(Re-)Connecting Inflation and the Labor Market: A Tale of Two Curves

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) (Re-)Connecting Inflation and the Labor Market: A Tale of Two Curves Hie Joo Ahn and Jeremy B. Rudd 2024-050 Please cite this paper as: Ahn, Hie Joo, and Jeremy B. Rudd (2025). “(Re-)Connecting Inflation and the Labor Market: A Tale of Two Curves,” Finance and Economics Discussion Series 2024-050r1. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2024.050r1. 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.

(Re-)Connecting Inflation and the Labor Market: A Tale of Two Curves⋆ Hie Joo Ahna, Jeremy B. Ruddb aFederal Reserve Board bFederal Reserve Board Abstract We propose an empirical framework in which shocks to worker reallocation, aggregate activity, and labor supply drive the joint dynamics of the labor market and inflation, and where reallocation shocks take two forms depending on whether they result from quits or from job losses. We find that these structural shocks, which affect the Beveridge curve, have different effects on inflation. Our model fully decomposes shifts of or along the empirical Beveridge curve in terms of the contribution of each shock and also allows us to estimate the Phillips correlation associated with each shock; observed Beveridge and Phillips correlations change over time depending on what types of structural shocks predominate in a given period. We find that reallocation shocks that accompany job losses were a key source of labor market dynamics and the steepening of the reduced-form Phillips curve during the Covid-19 pandemic, and were an important driver of the post-pandemic “soft landing.” Keywords: Beveridge curve, Phillips curve, monetary policy JEL classification: C11, C32, E24, E31, E52 Introduction The Covid-19 pandemic saw a sudden surge in the unemployment rate that was accompanied by a drop in labor force participation. While the unemployment rate declined gradually and the participation rate remained low in the wake of the pandemic recession, job vacancy postings rose to a record-high level and both wage and price inflation rose sharply and persistently. The unprecedented shocks to the labor market created equally unprecedented movements in the Beveridge curve. The empirical Beveridge curve shifted out dramatically—first with the rapid rise in unemployment, then with the prolonged increase in job vacancies—leading some to conclude that the pandemic induced a large and persistent reallocation shock in the labor market.1 Hidden in the outward shift, however, was a reduction in labor supply that shifted the Beveridge curve inward (Elsby et al., 2015); to make matters even more complicated, shocks to aggregate activity resulted in cyclical movements along the Beveridge curve. Emerging evidence also suggests that the pandemic might have strengthened the link between real activity and inflation.2 Plausibly, decreased labor supply combined with increased labor reallocation can be inflationary, with the former raising wage inflation and the latter raising the noncyclical portion of the unemployment rate (Lilien, 1982). As a result, the increased importance of these structural shocks might ⋆WegratefullyacknowledgeinvaluableguidancefromJamesHamiltonandusefulcommentsandsuggestionsfromGianni Amisano,TravisBerge,MarkBognanni,ThorstenDrautzburg,RochelleEdge,AndrewFigura,CallumJones,AntoineLepetit, LamNguyen,andSimonSmith. Allerrorsareourown. Theviewsexpressedarethoseoftheauthors,anddonotnecessarily reflecttheviewsoftheFederalReserveBoardoranypersonassociatedwiththeFederalReserveSystem. Email addresses: hiejoo.ahn@frb.gov(HieJooAhn),jeremy.rudd@frb.gov(JeremyB.Rudd) 1SeeBarreroetal.(2021). 2See,forexample,Hobijnetal.(2023). Preprint submitted to FEDS working paper series May 21, 2025

have changed the inflation–unemployment tradeoff that would be captured by a reduced-form Phillips curve. ThispossibilityshowsthatstructuralshocksinthelabormarketcanlinktheBeveridgecurveandthePhillips curve. Furthermore, joint consideration of the Beveridge and Phillips curves should help us to better identify the cyclical state of the economy than simply using the Phillips curve alone, especially given how flat this latter relation appears to have become in recent decades. Previous research has examined worker and job reallocation to assess the relative impact of reallocation andcyclicalshocksonlabormarketdynamics. (e.g., Lilien,1982; AbrahamandKatz,1986). Inawell-known paper, Blanchard and Diamond (1989) described labor market dynamics and the Beveridge curve in terms of shocks to aggregate activity, worker reallocation, and labor supply. They proposed an identification strategywhereapositiveaggregateactivityshockraisesjobvacanciesbutlowersunemployment(amovement along the Beveridge curve); a positive labor reallocation shock raises both unemployment and vacancies (an outward shift in the Beveridge curve); and a positive labor supply shock raises labor force participation and unemployment (also an outward shift in the Beveridge curve). In this sense, the aggregate activity (demand) shock drives the cyclical dynamics of the unemployment rate, while the labor reallocation shock and labor-supply shock are associated with changes in the noncyclical rate of unemployment (NCRU). AlthoughBlanchardandDiamond’sempiricalmodelisatypeofsign-restrictedVAR,asuitablestatistical approach for estimating these models was only developed recently.3 To explore the empirical link between the Beveridge and Phillips curves, we identify the structural shocks in a way that is similar to Blanchard and Diamond but use the estimation methodology of Baumeister and Hamilton (2015). While Blanchard and Diamond’s identifying restrictions are imposed on the impact matrix of the structural shocks, in Baumeister and Hamilton’s algorithm priors incorporating sign and exclusion restrictions are imposed on the parameters of the inverse of the impact matrix. Because of this difference, Baumeister and Hamilton’s method has some limitations when applied to Blanchard and Diamond’s model. We therefore modify Baumeister and Hamilton’s algorithm to allow the priors reflecting the identifying restrictions to be imposed directly on the structural parameters of the impact matrix. Our paper is the first to combine information from the Beveridge and Phillips curves to analyze the joint dynamics of labor markets and inflation within a Bayesian sign-restricted VAR framework. Though our model draws on Blanchard and Diamond’s characterization of labor market dynamics, our approach differs from theirs in two important ways. First, we introduce wage and price inflation into the model, which provides additional useful information about the state of the labor market. Second, we distinguish between two different types of reallocation shocks—one that primarily raises involuntary separations and another that raises quits—by using reported reasons for unemployment. We refer the former as the job-loss reallocation shock and the latter as the quits reallocation shock. The aggregate activity shock drives the cyclical tradeoff between real activity and inflation—the Phillips curve—and the size of that tradeoff is the slope of the Phillips curve. The effects of the other structural shocks on inflation are less clear. Increased reallocation, if driven by job quitters who seek out and then find better jobs with higher wages, puts upward pressure on inflation (all else equal).4 It also shifts the Beveridge curveout: Typically,jobquitterswillexperienceashortspellofunemploymentbecauseofjob-searchfrictions, and hence increased quits-related reallocation likely raises the frictional unemployment. In sharp distinction, if increased reallocation is primarily associated with job losses, the shock still shifts the Beveridge curve out but puts downward pressure on inflation: An increased unemployment rate owing to job-loss reallocation is likely to be related to worker–job mismatch, and thus to reflect a rise in structural unemployment. Similarly, 3More precisely, Blanchard and Diamond transformed their model into an over-identified system by imposing further restrictionsonthestructuralparameterssuchthattheimpliedimpulseresponseshadthedesiredsignpatterns. 4Other work has examined the relation between quits and inflation. Faccini and Melosi (2023) introduce a shock to the propensitytosearchonthejobthatissimilartoourquitsreallocationshock: Whenalargerfractionofworkersstartsearching onthejob,itshortensthetimethatittakestheeconomytoreachtheoptimalallocationoflaborandmakesoverheatingmore likely,creatingapositiveassociationbetweenquitsandinflation. MoscariniandPostel-Vinay(2023)alsostudytheinflationary effectofjob-to-jobtransitions,butintheirmodelabusiness-cycleshockisthekeysourceofmisallocationandquits. Inaddition, Birincietal.(2024)embedtheanalysisofon-the-jobsearchfromFacciniandMelosi(2023)inaHANKmodel. Butnoneof thesestudiesexplicitlyconnectstheBeveridgeandPhillipscurvesthroughtheunderlyingstructuralshockstothelabormarket. Inaddition,ouranalysiscoversalongersampleperiod. 2

increased labor supply also raises the NCRU and puts downward pressure on inflation.5 Hence, both of the reallocation shocks and the labor supply shock shift the Beveridge curve outward and all are associated with changes in the NCRU, but the quits reallocation shock has an effect on inflation that is opposite to the other two. We find evidence for two types of reallocation shocks that do in fact have different effects on inflation: The job-loss reallocation shock lowers wage and price inflation, while the quits reallocation shock raises it. A positive labor supply shock lowers wage and price inflation, similar to the job-loss reallocation shock. This finding implies that relying solely on a Phillips curve model may provide misleading inference about the NCRU. Specifically, if the model relates price inflation negatively to the unemployment-rate gap (defined as theunemploymentrateminustheNCRU),onlychangesintheunemploymentratedrivenbyquitsreallocation shockswillbedetectedaschangesintheNCRU(or,alternatively,intheunemploymentratethatisconsistent with the absence of price pressures). Furthermore, if frictional and structural unemployment affect inflation differently—as implied by the effects that the two reallocation shocks have on inflation—we won’t be able to correctly identify a change in the NCRU based on the Phillips curve alone. This highlights the importance of distinguishing among labor-market shocks, especially quits reallocation and job-loss reallocation shocks, and also of jointly analyzing the Beveridge and Phillips curves rather than relying on either one in isolation. Overall, we find that shocks to aggregate activity are the main source of labor market fluctuations over the business cycle (in line with Abraham and Katz, 1986), and that these shocks induce persistent changes in wage and price inflation. The result is an economically and statistically significant cyclical tradeoff between unemployment and inflation; that tradeoff is the slope of the Phillips curve. Notably, however, job-loss reallocation shocks and labor supply shocks each imply an unemployment–inflation tradeoff that is larger in size to the one observed for aggregate activity shocks, while quits reallocation shocks create a positive correlation between the unemployment rate and inflation. Hence, the observed relation between the unemployment rate and inflation depends on what sort of structural shocks drive the dynamics of the unemployment rate. We find that, while the aggregate activity shock is the main driver of variations in the unemployment rate, the non-negligible roles of job-loss reallocation shocks and labor-supply shocks are likely to overstate the estimated slope of the Phillips curve if one does not properly control for these shocks that affect the NCRU. Our decomposition further suggests that any change in the incidence or size of specific structural shocks likely causes an estimated reduced-form Phillips correlation to vary. In the specific case of the pandemic period, we find that job-loss reallocation shocks also became a key driver of labor market dynamics (in line with Barrero et al., 2021). In particular, these shocks were the main driver of the rapid outward shift in the Beveridge curve during the pandemic recession and its apparent inward shift during the subsequent recovery, implying a path for the NCRU that rises at the onset of the pandemic and declines after the pandemic recession. The shocks also contributed to a steepening of the observed reduced-form Phillips curve. Relative to the job-loss reallocation shocks, the quits reallocation shocks and labor supply shocks played a small role. Our analysis also speaks to the “soft landing” debate of 2022. According to Blanchard et al. (2022), the relatively flat empirical Beveridge curve during the pandemic implied that even a modest reduction in job vacancies would raise the unemployment rate significantly. By contrast, Figura and Waller (2022) claimed that increased unemployment inflows temporarily flattened the Beveridge curve, but that the slope of the structural Beveridge curve remained steep.6 Our model directly addresses this debate by estimating the changes in the vacancy and unemployment rates that were driven by the aggregate activity shock (as well as the other structural shocks, which shifted the curve), thereby recovering the slope of the structural Beveridge curve. Our findings support the conclusion of Figura and Waller, but are distinct in that our formal statistical model identifies and quantifies an important driver of labor market dynamics during this period—namely, job-loss reallocation shocks. 5Apositivelaborsupplyshockcanraisefrictionalunemploymentifitincreasesthenumberofjobseekerswhofindajob quickly,butitcanraisestructuralunemploymentifitresultsinmorejobseekersfacinglongjoblessspells. 6Weusetheterm“structuralBeveridgecurve”todistinguishitfromtheempiricalBeveridgecurve—theobservedlocusof vacancyandunemploymentrates. FiguraandWaller’s2022analysishasbeenextendedandpublishedasFiguraandWaller (2024). 3

