New Tests of the New-Keynesian Phillips Curve

New Tests of the New-Keynesian Phillips Curve Jeremy Rudd and Karl Whelan Division of Research and Statistics Federal Reserve Board (cid:3) June 26, 2001 Abstract Is the observed correlation between current and lagged inflation a function of backward-looking inflation expectations, or do the lags in inflation regressions merely proxy for rational forward-looking expectations, as in the new- KeynesianPhillips curve? Recent researchhas attempted to answer this question by using instrumental variables techniques to estimate \hybrid" speci(cid:12)cations for inflation that allow for e(cid:11)ects of lagged and future inflation. We show that these tests of forward-lookingbehavior have very low power against alternative, but non-nested, backward-lookingspeci(cid:12)cations, and demonstrate that results previously interpreted as evidence for the new-Keynesian model are also consistent with a backward-looking Phillips curve. We develop alternative, more powerful tests, which (cid:12)nd a very limited role for forward-looking expectations. JEL classi(cid:12)cation code: E31 Keywords: Inflation, Phillips curve ∗Correspondence: JeremyRudd,FederalReserveBoard,MailStop80,20thandCStreetsNW, Washington, DC 20551; e-mail: jeremy.b.rudd@frb.gov; tel.: (202) 452-3780; fax: (202) 872-4927. We are grateful to David Bowman, Doug Elmendorf, Vincent Hogan, David Lebow, Jesper Lind(cid:19)e, Dan Sichel, and Jonathan Wright for helpful comments on an earlier draft of the paper. The views expressed in this paperare our own and donot necessarily reflect the views of the Board of Governors or the sta(cid:11) of theFederal ReserveSystem.

1 Introduction An important stylized fact in macroeconomics is that the U.S. inflation process is well described by the reduced-form Phillips curve (cid:25) t = (cid:11)(L)(cid:25) t−1 +γ(L)x t+u t ; (1) where inflation, (cid:25) t, is modelled as a function of its lags and a measure of excess demand x t (such as the output gap or the unemployment rate). Despite this equation’s ability to characterize historical inflation behavior, there is no real consensus regarding its structural interpretation|in particular, the presence of lagged inflation has been interpreted in at least two very di(cid:11)erent ways. In the (cid:12)rst, more traditional view, agents formulate their expectation of this period’s inflation rate in a backward-looking manner, which causes past inflation rates to become directly incorporated into current wage and price contracts. Under this view, equation (1) is a structural relationship, with the lags of inflation proxying for E t−1 (cid:25) t. An alternative, more modern interpretation comes from the rational expectations staggered-contracting models of Taylor (1980) and Calvo (1983). As shown by Roberts (1995), these models imply a so-called new-Keynesian Phillips curve in which current inflation incorporates a forward-looking component: (cid:25) t = (cid:12)E t (cid:25) t+1+γx t : (2) This model provides a di(cid:11)erent interpretation of the reduced-form relationship betweencurrentandlaggedvaluesofinflation: Laggedinflationappearstomatteronly because it is correlated with the rational expectation of next period’s inflation rate. Although super(cid:12)cially similar to the traditional Phillips curve, the new-Keynesian Phillips curve carries very di(cid:11)erent implications for such practical questions as the optimal conduct of monetary policy and the cost of disinflation. Jordi Gal(cid:19)(cid:16) and Mark Gertler (1999) have recently attempted to distinguish between these two interpretations by estimating a \hybrid" speci(cid:12)cation of the form (cid:25) t = !fE t (cid:25) t+1+!b(cid:25) t−1+γx t : (3) This model can be re-written as (cid:25) t = !f(cid:25) t+1+!b(cid:25) t−1+γx t+(cid:15) t+1 ; (4) 1

where (cid:25) t+1 is actual (realized) inflation at time t+1 and (cid:15) t+1 is an expectational error. 1 Underrationalexpectations, thiserrorshouldbeunforecastableattimet,so !f canbeconsistentlyestimated usingvariablesdatedtorearlierasinstrumentsfor (cid:25) t+1. Employing this technique, Gal(cid:19)(cid:16) and Gertler estimated !f to be signi(cid:12)cantly larger than !b , and interpreted this result as implying that the purenew-Keynesian curve provides a good approximation to the true inflation process. Thispaperpresentsanewapproachtotestingthenew-KeynesianPhillipscurve. We motivate the usefulness of the new tests by (cid:12)rst documenting a potentially serious problem with Gal(cid:19)(cid:16) and Gertler’s estimates, which is that small speci(cid:12)cation errors in equation (4) can cause their procedure to yield highly misleading results. In particular, high estimates of !f and low estimates of !b turn out to be fully consistent with the true model’s being a purely backward-looking speci(cid:12)cation. Intuitively, this situation obtains when a variable z t that belongs in the true model for inflation is erroneously omitted from the test speci(cid:12)cation. In this case, the regression error in equation (4) is no longer a pure expectational error, because it alsoincludestheinfluenceofz t oninflation. Estimatesof!f willthereforebebiased upwards as long as (cid:25) t+1 and the variables used to instrument for it are both correlated with z t. In practice, it turns out to be highly plausible that these conditions will be met, and that the magnitude of the bias will be large. In light of this potential problem with tests based on estimation of equation (4), we develop an alternative testing procedure. Speci(cid:12)cally, we focus on direct estimation of the new-Keynesian model’s closed-form solution, which predicts that inflation should be determined by the expected present discounted sum of future values of the \driving" variable x t: X1 (cid:25) t = γ (cid:12)iE t x t+i : (5) i=0 (This result is obtained from repeated substitution of equation 2.) We also present other tests that augment this speci(cid:12)cation with lags of inflation. If Gal(cid:19)(cid:16) and Gertler’s test equation is correctly speci(cid:12)ed, then the present-value tests thatweconstructareequivalent to theirtests, andshouldyield similar results. However, our tests turn out to be far less likely to spuriously indicate the presence of forward-looking behavior when such behavior is not present. This is because in 1That is, (cid:15) t+1 is proportional to (E t π t+1 −π t+1). 2

this case, the term being instrumented for in the present-value test will not have a high correlation with variables that have been omitted from the inflation equation. Our new tests strongly reject the new-Keynesian interpretation of the reducedform Phillips curve. The reason for this result is quite simple. From equation (5), we can see that if the new-Keynesian model were correct, then the lags of inflation that enter the reduced-form Phillips curve must be proxying for the rational expectation of future values of the driving variable x t. However, our analysis rejects this conjecture. We show that lags of inflation actually have very little predictive power for futurevalues of x t, a resultthat weobtain whetherwe de(cid:12)nex t to betheoutput gap (as is commonly done) or labor’s shareof income (as has been proposedby Gal(cid:19)(cid:16) and Gertler). Not surprisingly, then, we (cid:12)nd that the estimated e(cid:11)ects of lagged dependent variables in inflation regressions are very similar whether or not we include measures of the expected present value of the driving variable. A key point is that these results imply that the new-Keynesian model should not be viewed as supplyinganalternativeexplanationfortheempiricalimportanceoflaggedinflation in reduced-form inflation equations; rather, this stylized fact should be considered strong evidence against the new-Keynesian model. We conclude from these results that Gal(cid:19)(cid:16) and Gertler’s estimates are likely the result of model misspeci(cid:12)cation, and also that the new-Keynesian model provides a poor description of the U.S. inflation process. 2 Assessing Previous Tests of the New-Keynesian Model In this section, we discuss the properties of instrumental variables (IV) estimates of equation (4). We begin by presenting a set of baseline results that, on the surface, appear to indicate that forward-looking behavior is an important component of the inflation process. We then demonstrate that these results could also be obtained even if the true process for inflation does not involve forward-looking behavior. 2.1 Empirical Estimates of the Hybrid Equation As we noted above, so long as equation (3) is a true representation of the inflation process,thentheonlyerrorsthatwillobtainfromestimatingequation(4)|inwhich realized (cid:25) t+1 is substituted for E t (cid:25) t+1|will be (rational) expectational errors. In 3

