American Economic Association

Can Rational Expectations Sticky-Price Models Explain Inflation Dynamics? Author(s): Jeremy Rudd and Karl Whelan Source: The American Economic Review, Vol. 96, No. 1 (Mar., 2006), pp. 303-320 Published by: American Economic Association Stable URL: http://www.jstor.org/stable/30034367 Accessed: 03/05/2010 22:40 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=aea. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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Can RationalExpectationsSticky-PriceModels Explain InflationDynamics? By JEREMY RUDD AND KARL WHELAN*

In recentyears,therehas been a trendin macroeconomics toward analyzing business cycles and stabilizationpolicy in the context of models that incorporateboth nominalrigiditiesand optimizing agentswith rational(i.e., model-consistent) expectations.'One importantway in which this "new-Keynesian"approachdiffers from earlier work in the Keynesiantraditioninvolves the way in which expectationsareassumedto affectpricesettingbehavior.In particular,ratherthanassuming adaptiveinflationexpectationson the partof wage- and price-setters,recent work draws on explicitmodels of pricestickiness(such as thatof GulliermoA. Calvo, 1983) in orderto motivate a forward-lookinginflation equation (a "newKeynesian Phillips curve") of the form (1)

ITt=

3E,, +I,+ yy,

where 0 is a parameterclose to or equal to one, and y, is a measure of the output gap. An importantimplicationof this model is that inflation should be independent of its own lagged values. As a result, this specificationhas often been criticized on the groundsthat it cannot account for the important empirical role played by lagged dependent variables in inflation regressions. In response to this critique, several researchershave suggested an alterna* Rudd:Divisionof Researchand Statistics,Boardof Governorsof the FederalReserveSystem,20th andC StreetsNW, Mailstop 80, Washington,DC 20551-0001 (e-mail:jeremy. [email protected]);Whelan: Departmentof Economic Analysis, Research, and Publications,Central Bank and Financial Services Authorityof Ireland,Dame Street, Dublin 2, Ireland (e-mail: [email protected]).We thank Dale Henderson,Frank Smets, William Wascher, and two anonymousreferees for helpful comments on earlierdrafts. The views expressed are our own and do not necessarily reflect the views of the Board of Governorsor staff of the FederalReserve System, or of the CentralBank and Financial Services Authorityof Ireland. 'See RichardClaridaet al. (1999) for a survey of much of this work, and Michael Woodford (2003) for a detailed treatment.

tive to the pure forward-lookingmodel that is intended to better capture observed inflation inertia.This "hybrid"specificationmodifies the new-Keynesian Phillips curve so that inflation depends on a weighted sum of its lag and its (rationally)expected future value, (2)

r, = (1 - 0)r7,_, + OE,rt,,,+ Yy,,

with the weights constrainedto sum to unity in orderto precludethe existenceof a long-runlevel tradeoffbetweeninflationand real activity.2 Withinthe class of papersemploying variants of this hybrid specification, the best-known studies have featuredmodels in which 0 : 1/2. For example, the well-known model of Jeffrey C. Fuhrerand George R. Moore (1995) employs an assumption that workers bargain over relative real wages in order to obtain an equation with 0 = 1/2.More recently, LawrenceJ. Christiano et al. (2005) have explicitly derived a specificationof this form using a variantof the Calvo model in which those firms that are unable to reoptimizetheir price insteadindex it to last period's inflationrate. In theirframework,0 equals P/(1 + 0) (where 0 is the factor used to discount firms' profits); this directly implies that 0 will be less than 1/2. In this paper,we assess whetherhybridmodels of this sort provide a good empiricalcharacterization of the behavior of U.S. inflation. For the case in which 0 o 1/2, our tests are based on the observationthat the hybrid specification implies an expressionfor the change in inflation of the form oo

(3)

AT,

=

C

,A, k=O

4EtYt+k

2 Examples of studies that use this pricing equation include Miguel Casares and Bennett T. McCallum (2000), Michael Ehrmann and Frank Smets (2003), and Glenn Rudebusch (2002).

303

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THEAMERICANECONOMICREVIEW

304

where A2 - 1. We focus on this prediction, ratherthan on the model's ability to fit the level of inflation, in order to derive tests that are capableof distinguishingthe hybridmodel from reasonable alternatives. In practice, inflation can be predictedwell from its own lagged value; hence, incorporatinglagged inflation into the inflation equation should allow the hybrid model to fit the level of inflationrelatively well. However, such a fit could also be obtained by any model that features an importantrole for lagged inflation-including models that rely on nonrational,backward-lookingexpectations. In contrast,the hybrid model's predictionsfor the are quite clear-cutand allow us evolution of to precisely Arrt distinguish this model from a traditional backward-lookingspecification. We consider two different methods for assessing whether this formulationof the hybrid model provides a reasonabledescriptionof the data. The first employs the well-known methodology of John Y. Campbell and Robert J. Shiller (1987), which entails using a VAR to forecast future values of the driving process y,. The second method involves estimating the equation using the generalized method of moments (GMM). Both methods turn out to yield useful insights--the firstinto the predictedtimeseries propertiesof Aur,that are implied by the model, and the second into the statisticalsignificance of the model's forward-looking component. While variantsof the hybrid specification in which 0 V/2have received a large amount of - recent work, there is no a priori attention in reason to rule out the possibility that price setting is characterized by a preponderanceof forward-looking behavior. We therefore also consider versions of the hybridmodel with 0 > 1/2, which imply the following closed-form solution: oo

(4)

=

,t

I Etyt+k + I t2"tT- . 0

-L1k

=

Here, the level of currentinflation is related to lagged inflation (with t2 < 1) and currentand expected future values of the driving term, where these receive a unit weight in all periods. Again, the presence of lagged inflation ensures that this model will be able to fit rw,relatively well; hence, the relevantquestionhere concerns

what contribution the forward-looking terms make to explaining inflationdynamics. Taken as a whole, our results suggest thatthe hybrid model provides a poor description of empirical inflation dynamics. Specifically, we find that the empiricalprocess for the change in inflationappearsto bear very little resemblance to a discounted sum of current and expected future Yt values. Moreover, we find that the coefficients on the discounted sum (A, or /1) are not significantlydifferentfrom zero for any variant of the hybrid model that we consider, implying thatinflationis unrelatedto the expectation of future values of the driving term, and indicating that the type of rational forwardlooking behavior hypothesized by the hybrid model is absent from the data. Importantly, these conclusions hold both when we use detrended output as y,, and when we use labor's share of income (real unit labor costs), as has been suggested by Jordi Gali and Mark Gertler (1999). The contents of the paper are as follows. Section I briefly discusses the nature of the "persistenceproblem"that is faced by the newKeynesian Phillips curve (and that motivates the use of hybridinflationequations).Section II introduces the hybrid model and discusses its closed-form solutions. Section III assesses the fit of the hybridmodel when 0 - 1/2.Section IV presentsGMM estimatesof this model, and also examines whether its performancecan be improved by incorporatinga more complex "ruleof-thumb"for backward-lookingagents. Finally, Section V considers the version of the model that obtains when 0 > 1/2, and Section VI concludes. I. The PersistenceProblem At first glance, it might appearas though the new-Keynesianinflation specification, (5)