In particular, because the job-loss reallocation shock was responsible for much of the rise in vacancy rates, the cyclical strength of the vacancy rate was smaller than what a simple read of the data suggested. Therefore, thedeclineinthevacancyratein2022and2023wasabletooccurwithoutthedramaticriseinthe unemployment rate that Blanchard et al. predicted. Also during this period, job-loss reallocation gradually unwound and put modest downward pressure on the unemployment rate; a soft landing with lower inflation was further facilitated because the job-loss reallocation shocks pushed down inflation through 2022. This episode once again underscores that it is vital to consider the underlying shocks driving the joint dynamics of the labor market and inflation to understand developments in the economy. 1. Data We use monthly data on the labor force participation rate and the unemployment rate by reason for unemployment from the Current Population Survey. We consider two reasons for unemployment: quits (job leavers), and any other reason including involuntary separation or entrance into the labor force.7 For data on vacancy postings, we use the composite measure of job vacancies compiled by Barnichon (2010), which covers the period prior to when the Job Openings and Labor Turnover Survey (JOLTS) data become available (December 2000). This series ends in September 2021; we extend it through December 2023 with the growth rate of total vacancy postings from JOLTS. Weusethe12-monthpercentchangeinaveragehourlyearningsofproductionandnonsupervisoryworkers (AHE)asourmeasureofwageinflation. TheAHEdataaremonthlyandavailablefromJanuary1965(among available monthly wage indicators, it is the only series that extends back into the 1960s). AHE is likely affected by shifts in the composition of the workforce; however, to the extent that these shifts are unrelated to underlying labor market strength, they can be captured with the model’s wage-specific disturbances.8 Forpriceinflation,weusethepercentchangeinthecoremarket-basedpersonalconsumptionexpenditures (PCE) price index from the Bureau of Economic Analysis (BEA). We use market-based core as our principal price measure because several nonmarket components of PCE are priced using input-cost indexes that are in turn derived from wage or compensation measures. Official data for this price index start in 1987; to compute a market-based series prior to 1987, we strip out the prices of core nonmarket PCE components from the published overall core PCE price index using BEA’s definitions and procedures. The market-based PCE inflation series that we use for the extended-sample VARs subtracts out Blinder and Rudd’s (2013) estimates of the effects of the Nixon-era price controls. 2. Linking the Beveridge Curve and the Phillips Curve WefirstdiscusstheidentificationofstructuralshocksinBlanchardandDiamond’smodel(BD,henceforth), and then extend that model to have two different reallocation shocks and to include wage and price inflation. 2.1. Identification of structural shocks BD identify three types of structural shocks—reallocation shocks, aggregate activity shocks, and labor supply shocks—based on how each shock affects unemployment, job vacancies, and labor force participation on impact (Table 1). Conceptually, a negative aggregate activity shock, which reflects weakened demand, reduces job vacancies while increasing unemployment, resulting in movement along the Beveridge curve. In contrast, a positive reallocation shock leads to job creation in some sectors and job destruction in others. This type of shock raises both job vacancies and unemployment, yielding an outward shift of the Beveridge curve. Both negative aggregate activity shocks and positive reallocation shocks reduce employment through job destruction, thereby also lowering labor force participation. Finally, a positive labor supply shock raises both unemployment and labor force participation by bringing more job seekers into the labor market or increasing the labor-force attachment of job seekers, with no impact effect on job vacancies. 7Laborforceentrycoversnewentrants(e.g.,collegegraduates)andre-entrants(e.g.,jobloserswholeaveandthenreturnto thelaborforce). UnemploymentbyreasonisavailablestartinginJanuary1967. 8Forthisreason,weviewpotentialcompositioneffectsinAHEaslessofaconcernforourexercise. 4

Table1: IdentificationofstructuralshocksintheBlanchard–Diamondmodel To a positive shock to ... Response of the following variables ... Reallocation Aggregate activity Labor supply Labor force participation − + + Vacancy + + 0 Unemployment + − + Notes: Thistablesummarizesthesignofeachshock’simpacteffectonthethreevariables: ‘+’and‘–’denotepositive andnegativeimpacteffects,respectively,and‘0’denotesnoimpacteffect. We extend BD’s model in two ways.9 First, while retaining BD’s identification of reallocation shocks, we further differentiate these shocks into job-loss reallocation shocks and quits reallocation shocks based on whether the shock affects the unemployment rate of job losers (Ul) or job leavers (Uq). We measure t t Ul with the unemployment rate excluding job leavers, and hence Ul is composed of those who experienced t t involuntary separation and entrants to the labor force. We interpret Ul as largely representing job losers, t because job losers are the majority of this category and the main driver of its cyclical fluctuations.10 These two types of reallocation shocks are likely to result in different labor market outcomes and to carry different implicationsfortheNCRU.Inparticular,thequitsreallocationshockcanresultinjobseekerswhoexperience frictional unemployment when switching jobs, while the job-loss reallocation shock can result in job seekers who experience a spell of structural unemployment owing to inefficient job searches or worker–job mismatch. The second extension is to include measures of wage growth (W ) and price inflation (Π ), which lets us t t examinetheeffectofeachstructuralshockoninflationwithinacoherentBayesianstructuralVARframework. This adds two more structural shocks (own shocks to wage and price inflation). Our extended model has a total of six structural shocks, and is a six-variable structural VAR. Table 2 summarizes the identifying restrictions.11 The elements in the first two columns identify the job-loss reallocation shock (ul) and the quits reallocation shock (uq). These two shocks still satisfy BD’s t t identification of the reallocation shock, lowering the labor force participation rate ([1,1] and [1,2] entries) but raising both the vacancy rate ([2,1] and [2,2]) and unemployment rate ([3,1] and [4,2]). The key difference is that on impact, the job-loss reallocation shock ul only raises the unemployment rate of job losers while the t quits reallocation shock uq only raises the unemployment rate of job leavers. These restrictions are captured t with the zero restrictions in the [4,1] and [3,2] entries. Following BD, the positive aggregate activity shock uc raises the labor force participation rate ([1,3]) and t vacancyrate([2,3])onimpact. AdeparturefromBD’smodelisthatapositiveaggregateactivityshocklowers the unemployment rate of job losers at impact ([3,3]) but the sign of the effect on the unemployment rate of job leavers is unrestricted ([4,3]). The reason is that unemployment inflows and job-finding probabilities for unemployed job leavers are both procyclical (Ahn, 2023); we therefore let the data give us the net cyclical effect. Finally,inlinewithBDapositivelaborsupplyshockus raisesthelaborforceparticipationrate([1,4])but t is assumed to have no impact effect on the vacancy rate ([2,4]).12 In the extended model, this positive labor 9Notethatweusetheunemployment,vacancy,andlaborforceparticipationrates,ratherthanthetransformedlevelsthat BlanchardandDiamondused. 10Thisgroupalsoincludesindividualswhosetemporaryjobshaveendedandthoseenteringorre-enteringthelaborforce; note that re-entrants also include individuals who previously left the labor force after losing a job (Ahn, 2023). We do not excludelabor-forceentrantsfromUl sothatthemodelparsesvariationsinthetotal unemploymentrateintothecontributions t ofstructuralshocks. TheonlineappendixconsidersexcludingtheentrantgroupfromUl andshowsthattheeffectsofthelabor t supplyshockslargelydisappearwhiletheidentificationoftheotherstructuralshocksremainsrobust. Thisobservationsuggests thatlaborsupplyshocksarenotwell-identifiedifwerelyprimarilyontheparticipationrate,butthatourbaselinemodeldoes sowhilemaintainingrobustidentificationofotherstructuralshocks. 11Inwhatfollows,tableentriesareidentifiedby[row,column]. 12Assumingazeroimpacteffectofthelaborsupplyshockonthevacancyrateisalsoinlinewiththeempiricaltreatmentof 5

