Rational Error Correction

RATIONALERRORCORRECTION P.A.Tinsley (cid:3) DivisionofResearch andStatistics Federal ReserveBoard WashingtonDC20551 ptinsley@frb.gov version: July1998 Abstract: Under general conditions,linear decisionrules of agents with rational expectations are equivalent to restricted error corrections. However, empirical rejections of rational expectation restrictions are the rule, rather than the exception, in macroeconomics. Rejections often are conditionedontheassumptionthatagentsaimtosmoothonlythelevelsofactionsoraresubjectto geometric random delays. Generalizations of dynamic frictions on agent activities are suggested that yield closed-form, higher-order decision rules with improved statistical fits and infrequent rejections of rational expectations restrictions. Properties of these generalized “rational” error corrections areillustratedfor producerpricinginmaufacturingindustries. Keywords: Companionsystems,error correction,producerpricing,rationalexpectations JEL classifications: C5, E3 (cid:3) I amindebtedforcommentsonanearlierversionofthispaperbycolleaguesandothers, especiallyF. Diebold, S. Johansen, and S. Kozicki. Views presented are those of the author and do not necessarily representthose of the FederalReserveBoard.

1 Intertemporal decision rules are indispensable in rational agent interpretations of macroeconomic behavior where a distinction is drawn between agent perceptions, summarized by agent forecasts of market events, and agent responses, subjected to dynamic constraints or “frictions.” In theoretical macroeconomic models since Lucas (1976), rational expectations have becomethebenchmarkstandardforrepresentingtheunobservedforecasts ofagents. Unfortunately,therecordforempiricalimplementationofrationalexpectationsmodelsremains dismal. A survey of existent journal publications by Ericsson and Irons (1995) summarizes an extensive accumulation of empirical evidence against rational expectations, including frequent rejections of rational expectations overidentifying restrictions. A review of policy simulation models used by central banks and international agencies, such as documented in Bryant et al. (1993), indicates that many key rational expectations specifications are either imposed or fit by roughempirical calibrations. Macroeconomists have adopted a variety of responses to the absence of strong empirical support for rational expectations. One is to maintain the rational expectations hypothesis, while aiming to interpret a more limited subset of empirical regularities as discussed by Kydland and Prescott (1991). Another approach is to view rational expectations as a limitingcase of complete information in a more general treatment of the information processing abilities of agents, such as the “bounded rationality” models of learning reviewed in Sargent (1993). Closely related is the positionthatrationalexpectationsare morelikelytoprevailat lowfrequencies,aviewcompatible withtestsoflong-runtheoreticalrestrictionsincointegratingrelationships,asdiscussedbyWatson (1994). Others reject the hypothesis of model-based rational expectations, such as the use in EricssonandHendry(1991)ofrule-of-thumbextrapolations. This paper examines an alternative explanation for the poor empirical properties of rational expectations models. Because most rational expectations restrictions are inherently dynamic due to the forecasting requirements of constraints on dynamic adjustment, a plausible source of difficulty could be the sharp friction priors typically imposed on agent responses. The standard dynamic specification in rational expectations models utilizes geometric lead response schedules to anticipated future events and geometric lag responses to recent “news.” Because this two-sided geometric response schedule is not a clear implication of economic theory, a generalized polynomial frictions specification is explored in this paper. Suggested interpretations of generalized frictions range from costs of adjusting weighted averages of current and lagged actionstoconvolutionsofgeometricrandom delaydistributionsofagent responses. Fordifference-stationaryvariables,thedecisionrulesbasedongeneralizedfrictionsareshown to be isomorphic to a class of “rational” error correction models.1 The parameters of the 1Discussion in this paper is aimed primarily at decision rules for difference-stationaryvariables, a specification thatisnotrejectedbystandardtestsoftheintegrationorderofmostmacroeconomicaggregatesinpostwarsamples.

2 decision rules are subject to tight cross-coefficient restrictions due to polynomial frictions and cross-equationrestrictions due to the assumptionof rational expectations. A closed-form solution that incorporates these restrictions is derived using two companion matrix systems, one a lead system for the forward planning required by polynomial frictions and the other a lag system associatedwiththeagents' forecast model.2 Rational error corrections under polynomial frictions inherit many desirable properties of atheoreticreduced-formtime-seriesmodels,includingseriallyindependentequationresidualsand small standard errors relative to many theory-based alternatives. However, whereas conventional error corrections and VARs have been criticized for in-sample overfitting attributable to large numbers of estimated parameters, rational error corrections are subject to many overidentifying restrictionswhichsubstantiallyreducethenumberoffree parameters. To provide concrete illustrations of the consequences of restrictive priors on adjustment costs, linear decision rules associated with alternative specifications of frictions are estimated for producer pricing in several manufacturing industries. The basic specification is based on the assumptions of difference-stationary producer prices and stationary price markups over costs of production,assumptionsthatare notrejectedforpostwarU.S.producerprices.3 Adifficultywithinterpretingmanyreportedrejectionsofrationalexpectationsoveridentifying restrictions is that the rejections could be due also to misspecified models. An advantage of the side-by-side comparisons reported in this paper is that several empirical problems, including rejectionsofrationalexpectationsrestrictions,areunambiguouslylinkedtotheuseofsecond-order Eulerequationsrather thanhigher-orderEulerequations. The paper is organized as follows. Section I summarizes dynamic properties of the restricted error correction that is implied by the standard decision rule with geometric response schedules. Section II derives the error correction format of rational decision rules implied by a generalized polynomial description of frictions. Several interpretations of polynomial frictions are suggested. Section III presents empirical estimates of decision rules for industry pricing, comparing the geometric responses of the standard decision rule with the responses of higher-order decision rules. Atractablemethodoftwo-stagemaximumlikelihoodestimationofrationalerrorcorrections 2Althoughthis paper developsa generalframeworkforformulatinghigher-ordererrorcorrectiondecision rules, thereareseveralprecedentsforrecastingdecisionrulesasrestrictederrorcorrectionsorapplyinghigher-orderlinear decision rules to macroeconomic aggregates. Nickell (1985) appears to be the first paper to explore similarities between second-orderdecision rules and error correction models. Generally, fourth-orderdecision rules have been confinedtoempiricalinventorymodels,suchasBlanchard(1983),Hall,Henry,andWren-Lewis(1986),Callen,Hall, and Henry (1990), and Cuthbertson and Gasparro (1993). Exceptions are the applications of fourth-orderdecision rulestoemploymentinthecoalindustrybyPesaran(1991)andtothepriceofmanufacturedgoodsinPrice(1992). 3State-independentfrictionsinpricingwouldnotbeexpectedtoholdoverallpossiblestates,suchasepisodesof extremehyperinflation. Nevertheless,lineartimeseriesmodelsappeartobeausefulwaytoanalyzestickypricingin periodsofmoderateinflation,vid. Sims(1992)andChristiano,Eichenbaum,andEvans(1994).

3 is derived in the Appendix, including corrections required for the sampling errors of the second stage. SectionIVconcludes. I.RationalError Correction under GeometricFrictions Just as one-sided polynomials in the lag operator are characteristic of atheoretic, linear time series models, two-sided polynomials in the lead and lag operators are a defining characteristic oflinearmodelsofrationalbehavior. Theprincipalvehicleofanalysisinthispaperisthedynamic first-orderconditionlinkingadecisionvariable, y t ,toitsequilibriumobjective, y (cid:3)t , E t f A ( L ) A ( B F ) y t (cid:0) A ( 1 ) A ( B ) y (cid:3)t g = 0 ; (1) where E t f : g denotes expectations based oninformationat the endof t (cid:0) 1 ; A ( L ) is a backwards scalar polynomialinthelagoperator, L j x t = x t (cid:0) j ,; A ( B F ) is aforward scalarpolynomialinthe leadoperator, F j x t = x t+ k ,;and B isadiscountfactor, 0 < B < 1 . In contrast to conventional backwards-looking time series models of the relationship between y t and y (cid:3)t ,anotablefeatureofequation(1)isthat theexpectationofthecurrent decisionvariable, E t f y t g = = E E t t f f A ( 1 X i= (cid:0) 1 1 ) A w ( B y i ) (cid:3)t+ A i ( g L ; ) (cid:0) 1 A ( B F ) (cid:0) 1 y (cid:3)t g ; (2) is a two-sided moving average of past and expected future values of the desired equilibrium, y (cid:3) . The agent response schedule, w i , determines the relative importance of past and future, where i= 1 P (cid:0) 1 w i = 1 . Manymodelsusedinmacroeconomicsassumetheserelativeimportanceweightsare adequatelyrepresented bytwo-sidedgeometricresponseschedules. In the case of linear decision rules, there are two prominent rationalizations of geometric responseschedules. Oneisthatchangesinthelevelofthedecisionvariablearesubjecttoquadratic adjustment costs, and this strictly convex friction induces geometric adjustments of the decision variable toward its equilibrium. The second interpretation is that each agent is subjected to a geometric distribution of random delays in adjustment, so that the level selected for a decision variableinagivenperiodisaweightedaverageofdesiredtargetsettingsovertheexpectedinterval between allowable resets. As noted by Rotemberg (1996), the assumptionof a geometric random delay distribution leads to aggregate behavior that is observationally equivalent to that generated bytheassumptionofquadraticcosts onadjustingtheleveloftheaggregatedecisionvariable. Under either interpretation, the required Euler equation is second-order, implying that the

