The slope of the Phillips Curve: evidence from U.S. States

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The author's accepted version of "The Slope of the Phillips Curve: Evidence from U.S. States" by Jonathon Hazell, Juan Herreño, Emi Nakamura and Jón Steinsson, dated January 4, 2022 and published in the Quarterly Journal of Economics, 137 (3), 1299-1344. The authors estimate the slope of the Phillips curve across U.S. states using newly constructed state-level price indexes for non-tradeable goods back to 1978. They report that the slope is small, was small even during the early 1980s, and has declined only modestly since then. The paper attributes the sharp drop in core inflation in the early 1980s mostly to shifting expectations about long-run monetary policy, and the greater stability of inflation between 1990 and 2020 to better-anchored long-run inflation expectations.

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Hazell, Joe , Herreño, Juan, Nakamura, Emi & Steinsson, Jón (2022) The slope of
the Phillips Curve: evidence from U.S. States. Quarterly Journal of Economics,
137 (3), 1299-1344. https://doi.org/10.1093/qje/qjac010

https://researchonline.lse.ac.uk/id/eprint/113326/

Version: Accepted Version

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The Slope of the Phillips Curve:
Evidence from U.S. States

Jonathon Hazell Juan Herreño Emi Nakamura Jón Steinsson∗
LSE UC San Diego UC Berkeley UC Berkeley

January 4, 2022

Abstract

We estimate the slope of the Phillips curve in the cross section of U.S. states using newly
constructed state-level price indexes for non-tradeable goods back to 1978. Our estimates in-
dicate that the slope of the Phillips curve is small and was small even during the early 1980s.
We estimate only a modest decline in the slope of the Phillips curve since the 1980s. We use
a multi-region model to infer the slope of the aggregate Phillips curve from our regional esti-
mates. Applying our estimates to recent unemployment dynamics yields essentially no miss-
ing 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 to shifting expectations about
long-run monetary policy as opposed to a steep Phillips curve, and the greater stability of in-
flation between 1990 and 2020 is mostly due to long-run inflation expectations becoming more
firmly anchored.

JEL Classification: E30

∗
We would like to thank Tomas Breach, Massimiliano Cologgi, and Victoria de Quadros for excellent research as-
sistance. We thank John Molino, our project coordinator at the BLS, for all his help throughout the project. We thank
Hassan Afrouzi, Regis Barnichon, Olivier Blanchard, Jordi Galı́, Yuriy Gorodnichenko, Pierre-Olivier Gourinchas, Greg
Kaplan, Chen Lian, Silvana Tenreyro, John Williams and seminar participants at various institutions for valuable com-
ments and discussions. We thank the National Science Foundation, the Alfred P. Sloan Foundation, the Smith Richard-
son Foundation, the Institute for New Economic Thinking, and the Julis-Rabinowitz Center for Public Policy and Fi-
nance for financial support.
1 Introduction

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:

πt = βEt πt+1 − κ(ut − unt ) + νt . (1)

According to this formulation, inflation πt is determined by three factors: expected inflation
Et πt+1 , the output gap — measured here as the difference between unemployment ut and the
natural rate of unemployment unt — and cost-push shocks νt . The slope of the Phillips curve κ
represents the sensitivity of inflation to the output gap (i.e., to an increase in demand).
The episode in US economic history that has perhaps most strongly influenced the profession’s
thinking regarding the slope of 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 evi-
dence for a relatively steep Phillips curve.
One way to formalize this conventional interpretation is to assume that inflation expectations
are adaptive: βEt πt+1 = πt−1 in equation (1). This yields the accelerationist Phillips curve:

∆πt = −κ(ut − unt ) + νt . (2)

Stock and Watson (2019) estimate κ in this equation and refer to it as the “Phillips correlation.”
They measure ∆πt by the annual change in 12-month core PCE inflation, and ut − unt by the CBO
unemployment gap, both at a quarterly frequency. Figure 1 reproduces this analysis. It suggests
that the slope of the Phillips 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 the Phillips 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 much less. The “missing disinflation” during and after the Great Recession
1
See also Ball and Mazumder (2011), Kiley (2015b), and Blanchard (2016).

1
6
1960-1983
5 1984-1999
2000-2019

Year-over-year change in inflation
4

-1

-2

-3

-4
-3 -2 -1 0 1 2 3 4 5
Unemployment Gap
Figure 1: Stock and Watson’s Changing Phillips Correlation
Note: Black solid line is a regression line for 2000-2019. Dark grey broken line is regression for 1984-1999. Light
gray dash-dot line is a regression line for 1960-1983. The year-over-year change in inflation is the 4 quarter
change in the (backward-looking) 4 quarter moving average of headline PCE inflation. The unemployment
gap is the 4 quarter (backward-looking) moving average of the gap between the unemployment rate and the
natural rate of unemployment. Authors’ calculations. The figure replicates Figure 1 from Stock and Watson
(2019).

then gave way to “missing reinflation” in the late 2010s as 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 to much of a rise in inflation . Some have
argued that the apparent flattening of the Phillips curve signals an important flaw in the Keynesian
model.
There is, however, an alternative interpretation of these facts that emphasizes the anchoring
of long-term inflation expectations in the United States (Bernanke, 2007; Mishkin, 2007). Figure
2 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 period of the Volcker disinflation. In sharp contrast, since 1998, long-term inflation expecta-
tions have been extremely stable.
An alternative to the standard narrative of the Volcker disinflation is that the decline in infla-
tion 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

2
10
Core CPI Inflation - Research Series
9 Long-Run SPF Forecast of CPI Inflation

0
1980 1985 1990 1995 2000 2005 2010 2015
Year
Figure 2: PCE Core Inflation and Long-Term Inflation Expectations
Note: The grey line plots 10 year ahead inflation expectation for the CPI. From 1990 onward, these come from
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’ research series. This research series uses current methods to calculate
inflation back in time.

in Figure 2. To see how this can be the case, it is useful to solve equation (1) forward and assume
for simplicity that unemployment follows an AR(1) process. This yields

πt = −ψũt + Et πt+∞ + ωt , (3)

where ũt denotes the deviation of unemployment from its long-run expected value, Et πt+∞ rep-
resents long-term inflation expectations, and the parameter ψ is proportional to κ in equation (1).
(Section 2 presents a more detailed derivation.) What this formulation of the Phillips curve makes
clear is that changes in beliefs about the long-run monetary regime feed strongly into current infla-
tion: the coefficient on Et πt+∞ in equation (3) is one. Furthermore, in the presence of substantial
variation in Et πt+∞ , the relationship between πt and ũt may be essentially uninformative about
the slope of the Phillips curve (ψ and κ). In particular, if changes in Et πt+∞ 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

3
explained by even a very large value of κ in the Phillips curve. In these episodes, it is clear that the
primary cause of the abrupt fall in inflation is an abrupt fall in Et πt+∞ associated with an abrupt
change in the policy regime. Volcker’s monetary policy constituted a sharp regime shift that was
imperfectly credible at the outset but became gradually more credible as time passed (Erceg and
Levin, 2003; Goodfriend and King, 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 pri-
mary cause of the rapid fall in inflation over this period rather than high unemployment working
through a steep Phillips curve.
This discussion highlights an important identification problem researchers face when they seek
to estimate the slope of the Phillips curve: inflation expectations may covary with the output gap.
Standard methods for estimating the Phillips curve aim to address this issue by controlling for
inflation expectations Et πt+1 when estimating equation (1). A challenge with this approach is that
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, and the 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 have the “right” as the “wrong” sign). Mavroeidis, Plagborg-Møller, and Stock
(2014) point to a weak instruments problem in driving these results: there simply isn’t enough
variation available in the aggregate data to separately identify the coefficients on unemployment
and expected inflation. They conclude: “the literature has reached a limit on how much can be
learned about the New Keynesian Phillips curve from aggregate macroeconomic time series. New
identification approaches and new datasets are needed to reach an empirical consensus.”
In addition to the identification problem discussed above, researchers seeking to estimate the
slope of the Phillips curve also face the classic simultaneity problem of distinguishing demand
shocks from supply shocks. Supply shocks (unt and νt ) yield positive comovement of inflation
and unemployment (stagflation). If the variation used to identify the slope of the Phillips curve is
contaminated by such shocks, the estimated slope will be biased towards zero and may even have
the “wrong” sign. Fitzgerald and Nicolini (2014) and McLeay and Tenreyro (2019) point out that
a central bank conducting optimal monetary policy will seek to offset aggregate demand shocks.
If the central bank is successful, the remaining variation in inflation will be only due to supply
shocks, a worst case scenario for the simultaneity problem.
Can cross-sectional data help overcome these problems? Several recent papers have argued

4
that they can. Fitzgerald and Nicolini (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 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 for-
mally how estimating the Phillips curve using regional data provides a solution to the problem of
shifting values of Et πt+∞ confounding the estimation of the slope of the Phillips curve. We de-
rive a regional Phillips curve in a “simple benchmark” multi-region model of a monetary union.
The model clarifies the interpretation of the slope of regional Phillips curves relative to that of the
aggregate Phillips curve. We also use the model to show that changes in the long-run monetary
regime are absorbed by time fixed effects when the regional Phillips curve is estimated using a
panel data specification. The intuition is that such long-run regime changes are common to all
regions and therefore “cancel out” across regions within 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 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 for the 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 the Phillips curve is small and was
small even during the 1980s. Combining our estimate of the slope of the Phillips curve with an
estimate of the persistence of fluctuations in unemployment, we find that a one percentage point
increase in unemployment reduces inflation by about 0.34 percentage points, i.e., ψ = 0.34 in
equation (3). This implies that only a modest fraction of the large changes in inflation in the early
1980s can be accounted for by the direct effect of increasing unemployment working through the
slope of the Phillips curve. In contrast, movements in long-run inflation expectations were large
over this period as is evident from Figure 2. In particular, long-run inflation expectations fell by
about 4 percentage points from 1981 to 1986, accounting for about 2/3 of the fall in core inflation
during this period. We conclude that a majority of the rapid decline in core inflation during the

5
Volcker disinflation arose from a rapid decline of long-term inflation expectations, associated with
a rapidly changing monetary regime.2
Our estimates of the slope of the Phillips curve imply essentially no “missing disinflation” dur-
ing the Great Recession or ”missing reinflation” in the late 2010s or late 1990s. In other words, our
cross-sectional estimates are consistent with the magnitude of movements in aggregate inflation
post 1990. We conclude that the stability of inflation since 1990 is due to long-run inflation expec-
tations becoming more firmly anchored. These conclusions echo those of Jorgensen and Lansing
(2019).
Our analysis uses new state-level consumer price indexes for the United States that we have
constructed back to the 1970s. Prior to our work, state level price indexes based on BLS micro price
data have not existed. The BLS has published city-level inflation series for a group of relatively
large cities. But it has refrained from reporting inflation indexes for smaller metropolitan areas
(and for states). Our new state-level price indexes use all the available underlying micro-data
gathered by the BLS. We also construct state-level price indexes for non-tradeables and tradeables.
We focus our analysis on the behavior of the prices of non-tradeable goods. This is important. For
prices set at the national level—as is more likely for tradeables—the slope of the regional Phillips
curve will be zero no matter how large the slope 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 the ag-
gregate 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 in equation (1). This means that they are not
directly comparable to much of the aggregate literature. We discuss how researchers can convert
estimates of ψ to κ and explain what other statistics this conversion depends on (primarily the
degree of persistence of the unemployment 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 indexes, allow us to leverage new forms
of variation in estimating the Phillips curve. We develop a new “tradeable demand spillovers”
instrument building on insights from Nguyen (2014). This instrument is based on the idea that
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 their model 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.
3
For example, Nishizaki and Watanabe (2000) find evidence of Phillips curve flattening in their baseline specification
with no time fixed effects but this evidence changes dramatically when time fixed effects are added.

6
supply shocks in tradeable sectors will differentially affect demand in non-tradeable sectors in
regions that are differentially exposed to the shocked tradeable sectors: e.g., an oil boom will
increase demand for restaurant meals in Texas. In carrying out our regional analysis, we are
careful 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 the regional Phillips curve for rents, and show that it is substantially steeper than the
regional Phillips curve for non-tradeables excluding housing. We use the combination of these
two estimates to predict 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 role in determining the slope of both the regional and
aggregate Phillips curves.
In addition to the papers discussed above, our work builds on the vast empirical and theoret-
ical literature on the Phillips curve. The literature on the Phillips curve originates with Phillips
(1958) and Samuelson and Solow (1960). Friedman (1968) and Phelps (1967) emphasized the im-
portance of including an inflation expectations term in the Phillips curve. Gordon (1982) empha-
sized the importance of supply shocks. Important early papers that estimate the New Keynesian
include Roberts (1995), Fuhrer and Moore (1995), Gali and Gertler (1999) and Sbordone (2002),
but see also papers cited in Mavroeidis, Plagborg-Møller, and Stock (2014). Important recent pa-
pers estimating the Phillips curve include Ball and Mazumder (2011, 2019), Coibion and Gorod-
nichenko (2015b), Stock and Watson (2019), Barnichon and Mesters (2019), and Del Negro et al.
(2020). Our paper is also related to a recent literature that assesses the missing disinflation during
the Great Recession (see, e.g., Del Negro et al., 2015; Christiano et al., 2015; Gilchrist et al., 2017;
Crump et al., 2019).
The paper proceeds as follows. Section 2 derives equation (3) and explains the problem of
regime change in estimating the Phillips curve. Section 3 describes our main framework for in-
terpreting the regional Phillips curve. Section 4 describes our new state-level inflation indexes.
Section 5 presents our empirical results. Section 6 concludes.
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 the difficulty of backing out the user cost of housing from actual house prices
in a theoretically appealing way. The expenditure weight of the CPI less food and energy is 77.7%, and 32.3 percentage
points out of this expenditure weight are rents.

