Breaks in the Phillips Curve: Evidence from Panel Data

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Breaks in the Phillips Curve: Evidence from Panel Data Simon C. Smith, Allan Timmermann, Jonathan Wright 2023-015 Please cite this paper as: Smith, Simon C., Allan Timmermann, and Jonathan Wright (2023). “Breaks in the Phillips Curve: Evidence from Panel Data,” Finance and Economics Discussion Series 2023-015. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2023.015. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Breaks in the Phillips Curve: Evidence from Panel Data Simon C. Smitha, Allan Timmermannb, Jonathan H. Wrightc aFederal Reserve Board bUniversity of California, San Diego cJohns Hopkins University Draft: March 25, 2023 Abstract We revisit time-variation in the Phillips curve, applying new Bayesian panel methods with breakpoints to US and European Union disaggregate data. Our approach allows us to accurately estimate both the number and timing of breaks in the Phillips curve. It further allows us to determine the existence of clusters of industries, cities, or countries whose Phillips curves display similar patterns of instability and to examine lead-lag patterns in how individual inflation series change. We find evidence of a marked flattening in the Phillips curves for US sectoral data and among EU countries, particularly poorer ones. Conversely, evidence of a flattening is weaker for MSA-level data and for the wage Phillips curve. US regional data and EU data point to a kink in the price Phillips curve which remains relatively steep when the economy is running hot. Keywords: Phillipscurve,Inflation,Unemployment,Paneldata,Structuralbreaks,Bayesian analysis JEL classifications: C11, C22, E51, E52 Email addresses: simon.c.smith@frb.gov (Simon C. Smith), atimmermann@ucsd.edu (Allan Timmermann), wrightj@jhu.edu (Jonathan H. Wright) Comments from Refet Gu¨rkaynak and seminar participants at the 2021 IAAE webinar are gratefully acknowledged. Any remaining errors are our own. The views expressed in this paper are those of the authors and do not necessarily reflect the views and policies of the Board of Governors or the Federal Reserve System.

1. Introduction “There was a time where there was a tight connection between unemployment and inflation. That time is long gone.” (Jerome Powell, 2021.)1 “... gradualism is a well-established principle for central banks in times of uncertainty. When faced with uncertainty about the resilience of the economy, it pays to move carefully.” (Christine Lagarde, 2022.)2 The Phillips curve is a key element of the new Keynesian macroeconomic model and is critical in how central banks think of the macroeconomy. Recently there has been much debate about a potential flattening of the Phillips curve, which could, in turn, hinder the central banks’ ability to control inflation. The goal of this paper is to apply Bayesian panel methods with breakpoints to disaggregate data in order to revisit time variation in the slope of the Phillips curve. There are a number of motivations for looking at disaggregate data, whether by industry, by region, or by country. First, there may be some cross-sectional heterogeneity which might shed light on the causes of changing Phillips curves. Second, since different regions and sectors experience different business cycles, there is extra information in disaggregate data that enables us to identify slope coefficients and regime changes more precisely than using aggregate data alone. For example, Bai et al. (1998) and Smith and Timmermann (2021) argue that panel data imposing common timing of breaks increases the precision of break date estimates, even when the effect of such breaks is allowed to vary across individual units or variables. Third, several recent papers (e.g. Hooper et al. (2020), Fitzgerald et al. (2020) and McLeay and Tenreyro (2020)) have pointed out that if the central bank is successfully targeting inflation, then this creates an endogeneity bias in the slope of the Phillips curve, biasing the coefficient towards zero. The use of disaggregate data in conjunction with the inclusion of time fixed effects avoids this problem, because the central bank does not specifically target inflation in any one particular region or sector.3 1This quote is from Federal Reserve Chair Jerome Powell’s press Conference, March 17, 2021; https://www.federalreserve.gov/mediacenter/files/FOMCpresconf20210317.pdf. 2This quote is taken from the speech “Monetary policy in an uncertain world” by Christine Lagarde, President of the ECB, at “The ECB and Its Watchers XXII conference, 17 March 2022.” 3Theproblemwouldnotbesolvedwithdisaggregatedatawithouttimefixedeffects,becauseinthatcase some of the identification would come from the time series dimension where there is endogeneity. 1

Building on these insights, in this paper we apply recent Bayesian panel break methods to study Phillips curve inflation dynamics. Our Bayesian panel break approach estimates the number of breaks to the Phillips curve and the time of their occurrence (location). Further, it endogenously identifies clusters of inflation series with common Phillips curves and for which the impact of breaks is similar. The latter feature allows us to examine evidence of convergence in Phillips curves. Finally, our approach allows us to identify lead-lag effects in the timing by which individual inflation series get affected by breaks. Existingworkon estimatingPhillipscurvesin panelsofdisaggregate datamostlyimposes the restriction of a common slope coefficient. An alternative is to estimate the Phillips curve for each industry or region separately, but this gives up a lot of information and tests conductedonunivariatePhillipscurvemodelsgenerallylacksufficientpowertodetectbreaks or to estimate their dates precisely. An advantage of our approach is that we can take a middle ground, and do partial pooling, while allowing for some cross-sectional variation in the slope coefficients. For example, our methodology allows us to consider groupings by industry or geographic region, with different slope coefficients applying to each group. We can impose the groupings a priori, or the grouping structure can be estimated as part of the modeling process. If the data support a homogeneous Phillips curve that is identical across all units, only a single group will be identified. Conversely, very strong heterogeneity in Phillips curves across industries or regions will lead to a model in which each group comprises a single unit. Our methodology endogenously determines whether any of these special cases or an intermediate scenario with multiple units in each cluster, is supported by the data, thus adapting to the degree of heterogeneity found in the data. The focus of our analysis is on understanding how the Phillips curve has changed over timeandidentifyingpossibledriversofsuchchange. Acomplexsetoffactorscouldbeatplay, including changes in unionization and wage indexation, exposure to international trade, and even economic integration. The impact of breaks to the Phillips curve is, therefore, likely to depend on the unit at which inflation is measured. To help identify these drivers, we therefore apply our estimation approach to a variety of data sets. Specifically, we consider US price Phillips curves using disaggregation at the industry and MSA level, and to wage Phillips curves at the state level. We also examine Phillips curves at the country level within the European Union. Turningtotheempiricalresults,inUSindustrydatacoveringthesample1959Q1-2022Q3, we find two regime changes in the Phillips curve; a steepening around 1972 and a flattening in 2001. Moreover, the recent flattening of the Phillips curve is more pronounced for goods 2

prices than for services prices. The steepening around 1972 comes after a period when inflation had been trending up for some years and when indexation of wage contracts, either implicit or explicit, became more common. This would in turn steepen the Phillips curve. Meanwhile, the subsequent flattening corresponds to a time of greater import penetration, especially from China, with China joining the World Trade Organization in 2001.4 While economic intuition might suggest that changes to the slope of the Phillips curve would occur gradually, China’s accession to the WTO may have caused more of a sudden break with quite sharp effects documented in studies such as Bena and Simintzi (2022). Declining unionization and the fact that inflation is stable at a low level creating less of a need for paying attention to inflation in wage setting are other possible explanations for the flattening of the Phillips curve.5 These regime changes that we detect are consistent with some of the existing literature (e.g. Hooper et al. (2020)), although Hazell et al. (2022) argue using state level data that the Phillips curve has been consistently flat. Our non-common break estimates suggest that the first break to the Phillips curve based ontheindustry-levelPCEseriesoccursbetween1972and1973whilethebreaktothePhillips curve based on the CPI series occurs between 1971 and 1974. A second break occurs between 2001 and 2002. In both cases, the timing of the breaks is, thus, quite precisely estimated.6 USregional(MSA)dataarenotavailableasfarbackintime, spanningtheshortersample 1980-2022. This means that we cannot examine the presence of Phillips curve breaks in the 70sforthisdata. Still,evenwiththisshortersamplecoverage,wemanagetoidentifyaregime change around 2000. Further, we find again that MSAs with above (below) median rates of import penetration from China have experienced a considerably stronger (weaker) flattening of their price Phillips curve. These findings are consistent with more goods competition from China explaining a part of the flattening of the price Phillips curve. Broadly similar patterns are found in the EU for a sample that begins in 1986 and ends in 2021. For this data we find evidence of a single break which we estimate occurs in 2004 at which point the slope of the Phillips curve flattens significantly. Using our clustering methodology, we find that the Phillips curve used to be particularly steep in poorer (mostly 4Auer et al. (2017), Stock and Watson (2020), Gilchrist and Zakrajˇsek (2019) and Firat (2020) all show how greater trade openness can flatten the Phillips curve. 5While we do find a flattening break in the wage Phillips curve (estimated over the sample 1980Q1 - 2019Q4), it is of smaller magnitude than for the price Phillips curve, and it comes in 1989, earlier than we find with most price data. 6The common break approach can be viewed as a simplification that places the break to the Phillips curve at a given date within the break interval identified by the noncommon break model. 3

East European) countries prior to the 2004 break, but has flattened by more in those countries, consistent with clear evidence of Phillips curve convergence across countries that, early in our sample, used to display a very different inflation-unemployment trade-off. We also study nonlinearity of the Phillips curve, which, as noted by Hooper et al. (2020), is much easier to do with disaggregate data since the national labor market has not really been tight since the late 1960s, whereas many individual MSAs have had tight labor markets in this time period. Since the current policy debate is focused on such tight values of the labor market, regional data seems likely to be helpful here. We consider a kink in the Phillips curve at a threshold of unemployment rates of 5 and 4.2 percent.7 Using these thresholds, we find that the Phillips curve is steeper in a tight labor market. Hooper et al. (2020), Babb and Detmeister (2017) and Leduc et al. (2019) also find that the Phillips curve is steeper in a tight labor market but do not consider subsample instability. Ignoring breaks has the effect of leading to underestimation of the additional steepness in tight labor markets in the most recent period. Next, we explore some aggregate implications of our Phillips curve estimates. Our estimates for both the US and the EU imply essentially no missing disinflation during the Great Recession and no missing reinflation during the subsequent recovery years. In addition, we find that a steeper (nonlinear) Phillips curve in hot labor markets combined with a higher natural rate of unemployment driven by unusually strong wage growth (Crump et al. 2022) can explain almost half of the surge in U.S. inflation between 2020 and 2022.8 Finally, we investigate the implications for optimal monetary policy of the break in the Phillips curve around the turn of the century using our MSA-level estimates. The break induces additional parameter uncertainty in our Bayesian framework which causes the central bank to respond more cautiously to deviations in the unemployment gap as the policy maker is uncertain about the link between economic slack and inflation, in line with Brainard (1967)’s “conservatism principle” that is encapsulated in the opening quote from ECB President Christine Lagarde. The policy-maker compensates for this caution by responding more aggressively to deviations in inflation from target. We find a similar pattern using our EU country-level estimates. These results are relevant for recent global monetary policy actions in response to the re-opening of the economy after the lockdowns induced by 7For comparison, Stock and Watson (2009) define a tight labor market as an unemployment gap below minus 1.5 percent while Babb and Detmeister (2017) use the same thresholds as we do. 8The kinked (nonlinear) Phillips curve effects are an important part of this explanation. Interestingly, our approach does not detect a break around the Covid lockdown, suggesting that the linear Phillips curve remained quite flat during this period. 4

the pandemic. Our analysis is related to a large body of research on time-variation in the Phillips curve. This literature can be divided into two broad categories.9 The first approach captures timevariation by assuming that the parameters of the Phillips curve follow a random walk (Ball and Mazumder 2011; Matheson and Stavrev 2013; Blanchard 2016; Inoue and Wang 2022). ThesecondapproachestimatestheparametersofthePhillipscurvesubjecttoanassumption that break dates are either pre-specified or determined based on the single breakpoint test of Andrews (1993) or based on informal modeling techniques, such as regressions with rolling windows (Roberts 2006; Coibion et al. 2013; Coibion and Gorodnichenko 2015; Leduc et al. 2017; Ball and Mazumder 2019; Gal´ı and Gambetti 2019; Gilchrist and Zakrajˇsek 2019; Del Negro et al. 2020; Fitzgerald et al. 2020; Hooper et al. 2020; Cerrato and Gitti 2022; Hazell et al. 2022).10 However, break tests conducted on univariate time series have low power, making it difficult to detect breaks in the Phillips curve on individual inflation series. Exploiting the rich information in the cross-section of panel data sets offers the opportunity for increased power and our study is the first to formally estimate multiple breaks in the Phillips curve in the context of such panel data.11 The remainder of the paper proceeds as follows. Section 2 introduces the panel data sets used in our analysis while Section 3 explains our Bayesian panel approach, including estimation, modelselectionandchoiceofpriors. Section4presentsourmainempiricalresults on breaks in the industry and regional Phillips curves, and Section 5 discusses aggregate implications of our results. Section 6 conducts a set of robustness exercises, while Section 7 concludes. Additional empirical results are described in an Appendix at the end of the paper. 2. Data This section introduces our data along with the data sources used in our empirical analysis. Wefirstdescribeourinflationexpectationsandaggregateunemploymentgapmeasuresbefore explaining the inflation rate, unemployment rate, and NAIRU measures which we use for the US Metropolitan Statistical Areas (MSAs) and industries, as well as for the EU. 9Appendix Table A1 contains a list of some of the main studies on variation in the Phillips curve. 10Barnichon and Mesters (2021) use a subsample split to track time-variation in the Phillips multiplier. 11Allowing for multiple breaks is also crucial when considering long time series samples for which stationarity is less likely to hold. 5

2.1. Inflation expectations, unemployment rates, and NAIRU We source four-quarter-ahead Consumer Price Index (CPI) inflation expectations from Blue Chip Economic Indicators. These data go back to 1985. Between 1980 and 1985, we use Producer Price Index (PPI) inflation expectations from the same source. Before 1980, we use data from Livingston which is only updated every six months and so we simply repeat observations in the two corresponding quarters, effectively assuming that inflation expectations remain the same in each 6-month period. Because U.S. inflation expectations are only measured for the aggregate price index as opposed to at the regional or sectoral level, we can only use these inflation expectations data in specifications without time fixed effects. We use the end-of-quarter monthly aggregate unemployment gap, measured as the difference between the unemployment rate from the U.S. Bureau of Labor Statistics (BLS) and the NAIRU estimate (from the Congressional Budget Office). These data begin in January 1949 and end in September 2022. We source the annual country-level unemployment rate and NAIRU estimates for the 28 EU member countries (the current 27 plus the UK which was a member until recently), and hencetheunemploymentgaps,forthesampleperiod1965-2021fromtheDGECFIN/AMECO— the European Commission’s macroeconomic database.12 For the regional analysis, we obtain annual unemployment rate data from 1980 to 2022 for 22 MSAs from the BLS. We also use the end of quarter monthly unemployment rate for all 51 states (including the District of Columbia), also obtained from the BLS. These data begin in January 1980 and end in December 2019. 2.2. Price data 2.2.1. MSA level We source monthly total CPIs for 22 MSAs from the BLS. We construct annual levels as the average of all monthly observations in the corresponding year.13 Next, we construct annual inflation rates as log(CPI /CPI )×100 in which CPI denotes the level for the ith MSA it it−1 it in year t. 12We thank Michele Lenza for helping us access these country-level NAIRU estimates. 13Data for all but a few MSAs are collected only in either odd or even months. See https://www.bls.gov/opub/hom/cpi/pdf/cpi.pdf for details of the complete methodology and https://www.bls.gov/cpi/additional-resources/geographic-sample.htm. for the geographic definitions. 6

