The Role of Time Series Analysis in Econometric Model Evaluation

This PDF is a selection from an out-of-print volume from the National Bureau of Economic Research

Volume Title: Evaluation of Econometric Models

Volume Author/Editor: Jan Kmenta and James B. Ramsey, eds.

Volume Publisher: Academic Press

Volume ISBN: 978-0-12-416550-2

Volume URL: http://www.nber.org/books/kmen80-1

Publication Date: 1980

Chapter Title: The Role of Time Series Analysis in Econometric Model Evaluation

Chapter Author: E. Philip Howrey

Chapter URL: http://www.nber.org/chapters/c11706

Chapter pages in book: (p. 275 - 307) EVALUATION OF ECONOMETRIC MODELS inThe Econometric Role of Time Model Series Evaluation Analysis ANNUNIVERSITYDEPARTMENTE. PHILIP ARBOR, OFMICHIGANHOWREY OF MICHIGAN ECONOMICS 1. Introduction facttime that series econometric analysis inmodels the evaluation are frequentlyThe of purpose econometric based of on this time models. paper series is data,Despite to consider classi- the the role of modern methods of methodsuationestimationcal regression inwhich particular, and draws andhypothesis related isheavily summarized testing. methods on time An Thisin seriesare approachthis paperalmost paper.methods is to alwaysorganized econometric generally, used as infollows. andmodel parameter spectral eval- In the next section the general ap- analysisfollowingmethodsThisproaches comparison including can ofsection classical play introduces anprovidesthe important econometricspower severalthe spectrum role motivation basicandin econometricand time concepts shows for sries the how ofanalysis modelview univariate these thatevaluation. are can time contrasted.time be used seriesseries The in inconcludestratesSectionmodel econometric the evaluation. 5 use containswith of amodeltime brief Sectionan series analysisevaluation.summary 4techniques is devotedof of aggregate the to potential to evaluate multivariate consumption role a simple of timetime datamodel. seriesseries which The analysis.methods illus-paper All rights of reproduclion in any form reserved.Copyright © 1980 by Academic Press, Inc. ISBN 0-12-416550-I 275 2762. Evaluation of Dynamic Econometric Models E. PHILIP HOWREY (1972,metricslationships pp. to 5_6)*the from problem as observed follows. of inference data. TheEconometrics is general succinctly approach issummarized concerned of classical withby Johnston drawingecono- inferences about economic re- form.nomicmathematical Next theory we areform, must not for assembleusually sufficient appropriateThe tofirst yield and step arelevant precisein the dataprocessmathematical from is the the specification of the model in . . . the a priori restrictions derived from eco- orconstitutesouttheeconomy whether datatests to onor aestimate sectorthesufficientlysomewhat estimated that the thedifferent parametersrealistic model model specificationpicture purports in of an the ofattempt model tothe describe.has economy andto bejudge finally Thirdlyestimated. being whether we studiedwe carry use it ficationGoldberger of the (1964,model. p. 4) also emphasizes the crucialmethodthe rules importance ofand measuring criteria of of the a statistical relationship speci- inferenceOnce of economic we in haveorder a theoryto specification develop from a rationala ofgiven a parent population we may rely on Thus the traditional econometric approach beginseconomicthetheoreticalsample population. with of theory. observations. theor empiricalpresumption Such a priori knowledgeIn many information cases about we isthe may a characteristicvalues rely alsoof parameters on feature previous ofof theminedspecifythat disturbance economic by a hypotheticalthe hypothetical theoryprocess. or model. "previous" model, Appropriate includingGrenander empirical estimation the & knowledge stochasticRosenblatt methods specificationis(1957, sufficient are pp. deter- 115-116) toof suggest that modern time series methods have been developed to deal withveryOne ratherpiricalspecified difficultylittle different datatheory schemes. into builtmanyformsituations. Inup ofconfidence such fromthe applicationsfields experience regionsit seems soof fortimemorethat the oneseries promising models is notis that led than to thereto use towell em-istest Company. Used with the permission* ofFrom McGraw-Hill "Econometric Book Methods" Company.sharply by defined J. Johnson. models Copyright whose © 1972 validity McGraw-Hill is questionable Book to say the least. TIME SERIES ANALYSIS IN ECONOMETRICS 277 believeseldomThesedemand testing relevantthat for the problems realistic generalto problems may methodsapproach have arising someto takenIn analysein engineering practice.theoretical by research stationary interest,and workers in time the but physicalseries. inthey these are We sciences there has been a strong fieldswithconsiststhesome earlieris a more finite similarin techniquesnot promising number specifying nonparametric of developedand parameters the in closermodel concept. by one contact verytheoreticians. considers much, with andrealitythe spectralinstead than someofdensity dealing of or . . . This approach PriestleyThe importance (1971, p. of 295). prior knowledge, or lack thereof,(or "identify") is also emphasized the transfer by functionwe of may a linear contrast time-invariant the problem "black of using box" input/output data to estimate twolinearin the problems relationshipcontext ofare control betweenoften statisticallysystems two economic . identical, time series. in the .former case one .. with the problem of estimating a . . Although the concludeWe need not that agree the appropriatewith Priestley's methods implicit of evaluationanalysispriorlyingusually knowledge.willthe of has "black-box",economicdepend substantial upon theory whereas knowledgehow to in the of latter the physical case one mechanism has virtually under- zero modelappropriatemuch prior that knowledgeis estimation specified. is and Supposeavailable. hypothesis thatA simple we testing wish example proceduresto test isthe sufficient hypothesis depend toon demonstrate thatthe the crucial point that If the wherealternativethere {u} is no is torelationshipa sequencethis null hypothesis of between independent isthe the two simpleidentically time distributedseries distributed { yj lagand model, N(O, t= ±1,±2,..., (1) 2) thewouldsimplyrandomtest correct ofsurely test variablesthe formulation.the hypothesis be hypothesis inefficient and x that andBut thatto uif employ area more independent a general more general alternative for all model t and is s,ifappropriate, (1)then is we indeed can a i = 0 in (1) may not beii =very informative about the 0 against the alternative that fr 0. It extentNospecifiedrelationship matterpossible model. what between that model none { of Yt}is thespecified and basic {x1} assumptions Thisinitially, since example it willit isof illustratesbeimportant the based model onthe toare an importanceverify violated.incorrectly to the of model evaluation or validation. 