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Unbiased Estimationof CoefficientVariance in a Time-VaryingParameter Model James H. STOCKand MarkW. WATSON

This articleconsiders inference about the varianceof coefficientsin time-varyingparameter models with stationaryregressors. The Gaussian maximumlikelihood (MLE) has a largepoint mass at 0. We thusdevelop asymptotically median unbiased estimatorsand asymptoticallyvalid confidenceintervals by invertingquantile functions of regression-basedparameter stability test ,computed under the constant-parameternull. These estimatorshave good asymptoticrelative efficiencies for small to moderateamounts of parametervariability. We applythese results to an unobservedcomponents model of trendgrowth in postwar U.S. per capita gross domesticproduct. The MLE impliesthat there has been no change in the trendgrowth rate, whereas the upperrange of the median-unbiasedpoint estimates imply that the annualtrend growth rate has fallenby 0.9% per annumsince the 1950s. KEY WORDS: Stochasticcoefficient model; Structuraltime series model; Unit movingaverage root; Unobserved components.

1. INTRODUCTION We considerthe problem of estimationof thescale pa- Sinceits introduction in the early 1970s by Cooleyand rameterr. If (as is common)et and nt are assumedto be Prescott(1973a,b, 1976), Rosenberg (1972, 1973), and Sar- jointlynormal and independentof {Xt, t-1) . . .; T}, then ris(1973), the time-varying (TVP), or "stochastic theparameters of (1)-(4) can be estimatedby maximum coefficients,"regression model has been used extensivelylikelihood implemented by the . However, the in empiricalwork, especially in forecastingapplications. maximum likelihood estimator (MLE) has theundesirable Chow (1984), Engle and Watson(1987), Harvey(1989), propertythat if r is small,then it has point mass at 0. In the Nicholsand Pagan (1985), Pagan (1980), and Stockand case Xt - 1, thisis relatedto theso-called pile-up prob- Watson(1996) haveprovided references and discussion of lemin the first-order [MA( 1)] modelwith a thismodel. The appealof the TVP modelis thatby permit- unitroot (Sargan and Bhargava 1983; Shephard and Harvey tingthe coefficients toevolve stochastically over time, it can 1990).In thegeneral TVP model(1)-l4), the pile-up proba- be appliedto timeseries models with parameter instability. bility depends on theproperties of Xt andcan be large.The The TVP modelconsidered in thisarticle is pile-upprobability is a particularproblem when 7 is small and thusis readilymistaken for 0. Arguably,small values yt= 3itXt+ ut, (1) of r are appropriatefor many empirical applications; in- deed,if 7 is large,then the distribution of the MLE canbe ,3t /3t-l+ vt, (2) approximatedby conventionalT1/2-asymptotic normality, butMonte Carlo evidence suggests that this approximation a(L)ut = Et, (3) is poorin manycases of empiricalinterest. (See Davis and and Dunsmuir1996 and Shephard1993 fordiscussions in the case of Xt = 1.) vt = Tvt, where vt = B(L)qt, (4) We thusfocus on theestimation of r whenit is small.In particular,we considerthe nesting where {(yt,Xt),t = 1,... ,T} are observed,Xt is an ex- ogenousk-dimensional regressor, /3t is a k x 1 vectorof r = A/T. (5) unobservedtime-varying coefficients, T is a scalar,a(L) is a scalarlag ,B(L) is a k x k matrixlag polyno- Orderof magnitudecalculations suggest that this might be an appropriatenesting for certain empirical problems of mial,and Et and qt are seriallyand mutually uncorrelated mean0 randomdisturbances. (Additional technical condi- interest,such as estimatingstochastic variation in thetrend tionsused forthe asymptotic results are givenin Section componentin thegrowth rate of U.S. realgross domestic product(GDP), as we discussin Section4. Thisis also the 2, wherewe also discussrestrictions on B(L) andE(rqtqr) thatare sufficientto identifythe scale factorT.) An im- nestingused to obtainlocal asymptoticpower functions of testsof 7r a fact thatif the researcher is in a portantspecial case of thismodel is whenXt = 1 and 0, suggesting B(L) = 1; followingHarvey (1985), we referto thiscase regionin which tests yield ambiguous conclusions about the as the"local-level" unobserved components model. nullhypothesis 7r 0, thenthe nesting (5) is appropriate. The maincontribution of this article is thedevelopment of asymptoticallymedian unbiased of A and JamesH. Stock is Professorof PoliticalEconomy, Kennedy School of asymptoticallyvalid confidence intervals for A in the model Government,Harvard University, Cambridge, MA 02138, and Research (1)-(5).These are obtained by inverting asymptotic quantile Associate at the National Bureau of Economic Research (NBER). Mark functionsof statisticsthat test the hypothesis A = 0. The W. Watsonis Professorof Economicsand PublicAffairs, Woodrow Wilson School,Princeton University, Princeton, NJ 08544, andResearch Associate at the NBER. The authorsthank Bruce Hansen,Andrew Harvey and two refereesfor comments on an earlierdraft, and Lewis Chan and Jonathan ? 1998 AmericanStatistical Association Wrightfor researchassistance. This researchwas supportedin part by Journalof the AmericanStatistical Association National Science Foundationgrant SBR-9409629. March1998, Vol.93, No. 441, Theoryand Methods