Table2: Identificationofstructuralshocksintheextendedmodel To a positive shock to ... Response of Job-loss Quits Aggregate Labor Wage- Pricethe following variables ... reallocation reallocation activity supply specific specific Labor force participation rate − − + + 0 0 Vacancy rate + + + 0 0 0 Unemployment rate (job losers) + 0 − + + ? Unemployment rate (job quitters) 0 + ? 0 0 0 Wage inflation ? ? ? ? + 0 Price inflation ? ? ? ? 0 + Notes: Thistablesummarizesthesignofeachshock’simpacteffectonthesixvariables: ‘+’and‘–’denotepositiveand negativeimpacteffects,respectively;‘0’denotesnoimpacteffect;and‘?’ denotesanunrestrictedeffect. supply shock raises the unemployment rate of job losers at impact ([3,4]) because the shock can encourage job losers or those who re-entered the labor force after a job loss to continue their job searches rather than exiting the labor force.13 We assume that the labor supply shock has no impact effect on the unemployment rate of job quitters ([4,4]). Note that we do not impose any sign restrictions on the impact effects of the structural labor-market shocks on wage and price inflation (see the fifth and sixth rows and first through fourth columns), which allows the model to estimate the sign and magnitude of each shock’s effect. The last two columns summarize the impact effects of own shocks to wage and price inflation. By definition, both shocks increase wage and price inflation at impact ([5,5] and [6,6]) but do not influence each other directly at impact ([6,5] and [5,6]). The wage-specific shock uw represents innovations like changes in t bargainingpower,minimumwageincreases,orwagemarkupshocks,whilethepriceshockuπ capturesfactors t such as cost-push shocks (e.g., supply bottlenecks) and changes in inflation expectations (e.g., from monetary policy actions). Both shocks are allowed to affect the unemployment rate of job losers: The wage shock is assumed to raise it ([3,5]), reflecting higher labor costs, while the price shock’s effect ([3,6]) is unrestricted, as a positive innovation to price inflation can raise it (supply disruptions) or lower it (a reduction in the real wage). For the other variables we impose zero restrictions on the nominal shocks’ impact effects, as most of the high-frequency variability of inflation reflects idiosyncratic movements that should have minimal within-month effects on job search and labor force entry.14 Foronietal.(2018). Itisimportanttonotethatthezeroimpacteffectdoesnotprecludeacasewhereapositivelaborsupply shockraisesvacancypostingsinthesubsequentmonthsaftertheimpact. However,itdiffersfromthepredictionofthesearch andmatchingmodelinPissarides(2000),whichsuggeststhatanincreaseinlaborsupplymakesiteasiertofillvacantjobs, causingapositivelaborsupplyshocktohaveanimmediatepositiveeffectonthevacancyrate. Weexplorethiscaseinour robustnessanalyses. Inthisalternativecase,bothpositivelaborsupplyshocksandreallocationshocksmovethevacancyrate andtheunemploymentrateinthesamedirection,buttheformerraiseslaborforceparticipation,whilethelatterlowersit. 13Apositivelaborsupplyshockcanalsoraisetheunemploymentrateofnewentrantstothelaborforce,arelativelysmall componentofUl. t 14Own shocks to wage and price inflation capture the nominal effects of cost-push shocks; they are identified from their limitedimpacteffectsonthelabormarket. Thelabormarketshocksthemselveswillcapturethereal effectsofcost-pushshocks; forexample,ifanoilshockhasrealeffectsonthelabormarketitcanshowupasanaggregateactivityshockorareallocation shock. Hence,theownshockstopricesandwagesinourmodelcontrolfortheportionofacost-pushshockthatisorthogonal tothefourlabormarketshocks,particularlytheaggregateactivityshock. Thistreatmentisinlinewithstandardmodelsofthe Phillipscurve,wheretheunemployment-rategapcapturesthestateofthebusinesscycleandcost-pushshocks(whichshiftthe Phillipscurve)aredirectlycontrolledforwithvariablessuchasimportorenergyprices. 6

2.2. Empirical model and estimation With the extended identification scheme, the dependent variables y , structural shocks u , and impact t t matrix H can be written as: y = [L ,V ,Ul,Uq,W ,Π ]′ t t t t t t t u = [ul,uq,uc,us,uw,uπ]′ t t t t t t t   −θ −θ +δ 1 0 0 l q +β l +β q 1 0 0 0   H=   1 0 −ϵ +ω +χ w χ p . (1)  0 1 ζ c 0 0 0    ϕ l ϕ q ϕ c ϕ s 1 0 ψ ψ ψ ψ 0 1 l q c s The signs of the parameters in H represent the sign restrictions that we impose on the model; when an explicit sign is absent, the parameter can be either positive or negative. WeincorporatetheextendedidentificationschemeintoaBayesiansign-restrictedVAR.Thereduced-form model is written as: y =Φx +ε (2) t t−1 t where the reduced form model is related to the structural model as follows: Φ = HB (3) ε = Hu (4) t t D = E(u u′) (5) t t Ω = E(ε ε′)=HDH′. (6) t t We include a constant and eight lags of y in x to adequately capture the dynamics of y . Since the t t−1 t structural shocks are uncorrelated with each other, D is a diagonal matrix. The reduced-form parameters are estimated from: (cid:32) T (cid:33)(cid:32) T (cid:33)−1 (cid:88) (cid:88) Φˆ = y x′ x x′ T t t−1 t−1 t−1 t=1 t=1 T (cid:88) Ωˆ = T−1 εˆεˆ′ T t t t=1 εˆ = y −Φˆ x . t t T t−1 This structural VAR is estimated with a Bayesian method using an algorithm that allows a researcher to impose restrictions directly on parameters in the impact matrix. In their 2015 paper, Baumeister and Hamilton (BH) propose an analytical framework for Bayesian inference in a structural VAR model, along with an estimation algorithm that forms priors incorporating sign and exclusion restrictions on the elements of interest in H−1. In spite of this flexible feature, the algorithm is not always applicable if a researcher has specific priors on parameters in the impact matrix H. One can still form priors on the parameters in H−1 after inverting the impact matrix.15 However, the priors on H often become intractable after the matrix is inverted; this is likely to occur when the impact matrix is large, or when the parameters in the impact matrix have complicated functional relationships. So it is more straightforward and convenient to form prior beliefs on the parameters of the impact matrix itself; we therefore modify the BH algorithm in a way that permits us to do so in a flexible manner.16 15BaumeisterandHamilton(2018)illustratesuchcases. 16Detailsareintheonlineappendix. 7

OurfullsampleperiodrunsfromJanuary1967toDecember2023. Thelargeswingsinmanyseriesduring thepandemiccandistortmodelestimatesthatincludesuchobservations. Weinterpretthepandemic-induced swings as resulting from large transitory shocks that had no material effect on the underlying dynamics of the economy, and so use the model parameters estimated from the pre-pandemic period (January 1967 to December 2019) to compute historical decompositions over the pandemic period (January 2020 to December 2023). In other words, the historical decompositions are extended through the pandemic period by using the pre-pandemic parameter values to back out the implied structural innovations and their propagation dynamics. This approach is in line with several recent studies that consider how to deal with the pandemic period in time-series models (Lenza and Primiceri, 2020; Ng, 2021; Cascaldi-Garcia, 2022), and we use it throughout the paper.17 2.3. Prior specification The priors and sign restrictions on the structural parameters of the impact matrix H are summarized in Table 3.18 If a parameter contributes to H with a specific sign, we impose that sign restriction (column [5]); “NA” indicates that no restriction is imposed. We characterize the prior for each structural parameter with a Student t distribution whose mode is close to the corresponding calibrated value in Blanchard and Diamond (1989) or to a value from more-recent related research, as noted in column [7].19 For most of the parameters, the scale parameter is set to 0.3 and 3 degrees of freedom are used (as Baumeister and Hamilton, 2022 note, this represents a fairly weak prior belief). Column [6] reports the 65% and 90% ranges of the prior distribution for the corresponding parameter. We highlight a departure from BD’s calibration. The slope of the structural Beveridge curve represents the cyclical tradeoff between job vacancies and unemployment that yields the same number of worker-job matches. Assuming a constant-returns-to-scale (CRS) matching function, the slope is determined by the matching elasticity with respect to vacancies (or unemployment). In this sense, the parameter −ϵ —the impact response of the unemployment rate to an aggregate activity shock that increases the vacancy rate by one percentage point—captures the tradeoff at the impact of an aggregate activity shock. Without the knowledge of how the tradeoff evolves through the propagation of an aggregate activity shock, our prior belief is that −ϵ approximates the average cyclical tradeoff between the vacancy rate and unemployment rate, and the prior mode −ϵ=−0.8 implies our prior belief that the matching elasticity with respect to vacancies is 0.45.20 We allow for considerable uncertainty in this parameter so that our prior distribution well covers the range of estimates documented in existing studies (0.3 to 0.5)—see Pissarides and Petrongolo (2001).21 In the rest of this section we focus on the parameters not discussed in BD, specifically the effects of the labor market shocks on wage and price inflation. We assume that an aggregate activity shock that lowers the 17We acknowledge a few caveats to this approach. First, our model does not estimate the direct effects of fiscal stimulus ormonetarypolicyduringthisperiod. Thoughthesepolicyshockscanbeparsedoutintooneormultiplestructuralshocks inthecontextofourmodel, adifferentframeworkisneededtoanalyzetheextenttowhichthepolicyshocksinfluencethe BeveridgeandPhillipscurves. Second,ourmodelcannotaccountforthepossibilitythatthepandemicpermanentlyalteredthe propagationofshocks,leadingtosystematicchangesinimpulseresponses. Fouryearsofmonthlyobservationsareinsufficientto inferpermanentchangesinthepropagationofstructuralshocks,thoughitmightbedoneusingatime-varyingparametermodel withmanymorepost-pandemicobservations. Seetheonlineappendixforadditionaldiscussion. 18ThepriordistributionssuggestthatthedeterminantofHispredominantlynegativebuttherighttailofthedistribution extendsto0.6andincludeszero. However,theprobabilitydensitywithintheregion[−0.5,0.6]isonlytwopercent. Wetherefore donotimposeanadditionalpriorthatpreventsthedeterminantfrombecomingzero(asconsideredinBaumeisterandHamilton, 2018);evenso,theposteriordistributionofthedeterminantisnonzero. 19Wealsousedtheempiricalresultsfromthisresearchtocalibratethepriormodesforafewparameters. (Seetheonline appendixforadetaileddiscussion.) 20AssumeaCRSmatchingfunctionoftheformH=MU(1−α)Vα,whereM representsmatchingproductivity,andU,V,and H denotethenumber(orshareofthelaborforce)ofunemployedworkers,vacancies,andworker-jobmatches. HoldingM and H constanttoshutdownshiftsofthestructuralBeveridgecurveandtorestricttheV −U trajectorytotheiso-matchcurve, the elasticity α can be approximated by noting that −1−α = dlog(V) ≈−1. BD’s benchmark calibration, by comparison, α dlog(U) ϵ impliesϵequals1.3,whichinturnimpliesthematchingelasticitywithrespecttovacanciesisabout0.6. 21AccordingtoamorerecentstudybyBersteinetal.(2023),theelasticitywithrespecttovacanciesrangesfrom0.15to0.70, widerthantherangedocumentedbyPissaridesandPetrongolo(2001). The90thpercentileintervalofourpriordistribution correspondstoanelasticityrangeof[0.2,0.6]. 8