this case, any variable dated t or earlier should be orthogonal to the regression model’s error term, and therefore constitutes a valid instrument for E t (cid:25) t+1. Table 1 reports the results that we obtain from (cid:12)tting several di(cid:11)erent versions of equation (4). Like Gal(cid:19)(cid:16) and Gertler, we estimate the equation with GMM, employing the same set of instruments that they did (namely, four lags of each of the following variables: price inflation, labor’s share of income, the output gap, the spread between long and short interest rates, compensation inflation, and commod- 2 ity price inflation). Coe(cid:14)cient standard errors are reported in parentheses in the table, and are estimated with a 12-lag Newey-West covariance matrix. Estimation is quarterly, and ranges from 1960:Q1 to 1997:Q4. ThefoursetsofresultsreportedinTable1usetwodi(cid:11)erentmeasuresofinflation andtwodi(cid:11)erentde(cid:12)nitionsofthedrivingvariablex t. Themeasuresofinflationthat weconsiderarethelog-di(cid:11)erenceoftheGDPchainpriceindexandthelog-di(cid:11)erence of the chain price index for nonfarm business (NFB) output. The NFB deflator is ourpreferredpriceconceptbecauseitexcludespricesforgovernmentandhouseholdsector output and so is more likely to reflect the type of optimizing private-sector behavior that is posited bythe sticky-price models that underliethenew-Keynesian model; we also report results for the GDP deflator to permit comparison with previousstudies. Thetwomeasuresofx t thatweemployaretheoutputgap(de(cid:12)ned as the quadratically detrended log of real GDP) and the share of labor income in production costs for nonfarm business. The output gap is a standard activity variable thatis commonly employed inPhillips curveregressions; however, Gal(cid:19)(cid:16) and Gertler havesuggestedtheuseofthelaborincomeshare,whichcanbeequatedwith 3 real marginal cost under certain restrictive assumptions. Because we estimate equation (4) directly, our approach di(cid:11)ers a little in its particulars from Gal(cid:19)(cid:16) and Gertler’s: They derive the new-Keynesian curve from a Calvo-style contracting model, and then use the estimated parameters from this modeltobackoutestimatesof!f ,!b ,andγ. Directestimationhastheadvantageof 2Wede(cid:12)nedthelong-shortspreadasthedi(cid:11)erencebetweenthe10-yearconstant-maturityTreasurybondyieldandtherateonthree-monthTreasurybills. Commoditypriceinflationwasde(cid:12)ned as thelog di(cid:11)erenceof theProducer Price Indexfor crude materials. 3Speci(cid:12)cally, we use the log of the ratio of the labor income share to its sample mean. This captures the notion that the driving variable should be expressed in terms of log deviations from its steady-state level. 4

allowing easier interpretation in that the new-Keynesian Phillips curve is consistent with a broad class of sticky-price models. In practice, however, this choice has little substantive implication: When we implement Gal(cid:19)(cid:16) and Gertler’s procedure, we obtain estimates for !f , !b , and γ that are essentially identical to those that we report here. Our estimates of the e(cid:11)ects of the driving variables are a little di(cid:11)erent from Gal(cid:19)(cid:16) and Gertler’s. Like them, we (cid:12)nd that the coe(cid:14)cients on the output gap in this speci(cid:12)cation are insigni(cid:12)cant and have the \wrong" sign. Unlike them, we do not (cid:12)nd the labor income share to be statistically signi(cid:12)cant either, although it does have the correct (positive) sign. However, concerning the main point of interest|the relative sizes of !f and !b |the results in Table 1 are very similar to those obtained by Gal(cid:19)(cid:16) and Gertler. Speci(cid:12)cally, we (cid:12)nd estimates of !f that are much higher than those for !b , particularly for our preferred price concept, the NFB deflator. 2.2 The E(cid:11)ect of Misspeci(cid:12)cation on the Hybrid Model Estimates Takenatfacevalue,theresultsinTable1suggestthatthenew-Keynesianconjecture| that the presence of lagged inflation in the reduced-form Phillips curve reflects its correlation with expected future inflation|is essentially correct, and that the pure new-Keynesian model with !b = 0 may be a good approximation to reality. This conclusion is reasonable as long as the test speci(cid:12)cation nests the true model. It turns out, however, that small misspeci(cid:12)cations can result in highly misleading parameter estimates. We can demonstrate this as follows. Assume that the true process for inflation is a backward-looking Phillips curve of the form (cid:25) t = (cid:12)(cid:25) t−1+(cid:21)x t+(cid:22)z t+u t ; (6) where z t denotes a vector of additional determinants of inflation. For simplicity, assumealso that thex t andz t terms are de(cid:12)nedsuch that (cid:21)and(cid:22) are bothpositive (note that (cid:12) is invariably positive in Phillips curve speci(cid:12)cations like this one). Suppose now that we (cid:12)t equation (4) using GMM, and that the instruments used to (cid:12)t the model include z t|that is, variables that actually belong in the true model for inflation. Such a situation is quite plausible in this context. Recall that the instruments used by Gal(cid:19)(cid:16) and Gertler included additional lags of inflation and 5

the outputgap and lags of commodity price inflation|variables that are commonly 4 employedinempiricalinflationequations. However,beyondthesespeci(cid:12)cvariables, it seems generally likely that instrument selection for this equation will be highly problematic. A good instrument for this problem|in the sense that it allows !f to be identi(cid:12)ed|will be a variable that is correlated with the portion of (cid:25) t+1 that is orthogonal to (cid:25) t−1 and x t. As a result, it is hard to think of a good instrument for (cid:25) t+1 (dated t and earlier) that could not also plausibly be included in a model for (cid:25) t. 5 Because equation (4) is a linear model, the GMM procedure employed here is 6 the same as two-stage least squares. Speci(cid:12)cally, an equivalent estimation scheme begins by using the (cid:12)tted values from a (cid:12)rst-stage regression of (cid:25) t+1 on (cid:25) t−1, x t, and z t in order to construct a proxy for expected future inflation. 7 These (cid:12)tted values (cid:25)^t+1, which can be written as (cid:25)^t+1 = (cid:14)^ 1 (cid:25) t−1+(cid:14)^ 2 x t+(cid:14)^ 3 z t ; (7) are then used in a second-stage regression that relates current inflation to expected future inflation (proxied for by (cid:25)^t+1), lagged inflation, and a driving variable x t: (cid:25) t = !^ f(cid:25)^t+1+!^ b(cid:25) t−1+γ^x t+(cid:15) t : (8) Combining the (cid:12)rst- and second-stage equations allows us to rewrite the hybrid model (8) as: (cid:16) (cid:17) (cid:16) (cid:17) (cid:25) t = !^ f(cid:14)^ 1+!^ b (cid:25) t−1+ !^ f(cid:14)^ 2+γ^ x t+!^ f(cid:14)^ 3 z t+(cid:15) t : (9) If we compare equation (9) with the assumed true model (6), it is apparent that we can express the probability limits of the estimated parameters from the hybrid 4SeeStaiger, Stock,and Watson (1997) and Gordon (1998) for typical implementations. 5In theory, one might imagine that tests of overidentifying restrictions could be used to detect invalid instruments. However, as we discuss below, it is unlikely that these tests will actually be able to do so. 6SeeHamilton (1994, pp 420-421) for a discussion. 7Note that the empirical estimates reported in the previous subsection were obtained using instrumentsdated time t−1 or earlier. (Whileit would havebeen valid to usetime-t variables as instruments,weemployedthemorerestrictedinstrumentsetinordertomatchtheprocedurethat Gal(cid:19)(cid:16) and Gertler followed in obtaining their results.) Here, we have assumed that the instrument set contains x t as well as elements of z dated time t or earlier; note that this is done merely to simplify theexposition, and makes no real di(cid:11)erence to theargument. 6