== 3Eirt + 1 + yyt,

would be difficultto distinguishfrom models in which inflationdependson its own lagged values. and so next peInflationis highly autocorrelated, riod's expectedinflationrateis likely to be highly correlatedwith last period'srate.Whencombined withthe assumptionof rationalexpectations,however,the new-Keynesianmodelmakesa very pre-

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305

cise predictionabout the natureof inflation of inflationenter empiricalPhillips curves only dynamics.Thiscanbe seenby applyingrepeated because they are proxying for expectations of substitutions to equation(5) to obtain future values of y,. =T, =

(6)

k=O

kEtyt+k.

The model predictsthat inflationdepends solely on currentand expectedfuturevalues of the output gap. Once we condition on this, no lagged variables-including lagged inflation-should have an impacton the currentlevel of inflation. There is, however, relatively wide agreement that this formulation does not sufficiently explain the role played by lagged inflation in reduced-forminflationregressions.JeremyRudd and KarlWhelan (2005a) provide an illustration of this problem by using the methodology of Campbelland Shiller (1987) to assess the fit of the new-KeynesianPhillips curve. Specifically, if we assume that y, is the first variable in a multivariateVAR of the form z, = Az,_ + E,,

(7)

then one can express the discounted sum of currentand future values of y, as

(8)

C

E

kEtyt+k = e'(I I -

A)-'z,

k=O

where e' denotes a vector with one in the first row and zeroes elsewhere. Rudd and Whelan demonstratethat the empirical fit of the newKeynesian Phillips curve is poor across a wide varietyof VAR specificationsfor z,. In addition, econometric specifications such as (9)

7, = ye'(I - PA)-'z, + A(L)rt,_1-

reveal that there is a statisticallysignificantand economically large role for lagged inflation, despite one's having proxied for the expected present value of the driving variable y,. This result is obtained whether one measures the outputgap as detrendedGDP or as labor's share of income, as suggested by Gali and Gertler (1999). The result is also robust to the use of VAR specificationsthat include inflation itself, so that one can rule out the possibility that lags

It is importantto stress that it is this resultthe failureof the pure forward-lookingmodel to account for the empirical importanceof lagged inflation-that defines the so-called persistence problem faced by the new-Keynesian Phillips curve. We make this observationbecause discussions of the empiricalperformanceof stickyprice models have commonly focused on the high autocorrelation of inflation, with the implication being that it is this propertyof the data that these models should seek to match.3However, despite their inability to account for the importantrole played by lagged inflation, empirical implementationsof the new-Keynesian Phillips curve still predict that inflation should be highly autocorrelated:as long as y, is highly autocorrelated(as is the case for detrendedoutput and the labor income share), the predicted inflationseries from the new-KeynesianPhillips curve will be highly autocorrelated. These findings suggest that it is the failure to capturethe inertia in inflation,given fundamentals, that characterizesthe pure forward-looking model's persistence problem. Put differently, the persistenceproblemstems from the fact that lagged inflation enters reduced-form inflation equations with large coefficients even after we have conditioned on driving variables (such as the output gap) that are themselves highly autocorrelated.This suggeststhathybridvariantsof the basic sticky-pricemodel,which directlyallow for a lagged inflationterm, may performbetter empirically.We now examinethese models. II. Closed-FormSolutionsto the HybridModel The approachwe take to evaluate the empirical relevance of the hybrid inflation equation (10)

+ yy, rr,= (1 - O)Tr,_,+ OE,Tr,+,

closely follows the approachdescribedin the previoussectionfor assessingthepureforward3 Fuhrerand Moore (1995), John B. Taylor (1999), and Luca Guerrieri(2002) providethreeexamples of papersthat discuss the new-Keynesian Phillips curve's "persistence problem"in terms of its ability to match high autocorrelations for inflation.

306

THEAMERICANECONOMICREVIEW

looking model. Specifically, we focus directly on the hybrid model's closed-form solutions, which express inflation in terms of its own lagged value and a composite forward-looking term of expected future output gaps. In this section, we first describe how to derive these expressions, and then contrast our method for evaluating the hybrid model with the procedures employed by Gali and Gertler(1999) and Fuhrer(1997) in previous work. A. Derivation of the Model's Closed-Form Solutions Begin by rewritingthe hybrid model (10) in terms of lead and lag operators: (11)

E F2

F+

L

-Yt

to applythe quadraticformula It is straightforward to show that one root of this characteristicpolynomial equals one, while the other equals (1 0)/0. Hence,the stochasticdifferenceequationimplied by the hybridmodel can be writtenas - 1)F

(12)

1- -0OL, ) y.t

Et[(F

-0Ytyt=

When 0 - 1/2, then (1 - 0)/0 - 1 and the unique stable solution is found by multiplying through by the forward inverse [F - (1 0)/0]- , which yields a solution of the form

(13) A = 1 /

k=

EtYt+k

Thus, hybrid models such as those of Fuhrer and Moore (1995) (which assumes 0 = 0.5) and Christiano et al. (2005) (which assumes 0 < 0.5) imply that the change in inflationshould be proportionalto a discountedsum of currentand expected future values of the output gap. Alternatively,when 0 > 1/2, the stable solution is found by multiplying through with the forward inverse (F - 1)-'. This results in a solution of the form

(14)

,-t=

r,_ +

Eyt+k.

