Inflation and Real Activity over the Business Cycle

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Inflation and Real Activity over the Business Cycle Francesco Bianchi, Giovanni Nicol`o, Dongho Song 2023-038 Please cite this paper as: Bianchi, Francesco, Giovanni Nicol`o, and Dongho Song (2023). “Inflation and Real Activity over the Business Cycle,” Finance and Economics Discussion Series 2023-038. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2023.038. 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.

Inflation and Real Activity over the Business Cycle Francesco Bianchi∗ Giovanni Nicolò Dongho Song Johns Hopkins University Federal Reserve Board Johns Hopkins University NBER and CEPR Carey Business School March 2023 Abstract We study the relation between inflation and real activity over the business cycle. We employ a Trend-Cycle VAR model to control for low-frequency movements in inflation, unemployment, and growth that are pervasive in the post-WWII period. We show that cyclical fluctuations of inflation are related to cyclical movements in real activity and unemployment, in line with what is implied by the New Keynesian framework. We then discuss the reasons for which our results relying on a Trend-Cycle VAR differ from the findings of previous studies based on VAR analysis. We explain empirically and theoretically how to reconcile these differences. JEL: E31, E32, C32. Keywords: Inflation, real activity, business cycles, trend-cycle VAR. ∗Bianchi: Department of Economics, Wyman Park Building, 3100 Wyman Park Drive, Baltimore, MD 21211 (francesco.bianchi@jhu.edu). Corresponding author: Nicolò, Federal Reserve Board, 20th Street and Constitution Avenue N.W., Washington, DC 20551 (giovanni.nicolo@frb.gov). Song: Johns Hopkins Carey Business School, 100 InternationalDrive,Baltimore,MD21202(dongho.song@jhu.edu). WewouldliketothankourdiscussantDomenico Giannone,GianniAmisano,GuidoAscari,DarioCaldara,ThomasDrechsel,FilippoFerroni,FrancescoFurlanetto, Etienne Gagnon, Ed Herbst, Margaret Jacobson, Ben Johannsen, Leonardo Melosi, Emi Nakamura, Anna Orlik, Luca Sala, Frank Schorfheide, Mark Watson, and all seminar participants at George Washington University, the 2022 Central Bank Research Association Annual Meeting, and the 2022 mid-year meeting of the NBER EFSF Workgroup on Methods and Applications for DSGE Models for useful comments and suggestions. The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Federal Reserve Board or the Federal Reserve System. 1

1 Introduction During the period of the Great Moderation, the evidence of an empirical relationship between real economic activity and inflation weakened, leading economists to rethink the foundations of New Keynesian models. Recent contributions offer explanations for the weakening of this empirical relationship (Del Negro et al., 2020 and references therein) or modifications to the New Keynesian model to improve its empirical fit (Gust et al., 2022). By contrast, other authors abandon the New Keynesian apparatus and develop business-cycle theories that abstract from discussing the implications for inflation (Beaudry et al., 2020; Basu et al., 2021) or in which shocks identified as the main drivers of business-cycle fluctuations are reminiscent of demand shocks but have no inflationary effects (Beaudry and Portier, 2013; Angeletos et al., 2018). An important empirical justification for these alternative theoretical frameworks is offered by Angeletos et al. (2020). The authors consider the entire U.S. post-WWII period and find a disconnectbetweenrealactivityandinflationatbusinesscyclefrequencies. Intheircleverempirical analysis, the authors follow an extensive empirical literature that uses vector autoregressions (VARs) as a “model free,” but still structural approach to the data. The authors use a VAR to identifya“business-cycle” shockthatexplainsthelargestpossibleshareofvariationinrealactivity or unemployment at business-cycle frequencies. This single shock explains most of the businesscycle movements in various measures of real activity, but close to nothing of the business-cycle variation of inflation. The authors conclude that their results are at odds with the premise of the New Keynesian framework in which demand shocks drive business-cycle fluctuations because nominal rigidities prevent an immediate adjustment in prices. As we argue next, the approach of using a VAR to identify shocks in the frequency domain has somelimitationswhenthegoalistoassessthebusiness-cyclerelationshipbetweenrealandnominal variables over the U.S. post-WWII period. The main reason is that a standard fixed-coefficient VAR might be unable to correctly disentangle business-cycle and low-frequency movements in those variables over a relatively short period of time that features structural breaks (Clarida et al. (2000), Sims and Zha (2006), Bianchi (2013), Bianchi and Ilut (2017)). In a VAR, a single set of parameters and reduced-form shocks need to accommodate the variation at all frequencies observedoverarelativelyshortperiod. Asaresult,aprocedurethatusestheestimatedparameters and reduced-form shocks to identify variation at business-cycle frequency might be biased. The problem is particularly severe if one of the variables of interest shows significant variation at low frequency, as it is the case with inflation. Ultimately, the identified shock might fail to capture a business-cycle relationship between the two variables even when such a relation is in fact in the data. To remedy this limitation of the VAR for the specific question at hand, we adopt a more flexible model that explicitly extracts business-cycle movements in the variables of interest. Specifically, we argue that a Trend-Cycle VAR (TC-VAR) model is better suited to analyze the relation between inflation and real activity at business-cycle frequencies. 2

We start by presenting simple, but insightful, evidence that serves as motivation for our analysis. We consider a measure of inflation—the GDP deflator—and two measures of real economic activity—the level of real GDP per capita and the unemployment rate—over the period between 1960 and 2019. Using a bandpass filter, we extract movements in those measures at frequencies between 6 and 32 quarters—labeled “business-cycle frequencies”—and between 8 and 30 years— labeled “medium-cycle frequencies”. After filtering out movements at high and low frequencies, the correlation of current inflation and real per-capita GDP (unemployment rate) over the business cycle is positive (negative) and roughly equal to about 0.2 (negative 0.4). The correlations become larger (in absolute value) when considering the relationship between current inflation and lagged measures of real economic activity, peaking at about 0.45 (negative 0.45) when considering real per-capita GDP (unemployment rate) lagged by four (two) quarters. In addition, over the medium cycle, these estimates can be up to nearly 50% larger (in absolute value) than those over the business cycle. This evidence is puzzling in light of the existing literature because it suggests that inflation is related to real activity at business cycle frequencies, at least to some extent. Motivated by this analysis, we adopt a more rigorous empirical framework and estimate a multivariate Trend-Cycle VAR model building on the work of Watson (1986), Stock and Watson (1988, 2007), Villani (2009) and, more recently, Del Negro et al. (2017). We consider the sample between 1960 and 2019 using seven time series. The first four time series are commonly used in previous studies: i) the growth rate of real GDP per capita; ii) the unemployment rate; iii) the effective federal funds rate; and iv) inflation. We then include three additional time series. First, to better capture low-frequency movements in inflation, we add ten-year-ahead inflation expectations. Second, we use the median one-year-ahead inflation expectations from the Survey of Professional Forecasters (SPF) as a variable that should respond to business cycle variation in inflation and be less affected by transitory shocks. Third, we include the median one-year-ahead expectations for the unemployment rate from the SPF to inform the estimates of the latent trend of the unemployment rate. Given that a TC-VAR already separates trends from cycles, we identify the shock that maximizes the variation of the latent cyclical component of the unemployment rate, and we study its contribution to the volatility of all cyclical components. A series of important results emerge from our analysis. First, under our baseline specification, the shock targeting the unemployment rate explains about 70% of the unemployment cycle and 31% of the volatility of the inflation cycle. This is a relatively large share, suggesting that it is important to account for the low-frequency movements in real and nominal variables when studying their cyclical relationship. Second, the unemployment-identified shock explains about 49% of the inflation expectations cycle. This result provides further support for the notion that business-cycle movements in inflation are in fact related to real activity, given that expected inflation is obviously related to actual inflation, but less affected by high-frequency fluctuations. Furthermore, the result is in line with the New Keynesian 3

framework, in which agents’ inflation expectations depend on the state of the economy. In line with this reasoning, when we focus on frequencies that correspond to fluctuations in the cyclical component with duration of at least 1.5 years, the results become stronger. In this case, the shock identified targeting the rate of unemployment explains a higher percentage of the business-cycle volatility of all inflation measures compared to when all the frequencies are considered: 34% for realized inflation and 52% for inflation expectations. Our results are robust to a number of alternative specifications. First, the results are very similar if we use GDP instead of unemployment to identify the real-activity shock. Second, the results are not sensitive to the choice of more conservative, though still realistic, priors on the standard deviation of shocks to the trend of the unemployment rate. Third, if we dogmatically model all latent trends as constant, rather than time-varying, we find evidence of a disconnect between nominal and real variables, thus recovering the results commonly found when adopting a standardVARmodel(Giannoneetal.,2019b; Angeletosetal.,2020). Crucially, thisfindingshows that the TC-VAR model is flexible enough to deliver results in line with the existing literature, but the data do not support this evidence, given that they present important trends that can confound the business-cycle analysis. The adoption of a TC-VAR has four clear advantages relative to the use of a VAR when identifying shocks in the frequency domain.1 First, the inference exercise automatically separates the trends from the cycles. Second, cyclical variation is controlled by a different set of parameters with respect to low-frequency variation. Third, we do not need to take a stance on the typical length of the business cycle. This is important in light of the fact that expansions have become progressively longer in the sample under consideration. Finally, by allowing for changes in trend growth, trend inflation, and variation in long-run unemployment, the model accommodates the notionthatwhatmattersforcyclicalmovementsininflationistheoutput(orunemployment)gap. We then verify that a standard VAR model cannot easily capture the uncovered business-cycle relationship between nominal and real variables even if we choose priors and sample periods to account for low-frequency movements in the data. Given our focus on the U.S. economy, we borrow the framework of Angeletos et al. (2020) who estimate a VAR model with a Minnesota prior on U.S. data. Extending their dataset by two years to the end of 2019, we estimate their baseline VAR model with a Minnesota prior as well as combining it with long-run priors à la Giannone et al. (2019a). For each specification, we follow their approach and identify a shock that targets the unemployment rate at business-cycle frequencies. The contribution of the shock to the variability of inflation at the same frequencies only marginally increases from about 8% when using a Minnesota prior to about 11% when also assuming long-run priors. Results more similar to our baseline specification emerge if (1) we focus on the Great Moderation period, a sample 1The approach of identifying shocks in the frequency domain starting from a VAR was pioneered by Uhlig (2003). Theapproachhaslatelyadoptedbymanyothers,includingGiannoneetal.(2019a),Angeletosetal.(2020) and Basu et al. (2021). 4

that presents less low-frequency variation in inflation, and (2) impose dogmatic long-run priors. However, when priors are optimized to maximize the fit of the VAR, the results are far from our baseline specification, even using the Great moderation period. This provides further evidence about the importance of tailoring the econometric approach to the research goal. We lay out theoretical arguments for why our results differ from recent contributions that challenge the validity of the New Keynesian paradigm. We demonstrate that a fixed-coefficient VAR estimatedoveraperiodoftimethatpresentsstructuralchangesismisspecified,ifthegoalistrying to assess the comovement at business-cycle frequency. The misspecification problem associated with the use of a VAR model to describe a data generating process characterized by both lowand high-frequency movements cannot be resolved. An econometrician would need infinite data to reconstruct the VAR representation of a TC-VAR model. Even in that case, the reduced-form innovations that she would recover would map into the innovations affecting the latent persistent and stationary components as well as the error associated with the estimates of the latent components. In reality, these issues are exacerbated by the fact that the VAR parameters estimated over a finite sample would be distorted because a single set of parameters needs to account for both trend and cycle fluctuations. We provide an illustration of these issues with a simple model of unemployment rate and inflation. We generate Monte Carlo simulations of the two series using a TC-VAR model. We then present three insights based on the estimation of a VAR on the simulated data and the subsequent identification of a shock targeting the unemployment rate at business-cycle frequencies. First, we consider a case in which we introduce a trend only in inflation and assume that the unemployment rate is stationary, persistent, and the only driver of cyclical movements in inflation. We show that the explanatory power of the unemployment-identified shock for business-cycle movements in inflation drops dramatically as the relevance of low-frequency movements in inflation rises. Second, even when we assume that inflation does not feature a trend and it is exclusively driven by cyclical variation in the unemployment rate, the identified shock explains small portions of the business-cyclemovementsininflationiftheunemploymentratefeatureslow-frequencymovements. Finally, inacasewithtrendsinbothinflationandunemployment, aswellasadependenceofcyclical inflation on both its own lag and cyclical unemployment, it becomes even more challenging to successfully recover the underlying business-cycle relationship between the two series. Overall, ourfindingshaveimplicationsforboththeNewKeynesianliteratureandtherecentand growing literature that proposes alternative explanations for the sources of business-cycle fluctuations. For the former, the results support the evidence of a relationship between real and nominal variables over the business cycle, while highlighting the importance of accounting for both lowfrequency movements in real and nominal variables and the observed high-frequency movements in inflation to properly quantify that relationship. For the latter, the results suggest that the alternative explanations for the drivers of business cycles should also propose transmission mech- 5

anisms which are consistent with the empirical evidence on the connection between movements in inflation and real economic activity.2 Our results are consistent with the findings of Hazell et al. (2022). These authors show that the relation between real activity and inflation, while tenuous, can be recovered from the data and has not undergone a structural change once controlling for long-term inflation expectations. Their evidence is based on a cross-sectional analysis of inflation across U.S. states, while we take a timeseriesapproach. Bothpapersemphasizetheimportanceofcontrollingforlow-frequencymovements in inflation when investigating the business cycle relation between inflation and unemployment (or realactivity). OuranalysisalsoconnectstoSargentandSims(1977)thatshowsthattwodynamic factors could explain a large fraction of the variance of a series of important macroeconomic variables, including output, employment, and prices. Two factors are in fact necessary to control for the low-frequency movements in nominal variables. The subsequent factor-analysis literature hasrepeatedlyconfirmedthiskeyinsight(Giannoneetal.,2006; Watson,2004; StockandWatson, 2011). Related branches of the literature use unobserved component models to estimate measures oflong-runinflation(StockandWatson,2007;Mertens,2016)andlong-runinterestrates(Laubach and Williams, 2003, 2015; Lubik and Matthes, 2015; Del Negro et al., 2017; Del Negro et al., 2019; Holston et al., 2017; Lewis and Vazquez-Grande, 2019; and Johannsen and Mertens, 2021). Ascari and Fosso (2021) use a methodology similar to the one adopted in this paper to study the role of imported intermediate goods in explaining the lack of sensitivity of inflation to businesscycle movements in the post-Millennial period. They find that the contribution of the shock decreases from about 60% for the period starting in 1960 and ending in 1984 to nearly 30% for the period between 1985 and 2019. In line with our results, these estimates are about four and six times larger than the counterparts of 17% and 5% that Angeletos et al. (2020) find for the pre- and post-Volcker periods using a VAR and interpret as evidence of a disconnect. Our paper provides an explanation to reconcile these differences. In addition, our findings support the evidence that a rise in unemployment during a recession is associated with a fall in inflation as highlighted by Stock and Watson (2010) for the U.S. and by Smets(2010)fortheeuroarea. Inlinewiththeseclaims, accountingforarelevantinflation-output relationship can improve inflation forecasts as shown in Stock and Watson (2008) for the U.S. and Smets (2010) and Giannone et al. (2014) for the euro area. Similar results hold also in datarich environments (Bańbura et al., 2015; Crump et al., 2021). Finally, our conclusions are also consistent with Justiniano et al. (2013). These authors use an estimated New Keynesian DSGE modeltoshowthatmostinflationfluctuationsaredemanddrivenandcomovewiththeoutputgap. Our results are obtained without imposing at the onset the New Keynesian framework, something that we find valuable if the goal is to provide external validation for such framework. 2ForanalternativetheorythatintegratesKeynesianeconomicswithgeneralequilibriumtheorywithoutrelying on nominal rigidities, refer to Farmer and Platonov (2019) and Farmer and Nicolò (2018, 2019). 6

