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The Timing of Monetary Policy Shocks

By GIOVANNI OLIVEI AND SILVANA TENREYRO*

A vast empirical literature has documented delayed and persistent effects of mon-etary policy shocks on output. We show that this finding results from the aggregationof output impulse responses that differ sharply depending on the timing of the shock.When the monetary policy shock takes place in the first two quarters of the year, theresponse of output is quick, sizable, and dies out at a relatively fast pace. Incontrast, output responds very little when the shock takes place in the third or fourthquarter. We propose a potential explanation for the differential responses based onuneven staggering of wage contracts across quarters. Using a dynamic generalequilibrium model, we show that a realistic amount of uneven staggering cangenerate differences in output responses quantitatively similar to those found in thedata. (JEL E23, E24, E58, J41)

An important branch of the macroeconomicsliterature is motivated by the questions of whether,to what extent, and why monetary policy matters.As concerns the first two questions, substantialempirical work has led to a broad consensus thatmonetary shocks do have real effects on out-put. Moreover, the output response is persis-tent and occurs with considerable delay: thetypical impulse response has output peaking sixto eight quarters after a monetary policy shock(see, for example, Lawrence Christiano, MartinEichenbaum, and Charles Evans 1999). As forthe third question, a large class of theoriespoints to the existence of contractual rigiditiesto explain why monetary policy might causereal effects on output. Theoretical models usu-ally posit some form of nominal or real rigidity

in wages or prices that is constant over time. Forexample, wage contracts are assumed to bestaggered uniformly over time or subject tochange with a constant probability at each pointin time (John B. Taylor 1980; Guillermo Calvo1983).1

This convenient simplification, however, maynot be a reasonable approximation of reality. Asa consequence of organizational and strategicmotives, wage contract renegotiations may oc-cur at specific times in the calendar year. Whilethere is no systematic information on the timingof wage contracts, anecdotal evidence supportsthe notion of “lumping” or uneven staggering ofcontracts. For example, evidence from firms inmanufacturing, defense, information technol-ogy, insurance, and retail in New England sur-veyed by the Federal Reserve System in 2003for the “Beige Book” indicates that most firmstake decisions regarding compensation changes(base pay and health insurance) during thefourth quarter of the calendar year. Changes incompensation then become effective at the verybeginning of the next year. The Radford Sur-veys of compensation practices in the informa-tion technology sector reveal that more than 90percent of companies use a focal base-pay ad-

* Olivei: Federal Reserve Bank of Boston, 600 AtlanticAvenue, Boston, MA 02210 (e-mail: [email protected]); Tenreyro: London School of Economics, St.Clement’s Building S. 579, London, WC2A 2AE, UK, CEP,and CEPR (e-mail: [email protected]). We would liketo thank Francesco Caselli and Jeff Fuhrer for insightfulconversations. Alberto Alesina, Robert Barro, Mark Bils,Jean Boivin, Fernando Broner, Janice Eberly, Raquel Fer-nandez, Doireann Fitzgerald, Chris Foote, Jordi Galı, MarcGiannoni, Peter Ireland, James Hamilton, Nobu Kiyotaki,Miklos Koren, Juanpa Nicolini, Torsten Persson, Chris Pis-sarides, Richard Rogerson, Garey Ramey, Valerie Ramey,Lars Svensson, Robert Tetlow, Jonathan Wright, and threeanonymous referees provided useful comments and sugges-tions. We also thank Ryan Chahrour for excellent researchassistance. The views expressed in this paper do not neces-sarily reflect those of the Federal Reserve Bank of Boston orthe Federal Reserve System.

1 State-dependent versions of price- and wage-settingbehavior have been developed in the literature (see MichaelDotsey, Robert King, and Alexander Wolman 1999). As weargue in the text, however, the probability of changingprices and wages over time may change for reasons notcaptured by changes in the state of the economy.

636

ministration, with annual pay-change reviews;pay changes usually take place at the beginningof the new fiscal year. According to the samesurveys, 60 percent of the information technol-ogy companies close their fiscal year in Decem-ber.2 To the extent that there is a link betweenpay changes and the end of the fiscal year, it isworth noting that 64 percent of the firms in theRussel 3,000 Index end their fiscal year in thefourth quarter, 16 percent in the first, 7 percentin the second, and 13 percent in the third.3

Finally, reports on collective bargaining activitycompiled by the Bureau of Labor Statisticsshow that the distribution of expirations andwage reopening dates tends to be tilted towardthe second semester of the year.4

If the staggering of wage contracts is notuniform, as the anecdotal evidence suggests, inprinciple monetary policy can have differenteffects on real activity at different points intime. Specifically, monetary policy should have,other things equal, a smaller impact in periodsof lower rigidity—that is, when wages are beingreset. This paper provides an indirect test for thepresence and the importance of the lumping oruneven staggering of contracts, by examiningthe effect of monetary policy shocks at differenttimes in the calendar year. In order to do so, weintroduce quarter-dependence in an otherwisestandard VAR model. Our goal is to assesswhether the effect of a monetary policy shockdiffers according to the quarter in which theshock occurs and, if so, whether this differencecan be reconciled with uneven staggering.

We find that there are significant differencesin output impulse responses depending on thetiming of the shock. In particular, after a mon-etary shock that takes place in the first quarter,the response of output is fairly rapid, with out-put reaching a level close to the peak effect fourquarters after the shock. The response is evenmore front-loaded and dies out faster when theshock takes place in the second quarter. In thiscase, the peak effect is attained three quarters

after the shock. In both cases, the response ofoutput to a monetary policy shock is economi-cally relevant. An expansionary shock in eitherthe first or the second quarter with an impacteffect on the federal funds rate of �25 basispoints raises output in the following 8 quartersby an average of about 25 basis points. In con-trast, the response of output to a monetary shockoccurring in the second half of the calendar yearis small, both from a statistical and from aneconomic standpoint. A 25-basis-points unex-pected monetary expansion in either the third orfourth quarter raises output in the 8 quartersfollowing the shock by less than 10 basis pointson average, with the effect not statistically dif-ferent from zero at standard confidence levels.The well-known finding that output takes a longtime to respond and is quite persistent may beinterpreted as the combination of these sharplydifferent quarterly responses.

The dynamics of output in response to amonetary policy shock at different times of theyear is mirrored by the dynamics of prices andnominal wages. The price and nominal wageresponses are delayed when the shock occurs inthe first half of the year, whereas prices andnominal wages respond more quickly when theshock occurs in the second half of the year.

The anecdotal evidence on wage-setting prac-tices provides an explanation for the qualitativedifferences in the quarterly impulse responses.It is important to gauge, however, whether un-even staggering can also explain the quantita-tive differences in the estimated responses. Toaddress this issue, we calibrate a variant of thestochastic dynamic general equilibrium modelpresented by Christiano, Eichenbaum, and Evans(2005). The crucial modification to the setup isthat we allow the probability of resetting wagesto differ across quarters. We show that a real-istic amount of uneven staggering in the modelcan quantitatively match the quarter-dependentimpulse responses estimated on actual data.

Our findings also speak to the long-standingissue of whether wages play an allocative role inthe economy. The larger response of economicactivity following a monetary policy shock inperiods when wages are more rigid suggests thatwages are allocative even in the short run.

The remainder of the paper is organized asfollows. Section I presents the empirical methodand introduces the data. Section II presents thedynamic effects of monetary policy on different

2 We thank Andy Rosen of Aon Consulting’s RadfordSurveys for providing us with the information.

3 This information is for the year 2003 and is availablefrom Standard and Poor’s COMPUSTAT. To compute thepercentages, we have weighted each firm in the Russel3,000 Index by the firm’s number of employees.

4 See Current Wage Developments, various issues, Bu-reau of Labor Statistics.

637VOL. 97 NO. 3 OLIVEI AND TENREYRO: THE TIMING OF MONETARY POLICY SHOCKS

macroeconomic aggregates and performs a setof robustness tests. Section III presents the the-oretical model and confronts it with the empir-ical findings. Section IV offers some concludingremarks.

I. Method

A. Empirical Model

Our empirical analysis for measuring the ef-fect of monetary policy shocks relies on a verygeneral linear dynamic model of the macro-economy whose structure is given by the fol-lowing system of equations:

(1) Yt � �s � 0

S

B�qt�sYt � s � �s � 1

S

C�qt�spt � s

� Ay�qt�vty;

(2) pt � �s � 0

S

DsYt � s � �s � 1

S

gspt � s � v tp.

