Does the Labor Share of Income Drive Inflation?

Does the Labor Share of Income Drive Inflation? Jeremy Rudd Karl Whelan Federal Reserve Board(cid:3) Central Bank of Ireland(cid:3)(cid:3) May 30, 2002 Abstract Woodford(2001)haspresentedevidencethatthenew-KeynesianPhillipscurve (cid:12)ts the empirical behavior of inflation well when the labor income share is used as a driving variable, but (cid:12)ts poorly when deterministically detrended output is used. He concludes that the output gap|the deviation between actual and potential output|is better captured by the labor income share, in turn implying that central banks should raise interest rates in response to increases in the labor share. We show that the empirical evidence generally suggests that the labor share version of the new-Keynesian Phillips curve is a very poor model of price inflation. We conclude that there is little reason to view the labor income share as a good measure of the output gap, or as an appropriate variable for incorporation in a monetary policy rule. (cid:3)Corresponding author. Mailing address: Mail Stop 80, 20th and C Streets NW, Washington, DC 20551. E-mail: jeremy.b.rudd@frb.gov. (cid:3)(cid:3)E-mail: karl.whelan@centralbank.ie. We are grateful to Dale Henderson, John Leahy, Argia Sbordone, and Michael Woodford for helpful comments on earlier drafts of the paper. The views expressedareourown anddonotnecessarily reflecttheviewsoftheBoard ofGovernors,thesta(cid:11) of theFederal ReserveSystem, or theCentral Bank of Ireland.

1 Introduction Recent years have seen an explosion in research aimed at assessing monetary policy rules using macroeconomic models built from explicit microfoundations. In many versions of thesemodels,pricingbehavioris describedbya\new-KeynesianPhillips curve," which relates inflation to expected future inflation and the output gap x t: (cid:25) t = (cid:12)E t (cid:25) t+1+γx t : (1) However, empirical implementations of this equation that use deterministically detrended output to measure the output gap are known to provide a poor description of the actual inflation process. One important problem can be seen from applying repeated substitution to equation (1), which yields (cid:88)1 (cid:25) t = γ (cid:12)kE t x t+k : (2) k=0 This equation implies that inflation is a purely forward-looking \jump" variable. As Fuhrer and Moore (1995) have noted, this prediction seems inconsistent with the empirical evidence from reduced-forminflation regressions, which indicates that inflationdependsimportantlyonitsownlags. Inaddition,equation (2)impliesthat higher inflation should Granger cause increases in detrended output, a prediction that is (cid:12)rmly rejected by the data. In a recent paper, Michael Woodford (2001) has argued that these empirical failures of the new-Keynesian Phillips curve stem from deterministically detrended GDP’s being an inappropriate measure of the output gap. He notes that the stickyprice models underpinningthe new-Keynesian Phillips curve imply that the correct drivingvariableinthisequationisactuallyrealmarginalcost(nominalmarginalcost divided by the price level); this in turn should be positively related to the di(cid:11)erence between actual output and potential output, de(cid:12)ned as the level of output that would prevail under flexible prices. Woodford presents evidence that the use of a direct measure of real marginal cost results in a better-(cid:12)tting inflation equation, implying that the new-Keynesian model can work well once one allows for the role of stochastic fluctuations in potential. In his empirical exercises, Woodford draws on the work of Sbordone (1998) and Gal(cid:19)(cid:16) and Gertler (1999), who suggest using average unit labor costs (nominal 1

