Consistent And Non-Degenerate Model Specification Tests .

annaleset de ificationSmoothNeural- n? 90 - atives*B. HillJonathan- Wetest thatABSTRACT.develop a regression model specificationdirects maximal power toward smooth transition functional forms, and isconsistent against any deviation from the null specification. We providenew details regarding whether consistent parametric tests of functionalform are asymptotically degenerate: a test of linear autoregressionagainstis never degenerate.STAR alternativesMoreover, a test of ExponentialSTAR has power attributes entirely associated with the choice of threshold.Ina simulation experiment inwhich all parameters are randomly selectedtest has power nearly identical to a most-powerfultest forthe proposedtrue STAR, neural network and SETAR processes,and dominates populartests. We apply the test to U.S. output, money, prices and interest rates.Tests de sp?cificationde mod?lecontredesalternativesd?g?n?r?set r?seaux de neuroneset nonconvergentsavec transition douceun test de sp?cification pour un mod?le deNousR?SUM?.d?velopponsr?gression qui dirige une puissance maximale vers des formes fonctionnelles? transition douce et qui est convergent contre toute d?viation par rapport ?l'hypoth?se nulle. Nous pr?cisions ?galement lad?g?n?ration asymptotiquedes tests param?triques convergents de formes fonctionnelles :un test d'unlin?aire contre une alternative STAR n'est jamaisprocessusautor?gressifuneuntestd'une fonction exponentielle STAR poss?deDeplus,d?g?n?r?.au choix du seuil. Dans une simulation o?enti?rement associ?epuissancele test propos? poss?detous les param?tres sont choisis al?atoirement,une puissance presqueidentique ? celle d'un test le plus puissant pour dede type STAR, r?seaux de neurones et SETAR, et dominevrais processustous les tests usuels. Nous appliquons ce test ? des donn?es am?ricainesde niveaude production, de monnaie, de prix et de taux d'int?r?ts.1 ThecommentsRouxS?bastienof twoandsuggestionsare gratefully acknowledged.Editorand Chiefrefereesanonymouson an earliery versionFor commentsof the paper Iwould liketo thankDick van Dijk, Phil Rothman, and the participants ofthe 12thannual meeting of theSociety forNonlinear Dynamics inEconomics, and the2004North AmericanEconometricSocietymeeting.1 Jonathan B. Hill : Dept. of Economics, Universityof North Carolina, Chapel Hill;jbhill@email.unc.edu;author wasaffiliatedhttp://www.unc.edu/ jbhill.with the Dept.of EconomicsTheir hospitality isgratefullyacknowledged.This paperat Floridawasthewritten whileInternationalUniversity.

146ANNALES D'ECONOMIE ET DE STATISTIQUE1 INTRODUCTION1.1 STAR MethodologiesSmooth Transition Autoregression (STAR) models have gained significant popularity as a means to transcend well known explanatory and forecasting limitations of linear and binary regime switchingmodels. See Chan and Tong [1986a,b],Terasvirta [1994], Luukkonen et al. [1988], Lin and Terasvirta [1994], van Duket al [2002], Lundberg et al. [2003] and Lundberg and Terasvirta [2005].as a two-regime autoregressionThe standard setupmodels a time series{yt}(i) yt a 2p ?fritt-i2/-ip?itt-ix Fu-rf.Y,c) zt/-iwhere F is a smooth function taking values on [0, 1], typically exp {- y(yt d c)2}exponential: F(yt d, y, c)logistic:F(yt d,y, c) -?-.l exp{-y(yt d-c)}The transition function F(yt d ,y,c)e[0,1] movesthe data generating processyr P ?ivt-ianda ZP21jpast informationyt d.The exponential F()basedon(?i ?/Xv*-/captures "inner" and "outer" regimes: if ?yt-i ?p 1 and yt ? a is close to c thenand(c . ?/) ., erF(yt d, y, c)ifyt d The logistic F()captures "upper" and "lower" regimes: if c yt d ? oo thenandF(yt d,y,c)- \,F(yt d,y, c) - 0asc v . - -oo. Thescaley 0 gauges the speed of transition: y 0 implies no transition inwhich case yt is alinearAR; and small (large) y 0 implies slow (fast) transition.Tests of linearity against STAR alternatives, however, have received almost noattention in the theory literature,although a standard practice dominates the appliedliterature.Since a testof ? 0 is not nuisance parameter-free the standard practiceis to exploit y 0. The hypothesis is indirectly tested by performing a truncated 0. This leads to aTaylor approximation of F(yt d, y, c) around 7simple second or thirdorder polynomial auxiliary regression in the spirit of Ramsey [1970]andstandardF-testsof parametriczero-restrictionsare usedto determinewhetherthe process is linearAR, or exponential or logistic STAR. See Luukkonen et al.[1988], Saikkonen and Luukkonen [1988], Lin and Terasvirta [1994], Terasvirta[1994], Gonzalez-Rivera[1998], Escribano and Jorda [2000], Rothman et al.TerasvirtaLundbergand[2001],[2002, 2005], and Lundberg et al. [2003], tonamea few.