Table3: Priorsforthestructuralparameters [2]Prior [3] [4] [5]Sign [1] [6]65(90)%range [7]Sourcesandempiricalgrounds mode Scale d.f. inH Panel A.Job-lossreallocationshock θl 0.5 0.3 3 – [0.28,0.86]([0.11,1.24]) BlanchardandDiamond(1989) BlanchardandDiamond(1989), βl 0.1 0.2 5 + [0.07,0.37]([0.02,0.56]) Foronietal.(2018) ϕl -0.2 0.3 3 NA [-0.53,0.13]([-0.91,0.51]) Authors’calibration ψl -0.2 0.3 3 NA [-0.53,0.13]([-0.91,0.51]) Authors’calibration Panel B.Quitsreallocationshock θq 0.9 0.3 3 – [0.60,1.24]([0.33,1.62]) BlanchardandDiamond(1989) βq 0.2 0.3 3 + [0.12,0.62]([0.04,1.02]) BlanchardandDiamond(1989) ϕq 0.2 0.3 3 NA [-0.13,0.53]([-0.51,0.91]) Authors’calibration ψq 0.2 0.3 3 NA [-0.13,0.53]([-0.51,0.91]) Authors’calibration Panel C.Aggregateactivityshock δ 0.4 0.3 3 + [0.21,0.77]([0.08,1.16]) BlanchardandDiamond(1989) BlanchardandDiamond(1989), ϵ 0.8 0.3 3 – [0.52,1.14]([0.26,1.52]) PissaridesandPetrongolo(2001) ζc 0.01 0.3 3 NA [-0.32,0.34]([-0.70,0.72]) Ahn(2023),author’scalibration ϕc 0.5 0.3 3 NA [0.17,0.83]([-0.21,1.21]) Gal´ıandGambetti(2020),Ahnetal.(2020) Hazelletal.(2022),Fitzgeraldetal.(2024), ψc 0.3 0.3 3 NA [-0.03,0.63]([-0.41,1.00]) BarnichonandMesters(2021) Panel D.Laborsupplyshock BlanchardandDiamond(1989), ω 0.2 0.3 3 + [0.12,0.62]([0.04,1.02]) Foronietal.(2018) ϕs -0.1 0.3 3 NA [-0.43,0.23]([-0.81,0.61]) Foronietal.(2018) ψs -0.1 0.3 3 NA [-0.43,0.23]([-0.81,0.61]) Foronietal.(2018) Panel E.Ownshockstowagesandprices χw 0.1 0.3 3 + [0.09,0.57]([0.03,0.97]) Foronietal.(2018),Gal´ıetal.(2012) χp 0.05 0.2 2 NA [-0.19,0.29]([-0.53,0.63]) Gal´ıetal.(2012) Notes: Thistablesummarizesthepriordistributionsofthestructuralparameters. “d.f.” standsfordegreesoffreedom. “NA” standsfornotapplicable. unemployment rate by approximately 1 percentage point (pp) likely increases price inflation by about 0.3 pp (themodeofψ )andwageinflationby0.5pp(themodeofϕ ).22 Thepriordistributionsoftheseparameters c c cover the range of estimates found in previous studies; without sign restrictions, these prior distributions extend into the negative region, allowing for the possibility that the slope of the Phillips curve can be zero.23 We assign smaller absolute values to the prior modes of the parameters that capture the effects of other labor-market shocks on inflation, reflecting our prior that aggregate activity shocks are likely to be the main source of the observed tradeoff between inflation and real activity. In addition, with limited evidence as to the differential effect of these structural shocks on wages and prices, we apply the same priors to both types 22Thepriordistributionofϵimpliesthatanaggregateactivityshockthatincreasesthevacancyratebyonepercentagepoint lowerstheunemploymentrateofjoblosersby0.9percentagepoint—thepriormean. Sincethismagnitudeissufficientlycloseto 1giventheuncertaintysurroundingthisparameter,weconsidertheimpactoftheaggregateactivityshockontheunemployment ratetobeapproximatelyonepercentagepoint. 23FurlanettoandLepetit(2024)provideacomprehensivesurveyofestimatedPhillipscurveslopes;thepriordistributionof ψc encompassestherangeofestimatesreportedthere. ForthepricePhillipscurve,BarnichonandMesters(2020)reporta slopevalueof−0.24andSmithetal.(2023)giveestimatesrangingfrom−0.25to−0.29. Fitzgeraldetal.(2024)findsthatthe empiricalestimateoftheslopebasedonMSAdatais-0.33,whereasthestructuralmodel-impliedslopesare−0.28forprices and−0.87forwagesover1977–2017,and−0.21and−0.60,respectively,over1999–2015. However,Hazelletal.(2022)finda slopeestimateveryclosetozero. ForthewagePhillipscurve,Gal´ıandGambetti(2020)findslopeestimatesthatrangefrom −0.7to−0.2duringthesampleperiod1964–2017,withapreferredestimateofabout−0.5,andAhnetal.(2020)estimate wagePhillipscurveslopesthatrangefrom−0.63to−0.37over1990–2019usingavarietyofcompensationmeasures. 9

of inflation. To characterize the inflationary effect of a quits reallocation shock that raises the unemployment rate of job leavers by 1 pp, we set the prior modes of ϕ and ψ to 0.2. To reflect the negative effect on q q wages and prices of a job-loss reallocation shock that raises the unemployment rate of job losers by 1 pp (see Davis and Wachter, 2011), we set the prior modes of ϕ and ψ to –0.2.24 The inflationary effect of a l l labor-supply shock that raises the labor force participation rate by 1 pp and the unemployment rate of job losers by 0.2 pp—the prior mode of ω—is assumed to be even closer to zero, with the prior modes of ϕ s and ψ set to −0.1. The prior mode of ω falls between the values from Blanchard and Diamond (1989) and s Foroni et al. (2018), and we allow for enough uncertainty such that the prior distribution encompasses the range of parameters considered and estimated by both studies. The prior modes of ϕ and ψ are also in line s s with the impulse responses found by Foroni et al. (2018). Finally, we assume that own shocks to wages have small positive impact effects on the unemployment rate of job losers (χ ). Interpreting these shocks as reflecting changes in wage bargaining power and wage w markups, we set the prior mode of χ to 0.1, which broadly aligns with the effect on the unemployment w rate of a wage-bargaining shock (Foroni et al., 2018) or a wage mark-up shock (Gal´ı et al., 2012).25 Since price-specific shocks can have positive or negative effects on unemployment, we assign a smaller prior mode for χ and use a lower degree of freedom (2), and a smaller scale parameter (0.2) to allow for a sufficient p degree of uncertainty, while preventing unreasonably large positive or negative values. 3. Estimation Results: Impulse Responses and Historical Decompositions 3.1. Impulse responses Figure 1 reports the impulse responses for each variable following each structural shock.26 (In the panel titles, “x<y” denotes the effect of shock y on variable x.) The first column reports the impulse responses following a job-loss reallocation shock.27 This shock raises the vacancy rate (first row) and has persistent positive effects on the unemployment rate of job losers (second row). The shock lowers the unemployment rate of job quitters for a few months after impact, but the statistical significance of the response declines rapidly (third row). The job-loss reallocation shock lowers wage growth persistently (fourth row); the shock’s effects on market-based core inflation are negative for about two years but are only statistically significant for a little over a year (fifth row). The second column reports the impulse responses following a quits reallocation shock. This shock also persistently raises the vacancy rate (first row) and the unemployment rate of job quitters (third row). The shock lowers the unemployment rate of job losers, but the effects are not statistically significant (second row).28 The quits reallocation shock results in a statistically significant increase in AHE growth and price inflation (fourth and fifth rows), which is noticeably different from the effects of a job-loss reallocation shock. The effects of a quits-reallocation shock on wage and price changes are what we would expect to see if the reallocation shock led to an increase in the NCRU. The third column reports the impulse responses that follow an aggregate activity shock. This shock has persistent effects on most of the variables: The aggregate activity shock raises the vacancy rate, AHE growth, and price inflation, and lowers the unemployment rate of job losers. These effects are statistically significant for some time. In particular, the responses of the vacancy rate and the unemployment rate of job losers peak 24Thesemagnitudesbroadlyalignwithregression-basedevidenceusingtheon-the-jobsearchrateandtheemploymentrateof badmatchesrecoveredfromFacciniandMelosi(2023). (Detailsontheregression-basedcalibrationareintheonlineappendix.) 25Foronietal.(2018)findthata1%riseinrealwagesduetoawage-bargainingshockraisesunemploymentby0.1pp,while Gal´ıetal.(2012)showthatawagemark-upshockofthesamemagnitudeincreasesunemploymentby0.2pp. 26Theposteriordistributionsofallstructuralparametersarereportedintheonlineappendix. 27Theonlineappendixreportsthefullsetofimpulseresponsesforallsixvariables. 28InFacciniandMelosi(2023),general-equilibriumeffectsreduceunemploymentafteranon-the-jobsearchshock(which issimilartoourquitsreallocationshock). Thislargelyalignswithourfindings: Whileourquitsreallocationshockinitially raisestheunemploymentrateamongjobquitterswithoutaffectingtheunemploymentrateofjoblosers,itsubsequentlylowers theunemploymentrateofjoblosers,althoughtheeffectisstatisticallyinsignificant. (NotethattheanalysisinFacciniand Melosi (2023) focuses on employment-to-employment flows, where workers switch jobs without experiencing any period of unemployment.) 10