7
2 The Power and Problem of Long-Run Inflation Expectations

To appreciate the value of using regional variation to estimate the slope of the Phillips curve, it is
useful to understand the central role of long-run inflation expectations in determining aggregate
inflation. To this end, we solve equation (1) forward to get

∞
X
πt = −κEt β j ut+j + ωt (4)
j=0

P∞ j n
where ωt ≡ Et j=0 β (κut+j + νt+j ). This equation illustrates how inflation at time t is deter-
mined 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 these concepts, we can
rewrite equation (4) as
∞
X κ
πt = −κEt β j ũt+j − Et ut+∞ + ωt , (5)
1−β
j=0

κ
Assuming that shocks to unt and νt are transitory, equation (1) implies that Et πt+∞ = − 1−β Et ut+∞ .
We can then rewrite equation (5) as

∞
X
πt = −κEt β j ũt+j + Et πt+∞ + ωt . (6)
j=0

Finally, let’s assume for simplicity that ũt follows and AR(1) process with autocorrelation coeffi-
cient equal to ρu . In this case Et ũt+j = ρju ũt and we can rewrite equation (6) as

πt = −ψũt + Et πt+∞ + ωt , (7)

where ψ = κ/(1 − βρu ).
5
While the most popular micro-foundation of the New Keynesian Phillips curve—and the one we develop in sec-
tion 3—is based on the price rigidity assumptions in Calvo (1983), this equation or something very similar arises from
several other micro-foundations. Roberts (1995) shows that the same Phillips curve arises 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 term is 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 contin-
uous 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), Gali, Gertler, and Lopez-Salido (2005), and Rudd and Whelan (2005) develop approaches to
estimating the Phillips Curve on aggregate data using versions of equation (4).

8
This way of writing the Phillips curve highlights the importance of long-run inflation expec-
tations in determining inflation at the aggregate level. Long-run inflation expectations Et πt+∞
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 determined
by the private sector’s beliefs about the long-run monetary regime being followed by the cen-
tral 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 dramatically without 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 the slope 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 Paul Volcker’s willingness to allow unemployment to rise to very high
values in the early 1980s—and the fact that Volcker was not forced to resign—signalled to the pub-
lic 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 controlled for. But in practice, controlling for in-
flation expectations is hard due to weak instruments (Mavroeidis et al., 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 past few decades? Figure 2 shows that
since roughly 1998, long-term inflation expectations have been firmly anchored at close to 2%.
This has led to a collapse of the covariance between Et πt+∞ and unemployment and therefore
eliminated any bias associated with poorly proxied variation in inflation expectations. A fall in
this bias 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 πt and
Et πt+1 in the data. Recall that the standard formulation of the New Keynesian Phillips—equation
(1)—implies that it is the gap between πt and βEt πt+1 —let’s call this the “inflation gap”—that
must be explained by demand pressure (the κut term) or supply shocks (κunt + νt ). Figure 3 plots
6
Equations (6) and (7) remain valid in the case where the coefficient on Et πt+1 in equation (1) is equal to one rather
than β. In this case, the long-run PhillipsPcurve is vertical, and long-run unemployment and inflation are unrelated.
That is, inflation still satisfies πt = −κEt ∞
j=0 ũt+j + ωt + Et πt+∞ , but Et ut+∞ is now independent of Et πt+∞ . The
forward sum is bounded even though β = 1, because ũt has zero unconditional mean.

9
CPI Core CPI
15 15

10 10

5 5

0 0
CPI Inflation Core CPI Inflation
1 Year SPF Forecast of PGDP Inflation 1 Year SPF Forecast of PGDP Inflation
-5 -5
1980 1990 2000 2010 1980 1990 2000 2010

Core PCE Core CPI Inflation Research Series
15 15

10 10

5 5

0 0
Core PCE Inflation Core CPI Inflation RS
1 Year SPF Forecast of PGDP Inflation 1 Year SPF Forecast of PGDP Inflation
-5 -5
1980 1990 2000 2010 1980 1990 2000 2010
Figure 3: πt vs Et πt+1
Note: 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, while 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.

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 πt − βEt πt+1 .
The measure of current inflation plotted in the top-left panel of Figure 3 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-right panel,
we measure current inflation by the 12-month change in core 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.

10
The measure of current inflation plotted in the bottom-left panel of Figure 3 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 back in time. 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 research
series 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 for overall housing inflation, including
for owner-occupied housing (“rental equivalence”). Before that time, housing services inflation in
the CPI was constructed from a weighted average of changes in house prices and mortgage costs
(i.e., interest rates). This 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
The overall message that emerges from Figure 3 is that the inflation gap for core inflation mea-
sured 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 both in the early
1980s and again in the Great Recession without much variation in the inflation gap. However, the
four panels in Figure 3 illustrate well that this conclusion is sensitive to the details of how inflation
is measured.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 (2015b) emphasize, and also
sensitive to the exact timing of the variables.

3 A Model of the Regional Phillips Curve

We now develop a two-region, New Keynesian, open economy model featuring tradeable and
non-tradeable sectors. We derive a 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 “difference out” changes in long run inflation expectations. The model
also illustrates the importance of using non-tradeable inflation when estimating the slope of the
7
These choices are consequential since the housing component of the CPI has a weight of roughly one-third in the
overall CPI. Appendix B.2 presents our attempt to replicate the pre-1983 BLS housing methodology on more modern
data. The main conclusion from this is that this methodology would have led to much more variable (and cyclical)
inflation over the past few decades.
8
We discuss this in more detail in appendix B.1.

11
Phillips curve using regional data.

3.1 Model Setup

Our model consists of two regions that belong to a monetary and fiscal union. We refer to the
regions as Home (H) and Foreign (F). The population of the entire economy is normalized to one.
The population of the home region is denoted by ζ. Labor is immobile across regions. Within 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 gov-
ernment.9 Throughout, we adopt the following conventions unless otherwise stated. Lower case
variables are the logs of upper case variables. Hatted variables denote the percentage deviation of
a variable from its steady state value. Steady state values are recorded without time subscripts.

3.1.1 Households

The representative household in the home region seeks to maximize the utility function

∞
X
E0 β t u(CHt , NHt ),
t=0

where CHt is per capita consumption of a composite consumption good, NHt is per capita em-
ployment, and β is the household’s subjective discount factor. We follow Greenwood, Hercowitz,
and Huffman (1988) in assuming that the function u(CHt , NHt ) takes the form

1+ϕ−1
1−σ−1
NHt
CHt − χ 1+ϕ−1
u(CHt , NHt ) = (8)
1 − σ −1

where ϕ is the household’s Frisch elasticity of labor supply, σ determines 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 of substitution (CES) index over
9
In other words, we are considering an economy in the cashless limit (Woodford, 1998, 2003).

12
T and non-tradeables C N given by
tradeables CHt Ht

η
N η−1 T η−1
1 1 η−1
CHt = φN CHt η + φTη CHt η
η
,

where η is the elasticity of substitution between tradeables and non-tradeables and φT and φN
are the household’s steady state expenditure shares on tradeable and non-tradeable goods, re-
N and C T are themselves composite goods described further below. Non-tradeable
spectively. CHt Ht

goods are only consumed in the region in which they are produced. In contrast, the market for
tradeable goods is completely integrated across regions. Hence, home and foreign households
may face different prices for non-tradeables, but face the same prices for tradeable goods. The
expenditure share on tradeable and non-tradeable goods must sum to one, i.e., φN + φT = 1.
N is given by
The composite non-tradeable good CHt

Z 1 θ
θ−1
θ−1
N N
CHt = CHt (z) θ dz
0

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

1 η−1 1
η
η−1 η−1
T H η TH η F η TF η
CHt = τHt CHt + τHt CHt , (9)

T H and C T F are home consumption of composite tradeable goods produced in the
where CHt Ht

home and foreign regions, respectively. We assume (for simplicity) that the elasticity of substi-
tution between home-produced and foreign-produces tradables is η (the same as the elasticity of
substitution between tradeables and non-tradeables). Demand for home-produced and foreign-
H and τ F , respectively. We normalize
produced tradeables is subject to shocks denoted by τHt Ht
H + τ F = 1. For simplicity, we do not allow for home bias in tradeable consumption. Thus, we
τHt Ht
H = τ H = ζ, i.e., the share of spending on goods from the home region in each region is equal
set τH F

to the size of the home region.

13
The home and foreign composite tradeable goods are CES indexes given by

Z 1 θ
θ−1 Z 1 θ
θ−1
θ−1 θ−1
TH TH TF TF
CHt = CHt (z) θ dz and CHt = CHt (z) θ dz .
0 0

T H (z) and C T F (z) are home consumption of varieties of tradeable goods produced in the
where CHt Ht

home and foreign region, respectively. The prices of these home-produced and foreign-produced
T (z) and P T (z), respectively.
tradeable good varieties are PHt Ft

Households maximize utility subject to a sequence of budget constraints

N N TH T TF T
CHt PHt + CHt PHt + CHt PF t + Et [MHt,t+1 BH,t+1 ] ≤ BHt + WHt NHt + ΞN T
Ht + ΞHt

where BHt is a random variable denoting payoffs of the state contingent portfolio held by house-
holds in period t; MHt,t+1 is the one-period-ahead stochastic discount factor of the home represen-
N , P T , and P T are price indexes that give the minimum cost of purchasing
tative household; PHt Ht Ft
N , C T H , and C T F , respectively; W
a unit of CHt Ht Ht Ht is the nominal wage received by workers in re-

gion H; and ΞHt and ΞHt are the profits of non-tradeable 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 Appendix A.1.
As we noted above, the problem of the foreign household is analogous. We therefore refrain from
describing 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.

3.1.2 Firms

There is a continuum of firms in each of the tradeable and non-tradeable 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 non-tradeable sector. The output of good z in the non-tradeable
N (z). The production function of firm z in this sector is
sector is denoted YHt

N N N
YHt (z) = ZHt NHt (z), (10)

N (z) is the amount of labor demanded by firm z and Z N is a productivity shock.
where NHt Ht

14
Firm z in the non-tradable sector maximizes its value:

∞
X N N N

Et MHt,t+j PH,t+j (z)YH,t+j (z) − WH,t+j NH,t+j (z)
j=0

given demand for its good, which is

N (z) −θ
N N PHt
YHt (z) = ζCHt N
.
PHt

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 non-tradeable sector, the output of firm z in the tradeable sector is denoted
T (z). Its production function is
YHt
T T T
YHt (z) = ZHt NHt (z)

T (z) is the amount of labor demanded by the firm producing good z and Z T is a pro-
where NHt Ht

ductivity shock.
Firm z in the tradeable sector maximizes its value:

∞
X T T T

Et MHt,t+j PH,t+j (z)YH,t+j (z) − WH,t+j NH,t+j (z)
j=0

given demand for its good. Demand in the tradeable sector comes from both the home and foreign
regions. Firm z’s demand is thus given by

T (z) −θ
T TH PHt
+ (1 − ζ) CFT tH

YHt (z) = ζCHt T
.
PHt

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 Appendix A.2. The
problems of foreign firms are analogous to those of home firms.

15
3.1.3 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 rule

r̂tn = ϕπ (πt − π̄t ) − ϕu (ût − ūt ) + εrt ,

where, as elsewhere in the paper, hatted variables denote deviations from a zero inflation steady
state and lower case variables are the logs of upper case variables. Economy-wide inflation πt is
a population weighted average of inflation in the two regions: πt ≡ ζπHt + (1 − ζ)πF t , where
πHt = pHt − pH,t−1 is consumer price inflation in the home region and πF t 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 a first order
ûHt = −n̂Ht and ûF t = −n̂F t . Economy-wide unemployment is a population weighted average
of unemployment in the two regions, so ût = ζ ûHt + (1 − ζ)ûF t .
Importantly, we allow the monetary authority to have a time-varying 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 unem-
ployment rate that is consistent with its long-run inflation target, i.e., ūt = (1 − β)π̄t /κ. We assume
that ϕπ and ϕu 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 zero
net supply. The government’s monetary policy, therefore, has no fiscal implications. An equi-
librium in this economy is an allocation that satisfies household optimization, firm optimization,
the government’s interest rate rule, and market clearing. We focus on the unique locally bounded
equilibrium of the model. Implicitly we rule out equilibria in which the inflation rate rises without
bound using the trigger strategy argument presented in Obstfeld and Rogoff (1983).
10
Prior work that allows for a time-varying inflation target includes Stock and Watson (2007), Ireland (2007) and
Cogley and Sbordone (2008).

16
3.2 Regional and Aggregate Phillips Curves

Taking a log-linear approximation of the model presented in section 3.1 around a zero-inflation
steady state with balanced trade yields the following regional Phillips Curve for the inflation of
non-tradeable goods:
N N
πHt = βEt πH,t+1 − κûHt − λp̂N N
Ht + νHt , (11)

and aggregate Phillips Curve for overall inflation:

πt = βEt πt+1 − κût + νt , (12)

N = pN − pN N N
where πHt Ht H,t−1 is home non-tradeable inflation, p̂Ht = PHt /PHt − 1 is the percentage
N is a
deviation of the home relative price of non-tradables from its steady state value of one, νHt
non-tradeable home supply shock, νt is a corresponding aggregate supply shock, and the parame-
ter κ = λϕ−1 , where λ = (1 − α) (1 − αβ) /α. We provide a detailed derivation of these equations
in Appendix A.
Equations (11) and (12) yield an important result: The slopes of the regional Phillips curve for
non-tradeables and the aggregate Phillips curve are the same in our model. These slopes are both
equal to κ. This result holds for the non-tradeable regional Phillips curve, but does not carry over
to the regional Phillips curve for overall consumer price inflation—which includes both tradeable
and non-tradeable inflation in the region. As we show in Appendix A.8, the slope of the regional
Phillips curve for overall consumer price inflation is smaller by a factor equal to the expenditure
share on non-tradeable goods.
Intuitively, the difference in the slope between the non-tradeable and overall regional Phillips
curves arises because all regions share the tradeable goods and these goods are priced nationally.
The tradeable goods therefore don’t contribute to difference in inflation across regions, which
means that the regional CPI is made up partly of goods whose regional prices are insensitive
to regional 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 non-tradeable 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 non-tradeable 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;

17
McLeay and Tenreyro, 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
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 in the
derivation of our result that the non-tradeable regional Phillips curve and the aggregate Phillips
curve have the same slope. We discuss this point at greater length in Appendix A.9. The form of
the Phillips curve in our model does not, however, depend on the structure of financial markets.
We have assumed complete financial markets across regions, but the Phillips curve is the same in
a model with incomplete markets across regions.
An important difference between equations (11) and (12) is the presence of the relative price of
non-tradeables term λp̂N
Ht in equation (11). This term implies that inflation in the non-tradeables

sector will be lower the higher is the relative price of non-tradeables. Conceptually, this term is
very important. It pushes relative prices towards 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 non-tradeable prices since 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 non-tradeable goods is driven by variation in the real wage deflated
by non-tradeable prices. Labor supply in the home region, however, is a function of the real
wage deflated by the home consumer price index. The real marginal cost variable in the home
non-tradeable Phillips curve therefore gives rise to an unemployment term and a relative price of
non-tradeables term.