2 Motivating Empirical Evidence In this section, we provide motivating evidence for our subsequent empirical analysis. We aim to show that inflation and real activity are related over the business cycle once we control for their trends. Inparticular,weunderscoretheimportanceofadoptingaunifiedempiricalframeworkthat distinguishes between business-cycle and low-frequency movements in these variables. Moreover, given that our goal is to study the relationship between inflation and real economic activity over the business cycle, we also want to control for higher-frequency movements. These are especially important for the dynamics of inflation. We consider the inflation rate—measured as the log difference in the GDP deflator—and two measures of real economic activity—the log level of real, per-capita GDP and the unemployment rate—over the period starting from 1955:Q1 to 2019:Q4. Using the bandpass filter proposed by Christiano and Fitzgerald (2003), we extract the corresponding filtered time series over two frequency bands: the business cycle—defined as the period between 6 and 32 quarters—and the medium cycle—defined as the period between 8 and 30 years. Panel (A) of Figure 1 plots the filtered time series at business-cycle frequencies for inflation and real per-capita GDP as well as for inflation and the rate of unemployment. Panel (B) of Figure 1 plots the corresponding filtered time series at medium-cycle, rather than business-cycle, frequencies. The plots offer two main empirical facts. First, after filtering out movements at high and low frequencies, inflation is correlated with both measures of real economic activity at both business-cycle and medium-cycle frequencies. As expected, inflation is positively correlated with real per-capita GDP and negatively with the unemployment rate. Second, movements in both measures of real economic activity lead changes in the inflation dynamics at business-cycle and medium-cycle frequencies. High (low) levels of real per-capita GDP (unemployment rate) are associated with subsequent high levels of inflation, and vice versa. Toformalizethenotionthatcyclicalfluctuationsininflationappeartocomovewithrealactivity, Table1reportsthecorrelationsbetweencurrent(filtered)inflationandcurrentandlagged(filtered) levelsofrealper-capitaGDPandunemploymentrateattime(t−j)forj = {0,2,6,8}. Weconsider bothbusiness-cycleandmedium-cyclefrequencies. Overthebusinesscycle, thepositive(negative) correlation of inflation peaks with real per-capita GDP (unemployment rate) lagged by four (two) quarters at about 0.45 (negative 0.45). Over the medium cycle, the correlation of inflation with real per-capita GDP (unemployment rate) lagged by eight quarters peaks at about 0.65 (negative 0.55). These correlations are larger (in absolute value) than the counterparts with real per-capita GDP and the rate of unemployment lagged by four quarters over the business cycle. The empirical evidence presented in this section motivates us to adopt a dynamic, multivariate framework that allows to study the relationship between inflation and real economic activity over the business cycle while controlling for low-frequency variation in those variables. We discuss the adopted framework in the next section. 7

Figure 1: Inflation and real economic activity at business- and medium-cycle frequencies (A) Business-cycle frequencies (6-32 quarters) Inflation and output 1.2 5 0.6 2.5 0 0 -0.6 -2.5 Inflation (LHS) Output (RHS) -1.2 -5 1950 1960 1970 1980 1990 2000 2010 2020 Inflation and unemployment rate 1.2 2 0.6 1 0 0 -0.6 -1 Inflation (LHS) Unempl. (RHS) -1.2 -2 1950 1960 1970 1980 1990 2000 2010 2020 (B) Medium-cycle frequencies (8-30 years) Inflation and output 1 6 0.5 3 0 0 -0.5 -3 Inflation (LHS) Output (RHS) -1 -6 1950 1960 1970 1980 1990 2000 2010 2020 Inflation and unemployment rate 1 4 0.5 2 0 0 -0.5 -2 Inflation (LHS) Unempl. (RHS) -1 -4 1950 1960 1970 1980 1990 2000 2010 2020 Notes: The inflation rate is defined as the log difference in the GDP deflator. For the two measures of real economic activity, we consider the log level of real, per-capita GDP and the unemployment rate. Data sample is from 1955:Q1 to 2019:Q4. Using the bandpass filter proposed by Christiano and Fitzgerald (2003), we extract the corresponding filtered time series over two frequency bands: the business cycle—defined as the period between 6 and 32 quarters—and the medium cycle—defined as the period between 8 and 30 years. Source: Federal Reserve Bank of St. Louis; authors’ calculations. 8

Table 1: Correlations of inflation with lagged measures of real economic activity Business-cycle frequencies (6-32 quarters) j = 8 j = 6 j = 4 j = 2 j = 0 Output 0.15 0.36 0.47 0.42 0.22 Unemployment rate 0.09 -0.13 -0.34 -0.44 -0.36 Medium-cycle frequencies (8-30 years) j = 8 j = 6 j = 4 j = 2 j = 0 Output 0.64 0.61 0.54 0.45 0.33 Unemployment rate -0.54 -0.50 -0.44 -0.36 -0.25 Notes: TheinflationrateisdefinedasthelogdifferenceintheGDPdeflator. Forthetwomeasuresofrealeconomic activity,weconsidertheloglevelofreal,per-capitaGDPandtheunemploymentrate. Datasampleisfrom1955:Q1 to 2019:Q4. Using thebandpass filter proposedby Christiano andFitzgerald (2003), we extract the corresponding filteredtimeseriesovertwofrequencybands: thebusinesscycle—definedastheperiodbetween6and32quarters— andthemediumcycle—definedastheperiodbetween8and30years. Weprovidethecorrelationsbetweencurrent (filtered) inflation and current and lagged (filtered) levels of real per-capita GDP and unemployment rate at time (t−j) for j ={0,2,6,8}. Source: Federal Reserve Bank of St. Louis; authors’ calculations. 3 The Trend-Cycle VAR Model In this section, we present the TC-VAR used to model the joint dynamics of the growth rate of real GDP per capita, the unemployment rate, the federal funds rate and the inflation rate as well as three expectations measures: the one-year-ahead unemployment rate expectations and the oneand ten-year-ahead inflation expectations. 3.1 Baseline specification Our baseline specification allows for four trends and six cycles. Real per-capita GDP growth g t and the unemployment rate u evolve according to t g = τ +(c −c ), (1.1) t g,t y,t y,t−1 u = τ +c , (1.2) t u,t u,t whereτ andτ arethetrendcomponentsofper-capitaGDPgrowthandunemployment,respecg,t u,t tively, whilec andc arethecyclicalcomponentsoflogper-capitaGDPandunemployment. It y,t u,t isworthemphasizingthatwemodelthecycleinrealGDP,asopposedtoGDPgrowth. Thisisconsistent with typical macroeconomic models in which output moves around a stochastic trend and with the notion that what matters for inflation and unemployment dynamics is the output gap, 9

not output growth. Additionally, one-year-ahead unemployment expectations share a common trend with the realized unemployment rate, while also following a separate cyclical component ue,1y = τ +ce,1y. (2) t u,t u,t We assume that the Fisher relation holds in the long run. This implies that the federal funds rate can be expressed as f = (τ +τ )+c , (3) t r,t π,t f,t where τ and τ are the trend of the real interest rate and inflation, respectively. Realized r,t π,t inflation and the one- and ten-year-ahead inflation expectations are decomposed as π = τ +c +(η −η ), (4.1) t π,t π,t π,t π,t−1 πe,1y = τ +ce , (4.2) t π,t π,t πe,10y = τ +δce +ηe,10y. (4.3) t π,t π,t π,t thus sharing a common trend τ . We allow for an i.i.d. measurement error in the log-level of π,t the GDP deflator, η , which implies that, after taking log difference, realized inflation features a π,t negativemovingaveragemeasurementerrorcomponent. Wealsoassumethatthereisonecommon cyclical component for expected inflation, ce , which is shared across the one- and ten-year-ahead π,t inflation expectation surveys. Because the ten-year inflation expectation πe,10y is fairly stable over t time, we estimate its loading with the beliefs that it is less than one δ < 1. This parameterization is consistent with the definitions of one-year-ahead and ten-year-ahead inflation expectations that measureexpectedaverageinflationoverthecorrespondinghorizons. Wealsoallowforidiosyncratic errors in the ten-year-ahead inflation expectations, ηe,10y. π,t For ease of exposition, we collect observables and state variables in vectors (cid:110) (cid:111)′ (cid:110) (cid:111)′ z = g ,u ,ue,1y,f ,π ,πe,1y,πe,10y , τ = {τ ,τ ,τ ,τ }′, c = c ,c ,ce,1y,c ,c ,ce , t t t t t t t t t g,t u,t r,t π,t t y,t u,t u,t f,t π,t π,t (cid:110) (cid:111)′ (cid:110) (cid:111)′ η = η ,ηe,10y , ε = {ε ,ε ,ε ,ε }′, ε = ε ,ε ,εe,1y,ε ,ε ,εe . t π,t π,t τ,t τ,g,t τ,u,t τ,r,t τ,π,t c,t c,y,t c,u,t c,u,t c,f,t c,π,t c,π,t The dynamics of the trend τ and cyclical component c are given as t t τ = τ +ε , (5.1) t t−1 τ,t c = Φ c +Φ c +...+Φ c +ε . (5.2) t 1 t−1 2 t−2 p t−p c,t 10

The measurement errors follow η = ε . t η,t We assume that, across time, shocks are independent and identically distributed as       ε 0 Σ 0 0 τ,t τ ε = ε  ∼ N 0, 0 Σ 0 , t  c,t    c  ε 0 0 0 Σ η,t η where the matrices Σ , Σ , and Σ are conforming positive definite matrices, Σ = E(Q Q′) τ c η s s s for s = τ,c,η, and N(·,·) denotes the multivariate Gaussian distribution. We do not restrict covariance matrix Σ to be diagonal. This implies that while the trends follow random walks, τ they are not assumed to be independent of each other. 3.2 Alternative specifications For robustness, we consider three alternative specifications to model the three measures of expectations, while leaving unchanged the decomposition assumed for the other variables. First, we consider a specification that is more parsimonious than the baseline one. We assume that realized unemployment rate and one-year-ahead unemployment rate expectations evolve as u = τ +c , (6.1) t u,t u,t ue,1y = τ +c +ηe,1y, (6.2) t u,t u,t u,t thus sharing a common cyclical component c . For the one-year-ahead unemployment rate exu,t pectations, we allow for an idiosyncratic error. Similar to the unemployment rate measures, we decompose the three inflation measures as π = τ +c +η −η , (7.1) t π,t π,t π,t π,t−1 πe,1y = τ +c +ηe,1y, (7.2) t π,t π,t π,t πe,10y = τ +ηe,10y. (7.3) t π,t π,t Therefore, we assume one common cyclical component for inflation, c , which is shared across π,t realized inflation and the one-year-ahead inflation expectations. Because ten-year-ahead inflation expectations are fairly stable over time, we treat them as a proxy for trend inflation. Lastly, we allow for idiosyncratic errors for both inflation expectations. Evidently, this alternative specification is more parsimonious than the baseline one because, while considering the same number of trends, it allows for four, rather than six, cyclical components. 11

In contrast to the first alternative specification, the second is more flexible than the baseline. Under this alternative specification, the decompositions of realized unemployment rate in (1.2) and one-year-ahead unemployment rate expectations in (2) are identical to those assumed under the baseline case. However, we consider the following more flexible decomposition for the two inflation expectations measures πe,1y = τ +ce,1y, (8.1) t π,t π,t πe,10y = τ +ce,10y. (8.2) t π,t π,t Hence, the two measures of inflation expectations follow separate, idiosyncratic cyclical components, while both sharing a common inflation trend with realized inflation decomposed as under the baseline in (4.1). As a result, this specification allows for seven, rather than six as in the baseline case, idiosyncratic cyclical components, one for each observable. The third and final alternative specification verifies the robustness of the results to the use of the one-year-ahead unemployment rate expectations for the estimation of the TC-VAR model. Specifically, this alternative specification excludes measurement equation (2) for the expectations of the one-year-ahead unemployment rate, while leaving all the other decompositions unchanged. 3.3 State-space representation Our baseline and alternative specifications can be cast into a stace-space representation.3 Using generic notation, let us begin with n observables which can be decomposed into n trends and n τ c cycles, where 0 < n ≤ n and 0 < n ≤ n. τ c Measurement equation. Allowingforavectorofobservationerrorsη , thevectorofobservables t z can be expressed as t z = Λx = Λ x +Λ x +Λ x , (9) t t τ τ,t c c,t η η,t where x = {x ,x ,x }′, x = τ , x = (cid:8) c ,c ,...,c (cid:9)′, x = {η ,η }′ and p t τ,t c,t η,t τ,t t c,t t t−1 t−(p−1) η,t t t−1 denotes the number of lags used to model the stationary cyclical component. The n×n matrix τ Λ captures(n−n )cointegratingrelationships,whileΛ = [Λ ,...,Λ ]andΛ = [Λ ,Λ ]. τ τ c c,0 c,p−1 η η,0 η,1 State-transition equation. The vector of states x evolves as t x = Φx +Rε , (10) t t−1 t 3Appendix A provides details on the construction of the matrices in (9) and (10) for our baseline specification described in Subsection 3.1. 12

or equivalently,         x I 0 0 x I 0 0 ε τ,t τ,t−1 τ,t x  = 0 Φ 0 x +0 R 0 ε ,  c,t  c  c,t−1  c  c,t x 0 0 Φ x 0 0 R ε η,t η η,t−1 η η,t where     Φ 1 Φ 2 ... Φ p I   I 0 0 ... 0    0   (cid:34) (cid:35) (cid:34) (cid:35) Φ =   0 I ... ... . . .  , R =   . . .  , Φ = 0 0 , R = I . c   c   η η   . . . ... ... ... 0    0   I 0 0     0 ... 0 I 0 0 The initial conditions are distributed as x ∼ N (cid:0) τ,V (cid:1) , x ∼ N (cid:0) 0,V (cid:1) , (11) τ,0 τ c,0 c where V is an identity matrix, and V is the unconditional variance of c consistent with (10) and τ c 0 thus a function of the VAR coefficients φ = {Φ ,...,Φ }′ and variance Σ . 1 p c 4 Inference We now describe the data and priors used in our analysis and the methodology used to assess the strength of the relation between inflation and real activity over the business cycle. 4.1 Data WeestimatetheTC-VARmodelusingthefollowingsevenquarterlytimeserieswhichareexpressed at annualized rates: i) the growth rate of real, per-capita GDP g ; ii) the unemployment rate u ; t t iii) the median four-quarter-ahead unemployment rate expectations, ue,1y, from the SPF; iv) the t effective federal funds rate (FFR) f by treating observations at the zero lower bound (ZLB) as t missing following Del Negro et al. (2017); v) the inflation rate π , measured as the log difference in t GDP deflator (PGDP); vi) the median four-quarter-ahead average PGDP inflation expectations, πe,1y, from the SPF; vii) a measure of average ten-year-ahead inflation expectations, πe,10y, which, t t following Del Negro and Schorfheide (2013), we construct by combining survey expectations on average ten-year-ahead CPI inflation from the SPF and Blue Chip Economic Indicators survey, and adjusting it for the historical difference between CPI and PGDP inflation. We use the period between 1955:Q1 and 1959:Q4 as pre-sample and estimate the TC-VAR model over the period from 1960:Q1 to 2019:Q4. Appendix B provides the definitions, data sources and transformations. 13