k=O

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In this case, inflationdependson its own lag and on an undiscountedsum of currentand expected future values of the output gap. These derivationsclearly show that the crucial feature of the hybrid model under rational expectations is the presence of a composite forward-lookingsum of expected outputgaps. It is this termthat distinguishesthese models from alternativesbased on purely backward-looking inflation expectations. Hence, our approachin this paperinvolves assessing the role played by this forward-lookingcomponent. For example, by specifying a forecastingmodel for the output gap, we can constructan empiricalproxy for the forward-lookingterm, which in turnwill permit us to determineits contributionto the model's fit. Alternatively,we can use GMM-basedtechniques to directly estimate the closed-form expressions (13) and (14), which will allow us to examine whether the composite sum is statistically significant. B. Comparisonwith Other Procedures It is useful to contrastbriefly our method of assessing the hybrid model with the approach taken in previous studies. Here we discuss the differences between our approachand two alternative procedures followed by Gall and Gertler(1999) and Fuhrer(1997). Comparisonwith Gall and Gertler (1999).These authorsfocus on estimates of 0 obtained from directly fitting equation (10) using GMM. Specifically, under this procedureET,'r+ +1 is replaced with 7+,, and the model is estimated using instruments for 7rT 1. If the model is correct and expectations are rational, then any estimation errorreflects the presence of an expectational error (rt+,, - Etrt+ 1)that should be unforecastableat time t or earlier. Thus, in theory, any variabledated t or earliercan serve as a valid instrument.Using this method, Gall and Gertlerfind that 0 is greaterthan one-half, and conclude that rationalforward-lookingbehavior plays an importantrole in determining U.S. inflation. It is possible to demonstrate,however, that a number of potential pitfalls can arise when GMM estimates of 0 from equations like (10) are used in order to assess the importance of forward-lookingbehavior in price setting. Al-

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though this particularapproachinvokes the assumption of rational expectations to justify its choice of instruments,it does not actually impose the assumption of fully rational (that is, model-consistent) expectations on the estimation. GMM is equivalent to two-stage least squares in this case, and in practice there may be little correlationbetween the fitted value of E,rr,, 1 from the first-stage regression and its value under model-consistent expectations. Thus, the constructedproxy for expected inflation can receive a significantcoefficient even if the model's key prediction regardingthe relationship between inflation and expectations of future output gaps is incorrect.4 Our approachdoes not suffer from this drawback:by directlyestimatingthe model's closedform solution, we ensure that model-consistent expectations are imposed. Moreover, focusing on the role played by expected futurevalues of y, permits us to highlight precisely the specific contributionof rationalforward-lookingexpectations to inflation dynamics. Comparisonwith Fuhrer (1997).-The methodologyemployedin thispapersharesa similarity with our own approach,in that Fuhrer'sestimation procedurealso imposesmodel-consistentexpectations on the inflation equation (which ensuresthatit will takethe formof eitherequation (13) or equation (14)). There is a fundamental difference,however,between our methodfor asbehavior sessingthe relevanceof forward-looking in price settingand the methodused by Fuhrer. Fuhrer's estimation procedure yields the value of the 0 parameterthat best fits the data. Based on the low values of 0 that he obtains,he concludes that forward-lookingbehavior plays essentially no role in observed inflationdynamics. It should be emphasized, however, that the estimate of 0 producedby this method does not necessarily allow one to discriminatebetween forward- and backward-lookingmodels of inflation. Indeed, this procedurecan yield significant positive estimates of 0 even when the true model for inflation features only backwardlooking behavior (in which case the term involving future output gaps is immaterial). To

4 See Ruddand Whelan (2005b) for a detaileddiscussion of this problem.

307

see this, suppose that the best-fitting specification is the one given by equation (14). In this case, the estimated value of 0 will be completely determinedby the estimated coefficient on lagged inflation.Given the empirical importance of lagged inflation, 0 will typically be estimated to be highly statistically significant (with a point estimate that will be greaterthan one-half so long as the coefficient on lagged inflation is less than one)-and this can be true even if the coefficient on the sum of future output gaps is itself statisticallyinsignificant. For this reason, we focus directly on the importance of the composite forward-looking term (i.e., the sum of currentand expected future output gaps). Furthermore,because some of our results are in fact consistent with significant positive estimates of 0, our rejection of forward-looking behavior in price setting is based on a reading of the empirical evidence that is differentfrom what is in Fuhrer'spaper.5 III. Fit of the HybridModelwith 0

V2 We now apply the Campbell-Shillermethodology to assess the fit of equation (13), which gives the closed-form solution of the hybrid model with 0 o1/2.As describedabove, we can assume that y, is the first variable in the vector z,, and calculate the discounted sum as 5

(15)

k=

k

o

1

0

)

0

eY +k : EYt

I-

- A )1 z

where zt is modelled as a VAR expressedin the companion form z, = 1 + Et. The "disAz/ count factor" associated with the infinite sum, 0/(1 - 0), is unknown,so the approachthat we take here involves using a grid search (over the interval zero to one) to obtain the value of the discount factor that yields the highest correla-

5 It is worth noting two other significant differences between the two papers. First, Fuhrer's paper uses detrended output as a proxy for the output gap; we use both detrendedoutputand the laborsharemeasurerecommended by Gall and Gertler (1999). Second, to apply Fuhrer's maximum-likelihood methodology, one must explicitly specify a driving process for the output gap proxy; in contrast, this is not requiredfor the GMM proceduresthat we consider in Sections IV and V.

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THEAMERICANECONOMICREVIEW

tion between the resulting discounted sum and the first difference of inflation. We consider two versions of the model. The first equates yt with a traditional output gap measure, defined here as the deviation of log real nonfarmGDP from a quadratictrend. The second follows Gali and Gertler(1999) in using (the log of) labor's share of income, again defined for the nonfarmbusiness sector. The motivation for this latter measure stems from the observationthat the sticky-pricemodels underpinning the new-Keynesian Phillips curve imply that the correctdriving variablefor inflation is actually real marginalcost. Because the theoretical restrictions required in order for real marginalcost to move with the traditionaloutput gap are restrictive, Gall and Gertler (and others) have instead proposed using average unit labor costs-nominal compensation divided by real output-as a proxy for nominal marginal cost. The resulting measure of real marginalcost is labor's share of income (nominal compensationdivided by nominal output). A. OutputGap Model To forecast future values of the output gap, we use a standardtwo-lag, three-variableVAR which includes the outputgap, the federalfunds rate, and inflation, which we measure as the log-difference of the price deflatorfor the nonfarm business sector.The sampleperiodextends from 1960:Q1to 2002:Q1.This simpleVAR forecasts the outputgap well and has been used in a numberof papers, includingJohn H. Cochrane (1994), Fuhrerand Moore (1995), and Julio J. Rotembergand MichaelWoodford(1997). The results from this exercise provide little support for the hybrid model. The model explains only about 3 /2 percentof the variancein the first-difference in inflation, and the grid search reveals that zero is the best-fittingnonnegative value of the discount factor, implying an equation that reduces to Aw, = yy,. In this model, then, expectationsof future outputgaps do nothing to improve the equation's fit. The model's poor fit is illustrated graphically in Figure 1. The top panel of the figure plots the time series for the first-differenceof inflation, along with the time series for the model's fitted values. Because the change in inflationis such a volatile series, it is somewhat difficult to assess