3.3 Estimating the Slope of the Phillips Curve with Regional Data

Next we solve the regional Phillips curve — equation (11) — forward to obtain

∞
X
N
β j κũH,t+j + λp̂N N N

πHt = −Et H,t+j + Et πt+∞ + ωHt , (13)
j=0

N =E
P∞ j N
where ũHt = uHt − Et uH,t+∞ and ωHt t j=0 β νH,t+j .

A major benefit of estimating the slope of the Phillips curve using regional data from a mone-
N
tary union is that variation in long-run inflation expectations — the Et πt+∞ term in equation (13)

18
— is constant across regions. This implies that variation in long-run inflation expectations will
be absorbed by time fixed effects in a panel specification. Intuitively, while short-run inflation
N ) will differ across regions due to differences in their economic circumstances,
expectations (Et πt+1
N ) are independent of the current business cycle. They are
long-run inflation expectations (Et πt+∞
determined by beliefs about the long-run monetary regimes. In a monetary union like the US,
these beliefs will vary uniformly across regions. This means that these expectations are “differ-
enced out” in a panel regression with time fixed effects.
The result that long-run inflation expectations are constant across regions (and sectors) in our
model relies on productivity and other drivers of real costs having a common trend in the long run.
If productivity growth (say) differs across regions even in the long run, this will lead to persistent
differences in non-tradeable inflation (a Balassa-Samuelson effect). However, if this difference is
constant over time, it will be absorbed by region fixed effects in a panel specification.
These observations imply that that we can adopt an empirical specification that replaces the
N
Et πt+∞ term in equation (13) with time and region fixed effects:

∞
X
N
β j κui,t+j + λp̂N N

πit = −Et i,t+j + αi + γt + ω̃it , (14)
j=0

where i denotes region, αi denotes a set of region fixed effects, and γt denotes a set of time fixed
N
effects. Variation in Et πt+∞ in equation (13) that is common across regions will be absorbed by
the time fixed effect.11 Constant differences across regions in Et πt+∞
N will be absorbed by the state
N
fixed effects. To the extent that there is remaining variation in Et πt+∞ across regions (e.g., due to
N.
changing trends), it will be a part of the error term ω̃it
It is useful to relate equation (14) to the empirical specifications used in the recent regional
Phillips curve literature. If we assume that both uHt and p̂N
Ht follow AR(1) processes with autocor-

relation coefficients equal to ρu and ρpN , respectively, equation (14) simplifies to

N
πit = −ψuit − δ p̂N N
it + αi + γt + ω̃it , (15)

where ψ = κ/(1−βρu ) and δ = λ/(1−βρpN ). This equation is similar to the empirical specification
used by much of the recent regional Phillips curve literature. Comparing equations (14) and (15),
11
The time fixed effects also absorb time variation in the long-run expected unemployment Et ut+∞ . We have there-
fore replaced ũi,t+j in equation (13) with ui,t+j in equation (14). This equation remains valid if β = 1. The forward
sum in the equation is still bounded, because uit has zero mean conditional on time fixed effects.

19
we see that an important difference between these two specifications is that the slope coefficient is
not the same. The slope coefficient in equation (14) is κ (which is the same as the slope coefficient
in equation (11) and (12)), while the slope coefficient in equation (15) is ψ = κ/(1 − βρu ). Since
unemployment is quite persistent, ψ is likely to be substantially larger 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 slope of the Phillips curve than more traditional estimation strategies based
on aggregate data (Fitzgerald and Nicolini, 2014; Babb and Detmeister, 2017; McLeay and Ten-
reyro, 2019; Hooper et al., 2019). Comparing equations (14) and (15) provides a simple explana-
tion 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 the Phillips curve estimated in the regional
literature has seemed large 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), the effects of
expected future unemployment on current inflation are captured by the inflation expectations
term Et πt+1 and the coefficient on current unemployment κ only reflects the effect of current
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 the
indefinite future on current inflation—i.e., the fact that high unemployment 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 the identification of the slope coefficient is less sensitive to the exact timing of changes
in inflation relative to inflation expectations. In Figure 3, we show that the difference between
inflation and inflation expectations is quite sensitive to the exact measure of inflation.
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 for
N
Et πt+1 when estimating the Phillips curve using aggregate data. Our analysis shows, however, that when researchers
choose to 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, e.g., by using the formula ψ = κ/(1 − βρu ) or a version of this formula appropriate for (say)
10-year ahead inflation expectations.
13
McLeay and Tenreyro (2019) control for inflation expectations at the Census Region level when they estimate the
regional Phillips curve. The variation across regions in these inflation expectations data is quite minimal. It may
therefore be that the variation in this variable is quite attenuated relative to actual variation in inflation expectations
across the MSA areas that form the regional units in their analysis.

20
We have so far manipulated the Phillips curve under the standard assumption of full-
information rational expectations. However, the arguments we make above — i.e., solving the
Phillips curve forward — rely only on the weaker assumption, that the law of iterated expecta-
tions holds. We elaborate on this point in 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 coeffi-
cient on unemployment is the same as in the aggregate Phillips curve, we have made a number of
strong assumptions (perfect labor mobility within region, no labor mobility across regions, GHH
preferences, production linear in labor, etc.). In the world, these assumptions are unlikely to hold
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 nontheless useful since it
highlights in a transparent way the importance of certain forces (e.g., inflation expectations). In
more general models in which no exact analytical mapping between the slope of the regional and
aggregate Phillips curves exists, our regional slope estimates can be used as empirical targets in
a moment matching exercise. Even away from the simple case we analyze where the aggregate
and regional slopes are equal, these moments are likely to provide valuable information about the
slope of the aggregate Phillips curve (Nakamura and Steinsson, 2018; Andrews, Gentzkow, and
Shapiro, 2020).

4 Data and Construction of State-Level Price Indexes

The BLS does not publish state-level price indexes. Prior work has used metropolitan level BLS
price indexes and cost of living estimates from the American Chamber of Commerce Realtors
Association (ACCRA) to construct state-level price indexes (see, e.g., Del Negro, 1998; Nakamura
and Steinsson, 2014). An important drawback of this approach is that the BLS imputes missing
data using data from other regions. Recent work has used scanner price data to construct state-
level price indexes (Beraja, Hurst, and Ospina, 2019). An important drawback of scanner data is
the short sample period available.
We construct new state-level price indexes for the US 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 in-
dexes on are available in the CPI Research Database at the BLS. The data for the period 1978-1987

21
were constructed by Nakamura et al. (2018). The micro-price data in the CPI Research Database
cover thousands of individual goods and services, constituting about 70% of consumer expendi-
tures. They are collected by BLS employees who visit outlets to record 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).

4.1 State-Level Price Index Construction

Our methodology for constructing price indexes 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 why
we do not simply employ the BLS’s own price index software, is that 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
1/τ
Pi,t
ri,t = . (16)
Pi,t−τ

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 was collected for this product. Several details are important.
First, it is important to use the effective price rather than the raw “collected price.” The difference
between the collected and effective prices is that the latter adjusts for changes in the number and
size of the items being priced (e.g. a 2L bottle of Diet Coke vs. a two-pack of 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 is sold. 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 the item that the price collector identifies in the store. We hold
all three of these parameters—outlet, quote, and version—fixed in constructing a product’s price

22
relative.
Third, we must decide what to do when prices are missing. Missing prices occur when the
product is unavailable due to a temporary 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 number of periods between successive price changes. There are
several other important details of our index construction procedure that we describe in 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 and state.
ELIs are relatively narrow product categories such as “Full Service Meals and Snacks” (restau-
rants) and “Motorcycles” defined by the BLS for the purpose of calculating the CPI.14 We then
calculate sectoral state-level price indexes by computing a weighted geometric average of the
ELI-state indexes across the ELIs within that state and sector. We use national weights from the
Consumer Expenditure Survey (CEX) for 1998 to perform this aggregation.15
Our empirical analysis focuses on non-tradeables but we also construct state-level price in-
dexes for tradeables—which we simply define as the complement of non-tradeables—and overall
state-level price indexes. We construct a price index for non-tradeables based on our own cate-
gorization of BLS’s ELI product categories. In doing this, we attempt to be conservative in our
definition of what constitutes a non-tradeable good, since including tradable goods could lead to
attenuation of the slope of the Phillips curve if tradeable goods are priced nationally. In contrast,
the main downside of excluding some non-tradeable goods is less precise estimates. The goods we
classify as non-tradeables account for roughly 44% of non-housing consumer expenditures. Im-
portantly, our index of non-tradeables does not include housing services or transportation goods
(mainly airline tickets).16 We estimate regional Phillips curves for housing services separately in
section 5 using different data. Appendix B.4 provides a detailed list of which ELI categories we
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 be weighted by sales shares.
16
We find that there is much more variablility across states in non-tradeable inflation than tradeable inflation. For
non-tradeables, the first principal component of state-level inflation captures only about 37% of the variance in the un-
derlying state-level series. In contract, for tradeables, the first principal component captures about 71% of the variance
in the underlying state-level series. This pattern is consistent with our argument in section 3.2 that many tradeable
goods are priced nationally, and do not respond to regional marginal costs.

23
classify as non-tradeable.
Our method for calculating state-level price indexes aims to approximate the non-shelter price
index published by the BLS. Appendix Figure C.1 illustrates our ability to match the official BLS
data by comparing the evolution of 12-month inflation at the aggregate level using our method-
ology with official CPI inflation excluding housing. The figure shows that we are able to approx-
imate the official BLS data very closely. This is true even for the pre-1988 period when we rely
on the micro-data recovered by Nakamura et al. (2018) which likely have greater measurement
error.17

4.2 Employment data

The measure of unemployment that we use as our measure of labor market slack in the Phillips
curve is the quarterly, seasonally adjusted, state unemployment rate from the Local Area Un-
employment Statistics (LAUS) published by the BLS. We also make use of employment data in
constructing our tradeable demand spillovers instrument discussed in section 5. This instrument
is a shift-share instrument, similar to the one used in Bartik (1991). It is constructed using em-
ployment 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 as well as 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 share in the same way as Mian and Sufi
(2014). Appendix B.5 discusses this in more detail.

5 Empirical Results

We now turn to our empirical results. We present estimates both of the structural parameter κ from
equation (14) and ψ from equation (15). Recall that κ is structural slope coefficient in the regional
Phillips curve for non-tradeables from our model, while ψ is the reduced form slope coefficient in
the type of regional empirical specification often run in prior work. To be able to estimate equation
(14), we replace expected future unemployment and relative prices with their realized values and
17
In the present draft, we drop Arizona due to anomalous trends that we have not been able to investigate due
Covid-19 related access restrictions at the BLS.
18
Industry-state employment data are available from the QCEW, at quarterly frequency for 2-digit SIC codes (1975-
2000) and 3-digit NAICS codes (1990-2017). Before 1990 we use 2-digit SIC codes to define industry, whereas after 2000
we use 3-digit NAICS Code. For the period 1990-2000, when both the NAICS and SIC code classifications are available,
we construct both versions of the instrument and use a simple average of the two.
19
Using the X-11 algorithm for seasonal adjustment yields virtually identical results.

24
an expectation error. We also truncate the infinite sum in equation (14) at j = T . Doing this yields

T
X T
X
N
πit = αi + γt − κ β j ui,t+j − λ β j p̂N N N
i,t+j + ω̃it + ηit , (17)
j=0 j=0

N denotes an expectations error that is orthogonal to information known at time t (and
where ηit
a truncation error). Equation (17) can now be estimated with standard GMM methods, i.e., by
instrumenting for the two forward sums. We do not attempt to estimate 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 4-quarter lagged unemploy-
ment ui,t−4 and the 4-quarter lagged relative price of non-tradeables p̂N
i,t−4 . Assuming rational
N . The
expectations, these lagged variables will be uncorrelated with the expectations error ηit
identifying assumption regarding supply shocks is that when one state experiences a boom or
bust relative to another state, it does not systematically experience non-tradeable supply shocks
relative to this other state. For example, when Texas experiences a recession relative to Illinois,
this is not systematically correlated with changes in restaurant technology in Texas relative to Illi-
nois. Notice, that national supply shocks are absorbed by the time fixed effects. So, only regional
non-tradeable 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 vari-
ation in demand for specific tradeable goods will differentially affect labor demand for non-
tradeable 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 non-tradeables in Texas. Building on this
idea, we construct a “tradeable demand spillovers” instrument as

X
Tradable Demandi,t = S̄x,i × ∆3Y log S−i,x,t , (18)
x

where S̄x,i is the average employment share of industry x in state i 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 shift-
share 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 both correlated with the shifts

25
15
CA
TX
PA

0
1980 1985 1990 1995 2000 2005 2010 2015
Date
Figure 4: Regional Business Cycles
Note: This figure plots the unemployment rate for California, Pennsylvania and Texas.