compensation divided by real output) as a proxy for nominal marginal cost. The resulting proxy for real marginal cost|and thus the output gap|is labor’s share of income (nominal compensation divided by nominal output). Woodford interprets the evidence as indicating that the labor share \is a much better measure of the true output gap, at least for purposes of explaining inflation variation." This idea carries substantial implications for the conduct of monetary policy. For example, it implies that central banks should not follow traditional Taylor rules, which recommend setting short-term interest rates with reference to the levels of inflation and deterministically detrended output. Rather, they should ignore standard measures of the output gap and instead raise interest rates in response to increases in labor’s share of income. The unconventional nature of such a policy rule is clear from Figure 1: Because the labor share has spiked upward in every modern U.S. recession, a policy rule based on this variable instead of detrended output would have called for higher interest rates during each of these episodes. Inthispaper,wepresentnewevidenceonthemeritsofthelaborshareversionof thenew-KeynesianPhillipscurve. Speci(cid:12)cally, were-examinetwopieces ofevidence cited by Woodford as illustrating how this model provides a good description of the the inflation process. We conclude that the case for this view is very weak. First,weprovideanewperspectiveontheempiricalresultsinWoodford’spaper, which were obtained by using a reduced-form VAR to calculate the E t x t+k terms in equation (2). Woodford reports that the predicted inflation series based on detrended output is negatively correlated with actual inflation, while the series based on the labor share (cid:12)ts well. However, we show that this latter result is not robust; in particular, we demonstrate that the (cid:12)t of the labor share version of the model is highly sensitive to small changes in the VAR used to forecast future values of the labor income share. For a broad range of VAR speci(cid:12)cations, the model’s (cid:12)t is actually very poor. In addition, the model’s key prediction|that inflation should Granger cause the labor share|is rejected, as is the idea that the labor share version of the new-Keynesian Phillips curve can account for the role of lagged inflation in reduced-form inflation regressions. Second, we re-examine the evidence presented in Sbordone (1998). This study used a di(cid:11)erent test equation from equation (2) and reported a good (cid:12)t for inflation when the labor share is used to proxy for real marginal cost, a result cited by 2

Woodford and others as demonstrating that the labor share version of the new- Keynesian Phillips curve provides a reasonable description of empirical inflation dynamics. We show, however, that Sbordone’s test equation would produce a good (cid:12)t for inflation even in cases where the new-Keynesian Phillips curve is clearly false; relatedly, we demonstrate that her method yields an inflation series that (cid:12)ts well regardless of which output gap proxy is used. Hence, these results cannot be interpreted as providing useful evidence in favor of the new-Keynesian model. We conclude that the new-Keynesian Phillips curve provides a poor description of the inflation process even when the labor income share is used as a proxy for real marginal cost. Hence, the relative (cid:12)ts of the various versions of the new- Keynesian Phillips curve should not be considered an appropriate metric for determining whether the true output gap is better measured by the labor share or by detrended output, and certainly cannot support the conclusion that the labor income share should supplant conventional gap measures in monetary policy rules. 2 Inflation and Expectations of Real Marginal Cost In this section, we describe empirical implementations of equation (2)|which expresses inflation as a function of a discounted sum of current and expected future output gaps|using both the labor share and detrended output proxies for the output gap. Our data are for the U.S. nonfarm business sector, and cover the period 1960:Q1 to 2001:Q1. The labor income share is plotted in the upper panel of Figure 1. Our detrended output series (shown in the lower panel of Figure 1) is de(cid:12)ned as the deviation of the log of real nonfarm GDP from a quadratic trend. The construction of an empirical inflation series consistent with equation (2) requiressomecharacterization ofhowagents formulateexpectationsoffuturevalues of x t. The procedure adopted by Woodford (2001) involves specifying x t as one of the variables in a multivariate VAR of the form Z t = AZ t−1+(cid:15) t : (3) This allows expected future values to be expressed in terms of variables observed today. Speci(cid:12)cally, thevector ofdiscountedsumsof thevariablesintheVARcan be written as e0 i (I −(cid:12)A) −1Z t (where e0 i is a unit vector that extracts the discounted 3

