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Median Unbiased Estimation of Coefficient Variance in a Time Median 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 estimator (MLE) has a largepoint mass at 0. We thusdevelop asymptotically median unbiased estimatorsand asymptoticallyvalid confidenceintervals by invertingquantile functions of regression-basedparameter stability test statistics,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 parameter (TVP), or "stochastic theparameters of (1)-(4) can be estimatedby maximum coefficients,"regression model has been used extensivelylikelihood implemented by the Kalman filter. 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 moving average [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 polynomial,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 estimators 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 univariate 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 statistic(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 growthrate 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 least squares (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 mean 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 matrix 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
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