∆3Y log S−i,x,t in the time series and correlated with the shares S̄x,i in the cross section. For exam-
ple, 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 the Phillips curve. Figure 4 depicts the evolution of the unemployment
rate for three states, California, Texas and Pennsylvania, over our sample period. While there
is certainly a great deal of comovement, this figure illustrates well that there is also substantial
cross-state variation. One example is that both the 1991 and 2007-2009 recessions affected Cali-
fornia much more than Texas and Pennsylvania. Another is that Texas experienced a recession in
the mid-1980s (widely thought to stem from a dramatic 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.
N = pN − pN
The dependent variable in our regressions is πit it i,t−4 , i.e., state-level non-tradeable

inflation over the previous 12 months. Studying inflation over four quarters allows us to reduce
measurement error and eliminate seasonality. In Appendix A.11, we show that using twelve-
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.

26
month inflation as our dependent variable implies that we need to divide our estimates of κ and
λ from equation (17) by four to account for the time aggregation. Recall that the inflation rate in
our model in section 3 is a quarterly inflation rate.
We truncate the discounted sums on the right-hand-side of equation (17) at T = 20 quarters.
Table C.4 presents robustness regarding this choice for our main specification. Our results are
similar for values between T = 20 and T = 40. In Appendix A.12, we estimate κ using equation
(17) with T = 20 on data simulated from our model from section 3. We find that our empirical
procedure is able to accurately estimate the true value 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 impact of this, we use a two-sample two
stage least squares 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 error at the state level
and apply a correction to our standard errors appropriate for two-sample 2SLS developed by
Chodorow-Reich and Wieland (2019).21
Our empirical specification for estimating ψ is

N
πit = αi + γt − ψui,t−4 − δpN
i,t−4 + εit . (19)

We use beginning-of-period unemployment and relative price of non-tradeables 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 equa-
tion (19) by OLS (i.e., instrumenting for lagged unemployment and relative price of non-tradeables
with themselves). The second approach replaces lagged unemployment among the instruments
with our tradeable-demand instrument.

5.1 Full-Sample Results

Table 1 presents estimates of κ and ψ for our full sample period of 1978-2018. Let’s start by con-
sidering the estimates of κ in Panel A. When we estimate equation (17) without fixed effects, our
estimate has the “wrong” sign, i.e., higher unemployment is associated with higher rather than
21
Our tradeable demand instrument uses all the information in national industry employment growth rates. So, our
standard errors are not subject to the concerns about inference with shift share instruments raised by Adao, Kolesar,
and Morales (2019).

27
Table 1: Slope of the Regional Phillips Curve
No Fixed No Time Lagged Tradeable
Effects Effects Unempl. Demand IV
(1) (2) (3) (4)

Panel A: Estimates of κ from equation (17)
κ -0.0037 0.0003 0.0062 0.0062
(0.0013) (0.0019) (0.0028) (0.0025)
Panel B: Estimates of ψ from equation (19)
ψ -0.103 0.017 0.112 0.339
(0.036) (0.027) (0.057) (0.126)

State Effects X X X
Time Effects X X
Note: This table presents estimates of κ and ψ from regression specifications (17) and (19), respectively. The outcome
variable is cumulative non-tradeable 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 non-
tradeables, in 100 x log points. For both variables, we truncate the discounted furture 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 non-tradeables. In the first three columns we instrument using the fourth lags of quarterly state unemployment
and the relative price of non-tradeables (this is OLS for ψ). In the fourth column, we replace lagged unemployment
with our tradeable demand instrument among the instruments. In all columns, we estimate κ by two-sample two stage
least squares, and apply the correction to our standard errors from Chodorow-Reich and Wieland (2019). The sample
period is 1978-2018. Standard errors are reported in parentheses, clustered by state. All regressions are unweighted.
The number of observations is 3323 in the first three columns of panel A, with slightly fewer in the last column due to
differencing. Likewise, the number of observations is 4490 in the first three columns of panel B.

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 we stress throughout the paper, time
fixed effects eliminate changes in long-run inflation expectations. Finally, using our tradeable de-
mand 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 is small in the sense that it is consistent with the modest response
of inflation to changes in unemployment seen in the aggregate time series since 1990. We develop
this implication in section 5.4. 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, and the present value of
relative prices, on the reduced form regressors. Lagged unemployment and tradeable demand

28
both strongly predict the present value of unemployment and weakly predict the present value
of relative prices. Lagged relative prices strongly predict the present value of relative prices and
weakly predict the present value of unemployment.
Table C.2 reports our estimates of λ for regression specification (17)—the coefficient on the
relative price of non-tradeables. 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 3, λ provides
an estimate of the degree of nominal rigidities. In a world with flexible prices, our estimate of λ
would be large. The fact that our estimate of λ is very small provides further support—over and
above our estimate of κ—for the notion that prices are quite rigid in the U.S. economy.
In our baseline results, we calibrate β = 0.99. It may, however, be that firms are considerably
less forward looking when they set prices than this calibration implies. Recent work has shows
that plausible deviations from full rationality or common knowledge yield a 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 less forward looking. Table
C.3 presents estimates of κ where we calibrate β to lower values. As we vary our quarterly cali-
bration 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 is small.
Our estimates of ψ in Panel B of Table 1 have a similar pattern to our estimates of κ discussed
above. The estimate without time or state fixed effects is negative and the estimate increases as
we include state and then time fixed effects. An important difference is that the absolute size of
our estimates of ψ are much larger than our estimates of κ. This reflects the fact that in equation
(19) the lagged unemployment rate is standing in for the entire future sum in equation (17). Since
unemployment is quite persistent, time variation in the future sum is much larger than time vari-
ation in the unemployment rate, which results in a much larger coefficient in equation (19) than in
equation (17).
Another difference is that ψ is much larger in column (4) than in column (3), while κ is virtually
identical. This reflects the fact 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 persistent variation 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 that is used to identify
them. More generally, κ is a structural parameter, while ψ is not. This implies that ψ may differ

29
Table 2: Has the Phillips Curve Flattened?
Lagged Unempl. IV Lagged Unempl. IV Tradeable Demand IV
Without Time Fixed Effect With Time Fixed Effect With Time Fixed Effect
Pre-1990 Post-1990 Pre-1990 Post-1990 Pre-1990 Post-1990
(1) (2) (3) (4) (5) (6)

Panel A: Estimates of κ from equation (17)
κ 0.0278 0.0002 0.0107 0.0050 0.0109 0.0055
(0.0025) (0.0017) (0.0080) (0.0040) (0.0062) (0.0028)
Panel B: Estimates of ψ from equation (19)
ψ 0.449 0.009 0.198 0.090 0.422 0.332
(0.063) (0.025) (0.113) (0.057) (0.232) (0.157)

Note: 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 price of non-tradeables (i.e., OLS in Panel B). In
columns (5) and (6), the instrument are the fourth lag of the tradeable demand instrument and the relative price of
non-tradeables. In all columns, we estimate κ by two-sample two stage least squares 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.

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 κ is policy invariant.

5.2 Subsample Results

We next analyze to what extent the Phillips curve was steeper during the period of the Volcker dis-
inflation than in subsequent years. Table 2 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 non-tradeables.
Consider first the specification without time fixed effects reported in columns (1) - (2). For
the pre-1990 sample, κ is estimated to be 0.0278, while ψ is estimated to be 0.449. In sharp con-
trast, for the post-1990 sample, κ is estimated to be 0.0002 and φ is estimated to be 0.009. The
difference across samples is roughly 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 early

30
part of the sample and the later part of the sample. For the pre-1990 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 roughly a factor of two and is not
statistically significant. The estimate for κ in columns (5) and (6) are very similar to the estimates in
columns (3) and (4), while the estimate of ψ in columns (5) and (6) show an even smaller difference
across sample periods.
As we emphasize in section 2, estimates of the Phillips curve based on time-series variation —
such as the estimates without time fixed effects in Table 2 — are likely to be heavily influenced
by time-series variation in long-run inflation expectations Et πt+∞ . In contract, the specifications
in Table 2 that include time fixed effects difference out the influence of long-run inflation expecta-
tions. The results in Table 2 therefore suggest that the apparent flattening of the Phillips curve in
the time series is largely due to inflation expectations becoming more firmly anchored over time.
In the early part of the sample, inflation expectations shifted a great deal and these shifts were
negatively correlated with the unemployment rate, which meant that shifts in inflation expecta-
tions masqueraded as a steep Phillips curve. The cross-sectional results in columns (3) - (6) of
Table 2 reveal that in fact the Phillips curve has always been quite flat (at least since 1978).
Figure 5 provides a visual representation of the results in Table 2. In the left panel, we plot
a binned scatterplot of state-level non-tradeable inflation against state-level unemployment after
removing state fixed effects and the effects of the relative price of non-tradeables. We plot the data
separately for the period 1978-1990 and 1991-2018. The plot also includes regression lines for each
subsample. The data in this panel does not account for time fixed effects and therefore includes
aggregate time-series variation. As a consequence, we see a huge flattening of the Phillips curve
in this case.
Contrast this with the right panel in Figure 5. This is an analogous figure to the left panel except
that we also demean by time fixed effects. These data therefore only reflect regional variation in
inflation. In this case, the difference in the slope of the Phillips curve between the early sample
and the late sample is modest. The modest flattening of the Phillips curve that we find over our
sample (once we account for time fixed effects) seems consistent with the fact that the frequency
of price change in the U.S. has declined by about 40% as inflation has fallen since the early 1980’s
(Nakamura et al., 2018).

31
No Time Effects Time Effects
2 2
1978-1990 1978-1990

State Non-Tradeable Inflation (%) State Non-Tradeable Inflation (%)
1.5 1991-2018 1.5 1991-2018

1 1

0.5 0.5

0 0

-0.5 -0.5

-1 -1

-1.5 -1.5

-2 -2
-4 -2 0 2 4 -4 -2 0 2 4
State Unemployment (pp) State Unemployment (pp)
Figure 5: Scatterplots—Non-Tradeable Inflation and Unemployment
Note: in the the left hand size panel we residualize state non-tradeable inflation and unemployment against state
fixed effects and the relative price of non-tradeables, before and after 1990. We then plot residualized inflation and
unemployment, before and after 1990, grouped by 20 bins of state unemployment. The right hand side panel carries
out the same 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.

5.3 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 3 reports three such estimates from
Rotemberg and Woodford (1997), Gali (2008), and Nakamura and Steinsson (2014). In all cases,
we have adjusted the reported value of κ in these papers by the elasticity of output with respect
to employment in the models used in these papers. As is well known, the value of κ in a New
Keynesian model is highly dependent on both the degree of nominal and real rigidities assumed.
The values for κ used in these papers ranges from about an order of magnitude larger than our
estimated value to a value roughly equal to our estimated value. The main difference between
Gali’s relatively high value and the much lower values in Rotemberg and Woodford (1997) and
Nakamura and Steinsson (2014) lies in the degree of real rigidity that the models used in these
papers imply. Gali’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 these
authors match moments that they target in their analysis. Similarly, our estimates imply that the
data we have analyzed is also more consistent with New Keynesian models that incorporate a

32
Table 3: Our Estimates Compared to Prior Work
κ

Rotemberg and Woodford (1997) 0.019
Gali (2008) 0.085
Nakamura and Steinsson (2014) 0.0077
Our Estimate
Full Sample IV Estimate 0.0062

Note: We adjust the estimates from Rotemberg and Woodford (1997), Gali (2008), and Naka-
mura and Steinsson (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.

large amount of real rigidity.

5.4 Aggregate Implications

A question that naturally arises regarding our cross-sectional estimates of κ is whether they can
explain the aggregate time-series 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 U.S.
has been surprising over the past 25 years (“missing disinflation” during the Great Recession and
”missing reinflation” during the late 1990s and late 2010s). Some researcher have recently argued
that cross-sectional variation suggests a steeper Phillips curve than time-series 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 2, we
made the simplifying assumption that the unemployment rate follows an AR(1). This assump-
tion allowed us to derive a simple aggregate relationship between the discounted future sum of
unemployment rates in equation (6) and the current unemployment rate—see equation (7). In
reality, however, the dynamics of the US unemployment rate differ substantially from an AR(1)
(see, e.g., Neftçi, 1984; Sichel, 1993; Dupraz, Nakamura, and Steinsson, 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

T
X
β j ũt+j = ζ ũt + α + t . (20)
j=0

The series we use for ũt in this regression is the difference between the aggregate unemployment

33
rate in the U.S. 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 the unemployment rate of 6.16 with a Newey-West standard error of
1.80. Using equation (20), we can rewrite equation (6) as

πt − Et πt+∞ = −κζ ũt + ωt . (21)

Our cross-sectional estimates of κ are for non-tradeables excluding housing services. As we
emphasize in section 2, the treatment of housing services has important implications for the be-
havior of inflation. Appendix Table C.5 presents estimates of κ and ψ using state-level annual rent
inflation data from the American Community Survey for the years 2001 to 2017. For our baseline
specification with state and time fixed effects, we estimate κ to be 0.0243. This estimate of κ is
roughly four times larger than our estimate of κ for non-housing non-tradeable goods reported in
Table 1. We account for this difference below by taking a weighted average of our full-sample κ
estimate for non-tradeables and this κ estimate for housing services.23
Figure 6 plots the left-hand side of equation (21) (black line) against the first term on the right-
hand side of equation (21) (gray line) 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 by our 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 disinflation and missing
reinflation. In fact, the amplitude of the gray line is very similar to that of the black line for the
Great Recession, the post-Great Recession recover, 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 Ball and Mazumder (2019).25
22
We are thus treating the CBO’s estimate of the natural rate of unemployment as a forecast of long run unemploy-
ment Et ut+∞ .
23
We use the shelter and non-shelter 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
Figure C.2 shows that modest flattening of the Phillips curve we estimate over our sample period has a minimal
effect on the fluctuations in the gray line in Figure 6. Figure C.3 shows that a disproportionate share of the systematic

34
4
Inflation less long-term inflation expectations
Philips Curve Fit
3

-1

-2

1980 1985 1990 1995 2000 2005 2010 2015
Date
Figure 6: Aggregate Phillips Curve
Note: 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 year ahead SPF inflation expectation for the CPI. The gray line plots the first
term on the right-hand 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 supply shocks in the early 1980s, for example, associated with the oil 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 our
estimates? Unemployment rose by about 5 percentage points between 1979 and 1982. Using a
weighted average of our slightly higher pre-1990 non-shelter 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 Figure C.4). Core CPI inflation first rose from 7% to 10%

variation in inflation and the fitted value predicted by our model comes from the housing services (rent) component of
the CPI. The figure is analogous to Figure 6 except that the black line excludes housing and the fitted value uses only
the κ estimate for the non-shelter component of inflation. We see that core inflation excluding housing services varies
much less systematically than core inflation including housing services.