Annual unemployment rates are constructed as the average of all monthly observations in the corresponding year. Our sample for these data begins in 1980 and ends in 2022, but for many MSAs the data only start in 1990. 2.2.2. Industry level We use quarterly Personal Consumption Expenditures price indexes (PCE) for 16 industry components, similartothoseanalyzedbyStockandWatson(2020), sourcedfromtheBureau of Economic Analysis (BEA).14 Our sample is 1959:Q1 - 2022:Q3. We construct annualized quarterly inflation rates as log(PCE /PCE )×400. i,t i,t−1 From the BLS, we source monthly CPI inflation for 31 “level 3” industries, as currently formulated, beginning in January 1954 and ending in September 2022, though not all series go all the way back. We construct our annualized quarterly inflation rate observations from end of quarter monthly observations as log(CPI /CPI )×400. i,t i,t−3 2.3. Implied national Phillips curve slopes Hazell et al. (2022) show that the regional Phillips curve slope can be divided by the expenditure share on nontradeables to obtain the national Phillips curve slope. We use the 31 CPI industry weights to compute the expenditure share on nontradeables. We follow Hazell et al. (2022) by assigning the following series to nontradeables: Full Service Meals and Snacks, Limited Service Meals and Snacks, Food at employee sites and schools, Food from vending machines and mobile vendors, Other food away from home, Electricity, Utility (piped) gas service, Water and sewer and trash collection services, Household operations, Medical care services, Transportations services, Recreation services, Education and communication services, Other personal services, and Shelter. The expenditure share on nontradeables is therefore 69.1 percent. Doing the same using the 16 PCE component weights, the expenditure share on nontradeables is 74.3 percent. 14The two categories – Housing and Household utilities – have since been replaced by one: Housing and utilities. 7

2.4. Wage data Following Hooper et al. (2020), we directly compute average hourly earnings (AHE) for each of the 50 states and the District of Columbia using the latest (2019) CEPR uniform extract from the Current Population Survey (CPS)15. Aggregating from monthly data, we construct quarterlydatafrom1980:Q1through2019:Q4, fromwhichweconstructquarterlyannualized wage inflation. 2.5. EU data We source headline (as well as total goods and total services) annual inflation rates for our 28 countries (the 27 current members and the UK) from the ECB statistical warehouse. Our sample begins in 1986 and ends in 2021. 2.6. Group structure We will be interested in group heterogeneity, with either the group allocation imposed according to pre-determined selection criteria, or determined by the Bayesian algorithm as part of the estimation process. The 16 PCE sectors are split into goods – Motor vehicles and parts, Furnishings and durable household equipment, Recreational goods and vehicles, Other durable goods, Food and beverages purchased for off-premises consumption, Clothing and footwear, Gasoline and other energy goods, and Other nondurable goods – and services – Housing and utilities, Health care, Transportation services, Recreation services, Food services and accommodations, Financial services and insurance, Other services, and NPISH. We also split the 28 EU countries into rich and poor countries with rich countries defined as countries with real GDP per capita deflated by PPP in 2019 above the EU average and poorcountriesdefinedastherest. TherichcountriesincludeLuxembourg,Ireland,Denmark, Netherlands, Austria, Germany, Sweden, Belgium, Finland, France, and UK.16 15The data are available from https://ceprdata.org/cps-uniform-data-extracts/. 16The poor countries therefore include Malta, Italy, Czech Republic, Spain, Cyprus, Slovenia, Slovakia, Romania, Portugal, Poland, Bulgaria, Estonia, Lithuania, Latvia, Hungary, Greece, and Croatia. 8

3. Methodology Our analysis examines three different Bayesian panel specifications. The first is our baseline pooled panel model with multiple breakpoints. This model applies the methodology developed by Smith and Timmermann (2021) to exploit information in the cross-section and obtain increased power to detect structural breaks. Breaks are assumed to be common, i.e., they hit every series in the cross-section at the same time and by the same amount. To summarize, this model assumes homogeneity both in the timing of any breaks and in their impact on individual variables. To gain further insight into the break dynamics, our second model relaxes the common break-timing assumption, allowing series to be hit at different times. We accomplish this using the methodology developed by Smith (2018) which is designed to detect lead-lag relations in the impact of breaks across different variables in the cross-section. This approach can, thus, shed light on the diffusion of breaks and the speed at which different sectors, regions, or countries are affected by breaks to their Phillips curves. Our third model endogenously estimates both the number of groups and the assignment of each series to a group using the methodology developed by Smith (2022). Relative to the baseline model that pools parameters across the entire cross-section, this model pools parameters across all series within a group, but allows the parameters to differ across groups. Thisprovidesaneffectivewaytoallowforheterogeneityintheimpactofbreaksonindividual variables.17 The baseline homogeneous (pooled) panel model arises as a special case of this specification when the data only identifies a single group. At the other extreme, a model where each series in the cross-section gets assigned to its own individual group would allow for complete heterogeneity. 3.1. Common breakpoint model The first–baseline–model we take to the data allows for an unknown number of K breaks occurring at unknown times τ = (τ ,...,τ ) which are assumed to be common to all i = 1 K 1,...,N series in the cross-section.18 Our first specification is for the Phillips curve at the MSA level. The data are annual, and the model for the kth regime takes the form (for 17We generally condition on the regimes identified by the baseline model when implementing this third model in our empirical analysis. 18ForsimplicityournotationusesN asthecross-sectionaldimension,butourapproachcanreadilyhandle unbalanced panels with a time-varying dimension, N . t 9

k = 1,...,K +1): π = α +γ +ρ π +λ URATE +(cid:15) , t = τ +1,...,τ (1) it i t k it−1 k it−1 it k−1 k in which π denotes the inflation rate for the ith series at time t, α and γ denote twoit i t way fixed effects, π is the lagged inflation rate for variable i, URATE denotes the it−1 it−1 unemployment rate for the ith series at time t−1, and (cid:15) is the residual for the ith series it at time t which is assumed to be normally distributed (cid:15) ∼ N(0,σ2 ), so we allow volatility it ik to vary across individual variables. The parameters ρ ,λ , and σ2 are all allowed to shift k k ik across regimes separated by a break, but the former two are assumed to be identical across all series within a given regime, effectively following step functions that shift at τ . Hall k (2023) argues that the Phillips curve is steeper in times of high volatility because volatile price determinants reduce price stickiness as a larger fraction of sellers elect to reset their prices. Allowing volatility to vary across breaks could thus be important in identifying shifts in the steepness of the Phillips curve slope. Our baseline model assumes that the residuals (cid:15) are cross-sectionally and serially unit correlated. This assumption means that we are not required to estimate the N(N − 1)/2 covariance terms in each break segment but may not be empirically valid in some empirical applications. Section 6 discusses how to test the validity of this assumption. More broadly, we can allow for cross-sectional correlation in (cid:15) through a common factor structure that it allows for heterogeneity in factor loadings across units but assumes that the idiosyncratic shocks that remain, after accounting for the common factors, are orthogonal across i. The specification in Equation (1) uses the unemployment rate rather than the unemployment gap as the slack measure. At the MSA level, there are no estimates of the natural rate of unemployment and while we could HP detrend the city-level unemployment data, such estimates would be sensitive to the bandwidth parameter. We instead rely on the two-way fixed effects to absorb variation in the natural rate across time and cities. Common time variation in inflation expectations in Equation (1) is also absorbed by the time fixed effects. The same model is applied to the EU-level data, except that the unemployment gap replaces the unemployment rate since we have NAIRU estimates for EU countries unlike for the US MSAs: π = α +γ +ρ π +λ UGAP +(cid:15) , t = τ +1,...,τ . (2) it i t k it−1 k it−1 it k−1 k For the US industry-level data, using either PCE or CPI, we do not observe industry- 10

level unemployment rates, let alone a NAIRU estimate.19 For this case, we substitute the aggregateunemploymentgap,UGAP ,forthedisaggregateunemploymentgapinEquation t−1 (2). This means we must drop the time fixed effects which are not separately identifiable from the aggregate unemployment gap. Finally, we include four-quarter-ahead CPI inflation expectations, BC , which are identified in the absence of time fixed effects, yielding the t−1 model: π = α +ρ π +λ UGAP +ψ BC +(cid:15) , t = τ +1,...,τ . (3) it i k it−1 k t−1 k t−1 it k−1 k Note that in this specification, the data are at a quarterly frequency. 3.2. Noncommon breakpoint model For parsimony, we only formally exposit the noncommon breakpoint model that generalizes the common breakpoint model detailed in Equation (1).20 The only difference is that the break timing, which was previously common (τ ), is now allowed to differ across series (τ ). k ik Formally, for the MSA-level annual data the model is (for regimes k = 1,...,K +1) π = α +γ +ρ π +λ URATE +(cid:15) , t = τ +1,...,τ . (4) it i t k it−1 k it−1 it ik−1 ik While this specification does not impose that the timing of the breaks is identical across all variables, we control the degree of heterogeneity in the timing of breaks across units by effectively only considering “local” variation in the break timing, i.e., breaks whose occurrence is close to the break date for the majority of variables. This prevents our approach from identifying idiosyncratic breaks in the individual series and enables us to use cross-sectional information to more accurately identify clusters of breaks whose impact can spread across units at different speeds. Intuitively, the approach works by identifying break windows rather than single break points. Variables can be hit at any time within a given break window. For example, a common break approach might identify a break around the Global Financial Crisis in September 2008 when Lehman Brothers failed. The break window approach, however, might identify a local break window of, say, 6-12 months during which firms were hit by the break at dif- 19BLS does have some data on industry-level unemployment, in the sense of breaking out unemployment by the sector of the unemployed worker’s last job, but this data only goes back to 2000. 20ThemodelsthatuseeitherEUdata–displayedinEquation(2)–orU.S.industry-leveldata–displayed in Equation (3) – generalize in the obvious way. 11

ferent times as the financial crisis cascaded through the economy. We control the degree of heterogeneity in the timing of breaks across series through the prior, detailed in Section 3.4. 3.3. Grouped heterogeneity model So far we assumed homogeneity in the regression coefficients and, consequently, in the effect of breaks on individual variables. However, in many cases both the slope coefficients and the impact of breaks may differ across sectors, regions, or countries. For such cases, it is important to allow for heterogeneous parameters. We accomplish this by assuming the existence of G groups or clusters of variables and allowing parameters to vary across groups k while they are the same within groups. Each unit (or variable) in the cross-section belongs to a single group (cluster) (i ∈ g ) and both the group membership and the number of k groups is allowed to vary across regimes. This approach offers a flexible specification. For example, we can allow for full heterogeneity in a given regime by setting G = N, whereas k homogeneity within the regime corresponds to G = 1. Values of G between these extremes k k indicate some degree of clustering within that regime. Moreover, variation across regimes in the number of clusters can provide important information about issues such as convergence (or lack thereof) in the Phillips curves across units. Using the model for the EU-level data as our lead example, we estimate the following model in each of the k = 1,...,K +1 regimes identified by the baseline model21 π = α +γ +ρ π +λ UGAP +(cid:15) , t = τ +1,...,τ , (5) it i t g it−1 g it−1 it k−1 k k k where (cid:15) ∼ N(0,σ2). The parameters ρ and λ are pooled across all series within the it i g k g k g th group, but differ across the G different groups. The number of groups and the series k k assigned to each group can be either specified a priori or alternatively determined as part of the estimation. In the latter case, our priors lean against identifying groups that contain only a single series, thus reducing the likelihood of simply identifying outliers in the data.22 21The models that use industry-level data are not formally exposited for simplicity, but follow the same structure. 22Our Bayesian approach has two features that help determine the number of groups. First, the marginal likelihood guiding the estimation prefers fewer groups and penalizes additional groups since these require estimatingmoreparameters. Second,weuseapriorthatanaverageoffiveseriescompriseagroupandapply a penalty to very small and very large groups although these are not ruled out. Still, our prior is towards not having groups with just a single member. In cases where we find empirically that some groups have just a single or very few members, the empirical evidence therefore strongly supports separating these units. 12

3.4. Prior distributions Our Bayesian panel break approach requires us to specify priors on the regime durations and regression parameters. Using the baseline model as our lead example, we next explain how these priors are set. We further specify our priors on the break lags in our second, noncommon breaks, model and our priors on the clustering (grouping) model. Finally, we discuss our prior choices. 3.4.1. Prior on regime durations Following Koop and Potter (2007), the regime durations, l = τ − τ , follow a Poisson k k k−1 prior distribution p(l | ζ ) = Po(ζ ), k = 1,...,K +1, (6) k k k where the intensity parameter ζ follows a conjugate Gamma prior distribution k p(ζ ) = Ga(c,d), k = 1,...,K +1, (7) k and c and d are the hyperparameters of ζ . These hyperparameters only determine the k average regime duration since the expected regime durations are allowed to differ across breaks, with each individual regime having its unique intensity parameter. 3.4.2. Priors on regression parameters For regimes k = 1,...,K+1, we follow conventional practice and specify an inverse gamma prior distribution over the residual variances p(σ2 ) ∼ IG(a,b), i = 1,...,N, (8) ik while we assume a Gaussian prior on the regression coefficients p(λ ) ∼ N(0,σ2), k λ p(ρ ) ∼ N(0,σ2). (9) k ρ 13