Boldface letters are used to indicate vectors ormatrices of variables or coefficients. In partic-ular, Yt is a vector of nonpolicy macroeconomicvariables (e.g., output, prices, and wages), andpt is the scalar variable that summarizes thepolicy stance. We take the federal funds rate asour measure of policy, and use innovations inthe federal funds rate as a measure of monetarypolicy shocks. Federal Reserve operating pro-cedures have varied over the past 40 years, butseveral authors have argued that funds-rate tar-geting provides a good description of FederalReserve policy over most of the period (see BenBernanke and Alan Blinder 1992; and Bernankeand Ilian Mihov 1998). Equation (1) allows thenonpolicy variables Yt to depend on both cur-rent and lagged values of Y, on lagged values ofp, and on a vector of uncorrelated disturbancesvy.5 Equation (2) states that the policy variablept depends on both current and lagged values ofY, on lagged values of p, and on the monetary

policy shock v p.6 As such, the system embedsthe key restriction for identifying the dynamiceffects of exogenous policy shocks on the var-ious macro variables Y: policy shocks do notaffect macro variables within the current period.Although debatable, this identifying assumptionis standard in several recent VAR analyses.7

The model in equations (1) and (2) replicatesthe specification of Bernanke and Blinder(1992), with the crucial difference that we allowfor time-dependence in the coefficients. Specif-ically, B(qt)s and C(qt)s are coefficient matriceswhose elements, the coefficients at each lag, areallowed to depend on the quarter qt that indexesthe dependent variable, where qt � j if t corre-sponds to the jth quarter of the year. The sys-tematic response of policy takes the time-dependence feature of the nonpolicy variablesinto account: substituting (1) into (2) shows thatthe coefficients in the policy equation are indi-rectly indexed by qt through their impact on thenonpolicy variables, Yt.

8

Given the identifying assumption that policyshocks do not affect macro variables within thecurrent period, we can rewrite the system in astandard VAR reduced form, with only laggedvariables on the right-hand side:

(3) Xt � F�L, q�Xt � 1 � Ut ,

where Xt � [Yt, p(t)]�, Ut is the correspondingvector of reduced-form residuals, and F(L, q) isa four-quarter distributed lag matrix of coeffi-cients that allows for the coefficients at each lagto depend on the particular quarter q indexingthe dependent variable. The system can then be

5 Note that the vector of disturbances vy, composed ofuncorrelated elements, is premultiplied by the matrix Ay(q)to indicate that each element of vy can enter into any of thenonpolicy equations. This renders the assumption of uncor-related disturbances unrestrictive.

6 Policy shocks are assumed to be uncorrelated with theelements of vy. Independence from contemporaneous eco-nomic conditions is considered part of the definition of anexogenous policy shock. The standard interpretation of v p isa combination of various random factors that might affectpolicy decisions, including data errors and revisions, pref-erences of participants at the FOMC meetings, politics, etc.(see Bernanke and Mihov 1998).

7 See, among others, Bernanke and Blinder (1992), Ber-nanke and Mihov (1998), Christiano et al. (1999, 2005),Jean Boivin and Marc Giannoni (2006), and Julio Rotem-berg and Michael Woodford (1997).

8 In this specification, the coefficients Ds and gs areconstant across seasons, neglecting differential policy re-sponses in different seasons beyond the indirect effectthrough Yt we already mentioned. We are unaware of anyevidence suggesting that policy responses to given out-comes vary by season.

638 THE AMERICAN ECONOMIC REVIEW JUNE 2007

estimated equation by equation using ordinaryleast squares. The effect of policy innovationson the nonpolicy variables is identified with theimpulse-response function of Y to past changesin v p in the unrestricted VAR (3), with thefederal funds rate placed last in the ordering.9

An estimated series for the policy shock can beobtained via a Choleski decomposition of thecovariance matrix of the reduced-form residuals.

One important implication of quarter depen-dence is that the effects of monetary policyshocks vary depending on the quarter in whichthe shock takes place. Denote by X(T) the skip-sampled matrix series, with X(T) � (X1,T, X2,T,X3,T, X4,T), where Xj,T is the vector of variablesin quarter j in year T, and j � 1, 2, 3, 4.10 Thenwe can rewrite the quarter-dependent reduced-form VAR (3) as follows:

(4) �0X�T� � �1�L�X�T � 1� � U�T�,

where �0 and �1(L) are parameter matricescontaining the parameters in F(L, q) in (3), andU(T) � (U1,T, U2,T, U3,T, U4,T), with Uj,T thevector of reduced-form residuals in quarter j ofyear T. The system in (4) is simply the reduced-form VAR (3) rewritten for annually observedtime series. As such, the reduced-form (4) doesnot contain time-varying parameters. Moreover,because the matrix �0 can be shown to be lower-block-triangular, it can be inverted to yield:

(5) X�T� � �0� 1�1�L�X�T � 1� � �0

� 1U�T�,

with �0�1 still being a lower bloc-triangular

matrix. The system (5) illustrates that when amonetary policy shock occurs in the first quar-ter, the response of the nonpolicy variables inthe next quarter will be governed by the re-duced-form dynamics of the nonpolicy vari-ables in the second quarter. The response twoquarters after the initial shock will be governed

by the reduced-form dynamics of the nonpolicyvariables in the third quarter, and so on.

B. Testing

The quarter-dependent VAR in (3) generatesfour different sets of impulse responses to amonetary policy shock, according to the quarterin which the shock occurs. It is then importantto assess whether the quarter-dependent im-pulse-response functions are statistically differ-ent from the impulse responses of the nestedstandard VAR with no time dependence. A firstnatural test for the empirical relevance of quar-terly effects consists of simply comparing theestimates obtained from the quarter-dependentVAR (3) with those obtained from the restrictedstandard VAR using an F-test, equation byequation. A rejection of the null hypothesis of noseasonal dependence would imply that the systemgenerates four different sets of impulse responses.The F-tests on the linear reduced-form VAR,however, do not map one for one into a test on thecorresponding impulse responses because the im-pulse-response functions are nonlinear combina-tions of the estimated coefficients in the VAR. Toassess the significance of quarter-dependence di-rectly on the impulse-response functions, we de-velop a second test that complements the F-test onthe linear VAR equations. Specifically, we con-sider the maximum difference, in absolute value,between the impulse responses of variable x in thequarter-dependent VAR and in the standard non-time-dependent VAR, to obtain the following sta-tistic:

D � supk

�x kq � xk�,

where xkq denotes the period k response in the

quarter-dependent model and xk the response inthe standard non-time-dependent model.11 Weresort to simulation methods for inference. Us-ing a bootstrap procedure, we calculate the dis-tribution of the D statistic under the assumptionthat there is no quarter-dependence. The boot-strap algorithm involves generating a randomsample by sampling (with replacement) from

9 The ordering of the variables in Yt is irrelevant. Sinceidentification of the dynamic effects of exogenous policyshocks on the macro variables Y requires only that policyshocks not affect the given macro variables within thecurrent period, it is not necessary to identify the entirestructure of the model.

10 If t � 1, ... , n, then the observations in X1,T are givenby t � 1, 5, 9, ... , n � 3, the observations in X2,T are givenby t � 2, 6, 10, ... , n � 2, and so on.

11 We compute the supremum of the difference inimpulse-response functions over 20 quarters following amonetary policy shock.


the residuals of the estimated non-time-depen-dent reduced-form VAR. Using fixed initialconditions,12 we recursively generate a new data-set using the estimated parameters from thestandard non-time-dependent VAR. We then es-timate new impulse responses from both thequarter-dependent and the standard VAR, andcompute a new value DS, where the superscriptS denotes a simulated value. The procedure isrepeated 2,000 times to obtain a bootstrap p-value, which is the percentage of simulated DS

exceeding the observed D.

C. Data and Estimation

Our benchmark analysis is based on quarterlydata covering the period 1966:Q1 to 2002:Q4. Thebeginning of the estimation period is dictated bythe behavior of monetary policy. Only after 1965did the federal funds rate, the policy variable inour study, exceed the discount rate and hence actas the primary instrument of monetary policy. Weuse seasonally adjusted data, but in the robustnesssection we also present results based on non–seasonally adjusted data. The nonpolicy variablesin the system include real GDP, the GDP deflator,and an index of spot commodity prices.13 As isnow standard in the literature, the inclusion of thecommodity price index in the system is aimed atmitigating the “price puzzle,” whereby a monetarytightening initially leads to a rising rather thanfalling price level. In Sections IIC and IID wediscuss alternative empirical specifications thatreplace some of the nonpolicy variables fromthe baseline specification with different vari-ables and/or include an expanded set of non-policy variables. The additional nonpolicyvariables entering these specifications are thecore Consumer Price Index (CPI), an index forwages given by compensation per hour in thenonfinancial corporate sector, and an index ofhours of production workers in the manufactur-ing sector.14

We estimate each equation in the VAR (3)separately by OLS, using four lags of eachvariable in the system. In our benchmark spec-ification, all the variables in the vector Y areexpressed in log levels. The policy variable, thefederal funds rate, is expressed in levels. Weformalize trends in the nonpolicy variables asdeterministic, and allow for a linear trend ineach of the equations of the VAR (3). In therobustness section we discuss findings whenGDP is expressed as the (log) deviation from asegmented deterministic trend, while the GDPdeflator and the commodity price index are ex-pressed in (log) first-differences.