1 sum of our output gap proxy). Given this discounted sum, we can then choose the value of γ that yields the best-(cid:12)tting inflation series. Results Using Baseline VARs: We start with the results for the detrended output version of the model. The VAR speci(cid:12)cation that we initially consider is identical to the one used by Woodford (2001) for this exercise; it consists of current and lagged values of detrended output, the labor income share, and unit labor cost 2 inflation. This system is then employed to generate a discounted sum of current and predicted values of detrended output (in constructing the discounted sum we assume a value for (cid:12) of 0.99, but our conclusions are robust to the use of other values). TheresultsfromthisexerciseareplottedintheupperpanelofFigure2; theyare essentially identical to results reported by Woodford for this version of the model. The expected discounted sum of detrended output values does a very poor job of explaininginflation;indeed,thisseriesisactuallynegatively correlatedwithinflation (hence, in Figure 2 we multiply it by an arbitrary positive constant). It is apparent from the (cid:12)gure that this model completely fails to predict the high inflation rates of the 1970s, or the low inflation rates of the 1990s. Moreover, this conclusion|that the expected discounted sum of detrended output values does poorly in explaining inflation|is robust across a wide range of VAR speci(cid:12)cations. Because thelaborincomeshareis amongthevariables includedin theVAR,itis asimplemattertousethissamesystemtoconstructtheexpecteddiscountedsumof labor shares. The resulting inflation series is plotted in the lower panel of Figure 2. The performance of this variant of the model is perhaps slightly better than the GDP-gap version inasmuch as the discounted sum of labor shares has the positive correlation withinflationpredictedbythetheory. However, themodelexplainsonly 1This formula relies on the fact thatE t Z t+k =AkZ t, and makes use of a matrix version of the standard geometric sum formula. See Sargent (1987, pp.311-312) for more details. 2Thespeci(cid:12)cVARsystememployedwasnotexplicitlydiscussedinWoodford’s2001paper(the speci(cid:12)cation used follows a similar exercise in Sbordone, 2001). We thank Professors Woodford andSbordoneforclarifyingthedetailsofthesecalculationsinaseriesofpersonalcommunications. Note that because equation (2) is actually derived as a loglinear approximation about a steady state,weincludeconstanttermsintheVARandestimationequations,andexpressallvariablesas logs or log-di(cid:11)erences. 4

a tiny fraction of the variation in inflation|the R2 for the model is 0.01. How can this (cid:12)nding be reconciled with the evidence presented in Woodford’s paper, which indicated that a discounted sum of labor shares tracks inflation relatively well? Itturnsoutthat thereason forthis discrepancystems fromWoodford’s useof adi(cid:11)erent VAR system to(cid:12)tthelabor shareversion of theinflation equation. When calculating the discounted sum of current and future labor income shares, Woodford employed a di(cid:11)erent VAR system from the one used to calculate the discounted sum of detrended output values; speci(cid:12)cally, the system used in the former case was a bivariate VAR containing the labor share and nominal unit labor cost growth. If we instead follow this procedure, we also obtain a (cid:12)tted inflation series (plotted in Figure 3) that tracks actual inflation more closely|the R2 is 0.44 for 3 this version of the model. An immediate conclusion that can be drawn from these exercises is that the (cid:12)t of the labor-share version of the new-Keynesian Phillips curve appears to be highly sensitive to how one speci(cid:12)es the forecasting VAR. However, in experimenting with variousVARspeci(cid:12)cationswehavefoundthatmostgenerateanexpecteddiscounted sum of labor income shares that has a very low correlation with inflation. Table 1 reports results based on several di(cid:11)erent VAR systems, including the bivariate system employed by Woodford (the second column of the table) and the speci(cid:12)cation used to generate our Figure 2 (the fourth column). The two other variables that we include in the additional VAR speci(cid:12)cations|namely, detrended hours and the consumption-output ratio|are used in the VARs that Sbordone (2001) considers. Several results from Table 1 are worth noting. (cid:15) Excludingdetrended outputfrom thethree-variable VAR|which is necessary inordertoobtainthewell-(cid:12)ttinginflationseriesshowninFigure3|isstrongly rejected on statistical grounds. Lags of detrended output receive statistically signi(cid:12)cant coe(cid:14)cients in the labor-share equation (see column 4). (cid:15) The improvement in (cid:12)t for inflation that occurs when we use Woodford’s bivariate system (described in column 2) stems from the small, positive co- 3This (cid:12)t is a bit lower than that reported in Woodford’s paper for the same exercise. We believe the di(cid:11)erence reflects our use of a more recent vintage of data (our data are current as of 2001); in particular, our dataset incorporates the substantial revisions made by the Bureau of Labor Statistics to thepublished labor income shares for thelate 1990s. 5