349 350 Journalof the AmericanStatistical Association, March 1998 teststatistics are based on generalizedleast squares (GLS) and residuals,which are readilycomputed under the null. As partof thecalculations, we obtainasymptotic representa- FT(s) = (SSRl,T- SSRl,[Ts] - SSRrTs]+1,T) tionsfor a familyof testsunder the local alternative(5). ? + - k)]. (9) Theserepresentations can be usedto computelocal asymp- [k(SSRl,[Ts] SSR[TS]+l,T)/(T toticpower functions against nonzero values of A. Section 2 presentsthese theoretical results. (For othertests in versionsof thismodel, see Franziniand Section3 providesnumerical results for the special cases Harvey 1983; Harveyand Streibel1997; King; and Hillier ofthe local-level model. Properties of the median 1985; Nabeya and Tanaka 1988; Nyblom1989; Reinsel and unbiasedestimators are compared to twoMLEs, whichal- Tam 1996; Shively1988.) ternativelymaximize the marginal and the profile (or con- The FT statisticis an empirical process, and infer- centrated)likelihoods; these MLEs differin theirtreatment ence is performedusing one-dimensionalfunctionals of of theinitial value for 3t. BothMLEs arebiased and have FT. We consider three such functionals:the maximum largepile-ups at A = 0. WhenA is small,the median un- FT (the Quandt [1960] likelihoodratio statistic), biasedestimators are more tightly concentrated around the QLRT = supSE(S,S1) FT(s); themean Wald statisticof An- truevalue of A thaneither MLE. drews and Ploberger(1994) and Hansen (1992), MWT = Section4 presentsan applicationto theestimation of a fZ FT(r) dr; and the Andrews-Ploberger(1994) exponen- long-runstochastic trend for the growth rate of postwar tial Wald statistic,EWT = ln{f1 exp( FT(r)) dr}, where real percapita GDP in theUnited States. Point estimates O < so < SI < 1. fromthe median unbiased estimators suggest a slowdownin Three assumptionsare used to obtainthe asymptoticre- theaverage trend rate of growth;the largest point estimate sults. For a stationaryprocess Zt, let ci ...i.(r, ..rn-1) suggestsa slowdownof approximately .9% perannum from denote the nth joint cumulantof zi1t1,...)zintn, where the1950s to the1990s. The MLEs suggesta muchsmaller rj = tj-tn,j = 1, . ..,n-1 (Brillinger1981), and let decline,with point estimates ranging from 0 to .2%. Section C(ri,..., rn-1) = Supil,.,in Cil ... in(rl, * * rn-1). 5 concludes. AssumptionA. Xt is stationarywith eighth order cumu- 2. THEORETICAL RESULTS lantsthat satisfy ,.rJ IC(rI,... ,r7) < 00 We assumethat a(L) has knownfinite order p andthus t = of considerstatistics based on feasible GLS. Specifically,(a) Yt AssumptionB. {Xt, 1,... ,T} is independent t1, ...,T}. is regressedon Xt byordinary (OLS), produc- {Ut, Vt, ingresiduals fit; (b) a univariateAR(p) is estimatedby OLS Assumption C. (Ct, r)' is a (k + 1) x 1 vectorof iid of fit on , and regression (1,It-i,... yielding&(L); errorswith 0 and fourmoments; Et and rt are inde- (c) t = &(L)ytis regressedon Xt &(L)Xt,yielding the pendent;a(L) has finite-orderp; and B(L) is one-summable GLS estimator,3 - (T-1 Et=L1XtX ) t1T-1 t=1xy, withB(1) 340. residualset andmoment V: AssumptionA requiresthat Xt have boundedmoments or,if nonstochastic,that it notexhibit a trend.The assump- = et It-3'Xt (6) tion of stationarityis made for conveniencein the proofs and could be relaxedsomewhat. However, the requirement and that Xt not be integratedof order 1 (I(1)) or higheris essentialfor our results. V = (T-1 E tk &2 (7) AssumptionB requiresXt to be strictlyexogenous. This assumptionpermits estimation of (1), underthe null f3t 3o0,by GLS. where&2 = (T - k)-1 ET= E2. If a(L) 1, thensteps (a) The assumptionthat a(L) has finite-orderp in assumption and(b) areomitted and the OLS andGLS regressionsof Yt C is made to simplifyestimation by feasibleGLS. The as- on Xt areequivalent. sumptionthat Et and rt are independentensures that ut and Two teststatistics are considered:Nyblom's (1989) LT vt have a zero cross-spectraldensity matrix. This is a basic statistic(modified to use GLS residuals)and the sequential identifyingassumption of the TVP model (Harvey 1989). GLS ChowF statistics,FT(s)(O < s < 1), whichtest for To constructthe Gaussian MLE, Et and rt are modeled as a breakat date [Ts],where [.] denotesthe greatest lesser independentiid normalrandom variables. integer.Let SSRt1,t2denote the sum of squaredresiduals The assumptionthat Xt is independentof the errors be in some For in fromthe GLS regressionof jt ontoXt overobservations can unappealing applications. example, some econometricapplications Xt is predeterminedbut not tl < t < t2,and let CT(S) = T-1/2 E[Ts] Xtet. The LT and strictlyexogenous, ut is plausiblyserially uncorrelated, but FT statisticsare thereis feedbackfrom Ut to futureXt. In lieu of assump- T tionsB and C, we thusintroduce an alternativeassumption to handleregressors that are predeterminedbut not exoge- LT-T-1 Z (T(t/T)V-1 T(t/T) (8) t= 1 nous. Stock and Watson: Time-VaryingParameters 351