Figure1: Impulseresponses Notes: Thisfiguredisplaystheestimatedimpulseresponsesfromourbaselinesystem(X-axis: numberofmonthsafterthe shockhits;Y-axis: themagnitudeoftheresponse). Theposteriormedianisshownasasolidline;thedottedlinesindicatezero; theshadedareagivesthe68percentposteriorcredibleset;andx<y indicatestheeffectofshocky onvariablex. VR :the vacancyrate;UR (Job Loss) : theunemploymentrateofjoblosersandentrantstothelaborforce;UR (Quits) : the unemploymentrateofjobquitters;AHE :Changeinaveragehourlyearnings;Mkt-Core(adj) : Market-basedcorePCEprice inflation(adjustedforeffectsofNixon-erapricecontrols). Ul: job-lossreallocationshock;Uq: job-quitsreallocationshock;Uc: t t t aggregateactivityshock;Us: laborsupplyshock. t approximately one year after the shock and remain statistically significant for about 2.5 to 3 years. Those of wage and price inflation peak at around 1.5 and 2 years, respectively, and remain statistically significant for 3 to 4 years after the shock. The response of wage inflation is a little faster and larger than that of price inflation; this result is consistent with the wage Phillips curve’s being steeper than the price Phillips curve. The fourth column reports the impulse responses following a labor supply shock. This shock raises both unemploymentrates,consistentwithincreasedlabor-forceattachmentandanincreasednumberofjobseekers drawn into the labor force. However, the effects are statistically significant for the unemployment rate of job losers (second row) but not for job leavers (third row). Effects of labor supply shocks on the vacancy rate are not statistically significant (first row). A labor supply shock lowers wage inflation (fourth row); it also 11

lowers market-based core inflation (fifth row). Though these effects are statistically significant, the statistical significance of labor-supply shock’s inflationary effects is not as strong as that of the other shocks. All told, the four labor-market shocks have unique and distinguishable effects on the joint dynamics of the unemployment and vacancy rates, as well as on wage and price inflation.29 3.2. Historical decompositions for the pre-Covid period We use historical decompositions to obtain the contributions of each structural shock to variations in the data.30 Figure 2 reports the historical decomposition of the vacancy rate (first row), the unemployment rates of job losers and quitters (second and third rows, respectively), AHE growth (fourth row), and market-based core inflation (fifth row) by each structural labor-market shock from September 1967 to December 2019. The aggregate activity shock is the main source of variations in the vacancy rate (row 1, column 3) and the unemployment rate of job losers (row 2, column 3), in line with Abraham and Katz (1986). During the Great Recession, the job-loss reallocation and labor-supply shocks were particularly important fortheNCRU.31 Job-lossreallocationshocksraisetheunemploymentrateofjoblosersbyaboutthree-quarters of a percentage point during the Great Recession (row 2, column 1); the labor supply shock raises the unemployment rate of job losers by half of a percentage point during this period (row 2, column 4). The sum of these magnitudes is broadly in line with the increase in the short-term natural rate that the Congressional Budget Office (CBO) and Federal Reserve Board staff identified during this episode. In this context, the rise in the unemployment rate from the job-loss reallocation shocks likely represents increased reallocational frictions, including mismatch between workers and jobs, while the portion driven by labor supply shocks might reflect the effect of the extended unemployment insurance (UI) benefits that were put in place during this period.32 If so, the portion of the unemployment rate driven by job-loss reallocation and labor supply shocks would represent a change in structural unemployment (and so the NCRU) over this period. The structural labor market shocks also affect wage and price inflation. The aggregate activity shock accounts for some cyclical variation in wage and price inflation (row 4, column 3; row 5, column 3). It is also noteworthy that quits reallocation shocks are important contributors to wage and price inflation from the 1970s to the 1990s and after the Great Recession (row 4, column 2; row 5, column 2). This finding aligns with Faccini and Melosi (2023), who argue that a shock to the propensity to search on the job—similar to our quits reallocation shock—creates a positive link between job quits and price inflation, with the low and stable price inflation following the Great Recession driven by reduced quits reallocation.33 3.3. Historical decompositions for the pandemic period Figure 3 reports the historical decompositions from our model for the period January 2020 to December 2023, with the values for December 2019 normalized to zero.34 In contrast to earlier periods, both the 29Theonlineappendixpresentsanumberofrobustnesschecks,includingimposingapositiveimpacteffectofthelaborsupply shockonthevacancyrate,excludinglaborforceentrantsfromtheunemploymentrate,andusinganalternativemeasureof inflation. Overall,ourmainresultsremainrobusttothesemodifications. 30AVARmodeldecomposesthevalueofyt intothecomponentattributabletostructuralshocks(orreduced-formforecast errors)sincethestartofthesampleandtheforecastfordatetformedonthebasisofinitialconditionsattime0(Hamilton, 2025). The first term is called the stochastic component, and a historical decomposition refers to the cumulative historical contributionofstructuralshockstothiscomponentofyt. Thesecondterm,obtainedfromdynamicforecastsbasedsolelyon theinitialconditionsandparametersoftheVARmodel,canexhibithighlypersistentmovementsiftheVARincludesvariables withhighpersistence(Giannoneetal.,2018). Becauseweareinterestedinthecontributionsofthestructuralshockstocyclical dynamics,wefocusontheseshocks’contributionswhenreportingthehistoricaldecompositions. 31Relativetotheseshocks,thequitsreallocationshock’scontributiontooverallunemploymentratefluctuationsismuted. Thequitsreallocationshockistheprimarydriveroftheunemploymentrateofjobquitters,butthiscategoryrepresentsasmall shareoftheoverallunemploymentpool. 32TheextendedUIbenefitslikelyhavemotivatedlong-termunemployedindividualstocontinuetheirjobsearches,thereby raisingtheunemploymentrate. 33Note that although the quits reallocation shock is identified from the unemployment rate of job quitters, it likely also capturesjob-to-jobtransitionsandcouldthereforehavesizableeffectsonwageandpriceinflation. Thecongruenceofourresults withFacciniandMelosi(2023)isnotableinasmuchasourestimationmethodologyisverydifferenttothatpaper(whichusesa relativelytightlyparameterizedDSGEmodeltoidentifytheinflationaryeffectsofachangeinon-the-jobsearch). 34Recallthattheparametervaluesthatweusecomefromamodelwhoseestimationperiodendsin2019. 12

Figure2: Historicaldecompositions: pre-pandemic Notes : ThisfiguredisplaystheposteriormediansofthehistoricaldecompositionsofthevariablesinourVARsystem priortotheCovid-19pandemic(X-axis: calendarmonth;Y-axis: themagnitudeofvariationsintheendogenousvariable). The shadedareagivesthe68percentposteriorcredibleset. Thebluelinesaretheportionsofthedatathatareexplainedbyeach shock. Theredlinesrepresentthesumofthehistoricaldecompositionsforagivenvariable. VR :thevacancyrate;UR (Job Loss) : theunemploymentrateofjoblosersandentrantstothelaborforce;UR (Quits) : theunemploymentrateofjob quitters;AHE :Changeinaveragehourlyearnings;Mkt-Core(adj) : Market-basedcorePCEpriceinflation(adjustedforeffects ofNixon-erapricecontrols). Ul: job-lossreallocationshock;Uq: job-quitsreallocationshock;Uc: aggregateactivityshock;Us: t t t t laborsupplyshock. aggregate activity shock and the job-loss reallocation shock make important contributions to the dramatic swings in the labor market variables and wage and price inflation seen during this period. The job-loss reallocation shocks (light-blue line with markers) cause the unemployment rate (upper-right panel) and the vacancy rate (upper-left panel) to rise sharply in April 2020 and to then decline rapidly over the second half of that year. This positive effect of the reallocation shocks on job postings is consistent with Barrero et al. (2021) and also with Haltiwanger (2021), who shows that new business formation spiked in the early phase of the pandemic. Hence, the job-loss reallocation shock might be raising the number of unemployed job losers in some sectors or firms, while at the same time creating new job vacancies elsewhere. From the second half of 2020, both the job-loss reallocation and aggregate activity shocks contribute to a persistent increase in the vacancy rate. The duration of the effect of the job-loss shocks is again consistent with Barrero et al. (2021), who conclude that the pandemic acted like a persistent reallocation shock. This observation suggests that the job-loss reallocation shock led to a rise in the NCRU. At the onset of the pandemic, negative aggregate activity shocks (yellow line) lowered the vacancy rate 13

Figure3: Historicaldecompositions: pandemicperiod Notes : ThisfiguredisplaystheposteriormediansofthehistoricaldecompositionsofthevariablesinourVARsystem duringtheCovid-19pandemic(X-axis: calendarmonth;Y-axis: themagnitudeofvariationsintheendogenousvariable). Vacancy rate: Thevacancyrate;Unemployment rate: Theunemploymentratesofjoblosersandleaverscombined;AHE: Changeinaveragehourlyearnings;Mkt-core(adj): Market-basedcorePCEpriceinflation(adjustedforeffectsofNixon-era pricecontrols). (upper-left panel) but sharply raised the unemployment rate of job losers (upper-right panel); for vacancies, aggregate activity shocks more than offset the positive effects from job-loss reallocation shocks. Positive aggregate activity shocks facilitated a rapid recovery of the vacancy rate and the unemployment rate of job losers in the second half of 2020; from 2021, the aggregate activity shock contributes to a higher vacancy rate and a lower unemployment rate. During this period of cyclical tightening, aggregate activity shocks acted to reduce the unemployment rate. These shocks pushed the unemployment rate about 2 percentage points lower than its pre-pandemic level (the yellow line in the upper-right panel), suggesting that labor market conditions in that year were tighter than those just before the pandemic.35 Meanwhile, job-loss reallocation shocks (light-blue line with markers) and shocks to wage and price inflation (dashed green line and dotted light-blue line, respectively) contributed to raising the unemployment rate. The upward pressure from the wage-specific shock might reflect increased wage growth expectations that were driven by expectations for higher price inflation or an increase in desired wages. In addition, the rise in price inflation (likely reflecting cost-push factors) negatively affected the labor market, increasing unemployment and reducing the vacancy rate, though these effects faded as supply disruptions normalized. 35Crumpetal.(2024)alsofindthattheirestimateoftheunemploymentrategapisbelowitspre-pandemiclevelin2022. However,in2023ourestimatesuggestslesstightnessinthelabormarketthanwhattheseauthorsfind(theirestimateforthe gapisafewpercentagepointsbelowitspre-pandemiclevel). 14