CONSISTENT AND NON-DEGENERATEMODEL SPECIFICATIONTESTSAGAINST SMOOTH TRANSITIONAND NEURAL NETWORK ALTERNATIVES147In order for the polynomial regression to have meaning in a STAR framework,however, the true data generating process is simply assumed to be a STAR. If noassumptions aremade the testmerely directs power toward low order polynomials.The test is thereforenot a true test against smooth transitionalternatives, per se.The nuisance "delay" parameter d remains in the polynomial regression. If d isnot simply assumed it is selected by minimizing theF-statistic /7-value. The statistic has a non-standard limitingnull distribution in the lattercase (Davies [1977],Stinchcombe andWhite [1992]), yet chi-squared or F- distributions are universallyused. Similarly, inmany instances the threshold c is simply fixed (e.g. GonzalezRivera [1998]).Finally, most smooth transitionmodels in the applied literature incorporate onlyone threshold variable yt d,and in some cases only time t (Lin and Terasvirta[1994], van Dijk et al. [2000], Lundberg et al. [2003]). Test consistency will requireeach stochastic variable thatenters into thenull specification (e.g. yt v .,y) toenter into theweight functionF(), cf. Bierens [1982, 1990] and Stinchcombe andWhite [1998].1.2 New STAR TestIn thispaper we develop a consistent1 parametric test of STAR functional form.Consistent parametric tests have been proposed by Bierens [1990], Bierens andPloberger [1997], Stinchcombe andWhite [1998], Dette [1999] and Hill [2007].See Yatchew [1992], Hardle and Hall[1993], Hong and White [1996], StuteHsiaoZinnandforandLi,[2003](semi) nonparametric methods.[1997]Inconsistency arises because only a finite number of moment conditions areactually tested.A failure to reject the null may simply be due to the fact that somealternative not covered by the test statistic is true. In a STAR framework, even ifweagree thatfinite-order polynomials adequately represents exponential and logisticfunctional forms, a failure to reject the testmay be due to some other smooth transitionmechanism (e.g. theNormal STAR: see Chan and Tong [1986b]).Our main contribution is a score test thatdirects power toward a general SmoothTransition Non-Linear Autoregression with Auxiliary variables (STARX). Singleequation ARX models have a myriad applications inmacroeconomics and finance(e.g. Baillie [1980]; Bierens [1987, 1991]; Pena and Sanchez [2005]). The test isconsistent against any deviation from the null, and nests specifications popularlyemployed in the STAR and Artificial Neural Network [ANN] literatures.ConsultHornik, Stinchcombe andWhite [1989], Bierens [1990], Hornik [1991] and Lee,White and Granger [1996] for details on ANN models and theirusage in economics. Whereas smooth transitionmodels have simple behavioral interpretations2,neural nets are typically employed to absorb evident and otherwise unexplainednonlinearity (e.g. Donaldson and Kamstra [1996]). A score testprovides an intuitive sample check that smooth transitionor neural net termshave not been omittedfrom a nonlinear ARX null specification.1. The power of the test statistic converges to one, as the sample size grows, under any deviation fromthe null.2. For example, as an exchange rate deviates from a target band, currency traders may expect openmarket transactions by a central bank to stabilize the rate. The planned transaction and its expectationtheby traders suggest tradersmay behave differently as the exchange rate increasingly deviates fromband.

148ANNALES D'ECONOMIE ET DE STATISTIQUEOur test is consistent because we enforce 7 0, permittinguncountably infinitelymany moment conditions based on flexible testweights. We simply testwhetherthe second regime termbelongs H0: ivs.0,i l.pH1: at leastone?. 0,and deliver a supremum testover 7 in order to elevate small sample power.In a second contributionwe prove consistency of a test against an ExponentialSTAR alternative is based on the threshold c. This suggests thepractice of fixing cmay curtail small sample testpower.Of separate interest,as a thirdcontributionwe prove a score testof linear autoregression against standardANN or STAR alternatives is never degenerate exceptin a trivial case. This provides farmore information concerning test degeneracythan previously characterized inBierens [1990] and de Jong [1996], and providesa natural setting for the optimal tests ofAndrews and Ploberger [1994,1995] whosimplyassumenon-degeneracy.There are, however, some notable limitations.Although we permit non-stationarytime series our testevidently cannot distinguish between non-stationarity (e.g. a unitroot or stochastic break) and nonlinearity. See Kapetanios, Shin and Snell [2000]and Kapetanios and Shin [2003] for tests in this genre. It also cannot handle someunbounded forms of global non-stationarity including linear trend invariance.A simulation study demonstrates our test dominates standard tests, and vastlydominates the STAR polynomial regression test. In fact, thepower of theproposedtest against STAR, ANN and SETAR alternatives nearlymatches thatof uniformlymost-powerful tests. Finally, we apply the test to a basket ofU.S. macroeconomicvariables.In Section 2 we detail the STARX framework. Sections 3 and 4 contain the scorestatistic and construct smooth transitionmoment conditions. Asymptotic theory isdeveloped in Section 5, and Section 6 characterizes testdegeneracy. Sections 7 and8 contain the simulation and empirical studies. Assumptions and proofs are in theappendices, and all tables are placed at the end.WriteA:-vectorssional\x\p: (Zjx,j\P)ypa and x, vectoridentity matrix.?- powersand\\x\\p: (?.y E\x,j\p)u"Xa representdenotes(xf,.,convergencexakk )'.Ik. For arbitrarydenotesin probability, - a -dimenconvergence in finite dimensional distributions; and weak convergence on a metricspace; [x] is the integerpart of x. C[A] denotes the space of continuous functionsendowed with theuniformmetric on some compact space A.2 STARX FRAMEWORK Let {Wt) : {yt,xt) be a k-vector stochastic process, where xt eRk l, k 1 areregressors that do not contain lags of vr Assume {Wt} lies in L2 (T, 3, P) with n tneprobability measure P and G-field 3, j({Wt} :t t), 3,o (U,e?3,)