Finally, in Appendix C, we report details about the Bayesian inference, including the settings and the Gibbs sampler adopted to estimate the TC-VAR model. 4.2 Priors and initial conditions Forourassumptionsaboutinitialconditionsandpriordistributions,wemainlyfollowtheapproach of Del Negro et al. (2017). We consider standard priors for covariance matrices Σ and Σ and for τ c the VAR coefficients φ = {Φ ,...,Φ }′ 1 p p(Σ ) = IW (cid:0) κ (κ +n +1)Σ (cid:1) , (12.1) τ τ τ τ τ p(Σ ) = IW (cid:0) κ (κ +n +1)Σ (cid:1) , (12.2) c c c c c p(ϕ|Σ ) = N (cid:0) ϕ,Σ ⊗Ω (cid:1) I(ϕ), (12.3) c c where ϕ = vec(φ), ϕ = vec (cid:0) φ (cid:1), IW = (κ(κ+n+1)Σ) corresponds to the inverse Wishart distribution with mode Σ and k degrees of freedom, and I(ϕ) is an indicator function that equals 0 if the VAR in (10) is explosive and 1 otherwise. Wecenterthepriordistributionfortheinitialconditionsofthetrendsτ in(11)tothepre-sample 0 mean of the corresponding variables. The priors for the initial conditions of the annualized trend of real per-capita GDP growth and of the unemployment rate are set to 1% and 5%, respectively. For the trend of the real interest rate and inflation, the priors for the initial condition are centered at 0.1% and 2.5%. Therefore, the vector of initial conditions for the trend components is τ = {1,5,0.1,2.5}′. To specify the prior for the covariance matrix of the shocks to the trends Σ in (12.1), we τ assume that those shocks are a priori uncorrelated. We then set the standard deviation for the expected change in the annualized trend of real, per-capita GDP growth to 1% over a time period of 40 years. For all the remaining variables, we assume a one-percent standard deviation for the expected change in their trends over 20 years. These assumptions imply that the prior covariance matrix of the shocks to the trends is diagonal with the following elements on the main diagonal diag (cid:0) Σ (cid:1) = [1/40,1/20,1/20,1/20]. As in Del Negro et al. (2020), we assume a tight prior by τ setting κ to 100. τ The shocks to the cyclical components Σ in (12.2) are also assumed to be uncorrelated a c priori. Wecalibratethestandarddeviationoftheshocksaffectingthestationarycomponentofthe (annualized) real, per-capita GDP growth and the unemployment rate to 5% and 1.1%, reflecting their pre-sample standard deviations. The standard deviation of the shocks affecting the cycle of the nominal interest rate and inflation are also set to their pre-sample standard deviations of 0.8% and 1.5% respectively. We also need to specify assumptions for the priors on the standard deviation of the cyclical component of the one-year-ahead unemployment rate expectations and the cyclical component that is common for the two inflation expectations measures. Because 14

these surveys are not available for the pre-sample period, we assume the standard deviations of the one-year-ahead unemployment rate expectation to be 0.9%, thus smaller than the pre-sample counterpart of 1.1% of realized unemployment rate. Similarly, we set the prior for the standard deviation of the common cyclical component of inflation expectations to 1.2%, therefore smaller than the pre-sample counterpart of 1.5% for realized inflation. As a result of these assumptions, the prior covariance matrix of the shocks to the cycles is diagonal and such that the values on the main diagonal approximately correspond to diag (cid:0) Σ (cid:1) = [25,1.3,0.8,0.6,2.2,1.4]. As in Kadiyala c and Karlsson (1997) and Giannone et al. (2015), we set κ = n +2. c c For the prior of the VAR coefficients ϕ in (12.3), we assume a conventional Minnesota prior with hyperparameter for the overall tightness equal to 0.2 in line with Giannone et al. (2015). Because the cyclical component in (5.2) is assumed to be stationary, we then center the prior for each variable’s own lag to 0, rather than 1, as in Del Negro et al. (2017). 4.3 Identifying shocks that drive business-cycle fluctuations AsdiscussedintheIntroduction, theTC-VARmodeldeliversadecompositionbetweentrendsand cycles. Given that the cyclical components are already cleaned of movements at frequencies other than business cycles, we do not need to remove the low-frequency variation by using spectral analysis. Instead,welookforthecombinationofreduced-formshocksthatexplainsthelargestpossible share of unemployment or output cycles, without having to take a stance on which frequencies correspond to the business cycle. In our baseline analysis, we ask how much the unemploymentidentified or output-identified shock contributes to the volatility of the cyclical component of the other variables, with a special focus on inflation and inflation expectations. As a robustness check, we also ask if the results are sensitive to further removing high-frequency movements in the cycles. In this second case, we ask how much the unemployment-identified shock contributes to the volatility of the cyclical component of the other variables at frequencies that correspond to fluctuations with duration of at least 1.5 years. Thus, in this second methodology we take into account that the cyclical component of the variables could present some residual high-frequency movements that are not related to the business cycle. 5 Results In this section, we present the main results of the paper. We first present the decomposition of the variables in trends and cycles. We then proceed to analyze how much the unemploymentidentified shocks can explain of the cyclical variation of inflation and inflation expectations and how it propagates through real and nominal variables. 15

Figure 2: Data, trends and cycles (A) Trends 15 12 12 20 10 9 9 15 5 10 6 6 0 5 -5 3 3 0 -10 0 0 -5 1960 1980 2000 2020 1960 1980 2000 2020 1960 1980 2000 2020 1960 1980 2000 2020 12 12 12 9 9 9 6 6 6 3 3 3 0 0 0 -3 -3 -3 1960 1980 2000 2020 1960 1980 2000 2020 1960 1980 2000 2020 (B) Cycles 25 4 4 15 10 0 5 0 0 0 -25 -5 -50 -4 -4 -10 1960 1980 2000 2020 1960 1980 2000 2020 1960 1980 2000 2020 1960 1980 2000 2020 9 4 4 6 2 2 3 0 0 0 -3 -6 -2 -2 1960 1980 2000 2020 1960 1980 2000 2020 1960 1980 2000 2020 Notes: Thefigureplotsthedata(redlines)usedfortheestimationoftheTC-VARmodelover1960-2019periodas wellastheposteriormedianoftheirlatenttrends(bluelines)inpanel(A)andlatentcycles(bluelines)inpanel(B) and the corresponding 68-percent posterior-coverage intervals (shaded blue areas). NBER recessions are denoted by shaded grey areas. Sources: Federal Reserve Bank of St. Louis; Federal Reserve Bank of Philadelphia; authors’ calculations. 5.1 Estimated latent trends and cycles Panel (A) of Figure 2 plots the data (red lines) used for the estimation of the VAR with common trends over the 1960-2019 period as well as the posterior median of their latent trends (blue lines) and the corresponding 68-percent posterior-coverage intervals (shaded blue area). Panel (B) of Figure 2 plots the posterior median of the latent cycles (blue lines) and the corresponding 68-percent posterior-coverage intervals (shaded blue area). The results confirm some stylized facts about the US economy that are commonly accepted. 16

Table 2: Variance contributions of unemployment shock All frequencies (0−∞ quarters) Unempl. rate Output Unempl. rate FFR Inflation Inflation exp.(1y) exp. 71.8 58.2 68.6 63.6 30.7 49.2 [60.0,84.5] [47.6,70.1] [56.0,80.3] [39.4,78.9] [11.8,50.5] [24.8,70.0] All-but-short-run frequencies (6−∞ quarters) Unempl. rate Output Unempl. rate FFR Inflation Inflation exp.(1y) exp. 72.8 57.8 71.7 64.4 34.4 52.3 [60.7,86.1] [46.9,70.3] [58.7,85.1] [39.4,79.9] [13.2,55.3] [27.2,74.1] Notes: Theshockisidentifiedbymaximizingitscontributiontothevolatilityofthecyclicalcomponentofrealized unemployment rate. We consider two cases. In the first case, the shock is chosen to maximize the fraction of the volatilityoverallthefrequenciesofthecycle,whileinthesecondcaseweexcludefrequenciesthatimplycyclesless than1.5years. Wereportthemediancontributionandthecorresponding68-percentposterior-coverageintervalof the identified shock to the variance of the cycle of all variables over the corresponding frequencies. Sources: Federal Reserve Bank of St. Louis; Federal Reserve Bank of Philadelphia; authors’ calculations. First, in the 1960s and 1970s the U.S. economy experienced an increase in trend inflation. This was possibly caused by the attempt of policymakers to counteract a break in productivity that manifesteditselfwithanincreaseinthenaturalrateofunemploymentortopartiallyaccommodate the inflationary pressure resulting for a large increase in spending that occurred starting from the mid-1960s. These two stylized facts are captured by an increase in the trend components of inflation and unemployment rate during those years. The appointment of Volcker marked a change in the conduct of monetary policy. Trend inflation declined, and so did the long-term inflation expectations. Note that even if we do not impose any restriction on the mapping from the trend component of inflation to long-term inflation expectations, the two variables largely coincide. Thus, including long-term inflation expectations helps in separating trend and cycle fluctuations. The behavior of the cycles is reported in panel (B) of Figure 2. From this figure, a clear pattern emerges, consistent with our understanding of how the economy behaves over the business cycle. The unemployment rate increases during recessions and smoothly declines over time as the economy recovers. Based on a cursory look at the cycles, inflation seems to behave as the New Keynesianframeworkwouldsuggest: Decliningduringarecession, whentheunemploymentrateis high, and increasing during an expansion when the unemployment rate is low. This is particularly visible when focusing on inflation expectations at the one-year horizon. The cyclical component of this variable behaves very much like inflation, but it is smoother. 17

Table 3: Variance contributions of GDP-identified business cycle shock All frequencies (0−∞ quarters) Output Unempl. rate Unempl. rate FFR Inflation Inflation exp.(1y) exp. 65.1 63.6 60.5 43.1 31.5 42.3 [57.6,75.9] [46.2,76.7] [43.1,73.3] [11.3,83.6] [10.3,62.1] [12.0,79.9] All-but-short-run frequencies (6−∞ quarters) Output Unempl. rate Unempl. rate FFR Inflation Inflation exp.(1y) exp. 65.4 63.0 61.7 44.9 35.9 49.0 [57.7,76.9] [45.9,77.7] [44.1,76.3] [11.3,85.0] [11.1,66.6] [13.6,84.0] Notes: The shock is identified by maximizing its contribution to the volatility of the cyclical component of real GDP (in loglevels). We consider two cases. In the first case, the shock is chosen to maximize the fraction of the volatilityoverallthefrequenciesofthecycle,whileinthesecondcaseweexcludefrequenciesthatimplycyclesless than1.5years. Wereportthemediancontributionandthecorresponding68-percentposterior-coverageintervalof the identified shock to the variance of the cycle of all variables over the corresponding frequencies. Sources: Federal Reserve Bank of St. Louis; Federal Reserve Bank of Philadelphia; authors’ calculations. 5.2 Inflation and unemployment over the business cycle We now move to formally study the relation between the real economy and inflation over the business cycle. We use the estimated TC-VAR to identify the unemployment cycle shock using the method described in Subsection 4.3. Specifically, the shock is identified by maximizing its contribution to the volatility of the cyclical component of unemployment. As explained above, we consider two cases. In the first case, the shock is chosen to maximize the fraction of the volatility over all the frequencies of the cycle, while in the second case we exclude frequencies that imply cycles less than 1.5 years. The top panel of Table 2 reports the median and the 68% posterior-coverage interval for the contribution of the identified shock to the variance of the cycle of all the other variables. In the second panel of the table we repeat the exercise by excluding frequencies that imply cycles less than 1.5 years. Not surprisingly, the shock can explain a large share of the fluctuations of the unemployment-rate cycle. However, the shock can also explain a sizable fraction of the cyclical component of realized inflation. In the baseline scenario, the unemployment-identified shock can explain around 30% of the inflation cycle. When excluding cycles shorter than 1.5 years, the unemployment-identified shock explains nearly 35% of inflation variability. These are large shares when considering that the unemployment shock does not explain the entirety of unemployment fluctuations either, but only around 72%. The results are even stronger when focusing on the cyclical component of inflation expectations: 18

Figure 3: Forecast error variances of unemployment shock 100 100 100 50 50 50 0 0 0 0 20 40 0 20 40 0 20 40 100 100 100 50 50 50 0 0 0 0 20 40 0 20 40 0 20 40 Notes: Thefigureshowsthecontributionoftheunemployment-identifiedshocktotheforecasterrorvarianceofthe cyclicalcomponentofallthevariablesatdifferenttimehorizons. Thefigureplotstheposteriormedian(bluelines) and the corresponding 68-percent posterior-coverage intervals (shaded blue area). Sources: Federal Reserve Bank of St. Louis; Federal Reserve Bank of Philadelphia; authors’ calculations. approximately 49% in the baseline scenario—or 52%, in the alternative scenario—of the businesscycle variability of underlying inflation is explained by the unemployment-identified shock. Given that the cycle of inflation expectations appears to be a smoother version of the cycle of realized inflation, this result corroborates the finding that inflation moves in a way consistent with the New Keynesian framework over the business cycle. When comparing the results of the baseline and alternative cases, we find that the contribution for inflation and inflation expectations goes visibly up when removing the short cycles. This implies that there is likely to be some residual high-frequency variation in these variables that it is not related to the business cycle. In addition, the shock explains about 64% of the volatility of cyclical nominal interest rate over both frequency bands. Combined with the previous two findings, this result implies that the shock also explains the volatility of the cyclical component of the real interest rate, defined as the FFR minus expected inflation. Finally, the identified shock explains a large portion of the volatility of cyclical real GDP (in loglevels) over all frequencies and also when excluding frequencies associated with the first 6 quarters of the cycles. The shock also explains a share of the volatility of the cyclical component of expected unemployment rate similar to the corresponding share for realized unemployment. All these findings support the evidence of a main driver of U.S. business-cycle fluctuations as found by Angeletos et al. (2020). The results are similar when using GDP to identify the business cycle shock. Table 3 reports the median contribution—and the corresponding 68-percent posterior-coverage interval—of the shock identified targeting the cycle of real GDP (in loglevels) to the variance of the cycles of all the variables. As before, we consider two cases. In the first case, we identify the shock and compute its contributions based on all frequencies of the cycles. In the second case, we consider 19