model in terms of the parameters of the true model: (cid:16) (cid:17) plim !^ b +!^ f(cid:14)^ 1 = (cid:12); (10) (cid:16) (cid:17) plim γ^+!^ f(cid:14)^ 2 = (cid:21); (11) plim !^ f(cid:14)^ 3 = (cid:22): (12) To determine the signs of the biases in the estimated coe(cid:14)cients, note that because inflation is highly autocorrelated, it is likely that the estimated e(cid:11)ects on (cid:25) t+1 of lagged inflation, thedrivingvariable x t, andthez t termswillhave thesamesigns as their respective e(cid:11)ects on (cid:25) t. In other words, the coe(cid:14)cients (cid:14)^ 1, (cid:14)^ 2, and (cid:14)^ 3 from the (cid:12)rst-stage regression will typically have the same (positive) sign as the coe(cid:14)cients (cid:12), (cid:21), and (cid:22) from the true model. It is evident from equation (12), therefore, that in general the estimated coe(cid:14)cient !^ f will be positive, even though the \true" model does not include a forward-looking component. Note also thattheestimated values of !^ b andγ^ willbebiased downwardrelative to the true coe(cid:14)cients on (cid:25) t−1 and x t. Intuitively, the reason for this bias is that the influence of lagged inflation and the output gap on current inflation is already partly captured by (cid:25)^t+1. If this term receives a positive weight, then (cid:25) t−1 and x t 8 will be crowded out of the second-stage regression. Forsimplicity, wehaveassumedherethattheinstrumentsfor(cid:25) t+1 wereidentical to the z t terms that were erroneously omitted from the estimated model. However, it is easy to see that the same conclusions will hold if we use instruments that are merely correlated with z t. Again, this is a plausible scenario: Even if we do not think that our instruments truly belong in the inflation speci(cid:12)cation, to be good instruments for this problem they will need to be correlated with the part of (cid:25) t+1 that is orthogonal to (cid:25) t−1 and x t. Almost by de(cid:12)nition, such variables are likely to be correlated with the omitted z t. An Analytical Example: The preceding argument establishes the direction of the biases that result from misspeci(cid:12)cation. It is also important to note that the magnitude of thesebiases maybelarge. For instance, considerthecase inwhich the 8These results help to explain Roberts’s (2001) (cid:12)nding that estimates from this equation are sensitive to theinclusion of lagged inflation in the instrument set. When this variable is excluded we haveδ^ 1 =0, so theestimated ωb will not bebiased down so much. 7

true model for inflation is given by (6), and x t and z t follow AR(1) processes with autoregressive parameters (cid:26) x and (cid:26) z. We can then derive analytical expressions for the probability limits of the estimated parameters; as we demonstrate in an appendix, these are given by: 1 plim !^ f = ; (13) (cid:12) +(cid:26) z (cid:12)(cid:26) plim !^ b = z ; (14) (cid:12) (cid:18) +(cid:26) z (cid:19) (cid:26) −(cid:26) plim γ^ = (cid:21) z x : (15) (cid:12)+(cid:26) z These results imply that plausible degrees of persistence in x t, z t, and (cid:25) t can yield estimates of !f that are large relative to those for !b , even when inflation is truly characterized by the backward-looking model (6). For example, if (cid:26) z = (cid:26) x = (cid:12) = 0:8, then the probability limits for the estimated parameters are !^ f = 0:625, !^ b = 0:4, and γ^ = 0|results that are quite similar to those reported in Table 1 9 (and in Gal(cid:19)(cid:16) and Gertler’s paper). It is particularly interesting to note that small or negative estimated coe(cid:14)cients on the driving variable, x t|such as we found for the output gap|could be obtained even if this variable has a large positive e(cid:11)ect in the true model. Tests of Overidentifying Restrictions: As we have described it, the problem with testing equation (4) is one of potential model misspeci(cid:12)cation|we may have omitted variables from the estimated model that belong in the correct speci(cid:12)cation, and then compounded this error by using these omitted variables (or variables that are correlated with them) as instruments. In principle, one might hope that the Hansen (1982) test of the model’s overidentifying restrictions could reveal the presenceofthisproblem(notethatforthemodelsestimatedinTable1,thistestdoes not reject the overidentifying restrictions). However, it is well known that this test tends to have low power; hence, at best the results from this test can provide only 9We used monte carlo simulations to con(cid:12)rm that for the sample sizes used in this paper, empiricalestimatesoftheparameterswilltendonaveragetobeclosetotheanalyticalprobability limits just derived. We also used monte carlo exercises to con(cid:12)rm that we obtain similar results when we employ a model for π t and higher-order AR processes for x t and z t that are calibrated from thedata. 8

10 limited evidence against misspeci(cid:12)cation. In fact, beyond this general problem, it turns out that the test will be particularly ine(cid:11)ective in detecting the type of misspeci(cid:12)cation that we are considering. To see this, note that for linear models the Hansen test is based on the size of the R2 from a regression of the model’s residuals (which in this case are equal to (cid:25) t −!^ f(cid:25) t+1 −!^ b(cid:25) t−1 −γ^x t) on the instrument set. 11 If the R2 is \too high," then this is interpreted as evidence that the instruments are invalid. Here, however, the e(cid:11)ect of the variables that are omitted from the estimated model are captured indirectly through the(cid:25) t+1 term. Since themanner in which these variables explain (cid:25) t+1 is likely to be similar to how they explain (cid:25) t, much of their influence on (cid:25) t will actually be incorporated in the estimated speci(cid:12)cation. As a result, in this case the model will typically \pass" a Hansen test. We have con(cid:12)rmed using monte carlo exercises that tests of the overidentifying restrictions invariably fail to detect the presence of this type of misspeci(cid:12)cation. Furthermore, in the appendix we show that for the example just discussed in which xandzareAR(1),theR2 underlyingtheHansenteststatisticshouldasymptotically tend toward zero, even though the model is misspeci(cid:12)ed by construction. Summary: The results of this section demonstrate that instrumental variables estimation of equation (4) cannot provide convincing evidence that inflation expectations are forward-looking in nature. Our results thus far do not preclude the possibility that the (cid:25)^t+1 series is indeed a good proxy for agents’ rational expectations of future inflation, and therefore that the high value of !f reported by Gal(cid:19)(cid:16) and Gertler (and in our Table 1) truly reflects the important role played by such expectations in the inflation process. However, they do imply that these results are also perfectly consistent with the traditional, backward-looking interpretation of the Phillips curve, and therefore that it is impossible to use this test to distinguish 12 this theory of inflation dynamics from one based on backward-looking behavior. 10Newey (1985) presentsgeneral results regarding thepower of theHansen test. 11This was noted by Newey (1985), who based his argument on an earlier result by Hausman (1984, pp 432-33). 12While we have focused on the problems related to instrumental variables estimation of equation (3),weshould notethattheissues highlighted herealso applytoanotherpopularmethodfor estimatingthisequation,inwhichasurveymeasureisusedasaproxyforE t π t+1(see,forexample, 9