The job-loss reallocation shock also raised the unemployment rate in 2021, but as this shock unwound it lowered the unemployment rate, leaving the portion of the unemployment rate attributable to job-loss reallocationslightlybelowitspre-pandemiclevelbytheendof2023. Primarilyledbythejob-lossreallocation shock, the boost to the NCRU gradually dissipated from mid-2021 through the recovery period. By the end of 2023, the portion of the unemployment rate reflecting aggregate activity shocks had largely returned to its pre-pandemic level.36 The evolution ofwage inflation over thepandemic periodis dominatedbylarge swings inthe contribution of own-shocks (the dashed green line in the lower-left panel), which in turn reflect a rapid change in the composition of employment: In the early phase of the pandemic, job losses were disproportionately concentrated among low-wage workers, which caused measured average wages to rise quickly. The sharp downward movement in wage growth in 2021 is the result of base effects (recall that these are 12-month changes), as the earlier upward spike in wage growth drops out of the 12-month moving window. These composition effects mostly play out by the end of 2022; thereafter, the contribution of these shocks declines steadily. The aggregate activity shocks lower wage inflation in 2020 and 2021, but push it up from 2022–on (yellowline),andthejob-lossreallocationshocksputpersistentdownwardpressureonwagegrowth(light-blue line with markers). These results highlight the important influence that the structural labor market shocks had on the evolution of wage inflation during the pandemic. Similarly,own-shocksarethedominantforcedrivingpriceinflationoverthisperiod(dottedlight-blueline). The importance of the own-shocks reflects the fact that typically only a modest fraction of the variability in price inflation can be explained by developments in the labor market. In this episode, the own-shocks to price inflation likely reflect import price growth and the presence of supply–demand imbalances in product markets. In turn, the own-shocks cause a reduction in inflation from 2022–on that likely captures the gradual resolution of these imbalances. Even so, the influence of some fundamentals is clearly apparent: Aggregate activity(yellowline)putsdownwardpressureoninflationuntil2021;inaddition,own-shockstowageinflation (green dashed line) contribute about 1.5 pp to core inflation by 2023, implying a non-negligible amount of pass-through from wages to prices. The contribution of job-reallocation shocks to price inflation is mostly negative until 2022. However, in 2023 decreased reallocation activity accompanying job losses puts small upward pressure on price inflation.37 The behavior of inflation and unemployment during this period therefore reflected a number of structural shocksthatinducedbothmovementsalongandshiftsoftheBeveridgeandPhillipscurves. Thismixofshocks allowed inflation to decline and the labor market to normalize without a large increase in the unemployment rate. 4. The Anatomy of the Beveridge and Phillips Curves In this section, we first decompose the movements in the unemployment rate and vacancy rate into portions attributable to each structural shock; we then compute the Phillips correlation conditional on each structural shock. 4.1. Structural shocks and the Beveridge curve AnoutwardshiftintheBeveridgecurvehasoftenbeeninterpretedasindicatinganincreaseintheNCRU. However, increasedunemploymentinflowscanalsoshiftthecurve, makingsuchaninterpretationproblematic (see Ahn and Crane, 2020 and Barnichon and Figura, 2010). Our empirical decomposition of Beveridge curve dynamics into the contributions of different structural shocks can be used to examine movements of or along the Beveridge curve. 36Thequitsreallocationshocksraisetheunemploymentrateofjobquittersin2021(notshown). Thisresultisconsistent withtheanecdotesofa“GreatResignation”thatoccurredduringtherecoveryfromthepandemicrecession;however,theeffect onthetotalunemploymentrateissmall. 37Thefactthatinflation’sdeclinedidnotrequireasequenceofdisinflationaryactivityorjob-lossreallocationshockssuggests thatlabor-marketnormalizationwasnotapreconditionforinflationtocomedown. 15

Figure4: TheanatomyoftheBeveridgecurve Aggregate activity 4 Panel A: Aggregate activity Panel B: Job-loss reallocation Reallocation (Job loss) 4 3 3.5 2 3 )% 1 2.5 ( e ta 0 )% ( e 2 r y c n a -1 ta r y c n 1.5 c a a c 1 V-2 a V 0.5 -3 0 -4 -0.5 -5 -1 -4 -3 -2 -1 0 1 2 3 4 -2 -1 0 1 2 3 4 5 6 Unemployment rate (%) Unemployment rate (%) Reallocation (Quits) Labor supply 1 Panel C: Quits reallocation 1 Panel D: Labor supply 0.8 0.8 0.6 0.6 0.4 0.4 )% )% ( e 0.2 ( e 0.2 ta ta r y 0 r y 0 c c n n a c-0.2 a c-0.2 a a V V -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 -1 -1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Unemployment rate (%) Unemployment rate (%) Notes: Thisfiguredisplaystheposteriormediansofthehistoricaldecompositionsforvacancyandunemploymentratepairs generatedbyfourstructuralshocks. Theredcirclesdenotepre-pandemicobservations;pandemic-periodobservationsstartas dark-bluedotsandendasyellowdots. TheX-axisistheportionoftheunemploymentratedrivenbyeachstructuralshock(%). TheY-axisistheportionofthevacancyratedrivenbyeachstructuralshock(%). Figure4displaystheestimatedportionsofthevacancyandunemploymentratesdrivenbyeachstructural shock; the vacancy rate (V) is plotted on the y-axis and the unemployment rate (U) is plotted on the x-axis. The red circles capture the trajectories of V and U for the pre-pandemic period. With a posterior median for −ϵ of −0.65, the aggregate activity shock and its propagation yield a downward-sloping V–U relation (panel A) that implies a CRS matching function elasticity of 0.40 with respect to vacancies for 1967M9–2019M12, and a similar value (0.36) for the period when JOLTS data are available (2000M12–2019M12).38 Meanwhile, 38OurempiricalanalogueofthestructuralBeveridgecurveistheV–U trajectorydrivenbyaggregateactivityshocksshown inpanelA.AssumingaCRSmatchingfunction(seefootnote20),thematchingelasticitywithrespecttovacanciesαcanbe recoveredfrom−1−α = dlog(Vˆ),whereVˆ andUˆ aretheportionsofthevacancyandunemploymentratesdrivenbyaggregate α dlog(Uˆ) activityshocks. ToobtainVˆ andUˆ,westartwiththeportionofthevariable’shistoricaldecompositionthatisattributableto aggregateactivityshocksandaddinthevariable’ssamplemean(recallthatthehistoricaldecompositionsaremean-zero). We 16

job-loss reallocation shocks yield an upward-sloping V–U relationship (panel B). By contrast, the quitsreallocation shock plays a limited role in observed V–U movements because job quitters represent a small share of the unemployment pool (panel C). Finally, labor supply shocks move the Beveridge curve sideways (panel D). All told, the structural labor market shocks generate distinct sets of V–U dynamics, suggesting that we should consider the contribution of each shock separately when using the Beveridge curve to evaluate the state of the labor market. The pandemic period is represented by the solid dots. The effects of the realized shocks in 2020 are captured with the dark-blue dots, the subsequent realizations with lighter blue dots, and the most-recent realizations with yellow dots. The dark-red dashed line traces out the trajectory induced by each shock. Our estimates demonstrate just how unprecedented the pandemic-induced shocks to the labor market actually were. Not only did this period see an extraordinarily large change in aggregate activity (panel A), it also saw a rise in job-loss reallocation that induced an outward shift in the Beveridge curve that was larger and more persistent than anything seen before (panel B). By contrast, the effects of the labor-supply shocks (panel D) were more in line with previous episodes. Our anatomy of the movements in the Beveridge curve during and after the pandemic directly speaks to the debate over the likelihood of a so-called soft landing in the labor market. Specifically, Blanchard et al. (2022) and Figura and Waller (2022) have expressed diametrically opposed views about the likely evolution of the labor market as the post-pandemic recovery matures. The former argue that the observed movements in vacancies and unemployment during the pandemic indicate that the structural Beveridge curve is flat (in V–U space), and so a reduction in labor demand will lower the vacancy rate but yield a very large increase in the unemployment rate. Meanwhile, Figura and Waller argue that the observed changes in vacancies and unemployment reflected a high number of job separations in the early phase of the pandemic, but that the downward slope of the structural Beveridge curve is steeper than what a simple read of the observed V–U trajectory would indicate, implying that a reduction in job vacancies could occur without a large increase in the unemployment rate. Wecanviewthisdebatethroughthelensofourformalstatisticalframework. Essentially, whatBlanchard et al. and Figura and Waller disagree about is the slope of the structural Beveridge curve—the locus of vacancy and unemployment rates that results from shocks to aggregate activity. However, identifying this slope requires controlling for the effects of other structural shocks. Our decomposition does exactly that, providingausefulwaytoevaluatethetwoviews. Ourestimatedmatchingelasticitywithrespecttovacancies of 0.40 suggests that the slope of the structural Beveridge curve is indeed steeper than the conjecture by Blanchard et al., broadly aligning with the perspective of Figura and Waller.39 Furthermore, our formal statistical approach uncovers the key structural shocks driving the soft landing, which neither paper explicitly identified or estimated. Looking at the historical decomposition in Figure 3, both the job-loss reallocation and aggregate activity shocks exerted comparable amounts of upward pressure on the vacancy rate from the second half of 2021. In other words, the cyclical strength in the vacancy rate was smaller than what a simple read of the data suggested. Therefore, in 2022 and 2023 the decline in the vacancyrateasdemandweakenedwasabletooccurwithoutthedramaticriseintheunemploymentratethat Blanchard et al. predicted. Also during this period, job-loss reallocation gradually unwound and put modest downward pressure on the unemployment rate; a soft landing with lower inflation was further facilitated because the job-loss reallocation shocks pushed down inflation through 2022. Thereafter, the unwinding of the price-specific shocks over the second half of 2023 put additional downward pressure on the unemployment rate that partly offset the upward pressure from aggregate activity. Ouranalysisofthisperioddemonstrateswhyitisimportanttoconsiderafullaccountingoftheunderlying structural shocks that affect labor-market outcomes when using the Beveridge curve to assess the current and prospective state of the labor market. thenregressthelogdifferenceoftheresultingvacancyseriesonthelogdifferenceoftheunemploymentseriesandaconstant. Theestimatedcoefficientforthelogdifferenceoftheunemploymentseries—ameasureoftheslopeofthestructuralBeveridge curve—is−1.51for1967M9–2019M12and−1.77for2000M12–2019M12,andtheestimatedconstantisclosetozeroinboth cases. (Detailsareintheonlineappendix.) 39Intheir2024paper,FiguraandWallerupdatetheirbaselineestimateto0.38. 17