278This process of model validation can be carried out by relaxing certain E. PHILIP HOWREY askingtotheinvolvesmaintained the hypothesis distributed if embeddinga more hypotheses that general lag the the example, more model informulation such general in we aa waymore could formulationsuch that general examineas they framework areis unnecessary.the testable. adequacy and This then Returning of usually testing(1) by originaltestingis unnecessary. and maintained hypothesis It might hypotheses searching. be objected are This subsequently that is certainlythis process rejected. possible, confounds But especially the hypothesisalternative if the Yt = + k=O kXt_k + Ut + (2) situationspreciselycases.of never examining becausein which they little maintained have prior been knowledge hypothesesTime developed, series is isavailable. methods equally at least unattractiveTocanin part,the be extent useful to deal in that formany with aneconometric model evaluation seriesofformulationwork,econometric a structural model,time series is modelthe econometricnecessary. time methods can series be Stated embeddedmodel canmodel be the place willused other in provide restrictionsato moreway determine aaround, general vehicle on ifa if timemoretothe the test more seriesassumptions general the generalvalidity frame- time ofassumean p,thosealternative q, andthat restrictions, r,economic thatto (1). x, andTheand theory Uhenceclassical are or independent otherthe econometric Returningadequacy prior knowledge for ofagain all approach the values toeconometric determinesthe of todistributed t(2) and would model.s,the and valueslag be that model, to suppose (2) is proposed as theseprocedures.leadsSuchthe (likelihood)disturbance adirectly complete This to methodsisterm parametricclassical not u to is issaya (likelihood)seriallytrivial specification that ineither independent this estimationthecase; of thetheory this model(normal) is orand indeed relatingthe hypothesis random application an {important yj variable. and testing oftx} isset variablestopica of sequence assumptions. in time and of seriesthat independent It(2) mightanalysis. is the be correct and appropriate, identicallyThe specification time for series distributed example, for approach some to N(O, finiteassume to modeling but that unknown {u} typically involves a slightly weaker 2) random randomunderliespursuedassumevalues variables ofthethat by the Box Parzen{u} parameters and &is a Jenkins(1974).p, sequence q, and p, In(1970) rq, arethisof and independent infinite. latter procedures. r. This case, This is identicallyanythe is Alternatively, essentially finitemodeling parameter distributed theapproach one approach modelmight (0, thato.2) is TIME SERIES ANALYSIS IN ECONOMETRICS 279 model.restrictionimposeviewed If noprioras moreanthat knowledgeapproximation the structure two processes ison even the to therelationship morehave correctly thelimited, representation between specified, it may Yt} be infinite-parameter and appropriate {x1} than to the polynomialswhere u of the form are mutually and serially uncorrelated processes and O1(L) are x, Yt= 011(L)u11 = 021(L)u1, + 012(L)u2,+ 022(L)u2, (4)(3) includesin the lag (1) operator and (2), L asdefined well as by all LkX the intermediate cases discussed above, as 0.3(L) = k=0= Xt_k. Notice that the system (3)(4) special cases. In particular, if 0 2(L) Oi (L)y = 0, ,(L)02 1(L)u,, + 0 ,(L)022(L)u2 = 021(L)x, + O11(L)022(L)u2. 0, then (4) can be rewritten as (5) embeddedtively,If 0(L), then 022(L), in (5) a moreand and (2) general021(L) are equivalent. timeare polynomialsseriesIt is model.now well ofIn degree particular,known p,that q, Zellner theand standard r, &respec- Palm linear econometric model can be Followingstudiedmultivariatecertain(1974), conditions byZellner Zellner,Quenouille autoregressive, (1975), the let standard z(1957), Wallisdenote moving-average linear Parzen(1977),a vector econometric (1969), and of variables others (ARMA) Hannan model have generated (1970),models isshown a special by'andofthat the caseothers. under type of matricesrandomwhere u} ofvariables is polynomialsa sequence of unobserved of in independent the lag operator disturbances, identically L, i.e., distributed and I(L) andN(0, ®(L) are = ®(L)u, ) vector (6) 'I(L) = (8)(7) where J? and ®, are matrices1 For simplicity, of coefficients. it is assumed that there are no purely deterministic variables inj=o the system. eL', 280 E. PHILIP HOWREY turetheand on systemexogenous the process. of equations variables, However, is a partitionedbasic it does Thefeature not multivariate according distinguishof econometric to ARMA between modeling. model endogenous imposes Suppose a considerable amount of struc- and the restrictions D21(L) [21(L)[11(L) G12(L)1 [p11 22(L)] [xj - [®21(L) 0, [®11(L) 0, and ®21(L) e22(L)]®12(L)1 [u2j[ui1 0 are imposed. (9) Then the system simplifies to Ij1(L)y + Iiz(L)x = ®2(L) = ®22(L)u2. (11)(10) TheoftheSuppose, secondequationsthe vector standard in equation,x addition,ofin thisexogenouslinear system which thatdynamic {u1jcontainsvariables describes simultaneousand the{u2} and howvector correspondsare the independent.equations exogenous to econometric the Then variablesstructural the firstmodel. areform set Yt of endogenous variables and ogenous.stochasticmodel{generated, u1} is and Thusto exogenous {u21}be is underusuallyused implies forthisvariables omitted ex specificationthat ante isfrom prediction,required. the the econometric simultaneousNotice a forecasting that model. the equations independence model However, econo-for any ifof the x1} is independent of {u11} and hence is ex- thestructuralhypothesesmetric endogenous model form about is ofavariables block theeconomic simultaneous recursive appear behavior. inmultivariateThe moreblock structuralThe than is distinguishingthe ARMA one fact form equation. that time of currentan series feature Thateconometric model.values is, of if thethe of model is typically used to test thenoperator need11(L) not is and written generally as will not be the identity matrix. If the 111(L) = j=0p11 (12) formissimultaneous generally of the system used block isfor obtained. ofpurposes equations The of conditional isreduced premultiplied form, prediction denoted by iIj0, and by then control. the reduced 11(L)y, + 12(L)x = 11(L)u11, (13) For illustrative purposes, consider the reduced-form model Y2tYit = 1Ylt-1 + 2Y2t-1 + 1t flYi,-i + f32x1 + u2, (15)(14) TIMEThiswith SERIES twomodel endogenousANALYSIS can be INwritten ECONOMETRICSvariables in operator Yi, and notation as Y2t and one exogenous variable x1. 281 one-periodwhich is a specific ahead forecasts example of of the general form given in (13). Conditional [fijL[1ci1L c2L1[y11 [011 Yit and Y2t are obtained][y2J - from this model by Jxl = [uJ' [u111 (16) extoYit-i,setting asante Y21-u1, = u2 = 0 in (14) and (15) and solving for ex post 1, forecastsand if x 1t' the predicted value of x1,. Such forecasts are referred is the actual realized value of this variable and Yit and Y2t in terms of forecastsandprediction and error variance for yFrom the expoint post of view of model evaluation, theforecasts interesting if the feature value assignedof the to x1 is a predictedforecasts. value The ofprediction x11.O2 The error variance for Y2t is a22 for + f3E(x1 - i)2 is a11, the variance of u1, for bothfor ex ante forecasts. ex exante post representationstransfermodelsmultivariate placefunctions ARMA restrictions of the model variables. on or thereduced-form Thetransfer reduced-form functions econometric andsystem univariatemodel implies is that ARMA a setthese of which relate each of the endogenous variables to lagged reduced-formvariables.2values of that These equationsendogenous transfer by variablefunctionsi1(L), the and adjointare to obtained current of T11(L). andby premultiplying lagged This yields exogenous the T1(L)12(L)x + Ti(L)1j(L)u1, 'P(L)x + reduced-formthesewheregiven predictions lagged model values will since generallyof y,they and are currentbe based inferior onand a tolaggedsmaller the predictions valuesinformation of x. from set.However, Thethe (L) = = 111(L)I. These transfer functions can be used to predict y1, = T(L)u11 (17) whereasfollows.thereduced-form endogenous the The transfer-function adjoint forecasts variables. of the use matrix forecastslagged Thevalues use transfer only of allthe thefunctions lagged endogenous values for the of variables, oneexpository of model can be obtained as c1L 2L1 is I11(L)=[f3L?1(L) [i -ri c2L 1 1 by Kmenta (1971, p. 591). 