Here σ2 and σ2 are hyperparameters that control the degree to which λ and ρ are shrunk λ ρ k k towards their prior means of zero.23 3.4.3. Priors on heterogeneity in break dates Our second specification allows for differences in the point in time when breaks affect the individual series within a break window, the length of which is estimated. Let τ denote the k date at which the kth break window begins. The lag with which the ith series is hit by the kth break is denoted ∆ = τ −τ which can be zero (hit immediately at the beginning of ik ik k the break window) as well as positive (hit with a lag). We specify a Poisson prior over such break delays p(∆ | δ ) ∼ Po(δ ), k = 1,...,K, i = 1,...,N. (10) ik k k We assume that the average expected lag with which the N series are hit by the kth break, δ , has a conjugate Gamma prior distribution. k p(δ ) ∼ Ga(e,f) k = 1,...,K. (11) k The hyper parameters e and f again control the average degree of heterogeneity in break dates across series and the lag in individual series’ break dates from the beginning of the break window τ is allowed to vary across breaks. Some breaks might spread very rapidly k across all series, while others may undergo a slower diffusion process. 3.4.4. Priors on heterogeneity and grouping structure Our third specification introduces heterogeneity through an endogenous break clustering structure. We accomplish this by placing a Poisson prior over the number of series included in the gth group, N 24 g p(N | ψ) ∼ Po(ψ), g = 1,...,G+1 (12) g 23The grouped heterogeneity model specifies a normal prior over the coefficients λ and ρ . g g 24This specification of multiple independent Poisson distributions is inferentially equivalent to a specification that uses a single Multinomial distribution. 14

where the expected number of series in every group, ψ, has a conjugate Gamma prior p(ψ) ∼ Ga(h,j). (13) The prior hyper parameters h,j control the average number of groups along with the difference in the number of groups across regimes. 3.5. Prior elicitation Our analysis calibrates the prior hyper parameters determining the regime duration so that breaks occur, on average, every twenty years. We achieve this by setting d = 2 and c = 40 or c = 160 for the annual and quarterly data, respectively. Our priors are thus set to focus on rare, “secular” breaks in the Phillips curve.25 We set a = 2 and b = 1. σ2 and σ2, which λ ρ control the degree to which λ and ρ are shrunk towards their prior means (zero), are both k k equal to 0.1. These are fairly uninformative priors which allow the autoregressive parameter and the slope of the Phillips curve to vary with the data. For the quarterly data, we set e = 8 and f = 1 such that the prior expected break lag for each series is eight quarters (two years). Similarly, for the annual data we set f = 1 and e = 2. Finally, to determine group size for our third specification, we set h = 5 and j = 1 to reflect our prior belief that there are, on average, five series in each group. This choice of prior on the groups thus leans towards not having a single series comprise a group. 3.6. Estimation Each of our models is estimated using a multi-step reversible jump Markov chain Monte Carlo algorithm (Carlin and Chib 1995; Green 1995). Estimation of the baseline model consistsofthreesteps. First,weestimatetheregressioncoefficientsfromtheirfullconditional distributions using a Gibbs step. Next, we estimate the break locations using a random-walk Metropolis-Hastings algorithm. Finally, the third step estimates the number of breaks using a reversible jump step. This latter step introduces the number of breaks K as a parameter and repeatedly attempts to “jump” to different values of K, with the proportion of iterations spent at each value of K approximating the posterior model probabilities.26 25Setting these priors to focus on breaks at higher frequencies (e.g., once every couple of years) tends to produce noisy regimes whose parameters and inflation dynamics are difficult to interpret economically. 26For full details on how our three models are estimated, we refer the reader to the articles cited in the first paragraph of Section 3 and only provide a brief discussion here for completeness. 15

Estimation of the second, noncommon breaks, model proceeds in the same manner as for the baseline model, except it includes an additional Metropolis-Hastings step that estimates the exact break location for each series in the cross-section. Finally, estimation of the third, grouping, model combines the first step of estimating the baseline model with a second reversible jump step that introduces the number of groups G as a parameter in the model and repeatedly attempts to ‘jump’ to different values of G, with the proportion of iterations spent at each value of G approximating the posterior model probabilities. The series are ordered with the first N series in group 1 and so on. 1 The ordering of the variables, and hence their group allocations, are further estimated using a random walk Metropolis-Hastings algorithm. 4. Empirical Results Having introduced our data and estimation approach, we next turn to the empirical analysis. We begin with the industry-level data before turning to the MSA and EU country data. 4.1. Industry-level data We separately analyze two panel data sets on industry-level inflation, namely 16 PCE inflation rates and 31 CPI series. 4.1.1. PCE inflation rates We first estimate Phillips curves on quarterly sectoral data spanning the sample 1959-2022. Both the number of breaks and their location is very precisely estimated from the data: Our model assigns nearly 100% probability to the presence of two breaks with negligible uncertainty as to the timing of these breaks.27 Table 1 displays the baseline results for the 16 PCE industry-level inflation rates. The first of the two breaks is a steepening in the Phillips curve around 1972. Prior to 1972, the estimated slope of the Phillips curve is -0.51. This slope estimate steepens notably 27One might be concerned as to whether the identified breaks could be sensitive to the omission of time fixed effects for the sectoral data. While we cannot test this directly, we estimated models with and without time fixed effects for our MSA and EU country data examined below. For both data sets, we found that the numberofbreaksandtheirlocationwerenotaffectedbythepresenceoftimefixedeffects. Whiletheprecision of our estimated break location may partly hinge on the assumption of no cross-sectional dependence in the residuals, we show in our robustness analysis that such dependencies are in fact quite weak in our data. 16

in the 1972-2001 regime to -0.87. Coupled with an AR coefficient of 0.37, this implies a dynamic slope of -1.38.28 Inflation volatility, computed as the square root of an industryweighted average of the individual σ2 estimates, is also notably higher in the 1972-2001 ik regime (2.90) than in the previous regime (1.60), consistent with major shocks to commodity prices and sharp shifts in inflation expectations accompanying the marked changes to the Federal Reserve’s monetary policy during this period. Our Bayesian panel model identifies a second break in the industry PCE data in 2001. Afterthisbreak,theslopeofthePhillipscurvebecomesinsignificantlydifferentfromzeroand inflation dynamics become notably less persistent with an AR(1) estimate of 0.12 compared with 0.37 in the regime prior to 2001. The estimated volatility of shocks to inflation is slightly lower in this regime (2.76 after 2001 versus 2.90 from 1972 to 2001) but remains well above that experienced in the first regime (1.60). The right-most column in Table 1 shows the equivalent panel estimate based on the full sample 1959-2022, i.e., for a conventional Phillips curve model with no breaks. At -0.24, the estimated full-sample slope shows that ignoring breaks results in a modestly steep Phillips curve. This estimate can be thought of as a weighted average of the slopes in the underlying regimes and so conceals the sharp differences in slope estimates across the more than six decades covered by our sample. Food and energy prices are known to be more volatile than prices in other (“core”) sectors. To examine the price dynamics in core industries, the second panel in Table 1 reports the Phillips curve slopes for a model estimated on all industries excluding food and energy. Excluding food and energy changes the slope during the 1972-2001 period from -0.87 to -0.53 which is notably flatter, but still quite steep. Moreover, this estimate continues to be steeper than that in both the first regime (1959-1972), which equals -0.35, and in the last regime (2001-2022) which equals 0.09 and is insignificant at the conventional 5 percent level. Excluding food and energy thus flattens the slope of the Phillips curve but the evidence of a steeper unemployment-inflation trade-off in the “middle regime” (1972-2001) continues to be strong. To help interpret the underlying drivers of these breaks, we also report results separately after pre-assigning the individual price indices into goods and services groups. For both of these groups we obtain a similar pattern in slope coefficients with a steepening in 1972 and a flattening in 2001. However, the shifts in the estimated slopes is much sharper among the 28The dynamic slope refers to the long-run effect of a sustained unit change in the unemployment gap. 17

goods sectors (third panel in Table 1) as compared to the services sectors (bottom panel). Specifically, for the goods sectors the slope coefficient steepens from -0.62 in the first regime (1959-1972) to -1.16 between 1972 and 2001, only to flatten to a statistically insignificant (and wrong-signed) value of 0.57 after 2001. For the services sectors, the corresponding slope estimates for the three regimes are -0.35, -0.59, and -0.02 with the last estimate again being statistically insignificant. In addition, the goods and services slope coefficients are significantly different from one another in the first two regimes (indicated through the bold font of the services slope), but not in the final regime. Across all data sets examined in Table 1, the full-sample estimates (reported in the right-most column) imply a markedly flatter Phillips curve than the curve implied by the estimates in the first two regimes, 1959-1972 and 1972-2001. The reason for this is that the Phillips curve essentially becomes flat in the last period (2001-2022) which, when pooled with the earlier samples, flattens the curve. Ignoring breaks would therefore lead to the wrong conclusion of a rather flat Phillips curve and conceal the more complex story that, while quite flat during the last twenty years, the Phillips curve has, historically, been quite steep, especially during the nearly three decades 1972-2001. The full-sample estimates also show that ignoring breaks conceals the significant differences between the goods and services slopes that we find prior to 2001. The disaggregate results in Table 1 assigns industries to a set of pre-determined groups. Our unobserved grouped heterogeneity model in Equation (5) instead endogenously assigns industries to groups. Table 2 displays parameter estimates, along with the posterior mode groupallocation, fromapplyingthisapproachtothe16industryPCEseries. Withinallthree regimes, our approach identifies two groups with very different Phillips curve estimates. The reallysteepPhillipscurveinthe1972-2001subsampleisconcentratedinagroup(“Group1”) that includes Gasoline and other energy goods along with Financial services and insurance and NPISH. In fact, the behavior of the slope coefficient for Gasoline and other energy goods is so different from that of the other industries that this is the only sector to be included in Group 1after2001anditisonlygroupedtogetherwithFinancialservicesandinsuranceandNPISH in the 1972-2001 regime. This narrow, unbalanced grouping only happens when the behavior of a very small number of individual industries is truly different from that of the remaining industries. This point is further highlighted by the extremely high volatility estimate (39.79) for the Gasoline industry in the 2001-2022 regime which is more than twenty times higher than that of the other industries (1.95). 18

Using our baseline panel break model, the black line in the top panel of Figure 1 plots the posterior mean of the Phillips curve slope within each of the three regimes with the blue bands denoting 95 percent posterior intervals. These bands are clearly narrower in the first regime and widest in the last regime after 2001. To illustrate the value of using crosssectional information to estimate the Phillips curve, the red dotted lines plot industry-level estimates of the slope coefficients estimated separately for the three regimes identified by our panel breakpoint model and reported in Appendix Table A2. Two points stand out. First, consistent with the estimates in Table 2, we see strong evidence of variation in the industrylevel Phillips curves both over time and across industries. The majority of industries have a significantly negative slope coefficient on the lagged aggregate unemployment gap in the first regime: 10 of 16 slope estimates are negative and significant (at the five percent level) for 1959-1972. In contrast, no more than four industries generate a significantly negative slope coefficient in either the final regime (2001-2022) or for the full sample (1959-2022). Second, we see that the individual industry PCE Phillips curves are imprecisely estimated with estimates covering a wide range of values that fall outside the 95% confidence band for our panel estimates. This demonstrates the value of using cross-sectional information to estimate the Phillips curve in a panel setting. Figure 2 displays the posterior mode break dates for the 16 PCE industries obtained from the generalized version of the baseline model displayed in Equation (3). This model allows the break timing to vary across industries as described in Section 3.2. Industries that are hit first appear further to the left while industries hit later show up on the right in this figure.29 The top and bottom panels show results for the 1972 and 2001 breaks, respectively. The earliest industries to be hit by the 1972 break are Financial Services, Food and Beverage, and NPISH. For these industries, the Phillips curve breaks in 1972Q3. Gasoline, Food Services and Other Services follow suit in 1972Q4. Slightly less than half of all the industries (six) are affected by the break in 1973Q3, a full year later than the first-hit industries. Overall, the impact of the 1972 break to the Phillips curve took six quarters to percolate through the economy. The 2001 break hits Gasoline very early (2001Q1) followed by Motor Vehicles and NPISH a year later (2002Q1) and the remaining 13 industries the following quarter (2002Q2). Thus, as for the first break an energy sector (Gasoline) is hit early, but there is far less dispersion in the timing of the break across industries for the 2001 break compared to the 1972 break. 29The vertical ordering of industries is arbitrary. 19

4.1.2. Univariate results It is important to emphasize that our ability to detect breaks in the Phillips curve is closely linked to our use of panel data in conjunction with the assumption that both the timing of breaks and their impact on individual series is relatively homogeneous, i.e., there is a strong common component in the breaks. To highlight this point, we undertook a set of univariate Phillips curve regressions on the individual inflation series using the breakpoint methodology of Chib (1998). We fail to identify a single break in any of the PCE inflation series. Next, we dispensed with the assumption of homogeneous slope coefficients, imposing only that the timing of the break is identical across all variables in the panel. Once again, we fail to find evidence of breaks to the Phillips curves. These results show that our ability to identify breaks in the Phillips curves hinges on the ability of our panel estimation approach to efficiently exploit multivariate information in a way that takes advantage of the relative homogeneity in both the timing and impact of the breaks across industries. This increases the power of the panel break tests compared with univariate approaches or approaches that rely on heterogeneous panels. Next, we evaluate the ability of the frequentist breakpoint approach of Bai et al. (1998) to detect breaks in the Phillips curve in the settings we consider. Since their approach only permits a single break, the most direct comparison is with the data sets for which we identify just one break, namely, the MSA- and EU country data. Their approach, which assumes heterogeneous slope coefficients, does not detect a break in either data set, echoing what we find when applying our approach with heterogeneous slope coefficients and no pooling across variables. Indeed, the ability of our approach to exploit cross-sectional commonalities in the timing and impact of breaks by (fully or partially) pooling parameters accounts for the additional power our approach has to identify breaks. 4.1.3. CPI industry inflation We next examine the results for the 31 CPI industry-level quarterly inflation rates (1954- 2022). Once again, there is very little uncertainty about the number and timing of breaks and our model identifies two breaks–corresponding to three regimes–with posterior modes in 1971 and 2001. Table 3 displays the baseline results for the 31 CPI industry-level quarterly inflation 20

rates (1954-2022). The first of the three regimes (1954-1971) has a Phillips curve slope of -0.77, somewhat steeper than that obtained for PCE inflation (-0.51) in the regime ending in 1972. The second regime (1971-2001), has a very steep Phillips curve with an estimated slope of -1.56, while the third regime has an almost completely flat Phillips curve with an insignificant slope estimate of 0.02. Autoregressive dynamics are generally quite weak with estimates of 0.09 in the first and last regime and 0.22 in the middle regime. Table 3 also reports results on the model that excludes food and energy prices (second panel). Here, we find a pattern of a Phillips curve that flattens across both breaks in 1971 and 2001. While the slopes of the CPI Phillips curves fitted to core and all prices are similar in the first and third regimes, the core CPI Phillips curve is noticeably flatter than the curve fittedtoallpricesinthemiddleregime(-0.49versus-1.56). Thethirdandfourthpanelsshow inflation estimates generated separately for goods and services. In the two regimes prior to 2001, the Phillips curve is significantly steeper for goods than for services. Conversely, the slope of the Phillips curve is insignificantly different from zero in the last regime for both goods and services. Table 4 displays the results from the unobserved grouped heterogeneity model that uses the 31 CPI industries.30 Our approach identifies a single group in the first regime but two groups in the second and third regimes. In the second regime (1971-2001), there is a group with an especially steep Phillips curve (slope estimate of -1.82 versus -0.26 for the other group) which includes energy and some food components of CPI. In the third regime (2001-2022), two groups are again identified, both with flat Phillips curves and only the first group generates a significant slope estimate. While certain food and energy items are again overrepresented in the second (smaller) group of industries identified for this regime, others are included in the first group and, as a result, the group structure in the third regime is quite different from that in the second.31 Appendix Table A3 examines the heterogeneity in further detail by estimating univariate Phillips curve time-series regressions separately for each of the three regimes identified by 30Some industries do not have inflation data in the early parts of our sample and so cannot be allocated to a group. These show up as missing observations in the first two regimes in the table. 31Our approach allows all parameters to shift across regimes and clusters. Which cluster a particular industry gets assigned to will therefore depend on its persistence, slope, and volatility parameter. For example, in the third regime, group 1 has an AR slope of 0.30 with a t-statistic of 12.51 while Group 2 has a comparatively modest AR slope of 0.08 with a t-statistic of 1.92. Group 1 therefore tends to consist of industries with more persistent inflation dynamics. Similarly, among the food industries, those allocated to group2(thehigh-volatilitycluster)havevolatilityestimatesof6.84(Meats),7.83(Dairy),7.80(Fruits),and 25.65(Foodatemployeesites). Theremainingfoodsectorshavemuchlowervolatilityestimatescloseto2.5. 21