AssumptionD. (6t,4)' is a (k + 1) x 1 vectorof iid ization.Henceforth, when k = 1, we thusset D = 1. When errorswith mean 0 and fourmoments; Et and nt are in- Xt = 1, underthis normalization, A is T timesthe ratio dependent;a(L) = 1;B(L) is one-summablewith B(1) 74 of thelong-run standard of A/3t to thelong run 0; nItis independentof {Xt, Xt?l, Xt?2,.. .}; and ut is in- standarddeviation of ut. dependentof {Xt,Xti l, Xt-2,.. Whenk > 1,Ql is identifiedupon making a singlesuit- ablenormalization; for example, that the of -Q is unity. Thispermits feedback from ut to future Xt, but not from However,the local-to-O variation in A/3tmakes it impossi- vt to Xt, and thusrules out Xt containinglagged Yt when ble to estimatethe free elements of Q consistentlywithout A#o. Our maintheoretical results are givenin thefollowing furtherrestrictions. In thiscase two typesof furtherre- theorem.Let ">" denoteweak convergenceon D[0,1], strictionssuggest themselves. First, Q maybe setequal to let W1 and W2 be independentstandard Brownian mo- a prespecifiedconstant matrix chosen by theresearcher in a mannerappropriate for the specific empirical model un- tions on [0, ]k, and let F = E{ [a(L)Xt] [a(L)Xt]'}, l = der Q maybe parameterizedas a andD= study.Second, B(1)E(7qntq)B(1)', r1/2n1/2/ of r and a 2 (whichare consistentlyestimable). As in the Theorem1. Let Ytobey (1)-(5), and suppose either that k = 1 case,a convenientparameterization sets Q - ar2Y-1, assumptionsA, B, andC holdor that assumptions A andD forthis implies that D = Ik. Thischoice of Q impliesthat hold.Then theregression coefficients evolve as mutuallyindependent randomwalks after rotating the regressors so thatthey are a. V-1/2(T X hO,where h0(s) = hA(s)-shA(1),where mutuallyuncorrelated. This is theparameterization used by hA(s) = Wi (s) + AD fsW2 (r) dr; Nyblom(1989) in his developmentof theLMPI testfor b. LT X f0h?(s)'h? (s) ds; and A O.0 Froma computationalperspective, this assumption c. FT => F*, where F*(s) = h0(s)'h?(s)/(ks(1 - s)). is attractivebecause it simplifiesthe calculation of median The proofis givenin theAppendix. unbiasedestimators of A andthe construction ofconfidence Limitingrepresentations of the QLR, meanWald, and intervals.From a modelingperspective, the restriction is exponentialWald statisticsare obtainedfrom part (c) of arguablyappealing because it makesthe magnitude of the TheoremI and the continuousmapping theorem. Thus timevariation comparable across variables when measured in standarddeviation units (or, for general a(L) and/(L), QLRT ?I F*(s), MWT f> JS F*(r) dr, and sup,,<,<,1 long-runstandard deviation units). With the additional re- EWT X> ln{<1 exp( 2F* (r))dr}. Notethat the limitingrep- strictionsa(L) =1 and B(L) = 1, this restrictionwas resentationfor LT can be writtenas LT =X k f (r(1 - usedby Stockand Watson (1996) in theirinvestigation of r))F* (r) dr. timevariation in empiricalmacroeconomic relationships. WhenA = 0, theprocess hO is a k-dimensionalBrown- Whetherthis assumption is desirablefor general Xt is a ianbridge, and the representations forthe statistics LT and matterof modeling strategy in a particularempirical appli- FT reduceto their well-known null representations as func- cation. tionalsof a Brownianbridge (Andrews and Ploberger 1994; Nabeyaand Tanaka 1988; Nyblom 1989). WhenA 7&0, thelimiting distributions of LT andFT de- 2.1 Local AsymptoticPower pendon twoparameters, A and D. The limitingrepresen- The representationscan be used to computethe distri- tationsin Theorem1 are usedfor three purposes: to com- butionof thetests under the local alternative(5) and thus putelocal asymptotic power functions, to constructmedian to computethe local asymptoticpower of testsof thenull unbiasedestimators of A,and to constructasymptotically T = 0. The varioustest statistics have limiting representa- validconfidence intervals for A. To do so, D musteither be tionsunder the local alternativethat are qualitativelysim- knownor be consistentlyestimable, so thatasymptotically ilar. This is interesting,because the FT-based statistics are A is theonly unknown parameter entering these distribu- typically motivated by considering the single break model, tions. whereasNyblom (1989) derivedthe LT statisticas the Thedetermination ofD raisesissues of identification and LMPI teststatistic for the seemingly rather different Gaus- modelingstrategy. Evidently a(L) andvar(ct) are not sep- sianTVP model. aratelyidentified, but this is resolvedwithout loss of gen- eralityby adopting the normalization ao0 1. Similarly,for standardreasons associated with moving average models, 2.2 Median Unbiased Estimationof A B(L) andEnitn' are not separately identified; we thusadopt Medianunbiased estimators of A can be computedfrom theconventional assumptions that Bo = Ik andthe roots of LT or froma scalarfunctional of FT. Consider,for exam- IB(z) areoutside the unit circle. Even with these assump- ple, the scalarfunctional g(FT), whichis assumedto be tions,however, inspection of (1)-(5) revealsthat A and Q? continuous.By thecontinuous mapping theorem, g(FT) => are notseparately identified: The parametercombinations g(F*), thedistribution of whichdepends on A and D. Let (A,Q) = (A,Q) and (A,Q) = (1,>2) are observationallymD(A) denote the median of g(F*) as a functionof A fora equivalentfor fixed (A, Q). givenmatrix of nuisance D. Supposethat mn(.) Whenk = 1, thisidentification problem can be solved is monotoneincreasing and continuousin A. Thenthe in- withoutloss of generalityby adopting an arbitrarynormal- versefunction mj-1 exists, and forD known,A can be es- 352 Journalof the AmericanStatistical Association, March 1998 timatedby done graphicallyby the methodof confidencebelts or by interpolationof a lookup table. The details parallel those (10) =mD(g(FT)). for constructionof confidenceintervals for autoregressive Asymptotically,A0 A mj-(g(F*)). By construction,Pr[% rootslocal to unity(Stock 1991) and are omitted. - < A] Pr[mj1(g(F*)) < Al = Pr[g(F*) < mD(A)] .5, so A9 is asymptoticallymedian unbiased. 3. NUMERICAL RESULTS FOR THE UNIVARIATE LOCAL-LEVEL MODEL In practiceD is not known,so the estimator(10) is in- feasible.However, as discussedearlier, D generallycan be In the univariateof local-level model, Xt 1 and consistentlyestimated for a givenchoice of Q. If in addition B(L) = 1, so thatyt is the sum of an I(0) componentand mjn (.) is continuousin D (which it is for the function- an independentrandom walk, whichunder the parameter- als consideredin this article),then (10) can be computed ization (5) has a small disturbancevariance. In thismodel withD replacedby a consistentestimator D, and thesame AYt followsa movingaverage (MA) process,with largest asymptoticdistribution obtains. Note, however,that this is MA root (1 - A/T)-> + o(T-1). In this section we first computationallycumbersome, as it requirescomputing the compare numericallythe power of the tests in Section 2 inversemedian function mn;1 (.) forevery estimate D under and of some otherpreviously proposed tests against the lo- consideration.(However, some simplificationsare possible cal alternative,then turn to an analysisof theproperties of because, as pointedout by a referee,the distributionof the median unbiased estimators.All computationsof asymp- teststatistics depends only on theeigenvalues of D.) When toticdistributions are based on simulationof the limiting Q is chosenso thatD= Ik, thelimiting distributions of LT representations,with T = 500 and 5,000Monte Carlo repli- and FT dependonly on A and k underthe local alternative. cations. It would be of interestto obtaintheoretical results com- paringthe efficiencyof median-unbiasedestimators based 3.1 AsymptoticPower of Tests on thevarious functionals of FT. However,the limiting dis- A great deal of work has been on tests of A = 0 in tributionsare nonstandardand do not appear to have any the local-levelmodel and of a unitMA root in the related simplerelation to each other.Thus theseefficiency compar- MA(1) model (see Nyblomand Makelainen 1983; Saikko- isons are undertakennumerically and are reportedin the nen and Luukonen1993; Shively1988; Stock 1994; Tanaka nextsection. 1990). In additionto the testsdiscussed in Section 2, local powerfunctions are computedfor two point-optimalinvari- 2.3 ConfidenceIntervals for A ant (POI) tests (Saikkonen and Luukkonen 1993; Shive- Suppose thatD = Ik, in whichcase the local asymptotic ly 1988), for A = 7 and A = 17, denotedby P01(7) and representatiornsin Theorem 1 dependonly on A and k. For P0(17). As a basis of comparison,we also computedthe a given scalar ,its representationcan thenbe asymptoticGaussian powerenvelope. used to computea familyof asymptotic5% criticalvalues Asymptoticpowers of various5% testsare summarized of testsof A = A0against a two-sidedalternative, and in turn in Figure 1. Evidently,for small values of A all tests ef- thesecritical values can be used to constructthe set of A0 fectivelylie on the asymptoticGaussian power envelope. thatare notrejected. This set constitutesa 95% confidence For more distantalternatives, the MW and L tests lose set forA0. This processof invertingthe test statistic can be powerand, to a lesserdegree, so do theEW and QLR. The asymptoticpower functionsof the EW and QLR testsare 0 essentiallyindistinguishable, consistent with findings else- where(Andrews, Lee, and Ploberger1996; Stock and Wat- son 1996) thatthese tests perform similarly.