2 What are referred to here as transfer functions are called fundamental dynamic equations = [s1L 1 - When282 (16) is premultiplied by IY1(L), the result is E. PHILIP HOWREY The transfer function for Y2tYit = 1Y2t I + 2fLY2t 2 + I32Xte - 1f32x11 + /31u1_iYit- 1 + 2f11Y1t-2 + 2fl2Xlt_l + Yit clearly illustrates the general proposition that lt + 22t-1, & + U21 (21)(20) whichtransfer-function clearly exceeds forecasts forecast areerror inferior is u1 +to reduced-form forecasts. The a (assuming both 2u2_ 1and a22 are nonzero).3 Noticewith variance a + ex post involvesoperatorsalso that Y2t-1,the in thereduced Y2ttransfer form functions. places restrictions The transfer function for 2 x11, and x . In addition, the composite disturbance on the orders of the polynomial Y2t, for example, term = U + fl1u11_1 - (22) moving-averagerestrictionsprocess.4is the sum Thus of on two therepresentationsthe moving-average transfer-function reduced-form for specificationA eachrelationships. link of between the variables the multivariate in the general model and univariate processes and is itself a moving-average places a number of testable autoregressive, islinear obtained,original dynamic system where econometric (6) A(L) is multiplied model byis provided b*(L), the by adjoint thefina( equations.5 = jF(L)j and F(L) = t*(L)®(L). In particular, if = F(L)u1 of 'I(L), the system If the (23) r'(L) denotes the jth row of F(L), then = f(L)u1 = z1 can be writtenk=1 as ]TJk(L)ukt. (24) processes,Since the disturbanceit is itself a moving-average process in this equationprocessThis model and, is provides hence, a the simple example of the more general problem considered a sum of moving-average process can be by Pierce with(1977).(1975). what Kmenta (1971, p. 592) calls the final formThe Forof term the a further finalequation equation discussion is taken of this from point, Zellner see Ansley, & Palm Spivey, (1974) and& Wrobleski should (1977) and Palm system. not be confused TIME SERiES ANALYSIS IN ECONOMETRICS written as A(L)z11 = (25)283 eachuniquestochasticrandomwhere variable {v}univariate variables. exogenous is hasa sequence autoregressivetheThe variablessame final of equationsautoregressive independent implies moving-average show that identicallythat the multivariate each partvariable (unless in the there model are hascommon a representation. Moreover, distributed N(0, () model with exogenousequationEqs.factors (14) inforand variable A(L) the (15), exogenousand Xtiit Fjk(L))is is necessary variable.6but generally toIn augmentorder Suppose different to obtainthe that model final the equations for the expository moving-average parts. model consisting of withequation a stochastic for the adjointrandomwhere u3}of variables. the is amatrix sequence The final of independent equations identically can be obtained as follows. The distributed N(0, cr33) (26) b(L)= 1c1L cc2L/31L 0 01 10 is (J)*(L) = f31L 11 1 - ci1L 2L 12 - cL1f32L c2/32L Thus the final equations, after cancellation of = iYi-i + c2fliYit2 + Uit + cL2U2_1 + c2132u31_1, [0 0 1 - 1L - 2fl1L2 common factors, are x1IY2t = U31. = 1Y2-1 + 2flh)'2t-2 + u1_1 + u2 - + /32U3 - c1/32u3_1, wouldleavethe theunivariatenot model exist for inmodels thethe ARMAXendogenous would include (see variables. Hannan6 the exogenous (1976))If variableform the exogenous variable x1 is deterministic, this step is not possible, and we would simply x1,as shown and univariate in (20) and ARMA (21). Thatmodels is, 284 E. PHILIP HOWREY alsotheerrorareThis predictioninferior that examplevariance the to reduced-form error illustratesfortransfer-function Yir variance using the model generalof(29) the forecasts.is transfer-functionimposes o result that Therestrictions univariate one-step forecast on aheadARMA the for order Yit.prediction forecasts Notice of the + 2a22 + cf3a33, which exceeds Ifrestrictionsof autoregressivethethe simultaneousvariables on thein and the transfer equations moving-averagemodel. functions specificationAs we partsand have final of is seen,the correct, equations univariate a structural these of representations therestrictions dynamic model. economic model imposes testable specified,modelofandshould these dynamiccan bethese besame satisfied used propertiesimplied characteristics. to byderive properties the of thedata.implications endogenous TheshouldMore next generally, be about twovariables.consistent sectionsprediction an If estimatedwith the are errordirectmodel concerned econometricvariances estimates is correctly with thosetimefully,attemptto be aspects series mosthowever, is madepromisingofmethods univariate enough to provide can for material andbe econometric used anmultivariate exhaustive isto included evaluate research time review toeconometric enableseries and of modelanalysis thethe readermodels.literature. evaluation. that to appear see Hope- how No 3. Univariate Time Series AnalysisEconometric research is primarily concerned with the estimation of inseriesrelationships connection is still veryamong with: important. variables.(a) development For testingNevertheless, example,development restrictions of benchmarkunivariate the analysisofimposed formulas methods forecasting of on individual theforare dataexpectationalof models, interest by time simultaneous variables, equation and (d)models, modeling to anddisturbancea brief the discussion power processes spectrum, of four in multipleimportant regression areas of univariate models. This timetestsdescriptive section series for serial analysis: is devotedmeasures correlation for in time a time series series, including the covariance function whichfor stationary samples time are series. obtained are covariance-stationary.Throughout this section,estimationidentification Ait iscovariance- assumed of the and covariance thatestimation the stochastic function of autoregressive, processesof a stochastic from moving-average process, and models stationaryTIME SERIES process ANALYSIS is IN a ECONOMETRICSprocess for which the mean, variance and auto- 285 tiondisturbancetooutcovariances whenbe any an trendapplied especially termdo in nottothe in economic dependmeanarestrictive time or series on variancetime calendarassumption regression series of time.thedata series.model.when directly In particular, appliedThis is not usually thought It is a restrictive assump- since most economic to stationaritythe stochastic rules thatformationseries.totime transform satisfies series It is has exhibitassumed thethe been stationarityraw pronounced applied seriesthroughout byassumption. to detrendingtrendsthe this data, sectionin theif in order that to obtain necessary, to producemean. aIt series is usually necessary an appropriate trans- a stationary important3.1. DESCRIPTIVE characteristics MEASURES of a FOR random TIMEThe sample.SERIES(sample) The mean and variance are frequently used to summarize sine qua non of time a samplethatseries arei, analysisdefined (or sampled. population) byis the Some (potential) is descriptive therefore existenceThe of autocovarianceinterest. of serial correlation function of the stationary measure of the correlation pattern in process x} with mean in the processes andprovides sometimes one way very to usefulsummarize way tothis summarize correlation the correlation y(s) = E(x+181 -. ILXXt - IL), s=O,±1,±2,..., pattern. An alternative (32) is definedprovided by by the power spectrum. The f(w) = y(s)exp(iois), power spectrum, provided it exists,7 pattern is (33) spectrumwhere exp(icos) derives from the fact that = cos(ws) - i sin(ws). The descriptivey(s) appeal of the = - 8 exp(iws)f(oi) dw power (34) y(s).absolutely The relationship summable (Fuller,in (34) is1976, the inverse8 Fourier transform.TheA sufficient function f(co)condition as defined for the in spectrum(33) is the toFourier exist transformis that the of autocovariance the p. 127). f autocovariance function function be 286and, in particular, E. PHJLIP HOWREY Thusfunction although and there the spectrum, is a one-to-one and hence relationship the same between information the autocovariance is conveyed by = 2it S f(w) dw. (35) f(w)terms(w,wboth, dw + theof measuresdco). sinusoidal spectrum the oscillationshas contribution an analysis per to unitAlthoughof the variance of variance time. it interpretation.Recallis notof {x} thatobvious, ofthe the timeThe theinterval quantityfunction argument w has an interpretation in is a perfectly periodic function with period p = 2it/) that is, y = acost) + bsint) + = Yt for (36) ofisi/pall measured a integer= cycle )./2ir pervalues of in year.9a years cycle of k. and thatThe the isfrequency completedperiodNow is of ten thispersuppose years, periodicunit theaof and time. frequencyfunction b are For independent, isexample, isthe one fraction tenth if zero-mean t random variables with Strictlyfunctionvariance speaking, 1. Then { theYJ ispower a random spectrum variable does with not mean exist zerofor this and process, covariance but y(s) = cos 2s. (37) we can consider the approximation'° f(co) = S = - /3scos,sexp(_kos) (38) inistheas a.95.