II. The Dynamic Effects of MonetaryPolicy Shocks

A. Results from the VAR Specification

In this section we present the estimated dy-namic effects of monetary policy shocks on realGDP, the GDP deflator, and the federal fundsrate. Impulse responses are depicted in Figures1 through 5, together with 95 percent and 80percent confidence bands around the estimatedresponses.15 We consider a monetary policyshock that corresponds to a 25-basis-point de-cline in the funds rate on impact. For ease ofcomparison, the response of the variables to theshock is graphed on the same scale across fig-ures. Figure 1 displays impulse responses to thepolicy shock when we do not allow for quarter-dependence in the reduced-form VAR, as iscustomary in the literature. The top panel showsthat the output response to the policy shock ispersistent, peaking seven quarters after theshock and slowly decaying thereafter. The re-sponse of output is still more than half the peakresponse 12 quarters after the shock. The centerpanel shows that prices start to rise reliablythree quarters after the shock, although it takes

12 The fixed initial conditions are given by the values thatthe variables included in the VAR take over the period1965:Q1 to 1965:Q4.

13 The source for real GDP and the GDP deflator is theBureau of Economic Analysis, Quarterly National Incomeand Product Accounts. The source for the spot commodityprice index is the Commodity Research Bureau.

14 The source for core Consumer Price Index (CPI),compensation per-hour in the nonfinancial corporate sector,

and aggregate weekly hours of production workers in themanufacturing sector is the Bureau of Labor Statistics.

15 While much applied work uses 95 percent confidenceintervals, Christopher Sims and Tao Zha (1999) note thatthe use of high-probability interval camouflages the occur-rence of large errors of over-coverage and advocate the useof smaller intervals, such as intervals with 68 percent cov-erage (one standard error in the Gaussian case). An intervalwith 80 percent probability corresponds to about a standarderror of 1.3 in the Gaussian case.


about one and a half years for the increase tobecome significant. The bottom panel, whichdisplays the path of the federal funds rate, illus-trates that the impact on the funds rate of apolicy shock is less persistent than the effect onoutput.

Figures 2 to 5 display impulse responseswhen we estimate the quarter-dependent reduced-form VAR (3). The responses to a monetarypolicy shock occurring in the first quarter of theyear are shown in Figure 2. Output rises onimpact and reaches a level close to its peakresponse four quarters after the shock. The out-put response dies out at a faster pace than in thenon-time-dependent VAR: 12 quarters after theshock, the response of output is less than a thirdof the peak response, which occurs 7 quartersafter the shock, as in the non-time-dependentVAR. Moreover, the peak response is now

more than twice as large as in the case with noquarter-dependence. The center panel showsthat, despite controlling for commodity prices,there is still a “price puzzle,” although the de-cline in prices is not statistically significant. Ittakes about seven quarters after the shock forprices to start rising. The federal funds rate,shown in the bottom panel, converges at aboutthe same pace as in Figure 1.

Figure 3 displays impulse responses to ashock that takes place in the second quarter. It isapparent that the response of output is fast andsizable. Output reaches its peak three quartersafter the shock, and the peak response is morethan three times larger than the peak response inthe case with no quarter dependence. Moreover,the response wanes rapidly, becoming insignif-icantly different from zero eight quarters after

FIGURE 1. 25-BASIS-POINT DECLINE IN FED FUNDS RATE

(No quarterly dependence: 1966:Q1 to 2002:Q4)FIGURE 2. 25-BASIS-POINT DECLINE IN FED FUNDS RATE IN Q1

(Quarterly dependence: Benchmark model1966:Q1 to 2002:Q4)


http://aea.literatumonline.com/action/showImage?doi=10.1257/aer.97.3.636&iName=master.img-000.jpg&w=191&h=319


the shock. The center panel shows that pricesstart rising three quarters after the shock. Thebottom panel illustrates that the large outputresponse occurs despite the fact that the policyshock exhibits little persistence.

The responses to a monetary policy shock inthe third and the fourth quarter of the yearcontrast sharply with the responses to a shocktaking place in the first half of the calendar year.Figure 4 shows the impulse responses to a shockthat occurs in the third quarter. The response ofoutput in the top panel is now small and insig-nificant, both from a statistical and from aneconomic standpoint. Interestingly, as the cen-ter panel illustrates, prices start to increase re-liably immediately after the shock. The outputand price responses to a shock in the fourthquarter are qualitatively similar. As Figure 5 il-

lustrates, the response of output is fairly weak,while prices respond almost immediately fol-lowing the shock.

The differences in output responses are sub-stantial from a policy standpoint. The policyshock raises output in the following eight quar-ters by an average of almost 25 basis points ineither the first or the second quarter. In contrast,the increase in output is less than 10 basis pointson average in both the third and the fourthquarter. Moreover, the differences documentedin Figures 1 to 5 are corroborated by formaltests on the importance of quarter-dependence.Equation-by-equation F-tests in the reduced-form VAR (3) yield p-values of 0.18 for theoutput equation, 0.04 for the price equation,0.03 for the commodity prices equation, and0.006 for the federal funds rate equation.

FIGURE 3. 25-BASIS-POINT DECLINE IN FED FUNDS RATE IN Q2(Quarterly dependence: Benchmark model

1966:Q1 to 2002:Q4)


1966:Q1 to 2002:Q4)




While indicative of the existence of seasonaldependence, these relatively low p-values donot necessarily translate into statistically differ-ent impulse responses for the correspondingvariables. For this purpose, we evaluate theD-statistic described in Section IB, which as-sesses whether the maximum difference be-tween the impulse response of a given variablein the quarter-dependent VAR and the corre-sponding response of that variable in the stan-dard non-time-dependent VAR is statisticallydifferent. Table 1 reports the bootstrapped p-values for the D-statistic in each quarter forGDP, the GDP deflator, and the federal fundsrate. The table shows that, according to this test,the output response in the first, second, and thirdquarter of the calendar year is statistically dif-ferent from the non-time-dependent output im-

pulse response at better than the asymptotic 5percent level. The null hypothesis of an outputresponse equal to the non-time-dependent re-sponse is rejected at the asymptotic 10 percentlevel in the fourth quarter. As may be inferredfrom Figures 1 through 5, the table shows thatthe evidence in favor of quarter-dependent priceimpulse responses is weaker than for output.Still, the test identifies statistical differences inthe third and fourth quarters: in the third quar-ter, the null hypothesis of a price response equalto the non-time-dependent response is rejectedat the asymptotic 5 percent level, and in thefourth quarter the null is rejected at the 10percent level.

B. The Distribution of Monetary PolicyShocks and the State of the Economy

We now consider whether the different im-pulse responses we obtain across quarters arethe result of different types of shocks. In prin-ciple, differences in the direction (expansionaryversus contractionary) of shocks could producedifferent impulse responses.16 To explore thisissue, we test for the equality of the distribu-tions of shocks across quarters by means of aKolmogorov-Smirnov test. The test consists ofa pairwise comparison of the distributions ofshocks between every two quarters, with thenull hypothesis of identical distributions. Wefind that we cannot reject the null hypothesis inany two quarters: the smallest p-value corre-sponds to the test for the equality of the distri-butions of shocks between the third and fourthquarters and is equal to 0.31; the largest p-valuecorresponds to the test between the second andthird quarters and is equal to 0.97. These find-ings suggest that differences in the direction of

16 See Harald Uhlig (2005).


1966:Q1 to 2002:Q4)

TABLE 1—DIFFERENCES IN IMPULSE RESPONSES

ACROSS QUARTERS

(p-values for D-Statistic)

Variable

Quarter

First Second Third Fourth

GDP 0.01 0.00 0.03 0.08GDP deflator 0.39 0.16 0.02 0.06Fed funds rate 0.22 0.00 0.12 0.02



the shocks across quarters are unlikely to pro-vide an explanation for the quarterly differencesin impulse responses documented in Figures 2through 5.

Another important issue is whether our find-ings are driven by the state of the economy. Inprinciple, a theoretical argument can be madethat an expansionary monetary policy shock hasa larger impact on output and a smaller impacton prices when the economy is running belowpotential, and, vice versa, a smaller impact onoutput and a larger impact on prices when theeconomy is running above potential. To explorethis issue, we partitioned the data according towhether the output gap was positive or negativeand estimated two different reduced-formVARs. The impulse responses for output andprices to a monetary policy shock from theVAR estimated using observations correspond-ing to a negative output gap were similar to theimpulse responses obtained from the VAR es-timated using observations corresponding to apositive output gap.17 These results suggest thatthe stage of the business cycle is unlikely to bea candidate for explaining the different impulseresponses across quarters.

There is, however, a more subtle way inwhich the state of the economy could influenceour findings. Robert Barsky and Jeffrey Miron(1989) trace a parallel between seasonal andbusiness cycles, and note that in seasonally un-adjusted data the first and third quarters resem-ble a recession (the third quarter being milder),whereas the second and fourth quarters resem-ble an expansion (the fourth being stronger).Our use of seasonally adjusted data should, inprinciple, control for the seasonal component ofoutput. And even if such a control were imper-fect, the pattern of impulse responses in Figures2 to 5 cannot be easily reconciled with theseasonal cycle. The response of output is, infact, large when the policy shock occurs in thefirst (recession) and second (expansion) quar-ters, and the response is weak when the shockoccurs in the third (recession) and fourth (ex-pansion) quarters. It is still possible, though,that the seasonal pattern of some activities thatare particularly sensitive to interest rate move-

ments, such as (consumption and producer) du-rables and structures, could affect some of ourfindings. In particular, Barsky and Miron (1989)show that in the first quarter of the year there isa pronounced seasonal slowdown in spendingfor both durables and structures. A potentialinterpretation for our fourth-quarter resultswould then be that at that time of the calendaryear, a policy shock has little impact on outputbecause the transmission mechanism is im-paired by the low level of interest-sensitiveactivities in the subsequent quarter. This inter-pretation, however, does not fit well with ourthird-quarter results. As Barsky and Miron(1989) also show, the seasonal level of spendingon producer and consumer durables is espe-cially high in the fourth quarter. If the previousreasoning is correct, we should observe a largeresponse of output to shocks taking place in thethird quarter, which is in conflict with the evi-dence. We thus view our findings as difficult toreconcile with an explanation that relies mainlyon seasonal fluctuations in output or in theinterest-sensitive components of output.