Figure 4: Impulse responses to unemployment shock 0.2 4 0.2 0 2 0 -0.2 0 -0.2 -0.4 -2 -0.4 0 10 20 0 10 20 0 10 20 1 0.4 0.4 0.3 0.3 0.2 0.2 0.5 0.1 0.1 0 0 0 -0.1 -0.1 0 10 20 0 10 20 0 10 20 Notes: The figure shows the response to the unemployment-identified shock of the cyclical component of all the variables over a period of 20 quarters. The figure plots the posterior median (blue lines) and the corresponding 68-percent posterior-coverage intervals (shaded blue area). Sources: Federal Reserve Bank of St. Louis; Federal Reserve Bank of Philadelphia; authors’ calculations. frequenciesthatimplyfluctuationsofatleast1.5years. Asinthecaseofunemployment,theshock can explain a large share of the cyclical fluctuations of real GDP. In line with the results for the unemployment-identified shock in Table 2, the shock also explains a sizable fraction of the cyclical component of realized and expected inflation as well as realized and expected unemployment rate. Additionally, the contribution of the output-identified shock for the variability of the realized and expected inflation increases noticeably when movements at frequencies shorter than 1.5 years are excluded. Given that the results are similar across the two specifications, in the rest of the paper we focus on the unemployment-identified shock. InFigure3, weshowthecontributionoftheunemployment-identifiedshocktotheforecasterror variance of the cyclical component of all the variables at different time horizons. The explanatory power of the shock reaches about 80% and 90% of the cyclical movements in unemployment and output after the first five years and about 70% of the movements in the nominal interest rate at any horizon. Moreover, the shock explains a large portion of the movements in inflation, reaching about30%afterfiveyears. Theresultforinflationandinflationexpectationsisconsistentwiththe motivatingevidenceofSection2thatshowsthatinflationcycleslagoutputcycles. Thispercentage rises to about 50% after the first few years for inflation expectations. These results are in line with the shock’s contributions reported in Table 2 and clearly point to the large explanatory power of the unemployment-identified shock for the business-cycle movements in all inflation measures. Finally, Figure 4 plots the median response—and corresponding 68-percent posterior-coverage intervals—of each cyclical component to the unemployment-identified shock over a period of 20 quarters. The resulting interpretation of the unemployment-identified shock is in line with a 20

Table 4: Variance contributions of unemployment shock: When trends are constant Unempl. rate Output Unempl. rate FFR Inflation Inflation exp.(1y) exp. 89.7 46.6 92.5 44.7 9.2 7.3 [86.7,92.4] [39.9,54.5] [89.5,94.6] [37.0,53.2] [4.9,14.9] [3.4,12.5] Notes: Theshockisidentifiedbymaximizingitscontributiontothevolatilityofthecyclicalcomponentofrealized unemployment rate over business-cycle frequencies (6-32 quarters). We report the median contribution and the corresponding 68-percent posterior-coverage interval of the identified shock to the variance of the cycle of all variables over the same frequencies. Sources: Federal Reserve Bank of St. Louis; Federal Reserve Bank of Philadelphia; authors’ calculations. demand shock in a canonical New-Keynesian model. The unemployment rate decreases about 2% percent on impact and subsequently returns to its initial level after about 2.5 years. The response of the one-year-ahead unemployment rate expectations is quantitatively and qualitatively similar to that of the unemployment rate. Considering real GDP (in loglevels), the shock causes an increase by nearly 3% on impact, and its effect gradually vanishes after about 2 years. When considering the nominal variables, the shock leads to a contemporaneous increase of about 0.2% (0.1%) in the cyclical component of realized (expected) inflation and a subsequent decline to the initial level of inflation after about 4 (5) years. In response to the identified shock that boosts real economic activity and increases realized and expected inflation, the nominal interest rate peaks at nearly .7% after about a year and gradually returns to its initial value thereafter. To summarize, the responses of the real side of the economy are consistent with the findings of Angeletos et al. (2020) who point to the presence of a main shock driving the fluctuations of real economic activity over the business cycle. However, differently from their findings, the unemployment shock that we identify has significant effects also on the nominal side of the economy. 6 Robustness checks In this section, we investigate the robustness of our results to different choice of priors and alternative specifications of the TC-VAR model. 6.1 Importance of time-varying trends Our previous results highlight the importance of properly capturing low-frequency movements in realandnominalvariables. Bytreatingthelatenttrendsastime-varyingobjects,ourmodelisable to assess the relationship between those variables at business-cycle frequencies. To elucidate this point,weconsideraconstrainedspecificationinwhichalltrendsareassumedconstant. Specifically, we leave unchanged all other assumptions about priors and initial conditions and simply set the 21

Table 5: Variance contributions of the unemployment shock under alternative priors Unempl. rate Output Unempl. rate FFR Inflation Inflation exp.(1y) exp. Conservative 68.6 59.3 66.0 57.2 30.5 45.3 priors [58.6,81.3] [46.9,74.2] [54.8,79.3] [26.3,81.7] [10.6,55.6] [15.3,76.2] Tight 67.0 62.6 65.0 48.0 28.4 38.7 priors [57.6,78.1] [45.9,79.2] [55.1,77.8] [12.8,79.7] [9.9,61.4] [9.4,78.4] Notes: Theshockisidentifiedbymaximizingitscontributiontothevolatilityofthecyclicalcomponentofrealized unemploymentrateoverallthefrequenciesofthecycle. Wereportthemediancontributionandthecorresponding 68-percent posterior-coverage interval of the identified shock to the variance of the cycle of all variables over the corresponding frequency. We consider two priors for the standard deviation of the expected change in the trend unemployment rate. The “Conservative” prior sets that standard deviation to 1% over a period of 30 years, while the “Tight” prior assumes the same standard deviation but over a period of 40 years. Sources: Federal Reserve Bank of St. Louis; Federal Reserve Bank of Philadelphia; authors’ calculations. standard deviation of shocks to the trends to zero. Table 4 reports the contributions of the identified shock to the volatility of the cyclical components of the unemployment rate as well as theremainingvariablesoverthesamefrequencies. Notethatnowtheshockonlyexplainsabout9% of the volatility of the realized inflation cycle and about 7% of the volatility of expected inflation. The estimate of the shock’s contribution to the volatility of inflation is well within the range of estimates provided in Angeletos et al. (2020). The similarity of our findings under this scenario to those of Angeletos et al. (2020) is not surprising. Once the latent trends are assumed to be constant, our model collapses to a VAR model, with the only distinction that output is modeled in deviations from a linear trend and the other variables are demeaned. 6.2 Alternative priors In this subsection, we show that our baseline results with time-varying trends are not sensitive to the choice of alternative priors for the standard deviation of the shock to the trend unemployment rate. In our baseline specification, we set the standard deviation for the expected change in the trend unemployment rate to 1% over a time period of 20 years. We now consider two alternative priors: a “Conservative” prior which sets that standard deviation to 1% over a period of 30 years anda“Tight” priorwhichassumesthesamestandarddeviationbutoveraperiodof40years. Table 5 reports the median contributions—and the associated 68-percent posterior-coverage intervals— of the unemployment-identified shock to the variability of each cyclical component under the two alternative priors. Setting an increasingly tighter prior on the standard deviation of the shock to the trend unemployment rate does not affect the results. The shock explains between about 60 - 70% of the cyclical components of the real variables. Similarly, the shock’s contributions on 22

the variability of the cyclical components of the nominal variables are also robust to the choice of either prior. 6.3 Alternative specifications We briefly discuss results for the three alternative specifications described in Subsection 3.2. In the interest of space, we only discuss the variance contribution of the unemployment shock to the inflation cycle. We report the details for these alternative specifications in Appendix D.2. Our results are robust to all considered alternative specifications. In the first alternative specification, we assume a more parsimonious specification than our baseline. Over the business-cycle frequencies, the identified shock explains nearly 40% of the common, cyclical component of inflation over all frequencies and nearly 45% over the frequencies that exclude the first 1.5 years. The second alternative decomposition is based on the assumption that, as under the baseline, realized unemployment rate and one-year-ahead unemployment rate expectations share a common trend and each measure follows an idiosyncratic cycle. However, this more flexible specification allows one- and ten-year-ahead inflation expectations to each follow an individual cycle while sharing a common trend as shown in (8). We find that the unemployment shock explains nearly 34% of the volatility of the cyclical component of one-year-ahead inflation expectations, a measure of underlying inflation. When excluding movements in cyclical components at frequencies higher than 1.5 years, the results point to an explanatory power of about 39%. The third and final decomposition verifies the robustness of the main results to the exclusion of one-year-ahead unemployment expectations. The results point to contributions of the identified shock on inflation and expected inflation of about 44% and 51%, respectively. Those contributions further increase to about 50% and 57% respectively when the shock is identified over frequencies that exclude cycles of less than 1.5 years. 7 Revisiting VAR models In this section, we show that the use of a standard VAR, as opposed to a TC-VAR, can lead to verydifferentconclusionsaboutthelinkbetweenrealactivityandinflationoverthebusinesscycle. This is because the VAR parameters need to account at the same time for the low-frequency and business-cycle frequency variation observed in the data with a finite number of observations. We check whether this discrepancy disappears when imposing alternative priors or when considering a sample that presents less low-frequency variation. We find that while these extensions help, the results are still quite different from our baseline analysis based on the TC-VAR. 23

Table 6: Variance contributions of unemployment shock: 1955-2019 Unemployment Output Investment Consumption Hours Minnesota 91.9 71.4 74.4 47.3 75.1 [88.3,94.5] [66.9,77.5] [69.4,79.5] [37.9,54.4] [70.7,78.8] Minnesota 92.4 72.8 74.6 49.7 75.9 + Long-run [88.7,94.7] [68.6,76.1] [70.7,77.6] [44.2,58.4] [73.1,79.4] TFP Labor share Inflation FFR Minnesota 19.2 15.1 7.6 36.3 [11.3,28.1] [9.8,22.7] [3.9,17.9] [26.5,45.4] Minnesota 18.5 14.9 11.5 38.2 + Long-run [10.2,24.4] [11.0,21.6] [5.9,17.8] [32.2,45.9] Notes: The shock is identified by maximizing its contribution to the volatility of the unemployment rate over business-cycle frequencies (6-32 quarters). We report the median contribution and the corresponding 68-percent posterior-coverage interval of the identified shock to the variance of all the variables over the same frequencies. Source: Angeletos et al. (2020); authors’ calculations. 7.1 Using a VAR to assess the link between inflation and real activity Alternative priors. In this subsection, we revisit the results obtained by Angeletos et al. (2020) by allowing for different priors and data periods. We start by extending the estimation sample of Angeletos et al. (2020) by two years so that the periods match those of our baseline analysis, i.e., 1955:Q1 to 2019:Q4. As detailed in Section I of their paper, the data consist of quarterly observations on the following macroeconomic variables: the unemployment rate; the real, percapita levels of GDP, investment, consumption; hours worked per person; the level of utilizationadjusted total factor productivity (TFP); the labor share; the inflation rate, as measured by the rate of change in the GDP deflator; and the nominal interest rate, as measured by the federal funds rate.4 As in their baseline specification, we proceed to estimate a VAR model with two lags using Bayesian methods. However, we consider two sets of priors. In one case, we follow their approach and use a Minnesota prior (“Minnesota”). In the other case, we combine a long-run prior à la Giannone et al. (2019a) with the Minnesota prior (“Minnesota+Long-run”). The cointegrating relationshipsthatweassumeamongthevariablesinthelongrunareinlinewiththoseofGiannone et al. (2019a) and are described in Appendix E.1. We allow for separate shrinkage for each active row of the matrix that captures the cointegrating relationship among the variables. In the 4We drop labor productivity in the VAR model because it can be measured by the ratio between output and hours worked. 24

Table 7: Variance contributions of unemployment shock: 1984-2008 Unemployment Output Investment Consumption Hours Minnesota 92.8 54.3 55.3 34.9 69.4 [85.5,95.6] [43.2,67.7] [44.8,67.9] [21.0,49.5] [61.3,79.5] Minnesota 92.8 52.7 55.9 33.1 69.6 + Long-run [87.7,95.3] [42.2,66.6] [45.8,66.1] [21.3,53.4] [61.7,78.1] TFP Labor share Inflation FFR Minnesota 9.4 11.6 15.8 63.2 [3.5,22.1] [3.7,21.3] [7.3,39.3] [56.9,73.9] Minnesota 9.6 9.9 16.3 63.5 + Long-run [3.4,21.5] [3.1,17.5] [7.7,29.1] [54.9,72.3] Notes: The shock is identified by maximizing its contribution to the volatility of the unemployment rate over business-cycle frequencies (6-32 quarters). We report the median contribution and the corresponding 68-percent posterior-coverage interval of the identified shock to the variance of all variables over the same frequencies. Source: Angeletos et al. (2020); authors’ calculations. estimation,weconsiderthesamenumberofdrawsandpre-sampleyearsasinourbaselineanalysis. Hyperparameters for priors are optimized following Giannone et al. (2019a). For each estimation, we then identify the shock targeting the unemployment rate at business-cycle frequencies using the method described in Subsection 4.3. Table 6 reports the contribution of that shock for each of the variables of the VAR model over the same frequencies. TheresultsinTable6indicatethat,evenwhencombiningtheMinnesotaandlong-runpriors,we still recover the evidence of a disconnect between real and nominal variables. While the identified shock explains nearly 8% of the volatility of inflation over the business-cycle when only using Minnesota priors, its contribution slightly increases to about 11% when those priors are combined with long-run priors à la Giannone et al. (2019a). These results are in line with those obtained from a vast variety of specifications that Angeletos et al. (2020) consider, including vector error correction models. Alternative samples. We now consider an estimation sample that is arguably less affected by time-varying trends. Specifically, our sample now starts in 1984:Q1 and ends in 2008:Q4. We estimate the VAR model with two lags and use the two alternative assumptions about the priors: onlya Minnesotaprioror combining the Minnesotaprior with long-runpriors. Table7 reports the contribution of the shock identified targeting the unemployment rate for the volatility of the other variables at business-cycle frequencies. As expected, the results in Table 7 show that restricting thesampletotheperiodoftheGreatModerationimprovesthecontributionoftheidentifiedshock on inflation. However, such contribution is overall small (about 16%) regardless of the priors. 25

Table 8: Inflation variance contribution of unemployment shock: 1984-2008 No long-run Optimized long-run Dogmatic long-run No Minnesota 25.2 26.2 33.1 [9.2,42.6] [13.3,52.4] [11.0,49.0] Optimized Minnesota 20.1 26.2 33.1 [11.0,31.5] [13.3,52.4] [11.0,49.0] Dogmatic Minnesota 2.0 26.2 – [0.4,5.6] [13.3,52.4] Notes: The shock is identified by maximizing its contribution to the volatility of the unemployment rate over business-cycle frequencies (6-32 quarters). We report the median contribution and the corresponding 68-percent posterior-coverage interval of the identified shock to the variance of all variables over the same frequencies. For long-run priors, we are imposing same degree of shrinkage for each active row of the matrix that captures the cointegrating relationship among the variables. Source: Angeletos et al. (2020); authors’ calculations. Dogmatic priors. As a final check, we explore if these results can be changed if we still use the 1984-2008 sample, but also choose the priors dogmatically. The objective is to understand if one can still use VAR models with the help of specific priors. We consider several different combinations of hyperparameters such that the two Minnesota and long-run priors can be either not imposed, optimized, or set dogmatically. For this exercise, it is important to notice that we are imposing the same degree of shrinkage for each active row of the matrix that captures the cointegrating relationship among the variables. Table 8 reports the contribution of unemployment-identified shock on inflation at business-cycle frequencies. The results are straightforward to understand. When only dogmatic Minnesota priors are imposed (see lower left corner of Table 8), the inflation variance contribution of the unemployment shock is nearly zero. It is not exactly zero because the shocks are correlated even when the individual series follow a random-walk process. However, the opposite case of dogmatic long-run priors (see upper right corner of Table 8) leads to a contribution of the unemployment shock to inflation volatility of about 33%.5 This is an interesting finding. It suggests that a VAR might recover results similar to the TC-VAR if (1) tight priors are chosen to reflect the long-run relationships suggested by economic theory and (2) a sample not subject to sever low-frequency variation is chosen. Not properly accounting for those features in the data could lead to imprecise conclusions about how real activity and inflation are connected over the business cycle. 5The fact that the number 26.2% in the middle of Table 8 does not coincide with 16.3% in Table 7 is because we are imposing the same degree of shrinkage for each active row of the matrix that captures the cointegrating relationship among the variables in Table 8 but not in Table 7 which allows for separate shrinkage. 26

