjusted, state unemployment rate from the Local Area
Unemployment Statistics (LAUS) published by the BLS. We also
make use of employment data in constructing our tradeabledemand spillovers instrument discussed in Section V. This instrument is a shift-share instrument, similar to the one used in Bartik(1991). It is constructed using employment shares of individual
industries at the state level.18 We seasonally adjust the resulting series by regressing it on an exponentially weighted moving
average of its lags and state by quarter-of-year fixed effects. We
use the variation not explained by the quarter-of-year dummies
as our instrument.19 We define the tradeable employment sharein the same way as Mian and Sufi (2014). Online Appendix B.5
discusses this in more detail.

V. EMPIRICAL RESULTS
We now turn to our empirical results. We present estimates ofthe structural parameter κ from equation (14) and ψ from equation (15). Recall that κ is the structural slope coefficient in theregional Phillips curve for nontradeables from our model, and ψ
is the reduced-form slope coefficient in the type of regional empirical specification often run in prior work. To estimate equation (14),we replace expected future unemployment and relative prices with
17. We drop Arizona because of anomalous trends that we have not been able
to investigate due to COVID-19 related access restrictions at the BLS.
18. Industry-state employment data are available from the QCEW, at quarterly frequency for two-digit SIC codes (1975–2000) and three-digit NAICS codes(1990–2017). Before 1990 we use two-digit SIC codes to define industry, whereas
after 2000 we use three-digit NAICS codes. For the period 1990–2000, when boththe NAICS and SIC code classifications are available, we construct both versionsof the instrument and use a simple average of the two.
19. Using the X-11 algorithm for seasonal adjustment yields virtually identicalresults.

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IV.B. Employment Data

THE SLOPE OF THE PHILLIPS CURVE

1327

their realized values and an expectation error. We also truncate
the infinite sum in equation (14) at j = T. Doing this yields(17)

πitN = αi + γt − κT

β j ui,t+ j − λNNNβ j p̂i,t+
j + ω̃it + ηit ,j=0

where ηitN denotes an expectations error that is orthogonal to information known at time t (and a truncation error). Equation (17)can now be estimated with standard GMM methods, that is, by
instrumenting for the two forward sums. We do not attempt toestimate β. Rather, we set it to a standard quarterly value of β =0.99.
We present results for two approaches to identifying the coefficients κ and λ in equation (17). Our first approach is to instrument
for the two forward sums with four-quarter lagged unemployment
ui, t−4 and the four-quarter lagged relative price of nontradeablesN. Assuming rational expectations, these lagged variables willp̂i,t−4
be uncorrelated with the expectations error ηitN . The identifying
assumption regarding supply shocks is that when one state experiences a boom or bust relative to another state, it does not
systematically experience nontradeable supply shocks relative tothis other state. For example, when Texas experiences a recession relative to Illinois, this is not systematically correlated withchanges in restaurant technology in Texas relative to Illinois. Notice that national supply shocks are absorbed by the time fixed
effects, so only regional nontradeable supply shocks are potential
confounders.
Our second approach to identification is to construct an instrumental variable that captures variation in demand. The idea
behind our instrumental variable is the notion that national variation in demand for specific tradeable goods will differentially affectlabor demand for nontradeable goods in states that produce those
tradeable goods. For example, an increase in oil prices will differentially affect labor demand in Texas (and other oil-producing
states). As a result, wages in Texas will rise, differentially affecting costs of nontradeables in Texas. Building on this idea, weconstruct a tradeable-demand spillovers instrument as(18)

Tradable Demandi,t =x

S̄x,i × 3Y log S−i,x,t ,Downloaded from https://academic.oup.com/qje/article/137/3/1299/6529257 by guest on 13 July 2022

j=0

T


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20. In a related approach, McLeay and Tenreyro (2019) use identified demand
shocks from government spending to estimate the slope of the regional Phillips
curve.