4.2. Structural shocks and the Phillips curve Changes in aggregate demand should induce movements along the Phillips curve rather than shifts in the curve. We can therefore get an idea of the slope of the Phillips curve by quantifying the movements in the unemploymentrateandinflationthatobtainfollowinganaggregateactivityshock. Asourresultsdemonstrate, one reason to take this approach is that other structural shocks can influence the size and direction of comovements in the unemployment rate and price inflation that can in turn affect the reduced-form correlation of these variables. To illustrate this point, we compute the correlation of the unemployment rate and market-based core inflation conditional on each structural shock. Using the impulse responses from the baseline model (Figure 1), we compute the ratio of the cumulative response of price inflation and that of the unemployment rate (we combine both types of unemployment) following each structural shock. We compute the cumulative responses over h periods, so these ratios give the average change in price inflation that is associated with a given change in the unemployment rate through horizon h following a structural shock.40 For the aggregate activityshock, thisratiocanbethoughtofasameasureoftheinflation–unemploymenttradeoff—thePhillips curve slope as it is typically defined. We also report analogous estimates for wage growth. Table 4 reports estimates of these “Phillips ratios” for the four labor market shocks. The aggregate activity shock results in a Phillips ratio of −0.10 for h=6 and −0.15 for h=12, indicating a statistically significant tradeoff between inflation and the unemployment rate (panel C). It is notable that the Phillips ratios are more negative for labor-supply shocks (panel D) and for job-loss reallocation shocks (panel A) than for aggregate activity shocks, implying that these two shocks can generate a larger negative reduced-form Phillips correlation than what the aggregate activity shock induces.41 Quite differently, a quits reallocation shock induces a positive Phillips ratio (panel B); a rise in unemployment associated with increased quits is actually inflationary. The quits reallocation shock can therefore flatten the slope of the reduced-form Phillips curve. For wage inflation, the Phillips ratios are more negative in response to aggregate activity shocks, job-loss reallocation shocks, and labor-supply shocks compared to their price counterparts. This finding aligns with evidence suggesting that the wage Phillips curve is steeper than the price Phillips curve (e.g., Gal´ı and Gambetti, 2020). An exception is the quits reallocation shock, where the price Phillips ratio is larger than the ratio for wage inflation.42 Our estimates suggest that the observed slope of the reduced-form Phillips curve will depend on the incidence and size of the structural labor-market shocks, and hence that the tradeoff between inflation and real activity can be obscured in periods where these other shocks are important. In this context, we can evaluate the contribution of each shock to the reduced-form correlation between price inflation and the unemployment rate by expressing the unconditional correlation as a weighted sum of the contributions of each shock’s Phillips ratio. Let U denote the aggregate unemployment rate, U ≡(Ul+Uq). We use the t t t t historical decompositions to break down the correlation between the stochastic components of Π and U , t t which we denote as Π˜ and U˜ , into the contributions of each shock (the stochastic components are the t t portions of Π and U that are explained by the structural shocks). Let ∆Π˜ denote changes in Π˜ between t t t+τ t t+τ −1 and t+τ, and let ∆U˜ represent changes in U˜ between t+τ −1 and t+τ. We denote the t+τ t 40Thisratioisconceptuallysimilartothe“Phillipsmultiplier”ofBarnichonandMesters(2021). 41OurpriorimpliesthattheaggregateactivityshockgeneratesalargerPhillipscorrelation(inabsolutevalue)atimpact than the other shocks. However, the posterior distributions suggest that the other shocks’ Phillips correlations are larger, whichsuggeststhattheshocks’identificationisdata-driven. OneinterpretationofthesteeperPhillipsratioforthejob-loss reallocationshockisthatthistypeofshockerodesthebargainingpowerofworkers,therebyloweringwagegrowthandputting downward pressure on price inflation. This sizable negative effect is in line with empirical evidence on the significant and persistentreductioninwagesandincomeofjoblosers(seeDavisandWachter,2011). 42Apossibleexplanationisthatapositivequitsreallocationshockcoulddrawlow-wageworkersintothejobseekerpool, leadingtoacompositionalshiftthatputsdownwardpressureonmeasuredaveragehourlyearnings. ThewagePhillipsratioof thequitsreallocationshock,measuredusingtheunemploymentrateofjobleavers,islargerpositivethantheestimatewiththe totalunemploymentratebutremainsstatisticallyinsignificantuntilh=12(bottompartofpanelB). 18

Table4: CumulativewageandpricePhillipscorrelationsbystructuralshock Wage Phillips ratio Price Phillips ratio Median 68% PI Median 68% PI A: Job-loss reallocation h=6 -0.77 (-1.27, -0.43) -0.50 (-1.04, -0.20) h=12 -1.01 (-1.90, -0.49) -0.56 (-1.39, -0.11) (Excluding job leavers) h=6 -0.76 (-1.23, -0.43) -0.49 (-1.01, -0.21) h=12 -1.00 (-1.85, -0.51) -0.56 (-1.36, -0.13) B: Quits reallocation h=6 0.36 (-0.23, 1.14) 0.65 (0.21, 1.28) h=12 0.58 (-0.76, 2.50) 1.33 (0.03, 3.63) (Job leavers) h=6 0.45 (-0.27, 1.17) 0.80 (0.27, 1.31) h=12 0.95 (-0.07, 1.96) 1.76 (0.93, 2.56) C: Aggregate activity h=6 -0.44 (-0.55, -0.32) -0.10 (-0.20, -0.00) h=12 -0.47 (-0.59, -0.36) -0.15 (-0.26, -0.05) D: Labor supply h=6 -0.48 (-0.76, -0.23) -0.24 (-0.47, -0.05) h=12 -0.61 (-0.95, -0.32) -0.30 (-0.63, -0.04) Notes: ThistablereportsthecumulativedynamicwageandpricePhillipscorrelationsconditionalonthestructural shocks. PanelsA,B,C,andDreportthewageandpricePhillipscorrelationsdrivenbyjob-lossreallocation,quitsreallocation, aggregateactivity,andlaborsupplyshocks. Rowsineachpanel(h)capturethemagnitudeofthecorrelationhmonthsafterthe shock’simpact. PIdenotesposteriorintervals. cumulative changes in ∆Π˜ and ∆U˜ between t and t+h as ∆Π˜ and ∆U˜ , respectively: t+τ t+τ t,t+h t,t+h h h ∆Π˜ = (cid:88) ∆Π˜ , ∆U˜ = (cid:88) ∆U˜ . t,t+h t+τ t,t+h t+τ τ=1 τ=1 Denoting the portions of ∆Π˜ and ∆U˜ attributable to structural shock j as ∆Π˜j and ∆U˜j , t,t+h t,t+h t,t+h t,t+h respectively, lets us write: ∆Π˜ t,t+h = (cid:88) J (cid:32) ∆Π˜j t,t+h (cid:33)(cid:32) ∆U˜ t j ,t+h (cid:33) = (cid:88) J cj (cid:32) ∆U˜ t j ,t+h (cid:33) +ϵ , (7) ∆U˜ ∆U˜j ∆U˜ h ∆U˜ t,t+h t,t+h j=1 t,t+h t,t+h j=1 t,t+h wherecj isthePhillipsratioofshockj (forj =1,2, ··· J, J=6)forhorizonhandϵ isanapproximation h t,t+h error.43 Note that the terms ∆U˜j and ∆U˜ in (7) are obtained from the historical decomposition. t,t+h t,t+h If we compute the covariance of both sides of equation (7) with ∆U˜ and divide through by the t,t+h 43Here,j=1representsthejob-lossreallocationshock,j=2thequitsreallocationshock,j=3theaggregateactivityshock, j=4thelabor-supplyshock,j=5theownshocktowageinflation,andj=6theownshocktopriceinflation. 19

variance of ∆U˜ , we obtain a decomposition of the Phillips correlation coefficient: t,t+h Cov(∆Π˜ t,t+h ,∆U˜ t,t+h ) = (cid:88) J cj (cid:32) Cov(∆U˜ t j ,t+h ,∆U˜ t,t+h ) (cid:33) + Cov(ϵ t,t+h ,∆U˜ t,t+h ) (8) Var(∆U˜ ) h Var(∆U˜ ) Var(∆U˜ ) t,t+h j=1 t,t+h t,t+h (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) βj et,t+h t,t+h 4 6 (cid:88) (cid:88) = cjβj + cjβj +e , (9) h t,t+h h t,t+h t,t+h j=1 j=5 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) labor−market other where the term on the left-hand side is the pseudo-unconditional Phillips correlation.44 The term βj on t,t+h the right-hand side of equation (8) captures the contribution of shock j to ∆U˜ , and the second term on t,t+h the right-hand side, e , is an approximation error. Equation (8) shows that the pseudo-unconditional t,t+h Phillips correlation approximates the sum of each shock’s Phillips ratio weighted by the contribution of each shock to ∆U˜ . Focusing on the contributions of labor-market factors, we express the right-hand side of t,t+h (8) as the sum of the contributions from the four labor-market shocks (the first term) and the remaining components (the second term), as shown in equation (9). We call the first term the labor-market Phillips ratio. Table 5 presents the resulting decompositions. We focus on h = 12 because the impulse responses indicate that the effects of aggregate activity shocks on the unemployment rate of job losers (and the total unemployment rate) and the vacancy rate peak approximately 12 months after the shock’s impact; hence h=12isthehorizonatwhichthetradeoffbetweenrealactivityandpriceinflationisadequatelyestimated.45 The four values in row (1) give cj and columns (d)–(f) report the βj estimates for each labor-market h t,t+h shock.46 The aggregate activity shock (uc) is the main driver of variations in the unemployment rate, while t the other shocks’ contributions are relatively small and change over time. It is notable that the contribution of the job-loss reallocation shock increases during the pandemic—row (5), column (d). Unlike the other labor-market shocks, the quits reallocation shock makes a small negative contribution to changes in the unemployment rate, reflecting its procyclical aspect and the small share of job leavers in total unemployment pool. We compare the labor-market Phillips ratio—the values not in parentheses in column (b)—with the pseudo-unconditional correlation—the values in column (a)—for the full sample period and three subsamples. The pseudo-unconditional correlation changes from a small positive value over 1980–1999 to a small negative value over 2000–2019, and both values are statistically insignificant, in line with the flattening of the Phillips curve (e.g., Atkeson and Ohanian, 2001). By contrast, all the estimates of labor-market Phillips ratio are statistically significant and larger negative than the unconditional correlations. Before the pandemic the labor-market Phillips ratio remains stable at around −0.20 with a sample average of −0.24. After adjusting the weights to account for the four labor-market shocks only, the weight-adjusted labor-market Phillips ratios—the values in parentheses in column (b)—become more negative, remaining stable at around −0.25 during the pre-pandemic period and with a sample average of −0.29. This apparent difference between the pseudo-unconditional correlation and the labor-market Phillips ratio is shown in column (c); it largely reflects factors that obscure the slope of the Phillips curve such as the prevalence of cost-push shocks during 44Thepseudo-unconditionalcorrelationdiffersfromtherawcorrelation,sincetheformeronlyusesthepartsofthevariables thatareexplainedbythestructuralshocks. SupplementingTable5,notethattherawunconditionalcorrelationsare−0.10 (1968–2023);0.00(1980–1999);0.00(2000–2019);and−0.19(2000–2023). 45Inthiscontext,h=12alignswiththefrequencydomainconsideredinpreviousPhillipscurvestudiesthatestimatethe pricechangesthataredirectlylinkedtochangesintheunemploymentratecausedbyaggregatedemandshocks. Intheonline appendix,wealsoreportresultsforh=6thatdemonstratetherobustnessofthemainresults. 46Thesumofthefourβ coefficientsisnotequaltoonebecausetheweightsoftheownshockstowageandpriceinflationare excluded;includingthemyieldsoneforeachsampleperiod. Theposteriormedianand68percentcredibilityintervalsforthe βj estimatesareprovidedintheonlineappendix. t,t+h 20