our panel break method as well as for the full sample. Many of the CPI series are not available in the first two regimes which limits the comparisons across time and industries. Nevertheless, for 19 of the 23 industry CPI series for which we have estimates for both the middle and last regime, the Phillips curve is steeper in the former (1971-2001) than in the latter (2001-2022) period. This again is strong evidence of a flattening of the Phillips curve at the industry level. Figure 3 displays the posterior mode break dates for the 31 CPI industries based on the model that allows the break timing to vary across industries. Our findings are in line with what we found for the PCE industries: For the 1971 break (top panel), Food prices (Meats, Poultry, Fish and Eggs and Fruits and Vegetables) are the first categories to be affected in 1971:Q3, followed by food items and various energy sectors whose break date is estimated to occur in 1972. For the majority of industries, the break date is 1974:03, a full three years after the first sectors are affected, suggesting that it took a very long time for this break to percolate throughout the economy. The 2001 break (bottom panel) initially affects fuel sectors (Motor Fuel and Utility (piped) gas service) in 2001:Q1, followed by Fuel oil and other fuels and various food industries. Once again, the impact plays out over three years with the vast majority of industries experiencing the break only in 2004:Q1. 4.2. MSA-level data The top panel of Table 5 displays the baseline results for the 22 annual MSA-level inflation rates (1980-2022).32 We identify a single break in 2000, with a marginal flattening of the Phillips curve which goes from a pre-break slope estimate of -0.29 to a post-break estimate of -0.25, with both beinghighlysignificant.33 Thepersistenceofthe inflationprocess, measured through the autoregressive parameter, increases significantly from 0.16 before the break to 0.35 afterwards.34 The MSA data suggests a much flatter slope of the Phillips curve in the pre-2000 period than that identified with either PCE or CPI sectoral data. There are a number of reasons for this. First, the sectoral Phillips curve is particularly steep in the period after the early seventies and the MSA data only starts in 1980. Consistent with this, sectoral Phillips curves 32The number of breaks and break dates are, once again, very precisely estimated. 33The slope, scaled by the expenditure share on non-tradeables (as discussed in subsection 2.3) flattens from -0.43 before the break to -0.37 after it. 34Our results are robust to the inclusion of regional inflation expectations from the Michigan survey. 22

areflatterifestimatedonlyondatastartingin1980. Second,theMSAdataareobservedonly at the annual frequency whereas sectoral data are quarterly, further attenuating the impact of the periods that experienced the steepest inflation-unemployment dynamics. Third, the MSA-level results apply to the slope of the regional Phillips curve and the implied national PC is steeper as we noted earlier. To examine a possible source of breaks to the Phillips curve, the middle panel in Table 5 displays results when, conditional on the regimes identified by the baseline model, we estimate the Phillips curve regression separately for MSAs located in states with below and above median rates of import penetration from China based on the state-level import penetration rates calculated by Riker (2022).35 We find that the flattening of the Phillips curve is concentrated in cities with above-median rates of import penetration. Specifically, whereas the slope of the Phillips curve changes only marginally from -0.19 to -0.18 for MSAs with below-median import penetration from China, it declines from -0.41 to -0.29 in cities with above-median import penetration from China. This finding lends credence to the role of international trade as an explanation for the flattening of the Phillips curve. Moreover, the slope coefficients for the two groups are significantly different from one another in both regimes and in the full-sample results.36 In Appendix Table A4 we report the results from a series of univariate MSA-level Phillips curve regressions on the two regimes identified by our benchmark model, i.e., 1980-2000 and 2001-2022, as well as for the full sample, 1980-2022. Importantly, when we conduct the break-point estimation for these univariate series, i.e., for individual MSAs, we fail to find significant evidence of breaks for any of the series. This reflects the weak power of break tests conducted on individual (univariate) time series which fail to exploit information in the cross-section to identify breaks. However, we can still use the breaks identified by our panel model to examine evidence of time-variation in Phillips curve slope estimates across time and cities. In fact, we observe stark differences over the two samples. In the early sample (1980-2000), 12 of 19 estimates are negative while, conversely, in the late sample (2001-2022) 18 of 22 slope estimates are positive.37 Moreover, 15 of 19 coefficient estimates increase in 35Riker (2022) estimates these values using a structural econometric model that exploits data on the location of import entry, domestic shipments, and distances between states. The MSAs that comprise the below median group are Detroit-Warren-Dearborn, MI, Dallas-Fort Worth-Arlington, TX, Denver-Aurora- Lakewood, CO, Philadelphia-Camden-Wilmington, PA-NJ-DE-MD, and St Louis, MO-IL. 36Many MSAs have missing data from the 1980s. This makes it difficult for our endogenous clustering approach to identify groups of cities with distinctly different Phillips curve dynamics, so we do not apply our grouping approach to the MSA-level data. 37For three of the 22 series we do not have sufficient data to estimate the regression in the first sample 23

the second regime compared to the first regime. This is strong evidence of a flattening of the Phillips curve estimated from MSA-level price data. OtherstudieshavefoundevidenceofaflatteningPhillipscurve. Usingasimplesubsample split on state-level data, Hazell et al. (2022) find that the Phillips curve flattens post-1990, but not significantly. We estimate our breakpoint approach using their data and corroborate theirfindingswithasinglebreak, exceptthebreakdateisestimatedtooccurin2000, aligned with the date estimated from our other data sets. In both regimes, we observe a flat Phillips curve with the slope estimate being marginally significant in the first regime (1978-2000) but insignificant in the second (2001-2017).38 Of course, their data is core CPI excluding shelter and so it is not surprising that the slope is much flatter than when we estimate it from the MSA-level data since shelter is one of the most cyclically sensitive categories (Stock and Watson 2020) and comprises more than 40 percent of core CPI. 4.3. Nonlinear Phillips Curve Testsforbreaksareconductedinthecontextof, andconditionalupon, themaintainedmodel specification, in our case a linear Phillips curve model. It is possible that our findings on the presenceofbreakstothismodelreflectomittednon-linearitiesintheinflation-unemployment trade-off. Previous studies such as Babb and Detmeister (2017) have in fact identified nonlinearities in the Phillips curve. To examine this possibility, while still allowing for the possibility of breaks, we generalize the linear Phillips curve specification in Equation (1) to allow the unemployment-inflation trade-off to have a kink at a pre-specified threshold, θ, so that for regimes k = 1,...,K +1 and t = τ +1,...,τ , k−1 k π = α +γ +ρ π +λ URATE +ω (URATE −θ)1 +(cid:15) . (14) it i t k it−1 k it−1 k it−1 URATEit−1<θ it Note that there is no discontinuity at the threshold point (θ) but the nonlinearity (“kink”) can differ across regimes. This allows us to examine if nonlinearities were more or less important in regimes with a steep or flat Phillips curve. and so are left with only 19 MSAs. 38Specifically, using the baseline model in Equation (1) on the state-level core CPI data from Hazell et al. (2022), we regress the 34 state-level nontradeables inflation rates from 1978 through 2017 on the lagged state-level unemployment rates and an autoregressive term while allowing for two-way fixed effects. The four-quarter inflation rate in each year is regressed on the four-quarter inflation rate in the previous year and the average monthly unemployment rate computed over the 12 months of the previous year. 24

Becauseweonlyhaveaggregatemeasuresofslackforthesectoral(PCEandCPI)data,we cannot estimate the model in Equation (14) on these data. Conversely, the MSA-level data has city-level unemployment rate data and so can be used to examine nonlinear (threshold) effects. The lower panel of Table 5 displays results from estimating Equation (14) with threshold valuesfortheunemploymentrate(θ)of5%(toppanel)and4.2%(secondpanel),respectively. These are the thresholds considered by Babb and Detmeister (2017).39 In both regimes we find evidence of a significant and economically large kink in the Phillips curve. For example, in the post-2000 regime, at an unemployment rate below 4.2 percent, the slope is -0.53 versus -0.23 at higher unemployment rates. The estimated size of the post-2000 kink is notably bigger than the kink estimated without allowing for any regime change (-0.30 versus -0.19). This is consistent with a time-varying non-linearity and shows that, at least for the MSA data, thePhillipscurvehasbecomenotablysteeperafter2000atlowlevelsofunemployment. This finding is clearly relevant to current policy debates about costs and benefits of a hot labor market. 4.4. Wage Phillips curves The top panel of Table 6 displays results for the wage Phillips curve when using the 51 state-level (including the District of Columbia) quarterly wage inflation rates from 1980 through 2019. We identify a single break in 1989Q4, earlier than the break dates found for price Phillips curves. This break results in a flattening of the wage Phillips curve with the estimated slope falling from -0.46 to -0.34 which is less dramatic than for some of the price Phillips curves. This smaller and earlier flattening of the wage Phillips curve–relative to the price Phillips curve–may reflect factors like declining unionization and better anchored inflation expectations, both of which were well underway around 1990. Conversely, the flattening of the price Phillips curve is likely to be more sensitive to the expansion of trade around the turn of the century. Table 6 also considers the same kind of non-linearity in the wage Phillips curve that we earlier examined for prices at the MSA level in Table 5, i.e., using thresholds of 5% (middle panel) and 4.2% (bottom panel). For both threshold values, we find very strong evidence of a much steeper wage Phillips curve in tight labor markets. For example, the slope of the 39We condition on the two regimes identified by the Phillips curve fitted to the MSA-level inflation data, but the break dates remain the same if we estimate the model augmented to allow for the kink. 25

Phillips curve after 1990 is -0.30 when the unemployment rate exceeds 4.2% but, at -0.93, is three times steeper when unemployment falls below this level. Moreover, for this data, the steep threshold effect holds in both the early and late regimes and is even slightly stronger in the earlier data. 4.5. EU Data Table 7 displays the baseline results for the 28 EU country-level annual inflation rates (1986- 2021). Our model identifies a single break in 2004. Before the break, the slope of the Phillips curve is -0.72 with an AR coefficient of 0.10, implying a dynamic slope coefficient of -0.8. After the break, the slope coefficient declines to -0.09 with an AR coefficient of 0.48, implying a dynamic slope coefficient of -0.17. Both estimates are significant, so the inflationunemployment trade-off remains valid, but the Phillips curve becomes much flatter after the break.40 Grouping the countries into “rich” and “poor” nations, based on whether GDP per capita deflated by PPP is above or below average, we find that the poor countries had a significantly steeper Phillips curve in the first subsample (1986-2003), whereas the slopes of the Phillips curve are the same, and flat, for the two groups in the second subsample (2004-2021). This finding is consistent with the greater goods and labor market integration of countries in Southern and Eastern Europe seen in recent years. To further track how the heterogeneity in Phillips curve slopes evolves over time, the black line in the lower panel of Figure 1 graphs the evolution of the posterior mean of the EU Phillips curve slope over time with blue bands denoting the 95 percent posterior interval and red lines tracking the univariate Phillips curves estimated country-by-country. Uncertainty about the panel estimate of the Phillips curve slope is much stronger in the first regime and considerably smaller after 2001. As for the PCE inflation series, we see that the individual country-level Philips curves are imprecisely estimated and often fall outside the 95% posterior interval. The lower panel of Table 7 displays results for the full sample, as well as separately in the two regimes identified by the baseline model, for a Phillips curve that uses either country-level total goods inflation, or total services inflation as the dependent variable. The flattening of the Phillips curve after 2004 is apparent for both goods and services inflation. 40Ourresultsarerobusttotheinclusionofcountry-levelinflationexpectationsfromConsensusEconomics when conditioning on those observations for which inflation expectations data from this data source are available. 26

Interestingly, while flatter in absolute terms, the Phillips curve remains significantly steeper for services inflation in the second regime (slope estimate of -0.15 versus -0.07), consistent with what we found for the US. For the full sample, the Phillips curve estimated on services inflation is significantly steeper than the curve estimated on goods inflation (slope estimates of -0.19 versus -0.11). Table 8 uses the group heterogeneity model that endogenously determines if there are differences in individual countries’ Phillips curves and how these are affected by breaks. We identify two groups of countries in the first regime that ends in 2003.41 One group (labeled group 1 in the table) has a relatively steep Phillips curve before 2004 with a slope of -0.39. This cluster mainly comprises countries on the European periphery, including Bulgaria, Estonia, Ireland and Portugal. The second group has a much flatter Phillips curve with an estimated slope of -0.05 which fails to be significant. In the post-2004 regime, we identify a single group, whose estimated slope is -0.09. To further understand this heterogeneity, Appendix Table A5 reports country-level estimates of the Phillips curve coefficients estimated separately for the two regimes (1986-2003, 2004-2021) and for the full sample (1986-2021). Only three countries (the Netherlands, Finland, and Cyprus) generate a significantly negative estimate over the full sample (1986- 2021), versus five countries in the early period and three countries for the regime that starts in 2004. This demonstrates two important points. First, Phillips curves are poorly identified using inflation series at the individual country-level. Possible explanations for this include non-stationarities in the data and the relatively small samples (at most 36 observations) available at the annual frequency. Second, break tests conducted at the univarate level tend to have weak power. As for the U.S. data, break tests conducted at the individual country level based on the break estimation methodology proposed by Chib (1998) fail to find significant evidence of breaks for any of the countries. This point is linked to the large estimation errors associated with the country-level Phillips curves and shows up in the form of quite large variation in coefficient estimates across the two regimes for individual countries.42 In summary, our estimates for the EU Phillips curve suggest that in addition to a flattening of the Phillips curves, there has been some “convergence” in Phillips curve slopes with the flattening being most pronounced in countries that previously had steep Phillips 41Inflation in Romania contains extreme outliers during the post-Communist transition, so this country is in a group of its own during the early sample. We simply mark it as missing in the table. 42We also fail to identify breaks in a panel break model with heterogeneous slope coefficients. Hence, it is exploiting cross-sectional information and pooling parameters that generates break detection power. 27

curves. European integration thus appears to have been associated with a convergence of the slopes of country-level Phillips curves, consistent with what we found for “rich” and “poor” countries in Table 7 above. To examine possible nonlinearities in the EU Phillips curve, the results displayed in the final panel of Table 7 allow for a single kink at an unemployment gap threshold below -1.5% as in Equation (14). Under normal labor market conditions, we find a significantly negative and very steep Phillips curve in the first regime (1986-2003) with a slope estimate of -1.17. Conversely, the Phillips curve in a tight labor market (UGAP < −1.5%) is poorly identified in this subsample, likely because Europe had so few cases with very tight labor markets in this time period. Turning to the second regime (2004-2021), there is a significant steepening of the Phillips curve which goes from -0.02 (flat) to -0.60 (steep) in tight labor markets. The full sample kink (-0.13) is insignificant, underscoring again the insights from considering nonlinearity and structural stability jointly. These findings are consistent with the US findings and support the presence of a Phillips curve trade-off over the last two decades but only in tight labor markets. Figure 4 displays the posterior mode break dates for the 28 EU countries based on the generalized version of the baseline model in Equation (2) that allows the break timing to vary across countries as described in Section 3.2. As in the earlier figures, circles to the left (right) indicate countries that are affected first (last) by a break. The vertical array of circles to the far left of the figure comprises all the advanced, early EU members which, thus, are affected first by the break to the Phillips curve in 2003. Countries such as Slovakia, Slovenia, and Hungary follow in 2004. With a five-year delay, Romania and Bulgaria are the last countries to exhibit flattening of their Phillips curves. The Bayesian algorithm has no knowledge of the timing of EU accession, but it is noteworthy that these two countries were the last to join the EU, in 2007. This points to EU membership, and the associated trade linkages and freedom of movement of labor, as possible factors associated with the observed flattening and convergence of the Phillips curves. 5. Aggregate Implications Inthissectionweexploreaggregateimplicationsofourfindings,includingevidenceofmissing disinflation, the recent inflationary surge, and implications of breaks in the Phillips curve for optimal monetary policy and the Brainard (1967) conservatism principle. 28