C. Robustness Checks

We now summarize results on the robustnessof our baseline specification along several di-mensions.18 Because the quarter-dependentreduced-form VAR (3) requires the estimationof a fairly large number of parameters, we in-vestigated whether our findings are sensitive tooutliers. For this purpose, we reestimated theVAR equation by equation using Peter Huber’s(1981) robust procedure. The Huber estimatorcan be interpreted as a weighted least squaresestimator that gives a weight of unity to obser-vations with residuals smaller in absolute valuethan a predetermined bound, but downweightsoutliers (defined as observations with residualslarger than the predetermined bound).19 With

17 These findings are available upon request. There is noestablished evidence in the extant empirical literature thatmonetary shocks have different effects according to thestage of the business cycle.

18 For reasons of space, we do not provide pictures of theestimated impulse responses in most cases. The pictures notshown in the paper are available from the authors uponrequest.

19 Denote by � the standard deviation of the residuals inany given equation of the VAR (3). For the given equation,the Huber estimator gives a weight of unity to observationswith residuals smaller in absolute value than c�, where c isa parameter usually chosen in the range 1 � c � 2, whileoutliers, defined as observations with residuals larger thanc�, receive a weight of (c�/�ui�), where ui is the residual for


this procedure, estimated impulse responses(not shown) turn out to be very similar to theones in Figures 2 through 5. A notable byprod-uct of the robust estimation procedure is thatp-values for the F-tests on quarter-dependenceare now well below 0.05 for all the equations inthe VAR.

Since our proposed explanation for the dif-ferent impulse responses relies on uneven stag-gering of wage contracts across quarters, weconsidered an alternative specification wherethe GDP deflator is replaced by a nominal wageindex in the vector of nonpolicy variables Y.Using wages in lieu of final prices does not alterour main findings. The estimated output re-sponses (not shown) are virtually the same as inthe benchmark specification, and the responseof wages closely mimics the response of pricesacross different quarters.

In our benchmark specification we control forseasonal effects by using seasonally adjusteddata. Still, because we are exploiting a time-dependent feature of the data, it is of interest tocheck whether our results are driven by theseasonal adjustment. To this end, we estimateimpulse responses to a monetary shock from thequarter-dependent reduced-form VAR (3) usingseasonally unadjusted data for the nonpolicyvariables Y.20 The results from this exercise areillustrated in Figures 6 through 9, which showthe responses of output, prices, and the federalfunds rate to a 25-basis-point decline in thefunds rate. The responses of output and pricesusing seasonally unadjusted data are remark-ably similar to the responses obtained in thebenchmark specification using seasonally ad-justed data, although the estimated responseswith seasonally unadjusted data are less precise.

The results continue to hold under a differenttreatment of the low-frequency movements inoutput and prices. Specifically, we considered aspecification for the quarter-dependent reduced-form VAR (3) in which variables in Y are notexpressed in log levels, but rather GDP is ex-pressed as a deviation from a segmented deter-

ministic trend,21 and prices are expressed in logfirst-differences. Such a specification is used inseveral papers in the literature (see, e.g., Boivinand Giannoni 2006). The estimated impulse re-sponses (not shown) are qualitatively similar tothose reported in the benchmark specification.In particular, output responds more strongly andmore quickly to a monetary policy shock in thefirst than in the second half of the year, whilethe opposite occurs for inflation.

observation i. The estimator is fairly insensitive to thechoice of c.

20 Since we do not have data on the seasonally unad-justed GDP deflator, we replace the GDP deflator with theseasonally unadjusted CPI. The CPI index is also used todeflate the seasonally unadjusted data for nominal GDP.

21 Specifically, we consider the deviation of log realGDP from its segmented deterministic linear trend, withbreakpoints in 1974 and 1995.

FIGURE 6. 25-BASIS-POINT DECLINE IN FED FUNDS RATE IN Q1(Quarterly dependence: NSA data system

1966:Q1 to 2002:Q4)



D. Additional Evidence and Interpretation

Overall, Figures 2 through 5 and the support-ing statistics uncover considerable differencesin the response of output across quarters. Theslow and persistent response of output to apolicy shock typically found in the literatureand reported in Figure 1 is the combination ofdifferent quarter-dependent responses. The dif-ference in the behavior of prices across quartersis somewhat less striking, given the imprecisionwith which the responses are estimated. It isinteresting, though, that when the policy shockoccurs in the third and fourth quarters, pricesrise more quickly than when the shock takesplace in either the first or the second quarter.

Since our explanation for the main empiricalfindings hinges on uneven staggering of wage

contracts, it is of interest to examine the quar-terly responses of relevant labor market vari-ables to a monetary policy shock. Specifically,we consider the responses of real wages andaggregate hours, estimated from a quarter-dependent VAR that includes real GDP, theGDP deflator, the real wage, hours, and thefederal funds rate.22 Figures 10 to 13 depict thequarter-dependent responses for the real wageand hours to a monetary policy shock.23 We

22 The real wage is computed as the ratio of compensa-tion per hour in the nonfinancial corporate sector and theGDP deflator. All the nonpolicy variables are expressed inlog levels.

23 The estimated responses (not shown) for GDP, theGDP deflator, and the federal funds rate are qualitativelysimilar to the responses depicted in Figures 2 to 5.


1966:Q1 to 2002:Q4)


1966:Q1 to 2002:Q4)




mentioned in the previous section that the re-sponses of the nominal wage are qualitativelysimilar to the responses of prices across differ-ent quarters. The additional information con-veyed by the present exercise is that theestimated responses for the real wage are mildlyprocyclical and similar across quarters. In con-trast, the estimated responses for hours exhibit anoticeable quarterly pattern. The pattern mimicsthe estimated responses for output, with largerand statistically more significant responseswhen the shock occurs in the first and secondquarters of the calendar year. In the next sec-tion, we provide an explanation for the contrast-ing dynamics of hours and the real wage.

Taken together, the VAR results and the an-ecdotal evidence on uneven staggering lendthemselves to the following interpretation. If a

large number of firms sign wage contracts at theend of the calendar year, then, on average, mon-etary policy shocks in the first half of the yearwill have a large impact on output, with littleeffect on nominal wages and prices. In contrast,


1966:Q1 to 2002:Q4)

FIGURE 10. 25-BASIS-POINT DECLINE IN FED FUNDS RATE IN Q1(Wage-hour data system 1966:Q1 to 2002:Q4)






monetary policy shocks in the second half of theyear will be quickly followed by nominal wageand price adjustments. The policy shock will be“undone” by the new contracts at the end of theyear and, as a result, the effect on output will besmaller on average. We next formalize this inter-pretation in the context of a dynamic stochasticgeneral equilibrium setup that can quantitativelyaccount for the quarter-dependent impulse re-sponses estimated in the data.

III. A Model of Uneven Staggering

The anecdotal evidence on uneven stagger-ing of wage-setting decisions provides an in-tuitively appealing explanation for the findingof quarter dependence in the response of theeconomy to monetary policy shocks. In thissection, we investigate whether this qualita-tive mechanism is also quantitatively rele-vant. In particular, we ask whether a variantof the model proposed by Christiano et al.(2005) that allows for a realistic degree of unevenstaggering of wage contracts can quantitativelymatch the quarter-dependent impulse responsesin the data.

Christiano et al.’s (2005) setup embeds asimple form of contractual rigidities based onCalvo (1983), whereby workers face a constantprobability of reoptimizing their nominal wageevery quarter. Here, we generalize the setup andallow the probability of changing wages to dif-fer across quarters in a calendar year. This mod-eling strategy is a simple way of introducingclustering of wage contracts at certain times ofthe calendar year, and it nests the standardCalvo-style sticky-wage framework as a specialcase when the probability of resetting wages isconstant across quarters.

The model’s key propagation mechanism inresponse to a temporary shock to the nominalinterest rate is as follows. After an expansion-ary monetary shock, aggregate demand goesup and so does the derived demand for labor.Consider first a period when a large fractionof the workers cannot readjust their wageoptimally. These workers will have to supplyas much labor as the firms demand at thepreset wage, leading to an increase in totalhours of work (and in the usage of otherinputs) and hence to an increase in aggregateactivity. Consider now a period in which alarge fraction of the workers can readjust their






wage optimally. Following an expansionarymonetary policy shock, firms can no longercount on as much labor at preset wages as inthe previous scenario, as they now meet anupward-sloping supply of labor for a largenumber of workers. The increase in aggregateactivity is therefore smaller than in the previ-ous case, and the economy operates at a levelcloser to the flexible-wage-price potential.