3 Tests Based on the Closed-Form Solution Inthissection,wedevelopnewteststhatpermitustodiscriminatebetweenthecompeting interpretations of the correlations summarized by the reduced-form Phillips curve. These tests achieve this goal by focusing more precisely on how inflation expectations are determined in the new-Keynesian model. 3.1 The Closed-Form Inflation Equation Consider again the new-Keynesian Phillips curve in its simplest form (equation 2, reproduced here): (cid:25) t = (cid:12)E t (cid:25) t+1+γx t : The tests that we have discussed thus far have characterized this model based only on a very weak property of rational expectations|speci(cid:12)cally, that the expectational error ((cid:25) t+1 −E t (cid:25) t+1) should be unforecastable by variables dated t or earlier. However, another feature of rational expectations is that they should be modelconsistent: Expectations for next period’s inflation rate should be consistent with the process for inflation as described by the model. This additional prediction yields testable implications beyond those that we have already considered. More importantly for our purposes, it turns out that tests of these implications are much less likely to mislead when the type of misspeci(cid:12)cation that we discussed above is present. Our new tests are derived from the following observation. If the new-Keynesian model is correct, then rational agents know that E t (cid:25) t+1 = (cid:12)E t (cid:25) t+2+γE t x t+1 ; E t (cid:25) t+2 = (cid:12)E t (cid:25) t+3+γE t x t+2 ; and so on. Repeated substitution of these and higher terms yields the following closed-form solution for current inflation: X1 (cid:25) t = γ (cid:12)iE t x t+i : (16) i=0 Roberts,1995). Supposethatinflationwereinfluencedbyavariablezthatwasobservedbyprivate agentsbutnotincludedintheestimatedmodel. Inthiscase,asurveymeasureofexpectedinflation would receive a large weight in equation (3) even if thenew-Keynesian model were false. 10

Our strategy is to test whether (16) provides a good characterization of the inflation process. If it does, then this would con(cid:12)rm that the lags of inflation in reduced-form Phillips curves are merely proxying for forward-looking expectations. In addition to estimating (16), we also estimate speci(cid:12)cations of the form X1 (cid:25) t = A(L)(cid:25) t−1+γ (cid:12)iE t x t+i : (17) i=0 This can be interpreted as combining elements of both the new-Keynesian Phillips curve and thetraditional backward-looking model; alternatively, it can beexplicitly derived from a modi(cid:12)ed staggered-pricing model in which the economy includes 13 both forward- and backward-looking agents. Whichever interpretation one wants to give this equation, if the pure new-Keynesian model is a good proxy for the empirical inflation process, then we should obtain relatively small coe(cid:14)cients on lagged inflation. When comparing the present-value tests with those reported in the previous section, the (cid:12)rst thing to note is that, if the results in Table 1 are to be interpreted as indicating an important role for forward-looking behavior (as opposed to model misspeci(cid:12)cation), then these tests should yield equivalent results. If the new-Keynesian model is correct, then the forward-looking behavior that showed up in the previous tests as a large coe(cid:14)cient on the proxy for E t (cid:25) t+1 should show up here in the form of a signi(cid:12)cant role for the present-value term. The only case in which this would not obtain would be if we had a poor proxy for the present-value variable; in the context of our estimation procedure, this would reflect the presence of a weak-instrument problem. However, as we will show below, the instruments for the present-value term have good (cid:12)rst-stage (cid:12)t. Thus, if we fail to (cid:12)nd a significant role for forward-looking behavior with this test, then we can reject the pure new-Keynesian model as a good approximation to reality. The second important aspect of these tests is that they provide a check on whether the results in Table 1 are due to the omission of one or more variables from the true model for inflation. When such variables are omitted, the presentvalue tests will not suggest forward-looking behavior if the true model is purely 13SeeGal(cid:19)(cid:16)andGertler(1999)foranexample. (Notethatinthiscaseβ inequation17represents the inverse of the unstable root of the expectational di(cid:11)erence equation implied by the hybrid model.) 11

backward looking. To see this, note that unlike the previous tests|in which we were instrumenting for (cid:25) t+1|we have no a priori reason to expect that variables omitted from the inflation equation will be highly correlated with the present-value term that we are instrumenting for here. The explanation is simple: If the new- Keynesianmodeliswrong,thentherelationshipbetweenthepresent-valuetermand theinstrumentsshouldbequitedi(cid:11)erentfromtherelationshipbetweeninflationand the instruments. Thus, even if we have omitted some variables that are useful for explaining inflation, this test will not automatically conclude that there is forwardlooking behavior if none actually exists. 3.2 Empirical Results ImplementingtheModel: Tomakeequations(16)and(17)empiricallytractable, we need to be able to construct a proxy for the in(cid:12)nite discounted sum of the expected future values of the driving variable. We do this by making use of the fact that X1 E t (cid:25) t+K+1 =γ(cid:12)−(K+1) (cid:12)iE t x t+i ; i=K+1 which allows us to truncate the in(cid:12)nite sum in equation (16) in some period t+K, and write the closed-form solution for inflation in terms of K+1 current and future values of the driving variable and a terminal inflation term: XK (cid:25) t = γE t (cid:12)ix t+i+E t (cid:12)K+1(cid:25) t+K+1 : (18) i=0 Similarly, we can make equation (17) tractable by noting that X1 E t (cid:25) t+K+1 = A(L)E t (cid:25) t+K +γ(cid:12)−(K+1) (cid:12)iE t x t+i ; i=K+1 which allows us to rewrite the closed-form solution for inflation as: XK (cid:25) t = A(L)(cid:25) t−1 +γE t (cid:12)ix t+i+E t (cid:12)K+1 [(cid:25) t+K+1 −A(L)(cid:25) t+K]: (19) i=0 PK Given a value for (cid:12), we can construct the actual (realized) values of (cid:12)ix t+i and i=0 the terminal inflation components. Again, if the new-Keynesian model is correct, thenthesetermswillequaltheirtime-texpectationstogetherwithanunforecastable 12

(rational) expectational error. Thus, we can also estimate equations (18) and (19) using instrumental variables techniques. For the results that we discuss, we used GMM to estimate γ based on an assumed value of 0.95 for (cid:12) and a value for K of 12 quarters. We also examined a number of alternative values for (cid:12) ranging from 0.80 to 0.99, as well as values for K that range from eight to twenty quarters; the results were essentially the same as 14 those described here. Because our dataset ends in 2000:Q2 and we require data for twelve quarters after the end of the estimation period, we estimate the model through 1997:Q2. The instrument set is the same as that employed in the previous tests. As before, we considered versions of the model that use either the GDP gap or the labor-income share as a driving variable, and that de(cid:12)ne inflation as the rate of change of either NFB or total GDP prices. Estimates: Table 2 summarizes the results obtained from (cid:12)tting equations (18) and (19). As a way of quantifying the new-Keynesian model’s ability to explain the data, we also report results from simple regressions of inflation on its own lags 15 only. If the new-Keynesian interpretation of the reduced-form Phillips curve is correct, then the inclusion of the present-value term should result in a substantial reduction in the coe(cid:14)cients on lagged inflation relative to those obtained from the purely backward-looking speci(cid:12)cations. Taken together, the results provide little support for the view that forwardlooking behavior is an important determinant of inflation dynamics. Beginning with the speci(cid:12)cation that uses the output gap as the driving variable, we see that γ (the coe(cid:14)cient on the present-value term) does not even receive the correct sign in the version of the model that excludes any lags of inflation. When we incorporate the lagged inflation term A(L)(cid:25) t−1, we obtain further evidence that the forward-looking behavior posited by the new-Keynesian Phillips curve is of limited empirical importance: Although γ receives the correct (positive) sign in 14For reference, Gal(cid:19)(cid:16) and Gertler’s empirical estimates of β range from around 0.93 for GDP price inflation to around 0.97 for NFB inflation. 15Ourspeci(cid:12)cimplementationofequation(19)hastwolagsofinflation(theresultsareessentially the same if we include additional lags). Similarly, the \own lags only" estimates reported in the table are from models that include two lags of inflation; the results are basically unchanged if we include additional lags, or additional drivingvariables such as theoutput gap. 13