5.1. Missing disinflation, reinflation and the recent inflationary surge The top panel of Figure 5 displays the Phillips curve fit from our linear MSA breakpoint model (black line). Specifically, in each year this is our prevailing regime-specific MSA regional linear Phillips slope coefficient divided by the nontradeables share and multiplied by the lagged national unemployment rate gap. The red line graphs the annual national headline CPI inflation rate minus long term inflation expectations, which are 10-year ahead SPF CPI inflation expectations.43 If there were missing disinflation during the Great Recession we would expect to see the black line run below the red line. Likewise, we would expect the black line to run above the red line if there were a missing reinflation during the recovery years following the Great Recession. In fact, the black line tracks the red line closely and so there appears to be little evidence of missing disinflation and missing reinflation according to the fit of our Phillips curve model. This echoes the results of Ball and Mazumder (2019) and Hazell et al. (2022). The dotted black line graphs the implied estimates from our nonlinear Phillips curve estimates in the second regime using MSA-level data and increasing the noncyclical rate of unemployment (NROU) in 2021 and 2022 to the estimates from Crump et al. (2022) who argue that NROU has risen since COVID-19 due to unusually strong wage growth.44 The dotted black line increases just under half as much as the red line between 2020 and 2022, suggesting that a steeper Phillips curve in hot labor markets combined with a higher NROU can explain a bit less than half of the recent inflationary surge. We repeat this exercise for the EU and display the results in the lower panel of Figure 5. The black line uses estimates from our linear breakpoint Phillips curve model. The red line uses the EU inflation rate and long term (five-year ahead) Eurozone inflation expectations from the ECB SPF which goes back to 2002 Q3. Prior to this, we use one-year ahead expectations, going back to 1999 Q1.45 We average expectations across the four quarters in a given year. Once again, we see little evidence of missing disinflation during the Great Recession or missing reinflation during the subsequent recovery. 43Missing observations prior to 1991 Q4 are filled using linear interpolation. 44Specifically, we use their 5.9 percent estimate in 2021 and a value of 5.6 percent in 2022 which is about the middle of their range of forecasts. 45Eurozone expectations data are sourced from the ECB statistical data warehouse. 29

5.2. Optimal monetary policy We next examine the implication of regime shifts in the Phillips curve for optimal monetary policy. We consider a standard model consisting of a Phillips curve and an IS curve: u = β u +β i +(cid:15) , t u t−1 i t−1 t,u π = γ u +γ π +(cid:15) , (15) t u t−1 π t−1 t,π where u is the unemployment gap, π is the deviation of inflation from steady state (all t t constants are dropped), and i is the central bank’s policy rate at time t. Following common t assumptions, the objective of the central bank is to minimize E(u2+π2) using a rule of the t t form i = ρ u +ρ π . By substitution this implies a VAR of the form: t u t−1 π t−1 x = Ax +(cid:15) (16) t t−1 t where (cid:18) (cid:19) β +β ρ β (ρ −1) A = u i u i π γ γ u π and (cid:15) = ((cid:15) ,(cid:15) )(cid:48) is N(0,Σ) with Σ = diag(σ2,σ2). t t,u t,π u π The loss function of the central bank can be written as ω +ω where Ω = [ω ] is the 11 22 ij unconditional variance of x and Ω solves the equation Ω = AΩA(cid:48) +Σ. We can plug in the t draws from the posterior for the parameters (β ,β ,γ ,γ ,σ2,σ2) and then find the choice u i u π u π of ρ and ρ that minimizes this loss. u π WeobtaintheIScurvebyestimatingaBayesiantimeseriesregression(withnobreaks)of the quarterly percent U.S. national unemployment gap on an intercept, its own one quarter lag, and the real federal funds rate lagged one quarter.46 The IS curve estimates (and tstatistics) for the intercept, autoregressive term, and lagged real federal funds rate are 0.25 (1.02), 0.69 (5.85), and 0.02 (0.33).47 We combine these estimates with our baseline Phillips curve estimates using MSA-level data for the US and country-level data for the EU to gauge how the break in the Phillips curve affects optimal monetary policy. To evaluate the policy impact of allowing for breaks, we compare results based on (i) the Phillips curve estimates 46WeusethenoncyclicalrateofunemploymentsourcedfromFRED.Therealfederalfundsrateismeasured as the nominal four-quarter average federal funds rate minus the four-quarter headline CPI inflation rate in percent. 47To isolate the impact of the break in the Phillips curve, we preclude breaks in the IS curve and hold the IS curve estimates fixed across our US and EU calculations. 30

from our baseline Bayesian breakpoint model, and (ii) the full-sample model that precludes breaks. We compute the optimal monetary policy coefficients by simulating 20,000 posterior draws and use a 200 × 200 grid search. The top panel of Table 9 displays results for the U.S.. Relative to the no-break framework, allowing for breaks induces more parameter uncertainty which, in turn, causes the policymaker to respond less aggressively to deviations in the unemployment gap (-0.68 versus -0.88) because they are less certain about the relation between the unemployment gap and inflation. This is in line with Brainard (1967)’s conservatism principle and the opening quote from the ECB president Christine Lagarde. The policymaker compensates for this additional caution by responding more aggressively to inflation deviations.48 Indeed, this has been the action taken by many central banks during the re-opening of the economy after pandemic-induced lockdowns. Specifically, for the break and no-break models we use the corresponding Phillips curve posteriors for our model or the same model that precludes breaks and combine this with the estimates of the U.S. IS curve. For the MSA data, the uncertainty surrounding the PC slope is nearly two-thirds (65%) higher than for the PC slope estimate from the no-break model. To further illustrate how parameter uncertainty depends on the presence of breaks to the Phillips curve, the top panel of Figure 6 displays density plots of the Phillips curve slope coefficients in the first (black line) and second regimes (green line) estimated from our baseline breakpoint model using MSA-level data against the corresponding plot from the same model that precludes breaks (red line). Parameter uncertainty is greater in both regimes as compared to the full-sample estimate. For the final row in the table, we simply allow the mean to shift according to how much the Phillips curve flattens across regimes (adding this mean difference to the posteriors), but keep the variance of the posteriors fixed. Here we see essentially no difference in optimal policy relative to the no-break case, suggesting that it is not the break in the mean of the Phillips curve slope itself, but the additional parameter uncertainty caused by the break, that affects optimal policy. The bottom panel displays results for the EU. Here we find the same pattern as for the US. Namely, that allowing for breaks causes the optimal monetary policy to respond less aggressively to deviations in the unemployment gap (-0.65 versus -1.01) and to compensate 48As noted by S¨oderstr¨om (2002), once parameter uncertainty in the persistence parameter is allowed for, the policymaker may respond more aggressively to additional uncertainty. Sack (2000) and Rudebusch (2001) also consider the effect of uncertainty on the aggressiveness of optimal monetary policy responses. 31

for this by responding more aggressively to deviations in inflation. For the EU data, the uncertainty surrounding the PC slope is 141% higher than for the PC slope estimate from the no-break model. The lower panel of 6 displays that parameter uncertainty is actually lower in the second regime compared to the full-sample estimate, and so the effect on optimal monetary policy is driven by the increased parameter uncertainty in the first regime. Once again, a break in the mean of the Phillips curve slope alone has no impact. The magnitude of our estimates of the effect of breaks in the Phillips curve on optimal monetary policy should be viewed as a lower bound because they are estimated ex-post using the full data sample. In practice, the central bank must set policy in real time. They would therefore face additional uncertainty regarding whether a break has occurred any time they observe a data realization in the tail of the distribution. In the no-break model, policy makers would only update their posteriors of the parameters of the distribution, but in our framework they now have to decide whether there has been a break. If a break has occurred, pre-break data becomes less informative, inducing a large spike in uncertainty. 6. Robustness checks In this section, we perform a number of robustness checks on our results. Specifically, we first consider the possible effect on our panel break estimates of serial correlation or crosssectionalerrordependenceintheresiduals. Next, weevaluatewhetherourpanelbreakmodel betterfitsthedatathanatime-varyingparametermodelwithsmoothly-evolvingcoefficients. Finally, we consider the effect of adopting alternative specifications for the priors. 6.1. Serial correlation Serialcorrelationintheresidualsofourmodelcouldpotentiallyresultinmisleadinginference. Across all four data sets, the top panel of Appendix Table A6 shows that the p value of the Durbin and Watson (1950) test statistic fails to reject the null hypothesis of no serial correlation in the Phillips curve residuals within every regime across the four data sets (CPI and PCE sectoral, MSA-level, and EU country-level) we consider in our analysis. 32

6.2. Cross-sectional error dependence Next, we consider the possibility of cross-sectional error dependence in the residuals from our model. In applications with reasonably large cross-sections, weak dependence or dependence that is confined to a relatively small number of series will not pose serious estimation and inferential problems and only pervasive cross-section dependence is problematic (Pesaran 2015). Moreover, if cross-sectional dependence is caused by unobserved common factors that are uncorrelated with the regressors, our estimator remains consistent, though some of the efficiency gains from pooling may be lost and the standard error estimates may be biased (Phillips and Sul 2003; Chudik and Pesaran 2013). We test for cross-sectional error dependence using the test statistic proposed by Juodis and Reese (2022) which is a bias-corrected version of the original CD test statistic proposed by Pesaran (2021). Results are reported in the lower panel of Appendix Table A6. We cannot reject the null hypothesis of no cross-sectional error dependence in any regime across the four data sets, although we are on the borderline of rejecting the null in the third regime for CPI data. If we exclude the Motor Fuel category in this regime when computing the test statistic, however, we cannot reject the null, suggesting that the cross-sectional error dependence is not pervasive. Reassuringly, the estimates from the CPI sectoral data in the third regime follows the same basic pattern as the PCE sectoral data (which has no crosssectional error dependence), namely a flattening curve in the final regime. We therefore conclude that any cross-sectional error dependence is insufficiently pervasive to cause serious inferential problems in our settings. 6.3. Time-varying parameters Finally, we compute Bayes factors for the baseline panel model with discrete breaks versus thesamemodelestimatedusingatime-varyingparameter(TVP)specification. Bayesfactors areconstructedusingthemarginallikelihoodofeachmodelcomputedusingthemethodology of Chib (1995), for our four price Phillips curve data sets (at the PCE and CPI sectorallevel, the MSA-level, and the EU country-level). Bayes factor values between 1 and 3 are inconclusive, values between 3 and 20 indicate positive evidence in favor of the restricted model, while values between 20 and 150 indicate strong evidence in support of the restricted model (Kass and Raftery 1995). The TVP model can be viewed through the lens of our breakpoint model, but imposes that a (typically small) break occurs every period. We do 33

not impose this assumption. Instead we estimate the number of breaks, specifying a prior on the regime duration that places relatively little weight on very short regimes and so our framework tends to reveal few (typically large) breaks. Across all four data sets we find Bayes factors above 20, suggesting strong evidence in favor of modeling time variation in the Phillips curve as discrete breakpoints rather than smoothly-evolving changes. 6.4. Alternative prior specifications Our analysis uses fairly uninformative priors on the key parameters of the Phillips curve such as ρ and λ , both of which are centered on zero. Effectively, this stacks the results k k against finding a steep Phillips curve, but we mitigate such effects by allowing for relatively large values of the prior variances σ2 and σ2. Because our priors are relatively uninformative, ρ λ changing the centering of λ has little impact on our results. k Priors can also be used to incorporate economic beliefs into the model. For example, truncating the prior on the slope of the unemployment rate at zero can be used to rule out positive values for the Phillips curve slope coefficient. Empirically, we find that truncating the prior has little impact on our baseline estimates of the price Phillips curve across all four data sets. Specifically, the truncation never binds for the MSA-level data and only binds for the CPI and PCE sectoral data sets in the final regime, causing their slope coefficients to turn negative (-0.17 and -0.07) but insignificantly different from zero. The truncation binds in each regime for the EU data but only on a relatively small number of posterior draws, causing the magnitude of the Phillips curve to steepen slightly without altering our conclusions in any way. Overall, truncating the Phillips curve slope coefficient at zero has little impact on inference in our study. Our prior on the break frequency is somewhat more informative and selected so that a break is expected to occur every 20 years. This means that our results tend to select relatively rare shifts in the Phillips curve which are likely to be of a more secular nature, representing “trend breaks”. 7. Conclusions In this paper, we examine time variation in the Phillips curve, applying new Bayesian panel methodswithbreakpointstopaneldatafromtheU.S.andtheEuropeanUnion. Ouranalysis uncovers a third reason for exploiting disaggregate data in the analysis of Phillips curves. 34