3.2 Estimatorsof A Each of thetests examined in Figure 1 has a powerfunc- tionthat depends only on A and has a medianfunction that is monotoneand continuousin A. Asymptoticallymedian un- 0 biased estimatorsof A based on each of thesetests thus can 0? /j be constructedas describedin Section2. In addition,results are reportedfor two versionsof theGaussian MLE thatdif- O X, ferin theirassumptions concerning the initialvalue of 1o. The firstestimator, the maximumprofile (or concentrated) likelihoodestimator treats as an unknownnui- (D (MPLE), Q0 sance parameterthat is concentratedout of the likelihood.

0 0 5 10 15 20 25 30 The second estimator,the maximummarginal likelihood estimator(MMLE), \ treatsIo as a N(,T ) randomvariable thatis independentof {Ut, vt, t =1 ,... ., T}, so thatf0o is in- Figure1. AsymptoticPower Functions ot 5% TestsotTr = 0 Against Alternativesw = A/T.- , envelope; ,L; -,MW; ,EW; tegratedout of the likelihood.When i oc, thisproduces -- -, QLR; ,P01(7J; ---, P01(17J. the "diffuseprior" (see Shephard1993 Stock and Watson: Time-VaryingParameters 353

Table 1. Pile-Up ProbabilityThat A = 0 forMLEs MLEs are biased and median biased. For A = 5, 77% of and VariousMedian-Unbiased Estimators the mass of the distributionof the MPLE is below the true A MPLE MMLE L MW EW QLR P01(7) P01(17) value and themedian is 0; theMMLE performsbetter, with 64% of its mass below A = 5 and a medianbias of approxi- 0 .96 .66 .50 .50 .50 .50 .50 .50 1 .91 .60 .47 .47 .47 .46 .47 .47 mately-1. The cdfsof themedian unbiased estimators are 2 .88 .57 .42 .42 .42 .43 .44 .43 fairlysimilar to each other,but markedly different than the 3 .81 .47 .34 .34 .34 .35 .35 .37 MLE. One apparentcost of unbiasnessis theirlonger right 4 .72 .40 .28 .28 .29 .29 .29 .30 tail relativeto the MLEs. 5 .65 .35 .24 .24 .24 .24 .24 .26 6 .56 .28 .19 .19 .19 .19 .18 .20 We comparethe estimatorsby computingtheir asymp- 7 .48 .24 .15 .16 .16 .16 .14 .15 toticrelative efficiencies (AREs). Because the distributions 8 .42 .19 .13 .13 .13 .13 .12 .13 are nonnormaland not proportional,conventional meth- 9 .37 .17 .11 .12 .12 .12 .09 .10 10 .30 .13 .09 .09 .09 .09 .07 .07 ods of computingAREs do not apply. Instead,we mea- 12 .24 .09 .06 .07 .07 .06 .05 .05 surethe ARE of theith estimator 4rj relativeto theMMLE, 14 .15 .06 .03 .04 .04 .04 .03 .02 TMMLE (denotedby AREi,MMLE), as the limitof the ratio 16 .13 .04 .03 .03 .03 .03 .01 .01 of observationsTMMLE/Ti needed for Pr[#ic = 18 .09 .03 .02 .03 .02 .02 .01 .01 G(r); Tj] 20 .07 .02 .01 .01 .01 .01 .01 .01 Pr[4MMLE E G(T); TMMLE], where Ti and TMMLE denote 25 .03 .01 .01 .01 .01 .01 .01 .01 thenumber of observationsused to compute$j and TMMLE. 30 .01 .01 .01 .01 .01 .01 .01 .01 The AREs reportedhere were for the sets G(r) = {x: NOTE: Entriesfor MPLE and MMLE forA = 0 are fromShepard and Harvey(1990). Entries .5r < x < 1.54}, so Pr[#ic GQr);Ti] = Pr[ITi - TirI < forother values of A are estimated using 5,000 replicationswith T = 500. To facilitatethe - pi(Tir), say, and similarlyfor TMMLE. Using (5), computations,the likelihoodswere computedon a discretegrid of 240 equally spaced values of .5Tjr] 0 < A < 60, and the MLEs were computedby a search over thisgrid. set A TMMLET; thenAREj,MMLE = limTMMLE/Ti can be computedby solvingPi(A/AREj,MMLE) = PMMLE(A)- and Shephardand Harvey1990). The MMLE is equivalent, In general,the ARE dependson A. afterreparameterization on a restrictedparameter space, to Table 2 reportsthese AREs for the MPLE and six me- the MA(1) MLE analyzedby Davis and Dunsmuir(1996), dian unbiasedestimators for various values of A; all AREs and theirlocal-to-unity asymptotic results apply here. are relativeto the MMLE. For example,when A = 4, the Pile-upprobabilities that A is estimatedto be exactly0 are ARE of theQLR-based medianunbiased estimator, relative reportedin Table 1. The mass of the medianunbiased esti- to the MMLE, is 1.02, whichindicates that in large sam- matorsat 0 is similarfor all estimators.The pile-upprob- ples theMMLE requires1.02 timesas manyobservations as abilityfor the MPLE remainslarge as A increases,both theQLR-based estimatorto achievethe same probabilityof in absolute terms(it is above 50% for A < 6) and rela- fallingin theset G(r). Evidently,MMLE dominatesMPLE tive to the medianunbiased estimators. As pointedout by forall values of A shownand is considerablymore efficient Shephardand Harvey(1990), thepile-up for the for small to moderatevalues of A. In contrast,the median MMLE is smallerthan for MPLE. unbiasedestimators perform slightly better than MMLE for Cumulativedistribution functions of the variousestima- small values of A (A < 4) and comparablyfor moderate tors for A 5 are plottedin Figure 2. As expected,both Table 2. AsymptoticRelative Efficiencies of Median-Unbiased EstimatorsRelative to the MMLE