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where S̄x,i is the average employment share of industry x in statei over time, and 3Y log S−i,x,t is the three-year growth in national
employment of industry x at time t excluding state i. This shiftshare instrument builds on Bartik (1991) and more closely on
Nguyen (2014). The identifying assumption in this case is that
there are no supply factors that are correlated with the shifts
3Y log S−i,x,t in the time series and correlated with the shares S̄x,iin the cross section. For example, costs will increase as a result of
an increase in oil prices. But if such cost increases are no larger
on average for restaurants in Texas than Illinois they will be
uncorrelated with our instrument.20
Our panel data approach implies that we are relying on
cross-state variation in unemployment to identify the slope of thePhillips curve. Figure IV shows the evolution of the unemployment rate for three states, California, Texas, and Pennsylvania,
over our sample period. While there is certainly a great deal ofcomovement, this figure illustrates well that there is also substantial cross-state variation. One example is that the 1991 and
2007–2009 recessions affected California much more than Texas
and Pennsylvania. Another is that Texas experienced a recession
in the mid-1980s (partly due to the savings and loan crisis andpartly to a large fall in oil prices) while most other states experienced a continued fall in unemployment. Estimates of unemployment at the state level may be plagued by measurement error.
Our IV estimation will address this insofar as the measurement
error is classical.
The dependent variable in our regressions is πitN = pitN −N
pi,t−4 , that is, state-level nontradeable inflation over the previous 12 months. Studying inflation over four quarters allows us toreduce measurement error and eliminate seasonality. In Online
Appendix A.11, we show that using 12-month inflation as our dependent variable implies that we need to divide our estimates ofκ and λ from equation (17) by four to account for the time aggregation. Recall that the inflation rate in our model in Section III isa quarterly inflation rate.
We truncate the discounted sums on the right-hand sideof equation (17) at T = 20 quarters. Online Appendix Table C.4 presents robustness regarding this choice for our main

THE SLOPE OF THE PHILLIPS CURVE

1329

Regional Business Cycles
This figure plots the unemployment rate for California, Pennsylvania, and Texas.

specification. Our results are similar for values between T = 20
and T = 40. In Online Appendix A.12, we estimate κ using equation (17) with T = 20 on data simulated from our model from
Section III. We find that our empirical procedure is able to accurately estimate the true value of κ in this setting for a very wide
range of true values of κ.
The forward sums in equation (17) imply that we lose five
years of observations at the end of our sample when we set T =20. To minimize the effect of this, we use a two-sample two-stage
least squares (2SLS) regression. We estimate the first stage on a
reduced sample without the last five years and the second stage
on the full sample. We cluster standard errors at the state level
and apply a correction to our standard errors appropriate for twosample 2SLS developed by Chodorow-Reich and Wieland (2020).21
21. Our tradeable-demand instrument uses all the information in national
industry employment growth rates. So our standard errors are not subject to theconcerns about inference with shift-share instruments raised by Adão, Kolesár,
and Morales (2019).

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FIGURE IV

1330

THE QUARTERLY JOURNAL OF ECONOMICSTABLE ISLOPE OF THE REGIONAL PHILLIPS CURVE
No fixed
effects(1)

Laggedunempl.
(3)

Tradeable
demand IV(4)

Panel A: Estimates of κ from equation (17)κ
− 0.0037
0.0003(0.0013)(0.0019)0.0062(0.0028)0.0062(0.0025)0.112(0.057)√√0.339(0.126)√√

Panel B: Estimates of ψ from equation (19)ψ
− 0.103
0.017(0.036)(0.027)√
State effects
Time effects

Notes. This table presents estimates of κ and ψ from regression specifications (17) and (19), respectively.
The outcome variable is cumulative nontradeable inflation over four quarters, measured in percentage points.
We include time and state fixed effects as noted at the bottom of each column. In Panel A, the regressors
are the discounted future sum of quarterly state unemployment, in percentage points, and the discounted
future sum of the relative price of nontradeables, in 100 × log points. For both variables, we truncate thediscounted future sum at 20 quarters. In Panel B, the regressors are the fourth lags of quarterly state
unemployment, measured in percentage points, and the relative price of nontradeables. In the first threecolumns we instrument using the fourth lags of quarterly state unemployment and the relative price of
nontradeables (this is OLS for ψ). In the fourth column, we replace lagged unemployment with our tradeabledemand instrument among the instruments. In all columns, we estimate κ by two-sample 2SLS, and apply thecorrection to our standard errors from Chodorow-Reich and Wieland (2020). The sample period is 1978–2018.
Standard errors are reported in parentheses, clustered by state. All regressions are unweighted. The numberof observations is 3,323 in the first three columns of Panel A, with slightly fewer in the last column due todifferencing. Likewise, the number of observations is 4,490 in the first three columns of Panel B.