these speci(cid:12)cations, the sum of the coe(cid:14)cients on lagged inflation (labelled \A(1)" inthetable)isabout0.9forbothGDPandNFBpriceinflation. And,inbothcases, these estimates arevirtuallyidentical tothecoe(cid:14)cients onlagged inflation fromthe autoregressions. Thus, we (cid:12)nd essentially no evidence that the role assigned to lags of inflation in reduced-form Phillips curves comes from their proxying for expected future output gaps, as the new-Keynesian model suggests. What about the version of the model that uses the labor income share as the driving variable? The pure new-Keynesian speci(cid:12)cation with this driving variable does a bit better inasmuch as the coe(cid:14)cient on the present-value term receives the correct sign. Nevertheless, even when we include this present-value term, the resulting weight given to lagged inflation remains very high, at about 0.8 for both GDP and NFB price inflation. These estimates are only a little lower than the 0.9 obtained for the pure autoregressive model, suggesting a very limited role for expected future values of the labor share. These results yield a picture of the merits of the new-Keynesian model that is starkly di(cid:11)erent from the one presented by Gal(cid:19)(cid:16) and Gertler. While they argued that \[b]ackward-looking price-setting, while statistically signi(cid:12)cant, is not quantitativelyimportant,"andconcludedthat\thenew-KeynesianPhillipscurveprovides a good (cid:12)rst approximation to the dynamics of inflation," our results imply that as a (cid:12)rst-order approximation to the inflation process, the model appears to fall well short of the mark. Moreover, expectations of future output gaps or labor shares appear to play a very small role in explaining why we (cid:12)nd such large coe(cid:14)cients on lagged inflation in empirical Phillips curves. 3.3 Why Does the New-Keynesian Model Fail? Theempirical estimates fromtheprecedingsubsection suggestthatforward-looking behaviorplayslittleornoroleinpricesetting. Inthissubsection,weexploreexactly whyitisthatthenew-Keynesianmodelfailstoprovideanadequatecharacterization of the inflation process. The Role of LaggedInflation: Thenew-KeynesianinterpretationofthereducedformPhillipscurve(equation1)isthatlagsofinflationaregivenasigni(cid:12)cantweight in the regression only because they contain information about future values of the 14

driving variable. Indeed, if one accepts this interpretation, then this informational content must be the reason that lags of inflation receive such a signi(cid:12)cant weight in theconstruction oftheempiricalproxyforE t (cid:25) t+1 usedintheregressionsinTable1. However, the reason the new-Keynesian model fails the present-value tests is that, as an empirical matter, this interpretation appears to be invalid. To illustrate this point, Table 3 summarizes the coe(cid:14)cients that we obtained PK from regressions of the present-value term (cid:12)ix t+i on lagged inflation, lags of the i=0 driving variable, and the other members of the instrument set used in estimating the models of the preceding subsection. (Note that the estimates using the full instrumentsetcorrespondexactlytothe(cid:12)rst-stageregressionsfromtheIVestimates of the present-value speci(cid:12)cations; as we claimed earlier, the (cid:12)rst-stage (cid:12)t for the present-value term is quite good.) The key result from Table 3 is that lags of inflation contain very little predictive power for the present value of the driving variable. For the output gap, lagged inflation receives the \wrong"(negative) sign; moreover, excluding it from thespeci(cid:12)cation has very little e(cid:11)ect on the regression’s (cid:12)t. For the labor-income share, lagged inflation receives the correct sign, but once again has almost no marginal predictive power for the present-value term. Hence, the level of current inflation does not appear to provide a strong signal of future values of the output gap or labor’s share. Thefact thatinflationis notapositive leadingindicator fortheoutputgap(and the problem this causes for formulations of the new-Keynesian Phillips curve that 16 use the output gap as a driving variable) was noted by Gal(cid:19)(cid:16) and Gertler. They also pointed out that inflation tends to be positively correlated with future values of the labor income share, and cited this as evidence supporting their choice of the labor share as a driving variable. As far as the sign of this relationship is concerned, this pattern is also apparent in Table 3. However, our results also show that|once one looks beyond bivariate correlations|the relationship between inflation and futurevalues of the labor share is a weak one. Indeed, with the full set of instruments, lagged inflation does not haveastatistically signi(cid:12)cante(cid:11)ectonthepresentvalueoffuturelaborshares. This explains whythe empirical models of theprevious section assigned a large weight to 16Seepages 201-202. 15

lagged inflation even after this present-value term was included as an explanatory variable. And, it explains why the large estimated weight on lagged inflation found in reduced-form Phillips curves cannot be interpreted as resulting from its role as a proxy for the rational expectation of future labor shares. The Role of Lagged x t: A notable feature of the results in Table 3 is that the lags of x t are by far the most important explanatory variables for the present value (this isparticularlytruefor theoutputgap). Thisstems fromthefactthatboththe output gap and labor share can be well-approximated as univariate autoregressive processes; in particular, for either series, the addition of lags of inflation (of any length) adds little explanatory power. These univariate approximations can be used to illustrate exactly why the new- Keynesian model cannot characterize inflation very well. Begin by noting that if a stationary variable w t is expressible as a univariate AR(p) process w t = (cid:18) 1 w t−1+(cid:18) 2 w t−2+(cid:1)(cid:1)(cid:1)+(cid:18) p w t−p+(cid:15) t = (cid:18)(L)w t−1+(cid:15) t ; then we can use a formula of Hansen and Sargent (1980) to express the time-t expectation of its present discounted value as X1 (cid:12)iE t w t+i = (cid:11)(L)w t i=0 where 2 0 1 3 pX−1 Xp 1 (cid:11)(L) = 1−(cid:18)((cid:12)) 4 1+ @ (cid:12)r−k(cid:18) r ALk5: k=1 r=k+1 Hence, if the new-Keynesian model is correct, then the model’s closed-form solution for current inflation can be well approximated as: (cid:25) t = γ(cid:11)(L)x t : (20) It should be obvious, however, that this will provide a very incomplete characterization of inflation dynamics|it is basically just the reduced-form Phillips curve (1) without any lagged inflation terms. Indeed, Table 4 shows that empirical versions of equation (20) do very poorly no matter what de(cid:12)nition of x t we use; in particular, adding (cid:11)(L)x t to an inflation autoregression barely changes the weight that 16

lagged inflation receives in the model. In essence, this is another way of stating the point made above|namely, that the rational expectation of E t (cid:25) t+1 implied by the new-Keynesian model cannot, in practice, explain more than a small fraction of the variation in current inflation. In connection with this point, it is useful to consider a result reported by Gal(cid:19)(cid:16) and Gertler in their paper; namely, that a calculated series of the same form as the right-hand side of equation (17), calibrated using their preferred parameter estimatesandaVARtogeneratetheE t x t+i values,appearedtoprovideareasonable 17 (cid:12)t to the inflation data. However, while Gal(cid:19)(cid:16) and Gertler’s calibration exercise usedtheestimates fromtheirGMM equations, ourestimation procedureisdesigned to (cid:12)nd the best-(cid:12)tting version of equation (17). And the best-(cid:12)tting estimates (the coe(cid:14)cients reported in Tables 2 and 4) have a substantially di(cid:11)erent economic interpretation, because they assign little importance to the expected future values of the driving variable. Comparison with Fuhrer’s (1997) Results: Our results provide a useful interpretation of Fuhrer’s (1997) estimates, which indicated that forward-looking behavior is a relatively unimportant component of the inflation process. Fuhrer used numerical methods to estimate a version of equation (3) under the assumption that the driving term x t (in his work, the output gap) was a function of its own lags, lagged inflation, and lags of the Federal funds rate. Like the Hansen-Sargent approximation that we have just presented, this procedure essentially amounts to computing the expected present value of the driving variable under the assumption that x t follows a speci(cid:12)c process, and then determining what fraction of inflation is accounted for by this present-value term. If the present value is well characterized as a function of lags of the driving variable (as the (cid:12)rst-stage regressions in Table 3 suggest) then this method will be equivalent to the Hansen-Sargent formulation summarized by (20). Thus, though the empirical methodology of our IV-based tests of the forward-looking model is quite di(cid:11)erent from Fuhrer’s in that it does not rely on specifying an explicit process for the driving variable, we believe that the same features of the data that cause us to reject the new-Keynesian curve are also behind Fuhrer’s very similar conclusion. 17Gal(cid:19)(cid:16)andGertlercalledthisseries\fundamentalinflation"andillustrateditinFigure2oftheir paper. 17