4 polynomial components of equation (1) are first-order polynomials, A ( L ) = ( 1 (cid:0) (cid:21) L ) and A ( B F ) = ( 1 (cid:0) (cid:21) B F ) , where 0 < (cid:21) < 1 . As in Tinsley (1970), the optimal decision rule that satisfiestheEulerequationandtherelevantendpoint(initialandtransversality)conditionsimplies partial adjustment of the decision variable to a discounted weighted average of expected forward positions of the desired equilibrium.4 This decision rule solution is obtained by multiplying the Eulerequation(1)bytheinverseoftheleadpolynomial, A ( B F ) (cid:0) 1 , E t f (cid:1) y t g = (cid:0) A ( 1 ) ( y t (cid:0) 1 (cid:0) y (cid:3)t (cid:0) 1 ) + E t f A ( 1 ) 1 X i= 0 ( (cid:21) B ) i (cid:1) y (cid:3)t+ i g ; (3) where (cid:21) B is thegeometricdiscountfactorovertheinfiniteplanninghorizon. Tocompletethederivationoftheconventionaldecisionrule,thedatageneratingprocessofthe forcingterm, y (cid:3)t ,mustbeprovided. Inlineardecisionrules,thedecisionvariable, y ,iscointegrated withtheconditionalequilibriumtarget, y (cid:3) ,whichis definedbyalinearfunctionof q variables.5 y (cid:3)t = X j q = 1 b j x j ;t : (4) Agent forecasts of the target, y (cid:3)t+ i , are assumed in this paper to be generated by p -order VARs in the q arguments, x 0t = [ x 1 ;t ; : : : ; x q ;t ] . Thus, the effective information set of agents is obtained by stacking p lagsoftheregressors,intoasinglevector, z 0t (cid:0) 1 = [ x 0t (cid:0) 1 ; : : : ; x 0t (cid:0) p ] . Thecompanionform of the agent VAR forecast model is denoted by E t f z t g = H z t (cid:0) 1 .6 Consequently, agent forecasts oftheforcingterm oftheEulerequationaregenerated by E t f y (cid:3)t+ i g = (cid:19) 0 (cid:3) H i+ 1 z t (cid:0) 1 ; (5) wherethe p q (cid:2) 1 selectorvector, (cid:19) (cid:3) ,containsthecoefficients ofthecointegratingrelationship,(4), thatdefines theequilibriumobjective. Substitutingforecasts oftheforcingterm from(5)intothedecisionrule(3)yields E t f (cid:1) y t g = = (cid:0) (cid:0) A A ( ( 1 1 ) ) ( ( y y t (cid:0) t (cid:0) 1 1 (cid:0) (cid:0) y y (cid:3)t (cid:0) (cid:3)t (cid:0) 1 1 ) ) + + A h ( 1 0 z (cid:3) ) t (cid:0) 0 (cid:19) (cid:3) 1 : 1 X i= 0 ( (cid:21) B H ) i [ H (cid:0) I n ] z t (cid:0) 1 ; (6) 4Historicalreferencesanddiscussionofissuesinformulatinglineardecisionruleswithdynamicforcingtermsare foundinTinsley(1970,1971). 5As needed, the q variablesincludedeterministictrend, intercept, andseasonaldummyseries. The conceptofa dynamic,frictionlessequilibriumisdiscussedbyFrisch(1936). 6CompanionformsforavarietyoflinearforecastingmodelsareillustratedinSwamyandTinsley(1980).

5 Asshowninthefirstlineof(6),giventhematrixoftheforecastmodelcoefficients, H ,andthe discount factor, B , a singlefriction parameter determines a geometric pattern of rational dynamic responses, (cid:21) = 1 (cid:0) A ( 1 ) . The second line of (6) indicates that the weighted sum of expected forwardchangesintheforcingterm, E t f (cid:1) y (cid:3)t+ i g ,canbereducedtotheinnerproductofarestricted coefficient vectortimestheindustryinformationset h (cid:3) = ( 1 (cid:0) (cid:21) ) [ H 0 (cid:0) I n ] [ I n (cid:0) (cid:21) B H 0] (cid:0) 1 (cid:19) (cid:3) ; (7) where equation (7) provides a transparent summary of rational expectations overidentifying restrictionsonthecoefficient vectoroftheagentinformationvector, z t (cid:0) 1 . Thus, there are two principal differences between the dynamic format of the “rational” error correction in equation (6) and the format of a conventional error correction with p lags of each regressor. First, only one lag of the decision variable is specified by the second-order decision rule in equation (6), whereas up to p (cid:0) 1 additional lags of the first-difference of the decision variable may appear in a conventional error correction. Second, as indicated in equation (7), the coefficient vector, h (cid:3) ,is completelydeterminedbycross-coefficient restrictionsduetothefriction parameter in the error correction coefficient, (cid:21) , and cross-equation restrictions due to the forecast model coefficients, H . By contrast, the coefficients of the information vector in a conventional errorcorrectionare unrestricted.7 Residual independence and rational expectations overidentifying restrictions are frequently rejected in macroeconomic studies of rational behavior.8 One interpretation of the often disappointingempiricalperformancesofconventionaltwo-rootdecisionrules,includingrejections ofrationalexpectationrestrictions,issimplythat expectationsofactual agentsmaynotbeformed under conditions required for rational expectations, such as symmetric access to full system information by all agents. However, the limited dynamic specifications illustrated in equation (6) suggest another contributing factor—the arbitrary prior that agents responses are adequately captured by two-sided geometric response schedules. The next section explores the dynamic formatsofhigher-orderEulerequationsandrationalerrorcorrectionsassociatedwithapolynomial generalizationofagent responseschedules. 7In conventional error corrections, an unrestricted coefficient vector is applied to the first-difference of the informationvector. 8Examples of studies that report rejections of restrictions imposed by rational expectations include Sargent (1978),Meese(1980),Rotemberg(1982),PindyckandRotemberg(1983),andShapiro(1986). Significantresidual autocorrelationsareindicatedforrationalexpectationsdecisionrulesinEpsteinandDenny(1983),AbelandBlanchard (1986), and Muscatelli (1989). See also the extensive rational expectations literature review in Ericsson and Irons (1995).

6 II.RationalError Corrections under PolynomialFrictions The standard second-order Euler equation provides a two-sided geometric description of agent responses to anticipated and past events. The geometric schedules are determined by the roots of the first-order component polynomials, A(L) and A(BF). This section discusses decision rules associated with higher-order Euler equations, where the degree of the component polynomials is increased to m > 1 . These m -order polynomialsare obtained by relaxing the modelingprior that agents aim to smooth only the levels of decision variables or, equivalently, that stochastic delays ofdecisionvariableadjustmentaregeneratedonlybygeometricdistributions. Thefirstsubsection derives the closed form of rational error correction decision rules associated with higher-order Eulerequations. Thesecondsubsectionbrieflyreviewssomecategoriesofpolynomialfrictionson agent actionsthatare consistentwith 2 m -order Eulerequations. II.1SolvingforRationalErrorCorrectionDecisionRuleswithHigher-OrderEulerEquations As demonstratedlaterinthissection,theEulerequationunderpolynomialfrictions isthesameas that initially shown in equation ( 1 ), except the factor polynomials are now m -order polynomials, A ( L ) = 1 + a 1 L + : : : + a m L m ,insteadoffirst-orderpolynomials. To obtainthe decisionrulein the case of 2 m -order Euler equations,multiplybythe inverseof theleadpolynomial, A ( B F ) (cid:0) 1 ,togive E t f A ( L ) y t g = (cid:17) E E t t f f A f ( t g B : F ) (cid:0) 1 A ( B ) A ( 1 ) y (cid:3)t g ; (8) The analytical solution for the forcing term of this equation, f t , is obtained by introducing a second companion system that describes the forward motion of the ( m (cid:0) 1 ) (cid:2) 1 lead vector, g t (cid:17) E t f [ f t+ m (cid:0) 1 ; : : : ; f t ] 0 g ,overtheplanninghorizon, g t = = E E t t f f G A g ( t+ B 1 ) A + ( A 1 ) ( B 1 X i= 0 ) A G ( 1 i (cid:19) m ) (cid:19) y m (cid:3)t+ (cid:3) y t i g g ; ; (9) wherethe m (cid:2) 1 selectorvector, (cid:19) m ,hasaoneinthe m thelementandzeroeselsewhere; (cid:19) 0m g t E t f f t g (cid:17) ; and G isthe m (cid:2) m bottomrowcompanionmatrixoftheleadpolynomial, A ( B F ) , G = 2 4 (cid:0) a 0 m B m (cid:0) a m (cid:0) 1 B m (cid:0) 1 I m : (cid:0) : : 1 (cid:0) a 1 B 3 5 :

7 Substituting the solution of the forcing term, f t , from equation (9) into (8) yields the generalized 2 m -orderdecisionrule, E t f (cid:1) y t g = = (cid:0) (cid:0) A A ( ( 1 1 ) ) y ( t 1 (cid:0) y t (cid:0) + 1 (cid:0) A (cid:3) y ( L (cid:3)t (cid:0) ) (cid:1) ) 1 + y t (cid:0) A 1 (cid:3) ( + L E ) (cid:1) t f y A t (cid:0) ( B + 1 ) A E ( 1 t f ) A 0 (cid:19) m ( B 1 X i= ) A 0 G ( 1 i ) (cid:19) (cid:19) m 0m (cid:3) y t+ 1 X i= i g [ I 0 ; m (cid:0) G ] (cid:0) 1 G i (cid:19) m (cid:1) y (cid:3)t+ i g : (10) Inthefirstlineof(10),thelagpolynomialispartitionedintoalevelanddifferenceformat, A ( 1 ) L + ( 1 (cid:0) A (cid:3) ( L ) L ) ( 1 (cid:0) L ) A ( L ) (cid:17) , where A (cid:3) ( : ) is an (m-1)-order polynomial whose coefficients are moving sums of the coefficients of A(.), as shown in the Appendix. The second line of (10) partitions the forward path of the target into an initial level and forward differences, and uses the identity, (cid:19) 0m [ I m (cid:0) G ] (cid:0) 1 (cid:19) m (cid:17) A ( B ) (cid:0) 1 ,toisolatetheerrorcorrection“gap,” y t (cid:0) 1 (cid:0) y (cid:3)t (cid:0) 1 . The final step inderivingthe decisionruleunder polynomialfrictions is toeliminateforecasts of the equilibrium path, E t f y (cid:3)t+ i g , using the companion form of the forecast model. Substituting forecasts of forward changes E t f (cid:1) y (cid:3)t+ i g , from (5) into (10) provides the closed form solution of thegeneralizedrationalerrorcorrectiondecisionrule9 E t f (cid:1) y t g = = (cid:0) (cid:0) A A ( ( 1 1 ) ) ( ( y y t (cid:0) t (cid:0) 1 1 (cid:0) (cid:0) y y (cid:3)t (cid:0) (cid:3)t (cid:0) 1 1 ) ) + + A A (cid:3) (cid:3) ( ( L L ) ) (cid:1) (cid:1) y y t (cid:0) t (cid:0) 1 1 + + A h ( B 0 z (cid:3) ) t (cid:0) A 1 : ( 1 ) (cid:19) 0m 1 X i= 0 [ I m (cid:0) G ] (cid:0) 1 G i (cid:19) m (cid:19) 0 (cid:3) H i [ H (cid:0) I p q ] z t (cid:0) 1 ; (11) There are twomajor differences in the dynamicformats of the polynomialfrictions versionof rational error correction in equation (11) and the conventional geometric frictions variant shown earlierin(6). First,useofthe m -ordercomponentpolynomial, A ( L ) ,introduces m (cid:0) 1 lagsofthe dependent variable, A (cid:3) ( L ) (cid:1) y t (cid:0) 1 to accompany the single lag of the decision variable in the error correction term.10 Second, in contrast to the single forward discount factor, (cid:21) B , employed by the decision rule under geometric frictions in (3), forecasts of anticipated changes in the equilibrium pathare nowdiscountedbythe m eigenvaluesembeddedintheleadcompanionmatrix, G . Justastherationalexpectationsrestrictionsweresummarizedbyacoefficientvectorunderthe geometric frictions prior, coefficient restrictions of the generalized rational error correction also canbecompactlystated. Asindicatedinthelastlineof(11),thesumoftheforward-lookingterms isagainequivalenttotheinnerproductofaweightingvector, h (cid:3) ,andtheinformationvector, z t (cid:0) 1 , 9Thedirectsolutionformatusingcompanionformsmaybecomparedwithalternativesolutionmethodsforlinear decision rules ranging from partialfractionsexpansionsof the characteristicrootsin Hansen and Sargent(1980)to SchurdecompositionsinAndersonandMoore(1985)andAnderson,Hansen,McGrattan,andSargent(1996). 10Notethat m willoftenbemuchsmallerthatthenumberoflagsofthedependentvariableinaconventionalerror correction, m < q .