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accuratelythe model's fit from this chart;hence, the lower panel of the figure presents a simple scatterdiagram.As can be seen from the almost randomdistributionof the data points, the ability of this model to predicteven the sign of the change in inflation is quite poor.6 B. Labor Share Model To test this version of the model, we augment our existing three-variableVAR with the log of the labor share. The results for this version of the hybridmodel are not much more encouraging. In this case, the grid searchreveals that the best-fitting hybrid model implies a value for 0/(1 - 0) of 0.97 (and thus 0 = 0.49), so the discounted sum does not vanish. As is illustratedin Figure 2, however, this model does an even worse job than the output gap model in fitting the first difference of inflation (its R2 is only 0.01). In addition, a simple regression of AwIr,on the discounted sum of labor income shares yields a t-statisticof only 1.40. Because the explanatoryvariablein this case is a generated regressor, and because we are arbitrarily treatingthe discount factor as known, this statistic cannotbe interpretedas being drawnfrom a standarddistribution (an issue that we will address in Section IV). But, together with the model's low R2, these results serve to question whetherthere is statisticalevidencefor any link betweenthe first-differenceof inflation and current and futurevalues of the laborincome share. These findingsunderscorea point made in the previous section; namely, that a positive estimate of 0 should not on its own be construedas evidence that the forward-looking component of these models adds anything to the models' overall fit (even when the estimate of 0 is obtainedfrom a procedurethatimposes the modelconsistent solution). While we do not have the space to report these results here, we note in passing that our finding that both the outputgap and labor share models fit poorly is robustto variouschanges in specification, including the use of alternative 6 The fact thatthe model cannotpredictthe magnitudeof these inflationchanges can also be seen from the scatterplot: while the x-axis, which plots actualchanges in inflation,has a range of 15 percentage points, the fitted values on the y-axis have a range of less than 2 percentagepoints.

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8 Change in inflation Fittedvalue

6 4 2 c

0

c

-2 -4 -6 -8

1960

1965

1970

1975

1980

1985

1990

1995

2000

0.8 0.6 0.4 0.2 -0.0 cD

cD

-0.2 -0.4 -0.6 -0.8 -1.0 -7.5

-5.0

-2.5

0.0

2.5

5.0

7.5

Actualchange in inflation,percentage points FIGURE 1. FIT FROM REGRESSING CHANGE IN INFLATION ON DETRENDED OUTPUT

inflation and output gap measures and estimation over pre- and post-1983 subsamples. C. Comparisonwith Reduced-FormRegressions Of course, because the first-differenceof inflation is such a volatile variable,we would not

necessarily expect such parsimoniousspecifications as these to fit very well. That said, a useful benchmarkthat illustratesjust how poor these models are can be obtained from a simple regression of Ari on a constant and its own lag. This regressionhas an adjustedR2 of 0.14; its fit is illustratedgraphicallyin Figure 3. While it is

THEAMERICANECONOMICREVIEW

310

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8 Change in inflation Fittedvalue

6 4 2 C")

0

C")

-2

-4 -6 -8 1960

1965

1970

1975

1980

1985

1990

1995

2000

0.6

0.4

0.2

a(

-0.0

a( -0.2

-0.4

-0.6 -7.5

-5.0

-2.5

0.0

2.5

5.0

7.5

Actualchange in inflation,percentage points LABORSHAREHYBRIDMODEL FIGURE2. FIT FORCHANGEIN INFLATION,

difficultto predictthe exact magnitudesof quarterly changes in inflation,this model does much betterthaneitherof the hybridmodels in matching the direction and size of these changes. The simple regressionachieves this improvement in fit by capturingan importantfeatureof inflationdynamicsthatis absentfrom the hybrid

model. The coefficient on the lagged change in inflation in this regression is -0.38, which reflects the fact thatthe first-differenceof inflation is negatively autocorrelated.In contrast,the discounted sum of the output gap (which here is merely the output gap itself) and of the labor income share are both highly positively auto-

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8 Changein inflation Fittedvalue

6 4 2 6 0 6

-2 -4

-6 -8 1960

1965

1970

1975

1980

1990

1985

1995

2000

3

2

1 (I (I

-1

-2

-3 -7.5

-5.0

-2.5

0.0

2.5

5.0

7.5

Actualchangeininflation, percentagepoints 3. FITFORCHANGE ININFLATION, FIGURE AR(1) MODEL

correlated,with first-orderautocorrelationcoefficients that exceed 0.9. Hence, the discounted sums fundamentallyfail to describe a key feature of the Aw, process. Table 1 reports some additional reducedform regressions for Avr,.Adding a second lag (column 2) raises the regression'sR2 a touch, to

0.15. More interestingly, the inclusion of the output gap also improves the fit of this regression: for the two-lag case, the R2 is 0.22 and the output gap's t-statistic equals 4.06. In contrast, the additionof the labor income share (column 4) yields essentially no improvement in this regression's fit. These patternsdemonstratethat

312

THEAMERICANECONOMICREVIEW

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TABLE 1--ESTIMATEDREDUCED-FORM MODELSFORArts

Included variables

A 1T_

Specification 1

2

3

4

5

-0.378**

-0.422**

-0.488**

-0.425**

-0.490**

(0.072)

(0.077) -0.119 (0.078)

(0.075) -0.179* (0.076) 0.122**

(0.077) -0.122 (0.078)

(0.077) -0.167* (0.071) 0.080**

AT,- 2

Yt

(0.030)

s,

2.302 (5.666)

Scorn Tt--1

(0.029)

0.031** (0.007) 0.006

cornm

(0.008) R2

0.138

0.145

0.218

0.141

0.315

Notes: y, detrendedoutput,s, - labor's shareof income, wrom - commodityprice inflation.Standarderrorsin parentheses; ** or * denotes significant at 1- or 5-percentlevel, respectively.