3.4 Robustness of the Results from the Present-Value Tests We believe that the present-value tests provide convincing evidence that the new- KeynesianPhillipscurvecannotserveasanadequateapproximationtotheempirical inflation process. We have, however, shown that despite the poor (cid:12)t of the new- Keynesian model, empirical estimation of the hybrid equation (4) may still assign a high coe(cid:14)cient to future inflation if some variables that belong in the true model for inflation are used as instruments for (cid:25) t+1. In addition, we have shown how the present-value test procedures can still recover the correct answer|that the new- Keynesian curve is a poor model|even if some of the instruments in this exercise belong in the true model for inflation. At this point, a reasonable question to ask is whether the estimated coe(cid:14)cients from the present-value regressions (reported in Table 2) are a(cid:11)ected by including some of the instruments in the estimation equation. This question is irrelevant for the key issue addressed in this paper; namely, whether the new-Keynesian model, with the output gap or labor share serving as the driving variable, explains the correlations described by reduced-form inflation regressions (clearly, neither version does). However, one may wonder whether our estimated coe(cid:14)cients on lagged inflationcouldbesigni(cid:12)cantlybiasedbyusingasinstrumentsvariablesthatactually belong in the estimated model. To address this issue, Table 5 reports the sum of the coe(cid:14)cients on the lagged inflation terms (the A(1) values) from regressions that augment the speci(cid:12)cation estimated in Table 2 with each of our instruments (with the exception of the lagged values of x t). In other words, these regressions add four lags of compensation inflation, commodity price inflation, and the spread between long- and short-term interest rates, as well as four lags of the labor share (for the equation using the outputgap as thedrivingvariable) andfourlags of theoutputgap (fortheequation using labor share as the driving variable). Asexpected,theseadditionalexplanatoryvariablesoftenreceivestatisticallysigni(cid:12)cant coe(cid:14)cients; nevertheless, the results do not change our main conclusions. The estimated A(1) values from these regressions are a bit smaller than those reportedin Table2. However, theyare stillquitelarge|in the0.6 to0.75 range|and most of the reduction comes from including lags of compensation inflation (a variable that is highly correlated with price inflation) in the model. Moreover, we still 18

obtain similar estimates of A(1) whether or not we include present-value terms. 18 These results tend to con(cid:12)rm our suspicion that some of the instruments used by Gal(cid:19)(cid:16) and Gertler likely belong in a well-speci(cid:12)ed empirical inflation regression. Futhermore, they provide additional con(cid:12)rmation for our conclusion that the signi(cid:12)cant role played by lags of inflation in reduced-form Phillips curves cannot be assigned totheir proxyingforexpected presentvaluesof theoutputgapor thelabor share of income. 4 Concluding Comments The goal of this paper has been to assess whether the new-Keynesian Phillips curve can account for the empirical correlations that are summarized by reduced-form inflation regressions. We have concluded that it cannot. Speci(cid:12)cally, we have focused on the new-Keynesian model’s prediction that inflation should be a function of an expected present discounted value of some \driving variable" x t that summarizes excess demand or marginal costs. We have found that such present-value terms can explain only a very small fraction of observed inflation dynamics, a conclusion that we obtain whether we use the output gap or the labor income share as the driving variable. In addition, the idea that the important role assigned to lags of inflation by reduced-form Phillips curves comes from their proxying for expected future values of x t is contradicted by the minor role that inflation plays in forecasting future values of the labor share or output gap. Our results yield an assessment of the new-Keynesian Phillips curve that is very di(cid:11)erent from the one found in Gal(cid:19)(cid:16) and Gertler’s recent work. We believe the reason for this di(cid:11)erence comes from the fact that their tests of the model have verylowpoweragainstalternative, butnon-nested,backward-lookingspeci(cid:12)cations. In connection with this point, we have shown that Gal(cid:19)(cid:16) and Gertler’s estimation procedure is likely to suggest that forward-looking behavior is very important even if the true model contains no such behavior. We suspect that the conclusions of this paper will be unsurprising to some students of the staggered-contracting literature. Indeed, we view our results as com- 18Note that these regressions achieve identi(cid:12)cation of the coe(cid:14)cient on the expected present valuesolely through thelags of x t in theinstrument set. Ascan be seen from Table 3, this avoids any problems associated with weak instruments. 19

pletely consistent with Fuhrer and Moore’s (1995) claim that the type of staggeredcontracting model from which the new-Keynesian Phillips curve is derived cannot deliver the degree of inflation persistence that we actually observe in the data. This problem is clearly evident in our own characterization of the model. Theresultsinthis paperalso provideaclear warningagainst theuseofthenew- Keynesian Phillips curve (or hybrid variants that place a large weight on forwardlooking expectations) for policy analysis. The understandable appeal of these speci(cid:12)cations, which allow nominal disturbances to have real economic e(cid:11)ects in the context of a rational-expectations framework, has made them the model of choice for modern theoretical examinations of the optimal conduct of monetary policy. However, the implications of these models are not innocuous|for example, the fact that they imply that current inflation summarizes the entire sequence of expected future output gaps for the economy is an extremely strong prediction that may well influence one’s assessment of the optimal choice of policy target (among other things). Given that this prediction is soundly rejected by the data, the use of these models for policy analysis strikes us as questionable at best. We concludebynotingthat, despiteour(cid:12)ndingthatthenew-KeynesianPhillips curve cannot explain the role played by lagged dependent variables in inflation regressions, our results should not necessarily be interpreted as implying that agents formulate expectations in a backward-looking manner, as in the traditional interpretation of the Phillips curve. We can think of two avenues through which future research may reconcile empirical inflation dynamics with rational behavior. First, it is possible that the important role assigned to lags of inflation could come from their proxying for expectations of some other drivingvariable that we have not considered. (That said, however, we hope that we have at least demonstrated that any future candidate driving variable should be tested in the context of the forwardlooking model’s closed-form solution.) Second, it may be that the research agenda oninflationdynamicsneedstomoveawayfromthesticky-pricemodelsthatunderlie the new-Keynesian Phillips curve, and towards other mechanisms that can generate the degree|and type|of inflation persistence that we observe in the data. Recent work by Sims (1998) and Mankiw and Reis (2001) represents two promising e(cid:11)orts in this direction. 20