Table 9: Variance contributions of unemployment shock: 1984-2008 Unempl. Output Unempl. FFR Inflation Inflation rate exp.(1y) exp.(1y) Minnesota 86.8 57.2 71.4 70.8 15.3 23.6 [79.0,93.0] [45.9,73.1] [61.0,81.6] [61.8,78.8] [5.2,26.2] [13.4,35.1] Minnesota 84.1 64.3 72.2 72.2 16.6 26.9 + Long-run [80.1,90.4] [49.9,74.2] [59.4,81.6] [64.7,78.2] [5.3,27.9] [13.5,47.1] Notes: The shock is identified by maximizing its contribution to the volatility of the unemployment rate over business-cycle frequencies (6-32 quarters). We report the median contribution and the corresponding 68-percent posterior-coverage interval of the identified shock to the variance of all variables over the same frequencies. Sources: Federal Reserve Bank of St. Louis; Federal Reserve Bank of Philadelphia; authors’ calculations. 7.2 Using the same data as in the TC-VAR As a final exercise, we bring our attention back to the data used in Section 5 and described in Subsection 4.1. We focus on the period of the Great Moderation between 1984 and 2008 and examine the extent to which the estimated VAR model can reproduce empirical evidence provided by our baseline TC-VAR model.6 We estimate the model using only a Minnesota prior (“Minnesota”) or combining it with long-run priors (“Minnesota+Long-run”). The cointegrating relationshipsthatweimposeacrossvariablesarereportedinAppendixE.2. Forbothspecifications, we identify the shock targeting the unemployment rate at business-cycle frequencies and report in Table 9 its contribution for other variables. In sum, while the long-run priors are better at separating long-run from short-run dynamics, the evidence suggests the need of going beyond the VAR dynamics to examine the (dis)connect between real activity and inflation over the business cycle. 8 VAR Results through the Lens of a Trend-cycle-VAR In this section, we provide theoretical arguments that motivate the adoption of a TC-VAR model, rather than a standard VAR model, for the results presented in Section 5. In Subsection 8.1, we show that a fixed-coefficient VAR model estimated over a period of time that presents structural changes is misspecified, if the goal is trying to assess the comovement at business-cycle frequency. The misspecification problem associated with the use of a VAR model to describe a data generating process characterized by both low- and high-frequency movements cannot be 6Relative to the baseline specification, we drop the ten-year-ahead inflation expectations as it is not straightforward to impose long-run relationships across two inflation expectations series. 27

easily resolved.7 Even if an econometrician could correctly reconstruct the VAR representation of the TC-VAR model, the parameter estimates of the misspecified model would confound lowfrequency movements associated with the trend with those at business-cycle frequencies related to the cycle. Moreover, the reduced-form innovations that she would recover would not only map into the innovations affecting the latent persistent and stationary components. By contrast, the reduced-form innovations would also capture the error associated with the estimates of the latent components. Of course, in reality, these issues would be exacerbated by the fact that the VAR parameters estimated over a finite sample would be distorted because a single set of parameters would need to account for both trend and cycle fluctuations. To build the intuition about the underlying issues, in Subsection 8.2, we generate data from a Monte Carlo simulation of a bivariate TC-VAR model and show that a VAR model does not succeed in capturing the assumed cyclical relationship between the two variables, even with a long data sample. Moreover, in Appendix F.3, we consider a univariate trend-cycle autoregressive (TC-AR) model of U.S. inflation based on Stock and Watson (2007) and derive analytically its infinite-order autoregressive (AR) representation. 8.1 Trend-Cycle models and their VAR(∞) representation Our goal is to map the state-space representation in (13) presented in Subsection 3.3 into a VAR model. In doing so, we follow the approach proposed in Fernández-Villaverde et al. (2007). For convenience, we report the state representation characterized in (9) and (10) here: z = Λx , (13.1) t t x = Φx +Rε , (13.2) t t−1 t where ε = Qw , E(w w′) = I, and E(ε ε′) = Σ. For all the specifications considered in our t t t t t t analysis, the overall number of shocks of the TC-VAR model is strictly larger than the number of observables. Equivalently, dim(w ) = (n +n +n ) > n = dim(z ). As a result, the ‘poor t τ c η t man’s invertibility condition’ proposed in Fernández-Villaverde et al. (2007) cannot be tested because it requires the number of shocks and observables to coincide. We therefore seek to find the ‘innovations representation’ of (13). Because the innovations representation results from the application of the Kalman filter to the state-space representation, we first ensure the suitability of the filter for our purpose and more specifically its asymptotic stability and convergence. Clearly, these properties of the filter depend on the properties of (13) and should not be taken for granted in our setup: In the transition 7Watson (1986) discusses the equivalence between an unobserved component model and its autoregressive, integrated, moving average (ARIMA) representation, thus pointing to the misspecification problem characterizing a VAR representation of a TC-VAR model. 28

equation(13.2),thecyclicalcomponentsareassumedtobestationary,whiletrendsfollowunit-root processes. However, wefollowAndersonandMoore(1979)whosuggesttoverifytwoconditions: i) the pair (Φ,RQ) is stabilizable; ii) the pair (Φ,Λ′) is detectable—or equivalently, the pair (Φ′,Λ′) is stabilizable.8 For all the specifications of the TC-VAR model that we consider in Section 3, both conditions are satisfied for each draw obtained from the model estimation. Having verified the suitability of the Kalman filter for our purpose, we follow Fernández- Villaverde et al. (2007) to derive the innovations representation. Specifically, we write the representation in (13) as x = Ax +Bw , (14.1) t+1 t t+1 z = Cx +Dw , (14.2) t+1 t t+1 where A = Φ, B = RQ, C = ΛA, D = ΛB and E(w w′) = I. Defining the linear projection of x t t t on zt ≡ {z }t as xˆ ≡ E (cid:0) x |zt(cid:1), the one-step-ahead error associated with the forecast of z as j j=1 t t t+1 Dˆ ν and the term updating the estimator of the state for the next period xˆ as Bˆ ν , t+1 t+1 t+1 t+1 t+1 the application of the Kalman filter to (14) delivers the innovations representation xˆ = Axˆ +Bˆ ν , (15.1) t+1 t t+1 t+1 z = Cxˆ +Dˆ ν , (15.2) t+1 t t+1 t+1 wherex ∼ (xˆ ,Ω ), thecovariancematrixΩ ispositivesemi-definite, andν isavectorofmean- 0 0 0 0 t zero, normal and uncorrelated white-noise innovations such that E(ν ν′) = I. Notably, under this t t representation, the number of shocks and observables coincide. Also, because the innovation ν is t fundamental for z by definition, it is uncorrelated with z and ultimately ν for any s ≥ 0. t t−s t−s The innovations representation in (15) shows that, with a finite sample {z }T where T < ∞, it t t=1 is not possible to derive a VAR representation because the matrices Bˆ and Dˆ depend on time t. t t As a result, we consider the limit case for T approaching infinity. Because the asymptotic stability and convergence of the Kalman filter hold, the matrices Bˆ and Dˆ also converge to their timet t invariant counterparts Bˆ and Dˆ.9 Therefore, we can derive the infinite-order VAR representation. Solving (15.2) for ν and combining with (15.1), we obtain t+1 (cid:104) (cid:16) (cid:17) (cid:105) I − A−BˆDˆ−1C L xˆ = BˆDˆ−1z (16) t+1 t+1 where L denotes the lag operator. The asymptotic properties of the Kalman filter also guarantee 8We provide the definition of stabilizability in Appendix F.1. For further details, refer to Burridge and Wallis (1983), Sargent (1988), Anderson et al. (1996) and Hansen and Sargent (2008, 2013) among others. We refer the reader to pp. 76-82 and Appendix C in Anderson and Moore (1979) for the details on the conditions mentioned here and the definition of stabilizability. 9Appendix F.2 provides the details on the equations for matrices Bˆ and Dˆ. 29

(cid:16) (cid:17) that all the eigenvalues of the matrix A−BˆDˆ−1C are all strictly inside the unite circle in modulus(AndersonandMoore,1979). Therefore, wesolveequation(16)forxˆ andplugthesolution t+1 in the time-invariant version of equation (15.2) to obtain the following VAR(∞) representation (cid:104) (cid:16) (cid:17) (cid:105)−1 z = C I − A−BˆDˆ−1C L BˆDˆ−1z +Dˆν t+1 t t+1 ∞ (cid:88) (cid:16) (cid:17)s = C A−BˆDˆ−1C BˆDˆ−1z +Dˆν , (17) t−s t+1 s=0 where we have used the fact that the inverted matrix in square brackets in the first equation is a square summable polynomial in L. Equation (17) allows us to reach three important conclusions. First, the state-space representation of the TC-VAR model in (13) maps into the infinite-order VAR representation in (17) under the assumption that infinite data are available. As a result, a finite-order VAR with finite data cannot capture the dynamics described by the decomposition of the observables z into trends t and cycles. Second, even if infinite data were available, equation (17) clarifies that estimates of the autoregressive parameters associated with the VAR(∞) representation confound movements of z that are driven by both the trend and cycle. Equivalently, the VAR(∞) representation t+1 cannot disentangle movements of z at low frequencies from those at cyclical frequencies. Fit+1 nally, asshowninFernández-Villaverdeetal.(2007), theinnovationsassociatedwiththeVAR(∞) representation, Dˆν , capture not only the shocks to the latent trends and cycles, Dw , but t+1 t+1 also the error associated with the estimate of those latent components, C(x −xˆ ). In Appendix t t F.3, we provide a simple analytical example based on an unobserved components model of U.S. inflation by Stock and Watson (2007) to show the intuition for these results. With these results, we are not arguing for the unconditional superiority of a TC-VAR over a VAR. Over the past four decades, economists have used VARs as extremely flexible econometric models capable of uncovering a variety of enlightening empirical results. Our point is that for the specific question of assessing the strength of the relation between inflation and real activity over the business cycle, a TC-VAR appears to be a more effective tool. 8.2 Monte Carlo simulations of a bivariate TC-VAR model To provide an intuitive example, let us assume that the data generating process for the unemployment rate and inflation rate, z = {u ,π }′, is described by the measurement equation z = τ +c , t t t t t t where the dynamics of the trend τ and cyclical component c can be modeled as: t t τ = τ +ε , (18.1) t t−1 τ,t c = Φ c +ε , (18.2) t 1 t−1 c,t 30

where τ = {τ ,τ }′, c = {c ,c }′, ε = {ε ,ε }′ and ε = {ε ,ε }′. In this t u,t π,t t u,t π,t τ,t τ,u,t τ,π,t c,t c,u,t c,π,t example, we assume that (18.2) is (cid:34) (cid:35) (cid:34) (cid:35)(cid:34) (cid:35) (cid:34) (cid:35) c ρ 0 c ε u,t uu u,t−1 c,u,t = + c −(1−ρ ) ρ c ε π,t ππ ππ π,t−1 c,π,t implyingthat, whilethecyclicalcomponentoftheunemploymentrateonlydependsonitslag, the cyclicalcomponentofinflationdependsonitslagaswellasonthelaggedcyclicalcomponentofthe unemployment rate. We also assume that the shocks are independent and identically distributed: (cid:34) (cid:35) (cid:32)(cid:34) (cid:35) (cid:34) (cid:35)(cid:33) (cid:34) (cid:35) (cid:34) (cid:35) ε 0 Σ 0 σ 0 σ 0 τ,t τ τ,u c,u ε = ∼ N , , Σ = , Σ = . t τ c ε 0 0 Σ 0 σ 0 σ c,t c τ,π c,π Within this framework, we consider two cases of interest and generate Monte Carlo simulations. Case I: Trend in inflation. In the first case, unemployment does not feature low-frequency variationanditisonlydrivenbyitsbusiness-cyclemovements,whileinflationalsofeatureschanges in the trend. We consider different degrees of low-frequency variation for inflation, while always maintaining the assumption that its cycle is affected by cyclical movements in unemployment. Specifically, for the unemployment rate, we set the autoregressive parameter ρ to 0.95 and uu normalize the standard deviation of the shock to the cycle σ to 1. Because we assume that c,u the unemployment rate is not subject to low-frequency movements, we turn off the shocks to its trend (σ =0). Focusing on inflation, we assume that its cyclical movements are only driven by τ,u the corresponding movements of the unemployment rate, thus assuming ρ = 0 and σ = 0. ππ c,π However,tocapturethedegreetowhichthelow-frequencymovementsdriveinflationdynamics,we consider three values for the standard deviation of the shock to the inflation trend (σ ). Relative τ,π to the normalized standard deviation of the shock to the cyclical component of the unemployment rate, the standard deviation of the shock to inflation trend is one order of magnitude smaller (σ τ,π = 0.1), equivalent (σ =1) or twice as large (σ =2). For each calibration, we produce several, τ,π τ,π long Monte Carlo simulations that we use iteratively to estimate a VAR model, identify the shock targetingtheunemploymentrateatbusiness-cyclefrequenciesandultimatelycomputethemedian contribution of the identified shock to the variance of both series over the same frequencies.10 Foreachcalibrationofthestandarddeviationoftheshocktoinflationtrend,Table10reportsthe median and 68% posterior-coverage intervals of the median contributions of the identified shock. As expected, the identified shock fully explains the cyclical movements of the unemployment rate regardless of the chosen calibration. However, the explanatory power of the unemploymentidentifiedshockforinflationdependsontheimportanceofthelow-frequencyvariationininflation. 10We generate 500 Monte Carlo simulation of 50,000 observations of which we keep the last 1,000 for each simulation. To estimate each VAR, we choose the lags to deliver the lowest Bayesian Information Criterion (BIC), useaMinnesotapriorasinAngeletosetal.(2020),set50,000drawsandkeepthelast1,000toidentifytheshock. 31