the ability of a standardreduced-formPhillips curve regression-which relates the level of inflation to its own lags (restrictingthe sum to one) and a measure of slack such as the output gap-to replicate importantaspects of the empirical behavior of inflation is not at all shared by these hybrid sticky-pricemodels.7 Finally, column 5 of Table 1 reports the effects of adding two lags of commodity price inflation to the basic reduced-form specification, where commodityprices are defined as the producer price index for crude materials. The purpose of adding this variable is to assess to what degree the observed negative autocorrelation in A r, reflects volatility in commodity prices. It seems unlikely that the kinds of frictions envisaged by sticky-pricemodels hold for these types of prices, which are often determined in auction markets. And, as might be expected for a competitively determinedprice, changes in commodity prices are quite random. As a result, one would expect the change in commodity price inflation to be negatively autocorrelated,and this patterndoes indeed hold in the data. Table 1 shows, however, that while including commodity prices improves the fit of the reduced-formregression, with the R2 rising

7 See Douglas Staigeret al. (1997) and RobertJ. Gordon (1998) for two typical implementationsof a reduced-form Phillips curve.

to 0.32 (see also Figure 4), it does little to alter the pattern of negative coefficients on the lagged changes in inflation. D. Results Using Annual Data An additionalfactor that could contributeto the negative autocorrelationthat we observe in Arrt is the presence of serially uncorrelated measurementerror(or some other type of transitory high-frequencyshock) in inflation.Noise of this sort would have an effect similarto that describedabove for commodityprices,and could act to obscureany relationshipbetweenthe firstdifferenceof inflationand the discountedsum of the drivingvariable. To test this possibility, we use annualdata to reestimatethe output gap and labor share variants of the hybrid model. When we do so, we find that none of our principal conclusions is altered; in particular,we still find that the expected discounted sum of the labor income shareexplains very little of the variancein AiTt, while the best-fittingvalue of the discount factor in the version of the hybridmodel that uses detrended GDP remains zero (thus implying that forward-lookingbehavioris completely absent from the model). The reason for the hybrid model's inability to fit annual data is closely related to the source of the model's failure in quarterly

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8 Changein inflation Fittedvalue

6 4

0

2 0

0

-2 -4 -6

-8

1960

1965

1970

1975

1980

1985

1990

1995

2000

4 3 2

1 0)

0

0)

-1 -2 -3 -4 -7.5

-5.0

-2.5

0.0

2.5

5.0

7.5

Actualchangein inflation, percentagepoints FIGURE4. FIT FORCHANGEIN INFLATION, REDUCED-FORM MODELWITHCOMMODITY PRICES

data. Recall that, in quarterly data, Airt was negatively autocorrelatedwhile the estimated discounted sum of the driving term was highly positively autocorrelated. Using annual data smooths away much of the highfrequency variation in Aw,, and leaves the first-difference of inflation essentially uncor-

related with its own lags. However, the discounted sums of both the output gap and labor's share remain strongly positively autocorrelated at an annual frequency. Hence, our finding that the hybrid model provides a poor characterization of the A7T,process does not depend on the use of quarterly data.

314

THEAMERICANECONOMICREVIEW

E. Summary The results of this section can be summarized as follows. * The popularclass of hybridmodels for which 0 : 1/2can generate predicted series for the level of inflation that are both highly correlated with actual inflation (for either driving variable,this correlationequals 0.85 in quarterly data) and highly autocorrelated. * There appears,however, to be very little evidence that the models' success in matching the level of inflation requires any of the rational forward-lookingbehavior posited by the hybrid models. In particular,the prediction of these models that distinguishes them from backward-lookingalternatives-that the change in inflation should move with a discounted sum of output gaps or labor income shares-is strongly rejected. * Moreover, these specifications completely fail to captureimportantfeatures of the data that can be summarizedby simple reducedform Phillips curves thatfeaturethe GDP gap and several lags of inflation. These results still leave some importantquestions unanswered. The first involves the certainty with which we can rule out the presence of forward-lookingbehaviorin the hybridinflation specifications:we have not yet been able to formallyassess the statisticalsignificanceof the discounted sum. The second issue relates to whether a patched-up version of the class of o -based, for example, hybridmodelswith 0 12 on an alternativerule-of-thumbfor backwardlooking agents-can do better in matching the data, perhaps thereby revealing an important behavior.Finally,thereis role for forward-looking the question of how models based on the assumption of 0 > 1/2perform.These topics are addressednext. IV. GMMEstimation

MARCH2006

used to derive statistical inferences about the model's parameters-in particular,we cannot determinewhetherthe discounted sums of output gaps or labor shares make a statistically significant contribution to observed inflation dynamics. GMM provides an alternative methodology that does not suffer from this problem. While GMM does not yield an explicit predictedseries for Ar, (and thus does not allow an assessment of the model's fit), it has the advantageof not requiringus to specify an explicit process for the driving term y,. And GMM allows us to estimate A, and A2consistently (with their standard errors)in the closed-form representation oo

(16)

= AI

AwT,

AkEyt+k. k

=

0

Note thatwe have deliberatelywrittenour equation for GMM estimation in this form as opposed to in the form of equation (13). This is because we are interestedin assessing directly whetherA1is statisticallysignificant,ratherthan in testing hypotheses about the coefficients y and 0. While one set of estimatesclearly implies values for the other (and we report both), the questionwe are asking is whetherthe composite forward-lookingterm has a statisticallysignificant effect on inflation-which in turn is a direct question about the statistical significance of A1. A. The Basic Hybrid Model GMM estimationof equation(16) requiresus to specify a set of instrumentsz, that are known by agents at time t. Underrationalexpectations, the orthogonalitycondition

(17)

E (ATr,

-

1 Akt +k Zt = 0 k=0

The usefulness of the Campbell-Shillerapproach comes from its ability to provide an explicit predictionfor the values of Air, that are implied by the hybrid model. However, one drawback of this method is that it cannot be

should hold in the data. One practicalissue that must be dealt with is the presence of an infinite sum in (17); we address this problem by following the approach of Rudd and Whelan

VOL.96 NO. I TABLE2--GMM

RUDD AND WHELAN:STICKY-PRICE MODELS OF HYBRIDINFLATION ESTIMATES EQUATION

Reduced-form parameters

Driving

variable(y,)

Structural parameters

A2

A,

Y 0 0.039 0.614 0.024 0.381** (0.035) (0.372) (0.027) (0.143) Labor income share 0.017 0.769 0.009 0.435** (0.030) (0.498) (0.019) (0.159)

Detrendedoutput

Notes: Table gives estimated parametervalues from the basic hybridmodel Arr, = A, I A'2EY,+i,with structural parametersimplicitly defined as At = y/(l - 0) and A2 0/(1 - 0). Standarderrorsin parentheses;** or * denotes significant at 1- or 5-percent level, respectively.

condi(2005b)andrewritingthe orthogonality tionsas

(18)

E (1 T, m

- AK+1 2

K[

K

k=O

t

+K+1

ktYt+k

Z

Z

0.