e(cid:14)cients that lagged unit labor cost growth receives when detrended output is excluded from the labor share equation. However, these coe(cid:14)cients are not statistically signi(cid:12)cant. If we omit unit labor cost growth from the VAR|thereby using a univariate regression to forecast future labor shares| the model’s (cid:12)tted inflation series has an R2 of only 0.16 (see column 1). (cid:15) In general, the labor share variant of the new-Keynesian Phillips curve explains a relatively small fraction of the observed variation in inflation. Moreover, this poor (cid:12)t obtains even for the VAR speci(cid:12)cations summarized in columns 4 through 7|all of which include unit labor cost growth|as well as for speci(cid:12)cations whose forecasting equation for the labor share (cid:12)ts as well as or better than the equation from Woodford’s bivariate VAR. Results from VARs Including Inflation: Whiletheprecedinganalysisexcluded lagged inflation from the VARs used to forecast future values of the labor share, there is no good reason to do so. In fact, it turns out that if the labor share variant ofthenew-KeynesianPhillipscurveistosuccessfullyexplaintheobserveddynamics of inflation, then it must be the case that lagged inflation is a very useful predictor of the labor income share. To see why this is so, note that as an empirical matter, U.S. inflation dynamics are well represented by a reduced-form regression of the form (cid:25) t = A(L)(cid:25) t−1 +(cid:11)y t ; (4) wherey tisusuallyde(cid:12)nedtobedetrendedoutputorarelatedmeasure. Estimatesof this reduced-form equation invariably (cid:12)ndthat the sumof the coe(cid:14)cients on lagged inflationislarge|typicallyaround0.9,andoftenstatisticallyindistinguishablefrom one. Hence, if the labor share version of the new-Keynesian model (2) is the correct structural description of inflation dynamics, then it must be that the role played by lagged inflation in the empirical model (4) stems purely from its serving as a proxyforexpected futurevaluesof thelaborshare(thetruedeterminants ofcurrent inflation). Table 2 summarizes the results obtained from using VAR speci(cid:12)cations that include lagged inflation. Importantly, we (cid:12)nd no evidence that inflation Granger 6

causes thelaborshare: For theseven speci(cid:12)cations reportedhere,thelowest p-value for an F-test of the hypothesis that lagged inflation can be excluded from the labor 4 shareequationequals0.199. Intermsofthe(cid:12)tofthenew-KeynesianPhillipscurve, the results in Table 2 are generally similar to those in Table 1: In most cases, the discounted sum explains only a tiny fraction of the observed variation in inflation. One new result worth noting from Table 2 is the R2 of 0.415 obtained by the (cid:12)ttedinflationseriesderivedfromthebivariateVARininflationandthelaborshare. Almost all of this (cid:12)t comes from the fact that, by construction, the series used to measurethediscountedsumoflaborsharesinthiscaseplacessmallpositiveweights on both lagged and contemporaneous inflation, the very variable we are attempting to explain. However, because the Granger causality tests indicate that there is no statistical reason to include inflation in this VAR, there is also no statistical reason to prefer the (cid:12)tted inflation series with an R2 of 0.415 to the other series with much poorer (cid:12)ts. The Role of Lagged Inflation: As a (cid:12)nal way of illustrating the inability of the labor share version of the new-Keynesian Phillips curve to explain inflation dynamics, note that if the model were correct, then there should be little role for lagged inflation in an equation like (cid:88)1 (cid:25) t = γ (cid:12)kE t x t+k +B(L)(cid:25) t−1 : (5) k=0 Inpractice, however, thisturnsoutnottobethecase. Eventheinclusionof thediscountedsumsthat,ontheirown,generatethebest-(cid:12)ttinginflationseries|i.e.,those based on the(s t ;(cid:1)ulc t) or (s t ;(cid:25) t)VARs|does little to reducetheestimated sumof the coe(cid:14)cients on lagged inflation in equation (5). (These coe(cid:14)cients are reported 5 in panel D of Tables 1 and 2.) 4These results run somewhat counter to the arguments of Gal(cid:19)(cid:16) and Gertler (1999) and Goodfriend and King (2001), who decide in favor of thehypothesis that inflation leads thelabor share. However, neither of these papers report Granger causality tests: Goodfriend and King report coe(cid:14)cients from a VAR in inflation and the labor share, but do not report standard errors, while Gal(cid:19)(cid:16)andGertler(1999)onlyreportcorrelationsbetweenvariousleadsandlagsofinflationandthe labor share. 5It is instructive to compare these results with the (cid:12)ndings of Rudd and Whelan (2001), who estimateequation(5)usinganinstrumentalvariablesapproachinwhichtherealized presentvalue 7