8 Twotypesofrestrictionsareimposedonthecoefficientvector, h (cid:3) ,underpolynomialfrictions: the cross-coefficient restrictions imposedby the component polynomialsof the Euler equation,as summarized by the forward companion coefficient matrix G ; and the cross-equation restrictions imposed by the agents' forecast model, summarized by the lag companion coefficient matrix H . To reveal these restrictions, successive column stacks are applied to simplify the solution for the coefficient vector, h (cid:3) .11 h (cid:3) = A ( 1 ) A ( B ) vec = A ( 1 ) A ( B ) ( (cid:19) 0m ( [ [ I H m 0 (cid:0) (cid:0) G I p q (cid:0) ] ] 1 1 X i= (cid:10) 0 ( [ H H i 0) 0 (cid:0) (cid:19) (cid:3) I 0 (cid:19) m p q ( G ] ) i 0) ( 1 [ G X i= 0 [ I m (cid:10) (cid:0) H G 0] ] i (cid:0) 1 ) 0(cid:19) m ) ; vec = A ( 1 ) A ( B ) ( (cid:19) 0m [ I m (cid:0) G ] (cid:0) 1 (cid:10) [ H 0 (cid:0) I p q ] ) [ I m p q (cid:0) G (cid:10) H 0] ( (cid:0) 0 (cid:19) (cid:19) m (cid:3) 1 ( (cid:19) m ) ; (cid:10) (cid:19) (cid:3) ) : (12) This definition of the restricted coefficient vector in (12) provides a closed form solution for the linear decision rule under polynomial frictions, and a summary of differences between the unrestricted regression coefficients in a conventional error correction and the tightly restricted coefficients oftheinformationvector, z t (cid:0) 1 ,inageneralizederror correction. Finally, equation (11) indicates the friction parameters of the generalized rational error correction are collected in A ( 1 ) and A (cid:3) ( : ) . This separable format is convenient for maximum likelihoodestimationbyaniterativesequenceoflinearregressions,as discussedintheAppendix. II.2Higher-OrderEulerEquationsduetoPolynomialFrictions As noted earlier, standard rationalizations of the linear, second-order Euler equation are based on theassumptionof: (1)quadraticcostsofadjustingthelevelofthedecisionvariable,or(2)adiscrete geometric distribution of random delays in adjustments of the decision variable. Polynomial extensionsofthesetwopriorspecifications arediscussed. Adjustmentcostsonweightedaveragesofdecisionvariables One class of generalized frictions is associated with agent efforts to smooth weighted averages of current and lagged values of decision variables. This smoothing is represented by quadratic penalties on C ( L ) y t , where C ( L ) is a m -order polynomial in the lag operator, C ( L ) = m Pj = 0 c j L j , and C ( 1 ) = 0 . Agents choose a sequence of decision variables that minimize the criterion, (cid:31) t , defined by a 11Thecolumnstackoftheproductofthreematricesisdenotedbyvec ( A B C ) = ( C 0 (cid:10) A ) vec B ,where (cid:10) denotes theKroneckerproduct.

9 second-orderexpansionofprofits orutilityaroundthepathofequilibriumsettings, (cid:31) t = E t f 1 X i= 0 B i [ c 00 ( y t+ i (cid:0) y (cid:3)t+ i ) 2 + ( C ( L ) y t+ i ) 2 ] g : (13) TheassociatedEulerequationisa 2 m -order equationintheleadandlagoperators, 0 = = E E t t f f [ A c 00 ( + B F s 0 ) A + ( X k L m = ) 1 y s t k (cid:0) ( L A k ( + B ) ( A B ( F 1 k ) ) y ) (cid:3)t ] g y ; t (cid:0) c 00 y (cid:3)t g ; (14) where the s k coefficients in the first line of ( 1 4 ) are defined by coefficients of the friction polynomial, C ( L ) . s k = m X j (cid:0) = k 0 c j c j + k B j ; k = 0 ; 1 ; : : : ; m ; BecausetheextendedEulerequationinthefirstlineof ( 1 4 ) issymmetricin L and B F ,theequation is unaffected if these two operator expressions are interchanged. This, in turn, implies that a solution of the characteristic equation of the Euler equation, say (cid:21) (cid:0) 1 , is also accompanied by the reciprocal solution, B (cid:21) . Consequently, the characteristic equation can be factored as shown in in thesecondlineofequation( 1 4 ). TheformatofthisEulerequationisthesameasthatshownearlier inequation( 1 ),exceptthefactorpolynomialsarenowexplicitlyidentifiedas m -orderpolynomials. The criterion and second-order Euler equation associated with the standard specification that quadraticcostsapplyonlytochangesinthelevelofthedecisioninstrumentarenestedinequations ( 1 3 ) and ( 1 4 ) ,respectively,forthestandardpriorassumptionthat m = 1 . Smoothinglevelsanddifferences In a frequent interpretation of higher-order Euler equations, the decision variable, y t , is an asset stock,andadjustmentcostsmaybeapplicablenotonlytochangesintheleveloftheassetbutalso changes in the first-difference. An example is optimal inventory planning, where y t indicates the inventory stock at the end of period t . The change in inventories, ( 1 (cid:0) L ) y t , equals production less sales. Given exogenous sales, the assumption of quadratic costs on the level of production impliesaquadraticpenaltyonchangesintheplannedlevelofinventory, c 1 ( ( 1 (cid:0) L ) y t ) 2 . Similarly, quadraticcostsassociatedwithchangesintherateofproductioncanberepresentedbyaquadratic smoothingpenaltyonchangesintheplannedfirst-differenceoftheinventorystock, c 1 ( ( 1 (cid:0) L ) 2 y t ) 2 . Thus,intheexampleofinventorymodeling,itisnotuncommontoassumepolynomialfrictionsof

10 thegeneral form, P 2 j = 1 c j ( ( 1 (cid:0) L ) j y t+ i ) 2 .12 A generalized criterion for smoothing levels and higher-order differences, ( 1 (cid:0) L ) k y t , in decisionvariablesis (cid:31) t = E t f 1 X i= 0 B i [ c 00 ( y t+ i (cid:0) y (cid:3)t+ i ) 2 + X k m = 1 c k ( ( 1 (cid:0) L ) k y t+ i ) 2 ] g ; (15) withtheassociated 2 m -orderEulerequation, E t f c 00 ( y t (cid:0) y (cid:3)t ) + m Pk = 1 c k [ ( 1 (cid:0) L ) ( 1 (cid:0) B F ) ] k y t g = 0 . Smoothingweightedaverages Asecondgeneralizationisthecasewherequadraticpenaltiesareassociatedwithweightedmoving averages of the decision instrument. For example, let y t denote new labor hires by a firm in period t . Suppose various job families within the firm require different durations of training by supervisors, and the number of employees occupied by training in a given period is represented by a fixed distributionof recent vintages of new hires, c 0 y t + : : : + c m (cid:0) 1 y t (cid:0) m + 1 . Costs associated with variations in the rate of training may be approximated by the quadratic penalty, c 0 ) L + : : : + ( c m (cid:0) 1 (cid:0) c m (cid:0) 2 ) L m (cid:0) 1 (cid:0) c m (cid:0) 1 L m ) y t ) 2 ( ( c 0 + ( c 1 (cid:0) ,whichisarestatementofthepolynomialfriction specificationinequation( 1 3 ).13 Another variation is the extension of quadratic penalties from smoothed one-period changes in the level of the decision variable, ( 1 (cid:0) L ) y t+ i , to smoothed changes in moving averages, L ) k (cid:0) Pj = 1 0 y t+ i (cid:0) j = ( 1 (cid:0) L k ) y t+ i ( 1 (cid:0) . Examples include seasonal or term contracts where some costs are associated with one-period averages, others with two-period averages, and so on. The criterion in thisinstancetakes theform, (cid:31) t = E t f 1 X i= 0 B i [ c 00 ( y t+ i (cid:0) y (cid:3)t+ i ) 2 + X k m = 1 c k ( ( 1 (cid:0) L k ) y t+ i ) 2 ] g ; (16) withtheassociated 2 m -orderEulerequation, E t f c 00 ( y t (cid:0) y (cid:3)t ) + m Pk = 1 c k [ ( 1 (cid:0) L k ) ( 1 (cid:0) B k F k ) ] y t g = 0 . Stochasticresponsedelays Giventhetractabilityoflinearfirst-orderconditions,thequadraticadjustmentcostspecificationis widely used to characterize optimal adjustment. Applications include decision variables such as nominal prices, extending from the seminal paper by Rotemberg (1982) to the recent example of HairaultandPortier(1993),althoughtheassumptionofstrictlyincreasingcostsinthesizeofprice 12SeeHall,Henry,andWren-Lewis(1986),Callen,Hall,andHenry(1990),andCuthbertsonandGasparro(1993). 13Inallexamples,notethatlinearcostcomponentscanbeaccommodatedbyredefiningtheequilibriumtarget, y (cid:3) t , oftherelevantdecisionvariable.