The estimates of A1and A2that we obtain using this procedureare reportedin Table 2. For the models that use labor's share as a proxy for Yt, the instrumentset zt consists of two lags each of the change in inflation,the outputgap, the labor share, and wage inflation (measuredas the logdifference in nonfarmcompensationper hour). When detrendedoutput is used as the driving term, we replace log-differenced hourly compensation-which makes no contribution to first-stage fit-with the federal funds rate, which is a highly significant predictor in the first-stageregressions. We set K equal to 12. The results confirm an empiricalfinding that was suggested by our VAR-basedexercises: for both the output-gapand labor-shareversions of the models, the estimated values of A, are not statisticallydifferentfrom zero. Hence, not only do the discounted sums of future labor shares and output gaps explain very little of the variation in A ,rt,they actually appear to have no statistically discernable influence on this variable whatsoever. This finding was robust to the value of K

315

used, as well as to differentdefinitionsof inflation and detrendedoutput, and estimation over pre- and post-1983 subsamples.In addition,this result was robust to the specific instrumentset used: A, was estimatedto be statistically insignificant across a wide range of instrumentsets that included various lags of additionalinstruments such as commodity price inflation, yield spreads, and short-terminterestrates. Table 2 also reportsthe estimates of y and 0 obtained from applying GMM estimation to equation (13). Both the output gap and labor share versions of the model imply estimates of 0 that are significantly greater than zero. It should be stressed,however, thatthe estimateof 0 obtained from this procedureis only a function of the estimated forwardroot A2 (because here 0 = A2/(1 + 2)). The fact that we obtain a significant value of 0 suggests that a discounted sum with a nonzero discount factor may yield the best-fitting model. But even this best-fitting discounted sum may make no significantcontributionto explainingthe change in inflation-and, indeed, in both of the cases considered here we are unable to reject the hypothesis thatthe coefficients on the discountedsums are zero. (This result closely parallelsthe VARbased resultsfor the laborsharemodel, in which the grid search selected a nonzero value of 0 even though the discounted sum made no contributionto the model's fit.) B. More General Hybrid Models Our earlierresultssuggest one potentialroute for improvingthe performanceof this model.Table 1 shows that an implicitassumptionunderlying the simple hybridspecification-namely, that a single lag of inflationwouldallow incorporating the model to matchthe empiricalnatureof inflation inertia-was incorrect.In particular,the negative autocorrelationof that the Arrtofimplies model for the level inflation should underlying include more thanone lag of the dependentvariable. One way to addressthis is to assumethatthe underlyingstructuralequationcontains an additional inflationlag, therebytakingthe form: (19)

rr, t=

T _T1+ 02 rt _2

+ (1 - 01 + YYt. 02)Etrt+,,

316 TABLE3--GMM

then, these results do little to endorse the presence of forward-lookingrational expectations, and thus the case for a more complex hybrid model featuringextra lags of inflation.

HYBRID ESTIMATES OF AUGMENTED INFLATION EQUATION

Driving variable (y,) Detrendedoutput Labor income share

A, 0.146"* (0.048) 0.024 (0.036)

A2

A3

-0.990** (0.050) 0.764 (0.465)

-0.364* (0.141) -0.392** (0.053)

Notes: Table gives estimatedvalues for the parametersfrom the augmented hybrid model Air, = Al

+

'=o

* denotes Standarderrors in parentheses;** orA'2Et,Y, hA3ATr-1. significant at 1- or 5-percent level, respectively.

Such a specificationcould be motivated,for example, by assuming a fraction of nonrational price-setterswho use the last two observationsof inflation to formulate their expectations, orwithinthe Christianoet al. (2005) framework-a morecomplex indexationrulefor those firmsthat do not set an optimalprice this period. Equation(19) has the following closed-form solution:

V. The HybridSpecificationwith 0 > 1/2 The versions of the hybrid model that we have consideredup to this point involve values of 0 that are less than or equal to one-half. We now examine the version of the model for which 0 > 1/2. Specifically, we examine the role played by the forward-lookingterm in the following closed-form solution for the level of inflation:

kk=O0 =

As before, and for the same reasons, we focus on estimating the equation in this form, rather than in the form given by equation (14). A. VAR-BasedMethod

oo

(20)

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THEAMERICANECONOMICREVIEW

A7T,= Al1

AkE,yt+k + A3ATt-

1

k=O

where the parametersA1, A2, and A3 represent highly nonlinear functions of the underlying parameters01, 02, and y. In Table 3, we report GMM estimates of A1, A2, and A3 that are obtainedusing the same procedureand the same instrument sets that were used inoestimating equation (18). Again, the key question is whether we obtain statistically significant and economically sensible values for A1and A2(i.e., whether allowing for extra lags of inflationimproves the case for the existence of a forwardlooking rationalexpectationsterm). As expected, Table 3 indicates that the coefficient on I is negative and highly statisAr,t_ But this exercise still fails to tically significant. produce any convincing evidence of forwardlooking behavior.For the output-gapversion of the model, the coefficient on the discounted sum, A1, is statisticallysignificant,but the estimated forward root, A2, is negative, which is not reasonable in this context. For the labor shareversion, the estimatedforwardroot is positive, but the coefficient on the discounted sum receives a t-statisticof only 0.65. On the whole,

Figure 5 summarizes the results obtained from applying the Campbell-Shillermethodology to assess the contributionof the forwardlooking term in equation(21). We again run the VAR specificationsdescribedin Section III and measurethe forward-lookingterm as

(22)

E E,y,+k = e'(I - A)-lz,. k=O

In this case, we do not need to estimate the best-fittingforwardroot because this model imposes the assumption that the forward root is one. Instead,we run a regressionof inflationon its own lag and our measure of the discounted sum to arrive at our estimate of PL2; i.e., we estimate ji, and A2from (23)

7T,= p~le'(I -A)-lz,

+ /2127T,_ 1

Ourresultssuggest an extremelylimitedrole for the forward-lookingterms in determining the behaviorof inflation.For the outputgap model, adding the discounted sum improves the fit somewhat, but the estimated coefficient jI has