In the model, monetary policy is neutral inthe long run. Consistent with our empiricalsetup, monetary policy targets a short-terminterest rate by means of a Taylor-type reac-tion function. Since interest rate targeting bythe monetary authority makes money demandirrelevant in determining the equilibrium evo-lution of the other variables in the model, weconsider a cashless version of the model.24 Asis now standard in the literature, the modelfeatures habit formation in consumption. Inaddition, the model incorporates investmentadjustment costs. The motivation for adjust-ment costs to consumption and investment islargely empirical. These costs attenuate theinitial impact of a monetary policy shock onthe economy, and make the effects of theshock long-lasting.

In what follows, we model the behavior offirms and households and the evolution of nomi-nal wages in the presence of uneven staggering.We then calibrate the model parameters andcompare the model-generated impulse responseswith the empirical responses from our quarter-dependent benchmark VAR specification andwith the additional responses reported in Sec-tion IID.

A. Firms

The final consumption good in the economy,Yt, is produced by a perfectly competitive firmthat uses a continuum of intermediate goods, Yjt,combined through the CES technology:

Yt � �� Yjt1/�f dj� �f

,

with �f � [1, �). The firm takes the final con-sumption good price, Pt, and the intermediate

good price, Pjt, as given. Each intermediategood j � (0, 1) is produced by a monopolistfirm using the technology:

Yjt � kjt�Ljt

1�� , � � �0, 1� if kjt�Ljt

1 � � �,

� 0 otherwise,

where Ljt and kjt are the labor and capital ser-vices used in the production of intermediate j,and � represents a fixed cost of production.Intermediate good firms rent capital at the rentalrate Rt

k and labor at the average wage rate Wt inperfectly competitive factor markets. Profits aredistributed to households at the end of eachperiod. Workers are paid in advance of produc-tion, hence the jth firm borrows its wage billWtLjt from a financial intermediary at the begin-ning of the period, repaying at the end of theperiod at the gross interest rate Rt.

Intermediate firm j sets price Pj,t following aCalvo-type rule, with a probability (1 � p) ofbeing able to reoptimize its nominal price ineach period. The ability to reoptimize is inde-pendent across firms and time. Firms that can-not reoptimize index their price to laggedinflation: Pj,t � �t�1Pj,t�1, where �t�1 � Pt/Pt�1.

B. Households

There is a continuum of infinitely livedhouseholds i, with i � (0, 1), which deriveutility from consumption and disutility fromlabor effort, and which own the economy’sstock of physical capital. Every period eachhousehold makes the following sequence of de-cisions. First, the household decides how muchto consume and invest in physical capital, andhow much capital services to supply. Second,the household purchases securities that are con-tingent on whether it can reoptimize its wage.25

Third, the household sets its wage after finding

24 See Michael Woodford (2003).

25 As in Christiano et al. (2005), this is assumed foranalytical convenience. Because the uncertainty overwhether a household can reoptimize wages is idiosyncratic,households work different numbers of hours and earn dif-ferent wage rates; this, in turn, could lead to potentiallyheterogeneous levels of consumption and asset holdings. Inequilibrium, the existence of state-contingent securities en-sures that households are homogeneous with respect toconsumption choice and asset holdings, though heteroge-neous with respect to the wage rate and the supply of labor.


out whether it can reoptimize. Household i’sexpected utility is given by26

(6)

Et � 1i � �

j � 0

�

�t � j[u(ct � j � bet � j) � z(hi,t � j)]� ,

where � � (0, 1) is the intertemporal discountfactor; ct is consumption at time t; u is the utilityderived from consumption, with u� � 0 andu 0; et� j � et� j�1 � (1 � )ct� j�1 rep-resents habit formation in consumption, with b, � [0, 1);27 hi,t is the number of hours workedat t; and z is the disutility of labor effort, withz� � 0 and z 0.28

The household’s dynamic budget constraintis given by

Vt � 1 � Pt �ct � it � a�ut �k� t � � Rt Vt � Rtkutk� t

� Dt � Ai,t � Wi,t hi,t � Tt , � t,

where Vt is the household’s beginning of periodt financial wealth, deposited at a financial inter-mediary where it earns the gross nominal inter-est rate Rt. The term Rt

kutk�t represents thehousehold’s earnings from supplying capitalservices, where utk�t denotes the physical stockof capital k�t, adjusted by the capital utilizationrate ut, whose level is decided by the householdat time t, and Rt

k denotes the correspondingreturn. Dt denotes firm profits, and Ai,t denotesthe net cash inflow from participating in state-contingent securities markets at time t. Wi,thi,t islabor income and Tt is a lump-sum tax used tofinance government expenditures.29 Finally, it

and a(ut)k�t represent, respectively, the pur-chases of investment goods and the cost ofsetting the utilization rate to ut, in units ofconsumption goods.

The evolution of the stock of physical capitalis given by

(7) k� t � 1 � �1 � ��k� t � F�it , it � 1 �,

where � is the depreciation rate, and the adjust-ment costs function F represents the technol-ogy transforming current and past investmentinto installed capital for use in the followingperiod.30 Capital services kt are related to thestock of capital k�t through kt � utk�t.

Financial intermediaries receive Vt from house-holds and lend all their money to intermediate-good firms, which use the money to buy laborservices Lt. Loan market clearing hence impliesWtLt � Vt.

C. Wage Setting with Nominal Rigidities andStaggering

The household enjoys monopoly power overits differentiated labor service hi,t, and sells thisservice to a representative competitive firm thattransforms the service into an aggregate laborinput Lt, using a CES technology with parame-ter �w � [1, �). The implied demand schedulefor hi,t is given by:

hi,t � � Wt

Wi,t� �w /��w�1�

Lt ,

where Wt is the aggregate wage level (the priceof Lt), which takes the form

Wt � ��0

1

Wi,t1/�1 � �w � di 1 � �w

.

Households take Wt and Lt as given, and setwages knowing that they will have to supplyas much labor as firms demand at the set

26 As a consequence of the model’s assumptions, theformula anticipates the result that all households consumethe same, despite working different hours, hence, the cor-responding omission of the subscript i in ct.

27 See Jeffrey Fuhrer (2000) for a thorough explanationof habit formation in consumption as represented in thisfunctional form, and the technical appendix in Christiano etal. (2005).

28 In the model’s calibration, we assume u� � log�and z� � �0�2.

29 We assume that the government budget is alwaysbalanced, that is, Tt � Ptgt, where gt denotes real govern-ment expenditures on the final good. In the present setup,government consumption does not substitute for house-holds’ private consumption; we assume gt is set at a levelproportional to the previous year’s quarterly average GDP.

30 In calibrating the model, we assume, as in Christianoet al., that F(it, it�1) � [1 � S(it/it�1)]it, with S�(1) � 0,� S(1) � 0, ut � 1 in steady state, and a(1) � 0. Wedenote �a � a(1)/a�(1).


wage. This way of modeling the labor marketimplies that wages play an allocative role atany point in time. We assume that wages areset following the mechanism proposed byCalvo (1983). However, instead of facing aconstant probability of resetting the wage inany given period, households face differentprobabilities over the course of a calendaryear. To keep a close parallel with the empir-ical exercise, we divide the year into fourquarters and assume that the probability ofresetting wages in quarter k is (1 � �k), withk � 1, ... , 4. When a household receives thesignal to change its wage, the new wage is setoptimally by taking into account the proba-bility of future wage changes. In particular, ifa contract is negotiated in the first quarter, theprobability that it is not renegotiated in thesecond quarter will be �2; the probability thatit is not renegotiated in the next two quarterswill be �2�3. Subsequent probabilities willbe, correspondingly, (�2�3�4), (�2�3�4�1),(�2

2�3�4�1), and so on. If a household cannotreoptimize its wage at time t, it uses a simpleautomatic rule of the form Wi,t � �t�1Wi,t�1. Theoptimal wage for labor-type i resetting the contractin the first quarter is then

(8) Wi,t � arg max�Wi,t�

Et � 1

��j�0

� k�1

4

�kj�4j��t�4jXt,4jWi,thi,t�4j

Pt�4j� z(hi,t�4j)�

� �2��t�4j�1Xt,4j�1Wi,thi,t�4j�1

Pt�4j�1� z(hi,t�4j�1)�

� �2�3�2��t�4j�2Xt,4j�2Wi,thi,t�4j�2

Pt�4j�2� z(hi,t�4j�2)�

� �2�3�4�3��t�4j�3Xt,4j�3Wi,thi,t�4j�3

Pt�4j�3

� z(hi,t�4j�3)��,where Xt,l is

(9) Xt,l � �t � �t�1 � �t�2 � ... � �t� l�1 if l 0,

� 1 if l � 0,

and �t is the marginal utility of (real) income attime t. The optimal wage in the first quarter max-imizes the expected stream of discounted utilityfrom the new wage, defined as the differencebetween the (real) gain derived from the hoursworked at the new wage, (Xt,t�Wi,thi,t�/Pt�) (ex-pressed in utils) and the disutility z(ht�) of work-ing, for all t� t.31 This expression is valid onlyfor wages set in the first quarter; expressions forthe optimal wage in the other quarters of thecalendar year can be derived in a similar fash-ion.32 Note that, since all labor types resettingtheir wages at a given quarter will choose thesame wage, we can drop the index i and simplyrefer to the optimal wage in period t as Wt.