Our empirical specification for estimating ψ is(19)

N
+ εit .
πitN = αi + γt − ψui,t−4 − δpi,t−4

We use beginning-of-period unemployment and relative price of
nontradeables as regressors for consistency with previous studies such as Ball and Mazumder (2019). We present results for two
identification approaches analogous to those we use for κ. The first
approach is to estimate equation (19) by OLS (i.e., instrumenting
for lagged unemployment and the relative price of nontradeables
with themselves). The second approach replaces lagged unemployment among the instruments with our tradeable-demand instrument.
V.A. Full-Sample Results
Table I presents estimates of κ and ψ for our full sample period of 1978–2018. Let’s start by considering the estimates of κ inPanel A. When we estimate equation (17) without fixed effects, ourestimate has the “wrong” sign, that is, higher unemployment is

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No timeeffects(2)

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1331

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associated with higher rather than lower inflation (κ = −0.0037).
Adding state fixed effects raises the estimate of κ to 0.0003. Adding
time fixed effects further raises the estimate of κ to 0.0062. As westress throughout the article, time fixed effects eliminate changes
in long-run inflation expectations. Finally, using our tradeabledemand instrument as opposed to instrumenting with lagged unemployment yields virtually the same estimate for κ of 0.0062.
The fact that our estimate of κ does not change between columns(3) and (4) suggests that the fixed effects we include are sufficient
to absorb supply shocks.
Our estimated slope of the Phillips curve is statistically significantly different from zero. In absolute size, however, the slope issmall in the sense that it is consistent with the modest response of
inflation to changes in unemployment seen in the aggregate timeseries since 1990. We develop this implication in Section V.D. Online Appendix Table C.1 presents estimates of the “first-stage” regressions for our IV estimates of equation (17). These first-stage
regressions show that our instruments are strong instruments.
We separately regress the present value of unemployment andthe present value of relative prices on the reduced-form regressors. Lagged unemployment and tradeable demand both strongly
predict the present value of unemployment and weakly predict thepresent value of relative prices. Lagged relative prices strongly
predict the present value of relative prices and weakly predict thepresent value of unemployment.
Online Appendix Table C.2 reports our estimates of λ for regression specification (17)—the coefficient on the relative price of
nontradeables. In our preferred specifications with time and state
fixed effects, we estimate values of λ between 0.002 and 0.003. In
the model we present in Section III, λ provides an estimate of thedegree of nominal rigidities. In a world with flexible prices, ourestimate of λ would be large. The fact that our estimate of λ is verysmall provides further support—over and above our estimate ofκ—for the notion that prices are quite rigid in the U.S. economy.
In our baseline results, we calibrate β = 0.99. It may be thatfirms are considerably less forward looking when they set prices
than this calibration implies. Recent work has shown that plausible deviations from full rationality or common knowledge yielda Phillips curve that is less forward looking (Angeletos and Lian
2018; Gabaix 2020). Also, a model with a combination of sticky
information and sticky prices yields a Phillips curve that is lessforward looking. Online Appendix Table C.3 presents estimates of1332

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V.B. Subsample Results
We analyze to what extent the Phillips curve was steeper
during the period of the Volcker disinflation than in subsequent
years. Table II presents estimates of κ and ψ for the periods 1978–
1990 and 1991–2018. We present these estimates for specifications
with and without time fixed effects. All specifications include state
fixed effects and control for the relative price of nontradeables.
Consider first the specification without time fixed effects reported in columns (1) and (2). For the pre-1990 sample, κ is estimated to be 0.0278, while ψ is estimated to be 0.449. In sharp
contrast, for the post-1990 sample, κ is estimated to be 0.0002