VOL.96 NO. 1

RUDD AND WHELAN:STICKY-PRICE MODELS

317

15.0 Inflation Fittedvalue

12.5 10.0 7.5 r

5.0 2.5 0.0 -2.5 1960

1965

1970

1975

1980

1985

1990

1995

2000

A. OutputGap Version

15.0

Inflation Fittedvalue

12.5 10.0 7.5 CL

5.0 2.5 0.0 -2.5 1960

1965

1970

1975

1980

1985

1990

1995

2000

B. LaborShare Version FIGURE 5. FITOFHYBRID MODEL WITH0 > 1/2

an incorrect(negative) sign. For the labor share case, the model has an R2 of 0.71, which is exactly the same as what is obtained from a regression of inflation on its own lag only.8

8 Experimentationwith various specificationsfor the labor share VAR showed that some yield series for e'(I -

It is worthemphasizinghere that focusing on the implied estimates of 0 will again yield a A)-lz, that can improve the model's fit somewhat. In each case, however, these VARs requiredexclusion restrictions that were strongly rejected by the data-in particular,exclusion of the output gap, which is invariablyhighly statistically significantin the labor share equation of the VAR.

318

THEAMERICANECONOMICREVIEW TABLE4-GMM

OF HYBRIDINFLATION ESTIMATES

EQUATION WITH 0

Driving variable

>

/2

Reduced-form parameters

Structural parameters

Y 0 A1k2 1 (yt) 0.617** Detrendedoutput -0.007 0.622** -0.004 (0.003) (0.026) (0.005) (0.069) 0.011 0.674** Labor income 0.017 0.485** share (0.010) (0.033) (0.014) (0.072) Notes: Table gives estimatedvalues for the parametersfrom T = t1 -`oEyt+i + the alternativebasic hybrid model 7r, 2 Tr- 1, with structuralparametersimplicitly defined as ,1= y/6 and /2 = (1 - 0)/0. Standarderrorsin parentheses; ** or * denotes significant at 1- or 5-percent level, respectively.

MARCH2006

odology and instrument set as before. (Note, though, that here we use lagged 7r,as an instrument in lieu of lagged Awr,.)Once again, we find that p1 is not statisticallysignificant,no matter which measure of the driving variable we use, which in turn implies that forward-lookingbehavior (as summarizedby the sum of current and expected future values of y,) plays no discernable empirical role in determining inflation.1oMoreover,this result obtains even if we allow the equationto include additionallags of inflation, as in

(24) 7r T/,= i=O

misleadingpictureof the importanceof forwardlooking expectations.For the laborsharemodel, our proceduregives an estimate for g2 of 0.85. Noting from equation(14) that92 = (1 - 0)/0, this is consistent with a value for 0 of 0.54. However, the estimated 0 from this procedure will be greater than or equal to 0.5 as long as our estimate of 9t2 is less than or equal to one. Thus, these high estimates of 0 tell us little about the importance of forward-looking behavior-instead, they tell us merely that the empirical inflation process is not explosive. These observationshelp to reconcile our assessment of the hybrid model with the more positive assessment of Gall and Gertler(1999). Figure 5 shows that models derived from the assumption of a preponderance of forwardlooking behavior in price setting (i.e., models with 0 > 0.5) are capable of fitting the level of inflationquite well.9 However, our calculations show that this good fit has little to do with the link between inflation and expected futureeconomic conditions, but instead reflects the substantial role that these models still allow for lagged inflation. B. GMMEstimation Table 4 presents the results that obtain from estimating equation (21) using the same meth-

9 The bottom panel of Figure 5 is analogous to Figure 2 of Gall and Gertler(1999), which compares actual inflation with what they term "fundamentalinflation."

E,y,t+

+ +1J27t-1 ++ AL3Tt-2,

in that g, remains statistically insignificant in this specification (see Table 5). We note that our results were again found to be robust across a wide range of variationsin the specification,includingchanges in the value of K, the use of alternativemeasures of price inflation and detrended output, and a sample break in 1983. In addition, these results are robustto the instrumentset used in that t1 was found to be statistically insignificantfor every instrumentset that passed the criterion for instrumentrelevancediscussed by JamesH. Stock and Motohiro Yogo (2002). VI. Conclusions The observation that lagged inflation plays an importantrole in empirical inflation regressions poses a major challenge to the rationalexpectations sticky-price models that underpin the new-KeynesianPhillips curve. Indeed,it has now become relativelywell acceptedthatpurely forward-lookingmodels of inflation cannot account for the degree of inflationinertiathat we actuallyobserve in the data, and thatthis failure significantly reduces these models' usefulness in assessing practical policy questions. In response, researchershave increasingly adopted 10

The implied values of 0 shown in Table 4 are statistically significant. Note again, however, that this tells us nothing aboutthe role played by forward-lookingbehavior: given the empirical importance of lagged inflation-and, hence, of h,2-in our estimated equation, we would invariably expect to obtain a statisticallysignificantvalue of 0.

VOL.96 NO. I

RUDD AND WHELAN:STICKY-PRICE MODELS

OF AUGMENTED TABLE5---GMM ESTIMATES HYBRID INFLATION EQUATIONWITH0 > 1/2

Driving variable (y,) Detrendedoutput Labor income share

,I

0.001 (0.005) 0.025 (0.015)

L2

113

0.429** (0.051) 0.337** (0.071)

0.325** (0.051) 0.247** (0.057)

Notes: Table gives estimatedvalues for the parametersfrom the alternativeaugmentedhybridmodel 7r,= I1o Eyr+i + -g1 1127 t-1 + k37rt-2. Standarderrorsin parentheses;** or * denotes significant at 1- or 5-percent level, respectively.

hybrid pricing specifications, in which lagged inflation is allowed to have an explicit role in pricing behavior. This class of model is widely seen as striking a reasonable compromise between the desire to fit a key empirical characteristic of the inflationprocess (its inertia),and the desire to preserve an important role for forward-looking,rational expectations in price setting. The goal of this paperhas been to determine whether this reformulationof the basic stickyprice model yields a pricing specificationthat is capable of capturingempirical inflation behavior. We have shown thatthe hybridspecification generatesprecise predictionsaboutthe inflation process that are easily tested-and firmly rejected. In fact, we find no evidence in postwar U.S. datathatinflationdynamicsreflectthe type of rational forward-looking behavior that the model hypothesizes. Hence, while the addition of a lagged inflation term permits the hybrid model to better capture certain features of the inflation process, ultimatelythis fix is cosmetic in that the feature of the model that truly distinguishes it from alternative models of inflation-such as a traditionalPhillips curve based on backward-lookingexpectations-appears to be empirically irrelevant. One conclusion thatcan be drawnfrom these results is that the hybrid model's approachto patching up the new-Keynesian Phillips curve-which involves a direct attemptto deal with its persistence problem-may merely be addressinga symptomof what is in fact a much more deeply rooted problem with this type of model. Specifically, our findings suggest that pricing models of this sort suffer from a more serious (and less easily addressed)weakness--