11 adjustments is often disputed. However, as noted by Rotemberg (1996), the aggregate response arising from the quadratic adjustment cost model is equivalent to the aggregate adjustment of agents subject to random decision delays drawn from an exponential distribution, as proposed by Calvo (1983). Although it appears to have received little attention outside the field of dynamic pricing, the stochasticdelay model would appear to be a useful framework for modeling adjustments in other market contexts when agent responses are dependent on unpredictable transmissionsofdecisions,suchas distributedproductionorcommunicationnetworks.14 In a discrete-time implementation of the stochastic delay approach, each agent j controls a decision variable, y j ;t , with the associated equilibrium trajectory, y (cid:3)j ;t . When adjustment of a decisionvariableoccurs, themovementtoequilibriumis completebutthe timingof adjustmentis stochastic. Theprobabilityofanagentadjustmentinthe i thperiodoftheplanninghorizon,having notadjustedinthe i (cid:0) 1 precedingperiods,is r i . Thescheduleoffutureadjustmentprobabilitiesis represented bythelead polynomial, r ( F ) = r 0 + r 1 F + r 2 F 2 + : : : , where the r i are nonnegative and r ( 1 ) = 1 . Usinga discrete geometricdistributionas the analogueof the exponentialresponse distribution in Calvo (1983), the generating function is r ( F ) = A ( 1 ) A ( F ) (cid:0) 1 , where A ( : ) is the first-orderpolynomial, A ( F ) = ( 1 (cid:0) (cid:21) F ) . Given the constraint that the decision variable must remain at the level selected, say ~y j ;t , until the next allowable adjustment period, the optimal setting that minimizes the expected sum of squareddeviationsfrom thediscountedpathofequilibriumsettingsis E t f ~y j ;t g = E t f r ( B F ) y (cid:3)j ;t g . Using simple sum aggregation, the aggregate of decision variables adjusted in t is E t f r ( B F ) y (cid:3)t g E t f ~y t g = . Theaggregatedecisionvariableisanormalizedaverageofcurrentandpastvintage decisionsthatsurvivein t . Intheexampleofageometricdelaydistribution,thesurvivalprobability in t ofapastdecisionvariablesettingfrom t (cid:0) i isproportionalto r i .15 Thus,thegeneratingfunction forthenormalizedsurvivalprobabilitiesoveraninfinitehorizonis r ( L ) ,andtheaggregatedecision variableinperiod t mayberepresentedby E t f y t g = = r r ( ( L L ) ) E E t t f f ~y r t ( g B ; F ) y (cid:3)t g : (17) The lag polynomial, r ( L ) , remains to the left of the expectations operator on the right-hand-side of ( 1 7 ) to ensure that the lagged expectations embedded in past decisions are represented in 14Effects of costly and stochastic communications in distributed production are discussed in Board and Tinsley (1996).SeealsoBertsekasandTsitsiklis(1989)forrepresentativeconfigurationsofcommunicationnetworks. 15Thehazardfunction,theratiooftheadjustmentprobability, r i ,tothesurvivalprobabilityatlag i ,isconstantfor theunivariategeometricdistribution,Johnson,Kotz,andKemp(1993).

12 the current aggregate, y t .16 Replacing the generating functions by the polynomial components yields the analogue to the familiar second-order decision rule for the aggregate decision variable, E t f A ( L ) y t g = E t f A ( 1 ) A ( B ) A ( B F ) (cid:0) 1 y (cid:3)t g . Aggregateddelayschedules Inprinciple,thechoiceoftheappropriatestochasticdelaydistributionshouldbeanempiricalissue. Remainingwithinthepolynomialfrictionsframeworkofthispaper,theapproachsuggestedbelow considers higher-order polynomial approximations of more general stochastic delay distributions. As indicated, these generalizations can be interpreted as convolutions of component geometric distributions. It is unlikelythat agents haveperfect informationabout thedistributionofdelays orstochastic congestion in future decisions. Suppose, for example, agents may be confronted by a “low-cost” response distribution, r 1 ( F ) , or a “high-cost” response distribution, r 1 ( F ) , where the expected response lag from the first distribution is smaller than that of the second. However, draws from either delay distribution are random. In this example, the generating function of the effective response probabilities is the product of the generating functions of the component response probabilities, r ( F ) = r 1 ( F ) r 2 ( F ) .17 More generally, in the case of random aggregation over m geometric response schedules, the aggregate reset of decision variables adjusted in t is E t f ~y t g = E t f m Qj = 1 r j ( B F ) y (cid:3)t g . As in the case of a univariate geometric distribution, a constant-hazard approximation permits the survival distribution of past vintage decisions to be represented by the polynomial generating function of thestochasticdelays, m Qj = 1 r j ( L ) .18 Thus,underan m -orderpolynomialstochasticdelaydistribution, 16AsdiscussedinTaylor(1993),somemodelspecificationsandinstrumentalmethodsofestimatingdecisionrules movetheequivalentofthelagpolynomial, r ( L ) ,insidetheexpectationoperator. 17If v and w arenon-negativeindependentrandomvariables,thegeneratingfunctionoftheconvolution, v + w ,is theproductofthegeneratingfunctionsof v and w ,Feller(1968). 18The mean absolute error of constant-hazard approximations of quarterly survival probabilities is about :0 2 percentage points for the first sixteen lags in the empirical decision rules using polynomial frictions discussed in the next section. The reason for the relatively modest approximation errors can be shown using a partial fractions representationoftheapproximationerror.Denotethepartialfractionsexpansionofan m -orderpolynomialgenerating function by: A ( 1 ) A ( L ) (cid:0) 1 (cid:17) m =Pj 1 (cid:11) j ( 1 (cid:0) (cid:21) j ) ( 1 (cid:0) (cid:21) j L ) (cid:0) 1 , where the (cid:21) j denote the characteristic roots of A ( L ) . Theerroroftheconstant-hazardapproximationofthesurvivalprobabilityatlag i is m =Pj 1 (cid:11) j (cid:22)(cid:21) (cid:0) 1 ( (cid:21) j (cid:0) (cid:22)(cid:21) ) (cid:21) (cid:0)i j 1 ,where (cid:22)(cid:21) (cid:17) S s ( 1 + S s ) (cid:0) 1 , and S s denotes the sum of the survival probabilities. In the case of the univariate geometric distribution, (cid:22)(cid:21) is equal to the single root, and the approximationerror is zero. In the case of convolutedgeometric distributionswith a single dominantroot, as with empiricalexamplesin this paper, (cid:22)(cid:21) is generally very close to the modulusof thedominantroot. In addition,errorcomponentsassociated withsmaller rootsdecayrapidlywith lag i becausethespreadforasmallerroot, (cid:21) j (cid:0) (cid:22)(cid:21) ,isscaledbypowersofthatroot, (cid:21) (cid:0)i j 1 .

13 theaggregatedecisionvariableinperiod t isrepresented by E t f y t g = m Y j = 1 r j ( L ) E t f m Y j = 1 r j ( B F ) y (cid:3)t g : (18) Substitutinginthecomponentpolynomialsofthegeneratingfunctionsforthestochasticdelay and survival probabilities yields a solution for the aggregate decision variable that is identical to thatderivedearlierforthedecisionruleunderpolynomialfrictions E t f A ( L ) y t g = E t f A ( 1 ) A ( B ) A ( B F ) (cid:0) 1 y (cid:3)t g ; (19) wherethecomponentpolynomials, A ( L ) and A ( B F ) ,are m -order. III.Empirical Examples ofRational Error Corrections forIndustry Pricing Empirical contrasts of second-order and higher-order rational error corrections are discussed in this section. The examples used are pricing decision rules of six SIC two-digit manufacturing industries: textiles, lumber, rubber & plastics, primary metals, motor vehicles, and scientific instruments.19 In addition to an expected difference in statistical fits, the rational expectation overidentifying restrictions are rejected by all but one of the second-order decision rules and by noneofthehigher-orderrules. III.1TheEquilibriumPrice Theequilibriumlogpriceoftheoutputofindustry j with s j identicalproducersis representedby p (cid:3)j = (cid:22) j + m c j ; (20) where (cid:22) j is the log markup by producers and m c j is the log of marginal cost. Ignoring strategic considerations, the markup is (cid:22) j = (cid:0) l o g (cid:16) 1 (cid:0) s 1 (cid:17) j j (cid:17) , where (cid:17) j is the price elasticity of demand, andthemonopolyandcompetitivesolutionsareobtainedas s j ! 1 or s j ! 1 . Grossproduction isCobb-Douglas inbothpurchasedmaterials andrentedservices ofprimaryfactors. Also,returns toscaleare constantsothatthelogofmarginalcostis proportionaltotheweightedaverageoflog inputprices, m c j / (cid:18) c j p c j + (cid:18) i j p i j + (cid:18) v j p v j ; (21) 19MotorvehiclesisalargesubsetoftheSICtwo-digitindustry,transportationequipment.