Two additional results are also worth highlighting. First, even the best-(cid:12)tting inflationseriesinTables1and2(cid:12)tfarlesswellthansimpleregressionsofinflationon its own lagged values (which typically receive an R2 of 0.7 and above). Second, the coe(cid:14)cients on detrended output in reduced-form regressions like (4) are also invariably statistically signi(cid:12)cant. Thislatter result illustrates an important point, which is that the failure of the detrended output version of the new-Keynesian Phillips curve should not, on its own, be used as evidence against the traditional proxies for the output gap as measures of resource utilization. The fact that deterministically detrended output works well in empirical reduced-form inflation regressions suggests that it may still represent a useful|albeit somewhat crude|measure of the output gap, and further suggests that the failure of the detrended output variant of the new-Keynesian Phillips curve may merely represent the inadequacy of the sticky-price framework that underpins the model. On balance, then, a closer examination suggests that the empirical results in Woodford (2001) do not provide persuasive support for the idea that the labor shareversion of thenew-KeynesianPhillips curveprovides agooddescription of the empirical behavior of price inflation. We next consider a second piece of evidence on this point that has been cited by Woodford and others|namely, the results reported by Sbordone (1998). 3 Interpreting Sbordone’s Evidence Sbordone (1998) examines the same new-Keynesian pricing equation that we have been studying, but does so in a somewhat di(cid:11)erent manner. Sbordone begins by re-writing equation (1) as p t −p t−1 = (cid:12)E t p t+1 −(cid:12)p t+γn t −γp t ; (6) where n t denotes nominal marginal cost. She then notes that this equation|which we have treated as a (cid:12)rst-order di(cid:11)erence equation in inflation with real marginal cost as the driving term|can instead be treated as a second-order di(cid:11)erence equaof x t is related to a set of predetermined instruments. (This method has the advantage of not requiring us to explicitly specify a process for the driving term x t.) In line with the conclusions presented here, Rudd and Whelan (cid:12)nd that the coe(cid:14)cients on lagged inflation are little a(cid:11)ected by including a discounted sum of thedriving variable. 8

tion in the price level with nominal marginal cost as the driving term. This pricelevel equation has a solution of the form: (cid:34) (cid:35) (cid:88)1 p t = (cid:21) 1 p t−1+(1−(cid:21) 1) (1−(cid:21) 2) (cid:21)i 2 E t n t+i ; (7) i=0 where (cid:21) 1 and (cid:21) 2 are obtained from the roots of the characteristic equation of (6). In her empirical implementation, Sbordone re-arranges equation (7) to obtain (cid:34) (cid:35) (cid:88)1 p t = (cid:21) 1 p t−1+(1−(cid:21) 1)n t+(1−(cid:21) 1) (cid:21)i 2 E t(cid:1)n t+i ; (8) i=1 and then constructs forecasts for (cid:1)n t (the rate of change of nominal marginal cost) 6 using a VAR that includes this variable. Sbordone assumes that the labor income share is the appropriate proxy for real marginal cost|implying that (cid:1)n t corresponds to the growth rate of nominal unit labor costs|and (cid:12)nds that her empirical implementation of equation (8) produces an inflation series that tracks observed inflation well. Citing the negative results previously reported by other researchers for the detrended output version of the new-Keynesian pricing equation, Sbordone interprets her equation’s ability to (cid:12)t inflation as indicating that the new-Keynesian model works much better when the labor share is used as an output gap proxy. However, we believe that the results from this exercise should not be considered evidence in favor of thenew-KeynesianPhillips curve, norshouldtheybeconsideredstrongevidenceinfavor ofusingthelaborsharemeasureoftheoutputgapover thedetrended output measure. To see why, (cid:12)rst observe that the price-level equation (8) implies an inflation equation of the form (cid:34) (cid:35) (cid:88)1 (cid:88)1 (cid:25) t = (cid:21) 1 (cid:25) t−1+(1−(cid:21) 1)(cid:1)n t+(1−(cid:21) 1) (cid:21)i 2 E t(cid:1)n t+i − (cid:21)i 2 E t−1(cid:1)n t+i−1 ; (9) i=1 i=1 in which inflation is related to its own lag, unit labor cost growth, and a term that is intended to capture updates to agents’ expectations of future unit labor cost growth (the expression in square brackets). Seen in this light, it is hardlysurprising 6Technically, because the term inside the square bracket in equation (8) starts at i = 1, we measure thisdiscounted sum using A(I−βA) −1Z t instead of (I−βA) −1Z t. 9