A MPLE L MW EW QLR P01(7) P01(17) 1 .13 1.00 1.00 1.00 1.00 1.07 1.07 2 .19 1.09 1.07 1.07 .96 1.07 1.02 o 3 .52 1.08 1.10 1.08 1.04 1.02 .94 4 .62 1.06 1.06 1.06 1.02 1.06 1.00 O~~~ _ 5 .65 .93 .93 .98 .97 1.11 1.14 6 .71 .94 .92 .99 1.00 1.03 1.08 n~~~~~~~~Q / /- 7 .76 .79 .79 .85 .88 .96 1.04 8 .77 .77 .77 .85 .85 .91 .98 9 .75 .69 .69 .75 .77 .86 .89 CD 10 .80 .65 .65 .71 .74 .80 .80 12 .67 .56 .56 .65 .67 .67 .67 0 / 14 .57 .50 .49 .57 .57 .57 .57 16 .50 .42 .42 .49 .50 .50 .50 18 .44 .38 .38 .44 .44 .44 .44 oiur /.Cmltv smttcDsrbtoso h asinME an ixMdinUnise stmtrso AWenA= . , PE 20 .40 .33 .33 .40 .40 .40 .40 --,ML;,L-,0W ,E ; QR-- 0() 25 .32 .28 .28 .32 .32 .32 .32 o-,P11) 30 .27 .22 .23 .27 .27 .27 .27 0 2 4 6 8 10 12 14 16 18 20 NOTE: The reportedAREs are the limitingratio of the numberof observations necessary forthe MLEto achievethe same probabilityofbeing in the region T ?t .5T as thecandidate estimator, as a functionofA = TIT, as describedin the text. AREs exceeding 1 indicategreater thanthe MLE. Entries are estimatesbased on interpolatingprobabilities from the values of A shownin column 1. These probabilitieswere estimated using 5,000 replications and T = 500 foreach valueof A~. 354 Journalof the AmericanStatistical Association, March 1998

Table 3. Lookup Table forConstructing Median-Unbiased Estimator Table 4. Postwar U.S. GDP Growth,1947:11-1995:IV: Tests of 1r= 0, of A forVarious Test StatisticsWhen Xt = 1 and D = 1 Median-UnbiasedEstimates, and 90% ConfidenceIntervals

A L MW EW QLR P017 P0117 Test Statistic(p-value) A (90% Cl) crA, (90% Cl) 0 .118 .689 .426 3.198 2.693 7.757 L .21 (.25) 4.1 (0, 19.4) .13 (0, .62) 1 .127 .757 .476 3.416 2.740 7.825 MW 1.16 (.29) 3.4 (0, 18.8) .11 (0, .60) 2 .137 .806 .516 3.594 2.957 8.218 EW .68 (.32) 3.1 (0, 17.0) .10 (0, .55) 3 .169 1.015 .661 4.106 3.301 8.713 QLR 3.31 (.48) .8 (0, 13.3) .03 (0, .41) 4 .205 1.234 .826 4.848 3.786 9.473 P01(7) 2.90 (.45) 1.7 (0, 12.9) .05 (0, .37) 5 .266 1.632 1.111 5.689 4.426 10.354 P01(17) 7.52 (.54) 0 (0, 11.3) 0 (0, .36) 6 .327 2.018 1.419 6.682 4.961 11.196 7 .387 2.390 1.762 7.626 5.951 12.650 -1 NOTE: ; is the estimateof the standarddeviation of A.3t in (11); thatis, ;A 8 = T A 8 .490 3.081 2.355 9.160 6.689 13.839 9 .593 3.699 2.910 10.660 7.699 15.335 10 .670 4.222 3.413 11.841 8.849 16.920 11 .768 4.776 3.868 13.098 10.487 19.201 whetherit has recentlybeen reversed are of considerable 12 .908 5.767 4.925 15.451 11.598 20.570 practicaland policy interest. Following Harvey (1985), we 13 1.036 6.586 5.684 17.094 13.007 22.944 examinethese issues using the local-level model in which 14 1.214 7.703 6.670 19.423 14.554 24.962 thegrowth rate of output is allowedto have a smallrandom- 15 1.360 8.683 7.690 21.682 16.153 27.135 16 1.471 9.467 8.477 23.342 18.073 30.030 walk component.This introducesthe possibilityof a per- 17 1.576 10.101 9.191 24.920 19.563 32.209 sistentdecline in meanoutput growth, consistent with the 18 1.799 11.639 10.693 28.174 21.662 35.426 productivityslowdown. 19 2.016 13.039 12.024 30.736 24.160 38.465 Thedata used are real quarterly values of GDP percapita 20 2.127 13.900 13.089 33.313 25.479 40.583 21 2.327 15.214 14.440 36.109 27.687 44.104 from1947:II-1995:IV. The datafrom 1959:I-1995:IV are 22 2.569 16.806 16.191 39.673 30.260 47.239 the GDP chain-weightedquantity index, quarterly, sea- 23 2.785 18.330 17.332 41.955 32.645 50.881 sonallyadjusted (Citibase series GDPFC). The from 24 2.899 19.020 18.699 45.056 35.011 54.426 1947:1-1958:IVare real GDP in 1987 dollars,seasonally 25 3.108 20.562 20.464 48.647 37.481 58.172 26 3.278 21.837 21.667 50.983 39.907 60.842 adjusted(Citibase series GDPQ, in releasesprior to 1996) 27 3.652 24.350 23.851 55.514 41.146 63.561 and proportionallyspliced to the GDP chain-weighted 28 3.910 26.248 25.538 59.278 43.212 66.782 quantityindex in 1959:1.These serieswere deflatedby 29 4.015 27.089 26.762 61.311 47.135 71.577 30 4.120 27.758 27.874 64.016 50.134 76.343 thecivilian population (Citibase series P16). ThisGDP se- rieswas transformedto (approximate) percentage growth NOTE: Entriesare the value ofthe teststatistic, for which the value of A givenin the firstcolumn at an annualrate, GYt, by settingGYt = 400Aln(real per is the median-unbiasedestimator. Care mustbe takento impose the normalizationD = 1 when using these estimates of A. Estimates of T are computed as AIT. Ifthe test statistictakes on capitaGDP). The modelis a value smallerthan that in the firstrow, then the median-unbiasedestimate is 0. Estimatesfor othervalues of the test statisticscan be obtained by interpolation.For example, suppose that QLR = 5.0 is obtainedempirically; using linearinterpolation, the median unbiased estimatorof A is 4 + (5.0 - 4.848)/(5.689 - 4.848). A softwareimplementation that handles general Xt for Table 5. Estimatesof Parametersin (1 1)-(13) forVarious the case D = Ik is available fromthe authorsby request. Allentries in the table were estimated Values of A and ImpliedSubsample TrendGrowth Rates using 5,000 replicationsand T = 500. Parameterestimates valuesof A (5 < A < 8). However,their performance dete- MMLE fixedA rioratesfor large values of A (A > 10). Parameter MPLE Estimates with One wayto calibratethe magnitude of A is to compare UA,8 0 .04 .13 .62 itto theasymptotic powers given in Figure1. WhenA = 4, 0a 3.85 (.17) 3.86 (.17) 3.85 (.17) 3.78 (.20) P1 .33 (.06) .34 (.07) .34 (.07) .32 (.08) thetests have rejection of approximately25%; P2 .13 (.06) .13 (.07) .13 (.06) .12 (.07) whenA = 7, therejection probabilities are approximately P3 --.01 (.07) -.01 (.08) -.01 (.07) -.01 (.08) 50%. ForA > 14,the power exceeds 80%. As an empirical p4 -.09 (.06) -.08 (.06) -.09 (.06) -.09 (.07) guideline,this suggests that the median unbiased estimators 00 1.80 (.46) 2.44 (.84) 2.67 (2.25) willbe roughlyas efficientas theMMLE whenthe results Estimatedaverage trendgrowths of stabilitytests are ambiguous.When there is substantial Date GY MPLE MMLE c,A = .13 <:Ap = .62 instability,the MMLE willbe moreefficient than the me- 1947-1995 1.80 1.80 1.80 1.80 1.80 dianunbiased estimators. 1947-1970 2.46 1.80 1.89 2.16 2.43 Table3 is a lookuptable that permits computing median 1970-1995 1.22 1.80 1.71 1.47 1.23 unbiasedestimates, given a valueof thetest statistic. The 1950-1960 2.75 1.80 1.91 2.25 2.27 = 1960-1970 2.39 1.80 1.84 1.98 2.39 normalizationused in Table 3 is thatD 1, and usersof 1970-1980 1.20 1.80 1.75 1.56 1.07 thislookup table must be sureto impose this normalization 1980-1990 1.58 1.80 1.70 1.45 1.50 whenusing the resulting estimator of A. 1990-1995 .62 1.80 1.68 1.36 1.04