14 where p c j is the log price of primary commodity production inputs, p i j is the log price of intermediate materials purchased from other industries, p v j is the log unit cost of value added in the j industry,and (cid:18) c j + (cid:18) i j + (cid:18) v j = 1 . As indicated in (21), input price regressors are specific to industry j and constructed from input-outputweightingsofindustryproducerprices. Industryproducerprices at theSIC two-digit levelofaggregationaregenerallyavailableonlyfromthemid-1980s,andindustrypricesinearlier periodswereassembledfromspecificcommodityprices,oftenatlowerlevelsofaggregation. The industry log unit cost of primary factors, p v j ;t , was estimated by the log of hourly earnings, w j ;t , less the log of trend productivity, (cid:26) j ;t . The latter was constructed from smoothed estimates of the logofindustryindustrialproductionless thelogofindustryemploymenthours. Industry producer prices in the U.S. do not reject the hypothesis of difference-stationarity over postwar samples. A common format was used to explore cointegration constructions of the equilibriumpriceofeachindustry p (cid:3)j ;t = c 0 + (cid:18) c j p c j ;t + (cid:18) i j p i j ;t + (cid:18) v j p v j ;t + c 1 (cid:26) t + c 2 t : (22) Given data limitations of the trend productivity estimates, both industry log productivity trends, (cid:26) j ;t and time trends, t , were added as additional regressors, and the intercept, c 0 contains both the log margin, (cid:22) j , as well as proportional mean errors in measurements of unit cost inputs. The relevant industry input share weights, ( (cid:18) c j ; (cid:18) i j ; (cid:18) v j ) , are displayed in the initial columns of Table 1. These share weights are not estimatedbut defined by benchmark input-outputestimates, obtained bymanipulatingtheBureau ofEconomicAnalysis(1991)industryuseandmaketables. As shown in the columns headed by f ( t ) in Table 1, additional trend productivity regressors wererequiredforcointegrationinthreeindustries. Asnotedabove,thelogpriceofprimaryfactors already incorporates trend productivity, p v j (cid:17) w j (cid:0) (cid:26) j . Because this assigns a weight of (cid:0) (cid:18) v to (cid:26) j , the additional positive coefficients lower the effective contribution of the trend productivity constructions. Finally,asshowninthelastcolumnofTable1,thehypothesisthatthecointegrating discrepancy, p j ;t (cid:0) p (cid:3)j ;t is I ( 1 ) is rejectedat the 9 0 % confidence levelorhigherforallindustries. III.2EmpiricalEstimatesofPricingDecisionRulesunder GeometricFrictions Table 2 presents summary statistics for second-order pricing rules of the six manufacturing industries, estimated under the prior of geometric frictions. Estimated parameters of the industry decision rules and VAR forecast model parameters were obtained by the maximum likelihood estimatordescribedintheAppendix.20 20Industry prices are quarterly averages of monthly, seasonally unadjusted series from the US Bureau of Labor database on commodity and industry producer prices for the 1954-1995sample. As noted earlier, the equilibrium

15 The estimated error correction coefficients, A ( 1 ) in Table 2, indicate the average quarterly reduction rates planned for the price “gap,” p t (cid:0) 1 (cid:0) p (cid:3)t (cid:0) 1 , of each industry. The proportion of explained variation in quarterly price changes can be substantial, with R 2 in three industries ranging from .3 to .6. The row in Table 2 labeled (cid:1) R 2 ( % ) indicates that for four industries, the modal source of explained variation is the sample variability of industry forecasts of future equilibrium prices, as captured by the rational forecast term, h 0 (cid:3) z t (cid:0) 1 . Table 2 also contains the estimated mean lag of producer responses to unexpected shocks and the estimated mean lead of responsestoanticipatedevents. Themeanleadoftheindustryplanninghorizonistypicallysmaller thanthemeanlagresponseduetodiscountingofforwardevents. Three characteristics of these estimated equations suggest significant dynamic specification problems. First,themeanlagresponsesappeartobeunusuallylargerelativetopreviousestimates of response lags for manufacturing prices.21 Second, serial independence of the residuals is rejected for all but one industry at 95% confidence levels. Although it is possible that producers mayhaveseriallycorrelatedinformationthathasnotbeenincludedintheindustryforecastmodels, it is plausible also that residual correlations could be due to misspecifications of the frictions in producerresponses. Afinal indicationofpotentialmisspecificationsisindicatedinthebottomrowofTable2. This row, labeled L R ( h (cid:3) j z t (cid:0) 1 ) , lists the rejection probabilities of a likelihood ratio test to determine if the data prefers an unrestricted forecast model of forward equilibrium price changes to the rational forecasts embedded in the geometric frictions version of rational error correction. With one exception (motor vehicles), the rational expectations overidentifying restrictions are rejected at 99%confidence levels. III.3EmpiricalEstimatesofPricingDecisionRulesunder PolynomialFrictions Estimates of the industry pricing rules under the polynomial generalization of frictions are listed in Table 3. Because the conventional two-root decision rule, m = 1 , is nested in the generalized frictions model, it is interesting to note that additional lags of the dependent variable are always priceforecastmodelforeachindustryisaVARcontainingtheequilibriumpriceandthepricesofproductioninputs. Althoughseasonalityisnotpronouncedinmostindustryprices(oneexceptionismotorvehicles, asnotedlater), all industry VARs contained at least four lags in regressors, and seasonal dummies were added to all VAR and error correction equations. To reduce space, estimates of equation intercepts and seasonal dummy coefficients are not reportedinthetables. Inallequationspresentedhere,thequarterlydiscountfactor, B ,wassetto :9 8 ,approximating thepostwarannualrealreturntoequityofabout 8 % ;empiricalresultsarenotnoticeablyalteredbymoderatevariations in B . 21In analysis of the Stigler-Kindahl data of producers' prices, Carlton (1986) reports an average adjustment frequencyofaboutonceayear. ReducedformregressionsbyBlanchard(1987,Table8)fortheU.S.manufacturing priceaggregateindicateameanlagofabouttwoquarters. Bycontrast,thelevelsfrictionmodelinRotemberg(1982) suggestsameanlagofabout12quartersfortheU.S.GDPdeflator.

16 significant in the industry pricing models, generally consistent with polynomial components of order m = 2 or m = 3 .22 In the case of motor vehicles, the preferred specification is 5 m = , requiring four lags of the dependent variable. This is due to a significant seasonal pattern in the producer price of motor vehicles which could not be adequately captured by fixed seasonal dummies. Without exception, all of the problems noted for the estimated decision rules under geometric frictions in Table 2 are eliminated under polynomial frictions. The percentage of explained variation, R 2 , is considerablyhigher for most industries inTable 3; meanlags are more plausible; the assumption of serially independent residuals is retained in all industries; and the rejection probabilities in the bottom row in Table 3 indicate that the rational expectations overidentifying restrictionsarenotrejectedatconfidencelevelsof95%orhigher. Thelatterisnoteworthybecause rejections of rational expectations overidentifying restrictions are often interpreted as evidence of non-rational forecasting by agents or of inadequate specifications of agent forecast models of forcing terms. Because the only difference between industry model specifications used in the side-by-side comparisons of Table 2 and Table 3 is the degree of the Euler equation polynomials, m ,theculprit,atleastintheseexamplesandforthestatisticalpropertiesconsidered,isrigidpriors onthespecificationofdynamicfrictions. More intuitive insights into the dynamic effects of the higher-order lag and lead polynomials areobtainedbyrearrangingtheEulerequationtodefinethecurrentperiodresponseweightstolags andexpectedleads oftheforcingterm, E t f p (cid:3)t+ i g ,impliedbytheindustrydecisionrules, E t f p t g = = E E t t f f A ( 1 X i= (cid:0) 1 1 ) A w ( L p i (cid:0) ) (cid:3)t+ 1 i g A ; ( B ) A ( B F ) (cid:0) 1 p (cid:3)t g ; (23) wherenegativesubscripts, i < 0 ,denoteresponsestolaggedeventsandpositivesubscripts, i > 0 , responses to anticipated events. The lag and lead weights of the six estimated industry decision rulesaredisplayedinthepanelsofFigure1. Thedottedlinesarethefrictionweightsgeneratedby thetwo-rootdecisionrules ( m = 1 )reported inTable 2andthesolidlinesare thefrictionweights associatedwiththe 2 m -root decisionrules( m > 1 )showninTable3. Several effects of the generalization of frictions are apparent from the plots of the industry friction weights in Figure 1. In each panel, the vertical line is positioned in the current period 22Thisisnotanisolatedfinding.Everymacroeconomicaggregatetowhichthegeneralizedfrictionsmodelwasfitin theFRB/USmacroeconomicmodelalsorejectedtheconventionalpriorthat m = 1 ,vid. BraytonandTinsley(1996). AsdiscussedinarecentliteraturesurveybyTaylor(1997),manystudiesofempiricalstaggeredcontractmodelsfor wagesdonotsupportgeometricresponseschedules, includingan estimatedbimodaldistributionofcontractlengths inLevin(1991).

17 ( i = 0 ). The mean lag of responses to unanticipated events is captured by the weighted average of lags using the friction weights to the left of center. The friction weights are nearly symmetric aboutthecurrentperiodwiththemeanlead,associatedwithweightstotherightofcenter,slightly smallerthanthemeanlagduetothediscountingoffutureevents. Thus,thenetmeanresponselag toperfectlyanticipatedeventsissmallformostindustries. Largermeanleadsrequirelongerplanninghorizonsandarecharacteristicoftheflatterfriction weightdistributionsindicatedbythedottedlinesinFigure1forthetwo-rootdecisionrules, m = 1 . Thus, vertical distances between the two sets of friction weight distributions in each panel are indicativeof differences between the industry mean leads of Table 2 and the corresponding mean leads ofTable3. AsshowninthepanelsofFigure1,relativelylow-orderfrictionpolynomials, A ( B ) A ( B F ) (cid:0) 1 , cangenerateavarietyofflexibleshapes,includingtheseasonalweightsatdistancesof (cid:6) 4 quarters indicated for the motor vehicles industry, SIC 371. Some estimated friction distributions are relatively flat for several quarters, while others fall off rapidly from the modal weight in the current quarter. The plots in Figure 1 do not support the two-sided geometric distribution prior that is consistent with two-root decision rules, m = 1 . In almost all industries, the drawback of a two-sided geometric response schedule is an inability to capture relatively stronger industry responsestoeventsinaone- ortwo-quarterneighborhoodofthecurrent quarter. Cyclical VariationsinPricingMargins Thus far, specification of the desired industry price settings has proceeded under the assumption thatrelevantargumentsaredifference-stationary,withestimationofindustry“target”paths, p (cid:3)t ,by cointegration. However, economic theory may suggest additional stationary variables as possible arguments of the desired target.23 If there is prior information that agents' perceptions of the forcing terms of Euler equations are significantly influenced by additional variables, this prior informationshouldbeintroducedintothemodeltoavoidpossibledistortionsinestimatedfrictions. An example is useful to illustrate how the distinction between friction and forecast parameters is maintainedfortrialargumentsoftheforcingterm byimposingdynamicfrictionrestrictions. In the present example of a price markup model, cyclical indicators such as industry capacity utilization rates may capture variations in planned margins, (cid:22) t , due to boom or bust pricing strategiesorcyclicalmovementsinthepriceelasticityofdemand. TheEulerequationforindustry price is restatedtoincludetheeffect ofcurrent andlagged industryutilizationrates, u t (cid:0) i ,24 onthe 23AsdiscussedbyWickens(1996),economictheoryisrequiredforstructuralinterpretationsofcointegrations. 24Industry utilization rates are constructed by the FRB staff from surveys of capacity utilization, see Raddock (1985). Logindustryutilizationrates arestationary, andtheerrorcorrectionresponsesofcapacityoutputareeither insignificantoranorderofmagnitudesmallerthanthatofindustryoutput. Samplemeansareremovedso u t canbe