Table 10: Variance contribution of unemployment shock (data simulated with σ = 0) τ,u Unemployment Inflation σ = 0.1 100.0 98.8 τ,π [100.0,100.0] [98.7,98.9] σ = 1 100.0 17.9 τ,π [100.0,100.0] [15.3,20.4] σ = 2 100.0 4.5 τ,π [100.0,100.0] [3.3,6.1] Notes: The shock is identified by maximizing its contribution to the volatility of the unemployment rate over business-cycle frequencies (6-32 quarters). We report the median and the corresponding 68-percent posteriorcoverage interval of the median contributions of the identified shock to the variance of all variables over the same frequencies. Tosimulatethedata,weusethefollowingcalibrations. Fortheunemploymentrate,wesetρ =0.95, uu σ =1 and σ =0. For the inflation rate, we set ρ =0 and σ =0 and σ ={0.1,1,2}. c,u τ,u ππ c,π τ,π Thehigherthelow-frequencyvariation,thelowerthedegreetowhichtheunemployment-identified shock explains business-cycle movements in inflation. This conclusion holds even with long data samples and in presence of strong assumptions about the cyclical relationship between the unemployment rate and inflation. In line with our results above, the identification of the shock at business-cycle frequencies does not succeed in extracting the cyclical relationship between unemployment and inflation because the fixed-coefficient VAR fails to separate cycle and trend innovations in inflation. Case II: Trend in unemployment. The second case considers the opposite instance relative to the first case. We introduce low-frequency movements in the unemployment rate, while the inflation rate only follows the cyclical component of the unemployment rate. In particular, the unemployment-rate cycle evolves as in the previous case—that is ρ = 0.95 and σ = 1. Howuu c,u ever, we also allow shocks to the trend unemployment rate and consider three calibrations for its standard deviation σ = {0.1,1,2}. For the inflation rate, we assume that its process is fully τ,u explained by the cyclical component of the unemployment rate. This assumption is evidently unrealistic but is chosen to ensure that the innovations to the cyclical component of the unemployment rate are the only source of fluctuations for inflation. Implementing this assumption requires turning off the shocks to both the trend and cyclical components of inflation—that is, σ = σ = 0—and setting the autoregressive parameter ρ to zero. As a result, the simulated τ,π c,π ππ inflation rate evolves as π = −c . t u,t−1 Table 11 reports the median and 68% posterior-coverage intervals of the median contributions of the identified shock. The unemployment-identified shock explains nearly the entirety of the business-cyclemovementsinunemploymentforallcalibrations, butsmallerportionsofmovements in inflation as the unemployment rate is increasingly driven by low-frequency movements. Thus, theshocksthatarerecoveredbytheproceduredonotcoincidewiththestructuralshocksproduced 32

Table 11: Variance contribution of unemployment shock (data simulated with σ = 0) τ,π Unemployment Inflation σ = 0.1 100.0 98.8 τ,u [99.9,100.0] [98.7,98.9] σ = 1 99.3 14.3 τ,u [98.7,99.7] [11.9,16.8] σ = 2 99.6 3.2 τ,u [99.0,99.9] [2.2,4.4] Notes: The shock is identified by maximizing its contribution to the volatility of the unemployment rate over business-cycle frequencies (6-32 quarters). We report the median and the corresponding 68-percent posteriorcoverage interval of the median contributions of the identified shock to the variance of all variables over the same frequencies. Tosimulatethedata,weusethefollowingcalibrations. Fortheunemploymentrate,wesetρ =0.95, uu σ =1 and σ ={0.1,1,2}. For the inflation rate, we set ρ =0 and σ =0 and σ =0. c,u τ,u ππ c,π τ,π by the true data generating process. By construction, the identified shocks explain a large fraction of unemployment variation at business cycle frequency, but this assessment is based on the VAR parameter estimates, not the true parameters of the TC-VAR generating the data. Notsurprisingly,theresultsgetevenworseiftheunderlyingtruedatageneratingprocessfeatures trendsinbothinflationandunemployment. WepresentresultsforthiscaseinAppendixF.4. Once more, these results confirm the importance of explicitly controlling for low-frequency movements in both inflation and unemployment rate to appropriately extract the relationship between these two series over the business cycles. 9 Conclusions In recent years, a series of papers have called into question the validity of the New Keynesian framework. One important argument is the observation that inflation seems to be unresponsive to business-cycle movements in real activity. In this paper, we used a Trend-Cycle VAR model to study the relation between inflation and the real economy over the business cycle. The Trend- Cycle VAR model has the virtue to remove low-frequency movements in inflation and real activity that can contaminate inference about the VAR parameters and innovations. We show that at business-cycle frequencies, fluctuations of inflation are related to movements in real activity, in line with what implied by the New Keynesian framework. We see three distinct directions for future research: first, to investigate the drivers of low-frequency movements in the macroeconomy; second, to allow for the possibility of multiple shocks at business-cycle frequencies to separate demand-driven and supply-driven fluctuations; third, to apply the same methods to study the cyclical behavior of other key macroeconomic variables that feature a strong trend component, such as the employment-to-population ratio (Fukui et al., 2023). 33

References Anderson, B. D. O. and Moore, J. B. (1979). Optimal Filtering. Prentice-Hall, Inc., New Jersey. Anderson, E. W., Hansen, L. P., McGrattan, E. R., and Sargent, T. J. (1996). Mechanics of Forming and Estimating Dynamic Linear Economies. In Amman, H. M., Kendrick, D. A., and Rust, J., editors, Handbook of Computational economics, volume 1, pages 171–252. North- Holland. Angeletos, G.-M., Collard, F., and Dellas, H. (2018). Quantifying Confidence. Econometrica, 86(5):1689–1726. Angeletos, G.-M., Collard, F., and Dellas, H. (2020). Business-Cycle Anatomy. American Economic Review, 110(10):3030–3070. Ascari, G. and Fosso, L. (2021). The Inflation Rate Disconnect Puzzle: On the International Component of Trend Inflation and the Flattening of the Phillips Curve. Norges Bank Research, Working Paper no. 17. Bańbura,M.,Giannone,D.,andLenza,M.(2015). Conditionalforecastsandscenarioanalysiswith vector autoregressions for large cross-sections. International Journal of Forecasting, 31(3):739– 756. Basu, S., Candian, G., Chahrour, R., and Valchev, R. (2021). Risky Business Cycles. NBER WP 28693. Beaudry, P., Galizia, D., and Portier, F. (2020). Putting the Cycle Back into Business Cycle Analysis. American Economic Review, 110(1):1–47. Beaudry, P. and Portier, F. (2013). Understanding Noninflationary Demand-Driven Business Cycles. NBER Macroeconomics Annual, 28(1):69–130. Bianchi, F. (2013). Regime Switches, Agents’ Beliefs, and Post-Wold War II U.S. Macroeconomic Dynamics. Review of Economic Studies, 80(2):463–490. Bianchi, F. and Ilut, C. (2017). Monetary/Fiscal Policy Mix and Agents’ Beliefs. Review of Economic Dynamics, 26:113–139. Burridge, P. and Wallis, K. F. (1983). Signal Extraction in Nonstationary Series. University of Warwick working paper no. 234. Carter, C. K. and Kohn, R. (1994). On Gibbs Sampling for State Space Models. Biometrika, 81(3):541–553. Christiano, L. and Fitzgerald, T. (2003). The Band Pass Filter. International Economic Review, 44(2):435–465. Clarida, R., Gali, J., and Gertler, M. (2000). Monetary policy rules and macroeconomic stability: evidence and some theory. The Quarterly journal of economics, 115(1):147–180. 34

Crump, R. K., Eusepi, S., Giannone, D., Qian, E., and Sbordone, A. M. (2021). A Large Bayesian VAR of the United States Economy. Federal Reserve Bank of New York Staff Report No. 976. Del Negro, M., Giannone, D., Giannoni, M., and Tambalotti, A. (2017). Safety, Liquidity, and the Natural Rate of Interest. Brookings Papers on Economic Activity, Spring:235–294. Del Negro, M., Giannone, D., Giannoni, M., and Tambalotti, A. (2019). Global trends in interest rates. Journal of International Economics, 118:248–262. Del Negro, M., Lenza, M., Primiceri, G. E., and Tambalotti, A. (2020). What’s up with the Phillips Curve? Brookings Papers on Economic Activity, Spring:301–357. Del Negro, M. and Schorfheide, F. (2013). DSGE Model-Based Forecasting. In Elliot, G. and Timmermann, A., editors, Handbook of Economic Forecasting, volume2, pages57–140.Elsevier, New York. Durbin, J. and Koopman, S. J. (2001). Time Series Analysis by State Space Methods. Oxford University Press, second edition. Farmer, R.E.A.andNicolò, G.(2018). KeynesianEconomicswithoutthePhillipsCurve. Journal of Economic Dynamics and Control, 89:137–150. Farmer, R. E. A. and Nicolò, G. (2019). Some International Evidence for Keynesian Economics without the Phillips Curve. The Manchester School, 89(S1):1–22. Farmer, R. E. A. and Platonov, K. (2019). Animal Spirits in a Monetary Model. European Economic Review, 115:60–77. Fernández-Villaverde, J., Rubio-Ramírez, J., Sargent, T. J., and Watson, M. W. (2007). ABCs (and Ds) of Understanding VARs. American Economic Review, 97(3):1021–1026. Fukui, M., Nakamura, E., and Steinsson, J. (2023). Women, Wealth Effects, and Slow Recoveries. American Economic Journal: Macroeconomics, 15(1):269–313. Giannone, D., Lenza, M., Momferatou, D., and Onorante, L. (2014). Short-term inflation projections: A Bayesian vector autoregressive approach. International Journal of Forecasting, 30(3):635–644. Giannone, D., Lenza, M., and Primiceri, G. E. (2015). Prior Selection for Vector Autoregressions. Review of Economics and Statistics, 97(2):436–451. Giannone, D., Lenza, M., and Primiceri, G. E. (2019a). Priors for the Long Run. Journal of the American Statistical Association, 114(526):565–580. Giannone, D., Lenza, M., and Reichlin, L. (2019b). Money, Credit, Monetary Policy, and the Business Cycle in the Euro Area: What Has Changed Since the Crisis? International Journal of Central Banking, 15(5):137–173. Giannone,D.,Reichlin,L.,andSala,L.(2006).VARs,commonfactorsandtheempiricalvalidation of equilibrium business cycle models. Journal of Econometrics, 132(1):257–279. 35

Gust, C., Herbst, E., and López-Salido, D. (2022). Short-Term Planning, Monetary Policy, and Macroeconomic Persistence. American Economic Journal: Macroeconomics, 14(4):174–209. Hansen, L. P. and Sargent, T. J. (2008). Robustness. Princeton University Press, Princeton, NJ. Hansen, L. P. and Sargent, T. J. (2013). Recursive Models of Dynamic Linear Economies. Princeton University Press, Princeton, NJ. Hazell, J., Herreño, J., Nakamura, E., and Steinsson, J. (2022). The Slope of the Phillips Curve: Evidence from U.S. States. The Quarterly Journal of Economics, 137(3):1299–1344. Holston, K., Laubach, T., and Williams, J. C. (2017). Measuring the Natural Rate of Interest: International Trends and Determinants. Journal of International Economics, 108(1):S59–S75. Johannsen, B. and Mertens, E. (2021). A Time Series Model of Interest Rates With the Effective Lower Bound. Journal of Money Credit and Banking, 53(5):1005–1046. Justiniano,A.,Primiceri,G.E.,andTambalotti,A.(2013). IsThereaTrade-OffbetweenInflation and Output Stabilization? American Economic Journal: Macroeconomics, 5(2):1–31. Kadiyala, R. K. and Karlsson, S. (1997). Numerical Methods for Estimation and Inference in Bayesian VAR-models. Journal of Applied Econometrics, 12(2):99–132. Laubach, T. and Williams, J. C. (2003). Measuring the Natural Rate of Interest. Review of Economics and Statistics, 85(4):1063–1070. Laubach, T. and Williams, J. C. (2015). Measuring the Natural Rate of Interest Redux. Working Paper 15, Hutchins Center on Fiscal & Monetary Policy at Brookings. Lewis, K. F. and Vazquez-Grande, F. (2019). Measuring the Natural Rate of Interest: A Note on Transitory Shocks. Journal of Applied Econometrics, 34(3):425–436. Lubik, T. A. and Matthes, C. (2015). Calculating the Natural Rate of Interest: A Comparison of Two Alternative Approaches. Economic Brief 15-10, Federal Reserve Bank of Richmond. Mertens, E. (2016). Measuring theLevel and Uncertainty ofTrend Inflation. Review of Economics and Statistics, 98(5):950–967. Sargent, T. and Sims, C. (1977). Business cycle modeling without pretending to have too much a priori economic theory. Federal Reserve Bank of Minneapolis, Working Papers no. 55. Sargent, T. J. (1988). Linear Optimal Control, Filtering, and Rational Expectations. Federal Reserve Bank of Minneapolis, Working Paper no. 224. Sims, C. A. and Zha, T. (2006). Were There Regime Switches in U.S. Monetary Policy? American Economic Review, 96(1):54–81. Smets, F. (2010). Commentary: Modeling Inflation After the Crisis. Macroeconomic Policy: Post-Crisis and Risks Ahead, Federal Reserve Bank of Kansas City Symposium. Jackson Hole, Wyoming, August 26-28. 36

Stock, J. H. and Watson, M. W. (1988). Testing for Common Trends. Journal of the American Statistical Association, 83(404):1097–1107. Stock, J. H. and Watson, M. W. (2007). Why Has Inflation Become Harder to Forecast? Journal of Money, Credit and Banking, 39(S1):3–33. Stock,J.H.andWatson,M.W.(2008). PhillipsCurveInflationForecasts. UnderstandingInflation and the Implications for Monetary Policy: A Phillips Curve Retrospective, Federal Reserve Bank of Boston Conference, June 9-11. Stock,J.H.andWatson,M.W.(2010). ModelingInflationAftertheCrisis. MacroeconomicPolicy: Post-Crisis and Risks Ahead, Federal Reserve Bank of Kansas City Symposium. Jackson Hole, Wyoming, August 26-28. Stock, J. H. and Watson, M. W. (2011). Dynamic Factor Models. In Clements, M. P. and Hendry, D.F.,editors,Oxford Handbook on Economic Forecasting,pages35–60.OxfordUniversityPress, Oxford. Uhlig, H. (2003). What moves real GNP? Unpublished. Villani, M. (2009). Steady-state Priors for Vector Autoregressions. Journal of Applied Econometrics, 24(4):630–650. Watson, M. (1986). Univariate Detrending Methods with Stochastic Trends. Journal of Monetary Economics, 18(1):49–75. Watson, M. (2004). Comment on “Monetary Policy in Real Time,” by Domenico Giannone, Lucrezia Reichlin, and Luca Sala. NBER Macroeconomics Annual, 19:216–221. 37