NOTE: Estimates were computed by Gaussian maximumlikelihood, with numerical standard 4. APPLICATIONTO TREND GROSS OF U.S. errorscomputed fromthe inverse of the outer productestimate of the Hessian. Unrestricted GROSS DOMESTIC PRODUCT MLEs (standard errorsin parentheses) are reportedin the firsttwo columns. (Because of the nonnormaldistribution ofthe MLE of A, the for aB is notreported.) The last The issuesof whetherthere has been a declinein the twosets of columns report estimates by restricted MLE, with A fisedto the indicated values. The columnlabeled GY inthe second part of the table is thesample mean of GY; theother entries long-rungrowth rate of outputin theUnited States, when are averagevalues Of 'pti T overthe indicated subsample for the indicated model, where t T thisdecline took place, how large the decline has been, and are theestimates of /3, obtained from the Kalman smoother. Stock and Watson: Time-VaryingParameters 355

0~ o -- l 0 ;0 2 A A AAA f]

- I~~~~~~~~~~~~~~~~~~~~~~r N -Y-r Vtl,TX tV t T f

I 1945 1950 1955 1960 1965 1970 1975 1 980 1985 1990 1995

Figure3. GrowthRate in U.S. Real per Capita GDP and EstimatedTrends Based On the Four Models in Table 5. , GY , MPLE;- MMLE;---, u6,8 = .13; -- -, u6, = .62.

GYt= t + Ut, 0(11) and indeedthe median-unbiased estimates are, with only one exception,nonzero. The medianunbiased estimates of A areall small,ranging from 0 (POI(17)) to 4.1 (L). These A =t= (A/T)rt, (12) correspondto pointestimates of uA,, the and ofA/3t, ranging from 0% to .13%.This of estimates is consistentwith intuition. For example, a valueof u(, = .1 = a(L)ut Et, (13) correspondsto a standarddeviation of /31995:IV-31947:II of 1.4 percentagepoints. wherethe order p = 4 is used fora(L). (The resultsare insensitiveto choiceof the AR orderor to substituting Estimatesof themodel parameters are presentedin the an ARMA(2,3) parameterizationfor a(L), thelatter being toppart of Table5, forvarious values of A:the MPLE and consistentwith Harvey's [1985] original unobserved com- theMMLE, themedian-unbiased estimate based on theL ponentsformulation.) Estimates of A areconstructed using (whichis thelargest of thepoint estimates in Table4), and thenormalization D = 1 (i.e.,ao2 - 0Q2/a(1)2) as discussed theupper end of the 90% confidenceinterval for A basedon in Section2. L (thelargest such value for the 90% confidenceintervals). It is worthdigressing to discussthe implications of this Consistentwith the large pile-up probability discussed in modelfor orders of integrationand unit roots. If thereis a Section3, AMPLE = 0. The MMLE producesa smallbut random-walkcomponent in GYt,then the logarithm of real nonzeroestimate of uA, equalto .04%,which corresponds percapita GDP is I(2). Thishypothesis is soundlyrejected to a pointestimate of A of 1.4. Estimatesof parameters byunit root tests applied to thesedata. However, when the of theut processchange little for this range of valueof varianceof A/t is small,the model implies that AGYt has uAo, althoughestimates of the initialvalue of thetrend a nearlyunit MA root.Because it is wellknown that tests growthrate increase (as do theirstandard errors) as uA, fora unitAR roothave a highfalse-rejection rate under the increases.These results are broadly consistent with other re- nullof a unitAR rootwhen there is a nearlyunit MA root sultsreported in theliterature. For example, Harvey (1985) (Pantula1991; Schwert 1989), these rejections are consis- reporteda MMLE pointestimate of uA, = 0 forannual tentwith the postulated model. U.S. real GNP data from1909-1970. Harvey and Jaeger Test statistics,median unbiased estimates, and equal- (1993) reporteda largerpoint estimate(&Afl = .36) con- tailedconfidence intervals for A andthe standard deviation structedfrom a frequencydomain estimator and quarterly of AO arepresented in Table4. Noneof the tests rejects at realGNP datafrom 1954-1989. the10% level. Of course, this could mean that the tests have Estimatesof the trendgrowth rates /tIT based on insufficientpower to detect a smallbut nonzero value of A- thesemodels over various time spans (computed using the

n0?