{a spectrum sharpfunctionyj The is duespike graphs of ofto at {Yt}. j3.variation inthe AsFig. frequencyThe the 1 indicategraphatparameter this )/2it.of frequency.that fp(CD) Thisf3 the approaches limiting indicatesis shownThis value is, that inone, of Fig. ofall course, f(w) the of1 forpowerthe approaches f precisely variation= .9spectrum and mable.longercycle cycles per time in the unit. discrete Cyclical data. variations with a periodWith shorter equallyAs than long spaced two as dataUnits points, of time the appear highest as observable frequency of oscillation is one-half < 1, the power spectrum will be defined since pscos) is absolutely sum- TIME SERIES ANALYSIS IN ECONOMETRICS20 287 l015 0 S what would beFig. expected 1. Power from spectrum the transformationdefinition 0of of { J3S Yt} coss). since Bold any curve: 13 = .90; light curve: 13 .0 .1 .2 FREQUENCY .3 .11 one realization =.5 .95. model,forexceptof the disturbances processfor illustrative is a in simple regressionpurposes. sinusoidal A models,moreProcesses oscillation." useful is modelsuch as for that time defined series, in especially (36) are not very important in economics, the first-order autoregressive svaluesvariables.where Ju}of p. Poweris For a sequence p spectra= 0, {Yt} of of independent is t} serially are shown0. uncorrelated Theidentically in power Fig. 2 spectrum distributedfor three in different thisrandom case is constant, indicating that all fre- Yt = pyt, + Ut, p1 < 1, so that y(s) = 0 for (39) Conversely,contributeissequence.quencies a decreasing of Thismore oscillation with function is to an pthe =important contribute -varianceof .7 frequency the spectrumbenchmark of equally { indicating yj than tois thecase.an do that varianceincreasing high-frequency With low-frequency p of = thefunction.7, theuncorrelated variations. variations spectrum of fre- Oncequency these values and high-frequency are fixed, y is determined variations" A realization by (36) are for of dominant. all{y,} values is obtained of t. by fixing the values of the random variables a and b. 288 \ ' B. PHILIP H0WREY LU \ 0 \ \ curve: p = .7. Fig. 2. Power spectrum ofj', = Yt-i + u. Light curve: p = .7; bold.0 curve: p .1 .2 FREQuENCY .3 .11 = 0; dashed autoregressive,obviousbusiness-cycleuncorrelated, to the contains moving-averageuntrainedvariation. strong eyeThese from seasonalrepresentation characteristicsA an graph examination fluctuations, of ofthe ofthe spectrum the of theseries series reveals itself, at the a glance whether or not the series is process, or the (sample) or exhibits strong may not be supposeeffectvisualautocovariance ofthataid linear in {yt} describing (moving-average)functionis related a of timeto the {x} series. series. Theby operations power Thus the spectrumon spectruma time series.also provides provides For example, a convenient way to examine the a useful Then the power spectrum of { yj is related to the spectrum of {x} by f(w) = ytG(w)f(co), = (41)(40) characteristicsThewhere function G(w) of theis referred transformed to as the series gain { of yj the will relationship. depend in Thepart cyclical G(co) = wjexp(iwj)I2. on the (42) TIME SERIES ANALYSIS IN ECONOMETRICS4 289 3 D 0 .0 .1 .2 FREQUENCY .3 .11 .5 thatcorrelation is used. patternAn examination of the original of the seriesfunction and G(w) in part of the on transformationthe transformation will Fig. 3. Gain function for Yt = - Consider,revealIn this case,the for nature G(co) example, isof the the smoothing centered first that difference is implicit transformation in the linear operation.defined by Yt=xt+iX1_i. (43) seriallyappliedthe graph correlated to of a whichserially and is uncorrelatedwouldshown containin Fig. process, 3. relatively If this the centered littleresulting long- first series differenceor short-run would were be G(w) = 2(1 - cos2co), (44) teristicspropertiesthevariations effect of macroeconometric ofbutof filteringseasonal rather pronounced onadjustment the models.'4 cyclical intermediate-run procedures,13Spectral characteristics methods and variation. theof have timedynamic beenseries,'2 charac- used the in this way in economics to study byprocedures, Chow (1975), see Nerlove Dhrymes (1964), (1970), Godfrey Howrey14' 12Spectral &See, (1971, Karreman for methods example,1972), (1967), have and Howrey Howreybeen Forand examplesGrangerused (1968) & Klein and(1978). of (1972). theFishman use of spectral methods.to study the properties of seasonal adjustment to study the dynamic properties of econometric models (1969). 3.2.290 TESTS FOR SERIAL CORRELATION E. PHILIP HOWREY however,correlationWatsonare an extremely testthe isinDurbinWatson probablyeconometrics. important the most test aspectAs Testswell is Durbin not known offor a model veryserial(1967) and powerful evaluation.correlationwidely and others used test of testThe havefor the for serialDurbin disturbancesnoted, serial in a regression model variables.wherecorrelation {v} Forisin aa sequencethismodel model, like of independentthe first-order and serial identically correlation distributed coefficient random is Ut = AUt_2 + Vt, (45) Watsonalternativenotifzero the necessarily but DurbinWatsontest the is is {u}rather appropriate. appropriatesequence powerful test leadsIndeed,is not toover assumeto serially therea rejectionmuch isthatuncorrelated. somewider a of first-order evidencethe range null On of hypothesis, autoregressivethatthe alternatives other the Durbin hand, it is is thatopedhavethan if thebeena thetest first-order timeproposed based series on autoregressivein the is the uncorrelated,cumulated literature. model.'5 Severalperiodogram. Durbin the spectrum tests (1969), for The forserialwill basic example, be correlation aidea constant ofhas this devel- based test on an estimate of the spectrum as in Fig. 2 (with p = 0) and the normalized cumulative spectrum, F(w) = 1 f() d2, (46) correlationualswill deviates trace is rejected. out significantly a 45° It shouldline. from If bean the obviousestimate 45° line, that F(w) thethis based nulltest, hypothesis aton least the regressionin principle,of no serial resid- y(0) 5: test,thethanis capable Durbin isthe that standard of ifperiodogram detectinglittle orDurbinWatson nothing a wider test oris range knownthe Thetest. Box of aboutdeparturespoint and Piercetheof thesenature from (1970) more theof potential null"portmanteau" general hypothesis depar- tests for serial correlation, such as toalternativestures satisfy from this theis requirementdesirable. null hypothesis, Test rather procedures 'well.a test that based is sensitive on spectrumSee, to for a estimatesexample,wide range Smith seem (1976).of TIME SERIES ANALYSIS IN ECONOMETRICS 291 estimator3.3. CONSISTENT of a covariance ESTIMATION matrix OF THE isThere COVARIANCE required are important to obtain MATRIX situations in econometrics in which an asymptotically a consistent whereefficient yexample, is a T estimator x the 1 vector model of ofthe observations parameters of a regression model. Consider, for y=X/3+u on the dependent variable, X is (47) Aitkentoindependently Taa21, Tx 