A The State-space Representation In Subsection 3.3, we discussed the state-state representation in (9) and (10) that we report below in (19) z = Λ x +Λ x +Λ x , (19.1) t τ τ,t c c,t η η,t x = Φx +Rε , (19.2) t t−1 t where Λ = [Λ ,Λ ,Λ ], Λ = [Λ ,...,Λ ] and Λ = [Λ ,Λ ]. For our baseline specification τ c η c c,0 c,p−1 η η,0 η,1 in Subsection 3.1, these matrices are constructed as     I 0 0 0 0 I 0 0 4×4 4×6 4×6 4×2 2×2 4×4 4×6 4×2  0 Φ Φ 0 0   0 I 0   6×4 1 2 6×2 2×2   6×4 6×6 6×2      Φ =  0 6×4 I 6×6 0 6×6 0 6×2 0 2×2 , R =  0 6×4 0 6×6 0 6×2 ,      0 0 0 0 0   0 0 I   2×4 2×6 2×6 2×2 2×2   2×4 2×6 2×2  0 0 0 I 0 0 0 0 2×4 2×6 2×6 2×2 2×2 2×4 2×6 2×2     ε Σ 0 0 τ,t τ 4×6 4×2 ε =  ε , Σ =  0 Σ 0 , t  c,t   6×4 c 6×2  ε 0 0 Σ η,t 2×4 2×6 η (cid:104) (cid:105) Λ = Λ Λ Λ Λ Λ , τ,0 c,0 c,1 η,0 η,1   1 0 0 0  0 1 0 0         0 1 0 0  I 5×5 0 5×1 (cid:34) (cid:35)   −1 0 Λ =  0 0 1 1 , Λ =  0 1 , Λ = 1×5 , τ,0   c,0   c,1   0 6×1 0 6×5  0 0 0 1  0 δ    0 0 0 1    0 0 0 1     0 0 0 0 4×1 4×1 4×1 4×1      1 0   −1 0  Λ η,0 =  , Λ η,1 =  .  0 0   0 0      0 1 0 0 38

B Data The following data series are from the Federal Reserve Economic Database (FRED) maintained by the Federal Reserve Bank of St. Louis: • Real GDP per capita, A939RX0Q048SBEA, quarterly frequency. We transform the series by taking quarterly growth rates at annual rate and express these rates in percentages. • Unemployment rate, UNRATE, monthly frequency. We transform the series by taking quarterly averages. • Inflation, GDPDEF, quarterly frequency. We transform the series for the GDP price index by taking quarterly growth rates at annual rate and express these rates in percentages. • Effective federal funds rate, FEDFUNDS, monthly frequency. Because the series is already expressed at annual rate, we take quarterly averages. The following data series are available from the Real-Time Data Research Center maintained by the Federal Reserve Bank of Philadelphia:11 • One-year-ahead inflation expectations, INFPGDP1YR, quarterly frequency. The series corresponds to the median forecast for one-year-ahead annual average inflation measured by the GDP price index. The series starts in 1970:Q2. • Ten-year-ahead inflation expectations. We follow Del Negro and Schorfheide (2013) to construct this time series. Specifically, we combine longer-run inflation expectations from the SPF and the Blue Chip Economic Indicators survey. We use the ten-year-ahead ConsumerPriceIndex(CPI)inflationexpectationsfromtheBlueChipsurvey—from1979:Q4to 1991:Q3andavailabletwiceayear—andthosefromtheSPF(INFCPI10YR)—availableeach quarter starting from 1991:Q4. To combine the measures, we subtract from the ten-yearahead CPI inflation expectations the historical average difference between CPI and GDP annualized inflation over the estimation period. • One-year-aheadunemploymentrateexpectations,UNEMP6,quarterlyfrequency. Theseries corresponds to the median forecast for one-year-ahead unemployment rate. The series starts in 1968:Q4. 11The data may contain missing observations. More details are available at the webpage: https://www.philadelphiafed.org/surveys-and-data/real-time-data-research/inflation-forecasts. 39

C Estimation Details C.1 Settings For all the considered specifications, we assume that the cyclical components evolve according to a VAR model with two lags (p = 2) following the baseline approach of Angeletos et al. (2020). For each estimation, we adopt the Gibbs sampler described in Appendix C.2. We use 50,000 draws to estimate the TC-VAR model. We then discard the first 25,000 draws and keep one in every 25 draws, thus leaving 1,000 draws, to use for the identification of the shocks. C.2 Gibbs Sampler We assume that some element of Σ is known. We collect parameters that need to be estimated η in Θ = {Σ }, Θ = {Φ ,Φ ,Σ }, Θ = {δ,Σ }. τ τ c 1 2 c e η We use the Gibbs sampler to estimate the model unknowns. We rely on the state-space representation of (9) and (10). For the jth iteration, • Run Kalman smoother to generate τj and cj conditional on Θj,Θj: This is explained in 1:T 0:T τ c Subsection C.2.1. • ObtainposteriorestimatesofΘj+1,Θj+1,Θj+1 fromtheMNIWconditionalonτj andcj : τ c e 1:T 0:T This is explained in Subsection C.2.2. C.2.1 Kalman smoother We rely on the state-space representation in equations (9) and (10). Conditional on the jth draw of Θj,Θj, we apply the standard Kalman filter as described in Durbin and Koopman (2001). τ c Suppose that the distribution of x |{z ,Θj,Θj} ∼ N(x ,P ). t−1 1:t−1 τ c t−1|t−1 t−1|t−1 Then, the Kalman filter forecasting and updating equations take the form x = Φx t|t−1 t−1|t−1 P = ΦP Φ′+RΣR′ t|t−1 t−1|t−1 x = x +(ΛP )′(ΛP Λ′)−1(cid:0) z −Λx (cid:1) t|t t|t−1 t|t−1 t|t−1 t t|t−1 P = P −(ΛP )′(ΛP Λ′)−1(ΛP ). t|t t|t−1 t|t−1 t|t−1 t|t−1 40

In turn, x |{z ,Θj,Θj} ∼ N(x ,P ). t 1:t τ c t|t t|t Next,thebackwardsmoothingalgorithmdevelopedbyCarterandKohn(1994)isappliedtorecursivelygeneratedrawsfromthedistributionsx |(X ,Z ,Θj,Θj)fort = T−1,T−2,...,1. The t t+1:T 1:T τ c last element of the Kalman filter recursion provides the initialization for the simulation smoother: x = x +P Φ′P−1 (cid:0) x −Φx (cid:1) (20) t|t+1 t|t t|t t+1|t t+1 t|t P = P −P Φ′P−1 ΦP t|t+1 t|t t|t t+1|t t|t xj ∼ N(x ,P ), t = T −1,T −2,...,1. t t|t+1 t|t+1 In sum, we obtain smoothed estimates of τj and cj . 1:T 0:T C.2.2 Posterior draw We treat the smoothed estimates of τj and cj as data points. The objective is to draw Θj+1 1:T 0:T τ and Θj+1. c VAR coefficients. For ease of exposition, we omit the superscript j below. For t ∈ {2,...,T}, we express the VAR as (cid:34) (cid:35) (cid:104) (cid:105) Φ′ c′ = c′ c′ 1 +ϵ′ , ϵ ∼ N(0,Σ ). (21) t t−1 t−2 Φ′ c,t c,t c (cid:124) (cid:123)(cid:122) (cid:125) 2 w′ (cid:124) (cid:123)(cid:122) (cid:125) t β Define X = [c ,...c ]′, W = [w ,...,w ]′, and ϵ = [ϵ ,...,ϵ ]′ conditional on the initial obser- 2 T 2 T c c,2 c,T vations. If the prior distributions for β and Σ are c β|Σ ∼ MN (cid:0) β,Σ⊗(V ξ) (cid:1) , Σ ∼ IW(Ψ,d), (22) β c then because of the conjugacy the posterior distributions can be expressed as β|Σ ∼ MN (cid:0) β,Σ⊗V (cid:1) , Σ ∼ IW(Ψ,d) (23) β c where β = (cid:0) W′W +(V ξ)−1(cid:1)−1(cid:0) W′X +(V ξ)−1β (cid:1) , (24) β β V = (cid:0) W′W +(V ξ)−1(cid:1)−1 , β β Ψ = (X −Wβ)′(X −Wβ)+(β −β)′(V ξ)−1(β−β)+Ψ, β d = T −2+d. 41

We follow the exposition in Giannone et al. (2015) in which ξ is a scalar parameter controlling the tightness of the prior information in (22). For instance, prior becomes more informative when ξ → 0. In contrast, when ξ = ∞, then it is easy to see that β = βˆ, i.e., an OLS estimate. In sum, we draw βj+1 and Σj+1 from (23). Hence, we obtain Θj+1 = {Φj+1,Φj+1,Σj+1}. c 1 2 c Trend component variances. Conditionalonτj , theobjectiveistodrawΘj+1 = {Σj+1}. For 1:T τ τ easeofexposition,weomitthesuperscriptj below. DefineX = [τ ,...,τ ]′ andW = [τ ,...,τ ]′. 2 T 1 T−1 Similarly as before we draw from Σ ∼ IW(Ψ,d), Ψ = (X −W)′(X −W)+Ψ, d = T −1+d. (25) τ Parameters for survey expectations. Conditional on τj and cj , the objective is to draw 1:T 1:T Θj+1 = {δj+1,Σj+1}. For ease of exposition, we omit the superscript j below. Define X = e η [πe,10y −τ ,...,πe,10y −τ ]′ and W = [ce ,...,ce ]′. We draw δj+1 and Σj+1 from posterior 1 π,1 T π,T π,1 π,T η distributions expressed in (23). 42

D Robustness Checks This appendix provides more details on the robustness checks presented in Section 6. D.1 Priors and time-varying trends Table5reportsthemediancontributions—andtheassociated68-percentposterior-coverageintervals— oftheunemployment-identifiedshocktothevariabilityofeachcyclicalcomponentunderthe“Conservative” and“Tight” priorsasdescribedinSubsection6.2. Theshockisidentifiedbymaximizing its contribution to the volatility of the cyclical component of realized unemployment rate over all the frequencies of the cycle, but those that imply cycles less than 1.5 years. The results show that, even in this case, the choice of priors does not influence the main conclusions of the paper. Table 12: Variance contributions of unemployment shock All-but-short-run frequencies (6−∞ quarters) Unempl. rate Output Unempl. rate FFR Inflation Inflation exp.(1y) exp. Conservative 69.4 59.2 68.5 57.5 33.6 48.2 [59.3,82.4] [46.1,75.1] [56.9,82.3] [26.0,82.1] [11.1,60.2] [16.3,80.5] Tight 67.7 63.3 67.2 48.2 30.7 41.5 [58.3,79.0] [45.4,80.1] [56.9,80.0] [11.8,80.3] [9.9,64.6] [10.0,82.6] Notes: Theshockisidentifiedbymaximizingitscontributiontothevolatilityofthecyclicalcomponentofrealized unemploymentrateoverallthefrequenciesofthecycle,butthosethatimplycycleslessthan1.5years. Wereport themediancontributionandthecorresponding68-percentposterior-coverageintervaloftheidentifiedshocktothe variance of the cycle of all variables over the corresponding frequency. We consider two priors for the standard deviation of the expected change in the trend unemployment rate. The “Conservative” prior sets that standard deviation to 1% over a period of 30 years, while the “Tight” prior assumes the same standard deviation but over a period of 40 years. Sources: Federal Reserve Bank of St. Louis; Federal Reserve Bank of Philadelphia; authors’ calculations. D.2 Alternative specifications Inthisappendix,wepresentdetailsonthethreealternativespecificationsconsideredinSubsection 3.2 and, for each alternative, we describe the assumptions about initial conditions and priors used for the model estimation, and report the corresponding results. D.2.1 Parsimonious specification Specificationandpriors. Inthisalternative,weconsideramoreparsimoniousspecificationthan the baseline. We assume that realized unemployment rate and one-year-ahead unemployment rate 43

expectations evolve as in (6) and reported below in (26) u = τ +c , (26.1) t u,t u,t ue,1y = τ +c +ηe,1y, (26.2) t u,t u,t u,t thus sharing the common cyclical component c . For the one-year-ahead unemployment rate u,t expectations, we allow for an idiosyncratic error. Similar to the unemployment rate measures, we decompose the three inflation measures as in (7) and reported below in (27) π = τ +c +η −η , (27.1) t π,t π,t π,t π,t−1 πe,1y = τ +c +ηe,1y, (27.2) t π,t π,t π,t πe,10y = τ +ηe,10y. (27.3) t π,t π,t Therefore, we assume one common cyclical component for inflation, c , which is shared across π,t realized inflation and the one-year-ahead inflation expectation. We assume that ten-year-ahead inflationexpectationsproxyforthetrendcomponentbutdoesnotincludeanycyclicalcomponent. Lastly, we allow for idiosyncratic errors for both inflation expectations. In this case, the vectors of observables, trends and shocks to the trends are unchanged relative to the baseline, while we re-define the other vectors as (cid:110) (cid:111)′ c = {c ,c ,c ,c }′, η = η ,ηe,1y,ηe,10y , ε = {ε ,ε ,ε ,ε }′. t y,t u,t f,t π,t t π,t π,t π,t c,t c,y,t c,u,t c,f,t c,π,t Evidently, this alternative specification is more parsimonious than the baseline because, while considering the same number of trends, it allows for four, rather than six, cyclical components. Initial conditions and priors. Under this alternative specification, we leave unchanged the assumptions on the initial conditions of the trends and the prior for the standard deviation of the shockstothetrendcomponents. Asaresult,thepriorcovariancematrixoftheshockstothetrends is diagonal with the following elements on the main diagonal diag (cid:0) Σ (cid:1) = [1/40,1/20,1/20,1/20]. τ Finally, we need to set the prior for the standard deviation of the shocks affecting the four cyclical components. For real GDP and nominal interest rate, we keep the same priors as under the baseline. For the common cyclical components of unemployment rate and inflation, we set those priors to the pre-sample standard deviations of the respective realized measures of unemployment rate andinflation. Therefore,thepriorcovariancematrixoftheshockstothecyclesisdiagonalandsuch that the value on the main diagonal approximately correspond to diag (cid:0) Σ (cid:1) = [25,1.3,0.6,2.2]. c Results. We estimate the alternative specification of the TC-VAR model using the state-space representation in (9) and (10) and subsequently identify the shock targeting the common cyclical component of unemployment rate. We report in Table 13 the contributions of the identified shock to the volatility of each cyclical components. Even in this case, the results are in line with those of 44

Table 13: Parsimonious specification: Variance contributions of unemployment shock All frequencies (0−∞ quarters) Unempl. rate Output FFR Inflation 74.7 81.6 44.7 39.2 [64.0,85.6] [70.2,90.2] [19.2,73.6] [24.3,54.2] All-but-short-run frequencies (6−∞ quarters) Unempl. rate Output FFR Inflation 75.8 82.7 44.6 44.9 [64.9,86.3] [71.1,90.9] [19.6,72.3] [28.0,59.8] Notes: The shock is identified by maximizing its contribution to the volatility of the cyclical common component of the unemployment rate. We consider two cases. In the first case, the shock is chosen to maximize the fraction of the volatility over all the frequencies of the cycle, while in the second case we exclude frequencies that imply cycleslessthan1.5years. Wereportthemediancontributionandthecorresponding68-percentposterior-coverage interval of the identified shock to the variance of the cycle of all variables over the corresponding frequencies. Sources: Federal Reserve Bank of St. Louis; Federal Reserve Bank of Philadelphia; authors’ calculations. the baseline specification. When the shock is identified over all frequencies, it accounts for about 39% of the cyclical volatility of the cyclical common component of inflation. When we identify the shock excluding frequencies that imply cycles less than 1.5 years, the same contribution of the shock rises to nearly 45%. D.2.2 Flexible specification Specification and priors. Under this alternative specification, the decompositions of realized unemployment rate in (1.2) and one-year-ahead unemployment rate expectations in (2) are identical to those assumed under the baseline. However, we consider the following more flexible decomposition for the two inflation expectations measures πe,1y = τ +ce,1y, (28.1) t π,t π,t πe,10y = τ +ce,10y. (28.2) t π,t π,t Hence, the two inflation expectation measures follow separate, idiosyncratic cyclical components, while both sharing a common inflation trend with realized inflation decomposed in (4.1). For this alternative specification, the vectors of observables, measurement error, trends and shocks to the trends are unchanged relative to the baseline. However, the vectors of the cyclical components 45