that Sbordone’s procedure yields a well-(cid:12)tting inflation series. As we have noted already, lagged inflationisahighlyusefulpredictorof currentinflation|inpractice, even a single lag explains a large fraction of the variation in the series. Moreover, contemporaneousunitlaborcostgrowthalsocontainssomeincrementalexplanatory power for inflation. Thus, even if the rational price-setting posited by the new- Keynesian Phillips curve were entirely incorrect|for example, because agents had backward-looking inflation expectations|we would still expect this procedure to give us an empirically reasonable inflation series. In addition, it is crucial to note that equations (2) and (9) represent two different ways of describing the same theoretical relationship. If the new-Keynesian pricing theory were correct, then equations (2) and (9) would both characterize the determination of prices equally well. Given that our previous results demonstrate that equation (2) generally does very poorly as a model of price inflation no matter which proxy for the output gap is used, the model should be viewed as a poor one, 7 irrespective of the (cid:12)ts generated by empirical implementations of equation (9). Finally, we note that, contrary to previous interpretations, only a very small part of the good (cid:12)t for inflation obtained under Sbordone’s method comes from her use of the labor income share as a proxy for real marginal cost. To illustrate this, we replicate the results from her estimation procedure, and compare them to the results from a parallel exercise in which we use detrended output as the real marginal cost proxy. For the labor share version of the model, nominal marginal cost n t equals unit labor costs, and so we can use the same three-variable VAR that we employed in the previous section in order to generate forecasts for (cid:1)n t (recall that this VAR included unit labor cost growth as one of the variables in the system). For the variant of the model that assumes real marginal cost to be proportional to detrended output, the corresponding measure of nominal marginal cost equals nominal detrended output (de(cid:12)ned as detrended log real GDP plus the log of the price level). Hence, to estimate this version of the model, we remove unit 7One possible critique of this position would be to argue that the empirical implementation of equation (2)|which requires using a VAR to forecast future labor shares|is somehow inferior to implementationsofequation(9). However,itappearsthattheoppositeisthecase. The R(cid:22)2 forthe labor-share equations in the VARs reported in the previous section are signi(cid:12)cantly higher than thosefornominalunitlaborcostinflation,whichisthevariablebeingforecastedwhenequation(8) or (9) is implemented using thelabor share as theoutput gap proxy. 10

labor cost growth from the VAR system and replace it with the (cid:12)rst di(cid:11)erence of this alternative n t measure. Once the two measures of the expected discounted sum of (cid:1)n t are in hand, we can then choose the values of (cid:21) 1 and (cid:21) 2 in equation (8) that 8 yield the best-(cid:12)tting series for inflation. The resulting inflation series are plotted in Figure 4; they demonstrate that Sbordone’s method produces an inflation series that (cid:12)ts well no matter which measure of marginal cost we use. Speci(cid:12)cally, for the labor’s share version of the model (the upper panel of Figure 4), we obtain an R2 for the (cid:12)tted inflation series of 0:80 (with (cid:21) 1 = 0:77 and (cid:21) 2 = 0:72). Likewise, for the model that uses detrended GDP (the lower panel), we obtain an R2 for the (cid:12)tted inflation series of 0:73 (with (cid:21) 1 = 0:92 and (cid:21) 2 = 0:94). While the labor share version of the inflation model (cid:12)ts slightly better than the GDP gap version, the principal message of these (cid:12)g- 9 ures is clearly that both series (cid:12)t well. Again, we emphasize that these results in no way contradict the conclusion of Section 2 that the model does a poor job in capturing the observed dynamics of inflation. Rather, they illustrate that Sbordone’s framework is not well designed for revealing the underlying weakness of the new-Keynesian Phillips curve. 4 Conclusions In this paper, we have assessed the claim that the new-Keynesian Phillips curve performs poorly when detrended real GDP is used as the driving variable, but (cid:12)ts well when real unit labor costs (labor’s share of income) is used. We (cid:12)nd that the robust conclusion that emerges is that neither variable allows the new-Keynesian model to (cid:12)t well. OurrelativelynegativeassessmentiscloselyrelatedtoFuhrerandMoore’s(1995) critique of standard sticky-price models, which highlighted the inconsistency between the forward-looking new-Keynesian inflation equation and the empirical (cid:12)nd- 8This di(cid:11)ers slightly from Sbordone (1998), who chooses these parameters to maximize the (cid:12)t ofthesimulatedprice-unitlaborcost ratio(i.e.,theinverseofthelaborshare). Whileweconsider our choiceof estimation procedureto besomewhat more natural in thecontextwe are discussing, our point|that the (cid:12)t for inflation undereither marginal cost proxy is good when this method is used|holds just as well if we use herapproach to estimate λ 1 and λ 2. 9Thisresult|thatthepredictedinflationseries (cid:12)twellwheneithermeasureofmarginal costis used|is robust across various speci(cid:12)cations of the VARsystem. 11