Defining variables with a hat as xt � (xt � x)/x,where x is the value of xt in the nonstochastic

steady state, we define wt Wt � Pt and wt Wt � Pt. Introducing the dummy variables qkt,with k � 1, ... 4, which take the value of one inquarter k and zero otherwise, we show in theAppendix that wage maximization in a givenquarter leads to the following recursive log-linear-ized expression for the optimal wage rate:

(10) wt � �t �

1 � �k � 1

4

�k�4

��q,�,��

� ��t ��w � 1

2�w � 1

� � �w

�w � 1wt � Lt � �t�

� � �k � 1

4

qkt�k � 1

��q,�,��

��q,�,��Et � 1�wt � 1 � �t � 1�,

31 Labor income is expressed in real terms (by dividingby the price level) and converted into utils using the mar-ginal utility of income �.

32 The corresponding expression for workers settingwages in the second quarter can be obtained by substituting�k�1 for �k, for k � 1, 2, 3 and substituting �4 for �1 inexpression (8). Similar substitutions lead to the formulae forthe optimal wage in the third and fourth quarters.


with the expressions for the parameters �(q,�,�)and �(q,�,�) provided in the Appendix. Accordingto this expression, the optimal (real) wage is anaverage between a constant markup �w over themarginal rate of substitution between consump-tion and leisure, and the optimal wage expected toprevail in the next period. The weights on the twoterms vary according to the quarter in which thewage is reset. In particular, the weight on thenext-period wage is larger the smaller the proba-bility of changing wages in the next period. Thedynamics for the aggregate wage level Wt is givenby the weighted average between the optimalwage of the labor types that receive the signal tochange and the wages of those who did not get thesignal. By the law of large numbers, the propor-tion of labor types renegotiating wages at quarterk will be equal to (1 � �k). Therefore, as shown inthe Appendix, the quarter-dependent law of mo-tion for the aggregate real wage in log-deviationsfrom the steady state is given by

(11) wt � �1 � � �k � 1

4

qkt�k� wt

� � �k � 1

4

qkt�k� �wt � 1 � �t � 1 � �t �.

D. Model Solution and Calibration

The model’s solution procedure involves takinga linear approximation about the nonstochasticsteady state of the economy. The log-linear rela-tionships characterizing the dynamics of the non-policy variables are given by equations (17)through (26) in the Appendix, together with thewage equations (10) and (11) in the text. To closethe model, we assume that the monetary authorityfollows a Taylor-type reaction function,

(12) Rt � �Rt � 1 � �1 � ��a��t � ayy� � �t ,

where Rt � ln Rt � ln R� is the nominal interestrate in deviation from its steady-state level, andyt is the output gap. The term �t is the policyshock, whose effect on the economy we want toevaluate. Because of the presence of the time-varying indicators qkt in the equations describ-

ing the wage dynamics, the system is nonlinear.To solve the model, we use the nonlinear algo-rithm proposed by Jeffrey Fuhrer and HoytBleakley (1996).

Table 2 summarizes the benchmark valuesused to calibrate the model. The parameters inthe policy reaction function (12) are in line withthe estimates reported in the literature (see, e.g.,Fuhrer 2000). The interest-smoothing parameter� is set at 0.92. The coefficients on output andinflation are calibrated at a� � 2 and ay � 0.6.We follow Christiano et al. (2005) closely in thecalibration of the remaining parameters. Thediscount factor � is set at 1.03�0.25, which im-plies a steady-state annualized real interest rateof 3 percent. The capital share � is set at 0.36.The depreciation rate � is equal to 0.025, whichimplies an annual rate of depreciation on capitalof 10 percent. As in Christiano et al. (2005), weset �a, the inverse of the elasticity of capitalutilization with respect to the rental rate ofcapital, at 0.1. We maintain a similar degree ofhabit persistence in consumption, with the pa-rameter describing the degree of habit persis-tence � set at 0.65; is set at 0.2, consistentwith evidence reported by Fuhrer (2000). In-vestment adjustment costs are calibrated so thata permanent 1 percent decline in the price ofcapital induces a 20 percent change in invest-ment. The parameter governing exogenousprice rigidity, p, is equal to 0.35. This impliesthat price contracts last, on average, about 4.5months, a value consistent with recent micro-economic evidence on the frequency of pricechanges (see Mark Bils and Peter Klenow2004). Christiano et al. (2005) show that the

TABLE 2—PARAMETER CALIBRATION

Parameter values

� 0.993�� 0.360� 0.025�w 1.150p 0.350� 0.650�a 0.100�1 0.600�2 0.970�3 0.450�4 0.300a� 2.000ay 0.600� 0.920


model’s response to a monetary policy shock isfairly insensitive to the value of this parameter;in particular, they show that the main resultshold when prices are fully flexible (p � 0).This is because wage contracts, not price con-tracts, are the important nominal rigidity forimparting empirically consistent dynamics tothe model.33

In calibrating the values for the probability ofresetting wages in a given quarter, we follow anapproach consistent with the parametrization inChristiano et al. (2005). In their setup, the prob-ability of wage changes across quarters is iden-tical and such that the average frequency ofwage changes is approximately 2.5 quarters.Our way of calibrating the �k’s ensures that theaverage frequency of wage changes is about 2.1quarters.34 To capture the seasonal effects, weset the values of �k so that �4 �3 �1 �2.This corresponds to a situation in which a largerfraction of wages is changed during the courseof the fourth and third quarters, a smaller frac-tion is changed in the first quarter, and an evensmaller fraction during the second quarter. Thisassumption is in line with anecdotal evidencesuggesting that wages are reset at the end or atthe very beginning of the calendar year, withfewer changes taking place during the secondquarter. The calibrated probabilities imply that24 percent of the wage changes take place in thefirst quarter, 2 percent in the second quarter, 32percent in the third quarter, and 42 percent in thefourth quarter.35 These frequencies are con-sistent with wage-setting practices of NewEngland firms surveyed in the Federal Re-serve System’s Beige Book and the additionalsources of anecdotal evidence discussed inthe introduction.36

E. Model Results

We now present the model-generated im-pulse responses to a 25-basis-point decline inthe nominal interest rate on impact for output,inflation, the nominal interest rate, hours, andthe real wage. To make the results comparableto the identifying assumption underlying ourempirical exercise, we assume that the shockoccurs at the end of period t, when all the periodt nonpolicy variables have been already set. Themodel responses are plotted against the corre-sponding empirical responses and the 95 percentconfidence bands from the previous section’s es-timated quarter-dependent VAR.

The responses of output are shown in Figure14. Policy shocks occurring in the first andsecond quarters of the calendar year have asignificant effect on output, since few house-holds are allowed to reset their wages optimally.In the third and fourth quarters, the response ofoutput is less than half the size of the responsein the first half of the year. This difference in theresponse of output occurs even though ourmodel features the same degree of real rigiditiesas in Christiano et al. (2005). In the model, realrigidities work in the direction of dampening theeffect of uneven staggering of wages, but ourcalibrated values for the �k’s still imply relevantdifferences in the response of output acrossquarters. At the same time, real rigidities help togenerate persistent responses to the policyshock. In line with Christiano et al. (2005), themodel-generated impulse response for outputpeaks three to four quarters after the shock,depending on the quarter in which the shockoccurs. This matches the response of output to apolicy shock in the second quarter, although theoutput response to a policy shock in the firstquarter is not as persistent as in the data.

The responses of inflation are displayed inFigure 15. Consistent with the data, the modelproduces impulse responses that are not strik-ingly different across quarters. In the two quar-ters after the shock, prices increase somewhatfaster when the shock occurs in the third orfourth quarter—a feature we also observe in thedata. The model responses fall comfortably

33 David Card (1990) also provides empirical evidenceon the importance of nominal wage rigidities in the trans-mission of aggregate demand shocks to real economicactivity.

34 In other words, �4 � �k�14 �k, where � is the constant

probability value used in Christiano et al. (2005), equal tothe geometric average of the quarter-dependent probabili-ties. Given our calibrated values for the �k’s in Table 2, thisimplies an average frequency of wage adjustment of 2.13(�1/(1 � �4 �k � 1

4 �k) quarters.35 In each quarter k, the proportion of wage changes

relative to the total number of changes in a given calendaryear is (1 � �k)/(4 � ¥k�1

4 �k).36 Findings about wage-setting practices of New En-

gland firms surveyed in the Federal Reserve System’s Beige

Book over the course of 2003 are available from the authorsupon request. Firms are identified by sector, but the identityof the firms is kept confidential.