1x estimator.the vectork matrix least ofof squares u, ofunobservedIf f3 Eobservations is aestimatornot k x known 1disturbances. vector onis the generallythe of preferred regressionexplanatory If inefficient variables relative distributed u '-. (0, parameters,) where and u is a is not equal to the whereAitken estimator is a consistent estimator of E. /= (X't'X)'X'1y, estimator is the feasible (48) context, u' = [u1 u2. The problem is to obtain a consistent estimator of . . UT] and . . . . In a time series = E(uu') = - y(0) l) (1) . . y(T-2)y(T 1) (49) where 'y(s) = E(u+11U). Thus if no restrictions y(T 1) y(T-2) are imposed on the covariance. . . y(0) thatassumedliteratureThematrix -y(s)traditional E, that is=there toa2p. {u1} impose aresolution This isT +generated restrictionsome k to parameters this restrictions problemby reduces to be the estimated number of a first-order autoregressive process so on as described in the econometrics . For example, it is frequently from T observations. parameters to be powerresult.16knownestimated spectrum As about Tand -> theateffectively the form harmonic of solvesthe disturbance frequencies theAn estimation alternative processwj = problem.2itj/T. is approach based The corresponding which is especially attractive when characteristic roots of are equal to the values of the on the following little is matrix of characteristic vectors is16 W See, for example, Fuller (1976, Chapter 4). = (wfk) with elements WJ, = T'12 292 E. PHILIP HOWREY exp(-2itzjk/T). In other words, urn WW = D, (50) estimatorwithunitarydiagonalwhere all W* matrix,elements elements of is the i.e., conjugatereplaced f(2xj/T). W" W by transpose = It theirWW' is easy complex = of I,to Wwhere verify and conjugates. DW* that is isa W diagonal the Henceis transposewhat amatrix isconsistent called of with W a is obtained from a consistent estimator of the spectrum usingspecificationSince consistent of the estimation disturbance of theprocess'7 spectrum a feasible does not Aitken require estimator a parametric can be = wDw*. (51) 3.4.obtained IDENTIFICATION in the absence AND of suchESTIMATION a specification.18 OF AuTo1GIssIvE,MOVING-AVERAGE MODELS relyaveragesetmethods on of a techniques finite modelshave parameter played in to which deal a timekey with the role. domain orders finiteThe Box univariatemodel. parameterof& Jenkinsthe Inautoregressive these time autoregressive,(1970) situations,series have methodsand developed spectral moving-moving- described a up to this point do not themodelaverage approachdegree relaxes operatorsof takenthe the polynomialcharacteristic by are Parzen assumed operatorsin assumption that toIn bebrief,a finite isunknown. known the of parameter BoxJenkinsclassical but This is modeleconometricslessspecification procedures extreme is retained. thanofthat for the univariate time series involve variousmaximumoperators.functions'9an examination diagnostic Oncelikelihoodto determine oftentative checks the estimates sample the andvalues order overfitting autocovariance of haveof the the been coefficientsautoregressive procedures assigned and partial are to and obtained.employedthese moving-averageautocovariance parameters, Finally,to make sureWatts that (1968, the Chaptermodel 6).that is identified17 is consistent withFor a the discussion data. of power spectrum estimation techniques, see for example, Jenkins & (1967)x+1, show it is possible to develop a19 regression analogue to Hannan'sTheThis partialis the estimator. basis autocovariance ofregarded the procedures as function a function suggested is theof bys. covariance Hannan (1963,1965). of x, and x1 As Amemiya & Fuller - given xj..... TIME SERIES ANALYSIS IN ECONOMETRICS 293 selectiontheinformal use of approachof likelihood the appropriate to theratio model model.tests selection and ThisAs posterior Zellner emphasis problem. (1975)odds on Zellner theratios has model remarked, hasto aid proposedselection in Box the and Jenkins employ a somewhat suremakesbothtimeaspect thatseriesa carefulprioriof anthese approaches important reasoning and techniques extensive tocharacteristic and modeling, serves usedata of analysis. tothe namely, underscoreof data, the Since itdata a provides model hasthethis notisdistinctive modeling a determined goodbeen wayoverlooked. procedurefeature tothrough make of 4. Multivariate Time Series AnalysisWithin the context of econometric modeling univariate time series ofdistributedareandanalysis the offor importancebasic modelingis usefullag concepts models. for disturbancefor estimating theof This multivariateestimation section processes. the finalbegins and time equations analysisMultivariate with series a brief ofanalysis.of transferan introductiontime econometric Followingseries functions methods to model some andthis metricsintroductory4.1. DISTRIBUTED are reviewed material, LAG including several MODELS applicationstests for causality. of particular interest in econo- where {x} and u} are mutually independentFor expository stationary purposes, stochastic consider processes. the bivariate distributed lag model Yt = fljx_ + u, t = 0, ±1, ±2, .., (52) werecases.modeldistributedTo ensure little Thisincludes theoreticalthatlag general coefficients{ the yj distributedhasformulation knowledge a finite {Ji} lag arevariance, ofor modelsabsolutely theprior model informationwe discussed imposesummable. might inbe theabout Sectionappropriate Thisrestriction the rather 2relationship as generalthatspecialif there the isforprovidebetween serially example, a {Yt} wayuncorrelated, want andof testing {x,},to test given sothe the that validity hypothesisthe one generalAlternatively, has of ato simpler that linearsearch the relationship forspecification.disturbancestatistical an appropriate analysisbetween process One model.might, {yt}of {u} this more general model would 294 E. PHILIP HOWREY neverandtotionsship {x,}. see investigateis it ifone-sided, Similarly,might the data be their i.e.,are appropriatea test consistentf3validity; of the tohypothesis in with impose other such cases such that assumptions. ittherestrictions might distributed be veryat lagthe important relation-outset and = 0 for j < 0, might be of interest. In some applica- It is not difficult to verify that the modelThe spectralimplies approachthe covariance to the relationshipsanalysisyyx(s)=>fljYxx(sj), of this model proceeds as follows.20 s=0,±l,±2,..., (53) covariancewhere y.(s) functionis the autocovariance of {u}. The spectral function and of cross-spectral x} and y,,,,(s) functions is the'y(s) auto- are = jk J3jI3yXX(s + k - I) + y,1(s), s = 0, ± 1, ± 2,. . . , (54) f(w)thus given by2' = S = co 53 (s)exp(iws) = B(w)f(w), it, (55) wheref(w)defined f(w) by and f,,,(w) are the power spectra of {x1} and u} and= B(co) is y(s)exp(iws) = IB(o2f(oi) + f,,,(w), it (56) tantis transformed numerical into simplifications a product relationship whenThe thefirst B(w)f(w).method point to of notice This moments resultsis that isthein used impor- convolution to relationship B(co) = /iexp(koj). f3jyjs -(57) j) In estimateaddition, the the parameters. inverse of (57) In particular, is (55) and (56) can be rewritten as f(w) B(co) f(w) - it- it it, (59)(58) 20 For a more detailed discussion of the spectral approach to the distributed lag models, = _- f' B(co)exp(iwj)dw, j = 0, ± 1, ± 2 (60) 1970,covarianceWahbasee Dhrymes Chapter (1969). function (1971)8). The results isand very Fishman can convenient be readily(1969),21 but generalized for can example. be relaxed to ThistheThe without multivariate section assumption undue is based distributed difficultythat in {x,} large is(Hannan,lag apart stationarycase. on stochastic process with an absolutely summable (60)However,TIME canSERIES be if ANALYSISreplaced /3, is zero by IN for ECONOMETRICS all j outside the interval - m + 1 m, then 295 Since consistent estimators of the spectrum and cross-spectrum basedf3 = on (2m) s= ni+ 