35
from 1979 to 1981 and then fell to 4% by 1986. Clearly, the direct causal effect of unemployment
working through the slope of the Phillips curve explains only a modest amount of this variation
in inflation. Over this same period, long-run inflation expectations first rose from 7% to 8% and
then fell to 4%. Our estimates, 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.

6 Conclusion

This paper 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 mid-1990s. Our estimates are consistent with the insensitivity
of inflation to unemployment during the both the Great Recession and during the low unemploy-
ment periods of the late 1990s and late 2010s.
To reach these conclusions, we estimate the Phillips curve in the cross-section of U.S. states.
We use newly constructed state-level price indexes for non-tradeable goods starting in 1978. We
map from our regional estimates to the slope of the aggregate Phillips curve using a multi-region
New Keynesian model. The model clarifies that the slope of the aggregate Phillips curve is equal
to the slope of the regional Phillips curve for non-tradeable goods. We also use the model to show
that regional data “difference 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 a
factor relating to the persistence of unemployment fluctuations to yield an estimate of the slope
of the Phillips curve. Finally, we develop 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 comes to managing inflation, the ele-
phant in the room is long-run inflation expectations. This view contrasts sharply with the con-
ventional view that managing inflation is about moving up and down a steep Phillips curve. A
crucial question for inflation dynamics is why long-run inflation expectations are sometimes so
firmly anchored but at other times move sharply? Beliefs about inflation in the long run are gov-

36
erned by beliefs about the long-run 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 well understood and its crucial
importance for the conduct of monetary policy implies that even more research should focus on
this question.

37
A Theoretical Appendix

A.1 Household Optimality Conditions

Households optimally trade off current consumption and current labor supply. This implies the
following labor supply curve must hold in our model:

un (CHt , NHt ) WHt
− = ,
uc (CHt , NHt ) PHt

where PHt denotes the lowest cost of purchasing a unit of the composite consumption good
CHt and subscripts on the utility function denote partial derivatives. Using expressions for
un (CHt , NHt ) and uc (CHt , NHt ), we can rewrite the home labor supply curve as

ϕ −1 WHt
χNHt = . (22)
PHt

Households optimally trade off current consumption and consumption in the next period. This
implies the following consumption Euler equation must hold in our model:

uc (CH,t+1 , NH,t+1 ) PHt
βRtn Et = 1. (23)
uc (CHt , NHt ) PH,t+1

where Rtn is the gross nominal interest rate, which is common to both regions in the monetary
union. Household optimization also implies a standard transversality condition must hold in
model and it implies that the stochastic discount factor takes a standard form.
Households choose how much to purchase of the various goods in the economy to minimize
the cost of attaining the level of consumption CHt they choose. This implies the following demand
curves for home and foreign tradeable and non-tradeable goods:

N −η
N PHt
CHt = φN CHt , (24)
PHt

T −η −η
PFTt

TH H PHt TF F
CHt = φT τHt CHt , and CHt = φT τHt CHt . (25)
PHt PHt

Utility maximization, furthermore, implies the following demand curves for each of the varieties

38
of goods produced in the economy:

−θ T (z) −θ −θ
N (z) PFTt (z)

N N PHt TH TH PHt TF TF
CHt (z) = CHt N
CHt (z) = CHt T
CHt (z) = CHt (26)
PHt PHt PFTt

The cost minimizing price indexes are given by

Z 1 1
1−θ Z 1 1
1−θ Z 1 1
1−θ
N N
PHt = PHt (z)1−θ dz , T
PHt = T
PHt (z)1−θ dz , PFTt = PFTt (z)1−θ dz ,
0 0 0

h i 1
N 1−η H T 1−η F 1−η 1−η
PHt = φN PHt + φT τHt PHt + φT τHt PFTt . (27)

A.2 Firm Optimality Conditions

Firms in our model must satisfy demand. For firms in the non-tradeable sector, this implies that

N (z) −θ
N PHt N N
ζCHt N
≤ ZHt NHt (z).
PHt

Optimal choice of labor by the firm z implies that

N N
WHt = SHt (z)ZHt , (28)

N (z) is the firm’s nominal marginal cost, i.e. the Lagrange multiplier on the constraint
where SHt
above.
Non-tradeable firms z that are able to reoptimize their price in period t set it to satisfy

∞
X
N∗ θ
α Etk N
MHt,t+k YH,t+k (z) PHt (z) − SN (z) = 0, (29)
θ − 1 H,t+k
k=0

N ∗ (z) is the price the firm chooses. Intuitively, the firm sets its price equal to a constant
where PHt
markup over a weighted average of current and expected future marginal cost taking into account
the probability that their price will remain unchanged in future periods. We can divide both sides
N
of this equation by PH,t−1 and rewrite it as

∞
" !#
N ∗ (z) N
PH,t+k
X
k N PHt θ N
α Et MHt,t+k YH,t+k (z) N
− M C H,t+k (z) N
= 0, (30)
PH,t−1 θ−1 PH,t−1
k=0

39
N
where M CH,t+k N
(z) = SH,t+k N
(z)/PH,t+k is real marginal cost in the non-tradeable sector.
Analogously to the non-tradeable sector, optimal firm labor demand in the tradeable sector is

T T
WHt = SHt (z)ZHt (z). (31)

T (z) is the tradeable goods firm’s nominal marginal cost. Optimal choice of a new reset
where SHt
price by tradeable goods firms implies

∞
X θ
T
αk Et MHt,t+k YH,t+k T
(z) PHt (z) − ST (z) = 0. (32)
θ − 1 H,t+k
k=0

T
We can divide both sides of this equation by PH,t−1 and rewrite it as

∞
" !#
T (z) T
PH,t+k
X PHt θ T
αk Et MHt,t+k YH,t+k
T
(z) T
− M C H,t+k (z) T
=0 (33)
PH,t−1 θ−1 PH,t−1
k=0

T
where M CH,t+k T
= SH,t+k T
(z)/PH,t+k is the real marginal cost in the tradeable sector.

A.3 Zero Inflation Steady State

In the next section, we take a log-linear approximation of the equilibrium conditions of our model
around a steady state with zero inflation and balanced trade. In this section, we solve for this
N = S T . The steady state of
steady state. The steady state of equations (28) and (31) are WH = SH H

equations (29) and (32) imply
WH θ−1
= . (34)
PH θ

The steady state of equation (22) implies

ϕ −1
χNH = µ−1 , (35)

where NH is the steady state per capita employment of households in the home region and µ =
θ/(θ − 1). The steady state of equation (23) implies βRn = 1.
Since all firms face the same marginal cost in the steady state we consider, all prices will be
equal and all relative goods prices will be one. This implies that in steady state the demand

40
curves for home tradeable and non-tradeable goods—equations (24) and (25)—imply

N
CH = φ N CH , (36)

TH H
CH = φT τH CH , (37)

TF F
CH = φT τH CH . (38)

N ≡ 1 N N (z)dz
R
We define total labor in the home non-tradeable and tradeable sectors as NHt 0 Ht
T ≡
R 1 T N
and NHt 0 NHt (z)dz, respectively. In steady state, total non-tradeable employment, NH , must

equal total non-tradeable consumption, ζφN C. This implies that

N N
NH = ζCH = ζφN C

where the second equality substitutes in equation (36). Similarly, steady state home tradeable
T , equals steady state total consumption for home tradeables. This implies that
employment, NH

T TH
NH = ζCH + (1 − ζ) CFT H .

Using equations (37) and (38), we get that

T H
NH = ζφT τH CH + (1 − ζ)φT τFH CF

Using that fact that CF = CH = C in a symmetric steady state, we get that

T H 1−ζ H
NH = ζφT C τH + τ
ζ F

H = τ H = ζ (no home bias in steady state), we get that
Finally, using the fact that τH F

T
NH = ζφT C.

41
A.4 Derivation of Regional Phillips Curves

A first order Taylor-series expansion of equation (30) around the zero inflation and balanced trade
steady state yields

∞ h
X i
pN ∗ N
Ht (z) − pH,t−1 = (1 − αβ) (αβ)k Et cmcN
H,t+k − pN
H,t+k − p N
H,t−1 .
k=0

Rearranging this equation yields

∗
mcN
N∗
pN N N N

Ht (z) − pH,t−1 = αβEt pH,t+1 (z) − pHt + (1 − αβ) c Ht + πHt . (39)

N given in the line above equation (27) implies
The expression for PHt

N 1−θ N 1−θ N ∗1−θ
PHt = αPH,t−1 + (1 − α) PHt

N ∗ denotes the reset price of firms that are able to change their price in period t. Here we
where PHt
exploit the fact that a random set of firms change their prices at time t and all of these firms set the
same price. A first order Taylor series approximation of this last expression is

N∗
pN N
Ht = αpH,t−1 + (1 − α) pHt

which implies that
N ∗
= (1 − α) pN N

πHt Ht − pH,t−1 . (40)

Manipulation of equations (39) and (40) yields that

N
πHt N
= βEt πH,t+1 mcN
+ λc Ht (41)

where
(1 − α) (1 − αβ)
λ= .
α

We can derive an analogous equation to equation (41) for the tradeable sector. In the tradeable
sector we have that
T
πHt T
= βEt πH,t+1 mcTHt ,
+ λc (42)

T = pT − pT
where πHt Ht H,t−1 is producer price inflation in the home tradeable sector.

42
Taking logs of equation (28) implies that

m̂cN N N
Ht = ŵHt − pHt − zHt .

Taking logs of labor supply—equation (22)—implies that

−1
ŵHt − pN
Ht = ϕ n̂Ht .

Combining these two equation yields

−1
m̂cN N N

Ht = ϕ n̂Ht + pHt − pHt − zHt . (43)

We can substitute this equation into equation (42) to get that

N N
πHt = βEt πH,t+1 + κn̂Ht − λp̂N N
Ht + νHt (44)

N = −λz N and κ = λϕ−1 . This is the regional non-tradeable Phillips Curve in our model.
where νHt Ht

An analogous sequence of steps yields

T T
πHt = βEt πH,t+1 + κn̂Ht − λp̂THt + νHt
T
. (45)

This is the regional tradeable Phillips Curve in our model.

A.5 Aggregate Phillips Curve Derivation

Aggregate non-tradeable inflation can be written as πtN = ζπHt
N + (1 − ζ) π N . Using the Phillips
Ft

curve for home non-tradeable inflation—equation (44)—and its foreign counterpart, we get that
satisfies
πtN = βEt πt+1
N
+ κn̂t + νtN − λ ζ p̂N N

Ht + (1 − ζ) p̂F t , (46)

Similarly, a weighted average of the Phillips curve for home tradeable inflation—equation (45)—
and its foreign counterpart yields

πtT = βEt πt+1
T
+ κn̂t + νtT − λ ζ p̂THt + (1 − ζ) p̂TF t .

(47)

43
First order expansions of equation 27 and its foreign counterpart around the zero inflation
steady state yield that
pHt = φN pN H T F T
Ht + φT τH pHt + φT τH pF t (48)

and
pF t = φN pN H T F T
F t + φT τF pHt + φT τF pF t . (49)

Then the aggregate price level, pt satisfies pt = ζpHt + (1 − ζ) pF t . Combining this equation with
H = τ H = ζ yields
the previous two equations and using the fact that τH F

pt = φN ζpN N T T T T

Ht + (1 − ζ) pF t + φT ζζpHt + ζ (1 − ζ) pF t + (1 − ζ) ζpHt + (1 − ζ) (1 − ζ) pF t

This equation simplifies to
pt = φN pN T
t + φT p t (50)

where we use the notation pN N N T T T
t = ζpHt + (1 − ζ) pF t and pt = ζpHt + (1 − ζ) pF t .

Equation (50) implies that
πt = φN πtN + φT πtT .

Combining this equation with equations (44) and (45) yields the aggregate Phillips curve

πt = βEt πt+1 + κn̂t + νt ,

where νt ≡ φN νtN + φT νtT and we make use of the fact that

N
ζ p̂Ht + (1 − ζ) p̂N
T T

F t + ζ p̂Ht + (1 − ζ) p̂F t = 0.

A.6 Deriving the Other Log-Linearized Equations

In this section, we will assume that supply shocks are zero for expositional simplicity. A log-linear
approximation of the home consumption Euler equation—equation (23)—yields

ucn ucn uc
ĉHt + n̂Ht = Et ĉH,t+1 + n̂H,t+1 + (r̂n − Et πH,t+1 ) , (51)
ucc ucc ucc C t

where we use the fact that home per capita consumption and labor equal aggregate per capita
consumption and labor at the steady state.