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κ where we calibrate β to lower values. As we vary our quarterly
calibration of β from 0.99 to 0.9, κ doubles in size. The absolute
size of the increase is small because our initial estimate of κ issmall.
Our estimates of ψ in Table I, Panel B have a similar patternto our estimates of κ discussed above. The estimate without timeor state fixed effects is negative, and the estimate increases as we
include state and then time fixed effects. An important differenceis that the absolute size of our estimates of ψ are much larger thanour estimates of κ. This reflects the fact that in equation (19) the
lagged unemployment rate is standing in for the entire future sumin equation (17). Since unemployment is quite persistent, time
variation in the future sum is much larger than time variation in
the unemployment rate, which results in a much larger coefficientin equation (19) than in equation (17).
Another difference is that ψ is much larger in column (4) thanin column (3), whereas κ is virtually identical. This reflects thefact that the tradeable-demand instrument we use in column (4)is more persistent than the unemployment rate itself. The coefficients in column (4) are therefore identified using more persistentvariation, which results in a larger value of ψ but not a larger
value of κ. This highlights an important advantage of estimating
κ as opposed to ψ: estimates of ψ are hard to interpret because
they are sensitive to the persistence of the variation used to identify them. More generally, κ is a structural parameter, and ψ isnot. This implies that ψ may differ depending on the setting being
considered (e.g., may be low in response to a policy change that
may be reversed due to a future change in government), while κis policy invariant.

THE SLOPE OF THE PHILLIPS CURVE

1333

TABLE II
HAS THE PHILLIPS CURVE FLATTENED?
Lagged unempl. IV
with time fixed effectTradeable-demand IV
with time fixed effectPre-1990(1)Pre-1990(3)

Post-1990(4)

Pre-1990(5)

Post-1990(6)

Panel A: Estimates of κ from equation (17)κ
0.02780.00020.0107(0.0025)(0.0017)(0.0080)

0.0050(0.0040)0.0109(0.0062)0.0055(0.0028)

Panel B: Estimates of ψ from equation (19)
ψ
0.449
0.009
0.198
(0.063)(0.025)(0.113)0.090(0.057)0.422(0.232)0.332(0.157)

Post-1990(2)

Notes. The table presents estimates of κ and ψ, before and after 1990. Columns (1), (3), and (5) present
results for the sample period 1978–1990; and columns (2), (4), and (6) for the sample period 1991–2018.
All specifications include state fixed effects. Specifications in columns (3)–(6) include time fixed effects. The
instruments in columns (1)–(4) are the fourth lag of the tradeable-demand instrument and the relative priceof nontradeables (i.e., OLS in Panel B). In columns (5) and (6), the instruments are the fourth lag of the
tradeable-demand instrument and the relative price of nontradeables. In all columns, we estimate κ by twosample 2SLS and apply the correction to our standard errors from Chodorow-Reich and Wieland (2019).
Standard errors are reported in parentheses, clustered by state. All regressions are unweighted.

and φ is estimated to be 0.009. The difference across samples isroughly a factor of 100 for κ and 50 for ψ. In other words, aggregate inflation became much less sensitive to unemployment after
1990 than it was during the Volcker disinflation.
Contrast this with the results in columns (3)–(4) where time
fixed effects are included in the regressions. In this case, the estimated values of κ and ψ fall only modestly between the earlypart of the sample and the later part of the sample. For the pre1990 sample, κ is estimated to be 0.0107 and ψ is estimated to be
0.198. For the post-1990 sample, κ is estimated to be 0.0050 and ψ
is estimated to be 0.090. The difference across samples is roughlya factor of two and is not statistically significant. The estimatesfor κ in columns (5) and (6) are very similar to the estimates incolumns (3) and (4), and the estimates of ψ in columns (5) and (6)show an even smaller difference across sample periods.
As we emphasize in Section II, estimates of the Phillips curve
based on time-series variation—such as the estimates without
time fixed effects in Table II—are likely to be heavily influenced
by time-series variation in long-run inflation expectations Et π t+∞ .
In contrast, the specifications in Table II that include time fixed
effects difference out the influence of long-run inflation expectations. The results in Table II therefore suggest that the apparent
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Lagged unempl. IV
without time fixed effect1334

THE QUARTERLY JOURNAL OF ECONOMICS

In the left panel, we residualize state nontradeable inflation and unemployment
against state fixed effects and the relative price of nontradeables, before and after
1990. We then plot residualized inflation and unemployment, before and after
1990, grouped by 20 bins of state unemployment. The right panel carries out thesame exercise after further residualizing against time fixed effects. In both panels,
inflation is cumulated over the previous four quarters and unemployment is lagged
by four quarters.