1 B(its/m)exp(imsj/m), m + 1 m. (61) definedstatisticsconsistentthe sample by estimationare auto- usually and of presented.B(w),cross-covariance J,,,(w), The andFor coherencef3, functions purposesis possible between are ofusing graphicalreadily (55), {yj (56),available,22 and presentation and{x1} (61). is of the results, several additional Thefrequency functioncoherence B(co)co atexplained frequencyis generally by wthe complex is linear the fraction valued. relationship ofInstead the between variance of graphing { of yj {theand Y} real at C(w) = 1 - J,,(co)/f(w). (62) whatand is imaginarythe called phase the function,parts gain of function, this function given separately, by the usual practice is to define G(co) = B(co)j, (63) G(co)2Thewhere interpretation Im is theB(co) factor and ofby Re the which B(co) gain the functiondenote variance the follows inimaginary {x} fromat frequency and (56) real which coparts is shows translated of B(w). that H(w) = tan' [Im B(w)/Re B(w)], (64) distributed{x}atby frequency the lead distributed or lag lag w modelindicatesoscillations lag modelintroduced the ininto extent varianceAsin to(1).an which illustrativeTransforming in { oscillations y} at frequencyexample to theat frequency distributedof co. these The phaserelationships, w lag in consider the Koyck at the same frequency. whereform, the model can be expressed as Yt = t} >/3X_ + Vt, (65) 22 See Jenkins & Watts (1968, Chapter 9), for example. j=O jut_j (66) 296 E. PHILIP HOWREY Thisand parametric specification implies that f3=j4)J, j=O,1 (67) andHence the gain and phase functions are B(w) G(co) = IiI/1= ifr/(1 - - e_ic0) = 4'(1 - 4)e°)/I1 - 4)e_b0)12. (69)(68) variationsfunctionThese two indicates functionsin {x} than thatare to graphed{ high-frequency j} responds on Fig. 4 forvariations. Ji The phase function H(co) = tan1[-4)sinco/(1 more strongly to low-frequency - 4)cosw)]. = 1 and 4) = 0.9. The gain (70) related,ofshows time. the that On disturbance {y1} the assumption lags behindspectrum thatx} isat the all disturbances frequencies, {u1}but by varying = at/Il - 4)e_i012. are serially uncor- amounts (71) 5.0 functionThus the shownspectrum in Fig. of disturbances 4. has the same general shape as the gain 3. 111 3.011.0 1.02.0 - 0.0 rn-u 0.0 phase. Fig. 4. Gain and.0 phase functions for .1 I .2 FREQUENCY y, = .9j', .3 + + u1. Bold curve: gain; dashed curve: .11 .5 I -3.111 TIME SERIES ANALYSIS IN ECONOMETRICS 297 sistenttheestimates adequacy with of the of auto-hypothetical a particular and cross-covariance model,parametricDirect the spectralspecification. estimatesfunctions statistics ofcan theIf be theshould gain,used data phase,to not areevaluate differ con- and disturbance spectrum based on incationquantitiesConversely,significantly Section is too and5 ofsignificantrestrictive. from thosethis paper.the implied differencescorrespondingAn example by the betweenmodel of thisquantities indicate type direct of estimatesimpliedthatcomparison the modelby of theis provided specifi-spectralmodel. 4.2.achievedof TESm variables FOR implicitly CAUSALITY as endogenous if not explicitly or exogenous.A distinguishing by imposing As shown certaincharacteristic previously,restrictions of econometric onthis the is models is the classification restrictionstatisticalUntilmultivariate quite testing. is recently, autoregressive,valid. Granger the validity (1969), moving-average ofIn Simssuch some (1972),restrictions econometric representation and otherswas applications not have of subjected the suggested system. it tois not obvious that an exogeneity ofcurrentThestatistical coefficients{Yt} basic value on procedures ideacurrent, of on isy future dependsthat lagged, forif values y} ontesting andis current causallyof future x.for If and causality thevalues related lagged set ofof in valuestocoefficientsx bivariateshould{x} ofin x,yieldthe thenrelationships.2on sense futureinsignificant a regression that values the of forregularported.x j is {x}.<0. significantly This Testsleast-squaresIn terms is for a testable causalityof different the regression distributed restriction can from be methods.24 carriedzero, lagon themodel out relationshipcausality using (52), an this assumptionestimate betweenrestriction of B(w) {is YJ isnot f3orand sup- =using 0 inor4.3. principleseasonally BAND LIMITED which unadjusted should REGRESSION databe used in parameter Innor econometrics is it clear estimation. that it seasonalis frequently It is notand always nonseasonal possible clear to use either seasonally adjusted equation model. 24 GewekePierce & (1978) Haugh has (1977) extended provide these an results excellent to the review case of conceptsa complete and dynamic tests of simultaneous causality. variations298 are related by the same model.25 More generally, there E. PHILIP HOWREY are those modelnotshort-runwho the sameargue variations model that different is appropriate in economic models for time allEngle frequencies. series. (1974) has Consider recently the proposed regression may be needed to explain long-run and a method to investigate whether or equations,tipliedwhere byy is the T xmatrix 1, Xis W, T thex k, matrix and introduced in (50), y=XJ3+u, u is T x 1. If this set of equations is mul- a transformed set of (72) vectoroscillationis obtained. /3 is as theIf the the same time regression for domain all T relationship observationsregression model is in invariant the transformed model. Engle 5=X[3+i7, assumes, then the coefficient over all frequencies of (73) 5.(1974) An Analysis provides of a Aggregatesimple test Consumption of this hypothesis. Data fortivariatepoints analysis that time in havepart series because been analysis. introduced it is soThe well aggregate inIt known connectionmay be andconsumption useful in with to consider an example to illustrate some of the basic part because it has been univariate and mul- function is chosen purposepotentialconsumptionstandardused repeatedly of problemstime this expenditure domain exerciseto test of newsimultaneous analysisis toestimationand illustrate disposable of quarterly equation techniques.26 the use personal biasof time We income.27 series techniques in postwar data on personal are ignored. The main begin with a fairly Initially, consumptionsumptionbeginmodel withevaluation. and a loosely expenditureincome formulated based depends on: (i) distributed theManylargely theoretical empirical lag investigations of the aggregateon disposableconsumption personal function income, relationshipproposition between that con- personal aon total personal of 93 consumptionobservations. expenditure272625 See,ForTheand adisposablefordata example, consist personal Zeliner income from 1954.1 more detailed discussion of these issues, see Plosser (1978). of real quarterly national income and product accounts observations & Geisel (1970). to 1977.1, noandTIME lagged (ii) SERIES the values empiricalANALYSIS of the findingIN dependent ECONOMETRICS that a or simple independent linear consumption variables does function not provide with 299 onan adequateincome, the explanation result of the data.28 If consumption is simply regressed = 15.86 + 0.89Y, (5.9) (213.2) R2 = .998, DW = 0.69, appropriate.usuallypossibilityparentheses.is obtained. taken that The as In the an lowthis indicationdisturbance valueand the of followingthethat process DurbinWatson a more equations,is general serially statistic distributed tcorrelated, statistics indicates lag and model this is are shown in a strong yieldsbe modified the result to include lagged valuesOn of theboth basis consumption of this preliminary and income result,which the regression equation might = 2.00 + 0.461'; - 0.35l';_ + 0.88C_1 correlationofNeither the consumption the of theDurbinWatson disturbance function processnor thus the satisfies Durbinis a potential theh statistic usual problem.29 criteria indicates employedThis that form serial in R2 = 0.999, (0.9) (7.0) DW = 1.96, (4.6) DH = 0.29. (11.1) disposableofworkingeconometric disturbances, hypothesis.3° income analysis the desired and wouldlevel of accordingly consumptionAnother way be isto accepted linearlyarrive at related (75) is toto specify a model in which, in the absence as a reasonable current consumptiontionActual plus consumption some in fractionthe previousin period ) of the quartert is discrepancy assumed plus ato random bebetween equal disturbance. to actual desired and Thus desired C8'==ct+f31';. consump- (76) model, see Kmenta (1971 Chapter 11).28 For a discussion of several plausible theoretical reasons for expecting a distributed lag C = C' + 2(C_ - C_1) + v, (77) shouldstatisticdependent be in introducedthisvariable case. is here,included since as thisa3029 regressor. equationTheIt is usualwell The was caveatsknown Durbin obtained aboutthat h statistic afterthe the DurbinWatson interpretationa preliminaryis a more appropriate test. test of the t statistics and otherstatistic coefficient is biased estimates toward 2 when a lagged 300 E. PHILIP H0WREY Thevariables.where estimated {v} These is aequation sequence assumptions (75) of is independent leadthe unrestricted to the model identically version of distributed (78). Imposing random the = - )) + fY - 2$1 + 2C_1 + Vt. (78) leadinginorrestriction othereconometric to summary (78).implied analysis, statistics. by (78) the has Thus, data very judging appear Alittle more impact fromto careful be theonconsistent theusuallook coefficient atcriteria thewith residuals theemployed estimates theory of the distributed lag function is autocorrelationthetionsrevealing,with fact (ACF thethat however. disturbanceandthe functionsDW PACF) andThe ofDRprocess, autocorrelationsuggest the statistics residual thatthe doestimatedsignificant series not and indicate partialare autocorrelationcorrelation shown thatautocorrelation therein Fig. remains is 5.a and problemDespite infunc- partial the isdisturbancesfunctionsresiduals. indicated.correlation indicateFigure isA functions notthird-order 6 adequate;that shows indeed for theautoregressive thean estimated aautoregressive residualsfirst-order autocorrelation processof autoregressive the process original for theof at anddisturbancemodel leastprocess partial third (74). for auto- orderterm These the function.in (74) implies The use that of lags an ofunrestricted up to order.11 third-order three are needed lag distribution in the consumption yields the .2 I / --I / / / / / / .' / / / / PACF. -.11Fig. 5. Estimated ACF and PACF for residuals of Eq. (75). Bold curve: ACF; dashed curve: 1 2 3 II 5 6 7 8LAG LI 10 11 12 13 111 15 18 TIME SERIES ANALYSIS IN ECONOMETRICS.6 301 -.2 0 .1 / / _-.-/ / ,/ \ / A \ I I, 8 LAG 9 10 11 12 13 It! 15 18 modelPACF. Fig. 6. Estimated ACF and PACF for residuals of Eq. (74). Bold curve: ACF; dashed curve: 123!! 58 = 2.89 + 0.511ç - 0.32Y_1 - 0.20Y_2 + 0.201'_3 (1.3)+0.86C_1 + 0.39C_2 - 0.45C_3, (7.5) R2 =(7.9) .999, (2.5) (3.5) DW = 2.07. (3.8) (2.3) (2.5) (79) Theordercarefulresiduals estimated model. analysis in this autocorrelation ofmodel the residuals do not and exhibit of partial the evidenceoriginal autocorrelation equation of serial (74) correlation. functions leads to of aThus third-the a modelthatapoint. second-ordera third-orderdid Standard not simply model. modeleconometric evolve isHowever, more in antechniques appropriate. ad aThe hocmore analysis way; carefulmight Indeed,rather, leadleading analysis it thean was investigatormoreto of suggested the the complicated third-orderdata toindicateson accept the model illustrates an important Foranalysisingbasis approachexample, of a is careful to anddetrendBox analysisthe & timeJenkins the ofseriesseries the (1970, residuals.toapproach Ainduce p. potentially 378) isstationarity statethat importantthe that first "when if thatstep difference theisin necessary.time processes betweenseries the usual econometric model- 302 E. PHILIP HOWREY tionthedifferencing."are modelnonstationary of the isserial estimated In correlationthis it is case assumedin terms first in thedifferences ofthat first disturbance stationarity differences appear process, can withto bebe appropriate inducedthesufficient.31 result by is recogni-suitable When AC = 1.44 + 0.46AY - 0.17AY1_2 + 0.42 AC2 orderdifferentIf this to is facilitaterewrittenfrom the coefficientscomparisons in terms of oflevels, with the unrestrictedthe the corresponding coefficients third-degree are time not serieslagdramatically model. results In R2(2.1) = .457, (7.9) DW = 2.10. (2.3) (4.1) (80) generaladequacyretained.presented (rather subsequently,of this than parametric rational) the first-differenced distributedspecification.The frequency lag model,If version the domain model the of resultproperties the is rewritten modelis of will (80) as be aprovide a further check on the 1.2 = 2.48 + (.46 + .02L2 ++(1 OiL4 + .42L2 + OiL6 + .18L4 + . + .07L6 + . . (81) 0.8 0,tl 0.0Fig. 7. Estimates of the gain of the consumptionincome relationship. Bold line:.0 direct .1 .2 PREUENCT .3 .11 .5 requiredestimate; todashed produce line: stationarity. model estimate.31 See Box & Jenkins (1970, pp. 174-175) for a discussion of the degree of differencing TIME SERIES ANALYSIS IN ECONOMETRICS50 303 '10 I / / / / I / I 0 .0 .1 .2 .3 .11 .5 Theline:and directgain the estimate;function implied dashed impliedresidual line: bymodelspectrumFig. this 8.estimate. Estimatesmodel is shown is of shown the as residual a as dashed a dashedspectrum line line ofin theFig. in consumptionincome Fig. 8. The 7 relationship. Bold FREQUENCY Thisrespondhypothesis.Engle,gain result function equally does is similar notThe indicatesstrongly appear residual to that to tothat both bereportedspectrum the consistent short-run model by for Engleimplies withandthis long-runmodelFriedman's(1974) that changes exhibitsand, changes permanent as inremarked a consumptioninrelative income. income by pre- areFigs.estimatesdominance broadly 7 and of 8.of consistent the Itboth is auto- clear low- andwithfrom and cross-covariance thehigh-frequencyvisual implications Directinspection estimates variation. functions,ofthat the while ofmodel, the arethe gain alsodirectthere function plotted areestimates some andon residual spectrum,32 based on resulttionthetheimportant gain conforms dataobtained function tends disparities. morefrom toimplied fall theclosely offIn parametric by particular, much (81).to the moreThus permanent model. the sharplythe gain direct function income at estimatehigh estimated hypothesisfrequencies of the directlygain than than func- the fromdoes estimationwindowEstimates with ofprocedure. thetruncation spectrum point 30. See Jenkins & Watts (1968) forThese a complete estimates discussion were obtained of the by replacing B(w) by E(w) = 7(w)/f(w) in (63) and (64). f(w) and cross-spectrum f,(w) were obtained using the Parzen 304 TABLE 1 E PHILIP HOWREY DISTRIBUTED LAG COEFFICIENTS AND ESTIMATED (-VALUES Coefficient Spectral estimates t Value Coefficient Regression estimates t Value AY13 .148.072.046 2.961.44 .92 .182.094.054 3.251.71 .96 AZ+1 .113 .204.384.197 4.087.693.942.26 .154 .207.383.183 3.576.483.142.63 A1_5L\}_4A1J .057.032 .059.020 1.141.18 .64.40 - .084- .047 .066.064 1.04 .97.69.54 currentestimatesspectral and estimates33 is thepast, fact changes that are the shown in lag income distribution inThe Table have distributed 1. ais The significanttwo-sided. strikinglag coefficients Future,effect feature on and of current approximatethese t statistics implied by the as well as hencesimilarandconsumption. thethese impliedwith are respect Asalso t avalues. checkshown to both on in these the The coefficientstable.results, conclusion The regression two and fromsets the estimates of thesestandard estimates estimates is clear. These results do not support were obtained, errors, and are very ofandindicates appearsthethe income hypothesisproblem. that