18 current pricetarget, E t f A ( B F ) A ( L ) p t (cid:0) A ( B ) A ( 1 ) [ p (cid:3)t + D ( L ) u t ] g = 0 ; (24) where p (cid:3)t continues to denote I(1) arguments of the equilibrium target, and D ( L ) is an m 0 -order polynomial in L . It is convenient to assume m 0 = 1 , which is the minimal order necessary to distinguish between pricing effects of changing utilization rates and effects of higher or lower levelsofcapacityutilization, D ( L ) = d 0 + d 1 L . Multiplying through by the inverse of the lead polynomial, A ( B F ) (cid:0) 1 , and substituting in forecasts from the agents' information set, z t (cid:0) 1 , defines the augmented rational error correction wheretheindustrypricedecisionrulenowcontainsaninfinite-horizonforecastofforwardindustry utilizationrates,discountedbythe m eigenvaluescontainedinthe frictionscompanionmatrix, G . Using the simplifying operations discussed earlier, the closed-form of the extended rational error correctionsolutionis E t f (cid:1) p t g = (cid:0) A ( 1 ) ( p t (cid:0) 1 (cid:0) p (cid:3)t (cid:0) 1 ) + A (cid:3) ( L ) (cid:1) p t (cid:0) 1 + h 0 (cid:3) z t (cid:0) 1 + X k 1 = 0 d k h 0 u k z t (cid:0) 1 ; (25) Theadditionalcoefficient vectors, h u k ( k = 0 ; 1 ) ,definediscountedsumsoftheexpectedforward pathoftheindustryutilizationrate, h h u u 0 1 = = A A ( ( 1 1 ) ) A A ( ( B B ) ) [ [ (cid:19) (cid:19) 0m 0m (cid:10) (cid:10) H I n 0] [ I m ] [ I m n n (cid:0) (cid:0) G G (cid:10) (cid:10) H H 0 (cid:0) ] 0 (cid:0) ] 1 ( (cid:19) m 1 ( (cid:19) m (cid:10) (cid:10) (cid:19) 0 ) ; u (cid:19) 1 ) ; u (26) where the n (cid:2) 1 selector vectors, (cid:19) u k , locate u t (cid:0) k in the information vector, z t , now extended to include current and lagged values of the industryutilization rate. Because industrylog utilization rates are stationary,the restricted coefficient vectors in (26) differsomewhat from that derivedfor forecasts ofdifference-stationarytrendprices, h (cid:3) ,in(12). Effects of adding industry utilization rates to the rational error correction are reported in Table 4. The rejection probabilities in the row labeled L R ( D ( L ) ) indicate that expected forward utilizationratesareasignificantdeterminantofpricingata90%levelofconfidenceinthreeofthe sixindustries. Cyclical markupeffects associatedwiththe levelof theindustryutilizationrate are indicated in the next row of Table 4, labeled D ( 1 ) . Procyclical margins are indicated for primary metals(SIC 33)andcountercyclicalmarginsformotorvehicles(SIC371). All significant features of the rational error corrections in Table 3 are retained in Table 4, including serially independent residuals and nonrejection of the RE overidentifying restrictions interpretedasindustryoutputdeviationsfromtrendorpreferredutilization.

19 which are now extended to include forecasts of forward utilization rates. Thus, the polynomial frictions description of industry pricing appears to be robust to the addition of a conventional determinantofcyclicalpricing. IV.ConcludingComments Aftertwodecades ofresearch inmacroeconomics,therational expectationsconjectureis afixture intheoreticalmacroeconomicmodelsbutisroutinelyrejectedinempiricalmacroeconomicmodels that test theassociatedoveridentifyingrestrictions. Rather than indictingthe rationalexpectations assumption, it appears that the main culprit may be the arbitrary tight prior used to characterize dynamicfrictionsinmacroeconomicmodels. The workhorse of macroeconomic descriptions of rational dynamic behavior is the conventionallineardecisionrulewithtwocharacteristicroots,whereonedeterminesthegeometric discount factor of anticipated events and the other provides a geometric description of lagged responses to unanticipated shocks. The two-sided geometric lead and lag response schedules are generally motivated by a geometric frictions prior where agents aim to smooth levels of activity or are subject to geometric random delays. Although it leads to tractable models of economic behavior, the geometric frictions prior is not based on compellingeconomic theory and is usually rejectedbymacroeconomicdata. An alternative specification of polynomial frictions is suggested in this paper which appears to eliminate many of the empirical drawbacks of the conventional frictions specification. The generalized frictions specification can be interpreted as the result of agents that smooth linear combinationsofcurrentandlaggedactionsoraconsequenceofconvolutedgeometricdistributions ofstochasticdelaysindecisions. Polynomial frictions lead to higher-order Euler equations whose decision rules are solved generallybynumerical techniques. Amethodofobtainingclosed-formsolutionsispresented that uses two simple, first-order companion systems: a lead system for the forward planning required whenagentactionsarerestrictedbyfrictions;andalagsystemfortheagents' forecastmodelofthe Eulerequationforcingterm. Atractablemethodofmaximumlikelihoodestimationbyasequence ofregressionsisoutlinedintheAppendix. Empirical models of producer pricing are estimated for six manufacturing industries. The second-orderdecisionruleimpliedbythegeometricfrictionspriorisnestedwithinthepolynomial frictions specification and rejected by the data for all industries. The decision rules based on geometric frictions had poor empirical properties, including overstatement of mean lags, strong residual correlations,and rejections of rational expectationrestrictions. Rational error corrections

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24 Appendix: EstimationofRational Error Corrections This appendix outlines the two-stage maximum likelihood estimator for rational error corrections used for the empirical examples in this paper. The estimator requires only a sequence of regressions. Because linear rational error corrections have closed-form solutions, analytical solutionsareavailableforall requiredgradients. EstimatingtheRationalErrorCorrectionDecisionRule The assumption of rational behavior leads to overidentified decision rules whose parameters are nonlinear functions of parameters in the companion matrix of the forecast model, H , and parameters in the companion matrix, G , of the frictions polynomial, A ( : ) . However, the forecast model of forcing terms often can be reasonably approximated by a system that is linear in the unknown parameters. There are significant computational advantages to adopting a two-stage maximum likelihood procedure when the forecast model can be estimated by standard linear estimators. Assuming the forecast model is linear in the unknown parameters and has normally distributed forecast errors, maximum likelihood estimates of the column stack of forecast model parameters, ^h x , are obtained by a GLS estimator, or even OLS regressions if the forecast model equationshaveidenticalregressors. Turning to the second stage of estimation, discussion will ignore estimation of unconstrained linearparameterssuchasinterceptsandfixedseasonals. Thevectorofunknownfrictionparameters is (cid:18) = [ (cid:0) A ( 1 ) ; a (cid:3)1 ; : : : ; a (cid:3)m (cid:0) 1 ] 0 . Using y t to denote the dependent variable ( (cid:1) p t for the industry price decisionrules) andassumingnormallydistributeddescriptionerrors, thelogofthe marginal likelihoodof (cid:18) is L y ( (cid:18) ) = (cid:0) ( T = 2 ) log (cid:27) 2 (cid:15) y (cid:0) ( T = 2 ) log ( 2 (cid:25) ) (cid:0) 2 1 (cid:27) 2 (cid:15) y T X t= 1 ( y t (cid:0) (cid:18) 0z y ;t (cid:0) 1 (cid:0) h 0 (cid:3) z t (cid:0) 1 ) 2 ; ( A : 1 ) where z y ;t (cid:0) 1 includesboththelaggederrorcorrection, p t (cid:0) 1 (cid:0) p (cid:3)t (cid:0) 1 ,and m (cid:0) 1 lagsofthedependent variable, and (cid:27) 2 (cid:15) y (cid:17) E [ ( y t (cid:0) (cid:18) z0 y ;t (cid:0) 1 (cid:0) h 0 (cid:3) z t (cid:0) 1 ) 2 ] . Maximum likelihood estimates of the friction parameters, ^(cid:18) , are defined by the zeroes of the gradients of (A.1) with respect to the friction parameters, g y ( ^(cid:18) ) ,giventhefirst-stageestimatesoftheforecastmodelcoefficients, ^h x . Theformat ofthesenonlinearequationsiswell-suitedforGauss-Newtonestimationusingiterativeregressions oftheform, ( y t (cid:0) ^ (cid:18) ( i) 0 z y ;t (cid:0) 1 (cid:0) h 0 (cid:3) ( ^ (cid:18) ( i) ; ^ h x ) z t (cid:0) 1 ) = (cid:16) z y ;t (cid:0) 1 + [ @ h 0 (cid:3) ( : j ^ h x ) ) = @ (cid:18) ] z t (cid:0) 1 (cid:17) 0 ( ^ (cid:18) ( i+ 1 ) (cid:0) ^ (cid:18) ( i) ) : ( A : 2 )