FIGURE 14. MODEL AND VAR IMPULSE RESPONSES OF OUTPUT TO A 25-BASIS-POINT DECLINE IN FED FUNDS RATE

Notes: Bold solid lines are the theoretical responses and solid lines-plus sign are the VAR responses. Broken lines indicatethe 95 percent confidence intervals around VAR estimates. Vertical axis units are deviations from the steady-state path.

FIGURE 15. MODEL AND VAR IMPULSE RESPONSES OF INFLATION TO A 25-BASIS-POINT DECLINE IN FED FUNDS RATE

Notes: Bold solid lines are the theoretical responses and solid lines-plus sign are the VAR responses. Broken lines indicatethe 95 percent confidence intervals around VAR estimates. Vertical axis units are deviations from the steady-state path(annualized percentage points).




within the 95 percent confidence bands in allfour quarters.

The responses of the federal funds rate,shown in Figure 16, are generally consistentwith the empirical responses, with the exceptionof the response to a second-quarter shock, whenthe theoretical response is weaker than its em-pirical counterpart.

As for the responses of the real wage, themodel’s setup with Calvo-style wage settingimplies that at each point in time the aggregatewage is a weighted average of the aggregatewage prevailing in the previous period (the“preset” wage) and of the “optimal” wage, thatis, the wage set by workers who at that partic-ular point in time are able to adjust their wageoptimally. Figure 17 shows the model responsesof the aggregate and the optimal real wage.There are noticeable differences in the responseof the optimal wage across quarters. Still, con-sistent with our empirical findings, the re-sponses of the real aggregate wage are fairlysimilar across quarters. The intuition for thisresult is the following. After an expansionarymonetary shock, aggregate demand goes up andso does the derived demand for labor. In a

period when a large fraction of workers cannotreadjust their wage optimally, these workerswill have to supply as much labor as firmsdemand at the preset wage. The fewer workerswho can adjust the wage optimally will raise thewage significantly following the shock.37 Spe-cifically, they will take into account that (a)activity is responding strongly to the shock,with firms demanding more labor from themany workers having a preset wage; and (b) infuture periods many other workers will resettheir wages at a higher level, given the persis-tent effect of the shock on aggregate demand.The aggregate wage is then determined by alarge fraction of preset wages and a small frac-tion of reoptimized wages responding stronglyto the shock.

In contrast, in a period in which a largefraction of the workers can adjust their wageoptimally, firms can no longer count on as much

37 Recall that due to the Dixit-Stiglitz formulation, dif-ferent types of workers are not perfect substitutes. Given thehigher levels of aggregate demand, the demand for theworkers who can reset wages also goes up, and hence theirwages.

FIGURE 16. MODEL AND VAR IMPULSE RESPONSES OF THE FED FUNDS RATE TO A 25-BASIS-POINT DECLINE IN FED FUNDS RATE

Notes: Bold solid lines are the theoretical responses and solid lines-plus sign are the VAR responses. Broken lines indicatethe 95 percent confidence intervals around VAR estimates. Vertical axis units are deviations from the steady-state path(annualized percentage points).



labor at preset wages as in the previous sce-nario. Now firms meet an upward-sloping sup-ply of labor for a large number of workers. Theincrease in aggregate activity following an ex-pansionary monetary policy shock is thereforesmaller, and the economy operates at a levelcloser to the flexible-wage-price potential. Be-cause aggregate activity stays closer to its long-run equilibrium following the shock, the optimalwage does not increase by as much as in theprevious case. The aggregate real wage is nowdetermined by a small fraction of preset wagesand a large fraction of reoptimized wages, withthe latter responding less strongly to the shockthan in the previous scenario.

In sum, the model predicts a quarter-dependentresponse for the optimal wage, but the responseof the aggregate wage needs not be strikinglydifferent across different quarters, precisely be-cause when the optimal wage responds more(less), the optimal wage has a smaller (larger)weight in the aggregate wage. In Figure 18 weplot the model-generated responses for the ag-gregate real wage, together with the correspond-ing empirical responses. As the figure shows,

the theoretical responses fall consistently withinthe empirical confidence bands.38

The adjustment of labor services plays a cen-tral role in the transmission mechanism in thepresence of nominal wage rigidities. After anexpansionary monetary policy shock, firms uselabor inputs more intensively, increasing thenumber of hours. The larger the extent of wagerigidity, the greater the intensity with whichlabor inputs are used. The model thus predictsthat in quarters when nominal wages are rela-tively rigid (the first two quarters of the calendaryear), labor hours will respond to a monetarypolicy shock more strongly than in quarterswhen nominal wages are relatively flexible (thethird and fourth quarters of the calendar year).In Figure 19, we plot both the theoretical andempirical impulse responses of hours to a mon-etary expansion. In line with the data, the the-

38 Note that as in Christiano et al. (2005), real wages areprocyclical. This is because prices, which are a markup overmarginal costs, are a weighted average of wages and theinterest rate R. In an expansion, R declines and hence theprice level increases less than wages.

FIGURE 17. MODEL RESPONSES OF THE AGGREGATE REAL WAGE (wt) AND THE OPTIMAL REAL WAGE �wt) TO A

25-BASIS-POINT DECLINE IN FED FUNDS RATE

Notes: Solid lines are the responses of the aggregate real wage and solid lines-plus asterisks are the responses of the optimalreal wage.



oretical response is strongly seasonal, being onaverage twice as large when the shock takesplace in the first two quarters. Though the the-oretical responses tend to be slightly smallerthan the empirical ones, they usually fall withinthe empirical confidence bands.

In sum, the model results indicate that a re-alistic amount of uneven staggering yields todifferent output and hours dynamics, dependingon the quarter in which the policy shock takesplace. The differences occur even if the modelembeds a significant (and empirically plausible)degree of real rigidities. The responses for sev-eral key model variables are not only qualita-tively, but also quantitatively, consistent withthe quarter-dependent patterns we have docu-mented in actual data.

IV. Concluding Remarks

The paper documents novel findings regard-ing the impact of monetary policy shocks onreal activity. After a monetary expansion thattakes place in the first quarter of the year, output

picks up quickly and tends to die out at arelatively fast pace. This pattern is even morepronounced when the monetary policy expan-sion takes place in the second quarter of theyear. In contrast, output responds little when themonetary expansion takes place in the third andfourth quarters of the year. The conventionalfinding that monetary shocks affect output withlong delays, and that the effect is highly persis-tent, may be interpreted as the combination ofthese different output impulse responses.

We argue that the differential responses arenot driven by different types of monetary policyinterventions, nor by different “states” of theeconomy across quarters. Encouraged by anec-dotal evidence on the timing of wage changes,which suggests that a large fraction of wages arereset toward the end of each calendar year, wepropose a potential explanation for the differen-tial responses based on contractual lumping anddevelop a theoretical general equilibrium modelbased on Christiano et al. (2005) featuring un-even staggering of nominal wage contracts. Themodel generates impulse responses that quanti-tatively match those found in the data.

FIGURE 18. MODEL AND VAR IMPULSE RESPONSES OF THE AGGREGATE REAL WAGE TO A 25-BASIS-POINT DECLINE IN FED

FUNDS RATE




While our model assumes uneven staggering,there are studies in the literature addressing theoptimality of uniform staggering versus syn-chronization of price (or wage) changes. Thegeneral finding of this literature is that synchro-nization is the equilibrium timing in many sim-ple Keynesian models of the business cycle.39

Yet, the new generation of Keynesian modelshas glossed over this finding and assumed uni-form staggering as both a convenient modelingtool and an essential element in the transmissionmechanism of monetary policy shocks. Thispaper provides an empirical setting to test the

hypothesis of uniform versus uneven staggeringof wage changes, and in so doing argues for theempirical and theoretical relevance of models inwhich wage changes are less staggered andmore synchronized.

We have addressed the robustness of ourfindings along several dimensions, but addi-tional evidence could corroborate our results.Other shocks, such as technology shocks, couldalso have a different impact across the calendaryear if wage staggering is not uniform. How-ever, while there is some consensus on how toidentify monetary policy shocks, the identifica-tion of other types of shocks remains conten-tious and would require additional variables inour VAR, thus reducing degrees of freedom atthe estimation stage. A more promising avenue,in our view, is the examination of internationalevidence. Other countries likely exhibit unevenstaggering of wage contracts, and the trans-mission mechanism of monetary shocksshould be affected accordingly. In this re-spect, Japan is particularly relevant, because alarge fraction of Japanese firms set wages in the

39 Lawrence Ball and Stephen Cecchetti (1988) showthat staggering can be the equilibrium outcome in somesettings with imperfect information, but even then such aresult is not necessarily pervasive, since it depends on thestructure of the market in which firms compete and on firmssetting prices for a very short period of time. In othersettings, staggering can be the optimal outcome for wagenegotiations if the number of firms is very small (see GaryFethke and Andrew Policano 1986). The incentive for firmsto stagger wage negotiation dates, however, diminishes thelarger the number of firms in an economy.