References [1] Calvo, Gulliermo A. (1983). \Staggered Prices in a Utility Maximizing Framework," Journal of Monetary Economics, 12, 383-398. [2] Fuhrer, Je(cid:11)rey C. (1997). \The (Un)Importance of Forward-Looking Behavior in Price Speci(cid:12)cations," Journal of Money, Credit, and Banking, 29, 338-350. [3] Fuhrer, Je(cid:11)reyC. andGeorge Moore (1995). \Inflation Persistence," Quarterly Journal of Economics, 440, 127-159. [4] Gal(cid:19)(cid:16), JordiandMarkGertler(1999).\InflationDynamics: AStructuralEconometric Analysis," Journal of Monetary Economics, 44, 195-222. [5] Gordon, Robert J. (1998). \Foundations of the Goldilocks Economy: Supply Shocks and the Time-Varying NAIRU," Brookings Papers on Economic Activity, 297-346. [6] Hamilton, James D. (1994). Time Series Analysis. Princeton, NJ: Princeton University Press. [7] Hansen, Lars Peter and Thomas J. Sargent (1980). \Formulating and Estimating Dynamic Linear Rational Expectations Models," Journal of Economic Dynamics and Control, 2, 7-46. [8] Hansen, Lars Peter (1982). \Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, 50, 1029-54. [9] Hausman, Jerry (1984). \Speci(cid:12)cation and Estimation of Simultaneous Equations Models," in Zvi Griliches and Michael Intriligator (eds.), Handbook of Econometrics, Amsterdam: North-Holland. [10] Mankiw, N. Gregory and Ricardo Reis (2001). \Sticky Information Versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve," working paper, Harvard University. [11] Newey, Whitney L. (1985). \Generalized Method of Moments Speci(cid:12)cation Testing," Journal of Econometrics, 29, 229-256. 21

[12] Roberts, JohnM. (1995). \New Keynesian Economics and the Phillips Curve," Journal of Money, Credit, and Banking, 27, 975-984. [13] Roberts, John M. (2001). \How Well Does the New-Keynesian Sticky-Price Model Fit the Data?" Federal Reserve Board, Finance and Economics Discussion Series, No. 2001-13. [14] Sims, Christopher A. (1998). \Stickiness," Carnegie-Rochester Conference Series on Public Policy, 49, 317-356. [15] Staiger, Douglas, James Stock, and Mark Watson (1997). \How Precise Are Estimates of the Natural Rate of Unemployment?" in Christina Romer and David Romer (eds.), Reducing Inflation, Chicago: University of Chicago Press. [16] Taylor, JohnB.(1980). \Aggregate Dynamics andStaggeredContracts," Journal of Political Economy, 88, 1-23. 22

A Properties of IV Estimates Under Misspeci(cid:12)cation In this appendix, we demonstrate the properties of the GMM estimates of equation (4) and the Hansen test of overidentifying restrictions for the speci(cid:12)c case in which the true model is (cid:25) t = (cid:12)(cid:25) t−1+(cid:21)x t+(cid:22)z t+u t ; where u t is white noise, and x and z are characterized by (cid:12)rst-order autoregressive processes: x t = (cid:26) x x t−1+(cid:15)x t ; z t = (cid:26) z z t−1+(cid:15)z t : Under these conditions, the process for (cid:25) t+1 can be written as (cid:25) t+1 = (cid:12)(cid:25) t+(cid:21)x t+1+(cid:22)z t+1+u t+1 = (cid:12)((cid:12)(cid:25) t−1+(cid:21)x t+(cid:22)z t+u t)+(cid:21)(cid:26) x x t+(cid:22)(cid:26) z z t+(cid:21)(cid:15)x t+1 +(cid:22)(cid:15)z t+1 +u t+1 = (cid:12)2(cid:25) t−1+(cid:21)((cid:12)+(cid:26) x)x t+(cid:22)((cid:12)+(cid:26) z)z t+(cid:21)(cid:15)x t+1 +(cid:22)(cid:15)z t+1 +(cid:12)u t+u t+1 : This expresses (cid:25) t+1 in terms of the variables in the (cid:12)rst-stage regression, and a set of stochastic processes that are orthogonal to each of them. Asymptotically, we can write the (cid:12)tted value for this model as (cid:25)^t+1 = (cid:14)^ 1 (cid:25) t−1+(cid:14)^ 2 x t+(cid:14)^ 3 z t ; where (cid:14)^ 1 = (cid:12)2; (cid:14)^ 2 = (cid:21)((cid:12) +(cid:26) x); (cid:14)^ 3 = (cid:22)((cid:12)+(cid:26) z): The second-stage regression can now be expressed as (cid:16) (cid:17) (cid:25) t = !^ f (cid:14)^ 1 (cid:25) t−1+(cid:14)^ 2 x t+(cid:14)^ 3 y t +!^ b(cid:25) t−1+γ^x t+(cid:29) t (cid:16) (cid:17) (cid:16) (cid:17) = !^ b +!^ f(cid:14)^ 1 (cid:25) t−1+ γ^+!^ f(cid:14)^ 2 x t+!^ f(cid:14)^ 3 z t+(cid:29) t (cid:16) (cid:17) (cid:16) (cid:17) = !^ b +!^ f(cid:12)2 (cid:25) t−1+ γ^+!^ f(cid:21)((cid:12)+(cid:26) x) x t+!^ f(cid:22)((cid:12) +(cid:26) z)z t+(cid:29) t : 23

So, we can write the probability limits of the parameter estimates as (cid:16) (cid:17) plim !^ b +!^ f(cid:12)2 = (cid:12); (cid:16) (cid:17) plim γ^+!^ f(cid:21)((cid:12) +(cid:26) x) = (cid:21); plim !^ f(cid:22)((cid:12)+(cid:26) z) = (cid:22); which implies that 1 plim !^ f = ; (cid:12) +(cid:26) z (cid:12)(cid:26) plim !^ b = z ; (cid:12) (cid:18) +(cid:26) z (cid:19) (cid:26) −(cid:26) plim γ^ = (cid:21) z x : (cid:12)+(cid:26) z This gives equations (13), (14), and (15) in the text. Finally, consider the properties of the Hansen test of the overidentifying restrictions. Drawing on an earlier result by Hausman (1984), Newey (1985) showed that forlinearmodelsthisteststatisticisequalto(T −k)R2 ,whereT isthesamplesize, k is the number of parameters being estimated, and the R2 is from a regression of themodel’s errors on theinstrumentset. Underthenull, this statistic is distributed as a (cid:31)2 with m−k degrees of freedom, where m denotes the number of variables in the instrument set. For the case considered here, this test procedure involves regressing (cid:15) t = (cid:25) t − !bf(cid:25) t+1 − !^ b(cid:25) t−1 − γ^x t on x t ;z t ; and (cid:25) t−1. Using the probability limits of the estimated parameters, we can calculate the probability limit for (cid:15) t as follows: (cid:18) (cid:19) (cid:15) t = (cid:25) t − (cid:12) + 1 (cid:26) z (cid:25) t+1 − (cid:12) (cid:12) + (cid:26) z (cid:26) z (cid:25) t−1 −(cid:21) (cid:26) (cid:12) z + − (cid:26) (cid:26) z x x t h i 1 = (cid:25) t − (cid:12) +(cid:26) z (cid:12)2(cid:25) t−1+(cid:21)((cid:12) +(cid:26) x)x t+(cid:22)((cid:12) +(cid:26) z)z t+(cid:21)(cid:15)x t+1 +(cid:22)(cid:15)z t+1 +(cid:12)u t+u t+1 (cid:18) (cid:19) (cid:12)(cid:26) (cid:26) −(cid:26) − z (cid:25) −(cid:21) z x x (cid:12)+(cid:26) z t−1 (cid:12)+(cid:26) z t = (cid:25) t −(cid:12)(cid:25) t−1 −(cid:21)x t −(cid:22)z t+ (cid:21)(cid:15)x t+1 +(cid:22)(cid:15) (cid:12) z t+ + 1 + (cid:26) z (cid:12)u t+u t+1 = u t+ (cid:21)(cid:15)x t+1 +(cid:22)(cid:15) (cid:12) z t+ + 1 + (cid:26) z (cid:12)u t+u t+1 (cid:0) (cid:1) = (cid:21)(cid:15)x t+1 +(cid:22)(cid:15)z t+1 + (cid:12)2 +(cid:12)+(cid:26) z u t+u t+1 : (cid:12)+(cid:26) z 24