Specifically, whereas break tests conducted on individual (univariate) inflation series have insufficient power to detect breaks in the Phillips curves for any of the sectoral, MSA, or country series, in contrast, exploiting commonalities in the timing and impact of breaks on the cross-section of variables allows us to uncover strong evidence of breaks to the Phillips curve. Our Bayesian panel estimation approach allows us to estimate the number of breaks, their location, as well as the magnitude of the shift in both mean and volatility parameters. We also consider an extension that allow breaks to affect individual inflation series at different points in time so as to identify “lead-lag” relations in the cross-sectional diffusion of breaks. Finally, we consider a “partial pooling” approach that endogenously forms groups or clusters of inflation series, allowing the Phillips curves to differ across clusters (but assuming homogeneity within a particular cluster). This approach is more flexible than conventional panel data methods and yet more efficient than estimating separate time series regressions for each region or industry. Because the grouping structure is identified as part of the estimation process, our approach can adapt to the degree of heterogeneity in Phillips curve dynamics observed across industries, cities, or countries. Though our empirical results depend on the specific data under consideration, we identify a number of consistent patterns. First, we find evidence for up to two breaks; one in the early 1970s and the other around 2000. The Phillips curve steepened after the first break, and flattened after the second. Second, the flattening around 2000 is greater for goods than for services, is greater for MSAs with above median rates of import penetration from China than for MSAs with below-median rates, is greater in price Phillips curves than in wage Phillips curves, and is greater in comparatively poor EU countries than in rich EU countries. Third, we identify a distinct pattern of convergence in EU country Phillips curve slope coefficients, consistent with greater geographic mobility. Finally, we discuss implications of breaks to the Phillips curve for optimal monetary policy. Breaks to the Phillips curve increases uncertainty about parameter values and we find that this has a notable effect on the optimal policy of a central bank policy maker. Specifically, accounting for breaks leads the policy maker to respond less aggressively to deviations in the unemployment gap but, conversely, respond more aggressively to deviations from target inflation. 35

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Table 1: Quarterly 16 PCE industry-level inflation rates (1959-2022) 1959-1972 1972-2001 2001-2022 1959-2022 All industries PC -0.51∗∗∗ -0.87∗∗∗ 0.24 -0.24∗∗∗ AR 0.31∗∗∗ 0.37∗∗∗ 0.12∗∗∗ 0.26∗∗∗ vol. 1.60 2.90 2.76 2.90 All industries (ex. food and energy) PC -0.35∗∗∗ -0.53∗∗∗ 0.09∗ -0.16∗∗∗ vol. 1.44 2.39 1.67 2.19 Goods PC -0.62∗∗∗ -1.16∗∗∗ 0.57∗ -0.19 AR 0.12∗∗∗ 0.36∗∗∗ 0.16∗∗∗ 0.24∗∗∗ vol. 1.89 3.37 3.96 3.55 Services PC -0.35∗∗∗ -0.59∗∗∗ -0.02 -0.21∗∗∗ AR 0.57∗∗∗ 0.36∗∗∗ 0.40∗∗∗ 0.43∗∗∗ vol. 1.39 2.31 1.44 2.02 Note: Thetoppanelofthistabledisplaysestimatesoftheslopecoefficientsonthelaggedaggregateunemployment gap (PC) and the autoregressive term (AR) from the baseline model displayed in Equation (3). Significance at the 10, 5, and 1 percent levels are denoted by ∗, ∗∗, and ∗∗∗. The reported volatility (vol.) istheweightedaverageofthesectoral-levelvolatilityestimates,weightedusingthe2022:Q1expenditureweights. Thismodelregressesthe16PCEsectorquarterlyinflationratesfrom1959through2022 on an autoregressive term, the lagged aggregate unemployment gap, and the lagged long-term inflation expectations, including industry fixed effects. We display results for the three regimes identified by the model, and for the full sample (by estimating the model but precluding any breaks). The second panel displaysresultswhenfoodandenergysectorsareexcludedfromthemodel. Thethirdandfourthpanels displayresultswhenestimatingthesamemodelseparatelyforgoodsandservicessectors, whileprecluding breaks and conditioning on either the regimes identified by the baseline model or on the full sample. The goods group consists of Motor vehicles and parts, Furnishings and durable household equipment, Recreational goods and vehicles, Other durable goods, Food and beverages purchased for off-premises consumption, Clothing and footwear, Gasoline and other energy goods, and Other nondurable goods. TheservicesgroupconsistsofHousingandutilities,Healthcare,Transportationservices,Recreationservices, Food services and accommodations, Financial services and insurance, Other services, and NPISH. Values in bold font denote that the services PC is significantly different from the goods PC at the 95% confidence level. 40

Table 2: Grouped heterogeneity estimates: Quarterly 16 PCE industry-level inflation rates (1959-2022) 1959-1972 1972-2001 2001-2022 1959-2022 Parameter Estimates Group 1 PC -0.54∗∗∗ -2.23∗∗∗ 3.19 0.21 vol. 3.09 9.31 39.79 11.09 Group 2 PC -0.39∗∗∗ -0.47∗∗∗ 0.07 -0.18∗∗∗ vol. 1.21 1.89 1.95 1.93 Equal-weighted average slope -0.43 -0.80 0.27 -0.13 Group Allocation Estimates Motor vehicles and parts 1 2 2 2 Furnishings and durable household equipment 2 2 2 2 Recreational goods and vehicles 2 2 2 2 Other durable goods 1 2 2 2 Food and beverages purchased for off-premises consumption 1 2 2 2 Clothing and footwear 2 2 2 2 Gasoline and other energy goods 1 1 1 1 Other nondurable goods 2 2 2 2 Housing and utilities 2 2 2 2 Health care 2 2 2 2 Transportation services 2 2 2 2 Recreation services 2 2 2 2 Food services and accommodations 2 2 2 2 Financial services and insurance 2 1 2 1 Other services 2 2 2 2 NPISH 1 1 2 2 Note: The top panel of this table displays estimates of the slope coefficient on the lagged aggregate unemployment gap (PC) from the model that estimates an unobserved grouping structure as described in Section 3.3. Significance at the 10, 5, and 1 percent levels are denoted by ∗, ∗∗, and ∗∗∗. This model regressesthe16PCEindustry-levelquarterlyinflationratesfrom1959through2022onanautoregressive term, the lagged aggregate unemployment gap, and lagged long-term inflation expectations, including industryfixedeffects. Wealsoreporttheindustry-weightedvolatility(vol.) estimatewithineachgroup, using the 2022:Q1 expenditure weights. The model is estimated within the three regimes identified by thebaselinemodeldisplayedinEquation(3)thatusesthe16PCEsectorinflationrates, andforthefull sample. The lower panel displays the corresponding posterior mode group allocations. 41

Table 3: 31 CPI industry-level quarterly inflation rates (1954-2022) 1954-1971 1971-2001 2001-2022 1954-2022 All industries PC -0.77∗∗∗ -1.56∗∗∗ 0.02 -0.34∗∗∗ AR 0.09∗∗ 0.22∗∗∗ 0.09∗∗∗ 0.15∗∗∗ vol. 1.57 3.39 4.79 4.14 All industries (ex. food and energy) PC -0.78∗∗∗ -0.49∗∗∗ 0.07 -0.10∗∗∗ vol. 1.31 1.91 2.33 2.42 Goods PC -0.85∗∗∗ -1.96∗∗∗ 0.23 -0.36∗∗ AR 0.03 0.29∗∗∗ 0.02 0.09∗∗∗ vol. 1.56 5.18 9.78 7.13 Services PC -0.54∗∗∗ -0.59∗∗∗ -0.14 -0.27∗∗ AR 0.34∗∗∗ 0.56∗∗∗ 0.40∗∗∗ 0.43∗∗∗ vol. 1.58 2.37 1.91 2.42 Note: The top panel of this table displays estimates of the slope coefficients on the lagged aggregate unemploymentgap(PC)andtheautoregressiveterm(AR)fromthebaselinemodeldisplayedinEquation (3). Significance at the 10, 5, and 1 percent levels are denoted by ∗, ∗∗, and ∗∗∗. The reported volatility (vol.) is the weighted average of the sectoral-level volatility estimates, weighted using the October 2022 relative importance weights. This model regresses the 31 CPI sector quarterly inflation rates from 1954 through2022onanARterm,thelaggedunemploymentgap,andlaggedlong-terminflationexpectations, includingindustryfixedeffects. Wedisplayresultsforthethreeregimesidentifiedbythemodel, andfor thefullsample(estimatingthemodelbutprecludinganybreaks). Theotherpanelsdisplayresultswhen excluding food and energy sectors, and running the models separately for goods and services sectors, conditioning on either the regimes identified by the baseline model or on the full sample. The services group consists of Full Service Meals and Snacks, Limited Service Meals and Snacks, Food at employee sitesandschools, Foodfromvendingmachinesandmobilevendors, Otherfoodawayfromhome, Utility (piped)gasservice,Shelter,Waterandsewerandtrashcollectionservices,Householdoperations,Medical care services, Transportation services, Recreation services, Education and communication services, and Other personal services. The remaining sectors comprise the goods group. Values in bold font denote that the unemployment gap slope for services is significantly different from that of goods at the 95% confidence level. 42

Table 4: Grouped heterogeneity estimates: 31 CPI industry-level quarterly inflation rates (1954-2022) 1954-1971 1971-2001 2001-2022 1954-2022 ParameterEstimates Group1 PC -0.77∗∗∗ -1.82∗∗∗ -0.12∗∗ -0.16∗∗∗ vol. 1.57 10.83 4.79 3.29 Group2 PC -0.26∗∗∗ 0.38 -0.50∗∗∗ vol. 0.72 18.27 11.71 Equal-weightedaverage PC -0.77 -1.14 0.01 -0.29 GroupAllocationEstimates CerealsandBakeryProducts 2 1 1 Meats,Poultry,FishandEggs 1 1 2 2 DairyandRelatedProducts 1 2 2 FruitsandVegetables 1 1 2 2 NonalcoholicBeveragesandBeverageMatls 1 1 1 2 OtherFoodAtHome 1 1 1 2 FullServiceMealsandSnacks 2 1 1 LimitedServiceMealsandSnacks 2 1 1 Foodatemployeesitesandschools 2 2 2 Foodfromvendingmachinesandmobilevendors 2 1 1 Otherfoodawayfromhome 2 1 1 Fueloilandotherfuels 1 1 2 2 Motorfuel 1 1 2 2 Electricity 1 1 1 2 Utility(piped)gasservice 1 1 2 2 Householdfurnishingsandsupplies 1 1 Apparel 1 1 1 1 Transportationcommoditieslessmotorfuel 2 2 Medicalcarecommodities 1 2 1 1 Recreationcommodities 1 1 Educationandcommunicationcommodities 1 1 Alcoholicbeverages 1 1 1 1 Othergoods 1 1 Shelter 1 1 1 1 Waterandsewerandtrashcollectionservices 2 1 1 Householdoperations 2 1 1 Medicalcareservices 1 2 1 1 Transportationservices 1 1 1 2 Recreationservices 1 1 Educationandcommunicationservices 1 1 Otherpersonalservices 1 1 Note: The top panel of this table displays estimates of the slope coefficient on the lagged aggregate unemployment gap (PC) from the model that estimates an unobserved grouping structure as described in Section 3.3. Significance at the 10, 5, and 1 percent levels are denoted by ∗, ∗∗, and ∗∗∗. The reported volatility (vol.) is the weighted average of the sectoral-level volatility estimates within each group, weighted using the October 2022 relative importance weights. This model regresses the 31 CPI industry-level quarterly inflation rates from 1954 through 2022 on an autoregressive term, the lagged aggregate unemployment gap, and lagged long-term inflation expectations, including industry fixedeffects. Themodelisestimatedwithinthethreeregimesidentifiedbythebaselinemodeldisplayed in Equation (3) that uses the 31 CPI sector inflation rates, and for the full sample. The lower panel displaysthecorrespondingposteriormodegroupallocations. Missinggroupallocationsindicatethatthe corresponding series had no inflation observations in the regime and so was not assigned to any group. 43

Table 5: Annual 22 CPI MSA-level inflation rates (1980-2022) 1980-2000 2001-2022 1980-2022 All MSAs PC -0.29∗∗∗ -0.25∗∗∗ -0.23∗∗∗ PC (scaled) -0.42 -0.37 -0.35 AR 0.16∗∗∗ 0.35∗∗∗ 0.32∗∗∗ vol. 0.63 0.64 0.65 Above and below median rate of import penetration from China PC (above) -0.41∗∗∗ -0.29∗∗∗ -0.28∗∗∗ PC (below) -0.19∗∗∗ -0.18∗∗ -0.18∗∗∗ Kink at 5% or 4.2% U rate PC -0.28∗∗∗ -0.22∗∗∗ -0.21∗∗∗ Extra PC (Urate <5%) -0.16 -0.27∗∗∗ -0.19∗∗∗ AR 0.15∗∗∗ 0.33∗∗∗ 0.29∗∗∗ PC -0.29∗∗∗ -0.23∗∗∗ -0.23∗∗∗ Extra PC (U <4.2%) -0.17 -0.30∗∗∗ -0.19∗∗ AR 0.15∗∗∗ 0.34∗∗∗ 0.31∗∗∗ Note: The top panel of this table displays estimates of the slope coefficients on the lagged MSA-level unemployment rates (PC) and the autoregressive term (AR) from the baseline model that includes twoway fixed effects displayed in Equation (1). Significance at the 10, 5, and 1 percent levels are denoted by ∗, ∗∗, and ∗∗∗. We also report the slope scaled by the expenditure share on nontradeables to map the regionalPCslopeintothenationalPCslopeassuggestedbyHazellet al.(2022). Thereportedvolatility (vol.) is the equal-weighted average of the series-specfic volatility estimates. We display results for the two regimes identified by the model, and for the full sample (by estimating the model but precluding breaks). The middle panel displays corresponding results when, conditional on the regimes identified by the baseline model and for the full sample, we estimate the regression separately for those MSAs that correspond to states with above or below median rates of import penetration from China based on the state-level import penetration rates estimated by Riker (2022) who estimates these values using a structural econometric model that exploits data on the location of import entry, domestic shipments, and distances between states. The MSAs that comprise the below median group are Detroit-Warren- Dearborn,MI,Dallas-FortWorth-Arlington,TX,Denver-Aurora-Lakewood,CO,Philadelphia-Camden- Wilmington,PA-NJ-DE-MD,andStLouis,MO-IL.ValuesinboldfontdenotethatthePCslopeforthe below median rate of import penetration group is significantly different from that of the above median group. The lower panel displays results, conditional on the regimes identified by the baseline model and for the full sample, from estimating the nonlinear Phillips curve in Equation (14), using a kink at unemployment rate values below either five or 4.2 percent. 44

Table 6: Wage Phillips curve: 51 state-level quarterly wage inflation rates (1980-2019) 1980:1-1989:4 1990:1-2019:4 1980:1-2019:4 Linear model PC -0.46∗∗∗ -0.34∗∗∗ -0.39∗∗∗ AR 0.03 0.04∗∗∗ 0.04∗∗∗ Nonlinear model Kink at 5% PC -0.41∗∗∗ -0.25∗∗∗ -0.33∗∗∗ Extra PC (U < 5%) -0.60∗∗∗ -0.50∗∗∗ -0.52∗∗∗ AR 0.02 0.04∗∗∗ 0.04∗∗∗ Kink at 4.2% PC -0.43∗∗∗ -0.30∗∗∗ -0.36∗∗∗ Extra PC (U < 4.2%) -0.85∗∗ -0.63∗∗∗ -0.69∗∗∗ AR 0.02 0.04∗∗∗ 0.04∗∗∗ Note: The top panel of this table displays estimates of the slope coefficient on the lagged state-level unemployment rate (PC) and the autoregressive term (AR) when regressing the 51 (including the District of Columbia) state-level quarterly wage inflation rates (growth rates of average hourly earnings of production and nonsupervisory workers) from 1980 through 2019 on an autoregressive term and the lagged state-level unemployment rates, including industry and time fixed effects. Significance at the 10, 5, and 1 percent levels are denoted by ∗, ∗∗, and ∗∗∗. We display results for the two regimes identified by the model. Average Hourly Earnings of production and nonsupervisory workers are at the quarterly frequency beginning in 1980:Q1 and ending in 2019:Q4, sourced from the CEPR extract of the underlying CPS data. The middle and lower panels display estimates when including a kinkpoint for unemployment rate values below 5 or 4.2 percent, and conditioning on either the full sample or the two regimes identified by the baseline model. 45