= - \\/- (U9 -;

<0 o 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 Date

Figure 4. Estimatedtrends of US. Real GDP GrowthBased on the Four Models in Table 5. MPLE; - , MMLE; -- = .13; - -, u6, = .62. 356 Journalof the AmericanStatistical Association, March 1998

Kalman smoother)are givenin the bottompart of Table 5, An Op (1) limitingrepresentation for T112( _- 30) can be ob- and these series are plottedin Figures 3 and 4. Figure 3 tainedusing the methodsin theproof of Theorem1, but showing includes the raw data (GYt); this series is omittedfrom the T112 rate (whichis all thatis requiredhere) can be verified Figure 4, in which the scale is enlarged.No large mean directlyusing Chebyschevs' inequality. shiftis evidentin the raw data, consistentwith the small Lemma A2. Let Zt be a mean 0 stationaryvector stochastic estimatesof uA, foundusing the variousmethods. The es- process with fourth-ordercumulants that satisfy E? rl ,r2 ,r3= oo0 timateof trendper capita GDP growthbased on theMPLE IC(rl, r2, r3)1 < oo. Let wt be either a scalar nonrandom se- is, of course,a horizontalline showingthe mean of theraw quence or a random that is independentof Zt for data. In contrast,the otherestimates reflect, to a varying which supi supt>1 EIzit I4 < oo and supt>I EIwt 14 < oo. Then degree,a slowdownin mean GDP growthover thisperiod. T-1 EZT71ztwt p 0 uniformlyin s. The pointestimate based on L impliesa slowdownin the annualtrend growth rate of approximately.9% per annum Proof Firstlet Zt be a scalar.For 3 > 0, fromthe 1950s to the 1990s. Finally,none of the meth- [Ts9] / r 4 ods detectsany substantialincrease in trendGDP growth Pr sup T-1 ZtWt > 6 < 6-4E max T-1 ztwt overthe 1990s relativeto the 1980s; indeed,all of thepoint estimatessuggest a modestdecrease. T r 4\ 5. DISCUSSION AND CONCLUSIONS < 6-4 E T-1 Eztwt ( r=l t=l The medianunbiased estimators developed here provide 4 empiricalresearchers with a device to circumventthe un- r < -4T+-3 max E tttj desirablepile-up problem and of theMLE in theTVP l

FT-based teststatistics are easily computedusing statistics T fromthe GLS regressionof yton Xt. Giventhese statistics, < 6-4T-3 supEIwt {T ZE(ztr1rr) the medianunbiased estimator can be obtainedby interpo- tI,t2 ,t3 ,t4= j latingthe entriesin a lookup table. Such a lookup table is providedhere for the univariatelocal levels model (Table T 3), and lookup tables in higherdimensions for the normal- izationD = Ik are available fromthe authorson request. In thespecial case of theunivariate local-level model, we examined six asymptoticallymedian unbiased estimators and two MLEs and foundconsiderable differences among tI ,t2,t3,t=-I them.The MLEs werebadly biased, particularly the MPLE. T~~~~ 2) When the varianceof the coefficientsis small,the median a < -4Tp-3isupEAwt 14is that , t3t1 unbiased estimatorsbased on the QLR and P01(17) test statisticshad good AREs. Because no asymptotictheory for _ =00 thePOI testsin theTVP modelappears to be availableout- side the case Xt = 1, and because the POI testsare some- Proof~~~~~~~~~~~~tof:Thoe 1 3 I 1 whatcumbersome to computeeven in thelocal-level model, by Zit~~~~~ t~~~~~~~~U 3T2IC(tY) theseresults provide support for using the QLR-based me- Animplication of assumptionsof~~~~~rA, o B, and C, or alternatively We frsthe t assumptions,A, Br , andvC. dian unbiased estimatorsin the generalTVP model when sproElwthe the coefficientinstability is small.

APPENDIX: PROOF OF THEOREM 1 asuniomptionsistencyD,his etensthatottb epaigz Beforeproving Theorem 1, we stateand provetwo prelim- inary lemmas. Let UttI - i.t_p)', A = (-ai, Lneimp l (atL)d assumptions,A, finaland ivrn, srovith -a2,..., -ap)', and A (Ut1U1<1(U' 1fi) usingthe usual matrixnotation. t=l t=

LemmaA]. Underassumptions A-C, T1/2(A - A) = Op(1). BrownianimotionProofof Paertmao ss. pin ,B n ratraieyo Proof. Theresult follows by showing T1 /2 (A-A) ?P 0,where WefrtpoetetermudrassumptionsDC. ta A,B and A = (Ut1 U_i)-1(Ut1 u), where Ut-I = (ut-i, ... . ut-PY. Afterstraightforward algebra, it is seen thatthis followsif ProofofPr ~~~ p -LIT P 0 and /12T -P* 0, where tIlT and tt2T are matriceswith (i,j) elements,it,i - T-12 Zt=1U ( - ) and wheet W,= (L andW2 areineenet_=Ej= k-imni_onal sthandr /12t,ij =T-112ZtI1 (f3-i-' -it_( _-3). Theselim- itsfollow using the Markov and Chebyschev inequalities and ap- plyingassumptions A-C, assumingthat T112 (p3- i3) => Op (1). Stock and Watson: Time-VaryingParameters 357

P yt =/36Xt + (Z=l Vr)'Xt ?t+ +?tt Accordingly, the argumentthat A6T P 0 followsthe argumentthat A3T 0 withthe same replacement. Thus A7T -A0, SO K;T(S) - Slk. CT (S) = 1T (S) + A\6T (S) + 3T (S) Similarcalculations imply that P2 soV P rPo. By collect- - fKT(S)1{1T(1) + /A2T(1) + 63T(1)}, (A.2) ing termsand using(A.2), it followsthat V"2- T(S) #> hx(s) - where shA(1), whereh, (s) = Wi (s) + ((r 1/2n1 /2/u) f' W2 (r) dr. [Ts] Proofof Part b 1T (S) = T-1 /2 ZitfLt, t=l This followsfrom the continuousmapping theorem.