inflation expectations becoming more firmly anchored over time.
In the early part of the sample, inflation expectations shifted agreat deal and these shifts were negatively correlated with the
unemployment rate, which meant that shifts in inflation expectations masqueraded as a steep Phillips curve. The cross-sectional
results in Table II, columns (3)–(6) reveal that in fact the Phillips
curve has always been quite flat (at least since 1978).
Figure V provides a visual representation of the results inTable II. In the left panel, we plot a binned scatterplot of statelevel nontradeable inflation against state-level unemployment after removing state fixed effects and the effects of the relative priceof nontradeables. We plot the data separately for 1978–1990 and
1991–2018. The plot also includes regression lines for each subsample. The data in this panel do not account for time fixed effects
and therefore include aggregate time series variation. As a consequence, we see a huge flattening of the Phillips curve in thiscase.
Contrast this with the right panel in Figure V. This is ananalogous figure to the left panel except that we also demean
by time fixed effects. These data therefore only reflect regionalvariation in inflation. In this case, the difference in the slope of

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FIGURE V
Scatterplots: Nontradeable Inflation and Unemployment

THE SLOPE OF THE PHILLIPS CURVE

1335

TABLE IIIOUR ESTIMATES COMPARED TO PRIOR WORK
κ
0.019
0.085
0.00770.0062

Notes. We adjust the estimates from Rotemberg and Woodford (1997), Galı́ (2008), and Nakamura andSteinsson (2014) by the elasticity of output with respect to employment in the model in these papers. For
Nakamura and Steinsson (2014), we use the calibration with GHH preferences.

the Phillips curve between the early sample and the late sample
is modest. The modest flattening of the Phillips curve that we findover our sample (once we account for time fixed effects) seems
consistent with the fact that the frequency of price change in the
United States has declined by about 40% as inflation has fallen
since the early 1980s (Nakamura et al. 2018).
V.C. How Do Our Estimates Compare to Prior Work?
It is instructive to compare our estimate of κ to values of κ arrived at by means of structural estimation or calibration of New
Keynesian models. Table III reports three such estimates from
Rotemberg and Woodford (1997), Galı́ (2008), and Nakamura andSteinsson (2014). In all cases, we have adjusted the reported valueof κ from these papers by the elasticity of output with respect toemployment. As is well known, the value of κ in a New Keynesianmodel is highly dependent on the degree of nominal and real rigidities assumed. The values for κ used in these papers range fromabout an order of magnitude larger than our estimated value to avalue roughly equal to our estimated value. The main difference
between Galı́’s relatively high value and the much lower valuesin Rotemberg and Woodford (1997) and Nakamura and Steinsson(2014) lies in the degree of real rigidity that the models used inthese papers imply. Galı́’s model is a relatively simple (textbook)version of the New Keynesian model, which does not incorporate
strong sources of real rigidity. Rotemberg and Woodford (1997)
and Nakamura and Steinsson (2014) use models with heterogeneous labor markets, which yields a much larger amount of real
rigidity. In both cases, the large amount of real rigidity helps theseauthors match moments that they target in their analysis. Similarly, our estimates imply that the data we have analyzed are also

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Rotemberg and Woodford (1997)
Galı́ (2008)
Nakamura and Steinsson (2014)
Our full sample IV estimate

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THE QUARTERLY JOURNAL OF ECONOMICS

more consistent with New Keynesian models that incorporate alarge amount of real rigidity.
V.D. Aggregate Implications

(20)

T


β j ũt+ j = ζ ũt + α + t .

j=0

The series we use for ũt in this regression is the difference between
the aggregate unemployment rate in the United States and the
CBO’s estimate of the natural rate of unemployment at each point
in time.22 We run this regression for the sample period 1979Q4–
2017Q4. This yields an estimate of ζ for aggregate variation in theunemployment rate of 6.16 with a Newey-West standard error of
22. We are thus treating the CBO’s estimate of the natural rate of unemployment as a forecast of long-run unemployment Et ut+∞ .