44
Next, we solve for the partial derivatives in the previous equation using the functional form
for preferences, equation (8). We have that

1+ϕ −1 −σ−1 −1
ucc C −σ −1 C C − χ N1+ϕ−1
= −σ−1
uc
1+ϕ−1
C − χ N1+ϕ−1
−1
!−1
N 1+ϕ
= −σ −1 C C −χ
1 + ϕ−1
−1
!!−1
N 1+ϕ
= −σ −1 C −1 C −χ
1 + ϕ−1
−1 −1 −1
= −σ −1 1 − C −1 χN 1+ϕ 1 + ϕ−1
−1 !−1
C −1 −1
= −σ −1 1 − χN ϕ 1 + ϕ−1

N
−1 !−1
−1 C −1 −1 −1

= −σ 1− µ 1+ϕ
N
−1 −1
= −σ −1 1 − µ−1 1 + ϕ−1 ,

where we use the steady state labor supply curve—equation (35)—and the fact that in the steady
state C = N . Furthermore, we have that

−1
!−σ−1 −1
N 1+ϕ χ −1
−1
1 + ϕ−1 N ϕ

ucn = −σ C −χ ×−
1 + ϕ−1 1+ϕ −1

−1
= −ucc χN ϕ

= −ucc µ−1 ,

where we again make use of equation (35).
Combining this last to equations with equation (51) yields

ĉHt − µ−1 n̂Ht = Et ĉH,t+1 − µ−1 n̂H,t+1 − σc (r̂tn − Et πH,t+1 )

−1
where σc = σ 1 − µ−1 1 + ϕ−1 .

45
Solving this last equation forward yields

∞
X
−1 n

ĉHt − µ n̂Ht = −σc Et r̂t+j − Et πH,t+1+j
j=0
X∞ ∞
X
n
= −σc Et r̂t+j + σc Et πH,t+1+j
j=0 j=0
∞
X ∞
X
n
= −σc Et r̂t+j + σc Et (pH,t+1+j − pH,t+j )
j=0 j=0
X∞
n
= −σc Et r̂t+j − σc pHt . (52)
j=0

Similarly, for foreign households we have

∞
X
ĉF t − µ−1 n̂F t = −σc Et n
r̂t+j − σc pF t . (53)
j=0

Combining equations (52) and (53) yields

ĉHt − µ−1 n̂Ht = ĉF t − µ−1 n̂F t + σc (pF t − pHt ) ,

which is the Backus-Smith condition for our model.
Define ξˆHt = log τHt − log τH and ξˆF t = log τF t − log τF . With this notation, log-linear approxi-
mations of (24) and (25) as well as their foreign counterparts yields

ĉN N

Ht = ĉHt − η pHt − pHt (54)

ĉTHt = ξˆHt + ĉHt − η pTHt − pTHt

(55)

ĉTF t = ξˆF t + ĉHt − η pTF t − pTHt

(56)

ĉN N

F t = ĉF t − η pF t − pF t (57)

ĉTF H T

t = ĉF t − η pHt − pF t (58)

ĉTF Ft = ĉF t − η pTF t − pF t .

(59)

Note that the expenditure share on tradeable and non-tradeable goods always sums to 1: τHt +

46
τF t = 1. This implies that
ξˆHt τH
H
+ ξˆF t τH
F
=0

which in turn implies that
ζ ˆ
ξˆF t = − ξHt .
1−ζ

First differencing equations (48) and (49) implies

N H T F T
πHt = φN πHt + φT τH πHt + φT τH πF t

and
πF t = φN πFNt + φT τFH πHt
T F T
+ φT τH πF t .

Note that without supply shocks, output and employment are equal. This implies that

N N T T
YHt = NHt , YHt = NHt , YFNt = NFNt , YFTt = NFT t .

We furthermore have that
N T
ζNHt = NHt + NHt .

This equation says that total labor supplied by households in the home region equals total labor
demanded by firms. The ζ on the left-hand-side reflects the fact that NHt is per capita labor supply.
A log-linear approximation of this last expression around the symmetric steady state yields

NN N NT
n̂Ht = n̂ Ht + n̂T
NN + NT N N + N T Ht
= φN n̂N T
Ht + φT n̂Ht

Similarly, in the foreign region we have that

n̂F t = φN n̂N T
F t + φT n̂F t .

Aggregate employment is
Nt = ζNHt + (1 − ζ) NF t .

47
Log-linearizing this equation around the symmetric steady state yields

ζN (1 − ζ) N
n̂t = n̂Ht + n̂F t
ζN + (1 − ζ) N ζN + (1 − ζ) N
= ζ n̂Ht + (1 − ζ) n̂F t ,

where N is steady state household labor supply, equal across the two regions at the symmetric
steady state.
N = ζC N . A log-linear
Market clearing conditions in the non-tradeable sector implies that NHt Ht

approximation of this expression yields that

n̂N N
Ht = ĉHt .

Using equation (54), we get that

n̂N N

Ht = ĉHt − η p̂Ht − p̂Ht

A similar set of steps for the foreign region yields

n̂N N

F t = ĉF t − η pF t − pF t .

In the tradeable sector market clearing implies

T TH
NHt = ζCHt + (1 − ζ) CFT tH
T −η T −η
H PHt PF t
= ζφT τHt CHt + (1 − ζ) φT τFHt CF t
PHt PF t

where the second line follows from equations (24) and (25). Log-linearizing around the symmetric
steady state implies

h i
n̂THt = ζ ĉHt − η pTHt − pHt + ξˆHt + (1 − ζ) ĉF t − η pTHt − pF t .

Similarly, in the foreign region we have

h i
n̂TF t = ζ ĉHt − η pTF t − pHt + ξˆF t + (1 − ζ) ĉF t − η pTF t − pF t .

48
Finally, we define deviations of unemployment from the steady state as

n̂Ht = log NHt − log NH ≈ (NHt − 1) − (NH − 1) = −(uHt − uH ) = −ûHt .

A.7 Log-Linearized Equations of the Model

For convenience we repeat the full set of log-linearized equilibrium conditions of the model.

• Parameters:
−1
– σc = σ 1 − µ−1 1 + ϕ−1

– κ = λϕ−1

– λ = (1 − α) (1 − αβ) /α

– µ = θ/ (θ − 1)
H = τH = ζ
– τH F

• The law of motion for tradeable demand is

ξˆHt = ρξ ξˆHt + εt

and
ζ ˆ
ξˆF t = − ξHt .
1−ζ

• The home non-tradeable Phillips Curve is:

N N
πHt = βEt πH,t+1 − κûHt − λp̂N N
Ht + νHt

• The home tradeable Phillips Curve is:

T T
πHt = βEt πH,t+1 − κûHt − λp̂THt + νHt
T
.

• The home Euler equation is:

ĉHt − µ−1 n̂Ht = Et ĉH,t+1 − µ−1 n̂H,t+1 − σc (r̂tn − Et πH,t+1 )

49
• The Backus-Smith condition is:

ĉHt − µ−1 n̂Ht = ĉF t − µ−1 n̂F t + σc (pF t − pHt )

• The foreign non-tradeable Phillips Curve is:

πFNt = βEt πF,t+1
N
+ κn̂F t − λp̂N N
F t + νF t

• The foreign tradeable Phillips Curve is:

πFT t = βEt πF,t+1
T
+ κn̂F t − λp̂TF t + νFT t

• Definitions of inflation:
πHt = pHt − pH,t−1

πF t = pF t − pF,t−1

N
πHt = pN N
Ht − pH,t−1

T
πHt = pTHt − pTH,t−1

πFNt = pN N
F t − pF,t−1

πFT t = pTF t − pTF,t−1

N H T F T
πHt = φN πHt + φT τH πHt + φT τH πF t

πF t = φN πFNt + φT τFH πHt
T
+ φF τFF πFT t

• The home resource constraint in the non-tradeable sector is:

n̂N N

Ht = ĉHt − η pHt − pHt

• The foreign resource constraint in the non-tradeable sector is:

n̂N N

F t = ĉF t − η pF t − pF t

50
• The home resource constraint in the tradeable sector is:

h i
n̂THt = ζ ĉHt − η pTHt − pHt + ξˆHt + (1 − ζ) ĉF t − η pTHt − pF t

• The foreign resource constraint in the tradeable sector is:

h i
n̂TF t = ζ ĉHt − η pTF t − pHt + ξˆF t + (1 − ζ) ĉF t − η pTF t − pF t

• Aggregate labor in the home region then satisfies the log-linear equations

n̂Ht = φN n̂N T
Ht + φT n̂Ht

• Aggregate labor in the foreign region satisfies

n̂F t = φN n̂N T
F t + φT n̂F t

• Monetary policy is
r̂tn = ϕπ (πt − π̄t ) + ϕn (n̂t − n̄t ) + εrt

• Aggregate employment satisfies

n̂t = ζ n̂Ht + (1 − ζ) n̂F t

• Aggregate inflation satisfies
π̂t = ζ π̂Ht + (1 − ζ) π̂F t

• The deviation of unemployment from its steady state value is

ût = −n̂t .

A.8 The Importance of Non-Tradeable Inflation

Here, we show that the slope of the regional Phillips Curve for overall regional consumer price
inflation is smaller than the slope of the aggregate Phillips Curve, by a factor equal to the expen-

51
diture share on non-tradeable goods. For simplicity, we present this derivation with all supply
shocks νt set to zero.
Consider the Phillips curves for home non-tradeables, home tradeables, and foreign trade-
ables:
N N
πHt = βEt πH,t+1 − κûHt − λp̂N
Ht

T T
πHt = βEt πH,t+1 − κûHt − λp̂THt

πFT t = βEt πF,t+1
T
− κûF t − λp̂TF t .

Substituting these three equations into the definition for home consumer price inflation

N H T F T
πHt = φN πHt + φT τH πHt + φT τH πF t

yields

H
κûHt − λ φN p̂N H T F F T

πHt = βEt πH,t+1 − φN + φT τH Ht + φT τH p̂Ht − φT τH κûF t − λφT τH p̂F t .

An analogous derivation yields the following Phillips curve for foreign consumer prices

πF t = βEt πF,t+1 − φN + φT τFF κûF t − λ φN p̂N F T H H T

F t + φT τF p̂F t − φT τF κûHt − λφT τF p̂Ht .

H =
Subtracting the second of these last two equations from the first (and using the fact that τH
τFH = ζ) yields

πHt − πF t = β (Et πH,t+1 − Et πF,t+1 ) − φN κ (ûHt − ûF t ) − φN λ p̂N N

Ht − p̂F t . (60)

The coefficient in a regional panel regression corresponds to the coefficient in a differenced equa-
tion like this one. Notice that the coefficient on unemployment is φN κ rather than κ. In other
words, the coefficient differs from the coefficient in the aggregate Phillips curve by the factor φN .

A.9 The Role of GHH Preferences

The key feature of GHH preferences that we exploit is that, with GHH preferences, there are
no wealth effects on labor supply either at the aggregate or the regional level. In contrast, with
separable preferences, wealth effects on labor supply are an important determinant of marginal

52
cost and therefore influence the Phillips curve.
To see this more clearly, consider the non-tradeable regional Phillips curve under separable
preferences:
N
πHt N
= βEt πH,t+1 − κûHt + λσ −1 ĉHt − λp̂N N
Ht + νHt , (61)

and the aggregate Phillips Curve under separable preferences:

πt = βEt πt+1 − κût + λσ −1 ĉt + νt . (62)

Relative to the GHH case, both the non-tradeable regional Phillips curve and aggregate Phillips
curve include a consumption term. These terms appear because of wealth effects on labor sup-
ply affect marginal cost in this model. These wealth effects complicate the comparison between
the regional and aggregate Phillips curve because the relationship between employment and con-
sumption is different at the aggregate level than at the regional level. At the aggregate level,
ĉt = n̂t + zt . This implies that we can replace the ĉt term with n̂t + zt in equation (62) and get
a consolidated coefficient of κ + λσ −1 on unemployment. At the regional level, however, this is
not possible because risk-sharing across regions implies that ĉHt 6= n̂Ht + zHt . This difference
implies that the slope of the non-tradeable regional Phillips curve will differ from the slope of the
aggregate Phillips curve when preferences are separable.

A.10 Relaxing Full Information Rational Expectations

To derive the solved forward Phillips Curve, equation (13) in the main text, we manipulated the
Phillips curve under the standard assumption of full-information rational expectations. However,
this type of derivation actually only relies on the weaker assumption that the law of iterated expec-
tations holds. Let’s consider the aggregate Phillips curve—equation (12)—for simplicity. Under
the assumption that the law of iterated expectations holds and the additional simplifying assump-
tion that the unemployment rate follows an AR(1) process, we can solve this equation forward to
get that
κ
πt = − ũt + Ft πt+∞ + ω̃t , (63)
1 − ρFũ β

where Ft denotes agents’ expectations conditional on information at time t, Ft πt+∞ is the agent’s
subjective forecast about the inflation target, ω̃t ≡ Ft ∞ j F
P
j=0 β νt+j , and ρũ is agents’ subjective

belief about the autoregressive coefficient governing the persistence of fluctuations in unem-

53
ployment. Notice that if ρFũ < ρũ , the Phillips Curve is less forward looking than the ratio-
nal expectations Phillips curve. Rational expectations is the special case where ρFũ = ρũ and
Ft πt+∞ = Et πt+∞ . Coibion and Gorodnichenko (2012, 2015a) provide evidence consitent with
the law of iterated expectations holding but full information rational expectations not holding.
See Adam and Padula (2011), Coibion and Gorodnichenko (2015a), Coibion, Gorodnichenko, and
Kamdar (2018) for further discussion of these issues.