that should any This,income thatfurther recognize of disposable course,is analysis causally explicitly is not incomeof dependent thean unexpectedrelationshipthe is ansimultaneous exogenouson consumption. result between in equationsvariable. this consumption This Rathernature result it case. There 6. statisticalmayConclusion be other evidence situations, would however,be extremely which valuable. are less clear cut for which such detailsclassical of the estimationeconometrics procedure. and has summarizedThis paper briefly hasThese shown some were ofhow obtained the time basic from series tech- (61) analysiswith differs in outlook from (iis/,n), in place of B(lrs/nl). See footnote 32 fbr niquesTIME SERIES of time ANALYSIS series INanalysis ECONOMETRICS that are useful for the evaluation of econo- 305 whichapproachmetrictradictions providesmodels. typicallyand ofthe An thethe time begins important basisassumptions series forwith theapproachdistinction a empiricalstrong of the toparametricmodel between modeling analysis. are theinvestigatedformulation Potentiallyis classical that the econometric ofbutblatant the diagnostic model con- ofmodel.typicallychecking simplifications Much isbegin not more pursuedwith that emphasis a may relatively vigorously. be isappropriate. placed weak,This Time on differencenonparametric seriesdata analysis models, in outlook toformulation on suggest the and other the approach of hand,types the provides the basis for using estimatesdynamicofintime the series evaluationstructural propertiesare methods compared econometric process. toare evaluate estimatedwith First, thosemodel econometric certain directly implied are measurable derived. from bymodels. the the Second, structural dynamicThreedata. Finally,thesteps model. characteristicscorresponding are the involvedDispari- direct indicationties between of modelthe direct inadequacy. estimates and implied characteristics provide an ACKNOWLEDGMENTS acknowledged.andreferees graphics for helpful assistance. comments Financial on anThe supportearlier author draft of is the ofindebted thisNational paper to Jan Scienceand Kmenta, to Mark Foundation James Greene Ramsey, foris gratefully editorial V. Kerry Smith, and two anonymous Ansley,Amemiya, Craig Takeshi, F., Spivey, & Fuller, W. Allen, Wayne & Wrobleski, A. A comparativedistributed William J.studylag On model. the of alternativestructure of estimatorsmoving average in a Econometrica, 1967, 35, 509-529. REFERENCES Box, George B.E. P., & Pierce,Jenkins, D. Gwilym A. Distribution M.Francisco,processes. of residual Cal.:Journal autocorrelations Holden-Day, of Econometrics, in autogressive- 1977, 6, 121-134. 1970. Time series analysis forecasting and control. San Dhrymes,Chow, Gregory Phoebus C. J. tion,integrated 1970, moving64, 1509-1526. average time series models. Analysis and control of dynamicEconometrics: economic Statistical systems. foundations and applications. Journal of the American Statistical Associa- New York: Wiley, New York: Harper 1975. Durbin,Dhrymes, J. PhoebusTests of serialJ. independence based onInternationalCal.:and the Row, Holden-Day, cumulated Statistical periodogram. Institute, 1967, 42, 1039-1047. 1970. Distributed lags: Problems1971. of estimation and formation. San Francisco,Bulletin of the 306 E. PHILIP HOWREY Fishman,Engle,Durbin, R. J. F. GeorgeTests Band of spectrumS. serial Spectral correlation regression. methods in International regressionin Press,residuals. based Economic1969. Biometrika, on the Review, periodogram 1969, 1974, 56, 15,of 1-15. least-squares 1-11. econometrics, Cambridge, Mass.: Harvard Univ. Godfrey,Fuller,Geweke, Wayne Michael John. A. Introduction TestingD., & Karreman, the toexogeneity statistical H. A specificationtimespectrumShubikequation series. analysis(Ed.), New model. in the EssaysYork: ofcompleteJournal seasonal Wiley,in mathematical of dynamic Econometrics,1976. adjustment. simultaneous economics In 1978,7, M. in 163-185.honor of Oskar Morgenstern. Princeton, Granger,Gold berger, CliveClive Arthur W. W. J. S.I. Investigating EconometricSeasonality: causal theory,Causation, relations Newmethods.N. York:interpretation, J.: by Princeton Econometrica, econometricWiley, 1964. Univ. and models 1969, implications.Press, 37, and1967. 424-438. cross-spectral Pp. In 367-421.Arnold Hannan,Grenander, E. J. RegressionU., & Rosenblatt, for time M. series. Statistical In Wiley,mentZellnerMurray analysis of 1957. (Ed.), Commerce,Rosenblatt of Seasonalstationary 1978.(Ed.), analysis time Pp. Proceedings 33-46.series. of economic New of York:the time series. Washington, D.C.: U.S. Depart- Hannan,Hannan, E. E.J. J.The MultipleThe identification estimation time series. of and relationships New parameterization York:33,symposium involving Wiley, 206-24. of1970. ondistributed ARMAX tune series andlags. analysis. state Econometrica, space New forms.York: 1965, Wiley, 1963. Pp. 17-37. Howrey, E. Philip. StochasticA spectrum properties analysis ofof thethe 39,Reviea',Econometrica, long-swingKleinGoldberger 133-66. 1968, hypothesis. 9,1976, 228-252. model. 44, International713-723. Econometrica, Economic 1971, Howrey, E. E. Philip, Philip. & Dynamic Klein, Lawrence properties R. Dynamic ofhuernational a condensedUniv.Bert properties G. Press, Hickman Economic version 1972. of nonlinear (Ed.), Pp. ofReview, the601-663. Econo,netric econometricWharton 1972, 13, model. models 599-618.models. In of cyclical behavior. New York: Columbia Johnston,Nerlove,Kmenta,Jenkins, J. GwilymJan.EconometricMarc. Elements Spectral M. & methods. ofWatts, analysis econometrics. Donald (2nd of seasonal ed.) G. New NewSpectral adjustmentYork: York:Cal.: analysis MacMillan,Holden-Day, McGraw-Hill, procedures. and its 1971. 1968.applications. 1972. Econo,netrica, San Francisco, 1964, 32, Palm,Parzen, F. On Emanuel. univariate Multiple time series time methods series modelling.andanalysis.Journal241-286. simultaneous of Vol.In Econometrics, P. II. R. equation New Krishnaiah York: 1977,econometric Academic (Ed.), 5, 379-388. Multivariate models. Press, 1969. Pp. 381-410. Pierce,Parzen, DavidEmanuel. A. Forecasting Some solutions in dynamic to the time models seriesmetrics,1974.Report with modeling stochastic No. 1975, 5, and Department3, regressions.349-374.prediction of problem. ComputerJournal ofTechnical Science,Econo- State University of New York at Buffalo, Pierce,TIME David SERIES A., ANALYSIS & Haugh, LarryIN ECONOMETRICS D. Causality in temporal systems: Characterization and a 307 Priestley,Plosser, Charles M. B. Fitting I. A time relationships series analysis between of time seasonalitymentZelinersurvey. series. of Journal(Ed.),Commerce, Bulletin in Seasonaleconometric of ofEconometrics, 1978.the analysis International Pp. 365-397. of 1977, economic 5, 265-293. time series. Washington, D.C.: U.S.models. Depart- In Arnold Statistical Smith,Sims,Quenouiile, C. V. A. Kerry. M.Money, H. The The income estimated analysis and of power causality. multiple of several time Americanalternatives.Institute, series, tests EconomicforLondon: 1971,Journal autocorrelation 44, C. of 295-321. the American Statistical Association, 1976, 71, 879-883. Review, Griffin,1972, 62, 1957. 540-552. with non-first-order Zellner,Wallis,Wahba, K. Grace.A. F. Time Multiple Estimation series time analysis of series coefficients and analysis econometric in and,netrica, a multidimensionalinetrica, the model final 1977, t969, formconstruction. 45, 31, 1481-1497.of distributed398-407. lag econometric models. Econo- In R. P. Gupta (Ed.), model. Econo- Zeilner,Zellner, A., A., & Palm,& Geisel, F. Time M. S. series Analysis analysis of distributed andJournaltionApplied simultaneous function lagof statistics. Econometrics, models estimation. equation Amsterdam: with Econometrica,1974, North-Holland2, 17-54. 1970, 38, Pub!., 865-888. 1975. Pp. 373-398. applications to consump- econometric models.