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A question that naturally arises regarding our cross-sectional
estimates of κ is whether they can explain the aggregate timeseries variation in inflation over our sample. A number of researchers and commentators have suggested that the stability of
inflation at the aggregate level in the United States has beensurprising over the past 25 years (missing disinflation during theGreat Recession and missing reinflation during the late 1990s andlate 2010s). Some researchers have recently argued that crosssectional variation suggests a steeper Phillips curve than timeseries variation for the past few decades. Here, we assess whether
this is the case for our estimates.
We start with the solved-forward aggregate Phillips curve—
equation (6). In Section II, we made the simplifying assumptionthat the unemployment rate follows an AR(1). This assumption
allowed us to derive a simple aggregate relationship between the
discounted future sum of unemployment rates in equation (6) andthe current unemployment rate—see equation (7). In reality, thedynamics of the U.S. unemployment rate differ substantially froman AR(1) (see Neftçi 1984; Sichel 1993; Dupraz, Nakamura, andSteinsson 2020). For this reason, we adopt an approach of estimating a scaling factor ζ that relates the current unemployment rate
to the discounted future sum in equation (6) using the following
regression

THE SLOPE OF THE PHILLIPS CURVE

1337

1.80. Using equation (20), we can rewrite equation (6) as(21)

πt − Et πt+∞ = −κζ ũt + ωt .

23. We use the shelter and nonshelter expenditure weights in the core CPI.
These are 0.42 and 0.58, respectively.
24. The coefficient κζ is calculated as 4 × (0.58 × 0.0062 + 0.42 × 0.0243) ×
6.16, where the factor of 4 accounts for time aggregation to annual inflation.
25. Online Appendix Figure C.2 shows that the modest flattening of thePhillips curve we estimate over our sample period has a minimal effect on thefluctuations in the gray line in Figure VI. Online Appendix Figure C.3 shows thata disproportionate share of the systematic variation in inflation and the fitted

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Our cross-sectional estimates of κ are for nontradeables excluding housing services. As we emphasize in Section II, the treatment of housing services has important implications for the behavior of inflation. Online Appendix Table C.5 presents estimatesof κ and ψ using state-level annual rent inflation data from theAmerican Community Survey for 2001 to 2017. For our baseline
specification with state and time fixed effects, we estimate κ tobe 0.0243. This estimate of κ is roughly four times larger thanour estimate of κ for nonhousing nontradeable goods reported inTable I. We account for this difference below by taking a weighted
average of our full-sample κ estimate for nontradeables and thisκ estimate for housing services.23
Figure VI plots the left-hand side of equation (21) (black line)
against the first term on the right-hand side of equation (21) (grayline) using our estimates of κ and ζ from above, which yields κζ= 0.34.24 We use the 10-year ahead SPF inflation expectations for
the CPI as our measure of long-term inflation expectations. The
gray line is the demand-induced variation in inflation predicted byour estimates. The figure indicates that the amplitude of inflation
fluctuations over the last few business cycles has been roughly
in line with what our cross-sectional estimates of κ suggest. In
particular, the disinflation during the Great Recession and reinflation during the 2010s lines up well with what our estimate of κ
implies. If the gray line had a larger amplitude than the black line
over the business cycle, this would indicate missing disinflationand missing reinflation. In fact, the amplitude of the gray line isvery similar to that of the black line for the Great Recession, thepost–Great Recession recovery, and the long 1990s expansion. By
this metric, there is thus no missing disinflation or missing reinflation over this period. These findings echo the results of Balland Mazumder (2019).25

1338

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The figure shows the fit of the aggregate Phillips curve for core inflation. The
black line is the difference between core inflation and the 10-years-ahead SPF
inflation expectation for the CPI. The gray line plots the first term on the righthand side of equation (21), which is the demand-induced variation in inflation
predicted by our estimates.

The most substantial deviation between the actual and fitted values arises during the Volcker period when actual inflation
relative to long-run expectations lies far above the fitted value.
While the conventional view is that the Phillips curve has broken down after 1990, we are finding the opposite: a poor fit of
our cross-sectional estimate of the Phillips curve when applied
to aggregate inflation dynamics over the Volcker period. A natural interpretation of this discrepancy is the presence of adverse

value predicted by our model comes from the housing services (rent) component of
the CPI. The figure is analogous to Figure VI except that the black line excludes
housing and the fitted value uses only the κ estimate for the nonshelter componentof inflation. We see that core inflation excluding housing services varies much lesssystematically than core inflation including housing services.