A.11 Time Aggregation

Here, we show how time aggregation associated with using four-quarter inflation as our depen-
dent variable implies that we should divide our estimate of κ by 4 since our model in section 3
is written in terms of quarterly inflation. Consider the non-tradeable regional Phillips Curve—
equation (13) from the main text:

∞
X
N
β j κũH,t+j + λp̂N

πHt = −Et H,t+j + Et πt+∞ , (64)
j=0

N equal to zero. We can rewrite this last
where for simplicity we have set the supply shock ωHt
equation as
p
pN N u
Ht − pH,t−1 = −κP VHt − λP VHt + Et πt+∞

where
∞
X
u
P VHt = Et β j ũH,t+j
j=0

∞
p
X
P VHt = Et β j p̂N
H,t+j .
j=0

This same equation hold for periods t, t − 1, t − 2, and t − 3:

p
pN N u
Ht − pH,t−1 = −κP VHt − λP VHt + Et πt+∞
p
pN N u
H,t−1 − pH,t−2 = −κP VH,t−1 − λP VH,t−1 + Et−1 πt+∞
p
pN N u
H,t−2 − pH,t−3 = −κP VH,t−2 − λP VH,t−2 + Et−2 πt+∞
p
pN N u
H,t−3 − pH,t−4 = −κP VH,t−3 − λP VH,t−3 + Et−3 πt+∞

54
Summing the preceding four equations together yields

pN N u u u u

Ht − pH,t−4 = −κ P VHt + P VH,t−1 + P VH,t−2 + P VH,t−3

p p p p
− λ P VHt + P VH,t−1 + P VH,t−2 + P VH,t−3

+ Et πt+∞ + Et−1 πt+∞ + Et−2 πt+∞ + Et−3 πt+∞ .

Taking expectations at time t − 4 then yields

Et−4 pN N u u u u

Ht − pH,t−4 = −κ Et−4 P VHt + Et−4 P VH,t−1 + Et−4 P VH,t−2 + Et−4 P VH,t−3

p p p p
− λ Et−4 P VHt + Et−4 P VH,t−1 + Et−4 P VH,t−2 + Et−4 P VH,t−3

+ 4Et−4 πt+∞ .

Adding and subtracting pN
Ht yields

pN N u u u u

Ht − pH,t−4 = −κ Et−4 P VHt + Et−4 P VH,t−1 + Et−4 P VH,t−2 + Et−4 P VH,t−3

p p p p
− λ Et−4 P VHt + Et−4 P VH,t−1 + Et−4 P VH,t−2 + Et−4 P VH,t−3

+ 4Et−4 πt+∞ − Et−4 pN N

Ht − pHt .

u and P V p are well approximated by univariate driftless random walks.
We now assume that P VHt Ht

We present empirical evidence supporting this assumption in section A.11.1 below. Given this

55
assumption, the preceding equation simplifies to

pN N u
Ht − pH,t−4 = −4κEt−4 P VHt
p
− 4λEt−4 P VHt

+ 4Et−4 πt+∞ − Et−4 pN N

Ht − pHt
∞
X ∞
X
= −4κ β j ũH,t+j − 4λ β j p̂N
H,t+j + 4Et−4 πt+∞
j=0 j=0
 
∞
X ∞
X
− 4κ Et−4 β j ũH,t+j − β j ũH,t+j 
j=0 j=0
 
∞
X ∞
X
− 4λ Et−4 β j p̂N
H,t+j − β j p̂N
H,t+j

j=0 j=0

− Et−4 pN N

Ht − pHt
∞
X ∞
X
= −4κ β j ũH,t+j − 4λ β j p̂N
H,t+j + 4Et−4 πt+∞ + υHt
j=0 j=0

where υHt is a rational expectations error uncorrelated with variables at time t − 4. This last
N = pN −
equation shows that estimating equation (17) with the dependent variable defined as πit it

pN
i,t−4 yields estimates of 4κ and 4λ provided that the forward sums in equation (17) are well

approximated by a random walk.

A.11.1 The Dynamics of the Present Value of Unemployment and Relative Prices
P∞ j
P∞ j N
The derivation above relied on the simplifying assumption that j=0 β ũH,t+j and j=0 β p̂H,t+j

follow univariate random walks. We can assess the accuracy of this assumption by running the
regressions
T
X T
X T
X
β j ui,t+j = αi + γt + ρuu β j ui,t+j−1 + ρup β j pN
i,t+j−1
j=0 j=0 j=0

T
X T
X T
X
β j pN
i,t+j = αi + γt + ρpu β j ui,t+j−1 + ρpp β j pN
i,t+j−1 .
j=0 j=0 j=0

As in the main text, we truncate the infinite sums at T = 20. Table A.1 presents results for these
regression. In both regressions, the coefficient on the lag of the dependent variable is very close
to 1, whereas the coefficient on the lag of the other future sum is near zero. This shows that
both the present value of both unemployment and the present value of relative prices are well

56
Table A.1: Random Walks in the Present Values of Unemployment and Relative Prices
PV of Unemployment PV of Relative Prices
(1) (2) (3) (4)
Lag PV of Unemployment 0.999 0.997 0.020 0.007
(0.001) (0.002) (0.005) (0.008)
Lag PV of Rel. Prices -0.002 -0.001 0.994 0.992
(0.001) (0.002) (0.001) (0.005)
State Effects X X X X
Time Effects X X
Note: In the first two columns, we regress Tj=0 β j ũi,t+j on its quarterly lag and the quarterly lag
P
PT
of j=0 β j pN N
i,t+j , where ũit is unemployment in state i in quarter t and pit is relative non-tradeable
PT j N
prices. In the last two columns we repeat the exercise with j=0 β pi,t+j as the outcome. The sample
period is 1978-2018. We set T = 20. Unemployment is in percentage points and relative prices are
in 100 x log points. The regression is unweighted. Standard errors are in parentheses. These are
two-way clustered by date and state. The number of observations is 3495.

approximated by univariate random walk processes at quarterly frequency.

A.12 Applying Our Estimation Procedure to Model Generated Data

Our empirical specification—equation (17)—approximates the structural regional Phillips curve
in the model we present in section 3—equation (14). The approximation is that we truncate the
present sums of unemployment and relative prices. We can assess the error associated with this
approximation by estimating κ with equation (17) using data generated by our model. This will
also more generally verify that our empirical method is able to identify κ when applied to data
generated from our model.
To simulate data from the model, we adopt a quarterly calibration. We simulate the model for
a wide range of values for the slope of the Phillips curve κ. We vary κ across these simulations
by varying the frequency of price change α. There are 12 remaining parameters in the model for
which we must select values. The values that we choose for these parameters are listed Table A.2.
We first calibrate 9 parameters to standard values, using external sources. We set the quarterly
discount factor to β = 0.99 consistent with an annual riskless real interest rate of 4 percent. We set
the elasticity of intertemporal substitution equal to σ = 1. We set the Frisch elasticity of labor sup-
ply to ϕ = 1 roughly in line with the mix of micro- and macro-economic evidence in Chetty et al.
(2011). We follow Itskhoki and Mukhin (2017) and Feenstra et al. (2018) in setting the elasticity of
substitution between tradeables and non-tradeables to η = 1.5. We set the elasticity of substitution
across varieties to θ = 4. We set the size of the home region to ζ = 0.05. This results in a home

57
Table A.2: Calibrated Parameters
Externally Calibrated Parameters
Intertemporal Elasticity of Substitution σ 1
Discount factor β 0.99
Elasticity of substitution between tradeables and non-tradeables η 1.5
Elasticity of substitution between varieties θ 4
Size of home region p 0.05
Steady state consumption share of non-tradeables φN
H 0.66
Taylor Rule coefficient on inflation ϕπ 1.5
Taylor Rule coefficient on unemployment ϕu 1.5
Frisch elasticity of labor supply ϕ 1

Parameters Calibrated From the Data
Persistence of tradeable demand ρξ 0.9

Parameters Targeting Moments from Data
Standard deviation of tradeable demand innovation σξ 2.1
Standard deviation of supply shock σν 2.1

Targeted Moments Model Data
Standard deviation of annual unemployment 2.1 2.1

regions that is in between the size of New York and Pennsylvania. We set the steady state share
of consumption of non-tradeables to φN = 0.66 following Nakamura and Steinsson (2014). We set
the parameters of the central bank’s interest rate rule to ϕπ = 1.5 and ϕu = 1.5.
We then calibrate the first order autocorrelation of tradeable demand to 0.9. We obtain this
value by regressing tradeable demand on its 16th lag, and taking the 16th root of the regression
coefficient. We assume that the supply shock is i.i.d. We choose the value of the remaining two pa-
rameters, the standard deviations of the innovations to the supply and tradeable demand shocks,
to jointly match two criteria. First, we require that the standard deviation of annual unemploy-
ment generated from the model equals its value in the data conditional on time and state fixed
effects. Second, we require that the contribution of supply and demand shocks to the variance of
unemployment is equal. To calculate the standard deviation of unemployment, we simulate data
from the model at a quarterly frequency, setting κ = 0.0062 (the estimate of κ in Column (4) of
Table 1). We time-aggregate the simulated data from the model, in order calculate the standard
deviation of unemployment in simulated data in the same way as in real world data.
We now use the model to show that our GMM procedure consistently estimates κ. We simulate
the model for a range of values for κ between 0.0005 and 0.1. For each value of κ, we simulate
the model 1,000 times. For each of these simulation, we generate data for 8,000 periods, roughly

58
Figure A.1: Estimates of κ By Two Stage Least Squares on Simulated Data from our Model
Estimates of By Two Stage Least Squares As Varies
0.1
Median Value
0.09 95th Percentile
5th Percentile
0.08 45 Degree Line
Our Estimate

Approximate Value of
0.07

0.06

0.05

0.04

0.03

0.02

0.01

0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

True Value of

the size of our quarterly dataset. We then estimate κ for each simulation using our two stage least
squares procedure.
Figure A.1 plots the median estimated κ as a function of the true value of κ along with the 5th
and 95th quantile of the distribution of the estimated κ’s as function of the true value of κ. For
visual aid, we also plot the 45 degree line and a horizontal line at our estimated value of κ.
Figure A.1 shows that our two stage least squares procedure consistently estimates κ. The
median estimate of κ always lies very close to the 45 degree line. This implies that inconsistency
due to truncating the present values of unemployment and relative prices is not important. Also,
our procedure is quite precise. In the region of the value of κ that we have estimated from the
data, the 5th and 95th percentiles are close to one another and to the median estimate of κ.

59
Table B.1: Slope of the Aggregate Phillips Curve
Pre-1990 Post-1990
(1) (2)
Core CPI 0.796 0.120
(0.120) (0.026)
Median CPI 0.386 0.247
(0.136) (0.032)
Shelter CPI 1.624 0.397
(0.350) (0.048)
PCE 0.416 0.040
(0.078) (0.019)
Core less Shelter CPI 0.221 -0.069
(0.103) (0.026)
Core CPI RS 0.182 0.149
(0.108) (0.027)

B Data Appendix

B.1 Sensitivity of the Phillips Curve Slope using Aggregate Data

Table B.1 presents estimates of the slope of the Phillips curve using aggregate data for several
different measures of inflation. We present estimates separately for the period 1978-1990 and
1991-2018. In each case, we run the regression

πt − Et πt+∞ = α + ψũt−4 + t , (65)

with 10-year ahead inflation expectations from the Survey of Professional Forecasters serving as
a proxy for Et πt+∞ , and the 4 quarter moving average of the CBO unemployment gap serving
as a proxy for ũt . We present results for six measures of inflation: the Core CPI, the Median CPI
produced by the Federal Reserve Bank of Cleveland, the CPI for shelter, the PCE, the Core CPI
less shelter, and the Core CPI research series. These regressions are run on quarterly data.
The results in B.1 show that the slope of the Phillips curve estimated using aggregate data is
highly sensitive to seemingly minor changes in the inflation measure used. This is particularly the
case in the pre-1990 sample where the slope estimates vary by roughly a factor of 10 from 0.182 to
1.624. The estimates for the post-1990 sample also vary a great deal, but somewhat less than the

60
pre-1990 estimates.
Table B.1 also illustrates that inference about the degree to which the Phillips curve flattens
based on aggregate data is highly sensitive to the inflation measure used. For some measures, the
Phillips curve flattens a great deal (e.g., Core CPI and CPI for shelter). But for others it does not
flatten much at all (e.g., median CPI and Core CPI research series).
There has been extensive discussion in the literature behind this. Stock and Watson (2019)
discuss how certain sub-indices — such as shelter — are more cyclical than others. Ball and
Mazumder (2019) argue that the Median CPI has advantages arising from the elimination of large
fluctuations in certain components of the CPI. The difference between CPI inflation and PCE in-
flation arises to a significant degree from differences in the treatment of housing services in the
early 1980s and the fact that the BLS does not revise the CPI, while the PCE is revised.

B.2 CPI Inflation Using Pre- and Post-1983 Housing Methodology

The BLS made a significant change to the methods used to calculate inflation for owner-occupied
housing in 1983. This was important given the sizable weight of owner-occupied housing in the
CPI (22.8%). Before 1983, the component of the CPI having to do with owner-occupied housing
was constructed from a weighted average of changes in house prices and mortgage costs (i.e.,
interest rates). More specifically, it was made up of home purchases (9.9 percentage points); mort-
gage interest cost (6.5 percentage points), other financing, taxes and insurance (2.7 percentage
points); and maintenance and repairs (3.7 percentage points). For further discussion, see Bureau
of Labor Statistics (1982) and Poole, Ptacek, and Verbrugge (2005).
In 1983, the BLS shifted to using changes in rents as a proxy for inflation of owner occupied
housing. Figure B.1 plots CPI inflation from 1972 to 2018 (gray line). It also plots our attempt
at estimating what CPI inflation would have been had the BLS not changed the methodology for
calculating the shelter component in 1983 (black line). Evidently, the pre-1983 methodology yields
a much more variable (and cyclical) measure of inflation over the last few decades. The difference
between the gray line and the black line in Figure B.1 prior to 1983 gives a sense for how accurately
we can replicate the BLS’s pre-1983 methodology.

B.3 Price Index Construction

Here we discuss several details of our procedure for constructing state-level price indexes.