ing that lags of inflation play an important role in inflation regressions. One way to reconcile these (cid:12)ndings would be to show that lagged inflation proxies for future values of the output gap; alternatively, one could argue that lags of inflation are proxying for expectations of future labor shares. However, the evidence presented in this paper suggests that neither possibility is correct. We (cid:12)nd no evidence that inflation Granger causes the labor share of income, and the discounted sum of current and expected future labor shares generally explains very little of the empirical variation in inflation. Thus, we believe that the evidence provides a (cid:12)rm answer to the question posed in the title: The labor share of income does not appear to drive inflation. Hence, there is little reason to view the labor income share as providing a good measure of the output gap (particularly given that this would imply that every postwar U.S. recession has actually been a boom relative to the prevailing level of potential output). Similarly, a compelling case cannot be made for replacing conventional output gap measures with the labor share in a monetary policy rule. Finally, we note that our conclusions should not be interpreted as implying that forward-looking inflation models based on real marginal cost cannot work, since it may be that both of the driving variables considered here are actually very poor proxies for marginal cost. For example, Rotemberg and Woodford (1999) detail a number of reasons|such as the existence of overhead labor, overtime premia, and adjustment costs for labor|why real marginal cost could be procyclical even though real unit labor costs are not. Thus, the increases in average cost that are observedduringrecessionsarelikelytobepoorindicatorsofmarginalcostpressures. On balance, then, we conclude that it remains possible that some forward-looking model based on a measure of real marginal cost provides a good description of the inflation process, but this conjecture can by no means be considered proven. 12

References [1] Fuhrer, Je(cid:11)rey C. and George R. Moore (1995). \Inflation Persistence," Quarterly Journal of Economics, 110, 127-159. [2] Gal(cid:19)(cid:16), JordiandMarkGertler(1999).\InflationDynamics: AStructuralEconometric Analysis," Journal of Monetary Economics, 44, 195-222. [3] Goodfriend, Marvin and Robert King (2001). \The Case for Price Stability," NBER Working Paper No. 8423. [4] Rotemberg, Julio and Michael Woodford (1999). \The Cyclical Behavior of Prices andCosts," in JohnTaylor and Michael Woodford (eds.), The Handbook of Macroeconomics, North-Holland. [5] Rudd, Jeremy and Karl Whelan (2001). \New Tests of the New-Keynesian Phillips Curve," Federal Reserve Board, Finance and Economics Discussion Series Paper No. 2001-30. [6] Sargent, Thomas J. (1987). Macroeconomic Theory (2nd edition), Academic Press. [7] Sbordone, Argia (1998). \Prices and Unit Labor Costs: A New Test of Price Stickiness," Journal of Monetary Economics, forthcoming. [8] Sbordone, Argia (2001). \An Optimizing Model of U.S. Wage and Price Dynamics," March draft, Rutgers University. [9] Woodford, Michael (2001). \The Taylor Rule and Optimal Monetary Policy," American Economic Review, 91(2), 232-237. 13

Table 1: Results from Alternative VAR Forecasting Models for Labor’s Share VAR speci(cid:12)cations       s   s s t [s t] (cid:34) (cid:1)u s t lc t (cid:35) (cid:34) y s t t (cid:35)  (cid:1)u y s t lc t       (cid:1)u y t t lc t         (cid:1) y u t t lc t           (cid:1) h u y t lc t       t h c =y t t t t c =y t t A. R2 from inflation equation 0.162 0.437 0.129 0.014 0.040 0.040 0.001 B. Exclusion restriction p-values (s t equation) s t 0.000 0.000 0.000 0.000 0.000 0.000 0.000 (cid:1)ulc t 0.288 0.901 0.483 0.559 0.480 y t 0.000 0.000 0.019 0.018 0.614 h t 0.252 0.133 c t =y t 0.118 0.063 C. R(cid:22)2 from labor’s share VAR equation 0.848 0.849 0.864 0.862 0.863 0.864 0.866 D. Sum of lagged (cid:25) coe(cid:14)cients in inflation equation (if included) 0.871 0.766 0.855 0.890 0.883 0.882 0.898 Key: s t (cid:17) labor’s share, (cid:1)ulc t (cid:17) unit labor cost growth, y t (cid:17) detrended output, h t (cid:17) detrended hours, c t =y t (cid:17) detrended consumption-output ratio. See text for additional details. 14