Table5: Decompositionofreduced-formPhillipscorrelationsbystructuralshock(h=12) (a) Reduced (b) LM (c) a-b (d) ul (e) uq (f) uc (g) us t t t t (1) Phillips ratio -0.56 1.33 -0.15 -0.30 (β estimates) (2) Total -0.07 -0.24 (-0.29) 0.17 0.24 -0.01 0.60 0.03 (1968 – 2023) [-0.11,-0.03] [-0.45,-0.08] ([-0.55,-0.10]) (3) 1980 – 1999 0.07 -0.18 (-0.25) 0.26 0.10 -0.02 0.67 0.01 [-0.04,0.19] [-0.35,-0.05] ([-0.48,-0.07]) (4) 2000 – 2019 -0.05 -0.19 (-0.24) 0.14 0.13 -0.01 0.59 0.10 [-0.11,0.02] [-0.33,-0.06] ([-0.44,-0.07]) (5) 2000 – 2023 -0.19 -0.32 (-0.35) 0.14 0.38 -0.01 0.57 0.04 [-0.20,-0.16] [-0.60,-0.12] ([-0.66,-0.13]) Notes: Column(a)reportsthepseudo-unconditionalcorrelation. Column(b)presentsthelabor-market(LM)Phillips ratio: numberswithoutparenthesesaretheweight-unadjustedestimates,whilethoseinparenthesesaretheweight-adjusted estimates. Theseestimatesrepresenttheposteriormedian. Numbersinsquarebracketsindicate68%posteriorintervals(PI), andthoseinparenthesesarethePIfortheweight-adjustedestimates. Column(c)reportstheposteriormedianfromcolumn(a) andtheweight-unadjustedestimatefromcolumn(b). Theestimatesforrow(1)arefromTable4. Theβ estimatesintheright sectioncapturetherelativecontributionofeachstructuralshocktovariationsintheunemploymentrate. Wereportestimates basedontheposteriormedianofthehistoricaldecomposition. the earlier period and anchored inflation expectations in the later period. The effects of these obscuring factors change significantly between the two subperiods, and drive the evolution of the pseudo-unconditional correlation. Expanding the sample period to include the pandemic (2000-2023), however, the increased importance of the job-loss reallocation shock resulted in a more negative labor-market Phillips ratio of −0.32 (with a weight-adjusted ratio of −0.35), which in turn caused the reduced-form correlation to become more negative (the −0.19 value in column (a)). Our analysis carries important implications for the empirical Phillips curve literature. Identifying the Phillips curve slope depends on how effectively we can isolate the portion of inflation that is linked to cyclical changes in the unemployment rate, and how well we can control for other factors that obscure the relationship. Our approach does so by identifying the structural labor-market shock that is relevant for the Phillips curve; specifically, we carefully uncover the portion of the unemployment rate driven by aggregate activity shocks while controlling for shocks that affect the NCRU, such as job-loss and quits reallocation shocks and labor-supply shocks.47 By bringing this valuable identifying information from the Beveridge curve to bear, our methodology refines the estimate of the Phillips curve slope by specifically identifying the effects of aggregate demand on inflation and the unemployment rate. The literature on the Phillips curve disagrees on the magnitude of its slope, with estimates ranging from nearly flat (e.g., −0.0062 from Hazell et al., 2022) to steeper values around −0.3 (e.g., −0.24 from Barnichon and Mesters, 2020 and −0.33 from Fitzgerald et al., 2024).48 Our estimate—the Phillips ratio for aggregate activity shocks—is −0.15, placing it midrange of previous estimates. The difference between our estimate and the steeper estimates from previous studies primarily reflects the effects of the job-loss reallocation and labor-supply shocks, both of which have a more negative Phillips ratio but influence the NCRU. Since existing studies do not account for these labor-market shocks, their estimates are conceptually 47PreviousstudiesconsiderlimitedvariationintheNCRU,relyingontrendsfromstatisticalfiltersorcross-sectionalfixed effects(e.g.,BarnichonandMesters,2020;Hazelletal.,2022). 48McLeayandTenreyro(2020)estimateaslopeof−0.38usingregionaldata,similartoFitzgeraldetal.(2024),butMcLeay andTenreyro’ssampleperiod(1990–2017)isshorterthanthatofFitzgeraldetal. (1977–2018). Inaddition,Smithetal.(2023) findamodestflatteningintheslopefrom−0.29to−0.25beforeandafter2000. 21

similartoourlabor-marketPhillipsratios(weight-adjustedorunadjusted), whichareinfactsteeperthanour aggregate-activity Phillips ratio and which also show sub-sample stability in line with Hazell et al. (2022) and Fitzgerald et al. (2024). In this sense, some existing estimates likely overstate the steepness of the structural Phillips curve.49 The preceding results show why it is important to distinguish different labor-market shocks when estimatingtheslopeofthePhillipscurve. Tofurtherillustratetheimportanceofidentifyingtheseshocks, the online appendix reports results from three exercises: (1) estimating the Phillips ratio for the unemployment rate from the generalized impulse responses of the reduced-form VAR; (2) removing sign restrictions from the baseline model; and (3) using the total unemployment rate rather than the two separate types of unemployment. The first two analyses demonstrate that neither the labor market shocks of interest nor the Phillips correlation coefficients for each shock can be reliably recovered without the identification of structural shocks based on the sign restrictions. The third analysis demonstrates the necessity of considering two distinct unemployment rates to identify the quits reallocation shock, whose effects on price and wage inflation are fundamentally different from those of the job-loss reallocation shock. As a corollary, our analysis implies that the reduced-form Phillips curve over a particular span of time will depend on the composition of the structural shocks that are realized over that period. Depending on that set of realizations, therefore, the slope of the reduced-form Phillips curve can change over time in ways that are unconnected with the actual tradeoff between real activity and inflation. As an illustration, after the pandemic recession the observed rise in price and wage inflation given conventional estimates of productand labor-market slack led some researchers to conclude that the inflation–activity tradeoff had worsened. Our analysis provides an alternative interpretation: The apparent non-linearity in the reduced-form Phillips curve in recent years might actually reflect the greater size and prevalence of job-loss reallocation shocks that accompanied the unprecedentedly large aggregate activity and price-specific shocks of this period. 5. Conclusion We have demonstrated that common structural shocks drive the dynamics of inflation and the labor market, leading to a deep connection between the Beveridge and Phillips curves that can be uncovered by analyzingbothrelationsjointly. Wefindthatovertime, changesinthesizeandincidenceofdifferenttypesof structurallabormarketshockscanalterthereduced-formrelationshipsamongvacancies, unemployment, and inflation, highlighting the importance of considering these structural shocks when using observed Beveridge and Phillips relations to evaluate and forecast the state of the economy. As an application, we used our structural accounting to analyze the pandemic recession and recovery, and found that labor reallocation shocks associated with job losses played a critical role in driving labor market dynamics and the steepening of the reduced-form Phillips curve during this period. Hence, the post-pandemic “soft landing” was not the result of an atypically favorable tradeoff between vacancies and unemployment but instead resulted from an interplay of shocks that induced both shifts of and movements along the Beveridge curve. References Abraham,K.,Katz,L.F.,1986. Cyclicalunemployment: Sectoralshiftsoraggregatedisturbances? JournalofPoliticalEconomy 94,507–522. Ahn, H.J., 2023. The role of observed and unobserved heterogeneity in the duration of unemployment. Journal of Applied Econometrics38,3–23. Ahn,H.J.,Chen,H.,Kister,M.,2020. Anewindicatorofcommonwageinflation. JournalofMoney,CreditandBankingn/a. Ahn,H.J.,Crane,L.D.,2020. DynamicBeveridgeCurveAccounting. FinanceandEconomicsDiscussionSeries2020-027.Board ofGovernorsoftheFederalReserveSystem(U.S.). 49Our slope estimate differs from Hazell et al. (2022), who find a near-zero Phillips curve slope. One explanation is that Hazelletal. usethepricesofnontradablesexcludinghousingservicesprices,whileweuseameasureofinflationthatincludes housingservicesprices(definedconsistentlyovertime),whicharemorecyclicallysensitivethanotherPCEpricecomponents suchasgoodsprices. Hazelletal. dofindthattheslopeofthePhillipscurvebecomesapproximatelyfourtimeslargerwhena measureofshelterrentsisincluded—thoughthatslopeisstillquiteflat. 22

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