FIGURE 19. MODEL AND VAR IMPULSE RESPONSES OF HOURS TO A 25-BASIS-POINT DECLINE IN FED FUNDS RATE




spring season (during the wage-setting processknown as Shunto, or “spring offensive”). Pre-liminary findings for Japan indicate that mone-tary policy shocks that take place in the third

quarter (after the Shunto) have a large impact onoutput, whereas monetary policy shocks occur-ring in the second quarter (during the Shunto)have virtually no output effect.

APPENDIX

This Appendix is composed of two parts. The first part details the derivation of the equationsgoverning the wage dynamics (equations (10) and (11) in the text). The second part presents themodel’s log-linearized relationships for the nonpolicy variables which, together with the policyequation (12) in the text, summarize the model’s dynamics around its steady state.

Wage-Setting Process

We let period t correspond to the first quarter (and so does t � 4j, @j) while t � 1 � 4j, @jcorresponds to the second, t � 2 � 4j, @j corresponds to the third, and t � 3 � 4j, @j correspondsto the fourth quarter. Since all labor types resetting their wages in a given quarter will choose thesame wage, we can drop the i’s from the notation and simply refer to the optimal wage in period tas Wt. The first-order condition for the optimal wage Wt in the first quarter satisfies

Et � 1��

j � 0

� i � 1

4

�ij�4jht � 4j�t � 4j��Wt � Xt,4j

Pt � 4j� �w

zh,t � 4j

�t � 4j

� �2�ht � 4j � 1�t � 4j � 1�Wt � Xt,4j � 1

Pt � 4j � 1� �w

zh,t � 4j � 1

�t � 4j � 1

� �2�3�2ht � 4j � 2�t � 4j � 2�Wt � Xt,4j � 2

Pt � 4j � 2� �w

zh,t � 4j � 2

�t � 4j � 2

� �2�3�4�3ht � 4j � 3�t � 4j � 3�Wt � Xt,4j � 3

Pt � 4j � 3� �w

zh,t � 4j � 3

�t � 4j � 3 �

� 0,

where zh,t� l is the derivative of z with respect to ht� l and Xt,l is defined in (9). Log-linearizing thisexpression around its steady state, we obtain

Wt � Pt � 1 � �Et � 1��j � 0

� i � 1

4

�ij�4j � ��t � 4j �

�w � 1

2�w � 1 � �w

�w � 1(Wt � 4j � Pt � 4j) � Lt � 4j � �t � 4j�

� �2�� t � 4j � 1 ��w � 1

2�w � 1 � �w

�w � 1(Wt � 4j � 1 � Pt � 4j � 1) � Lt � 4j � 1 � �t � 4j � 1�

� �2�3�2� �t � 4j � 2 �

�w � 1

2�w � 1 � �w

�w � 1(Wt � 4j � 2 � Pt � 4j � 2) � Lt � 4j � 2 � �t � 4j � 2�

� �2�3�4�3��t � 4j � 3 �

�w � 1

2�w � 1 � �w

�w � 1(Wt � 4j � 3 � Pt � 4j � 3) � Lt � 4j � 3 � �t � 4j � 3��,

where xt � (xt � x)/x and x is the nonstochastic steady-state level of xt and


��1 � �j � 0

� i � 1

4

�ij�4j�1 � �2� � �2�3�

2 � �2�3�4�3� �

�1 � �2� � �2�3�2 � �2�3�4�

3�

1 � �1�2�3�4�4 .

We can then write Wt in recursive form as

(13) Wt � Pt � 1 � �� t ��w � 1

2�w � 1 � �w

�w � 1(Wt � Pt) � Lt � �t�

� ��2

1 � ��3(1 � ��4 � �2�4�1)

1 � ��2(1 � ��3 � �2�3�4)Et � 1(Wt � 1 � Pt ).

The aggregate wage level is a weighted average of the optimal wage set by the workers whoreceived the signal to reoptimize, and the wage of the workers who did not get the signal. Asmentioned in the text, the workers who do not reoptimize have their wage indexed to the previousperiod rate of inflation. If we keep our convention that t (and {t � 4j}) corresponds to the firstquarter, the proportion of workers changing wages at t will be equal to (1 � �1). Then, the expressionfor the aggregate wage level is given by

(14) Wt � �1 � �1 �Wt � �1 �Wt � 1 � �t � 1 �.

Equations (13) and (14) describe the law of motion for the optimal wage and the aggregate wage inthe first quarter of the calendar year. To generalize these expressions to any quarter of the calendaryear, we introduce the dummy variables qkt, with k � 1, ... , 4 which take on the value one in the kthquarter and zero otherwise. We can then write the equations governing the wage dynamics as

(15) Wt � Pt � 1 �

1 � �j � 1

4

�j�4

��q,�,�� t �

�w � 1

2�w � 1 � �w

�w � 1(Wt � Pt) � Lt � �t�

� � �k � 1

4

qkt�k � 1

��q,�,��

��q,�,��Et � 1 �Wt � 1 � Pt �,

and

(16) Wt � �1 � � �k � 1

4

qkt�k� W � � �k � 1

4

qkt�k� �Wt � 1 � �t � 1 �,

where

��q,�,�� 1 � ��q1t�2 �1 � ��3 � �2�3�4 � � q2t�3 �1 � ��4 � �2�4�1 �

� q3t�4 �1 � ��1 � �2�1�2 � � q4t�1 �1 � ��2 � �2�2�3 �};

��q,�,�� 1 � ��q1t�3 �1 � ��4 � �2�4�1 � � q2t�4 �1 � ��1 � �2�1�2 �

� q3t�1 �1 � ��4 � �2�4�3 � � q4t�2 �1 � ��3 � �2�3�4 �}.


Using the definitions of wt and wt in the text (wt Wt � Pt and wt Wt � Pt), equations (15)and (16) become, respectively,

wt � �t �

1 � �j � 1

4

�j�4

��q,�,�� t �

�w � 1

2�w � 1 � �w

�w � 1wt � Lt � �t�

� � �k � 1

4

qkt�k � 1

��q,�,��

��q,�,��Et � 1 �wt � 1 � �t � 1 �,

and

wt � �1 � � �k � 1

4

qkt�k� wt � � �k � 1

4

qkt�k� �wt � 1 � �t � 1 � �t �,

which correspond to equations (10) and (11) in the text.

The Model’s Log-Linearized Equilibrium Equations

We now briefly describe the set of log-linearized equations which, together with the wageequations (10) and (11), and the Taylor rule (12), characterize the dynamics of the model.

The inflation dynamics implied by Calvo-pricing takes the form

(17) �t �1

1 � ��t � 1 � Et � 1� �

1 � ��t � 1 �

(1 � �p)(1 � p)

(1 � �)pst� ,

where the marginal cost, st, is given by

(18) st � ��wt � Rt � Lt � kt � � �1 � ��wt � Rt �.

Firms that do not reoptimize in a given period change prices according to the most recent rate of inflation,hence the presence of lagged inflation in the equation. Because in the Calvo setup firms cannot adjustprices optimally every period, inflation is a function not only of current marginal costs, but also of theexpected present discounted value of current and future marginal costs (entering through �t�1).

The Euler equation for consumption is given by

(19) Et � 1�� t � 1 � �c� ct �b

1 � et� � (b � )��c� ct � 1 �

b

1 � et � 1�� t ,

where �c � (1 � /1 � � b)(1 � � /1 � � � �b), et�1 � et � (1 � )ct, and

(20) Et � 1�t � 1 � �t � Et � 1 �Rt � 1 � �t � 1 �.

The presence of habit formation implies that consumers’ current utility is determined by currentconsumption relative to a reference level of consumption. The parameter indexes the persistenceor “memory” in the habit-formation reference level. In essence, the Euler equation says that theexpected level of consumption next period relative to its reference level depends on currentconsumption relative to its reference level on a present-discounted stream of expected real interestrates.


The specification for investment takes the form

(21) Et � 1 � ı t � ı t � 1 � � �Et � 1 � ı t � 1 � ı t � �1

�Pt

k,

where Ptk is the hypothetical spot price of a unit of capital stock installed at time t. In turn, the price

of capital is determined according to a familiar asset-pricing equation whereby the price of capitalat time t depends on the price of capital next period plus its period t dividend, equal to the rentalrate of capital rt

k � wt � Rt � Lt � kt. Forward iteration of the equation for the price of capitalyields the following relationship:

(22) Et � 1Ptk � �1 � ��1 � ��Et � 1 �

i � 0

�

�i�1 � ��irt � 1�ik � Et � 1 �

i � 0

�

�i�1 � ��i�Rt � 1�i � �t � 1�i�.

This relationship can be substituted into (21) to obtain an expression for the change in investmentin terms of a present discounted value of the rental rate of capital and of the real interest rate.

The Euler equation for the household’s capital-utilization decision becomes

(23) Et � 1 ut � Et � 1 �kt � k� t � �1

�art

k.

This expression says that the expected marginal benefit of raising utilization must equal theassociated expected marginal cost.

Finally, the log-linearized versions of the capital accumulation equation and the economy-wideresource constraint are given by

(24) k� t � �1 � ��k� t � 1 � ��t ,

and

(25) scct � sggt � si� �t � � 1

�� (1 � �) 1

�(kt � k� t)� � yt ,

where output yt is equal to

(26) yt � �kt � �1 � ��Lt .

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