319

namely, their reliance on a strict form of rational expectations. The new-Keynesian inflation equation makes three assumptions about pricesetting behavior: first, that prices are sticky; second, that agents optimize their behavior given that their prices are fixed; and third, that agents' expectations are formulatedin a rational-i.e., model-consistent-manner. Empirical studies suggest that a significant degree of price stickiness is present in the U.S. economy, and thusthatfirmsalmostsurelyattemptto make some predictionabout futureinflationwhen determining their currentprice. What appears to be less reasonable,however, is the assumption that these predictionsare formulatedin the manner implied by the new-Keynesianmodel under rational expectations. Put differently,it may well be thatEtrt,, has an importantinfluence on currentinflation.But if this is so, the evidence indicates that this expectationis not determinedin the mannerthat the current generation of rational expectations sticky-price models would predict. This conclusion does not rule out a role for some sort of rational optimizing behavior in explaining inflation dynamics; indeed, there may be an rationalefor why the reducedoptimization-based form Phillips curve models discussed in this paperfit so well. For example, in the absence of any agreement among economists on what the correct models for inflation (or the rest of the economy) actually are, and given most individuals' limited ability to understand or model these uncertainties,a procedurein which agents base their expectations for future inflation on extrapolationsof the recent past may itself constitute a form of optimizing behavior. We conclude, then, that further research in this area is probablybest aimed toward developing models that deviate from the standard rationalexpectations frameworkin favor of alternative descriptions of how agents process informationand develop forecasts. Work in this vein by ChristopherA. Sims (1998, 2003) and N. Gregory Mankiw and Ricardo Reis (2002) may proveto be a promisingstartin this direction. REFERENCES Calvo,GuillermoA. "StaggeredPrices in a Utility-Maximizing Framework." Journal of

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MonetaryEconomics, 1983, 12(3), pp. 38398. Campbell,JohnY. and Shiller,RobertJ. "Cointegration and Tests of Present Value Models." Journal of Political Economy, 1987, 95(5), pp. 1062-88. Casares,Migueland McCallum,BennettT. "An Optimizing IS-LM Frameworkwith Endogenous Investment."National Bureau of Economic Research, Inc., NBER Working Papers:No. 7908, 2000. Christiano, Lawrence J.; Eichenbaum,Martin and Evans, Charles L. "Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy." Journal of Political Economy, 2005, 113(1), pp. 1-45. Clarida,Richard;Gall, Jordi and Gertler,Mark. "The Science of Monetary Policy: A New KeynesianPerspective."Journal of Economic Literature,1999, 37(4), pp. 1661-1707. Cochrane,JohnH. "Shocks."Carnegie-Rochester Conference Series on Public Policy, 1994, 41(0), pp. 295-364. Ehrmann,Michaeland Smets,Frank."Uncertain Potential Output:Implications for Monetary Policy." Journal of Economic Dynamics and Control, 2003, 27(9), pp. 1611-38. Fuhrer, Jeffery C. "The (Un)Importance of Forward-Looking Behavior in Price Specifications." Journal of Money, Credit, and Banking, 1997, 29(3), pp. 338-50. Fuhrer,Jeffery C. and Moore,GeorgeR. "Inflation Persistence."QuarterlyJournal of Economics, 1995, 110(1), pp. 127-59. Gali,Jordiand Gertler,Mark."InflationDynamics: A Structural Econometric Analysis." Journal of MonetaryEconomics, 1999, 44(2), pp. 195-222. Gordon, Robert J. "Foundationsof the Goldilocks Economy: Supply Shocks and the Time-VaryingNAIRU."BrookingsPapers on EconomicActivity,1998, 0(2), pp. 297-333. Guerrieri,Luca. "The Inflation Persistence of Staggered Contracts."US Federal Reserve Board, InternationalFinance Discussion Paper Series: No. 2002-734, 2002. Mankiw,N. Gregoryand Reis, Ricardo."Sticky Informationversus Sticky Prices: A Proposal

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to Replace the New Keynesian Phillips Curve." Quarterly Journal of Economics, 2002, 117(4), pp. 1295-1328. Rotemberg,Julio J. and Woodford,Michael."An Optimization-BasedEconometricFramework for the Evaluationof MonetaryPolicy,"in Ben S. Bernanke and Julio J. Rotemberg, eds., NBER macroeconomicsannual 1997. Cambridge,MA: MIT Press, 1997, pp. 297-346. Rudd,Jeremyand Whelan,Karl. "Does Labor's Share Drive Inflation?"Journal of Money, Credit and Banking, 2005a, 37(2), pp. 297312. Rudd,Jeremyand Whelan,Karl. "New Tests of the New-Keynesian Phillips Curve."Journal of Monetary Economics, 2005b, 52(6), pp. 1167-1181. Rudebusch,Glenn D. "Assessing Nominal Income Rules for MonetaryPolicy with Model and Data Uncertainty."Economic Journal, 2002, 112(479), pp. 402-32. Sims, ChristopherA. "Stickiness." CarnegieRochester ConferenceSeries on Public Policy, 1998, 49(0), pp. 317-56. Sims, ChristopherA. "Implicationsof Rational Inattention."Journal of Monetary Economics, 2003, 50(3), pp. 665-90. Staiger, Douglas; Stock, James H. and Watson, Mark W. "How Precise Are Estimates of the Natural Rate of Unemployment?"in Christina D. Romer and David H. Romer, eds., Reducing inflation:Motivation and strategy. Chicago: University of Chicago Press, 1997, pp. 195-242. Stock, James H. and Yogo, Motohiro."Testing for Weak Instrumentsin Linear IV Regression." National Bureau of Economic Research, Inc., NBER Technical Working Papers:No. 284, 2002. Taylor, John B. "Staggered Price and Wage Setting in Macroeconomics,"in John B. Taylor and Michael Woodford, eds., Handbook of macroeconomics. Vol. lB. Amsterdam: Elsevier Science, North-Holland, 1999, pp. 1009-50. Woodford,Michael.Interest and prices: Foundations of a theory of monetary policy. Princeton:PrincetonUniversity Press, 2003.