[Ts] t Proofof Part c

42(S= T / tt Vr This followsby straightforwardbut tedious manipulations using t=l r=l the previouslimiting results. Next,turn to the proofunder assumptions A and D. Underas- [Ts] sumptionD, a(L) = &(L) = 1, so Xt = Xt = Xt andet = Yt- 3T (S) = T-1 /2 : xt(t ,3'Xt (where,3 remainsthe OLS estimator).The proofunder these t=l conditionsfollows the foregoingproof but is simpler.In particu- lar,(A.2) nowholds with (lT(S) = T-1 /2 EtT=s and Xtet,)42T(S) T 3/2 TStj XtX't r 1/r,3T(S) 0, and IKT(S) - - [Ts] - I T _- [T-1 E[T7s]XtX'][T-1 ET1 XtX']-1. The limitof 41T fol- fKT (S) = T-1E Z ' T ic lows from(A.1). Write42T(S) as 42T(S) = T-E '11(XtXt - r)(T-1/2 Zt=1 vr) + rT-3/2 r. The firstterm in Limitsare obtainedfor these terms in turn.All limitsare uniform Z:1T'sZEt=l P in s E [0,1]. this expression 0 as a consequence of Lemma A.2 and the independenceof {vt,t 1,..., T} and {Xt,t = 1,...,T}, as 1. Write(1T(S) AlT(S) + A2T(s) + ( T(s), where discussedearlier for the termA3T. The limitof the second term followsfrom (A. 1) and the continuousmapping theorem. The ar- AlT(s) - j= o &i >7/T2 (a, - ai)(T1 >s? xt-j)t-i)) gumentgiven earlierfor $KT(s) P slk applies under these as- A2T(S) = T1/2 Et= 1(it - Xt and -lT(1) = T1/2 ZI11 sumptions,and V -Ao- 'r. This provespart (a) underassumptions Xt?t. AssumptionsA, B, andC implythat T-1 E>Ts]Xt_jut_i A and D; the proofof parts(b) and (c) followsaccordingly. satisfiesthe conditionsof Lemma A.2 with zt = Xt-jut-i and wt - 1; so, because p is fixed,AlT P 0. By an anal- [ReceivedAugust 1996. Revised June 1997.] ogous argument,A2T P 0. Using the limitin (A.1), we have (IT(S) =X U,rl/2 W (S). REFERENCES 2. Write 42T(S) A3T(S) + A4T(S) + (2T (S), where Andrews,D. W. K., Lee, I., and Ploberger,W. (1996), "OptimalChange- *~~~~-/ E[Ts] 1XXi-/ point Tests for Normal ,"Journal of , T - - A3T(S) >tk(XtXt r) Et=1 V, A4T(S) T 70, 9-38. Zt= (XtXt - XtXtk) Et r, and 42T(s)Ts = 3/2 >7"1 Andrews,D. W. K., and Ploberger,W. (1994), "OptimalTests When a Nui- P sance Parameteris PresentOnly Underthe Alternative,"Econometrica, IZJ,r To show A3T P and A4T 0, considerfor nota- Er= 0 62, 1383-1414. tionalsimplicity the case k = 1. (The argumentfor k > 1 is sim- Brillinger,D. R. (1981), TimeSeries: Data Analysisand Theory,San Fran- ilar.) Note that T2 maxt1...t4E Zr-=j vr1 r4Z = lVr4j r cisco: Holden-Day. ... < supSI...s4 EjW2(si) W2(s4)jfQ2 oo. Because Xt is station- Chow,G. C. (1984), "Randomand ChangingCoefficient Models," in Hand- book ofEconometrics, eds. Z. Grilichesand M. D. Intrilligator,Amster- ary with absolutelysummable eighth-order , Xt2-r dam: North-Holland,chap. 21. is stationarywith absolutelysummable fourth-order cumulants. Cooley, T. F., and Prescott,E. C. (1973a), "An Adaptive Regression Thus A3T satisfiesthe conditionsof Lemma A.2 with Zt Model," InternationalEconomic Review, 14, 364-371. P t - r and wt = T-12 r 1 Vr, SO A3T 0. Turningto (1973b), "Testsof an AdaptiveRegression Model," Reviewof Eco- nomicsand Statistics,55, 248-256. A4T, A4T =~0= 70 (&i+ai)T / (j -aj)A4T,ij(S), where E -~ (1976), "Estimationin the Presenceof StochasticParameter Vari- A4T,ij(S) [T-3/2 E[Ts] Xt_j Xt_jT-1/2 =3 I vr]. An argu- ation,"Econometrica, 44, 167-184. mentanalogous to thatused forA3T showsthat A4T,ij P 0 and, Davis, R. A., and Dunsmuir,W. T. M. (1996), "MaximumLikelihood Es- timationfor MA(1) Processes witha Root on or Near the Unit Circle," -P because p is finite,A4T 0. The limitof $ijT followsfrom (A.1). EconometricTheory, 12, 1-29. Thus 42T (s) => rPl /2 f0oW2 (r) dr. Engle,R. F., and Watson,M. W. (1987), "The KalmanFilter Model: Appli- 3. Write43T(S) = -AZEP=0 i=o s3T,ilJ(S),j ( where cationsto Forecastingand RationalExpectations Models," in Advances in Econometrics:Fifth World Congress of the EconometricSociety, ed. 63T,i1j(s) (T-3/2 E[Ts] Xt_iX/_jVt_i) As before,consider T. Bewley,Cambridge, U.K.: CambridgeUniversity Press. the case k 1. Now, T1/2 3T,i1j(s) satisfiesLemma A.2 with Franzini,L., and Harvey,A. C. (1983), "Testingfor DeterministicTrend = - and Seasonal Componentsin Models,"Biometrika, 70, 673- Zt Xt0X- 3_jvt-i and wt 1; thus63T 0. 682. 4. Let AlT(s) = T1 X71](XtXt - XtXt/) and A,T(s) = Hansen,B. E. (1992), "Tests forParameter Instability in Regressionswith I(1) Processes,"Journal of Businessand EconomicStatistics, 10, 321- T-1 >3Ts](X)tX)t - F), and let A7T Al^T ? AlT = 336. AlT 0 T-1 >3TS7](X0X - F). The argumentthat P~ followsthe Harvey,A. C. (1985), "Trendsand Cycles in MacroeconomicTime Series," argumentthat A4T P~ 0 withT-~12 >3r=l Vr replacedby 1, and Journalof Businessand EconomicStatistics, 3, 216-227. 358 Journalof the AmericanStatistical Association, March 1998

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