61
Pre-1983 Methods
Post-1983 Methods
15

-5
1975 1980 1985 1990 1995 2000 2005 2010 2015
Date
Figure B.1: CPI Inflation Using Pre- and Post-1983 Housing Methodology
Note: This figure plots overall CPI inflation in the US (gray line) and our attempt at estimating what CPI
inflation would have been had the BLS not changed the methodology for calculating the shelter component
in 1983 (black line). We present these results for the sample period 1972 to 2018. The difference between
the gray and the black line before 1983 gives a sense for how accurately we can replicate the BLS’s pre-1983
methodology.

B.3.1 Sample Restrictions

We restrict the sample we use in several ways. First, we exclude from our sample price relatives
involving a product replacement when the size of the new product is unobserved. This reduces
sampling error in our price indexes. Second, we Winsorize price relatives that are larger than 10
or smaller than 0.1. Third, we drop quote lines that include collected prices that are smaller than
a tenth of a cent. A quote line includes all versions of a particular “quote-outlet” pair. Recall that
a “quote-outlet” pair represents a specific product in a specific location, such as a 2L bottle of Diet
Coke from the Westside Market at 110th Street in New York City.
Fourth, we drop observations associated with clearance sales at the end of a quote line. Intu-
itively, if products systematically go on sale, and then disappear from the data, this can lead to
a sharply declining price index (e.g., for women’s dresses) unless the product that exits is linked
with a new comparable product (next season’s similar women’s dress). To be precise, we drop ob-

62
servations when they are flagged as on temporary sale and are not observed with a regular price
afterwards. In contrast, if we observe a price for the same quote line at a later point following the
sale, we will include the sale observations even if there has been a version change. In the case of a
version change, we compute the effective price change by adjusting for quality as in equation (66)
below.

B.3.2 Quality Adjustments

When a BLS price collector identifies a version change of a particular product (e.g., a new version
of the same rain coat), they determine whether the substitution is “comparable.” If they deem it
to be comparable, they assess whether a quality adjustment is necessary. Specifically, the price
collector uses the code CP for a comparable substitution, the code QC for a substitution that is
considered comparable after quality adjustment, and SR for non-comparable substitutions. For
observations that are considered QC, the analyst will record a quality adjustment factor. This
information is then used in the construction of the price relative for that product.
We follow an analogous procedure. We include price relatives at the time of version changes
in our index construction only if the version change is comparable (i.e., CP or QC). In the case of
QC substitutions, we make use of the reported quality adjustment using the formula

1/τ
Pit
rit = , (66)
Pi,t−τ + QAi,t−τ,t

where QAi,t−τ,t is the quality adjustment entered for the substitution.

B.3.3 Aggregation

Armed with these price relatives, we first aggregate to the product category level (ELI) within
each state using a simple geometric average

Y
Rj,x,t = ri,t ,
i∈j,x

where j is an ELI and x is a state.
Finally, we aggregate the ELI price relatives Rj,x,t within sectors in each state using a weighted
geometric average
Yh P i
Rs,x,t = (Rj,x,t )Wj / m∈s,x Wm ,

63
where s denotes sector, and Wj is the expenditure weight of each ELI. These sectors can be defined
broadly as all of non-tradeables or even the entire non-shelter CPI. We use expenditure weights
that are constant across states and time. Specifically, we use the CPI expenditure weights for 1998.

B.4 Definition of Non-Tradeables Inflation

Below we list the ELIs that we categorize as non-tradeables. We define non-tradeables in a rela-
tively conservative manner since including tradeable goods in our definition of what constitutes a
non-tradeable good can lead to attenuation in the slope of the Phillips curve (if tradeable goods are
price nationally). Our definition of non-tradeables is similar to the BLS service aggregation. It dif-
fers in two ways. First, we include ELIs in the Food Away from Home category as non-tradeables.
Second we exclude several ELIs in Transportation Services, Utilities, and Truck Rentals. An impor-
tant example is airline tickets. These have highly variable prices and are collected using a different
procedure than other services in the CPI Research Database. See Nakamura and Steinsson (2008)
for more discussion of the behavior of transportation services prices.

• education services

– college tuition and fixed fees

– elementary and high school tuition and fixed fees

– day care and nursery school

– technical and business school tuition and fixed fees

• telephone services

– main station charges

– interstate telephone services

• food away from home

– lunch

– dinner

– candy, gum, etc.

– breakfast or brunch

– full service meals and snacks

64
– limited service meals and snacks

– food at employee sites and schools

– food from vending machines and mobile vendors

– board, catered events, and other food away from home

– beer, ale, and other alcoholic malt beverages away from home

• other personal services

– beauty parlor services for females

– legal fees

– funeral expenses

– household laundry and dry cleaning, excluding coin-operated

– shoe repair and other shoe services

– clothing rental

– replacement of setting for women’s rings

– safe deposit box rental

– ax return preparation and other accounting fees

– care of invalids, elderly and convalescents in the home

• housing services

– housing at school, excluding board

– lodging while out of town

– tenants’ insurance

– electricity

– utility natural gas service

– residential water and sewer service

– garbage/trash collection

– gardening or lawn care services

– moving, storage, freight express

65
– repair of household appliance

– reupholstery of furniture

– inside painting and/or papering

• medical services

– general medical practice

– dentures, bridges, crowns, implants

– optometrists/opticians

– services by other medical professionals

– hospital room inpatient

– nursing and convalescent home care

• recreational services

– community antenna or cable tv

– prerecorded - video tapes and discs

– other entertainment services

– pet services

– veterinarian services

– photographer’s fees

– film processing

– fees for participant sports

– admission to movies, theaters, and concerts

– admission to sporting events

– fees for lessons or instructions

• transportation services

– used cars

– truck rental

– other vehicle rental

66
– painting entire automobile

– vehicle inspection

– automotive brake work

– automobile insurance

– drivers license

– local automobile registration

– vehicle tolls

– automobile service clubs

– intercity bus fare

– intercity train fare

– passenger ship fares

– intracity mass transit

– taxi fare

B.5 Definition of Tradeable Employment Shares

We follow Mian and Sufi (2014) in defining the tradeable employment share as the share associated
with the following sectors: “agriculture, forestry, fishing and hunting,” “mining, quarrying, and
oil and gas extraction,” and manufacturing (SIC sectors A, B and D; and NAICS sectors 11, 21,
and 31-33). The QCEW censors data if there are fewer than three establishments in the industry-
state, or if one firm constitutes more than 80 percent of industry-state employment. 5% of NAICS
3 digit state-by-industry cells are censored, while 10% of SIC 2 digit state-by-industry cells are
censored. If an industry-state observation is missing or censored in a given quarter, we exclude
this observation when we calculate the instrument.
Anthracite mining is discontinued after 1987 in the SIC. We drop this industry. We also drop
observations from California before 1978, due to the exceptionally volatile share of agricultural
employment in California during 1976-1978.

67
15
Replication
Official

-5
1980 1985 1990 1995 2000 2005 2010 2015
Date
Figure C.1: Aggregate Non-Shelter Inflation
Note: The figure plots the 12-month non-shelter inflation rate for the US published by the Bureau of Labor
Statistics (official) as well as the corresponding inflation rate using our methods (replication).

4
Pre-1990
Post-1990
3

-1

-2

1980 1985 1990 1995 2000 2005 2010 2015
Date
Figure C.2: Effect of Flattening on Aggregate Fit
Note: This figure plots the variation in inflation caused by changes in unemployment working through the
slope of the Phillips curve according to our pre-1990 and post-1990 estimates of κ. In both cases we weight
our non-shelter estimates of κ with our estimate of κ for rents.

68
Core Ex Housing
4
Inflation less long-term inflation expectations
Philips Curve Fit
3

-1

-2

1980 1985 1990 1995 2000 2005 2010 2015
Date
Figure C.3: Aggregate Phillips Curve Excluding Housing
Note: This figure shows the fit of the aggregate Phillips curve for core inflation excluding housing.

4
Inflation less long-term inflation expectations
Philips Curve Fit
3

-1

-2

1980 1982 1984 1986 1988 1990
Date
Figure C.4: Fit of the Aggregate Phillips Curve During the Volcker Disinflation
Note: This figure shows the fit of the aggregate Phillips curve for core inflation over the period 1980-1990. The
gray line uses a weighted average of our pre-1990 non-shelter estimate for κ and our estimate of κ for rents.

69
Table C.1: First Stage Regressions with Future Sum of Unemployment and Relative Prices
(1) (2) (3) (4)

Panel A: Future Sum of Unemployment
Lagged Unemployment 7.029 3.661 5.477
(0.635) (0.474) (0.510)
Lagged Tradeable Demand -4.465
(0.594)
Lagged Relative Price 0.181 -0.178 0.259 0.833
(0.160) (0.202) (0.565) (0.516)
Panel B: Future Sum of Relative Price of Non-Tradeables
Lagged Unemployment 0.520 1.713 -1.973
(0.889) (0.684) (1.096)
Lagged Tradeable Demand 1.011
(1.028)
Lagged Relative Price 18.572 16.895 13.081 13.710
(0.194) (0.288) (1.000) (1.283)

State Effects X X X
Time Effects X X
Note: This table presents results for the first stage regressions for our estimation of κ. In Panel A, the outcome is the
discounted future sum of quarterly state unemployment, in percentage points, truncated at 20 quarters. In Panel B the
outcome is the discounted future sum of the relative price of non-tradeables, in 100 x log points, truncated at 20 quarters.
In the first three columns, the regressors are the fourth lags of unemployment and the relative price of non-tradeables,
in 100 x log points. In the final column, the regressors are the fourth lags of tradeable demand and the relative price of
non-tradeables. Standard errors are reported in parentheses, clustered by state. The fixed effects included for each column
are reported at the bottom of the table. All regressions are unweighted and have 3323 observations.

70
Table C.2: Estimates of λ from Regression (17)
No Fixed No Time Lagged Tradeable
Effects Effects Unempl. Demand IV
(1) (2) (3) (4)

λ 0.0010 0.0022 0.0029 0.0020
(0.0001) (0.0002) (0.0009) (0.0007)

State Effects X X X
Time Effects X X
Note: This table presents estimates of λ, the coefficient on the present value of relative prices from regression
equation (17). The outcome variable is cumulative non-tradeable inflation over four quarters, measured in
percentage points. The regressors are discounted future sums of quarterly state unemployment, in percentage
points, and the relative price of non-tradeables, in 100 x log points. Each of these is truncated at 20 quarters. In
the first three columns we instrument using the fourth lags of quarterly state unemployment and the relative
price of non-tradeables (this is OLS for ψ). In the fourth column, we replace lagged unemployment with our
tradeable demand instrument among the instruments. In all columns, we estimate λ by two-sample two stage
least squares, and apply the correction to our standard errors from Chodorow-Reich and Wieland (2019). The
sample period is 1978-2018. Standard errors are reported in parentheses, clustered by state. Fixed effects
for each column are reported at the bottom of the table. All regressions are unweighted. The number of
observations is 3323 in the first three columns with slightly fewer in the last column due to differencing.

Table C.3: Estimate of κ as Calibrated Value of β Varies
β = 0.99 β = 0.95 β = 0.90
(1) (2) (3)

κ 0.0062 0.0084 0.0116
(0.0025) (0.0033) (0.0046)
State Effects X X X
Time Effects X X X
Note: This table presents estimates of κ from regression equation (17), with different calibrated val-
ues of β. The outcome variable is cumulative non-tradeable inflation over four quarters, measured
in percentage points. The regressors are discounted future sums of quarterly state unemployment, in
percentage points, and the relative price of non-tradeables, in 100 x log points. Both sums are trun-
cated at 20 quarters. In all columns, we estimate κ by two-sample two stage least squares, and apply
the correction to our standard errors from Chodorow-Reich and Wieland (2019). We include time
and state fixed effects. The sample period is 1978-2018. Standard errors are reported in parentheses,
clustered by state.

71
Table C.4: Estimates of κ for Different Truncation Lengths of Discounted Sums
T = 10 T = 20 T = 30 T = 40
(1) (2) (3) (4)

κ 0.0100 0.0062 0.0044 0.0051
(0.0038) (0.0025) (0.0019) (0.0021)
State Effects X X X X
Time Effects X X X X
Note: This table presents estimates of κ from regression specification (17), for different trunca-
tion lengths of the discounted sums on the right-hand-side. The outcome variable is cumulative
non-tradeable inflation over four quarters, measured in percentage points. The regressors are the
present values of quarterly state unemployment, in percentage points, and the relative price of non-
tradeables, in 100 x log points. We vary to truncation point for these sums between T=10 and t=40
across the columns in the table. We include time and state fixed effects. The sample period is 1978-
2018. Standard errors are reported in parentheses, clustered by state. We use a two sample two stage
least squares regression and apply the correction to our standard errors from Chodorow-Reich and
Wieland (2019).

Table C.5: Slope of the Regional Phillips Curve: Rents
No Fixed No Time Lagged
Effects Effects Unempl.
(1) (2) (3)

Panel A: Estimates of κ from equation (17)
κ 0.0074 0.0179 0.0243
(0.0006) (0.0014) (0.0053)

Panel B: Estimates of ψ from equation (19)
ψ 0.268 0.356 0.603
(0.041) (0.044) (0.124)

State Effects X X
Time Effects X
Note: The table presents estimates of ψ, and κ for rents. The outcome variable is the state-level
annual rent inflation rate, measured in percentage points from the American Community Survey
for the years 2001 to 2017 that we gathered from IPUMS USA . In Panel A, the regressor of inter-
est is the discounted future sum of annual state unemployment, measured in percentage points. In
Panel B, the regressor of interest is lagged state unemployment, measured in percentage points. We
estimate κ by two-sample two stage least squares, and apply the correction to our standard errors
from Chodorow-Reich and Wieland (2019). Standard errors are reported in parentheses and clus-
tered by state. Controls for each column are reported at the bottom of the table. All regressions are
unweighted.

72
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