Table 7: Annual 28 EU countries inflation (1986-2021) 1986-2003 2004-2021 1986-2021 All countries PC -0.72∗∗ -0.09∗∗∗ -0.15 PC (scaled) -1.03 -0.13 -0.21 AR 0.10 0.48∗∗∗ 0.54∗∗∗ vol. 2.54 1.08 1.84 Rich vs poor PC (rich) -0.21∗∗∗ -0.06 -0.12∗∗∗ PC (poor) -0.73 -0.07∗ -0.13 Goods vs services PC (servs.) -0.34∗∗∗ -0.15∗∗∗ -0.19∗∗∗ PC (goods) -0.33∗∗∗ -0.07∗∗ -0.11∗∗∗ Kink at -1.5% PC -1.17∗∗∗ -0.02 -0.14 Extra PC (UGAP < -1.5%) 3.30 -0.58∗∗∗ -0.13 AR 0.09 0.45∗∗∗ 0.54∗∗∗ Note: The top panel of this table displays estimates of the slope coefficient on the lagged countrylevel unemployment gaps (PC) and the autoregressive term (AR) from the baseline model displayed in Equation (2). Significance at the 10, 5, and 1 percent levels are denoted by ∗, ∗∗, and ∗∗∗. We also report the slope scaled by the expenditure share on nontradeables to map the regional PC slope into the national PC slope as suggested by Hazell et al. (2022). The reported volatility (vol.) is the weighted average of the country-level volatility estimates, using HICP country weights. This model regresses the 28 EU (including the UK) country-level annual inflation rates from 1986 through 2021 on anautoregressivetermandthelaggedcountry-levelunemploymentgaps,includingtwo-wayfixedeffects. Wedisplayresultsforthetworegimesidentifiedbythemodel,andforthefullsample(byestimatingthe model but precluding any breaks). The second panel displays results when estimating the same model separately for rich and poor countries – while precluding breaks and conditioning on either the regimes identified by the baseline model or on the full sample. Rich countries are defined as countries with real GDP per capita deflated by PPP in 2019 above the EU average and poor countries are defined as the rest. The third panel displays results when using either total services or total goods inflation, rather than total inflation for each country. Values in bold font denote that the PC for goods (poor countries) issignificantlydifferentfromthatofservices(richcountries)atthe95%confidencelevel. Thefinalpanel displaysresults,conditionalontheregimesidentifiedbythebaselinemodelandforthefullsample,when including a kink at an unemployment gap below minus 1.5 percent as displayed in Equation (14). 46

Table 8: Grouped heterogeneity estimates: Annual 28 EU countries inflation (1986-2021) 1986-2003 2004-2021 Parameter Estimates Group 1 PC -0.39∗∗ -0.09∗∗∗ vol. 1.86 1.08 Group 2 PC -0.05 vol. 0.72 Group Allocation Estimates Germany 2 1 Belgium 2 1 Bulgaria 1 1 Cyprus 1 1 Croatia 2 1 Czech Republic 1 1 Denmark 2 1 Estonia 1 1 Spain 2 1 Finland 2 1 France 2 1 Greece 2 1 Hungary 2 1 Ireland 1 1 Italy 2 1 Lithuania 2 1 Latvia 2 1 Luxembourg 2 1 Malta 2 1 Netherlands 2 1 Austria 2 1 Poland 1 1 Portugal 1 1 Romania 1 Sweden 1 1 Slovenia 2 1 Slovakia 1 1 United Kingdom 2 1 Note: The top panel of this table displays estimates of the slope coefficient on the lagged EU countrylevel unemployment gaps (PC) from the model that estimates an unobserved grouping structure as described in Section 3.3. Significance at the 10, 5, and 1 percent levels are denoted by ∗, ∗∗, and ∗∗∗. Thereportedvolatility(vol.) istheweightedaverageofthecountry-levelvolatilityestimateswithineach group, using HICP country weights. This model regresses the EU country-level annual inflation rates from 1986 through 2021 on an autoregressive term and the lagged country-level unemployment gaps, and includes two-way fixed effects. The model is estimated within the two regimes identified by the baseline model displayed in Equation (2). The lower panel displays the corresponding posterior mode group allocations. Due to high volatility and extreme outliers, Romania was omitted from the analysis in the first regime. 47

Table 9: Optimal monetary policy Model specification Optimal coefficient Unemployment Inflation US No break -0.88 0.68 Break -0.68 1.10 Break in PC mean holding distribution fixed -0.82 0.77 EU No break -1.01 1.19 Break -0.65 1.26 Break in PC mean holding distribution fixed -0.97 1.19 Note: The top panel of this table displays the optimal monetary policy rule coefficients on the unemployment gap and inflation simulated using the U.S. IS curve estimates detailed in Section 5.2 and the MSA Phillips curve estimates from our panel break model in which, following Hazell et al. (2022), the Phillips curve slope is scaled by the nontradeables share to back out the implied national Phillips curve slope. We display estimates for specifications that preclude breaks, allow for breaks in all parameters, and allow for breaks in the mean of the Phillips curve coefficient holding the distribution fixed. We use 20,000 posterior draws and a 200 × 200 grid search in our simulation. The lower panel displays corresponding results for the EU. To isolate the effect of the Phillips curve, we hold the IS curve estimates fixed across the US and EU simulations. 48

3 2 1 0 1− 2− 3− Year epols evruc spillihP 1960 1980 2000 2020 (a)PCE 4 2 0 2− Year epols evruc spillihP 1990 2000 2010 2020 (b)EU Figure 1: The black line in the top panel of this figure graphs the evolution of the posterior mean Phillips curve slope over time estimated from our baseline breakpoint model displayed in Equation (3) using the PCE sectoral data. The blue bands cover the corresponding 95 percent posterior interval of the estimates. The red dotted lines graph the OLS time series estimates for each individual sector, conditioning on each of the regimes identified by our breakpoint model. For illustrative clarity, the red dotted lines are not allowed to overlay the blue shaded area. The lower panel displays the same information but uses the EU data and the breakpoint model displayed in Equation (2). 49

Break Date rotceS ECP NPISH Other servs. Fin. servs. Food servs. Gasoline Food and bvg. Motor vehicles 1972:3 1972:4 1973:1 1973:2 1973:3 1973:4 (a)1972Breakpoint Break Date rotceS ECP NPISH Gasoline Motor vehicles 2001:1 2001:2 2001:3 2001:4 2002:1 2002:2 (b)2001Breakpoint Figure2: Thisfiguredisplaystheposteriormodebreakdatesestimatedfromthemodelthatregressesthe16 PCE industry-level quarterly inflation rates from 1959 through 2022 on an autoregressive term, the lagged aggregate unemployment gap, and lagged long-term inflation expectations, including industry fixed effects, and allowing the timing of the breaks to vary across industries as described in Section 3.2. The top panel displays results for the 1972 break, and the lower panel displays results for the 2001 break. 50

Break Date rotceS IPC Utility (piped) gas service Electricity Motor fuel Fuel oil and other fuels Other food at home Fruits and Vegetables Nonalcoholic Beverages Meats, Poultry, Fish and Eggs 1971:3 1972:1 1972:3 1973:1 1973:3 1974:1 1974:3 (a)1971Breakpoint Break Date rotceS IPC Utility (piped) gas service Fuel oil and other fuels Motor fuel Cereals and bakery products Fruits and vegetables Dairy and related products Food at employee sites and schools 2001:1 2001:3 2002:1 2002:3 2003:1 2003:3 2004:1 (b)2001Breakpoint Figure 3: This figure displays the posterior mode break dates estimated from the model that regresses the 31CPIindustry-levelquarterlyinflationratesfrom1954through2022onanautoregressiveterm,thelagged aggregate unemployment gap, and lagged long-term inflation expectations, including industry fixed effects, and allowing the timing of the breaks to vary across industries as described in Section 3.2. The top panel displays results for the 1971 break, and the lower panel displays results for the 2001 break. 51

Break Date yrtnuoc UE Slovakia Slovenia Romania Poland Latvia Lithuania Hungary Estonia Czech Republic Bulgaria 2003 2004 2005 2006 2007 2008 Figure 4: This figure displays the posterior mode break dates estimated from the model that regresses the 28EUcountry-levelannualinflationratesfrom1986through2021onanautoregressivetermandthelagged country-level unemployment gaps, including industry and time fixed effects, and allowing the timing of the breaks to vary across countries as described in Section 3.2. 52

4 2 0 2− Year tnecreP Inflation less long term inflation expectations Phillips curve fit (linear) Phillips curve fit (nonlinear & COVID u star) 1980 1990 2000 2010 2020 (a)HeadlineCPI 5.1 0.1 5.0 0.0 5.0− 0.1− 5.1− Year tnecreP Inflation less long term inflation expectations Phillips curve fit 2001 2011 2021 (b)EU Figure 5: The top panel of this figure displays the linear Phillips curve fit from our MSA breakpoint model (solid black line). Specifically, in each year this is our prevailing regime-specific MSA regional Phillips slope coefficientdividedbythenontradeablessharemultipliedbythelaggednationalunemploymentgap. Thered linegraphstheannualnationalheadlineCPIinflationrateminuslongterminflationexpectations,whichare 10-year ahead SPF CPI inflation expectations back to 1991 Q4. Missing observations prior to 1991 Q4 are filled using linear interpolation. The dotted black uses our implied national PC slopes from the nonlinear Phillips curve estimated in the second regime using MSA-level data and replacing the noncyclical rate of unemployment in 2021 and 2022 with the higher estimates from Crump et al. (2022). The lower panel plots the same information for the EU. Specifically, the black line is our estimated prevailing regime-specific EU linearPCslopecoefficientmultipliedbythelaggedEUunemploymentgap. TheredlineusestheEUinflation rate and long term (five-year ahead) Eurozone inflation expectations from the ECB SPF which goes back to 2002 Q3. Prior to this, we use one-year ahead expectations, going back to 1999 Q1. Eurozone expectations dataaresourcedfromtheECBstatisticaldatawarehouse. Weaverageexpectationsacrossthefourquarters in a given year. 53

−0.5 −0.4 −0.3 −0.2 −0.1 21 01 8 6 4 2 0 Phillips curve slope ytisneD regime 1 regime 2 full sample (a)MSA −1.5 −1.0 −0.5 0.0 41 21 01 8 6 4 2 0 Phillips curve slope ytisneD regime 1 regime 2 full sample (b)EU Figure6: ThetoppanelofthisfiguredisplaysdensityplotsofthePhillipscurveslopecoefficientsinthefirst (black line) and second regimes (green line) estimated from our baseline breakpoint model using MSA-level data against the corresponding plot from the same model that precludes breaks (red line). The lower panel displays the same information estimated using the EU data. 54

Appendix A. Appendix Tables Table A1: Summaries of existing papers on instability in the Phillips curve Authors Sample Method Finding Notes BallandMazumder(2011) 1960-2010 RandomWalkparameter Steepening around 1970, Lowerandmorestableinflation flatteningin80s both flatten curve. Paper uses medianandcoreCPI Slope coefficient linear function of level and variance BallandMazumder(2019) 1985-2015 SupWaldtest Flatteningbreakin1995 Breakidentifiedindirectlyfrom expectations formation. Paper usesmedianCPI. PerronandYamamoto 1960-1997 SupWaldtest Breakin1991 UsesGDPdeflator. Matheson&Stavrev(2013) 1961-2012 RandomWalkparameter Flatteningin80s UsesheadlineCPIinflation. GaliandGambetti(2019) 1964-2017 Regimeswithfixeddates Flatteningin2007 WagePhillipscurve LeducandWilson(2017) 1991-2015 Regimeswithfixeddates Flatteningin2009 WagePhillipscurve Hooperetal. (2019) 1961-2018 Regimeswithfixeddates Flatteningin1988 Uses headline and core PCE andaveragehourlyearningsand MSApaneldata. Coibion & Gorodnichenko 1961-2007 Regimeswithfixeddates Possible break in 1985; No break if augmented with (2015) mixedevidence household expectations. Uses variousaggregateinflationmeasures(CPI,coreCPI...) Coibionetal. (2013) 1968-2013 Regimeswithfixeddates Flatteningbreakin1985 BreakinpricePhillipscurvenot wagePhillipscurve Roberts(2006) 1960-2002 Regimeswithfixeddates Flatteningbreakin1983 UsescorePCEinflation. Hazelletal. (2002) 1978-2018 Regimeswithfixeddates Breakin1990butnotsig- Statelevelpaneldata nificant CerratoandGitti(2022) 1990-2022 Regimeswithfixeddates Flattening in pandemic; MSAlevelpaneldata steepenedafter Fitzgeraldetal. (2020) 1977-2018 Regimeswithfixeddates Nosignificantbreak MSAlevelpaneldata Williams(2006) 1980-2016 Recursiveregressions Flatteninginthe90s CoreCPIandPCE DelNegroetal. (2020) 1964-2019 Regimeswithfixeddates Breakin1990 EstimatedinVAR Barnichon&Mesters(2021) 1969-2007 Regimeswithfixeddates Breakin1990 Phillips multiplier not slope of curve. UsesheadlinePCE Gilchrist&Zakrajsek(2019) 1962-2017 Sup-Waldtest Mixed results; possible Panelandaggregateddata(CPI breakin80s andPPI) Interact gap with trade share Inoueetal. (2002) 1970-2021 IV estimation with ran- Flattening until early UsescorePCE domwalkparameters 2000s;thensteepening Blanchard(2016) 1960-2014 Randomwalkparameter Flatteninginthe1980s UsesheadlineCPI 55