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FIGURE VI
Aggregate Phillips Curve

THE SLOPE OF THE PHILLIPS CURVE

1339

VI. CONCLUSION
This article provides new estimates of the slope of the Phillips
curve. We estimate that the slope of the Phillips curve is small,and was small even during the Volcker disinflation of the early
1980s. Our results indicate that shifts in expectations about
the conduct of monetary policy explain much of the drop of inflation in the early 1980s and more firmly anchored inflation
expectations explain the stability of inflation since the mid1990s. Our estimates are consistent with the insensitivity of inflation to unemployment during both the Great Recession andduring the low unemployment periods of the late 1990s andlate 2010s.
To reach these conclusions, we estimate the Phillips curve inthe cross-section of U.S. states. We use newly constructed statelevel price indices for nontradeable goods starting in 1978. We
map from our regional estimates to the slope of the aggregate
Phillips curve using a multiregion New Keynesian model. The
model clarifies that the slope of the aggregate Phillips curve isequal to the slope of the regional Phillips curve for nontradeable
goods. We also use the model to show that regional data difference

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supply shocks in the early 1980s, for example, associated with theoil price shocks.
How much of the fall in inflation during the Volcker disinflation can be attributed to the causal effect of higher unemployment
working through the slope of the Phillips curve according to ourestimates? Unemployment rose by about 5 percentage points between 1979 and 1982. Using a weighted average of our slightly
higher pre-1990 nonshelter estimate for κ and our estimate of κ
for shelter, we find that this increase in unemployment caused
inflation to fall by only about 2 percentage points (see gray line
in Online Appendix Figure C.4). Core CPI inflation first rose from
7% to 10% from 1979 to 1981 and then fell to 4% by 1986. Clearly,
the direct causal effect of unemployment working through theslope of the Phillips curve explains only a modest amount of thisvariation in inflation. Over this same period, long-run inflation
expectations first rose from 7% to 8% and then fell to 4%. Ourestimates, therefore, suggest that the bulk of the variation in
inflation over the early 1980s is due to changes in long-run inflation expectations, with supply shocks also playing an important
role.

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THE QUARTERLY JOURNAL OF ECONOMICSLONDON SCHOOL OF ECONOMICS, UNITED KINGDOMUNIVERSITY OF CALIFORNIA, SAN DIEGO, UNITED STATESUNIVERSITY OF CALIFORNIA, BERKELEY, UNITED STATESUNIVERSITY OF CALIFORNIA, BERKELEY, UNITED STATESSUPPLEMENTARY MATERIAL
An Online Appendix for this article can be found at The
Quarterly Journal of Economics online.
DATA AVAILABILITY
Code replicating the tables and figures in this article canbe found in Hazell et al. (2022) in the Harvard Dataverse,
https://doi.org/10.7910/DVN/OQNZYE.

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out the effects of the long-run monetary regime, which otherwise
confound estimates of the slope of the Phillips curve. Guided by
the model, we show that the conventional empirical specification
used to estimate regional Phillips curves must be scaled by afactor relating to the persistence of unemployment fluctuations
to yield an estimate of the slope of the Phillips curve. Finally, wedevelop a new tradeable-demand spillover instrument that allows
for flexible patterns of supply shocks at the local level.
An important lesson from our analysis is that when it comesto managing inflation, the elephant in the room is long-run inflation expectations. This view contrasts sharply with the conventional view that managing inflation is about moving up anddown a steep Phillips curve. A crucial question for inflation dynamics is why long-run inflation expectations are sometimes sofirmly anchored but at other times move sharply? Beliefs about
inflation in the long run are governed by beliefs about the longrun behavior of the monetary authority and ultimately the political process that shapes the long-run behavior of the monetary authority. Since this is fundamentally a very low-frequency
phenomenon, it is not easily pinned down by half a century or
so of data from a single country. While much interesting research has sought to understand the behavior of long-run inflation expectations, we believe it is still not sufficiently wellunderstood and its crucial importance for the conduct of monetary policy implies that even more research should focus on thisquestion.

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