This series is uncorrelated with z t, x t, and (cid:25) t−1, so we should not expect to obtain a high R2 from this regression. Indeed, asymptotically this R2 should tend towards zero. As a result, the test will not beable to detect the presenceof misspeci(cid:12)cation. Thereason for this is that the rolethat z t plays in theregression is indirectlypicked up through its e(cid:11)ect on (cid:25) t+1. Thus, the misspeci(cid:12)ed regression has not \left out" the e(cid:11)ect of z t, thereby pushing it into the regression residual|which is what the Hansen test is attempting to detect. Rather, it has just incorrectly attributed the e(cid:11)ect of z t to (cid:25) t+1. 25

Table 1: \Hybrid" Inflation Equation|GMM Estimates Driving variable (x t) !f !b γ A. NFB price inflation GDP gap 0.777 (cid:3)(cid:3) 0.218 (cid:3)(cid:3) −0.007 (cid:3) (0.044) (0.039) (0.003) (cid:3)(cid:3) (cid:3)(cid:3) Labor’s share 0.748 0.237 0.002 (0.039) (0.036) (0.004) B. GDP price inflation GDP gap 0.605 (cid:3)(cid:3) 0.393 (cid:3)(cid:3) −0.000 (0.026) (0.027) (0.002) (cid:3)(cid:3) (cid:3)(cid:3) Labor’s share 0.594 0.397 0.004 (0.022) (0.023) (0.003) Note: Standard errors in parentheses. ∗∗/∗/a denotes signi(cid:12)cant at 1/5/10 percent level, respectively. Estimation from 1960:Q1to 1997:Q4. 26

Table 2: Closed-Form Inflation Equations|IV Estimates ((cid:12) (cid:17) 0:95) Driving variable (x t) γ A(1) A. NFB price inflation GDP gap −0.011 (cid:3)(cid:3) (0.002) (cid:3)(cid:3) (cid:3)(cid:3) GDP gap 0.002 0.877 (0.001) (0.025) (cid:3)(cid:3) Labor’s share 0.012 (0.003) (cid:3)(cid:3) (cid:3)(cid:3) Labor’s share 0.005 0.795 (0.001) (0.027) (cid:3)(cid:3) Memo: Own lags only 0.892 (0.043) B. GDP price inflation GDP gap −0.010 (cid:3)(cid:3) (0.002) (cid:3)(cid:3) (cid:3)(cid:3) GDP gap 0.002 0.877 (0.001) (0.040) (cid:3)(cid:3) Labor’s share 0.011 (0.003) (cid:3)(cid:3) (cid:3)(cid:3) Labor’s share 0.004 0.806 (0.001) (0.032) (cid:3)(cid:3) Memo: Own lags only 0.918 (0.038) Note: Standard errors in parentheses. ∗∗/∗/a denotes signi(cid:12)cant at 1/5/10 percent level, respectively. A(1) is the sum of the coe(cid:14)cients on two lags of inflation. 27

Table 3: First-Stage Regressions for Present-Value Terms (K (cid:17) 12, (cid:12) (cid:17) 0:95) NFB inflation GDP inflation Sum of coe(cid:11)s. Sum of coe(cid:11)s. x lags (cid:25) lags R(cid:22)2 x lags (cid:25) lags R(cid:22)2 1. x t (cid:17) GDP gap Full instrument set 5.88 (cid:3)(cid:3) −19.05 (cid:3)(cid:3) 0.57 5.98 (cid:3)(cid:3) −21.71 (cid:3)(cid:3) 0.56 (0.72) (4.76) (0.72) (5.96) Full set less lagged (cid:25) t 6.12 (cid:3)(cid:3) 0.53 6.12 (cid:3)(cid:3) 0.53 (0.74) (0.74) Lagged x t and (cid:25) t only 5.21 (cid:3)(cid:3) −11.18 (cid:3)(cid:3) 0.50 5.28 (cid:3)(cid:3) −13.25 (cid:3)(cid:3) 0.50 (0.53) (2.30) (0.54) (2.54) Lagged x t only 4.14 (cid:3)(cid:3) 0.41 4.14 (cid:3)(cid:3) 0.41 (0.52) (0.52) Lagged (cid:25) t only −10.62 (cid:3)(cid:3) 0.08 −10.70 (cid:3)(cid:3) 0.07 (2.68) (3.04) 2. x t (cid:17) Labor’s share (cid:3)(cid:3) (cid:3)(cid:3) Full instrument set 7.60 0.69 0.79 7.63 2.63 0.79 (0.52) (2.34) (0.51) (2.89) Full set less lagged (cid:25) t 7.42 (cid:3)(cid:3) 0.79 7.42 (cid:3)(cid:3) 0.79 (0.50) (0.50) Lagged x t and (cid:25) t only 5.37 (cid:3)(cid:3) 4.47 (cid:3)(cid:3) 0.48 5.11 (cid:3)(cid:3) 6.54 (cid:3)(cid:3) 0.50 (0.62) (1.54) (0.61) (1.72) Lagged x t only 5.96 (cid:3)(cid:3) 0.46 5.96 (cid:3)(cid:3) 0.46 (0.59) (0.59) Lagged (cid:25) t only 9.74 (cid:3)(cid:3) 0.15 12.67 (cid:3)(cid:3) 0.20 (1.83) (1.96) Notes: Standard errors in parentheses. ∗∗/∗/a denotes signi(cid:12)cant at 1/5/10 percent level. 28

Table 4: Closed-Form Inflation Equations|Hansen-Sargent Estimates ((cid:12) (cid:17) 0:95) Driving variable (x t) γ A(1) R(cid:22)2 A. NFB price inflation a GDP gap 0.005 0.01 (0.003) (cid:3)(cid:3) (cid:3)(cid:3) GDP gap 0.005 0.893 0.75 (0.001) (0.043) (cid:3)(cid:3) Labor’s share 0.016 0.12 (0.003) (cid:3)(cid:3) Labor’s share 0.002 0.873 0.73 (0.002) (0.048) B. GDP price inflation (cid:3) GDP gap 0.005 0.03 (0.002) (cid:3)(cid:3) (cid:3)(cid:3) GDP gap 0.003 0.911 0.82 (0.001) (0.035) (cid:3)(cid:3) Labor’s share 0.015 0.17 (0.003) (cid:3)(cid:3) Labor’s share 0.002 0.893 0.82 (0.001) (0.039) Notes: Standard errors in parentheses. ∗∗/∗/a denotes signi(cid:12)cant at 1/5/10 percent level, respectively. A(1) is the sum of the coe(cid:14)cients on two lags of inflation. 29

Table 5: Lagged Inflation Term: Closed-Form Models with Additional Correlates A(1) Driving variable (x t) no PDV with PDV A. NFB price inflation (cid:3)(cid:3) (cid:3)(cid:3) GDP gap 0.699 0.672 (0.073) (0.075) (cid:3)(cid:3) (cid:3)(cid:3) Labor’s share 0.685 0.752 (0.072) (0.046) B. GDP price inflation (cid:3)(cid:3) (cid:3)(cid:3) GDP gap 0.630 0.607 (0.075) (0.097) (cid:3)(cid:3) (cid:3)(cid:3) Labor’s share 0.619 0.710 (0.073) (0.047) Notes: Standard errors in parentheses. ∗∗/∗/a denotes signi(cid:12)cant at 1/5/10 percent level. 30