25 AdjustingfortheSamplingVariabilityoftheForecastModelParameters As noted, forecast model parameters estimatedin the first step, ^h x , are used to initializegradients of the friction parameters in the second step. A well-known, but often ignored, consequence is thatthesecond-stepestimatorunderstatesthesamplingerrorsoftheestimatedfrictionparameters, ^(cid:18) , unless account is taken of the sampling uncertainty of the first-step estimates of the forecast modelparameters. MurphyandTopel(1985)citeempiricalexampleswhereuncorrectedt-ratiosof two-stagemaximumlikelihoodestimatorsareoverstatedbymorethan 1 0 0 % . Analyticalsolutions of the correction for “generated regressor” bias in two-stage maximum likelihood estimation of rational error corrections are presented below, along with empirical estimates of the t -ratio corrections fortheexamplesofindustrypricingdecisionrules. Using more compact notation, let (cid:12) denote the concatenation of the unknown parameters of the forecast model, h x , and the friction parameters, (cid:18) . Similarly, g ( ^(cid:12) ) will denote the stack of the relevant likelihood gradients, g x ( ^h x ) and g y ( ^(cid:18) ) . For a T -period sample, the mean-value theorem impliesthat the p T -normalizationof the difference between thesample estimate, ^(cid:12) and theplim, (cid:12) o ,isprovidedbyrearrangingthefirst-orderexpansionofthegradientsaround g ( (cid:12) o ) , p T ( ^(cid:12) (cid:0) (cid:12) o ) = [ (cid:0) 1 T r g ( (cid:12) (cid:3) ) ] (cid:0) 1 [ p 1 T g ( (cid:12) o ) ] ; where r denotes the gradient of g wrt (cid:12) 0 , and the rows of r g ( (cid:12) (cid:3) ) are evaluated at (cid:12) (cid:3) , on the segment connecting ^(cid:12) and (cid:12) o . If the normalized Hessian approaches a fixed limit, p l i m [ (cid:0) 1 T r g ( (cid:12) (cid:3) ) ] ! M , and the marginal likelihood functions satisfies standard regularity conditions then, as demonstrated in White (1994), the likelihood estimates are distributed asymptoticallyasthenormaldistribution, p T ( ^(cid:12) (cid:0) (cid:12) o ) a (cid:24) N ( 0 ; M (cid:0) 1 V ( M 0) (cid:0) 1 ) ; ( A : 3 ) where V denotestheexpectedvalueofthegradientcovariance, V = E f [ p 1 T g ( (cid:12) o ) ] [ p 1 T g ( (cid:12) o ) ] 0 g . An advantageof the two-stage estimationapproach is that the structure of the samplingerrors in ( A : 3 ) issubstantiallysimplifiedwhentheindustryforecastmodelisnotafunctionofthefriction parameters, (cid:18) . To see this, partition M and V to reflect the separate contributions of the forecast coefficients, h x ,andtheadjustmentcostparameters, (cid:18) . M = 2 4 M M 0 h ;h x x 0 (cid:18) ;h x M M 0 h ;(cid:18) x 0 (cid:18) ;(cid:18) 3 5 ; V = 2 4 V V 0 h ;h x x 0 (cid:18) ;h x V V 0 h ;(cid:18) x 0 (cid:18) ;(cid:18) 3 5 : Because the parameters of the forecast model, h x , are not functions of the friction parameters, (cid:18) , the upper right hand partition of M is zero, M h x ;(cid:18) 0 = 0 . This, in turn, implies that the inverse of

26 M requiredforequation ( A : 3 ) simplifiesto M (cid:0) 1 = 2 4 (cid:0) M M 1 0 (cid:0)(cid:18) ;(cid:18) M (cid:0)h 1 0 ;h x x 0 (cid:18) ;h x M 1 0 (cid:0)(cid:18) ;(cid:18) M 0 1 0 (cid:0)(cid:18) ;(cid:18) 3 5 : Substituting this partitioned inverse into ( A : 3 ) yields the following expression for the asymptotic covarianceoftheadjustmentcost parameters, v a r ( ^(cid:18) (cid:0) (cid:18) o ) = M 1 0 (cid:0)(cid:18) ;(cid:18) + M 1 0 (cid:0)(cid:18) ;(cid:18) [ M (cid:0) (cid:18) ;h V 0 M x 0h x ;(cid:18) 1 0 (cid:0) M h ;h x x 1 0 (cid:0) M h ;h x 0(cid:18) ;h 0 M x 0 x 0(cid:18) ;h 0 x (cid:0) M (cid:18) ;h 0 x M (cid:0)h 1 ;h x 0 x V h x ;(cid:18) 0 ] M 1 0 (cid:0)(cid:18) ;(cid:18) : ( A : 4 ) The last two terms in ( A : 4 ) are zero if the residual of the rational error correction is uncorrelated withresidualsoftheVARforecast model, V h x ;(cid:18) 0 = 0 . Thenonzeropartitionsof M arereplaced withthefollowingsampleestimates: M M M h (cid:18) (cid:18) ;h x 0 ;(cid:18) 0 ;h x 0 x = = = E E E [ [ [ (cid:0) (cid:0) (cid:0) 2 @ L x 0 @ h @ h x x 2 @ L y 0 ] @ (cid:18) @ (cid:18) 2 @ L y 0 ] @ (cid:18) @ h x ] = = = E E E [ [ [ g g g x y y ( ( ( h (cid:18) (cid:18) x ) ) ) g g g y y x ( ( ( (cid:18) h h ) x ) x ] : 0 ) ]0 ]0 : : ( A : 5 ) Components required by ( A : 5 ) can be assembled from gradients and covariances produced in the first and second stages of the estimation. Analytical solutions of required gradients are indicated below. EmpiricalEstimatesofSamplingErrorBias The magnitude of “generated bias” adjustments in the t -ratios of the estimated industry pricing rules can be gauged from Table 4. Along with the adjusted t -statistics in parentheses, ( : ) that are reported in all three tables, Table 4 also lists the unadjusted t -statistics from the second step estimation in brackets, [ : ]. Although the extent of adjustment varies, the maximum downward adjustmentof t -ratiosisaround20%. Anintuitiveexplanationforthegenerallymodestreductions instatisticalsignificanceis that muchoftheforecast model uncertaintyis “averaged out”overthe forecast horizonbythediscountedaverageofequilibriumpricechangeforecasts, h 0 (cid:3) z t (cid:0) 1 . AnalyticalGradientSolutions Nonlinear interactions of the frictions parameters and the forecast model parameters are confined to the scalar summary of industry forecasts in the decision rule, h 0 (cid:3) z t (cid:0) 1 . Using n (cid:17) p q to reduce

27 subscriptclutter. the n (cid:2) 1 coefficientvector h (cid:3) iscompletelydefinedby H ,the n (cid:2) n companion matrix of the industry forecast model, and the m friction arguments, (cid:18) . From (12), the restricted coefficient vector, h (cid:3) ,canbepartitionedintofourcomponents h (cid:3) = = [ [ A A ( ( B B ) ) ] ] [ A [ A ( 1 ( 1 ) ] [ ) ] 0 (cid:19) m [ W ( I 1 m ] [ (cid:0) W 2 G ] (cid:0) ) [ w 1 1 ] (cid:10) ; ( H 0 (cid:0) I n ) ] [ ( I n m (cid:0) G (cid:10) H 0) (cid:0) 1 ] [ (cid:19) m (cid:10) (cid:19) (cid:3) ] ; ( A : 6 ) where the first four partitions on the last line of ( A : 6 ) are functions of the unknown frictions parameters, (cid:18) . To indicate the partial derivatives with respect to the j th argument of the parameter vector, (cid:18) ( j ) , the following notation conventions are used. Element-by-element differentiation of vectors and matrices is represented by the relevant subscripted (cid:19) n ( j ) vector(s). For example, the partial derivativeof the n (cid:2) 1 vector b with respect toits second element is (cid:19) n ( 2 ) @ b = @ b ( 2 ) = [ 0 ; 1 ; 0 ; : : : ; 0 ] 0 = . Similarly, the partial derivative of the m (cid:2) n of matrix B with respect to its i j th element is denoted by @ B = @ B ( ij ) = (cid:19) m ( i) (cid:10) (cid:19) 0 n ( j ) . Two m (cid:2) m transformation matrices are useful also: T 1 is an upper-triangular matrix for effecting moving-summations,with (cid:0) 1 in each element of the first rowand 1 in each of the remainingnonzero elements; T 2 is a reverse-diagonal discountingmatrix with (cid:0) B m (cid:0) i+ 1 inthe m (cid:0) i + 1 thelementofthe i throw. As showninthemaintext,the bottomrowof the m (cid:2) m companionmatrix, G ,of thefriction lead polynomial contains powers of the discount factor, B , and the parameters of the friction lag polynomial, a 0 = [ a 1 ; a 2 ; : : : ; a m ] . The relationship between these parameters and the friction parameters, (cid:18) , in the decision rule likelihood, ( A : 1 ) , is given by (cid:18) = T 1 a (cid:0) (cid:19) m ( 1 ) . Denoting g 0 as thebottomrowofG,thenitisseenthat g 0 = a T0 02 = ( (cid:18) 0 + (cid:19) 0 m ( 1 ) ) T 0 (cid:0) 1 1 T 02 . Inturn,the m (cid:2) m matrix G is defined by G = T 3 + (cid:19) m ( m ) (cid:10) g 0 , where T 3 is constructed by bordering I m (cid:0) 1 by a column of zeroesontheleftandarowofzeroesonthebottom. Itfollowsthatthepartialderivativeof G with respect tothe j th element of (cid:18) is, @ G = @ (cid:18) ( j ) = (cid:19) m ( m ) (cid:10) (cid:19) 0 m ( j ) T 0 (cid:0) 1 1 T 02 . Usingthesedefinitions,analyticalexpressionsforthegradientsrequiredbytheGauss-Newton regressionin ( A : 2 ) are constructedfromthefollowingpartialderivatives: @ @ @ @ A A W W ( B ( 1 = 1 = 2 ) ) = @ @ = @ (cid:18) (cid:18) @ (cid:18) (cid:18) ( j ( ) j ( ) ( j j ) ) = = = = (cid:0) 0 (cid:19) m ( 0 (cid:19) m m ( 0 (cid:19) m m ( [ I m n m ) T ) [ I ) (cid:0) T 2 (cid:0)1 m G (cid:0) T 1 1 (cid:19) m (cid:0) (cid:10) 1 ( G H (cid:19) ; m j ( ) ; j ) 1 ] [ @ G (cid:0) 1 ] [ ( 0 (cid:0) @ = G @ (cid:18) = ( @ j ) (cid:18) ] [ I j ( ) m ) (cid:10) (cid:0) H G ] (cid:0) ] [ 0 1 I m (cid:10) n ( (cid:0) H G 0 (cid:0) (cid:10) I H n ) ; ]0 (cid:0) 1 : Finally, the gradient of the likelihood of friction parameters with respect to the vector of forecast model parameters, g y ( h x ) (cid:17) @ L y = @ h 0x , is required for the construction of M (cid:18) ;h 0 x . Let h x denotethecolumnstackofthefirst n x rowsofthecompanionmatrixofthefullforecastmodel, H .