Table 2: Results from VAR Forecasting Models That Include Inflation VAR speci(cid:12)cations       s   s s t     s t t  (cid:25)  (cid:34) (cid:25) s t t (cid:35)   (cid:1) (cid:25) u s t t lc t    (cid:25) y s t t t      (cid:1) (cid:25) u y t t lc t           (cid:1) (cid:25) y u t t lc t             (cid:1) (cid:25) u y t t lc t              (cid:1) h y u t t lc t        t h c =y t t t t c =y t t A. R2 from inflation equation 0.415 0.364 0.001 0.026 0.113 0.121 0.040 B. Exclusion restriction p-values (s t equation) s t 0.000 0.000 0.000 0.000 0.000 0.000 0.000 (cid:25) t 0.199 0.401 0.921 0.631 0.501 0.925 0.965 (cid:1)ulc t 0.518 0.635 0.564 0.662 0.555 y t 0.000 0.000 0.018 0.019 0.372 h t 0.463 0.273 c t =y t 0.297 0.186 C. R(cid:22)2 from labor’s share VAR equation 0.849 0.849 0.862 0.863 0.863 0.864 0.864 D. Sum of lagged (cid:25) coe(cid:14)cients in inflation equation (if included) 0.780 0.800 0.897 0.887 0.869 0.861 0.883 Key: s t (cid:17) labor’s share, (cid:25) t (cid:17) inflation, (cid:1)ulc t (cid:17) unit labor cost growth, y t (cid:17) detrended output, h t (cid:17) detrended hours, c t =y t (cid:17) detrended consumption-output ratio. Lag lengths chosen using Schwarz criterion. See text for additional details. 15

Figure 1 Output Gap Concepts, U.S. Nonfarm Business Sector (NBER Recession Dates Shaded) A. Labor Income Share 0.68 0.67 0.66 0.65 0.64 0.63 0.62 0.61 0.60 1960 1965 1970 1975 1980 1985 1990 1995 2000 B. Quadratically Detrended Log GDP 0.08 0.06 0.04 0.02 0.00 -0.02 -0.04 -0.06 -0.08 -0.10 -0.12 1960 1965 1970 1975 1980 1985 1990 1995 2000 16

Figure 2 Actual and Predicted Inflation--Present-Value Method (VAR models include GDP gap, labor’s share, and unit labor cost growth) A. Present Value of GDP Gaps from VAR System 0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.005 0.000 -0.005 1960 1965 1970 1975 1980 1985 1990 1995 2000 B. Present Value of Labor Income Shares from VAR System 0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.005 0.000 -0.005 1960 1965 1970 1975 1980 1985 1990 1995 2000 Note: Actual inflation given by solid line; predicted inflation given by dashed line. 17

Figure 3 Actual and Predicted Inflation--Present-Value Method (alt. VAR) (VAR model includes labor’s share and unit labor cost growth only) Present Value of Labor Income Shares from VAR System 0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.005 0.000 -0.005 1960 1965 1970 1975 1980 1985 1990 1995 2000 Note: Actual inflation given by solid line; predicted inflation given by dashed line. 18

Figure 4 Actual and Predicted Inflation--Sbordone Method (VAR models include GDP gap, labor’s share, and ULC or nominal GDP gap growth) A. Price-Level Equation Using Expected ULC Growth 0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.005 0.000 -0.005 1960 1965 1970 1975 1980 1985 1990 1995 2000 B. Price-Level Equation Using Expected Nominal GDP Gap Growth 0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.005 0.000 -0.005 1960 1965 1970 1975 1980 1985 1990 1995 2000 Note: Actual inflation given by solid line; predicted inflation given by dashed line. 19