Table A2: Time series regressions: 16 PCE industry-level quarterly inflation rates (1959-2022) 1959-1972 1972-2001 2001-2022 1959-2022 1959-1972 1972-2001 2001-2022 1959-2022 Motor vehicles and parts Furnishings and durable household equipment PC -0.18 0.12 0.63∗∗ 0.31∗∗ -0.33∗∗∗ -0.11 -0.06 -0.09 corr -0.06 0.18 0.29 0.19 -0.64 0.10 -0.14 -0.13 Recreational goods and vehicles Other durable goods PC -0.13 -0.10 0.08 -0.03 -0.59∗∗ -0.22 0.08 -0.11 corr -0.22 0.19 -0.09 -0.14 -0.36 -0.03 0.04 -0.09 Food and beverages purchased for off-premises consumption Clothing and footwear PC -0.97∗∗∗ -0.54∗ -0.42∗∗∗ -0.38∗∗∗ -0.69∗∗∗ -0.00 0.62∗∗ -0.01 corr -0.38 -0.31 -0.24 -0.24 -0.76 -0.00 0.24 -0.05 Gasoline and other energy goods Other nondurable goods PC -1.03∗ -3.49∗∗∗ 3.19 0.36 -0.38∗∗∗ -0.25∗ -0.07 -0.13∗∗ corr -0.19 -0.29 0.12 -0.01 -0.59 0.07 -0.05 -0.04 Housing and utilities PC -0.17∗ -0.02 -0.09 -0.05 corr -0.30 0.27 -0.59 -0.01 Health care Transportation services PC -0.84∗∗∗ -0.12 -0.06 -0.09∗ -0.56∗∗ -0.07 -0.12 -0.10 corr -0.65 0.31 -0.10 -0.06 -0.41 0.02 -0.11 -0.09 Recreation services Food services and accommodations PC -0.29∗∗ -0.16 -0.25∗∗∗ -0.14∗∗∗ -0.58∗∗∗ -0.16 0.06 -0.04 corr -0.34 -0.07 -0.35 -0.22 -0.64 -0.13 -0.16 -0.19 Financial services and insurance Other services PC -0.24 1.05∗∗ 0.41∗ 0.25 -0.35∗∗ 0.01 -0.06 -0.06 corr -0.34 0.18 0.17 0.07 -0.42 0.13 -0.18 -0.07 NPISH PC -0.37 -1.11∗∗∗ -0.00 -0.35∗ corr -0.25 -0.41 -0.14 -0.28 Note: This table displays estimates of the slope coefficient on the lagged aggregate unemployment gap (PC) when estimating OLS time series regressions of each of the 16 PCE sector quarterly inflation rates from 1959 through 2022 on an intercept, an autoregressive term, the lagged aggregate unemployment gap, and the lagged long-term inflation expectations. Significance at the 10, 5, and 1 percent levels are denoted by ∗, ∗∗, and ∗∗∗. We estimate the model conditioning on each of the three regimes identified by the baseline PCE model displayed in Equation (3), and for the full sample. Within each of the three regimes, and for the full sample, we also report the correlation between the industry’s inflation rate and the aggregate unemployment gap. 56

Table A3: Time series regressions: 31 CPI industry-level quarterly inflation rates (1954-2022) 1954-1971 1971-2001 2001-2022 1954-2022 1954-1971 1971-2001 2001-2022 1954-2022 1954-1971 1971-2001 2001-2022 1954-2022 CerealsandBakeryProducts Meats,Poultry,FishandEggs DairyandRelatedProducts PC -0.18 -0.35∗ -0.34∗∗∗ -4.54 -1.48 -0.21 -0.66 -1.14 -0.32 -0.42 corr 0.15 -0.27 -0.24 -0.42 -0.09 -0.11 -0.09 -0.14 -0.13 -0.13 FruitsandVegetables NonalcoholicBeveragesandBeverageMatls OtherFoodAtHome PC -0.99 -2.19∗∗ -0.41 -0.88∗∗ -1.47 -0.33 -0.48∗∗ -0.48 -0.83 -1.36∗∗ -0.22 -0.56∗∗∗ corr -0.09 -0.09 -0.09 -0.07 -0.19 0.01 -0.31 -0.05 0.01 -0.21 -0.21 -0.19 FullServiceMealsandSnacks LimitedServiceMealsandSnacks Foodatemployeesitesandschools PC 0.64 -0.07 -0.04 -1.85 -0.08 -0.05 7.39∗∗∗ -0.00 0.33 corr 0.36 -0.33 -0.27 -0.44 -0.19 -0.12 0.64 -0.14 -0.13 Foodfromvendingmachinesandmobilevendors Otherfoodawayfromhome Fueloilandotherfuels PC -1.55∗∗∗ -0.31 -0.17 -1.43 -0.32∗∗ -0.30∗∗ -0.39 -6.91∗∗∗ 1.50 -0.51 corr -0.51 -0.14 -0.06 -0.20 -0.26 -0.26 -0.19 -0.32 0.06 -0.06 Motorfuel Electricity Utility(piped)gasservice PC -0.15 -5.89∗∗∗ 3.15 0.01 -0.01 -1.56∗∗∗ -0.28 -0.27 0.29 -0.89 -0.74 -0.41 corr 0.02 -0.26 0.11 0.00 -0.04 -0.02 -0.25 -0.03 0.07 0.06 -0.11 -0.03 Householdfurnishingsandsupplies Apparel Transportationcommoditieslessmotorfuel PC -0.04 -0.04 -1.00∗∗∗ -0.66∗∗∗ 0.72∗∗∗ -0.12 1.24∗ 1.24∗ corr -0.19 -0.19 -0.66 -0.00 0.25 -0.04 0.17 0.17 Medicalcarecommodities Recreationcommodities Educationandcommunicationcommodities PC -0.08 -0.21 0.02 0.02 0.17 0.17 0.16 0.16 corr 0.19 0.36 -0.05 0.15 0.02 0.02 0.11 0.11 Alcoholicbeverages Othergoods Shelter PC -0.35 -0.94∗∗∗ -0.19∗∗ -0.41∗∗∗ -0.13 -0.13 -1.17∗∗∗ -1.75∗∗∗ -0.17∗∗ -0.56∗∗∗ corr -0.05 -0.11 -0.27 -0.19 -0.24 -0.24 -0.63 -0.15 -0.64 -0.22 Waterandsewerandtrashcollectionservices Householdoperations Medicalcareservices PC -1.17 0.23∗∗∗ 0.28∗∗∗ -4.95 -0.73∗∗∗ -0.71∗∗∗ -0.53∗∗∗ -0.35∗∗ -0.20∗∗ -0.29∗∗∗ corr -0.17 0.27 0.37 -0.41 -0.32 -0.32 -0.49 0.23 -0.32 -0.09 Transportationservices Recreationservices Educationandcommunicationservices PC -0.60∗ -0.24 0.34 -0.08 -0.35∗∗∗ -0.35∗∗∗ 0.31∗∗ 0.31∗∗ corr -0.37 0.19 0.05 0.00 -0.27 -0.27 0.33 0.33 Otherpersonalservices PC -0.19 -0.19 corr -0.26 -0.26 Note: This table displays estimates of the slope coefficient on the lagged aggregate unemployment gap (PC) when estimating OLS time series regressions of each of the 31 CPI sector quarterly inflation rates from 1954 through 2022 on an intercept, an autoregressive term, the lagged aggregate unemployment gap, and the lagged long-term inflation expectations. Significance at the 10, 5, and 1 percent levels are denoted by ∗, ∗∗, and ∗∗∗. We estimate the model conditioning on each of the three regimes identified by the baseline CPI model displayed in Equation (3), and for the full sample. Within each of the three regimes, and for the full sample, we also report the correlation between the industry’s inflation rate and the aggregate unemployment gap. Missing values indicate that the industry has insufficient inflation observations in the corresponding regime to either estimate the regression or compute the correlation. 57

Table A4: Time series regressions: 22 CPI MSA-level annual inflation rates (1980-2022) 1980-2000 2001-2022 1980-2022 1980-2000 2001-2022 1980-2022 Urban Alaska Atlanta-Sandy Springs-Roswell, GA PC -0.44 1.08∗∗∗ 0.09 0.18 0.01 0.02 corr -0.41 0.49 0.03 0.37 -0.20 -0.13 Boston-Cambridge-Newton, MA-NH Baltimore-Columbia-Towson, MD PC 0.27 -0.11 -0.04 -0.14 0.11 -0.04 corr 0.56 -0.25 -0.17 0.26 -0.07 0.13 Chicago-Naperville-Elgin, IL-IN-WI Detroit-Warren-Dearborn, MI PC -0.09 0.08 0.03 0.03 0.07 0.05 corr 0.23 -0.04 0.11 0.11 -0.09 -0.01 Denver-Aurora-Lakewood Houston-The Woodlands-Sugar Land, TX PC 0.00 0.09 0.07 0.52∗∗∗ -0.04 -0.00 corr 0.26 -0.05 -0.02 0.83 -0.09 -0.11 Los Angeles-Long Beach-Anaheim, CA Miami-Fort Lauderdale-West Palm Beach, FL PC 0.23 0.61∗∗ 0.54∗∗∗ -0.18∗ 0.08 0.02 corr 0.41 0.40 0.39 -0.40 -0.17 -0.21 Minneapolis-St Paul-Bloomington, MN-WI Dallas-Fort Worth-Arlington, TX PC 0.33 0.02 -0.02 -0.27 0.07 -0.00 corr 0.43 -0.19 -0.15 0.11 -0.05 -0.00 New York-Newark-Jersey City, NY-NJ-PA Philadelphia-Camden-Wilmington, PA-NJ-DE-MD PC -0.03 0.13 0.08 -0.04 0.09 0.05 corr -0.11 0.06 0.06 0.01 -0.06 -0.05 Phoenix-Mesa-Scottsdale, AZ Riverside-San Bernardino-Ontario, CA PC 0.18 0.18 0.74∗∗∗ 0.74∗∗∗ corr -0.24 -0.24 0.66 0.66 San Diego-Carlsbad, CA San Francisco-Oakland-Hayward, CA PC -0.24 0.07 -0.03 -0.65∗∗∗ -0.07 -0.11 corr 0.02 -0.19 -0.08 -0.90 -0.33 -0.40 St Louis, MO-IL Seattle-Tacoma-Bellevue WA PC -0.05 0.09 0.03 -0.03 0.17 0.08 corr 0.38 0.04 0.25 -0.28 -0.21 -0.24 Tampa-St Petersburg-Clearwater, FL Washington-Arlington-Alexandria, DC-VA-MD-WV PC 0.01 0.01 -0.39 0.21 0.04 corr -0.20 -0.18 -0.05 -0.02 -0.06 Note: ThistabledisplaysestimatesoftheslopecoefficientonthelaggedMSA-levelunemploymentrate (PC)whenestimatingOLStimeseriesregressionsofeachofthe22MSA-levelannualinflationratesfrom 1980 through 2022 on an intercept, an autoregressive term, and the lagged MSA-level unemployment rate. Significanceatthe10,5,and1percentlevelsaredenotedby∗,∗∗,and∗∗∗. Weestimatethemodel conditioning on each of the two regimes identified by the baseline MSA model displayed in Equation (1), and for the full sample. Within each of the two regimes, and for the full sample, we also report the correlation between the MSA’s inflation rate and its unemployment rate. Missing values indicate that the MSA has insufficient inflation observations in the corresponding regime to either estimate the regression or compute the correlation. 58

Table A5: Time series regressions: 28 EU country-level annual inflation rates (1986-2021) 1986-2003 2004-2021 1986-2021 1986-2003 2004-2021 1986-2021 Germany Belgium PC -0.49∗∗ 0.37 -0.12 -0.22 0.10 -0.14 corr -0.69 0.29 -0.23 -0.46 0.09 -0.09 Bulgaria Cyprus PC -0.40 0.29 0.05 -0.44 -0.31∗∗∗ -0.33∗∗∗ corr -0.40 -0.12 -0.21 -0.23 -0.64 -0.58 Croatia Czech Republic PC -3.31 -0.13 -0.09 -0.11 -0.92 -0.26 corr 0.13 -0.16 -0.04 -0.25 -0.34 -0.29 Denmark Estonia PC -0.09 -0.11 -0.08 1.31 0.15 0.24 corr -0.51 -0.21 -0.30 0.01 -0.00 0.02 Spain Finland PC -0.09 -0.14 -0.07 -0.26∗∗∗ -0.61 -0.18∗∗ corr -0.14 -0.44 -0.25 -0.85 -0.52 -0.59 France Greece PC -0.20 -0.46 -0.20 -0.43 -0.25∗∗∗ -0.06 corr -0.51 -0.36 -0.31 -0.59 -0.84 -0.46 Hungary Ireland PC 2.11 -0.29 -0.25 -0.49 0.18 0.09 corr 0.88 0.03 0.09 -0.35 -0.01 -0.09 Italy Lithuania PC -0.21 -0.45∗∗∗ -0.11 -0.01 -0.26 -0.27 corr -0.46 -0.68 -0.07 -0.40 -0.43 -0.45 Latvia Luxembourg PC -0.38 -0.40 -0.42 -1.44 0.07 -0.12 corr -0.19 -0.52 -0.52 -0.59 -0.04 -0.14 Malta Netherlands PC 4.78 0.51 0.55 -0.61∗∗ -0.39∗ -0.49∗∗∗ corr 0.89 0.13 0.12 -0.56 -0.46 -0.53 Austria Poland PC -0.84∗∗ 0.41 0.08 -0.97 -0.16 -0.26∗ corr -0.48 0.18 -0.11 -0.76 -0.37 -0.43 Portugal Romania PC -0.19 0.02 -0.15 2.21 0.10 6.43 corr -0.36 0.09 -0.17 -0.12 -0.18 0.01 Sweden Slovenia PC -0.41 0.04 -0.25 1.23 -0.25 -0.03 corr -0.54 -0.12 -0.41 0.36 -0.40 -0.20 Slovakia United Kingdom PC -0.95 -0.05 -0.21 -0.49∗∗ 0.09 -0.24 corr -0.67 0.06 0.07 -0.53 0.33 -0.22 Note: ThistabledisplaysestimatesoftheslopecoefficientonthelaggedEUcountry-levelunemployment gap(PC)whenestimatingOLStimeseriesregressionsofeachofthe28EUcountry-levelannualinflation rates from 1986 through 2021 on an intercept, an autoregressive term, and the lagged country-level unemployment gap. Significance at the 10, 5, and 1 percent levels are denoted by ∗, ∗∗, and ∗∗∗. We estimatethemodelconditioningoneachofthetworegimesidentifiedbythebaselineEUmodeldisplayed in Equation (2), and for the full sample. Within each of the two regimes, and for the full sample, we also report the correlation between the country’s inflation rate and its unemployment rate gap. 59

Table A6: Testing for error dependence PCE CPI EU MSA Durbin-Watson p-value Regime 1 0.41 0.68 0.93 0.47 Regime 2 0.65 0.64 0.28 0.15 Regime 3 0.23 0.44 CD test statistic Regime 1 0.91 0.44 1.58 1.75 Regime 2 1.52 1.44 0.32 0.60 Regime 3 0.11 1.97 Note: The top panel of this table reports the p-values from a Durbin-Watson test for serial correlation in the residuals from our baseline panel breakpoint models across every regime and all four data sets we consider for the price Phillips curve. Here, we exclude observations for Romania when computing the p-value of the DW test in the first regime due to extreme and volatile outliers. The lower panel displays the bias corrected CD test statistic, which has a standard Normal distribution, proposed by Juodis and Reese(2022)ineachregimeacrossthesamefourdatasets. Thefirsttentimeperiodsofthefirstregime are excluded when computing the test statistic for the MSA data because more than half of the series have missing observations. 60