28 Theorderedelementsofthis ( n x n ) (cid:2) 1 vectorarereferencedby h x ;( k ij ) ,where k ij (cid:17) ( j (cid:0) 1 ) n x + i for 1 (cid:20) i (cid:20) n x and 1 (cid:20) j (cid:20) n . Using this subscripting convention, the required elements of @ h (cid:3) = @ h x are @ @ W W 1 2 = = @ @ h h x x ;( ;( k k ij ij ) ) = = (cid:19) 0 m [ I ( m m n ) [ I (cid:0) m G (cid:0) (cid:10) G H ] (cid:0) 1 ]0 (cid:0) (cid:10) 1 [ [ G (cid:19) n ( (cid:10) j ) ( (cid:10) (cid:19) n 0 (cid:19) n j ( ) i ( ) (cid:10) ] : (cid:19) 0 n ( i) ] [ I m n (cid:0) G (cid:10) H ]0 (cid:0) 1 ;

Table 1 CointegrationConstruction ofIndustry PriceTrenda p (cid:3)t / (cid:18) c p c t + (cid:18) i p i t + (cid:18) v p v t + f ( t ) : industrycostshareb f(t) industry SIC (cid:18) c (cid:18) i (cid:18) v (cid:26) t SEEc ADFd textiles 22 .051 .339 .610 .359 -.0027 .036 -5.45 (cid:3) (cid:3) (cid:3) (5.9) (-4.6) lumber 24 .113 .145 .741 .0025 .082 -3.47 (cid:3) (cid:3) (19.4) rubber& 30 .011 .449 .540 .278 -.0005 .024 -3.85 (cid:3) (cid:3) plastics (10.1) (1.6) primary 33 .071 .168 .761 .875 -.0019 .053 -4.59 (cid:3) (cid:3) (cid:3) metals (6.9) (-2.6) motor 371 .005 .410 .585 .0023 .036 -3.10 (cid:3) vehicles (14.2) scientific 38 .007 .341 .652 .0036 .046 -3.44 (cid:3) (cid:3) instruments (48.7) aIndustry-specific log input prices aggregated with industry input share weights into primary commodity, p c , intermediate material, p i , and value-added, p v , input categories. The log of the unit cost of industry value added isestimatedbythelogoftheindustrylaborcompensationrate, w ,lessthelogofindustrytrendproductivity, (cid:26) . bConstructedfromthe1982USindustryinput-outputaccounts,BureauofEconomicAnalysis(1991). cSamplespan1957q1-1995q4. dRejection of the hypothesisthat the cointegratingdiscrepancy, p t (cid:0) p (cid:3) t , is I(1) at 90% (*), 95% (**), and 99% (***)confidencelevels,usingcriticalvaluesinMacKinnon(1991).

Table 2 Industry PriceAdjustment Under GeometricFrictionsa (cid:1) p t = (cid:0) A ( 1 ) [ p t (cid:0) 1 (cid:0) p (cid:3)t (cid:0) 1 ] + h 0 (cid:3) z t (cid:0) 1 + a t : manufacturingindustries SIC 22 SIC 24 SIC 30 SIC 33 SIC 371 SIC 38 rubber& primary motor scientific textiles lumber plastics metals vehicles instruments friction parametersb m=1 m =1 m=1 m= 1 m=1 m =1 A ( 1 ) .09 .05 .21 .13 .13 .08 (2.1) (1.2) (4.4) (3.0) (5.0) (3.9) R 2 .10 .13 .44 .11 .62 .28 S E E .018 .030 .011 .022 .010 .010 B G ( 1 2 ) c .00 .03 .00 .00 .00 .06 frictionweights meanlag(qtrs) 10.2 19.4 3.8 5.9 6.6 10.9 meanlead(qtrs) 8.3 13.7 3.4 6.9 5.7 8.7 pricetrend, (cid:1) p (cid:3) , expectations (cid:1) R 2 ( % ) d 98% 7% 98% 86% 10% 70% L R ( h (cid:3) j z t (cid:0) 1 ) e .00 .01 .00 .00 .31 .01 aSamplespan1957q1-1995q4. bmdenotestheorderofthepolynomial, A ( L ) (cid:0) 1 ,thatgeneratesfrictionweights. cRejectionprobabilityofseriallyindependentresiduals,Breusch-Godfreytest(12lags). dProportionof R 2 duetoforwardpricetrendexpectations, h 0 (cid:3) z (cid:0)t 1 (cid:17) E t f A ( 1 ) 1 =Pi 0 ( (cid:21) B ) i (cid:1) p (cid:3) +t i g . eRejectionprobabilityofREoveridentifyingrestrictionsonindustryexpectationcoefficientvector, h (cid:3) .

Table 3 Industry PriceAdjustment Under PolynomialFrictions a (cid:1) p t = (cid:0) A ( 1 ) [ p t (cid:0) 1 (cid:0) p (cid:3)t (cid:0) 1 ] + A (cid:3) ( L ) (cid:1) p t (cid:0) 1 + h 0 (cid:3) z t (cid:0) 1 + a t : manufacturingindustries SIC 22 SIC 24 SIC 30 SIC 33 SIC 371 SIC 38 rubber& primary motor scientific textiles lumber plastics metals vehicles instruments friction parametersb m=3 m =2 m=3 m= 3 m=5 m =2 A ( 1 ) .09 .09 .15 .14 .10 .06 (2.8) (2.5) (4.3) (4.2) (4.7) (3.4) A (cid:3) ( 1 ) .58 .35 .45 .56 .38 .36 (8.0) (4.3) (5.2) (6.9) (3.4) (4.8) R 2 .55 .23 .60 .43 .70 .38 S E E .012 .028 .009 .018 .009 .009 B G ( 1 2 ) c .77 .49 .94 .93 .98 .98 frictionweights meanlag(qtrs) 3.4 6.6 2.7 2.1 5.1 9.2 meanlead(qtrs) 3.3 5.8 2.6 2.1 5.0 7.8 pricetrend, (cid:1) p (cid:3) , expectations (cid:1) R 2 ( % ) d 2% 1% 12% 6% 5% 18% L R ( h (cid:3) j z t (cid:0) 1 ) e .07 .26 .15 .71 .31 .18 aSamplespan1957q1-1995q4. bmdenotestheorderofthepolynomial, A ( L ) (cid:0) 1 ,thatgeneratesfrictionweights. cRejectionprobabilityofseriallyindependentresiduals,Breusch-Godfreytest(12lags). dProportionof R 2 duetoforwardpricetrendexpectations, h 0 (cid:3) z (cid:0)t 1 . eRejectionprobabilityofREoveridentifyingrestrictionsonindustryexpectationcoefficientvector, h (cid:3) .

Table 4 IndustryPrice Adjustment withCyclicalMargins a (cid:1) p t = (cid:0) A ( 1 ) [ p t (cid:0) 1 (cid:0) p (cid:3)t (cid:0) 1 ] + A (cid:3) ( L ) (cid:1) p t (cid:0) 1 + h 0 (cid:3) z t (cid:0) 1 + X k 1 = 0 d k h 0u k z t (cid:0) 1 + a t : manufacturingindustries SIC22 SIC 24 SIC 30 SIC 33 SIC 371 SIC 38 rubber& primary motor scientific textiles lumber plastics metals vehicles instruments friction parametersb m=3 m =2 m=3 m=3 m=5 m= 2 A ( 1 ) .09 .08 .15 .15 .11 .06 [2.9]c [2.2] [4.6] [4.4] [5.7] [3.6] (2.9) (2.2) (4.2) (4.3) (4.5) (3.2) A (cid:3) ( 1 ) .59 .34 .44 .49 .21 .36 [8.1] c [4.3] [6.5] [5.7] [1.8] [5.1] (7.6) (4.2) (5.2) (5.5) (1.6) (4.8) R 2 .57 .23 .60 .45 .71 .38 S E E .012 .028 .009 .018 .009 .009 B G ( 1 2 ) d .88 .42 .72 .95 .81 .99 capacityutilization expectations L R ( D ( L ) j z t (cid:0) 1 ) e .07 .60 .95 .04 .05 .77 D ( 1 ) 0.22 0.11 -0.03 0.32 -0.23 0.48 [0.4] c [0.1] [-0.3] [1.6] [-2.4] [0.6] (0.4) (0.1) (-0.3) (1.5) (-1.9) (0.6) L R ( h (cid:3) ; h u k j z t (cid:0) 1 ) f .05 .47 .06 .87 .09 .09 aSamplespan1957q1-1995q4. bmdenotestheorderofthepolynomial, A ( L ) (cid:0) 1 ,thatgeneratesfrictionweights. ct-statisticsinbrackets,[.],arenotadjustedforgeneratedregressorbias;seediscussioninAppendix. dRejectionprobabilityofseriallyindependentresiduals,Breusch-Godfreytest(12lags). eRejectionprobabilityofnopricingresponsetoexpectationsofforwardutilizationrates, h 0 u k z (cid:0)t 1 ( k = 0 ; 1 ) . fRejectionprobabilityofREoveridentifyingrestrictionsonindustryexpectationcoefficientvectors, h (cid:3) and h u k .

Figure1: QuarterlyLagandLeadFrictionWeightsofProducer PricingDecisionRulesa STHGIEW SIC 22 Dotted: m=1 Solid: m=3 -12 -8 -4 0 4 8 12 51.0 01.0 50.0 0.0 SIC 24 Dotted: m=1 Solid: m=2 -12 -8 -4 0 4 8 12 STHGIEW 51.0 01.0 50.0 0.0 SIC 30 Dotted: m=1 Solid: m=3 -12 -8 -4 0 4 8 12 51.0 01.0 50.0 0.0 SIC 33 Dotted: m=1 Solid: m=3 -12 -8 -4 0 4 8 12 STHGIEW 51.0 01.0 50.0 0.0 SIC 371 Dotted: m=1 Solid: m=5 -12 -8 -4 0 4 8 12 LAGS LEADS 51.0 01.0 50.0 0.0 SIC 38 Dotted: m=1 Solid: m=2 -12 -8 -4 0 4 8 12 LAGS LEADS 51.0 01.0 50.0 0.0 aGeneratedbyEulerequationscontaining 2 m -orderpolynomialsinlagandleadoperators.