Table 14: Flexible specification: Variance contributions of unemployment shock All frequencies (0−∞ quarters) Unempl. rate Output Unempl. rate FFR Inflation Inflation Inflation exp.(1y) exp.(1y) exp.(10y) 67.6 60.1 65.5 51.2 17.5 33.8 12.3 [58.2,78.6] [49.0,71.6] [54.8,76.7] [27.3,71.7] [7.6,33.7] [16.5,54.7] [4.8,26.9] All-but-short-run frequencies (6−∞ quarters) Unempl. rate Output Unempl. rate FFR Inflation Inflation Inflation exp.(1y) exp.(1y) exp.(10y) 68.6 60.0 68.8 51.8 20.2 38.9 18.4 [58.8,80.0] [48.5,71.9] [57.2,80.5] [26.4,73.0] [8.4,38.0] [19.2,62.0] [7.3,40.3] Notes: Theshockisidentifiedbymaximizingitscontributiontothevolatilityofthecyclicalcomponentofrealized unemployment rate. We consider two cases. In the first case, the shock is chosen to maximize the fraction of the volatilityoverallthefrequenciesofthecycle,whileinthesecondcaseweexcludefrequenciesthatimplycyclesless than1.5years. Wereportthemediancontributionandthecorresponding68-percentposterior-coverageintervalof the identified shock to the variance of the cycle of all variables over the corresponding frequencies. Sources: Federal Reserve Bank of St. Louis; Federal Reserve Bank of Philadelphia; authors’ calculations. and the associated disturbances are re-written as (cid:110) (cid:111)′ (cid:110) (cid:111)′ c = c ,c ,ce,1y,c ,c ,ce,1y,ce,10y , ε = ε ,ε ,εe,1y,ε ,ε ,εe,1y,εe,10y , t y,t u,t u,t f,t π,t π,t π,t c,t c,y,t c,u,t c,u,t c,f,t c,π,t c,π,t c,π,t therefore allowing for seven, rather than six, idiosyncratic cyclical components. Initial conditions and priors. Under this alternative specification, we keep the same assumptions on the initial conditions of the trends and the standard deviation of the shocks to the trends as under the baseline, that is diag (cid:0) Σ (cid:1) = [1/40,1/20,1/20,1/20]. τ For the shocks to the cyclical components, we assume the same priors as under the baseline for real GDP growth, realized and expected unemployment rate, nominal interest rate and inflation. Because the measures of one- and ten-year-ahead inflation expectations are not available for the pre-sample period, we set the standard deviations of the shocks to those cyclical components to about 1.2% and 1% respectively, thus smaller than the pre-sample counterpart of 1.5% for realized inflation. As a result of these assumptions, the prior covariance matrix of the shocks to the cycles is diagonal and such that the values on the main diagonal approximately correspond to diag (cid:0) Σ (cid:1) = [25,1.3,0.8,0.6,2.2,1.4,1.0]. c Results. We estimate the proposed alternative specification of the TC-VAR model and identify the shock targeting the cyclical component of realized unemployment rate. In Table 14, we report the shock’s contributions to the volatility of the cyclical components of all variables. The results show that the findings under the baseline specification carry over. In fact, the shock explains 46

Table 15: No unemployment rate expectations: Variance contributions of unemployment shock All frequencies (0−∞ quarters) Unempl. rate Output FFR Inflation Inflation exp. 71.2 80.4 56.2 44.5 51.5 [61.2,81.5] [69.4,88.1] [21.8,81.9] [29.8,57.7] [25.3,71.2] All-but-short-run frequencies (6−∞ quarters) Unempl. rate Output FFR Inflation Inflation exp. 72.2 81.2 55.7 49.8 57.4 [62.2,82.4] [70.5,88.7] [21.9,81.6] [34.1,62.7] [28.5,77.6] Notes: The shock is identified by maximizing its contribution to the volatility of the cyclical component of the unemployment rate. We consider two cases. In the first case, the shock is chosen to maximize the fraction of the volatilityoverallthefrequenciesofthecycle,whileinthesecondcaseweexcludefrequenciesthatimplycyclesless than1.5years. Wereportthemediancontributionandthecorresponding68-percentposterior-coverageintervalof the identified shock to the variance of the cycle of all variables over the corresponding frequencies. Sources: Federal Reserve Bank of St. Louis; Federal Reserve Bank of Philadelphia; authors’ calculations. nearly 34% of the volatility of the cyclical component of one-year-ahead inflation expectations which we consider as a measure of underlying inflation. When excluding movements in cyclical components at frequencies higher than 1.5 years, the results point to an increase to about 39% in the explanatory power of the identified shock for the cyclical component of underlying inflation. D.2.3 Specification with no unemployment rate expectations Specification and priors. Under the third and final alternative specification, we exclude the measurement equation (2) for the expectations of the one-year-ahead unemployment rate, while leaving all the other decompositions as under the baseline. As a result, only the vectors of observables, cyclical components and the associated disturbances are modified as follows z = (cid:110) g ,u ,f ,π ,πe,1y,πe,10y (cid:111)′ , c = (cid:8) c ,c ,c ,c ,ce (cid:9)′ , ε = (cid:8) ε ,ε ,ε ,ε ,εe (cid:9)′ . t t t t t t t t y,t u,t f,t π,t π,t c,t c,y,t c,u,t c,f,t c,π,t c,π,t Initial conditions and priors. Even under this alternative specification, we keep the same assumptions on the initial conditions of the trends and the standard deviation of the shocks to the trends as under the baseline, that is diag (cid:0) Σ (cid:1) = [1/40,1/20,1/20,1/20]. τ For the shocks to the cyclical components, we simply drop the assumption on the standard deviationforthecyclicalcomponentoftheone-year-aheadunemploymentrateexpectations, while keeping the other assumptions as under the baseline. The resulting prior covariance matrix of the 47

shocks to the cycles is diag (cid:0) Σ (cid:1) = [25,1.3,0.6,2.2,1.4]. c Results. After estimating the TC-VAR model under the assumptions of the proposed alternative specification, we identify the shock targeting the cyclical component of realized unemployment rate. Table 15 reports the contributions of the identified shock to the volatility of the cyclical components of all variables. The results verify the robustness of our main findings to the exclusion of the one-year-ahead unemployment rate expectations for the estimation of the TC-VAR model. In fact, the shock explains about 44% and 51% of the volatility of the cyclical component of realized inflation and one-year-ahead inflation expectations respectively. Excluding movements in cyclical components at frequencies higher than 1.5 years, the contribution of the shock to the cyclical components of realized and expected inflation reaches about 51% and 57% respectively. 48

E Details on Long-run Priors E.1 Long-run priors for Angeletos et al. (2020) As detailed in Section I of Angeletos et al. (2020), their data consist of quarterly observations on the following macroeconomic variables: real, per-capita levels of GDP (Y ), investment (I ), t t consumption (C ); unemployment rate (u ); hours worked per person (h ); the level of utilizationt t t adjusted total factor productivity (TFP ); the labor share (w h /Y ); the inflation rate (π ), as t t t t t measured by the rate of change in the GDP deflator; and the nominal interest rate (R ), as t measured by the federal funds rate. When estimating the VAR model using the long-run priors, weconsiderthearbitraryorderingoftheobservablesz = {Y ,I ,C ,u ,h ,TFP ,w h /Y ,π ,R }′. t t t t t t t t t t t t We assume that the following matrix H captures the cointegrating relationships in the long run   1 1 1 0 0 0 0 0 0  −1 1 0 0 0 0 0 0 0       −1 0 1 0 0 0 0 0 0     0 0 0 1 0 0 0 0 0      H =  0 0 0 0 1 0 0 0 0 .    0 0 0 0 0 1 0 0 0       0 0 0 0 0 0 1 0 0     0 0 0 0 0 0 0 1 1    0 0 0 0 0 0 0 −1 1 E.2 Long-run priors for VAR model in Subsection 7.2 When estimating the VAR model using the long-run priors, we consider the arbitrary ordering of (cid:110) (cid:111)′ the observables z = g ,u ,f ,π ,πe,1y,ue,1y . We assume that the following matrix H captures t t t t t t t the cointegrating relationships in the long run   1 0 0 0 0 0    0 1 0 0 0 1     0 1 0 0 0 −1  H =  . (29)    0 0 1 −1 0 0     0 0 1 1 1 0    0 0 0 −1 1 0 49

F Supplementary Material to Section 8 F.1 Definition of stabilizability Definition 1 The pair (A,B) is stabilizable if any of the following conditions holds: • There exists no left eigenvector of A associated with an eigenvalue having nonnegative real part that is orthogonal to the columns of B;  ν∗A = λν (Re[λ(A)] ≥ 0) ⇒ ν = 0. ν∗B = 0 • rank [λI −A B] = dim(A) for all Re[λ(A)] ≥ 0. F.2 Matrices for time-invariant innovations representation As shown in Fernández-Villaverde et al. (2007), the time-invariant matrices Bˆ and Dˆ satisfy the following equations: Ω = AΩA′+BB′− (cid:0) AΩC′+BD′(cid:1)(cid:0) CΩC′+DD′(cid:1)−1(cid:0) AΩC′+BD′(cid:1)′ , (30.1) K = (cid:0) AΩC′+BD′(cid:1)(cid:0) CΩC′+DD′(cid:1)−1 , (30.2) DˆDˆ′ = DD′+CΩC′, (30.3) Bˆ = KDˆ. (30.4) F.3 The AR representation of a univariate TC-AR model In this appendix, we provide an intuitive, analytical example that considers the unobserved components model used by Stock and Watson (2007) to test whether the U.S. inflation process experienced a structural change since the beginning of the Great Moderation. Inflation is described by the following state-space representation π = Λ τ +Λ ε , (31.1) t τ π,t η η,π,t τ = Φ τ +Rε , (31.2) π,t τ π,t−1 τ,π,t where Φ = 1, R = 1, Λ = 1, Λ = 1, τ τ η and ε = Qw , Q = σ , ε = σ w such that E(w w′) = I where w = {w ,w }′. τ,π,t τ,π,t τ η,π,t η η,π,t t t t τ,π,t η,π,t Definingz = π andx = τ ,therepresentationin(31)coincideswith(9)and(10)where,following t t t t the notation in Anderson and Moore (1979), we appended the measurement error ε directly in η,π,t 50

(9) as opposed to redundantly defining it as η in the transition equation (10). Thus, we verify π,t two conditions: i) the pair (Φ ,RQ) is stabilizable; ii) the pair (Φ′,Λ′) is stabilizable. Because τ τ τ Φ has only one (unit-root) eigenvalue, then τ rank[I −Φ RQ] = rank[0 σ ] = 1, τ τ and rank (cid:2) I −Φ′ Λ′(cid:3) = rank[0 1] = 1. τ τ Therefore, the asymptotic properties of the Kalman filter hold, and we can derive the AR(∞) representation of (31). In particular, we write (31) as in (14) where (cid:104) (cid:105) (cid:104) (cid:105) A = 1, B = σ 0 , C = 1, D = σ σ , τ τ η assuming that the shock to the trend is ε = σ w and the measurement error is η = τ,π,t τ τ,π,t π,t σ w and E(w w′) = I where w = {w ,w }′. Defining σ ≡ Ω as the variance of the η η,π,t t t t τ,π,t η,π,t τ (cid:98) error associated with the estimate of the trend, (τ −τ ), we can use the equation (30) in π,t (cid:98)π,t Appendix F.2 to derive the time-invariant matrices Bˆ and Dˆ as 1 (cid:16) (cid:113) (cid:17) σ2 = −σ2+ σ4+4σ2σ2 > 0, (32.1) τ (cid:98) 2 τ τ τ η K = 1−δ, (32.2) DˆDˆ′ = (cid:0) σ2+σ2+σ2(cid:1) , (32.3) τ τ η (cid:98) Bˆ = (1−δ) (cid:0) σ2+σ2+σ2(cid:1)1/2 , (32.4) τ τ η (cid:98) where δ = σ2/ (cid:0) σ2+σ2+σ2(cid:1) < 1. Finally, using (17) and (32), we map the state-space represen- η τ τ η (cid:98) tation in (31) into the infinite-order autoregression, AR(∞), ∞ (cid:88) (cid:16) (cid:17)s π = C A−BˆDˆ−1C BˆDˆ−1π +Dˆν t+1 t−s t+1 s=0 ∞ (cid:88) = C(A−KC)sKπ +Dˆν , t−s t+1 s=0 ∞ = (1−δ) (cid:88) δsπ + (cid:0) σ2+σ2+σ2(cid:1)1/2 ν , (33) t−s τ τ η t+1 (cid:98) s=0 where ν ∼ N (0,1). Equation (33) shows that, even with infinite data, the estimation of the t AR(∞)representationleadstoparameterestimatesandVARresidualsthatconfoundthestandard deviation of both the measurement error σ and the innovations to the trend σ with the standard η τ deviation σ resulting from the error associated with the estimate of the trend, (τ −τ ). τ π,t (cid:98)π,t (cid:98) 51

F.4 Bivariate TC-VAR model: An alternative case In this appendix, we consider an alternative case for the model presented in Subsection 8.2. We introduce trends in both inflation and unemployment and assume that cyclical inflation is persistent and not exclusively driven by cyclical unemployment rate. Specifically, for the unemployment rate,wesetρ = 0.95andσ = 1foritscyclicalcomponentandσ = 1foritstrendcomponent, π c,u τ,u implying the presence of low-frequency movements. For inflation, we set ρ = 0.5 and σ = 0, π c,π resulting in a cyclical component of inflation that is driven by its lag and cyclical unemployment rate. Additionally, σ = 1, thus introducing an inflation trend. τ,π Table 16: Variance contribution of unemployment shock Unemployment Inflation 100.0 2.2 [99.9,100.0] [1.1,3.7] Notes: The shock is identified by maximizing its contribution to the volatility of the unemployment rate over business-cycle frequencies (6-32 quarters). We report the median and the corresponding 68-percent posteriorcoverage interval of the median contributions of the identified shock to the variance of all variables over the same frequencies. Tosimulatethedata,weusethefollowingcalibration. Fortheunemploymentrate,wesetρ =0.95, uu σ =1 and σ =1. For the inflation rate, we set ρ =0.5 and σ =0 and σ =1. c,u τ,u ππ c,π τ,π Table 16 reports the median—and the corresponding 68-percent posterior-coverage interval—of the median contributions of the identified shock to the variance of all variables over businesscycle frequencies. The contribution of the unemployment-rate shock to inflation at business-cycle frequencies is only 2.2%. Intuitively, under this alternative calibration, the inflation rate depends not only on its trend and the persistence of its cyclical component but also on the business-cycle andlow-frequencymovementsinunemploymentrate. Consequently,theestimatedVARconfounds all these effects, implying no explanatory power of the unemployment-identified shock on inflation at business-cycle frequencies. To conclude, even if the long simulations were generated under the assumption that the cyclical components of unemployment rate and inflation were related, the identified shock does not capture this feature of the simulated data. 52