Appl. Math. J. Chiese Uiv. 008, 3(): 97-0 A ote o self-ormalized Dickey-Fuller test for uit root i autoregressive time series with GARCH errors YANG Xiao-rog ZHANG Li-xi Abstract. I this article, the uit root test for AR(p) model with GARCH errors is cosidered. The Dickey-Fuller test statistics are rewritte i the form of self-ormalized sums, ad the asymptotic distributio of the test statistics is derived uder the weak coditios. Itroductio Time series is a class of importat model i moder statistics, it is widely applied i ecoomics ad fiace. As we kow, may time series records preset ostatioary properties ad the uit root pheomeo widely exists. Testig uit root hypothesis is recogized to be of icreasig importace, much work has bee doe i this directio. For the first order autoregressive process with i.i.d errors, Dickey ad Fuller [4] showed that the limitig distributio of the test statistic is a quotiet of weighted sums of idepedet stadard ormal variables. Cha ad Tra [3] showed that if the residuals are i the domai of attractio of a strictly α-stable law with <α< ad zero mea, the the test statistic coverges i distributio to the fuctio of a α-stable Lévy process. Some related results o this issue ca be foud i [], [] ad [] etc. Motivated by practical applicatios i statistics ad ecoometrics, may authors bega to cosider various uit root processes with o-i.i.d errors. Recetly, ostatioary autoregressive model with GARCH errors have received icreasig attetio i the literature. I this article, we cosider the p-th order autoregressive process (AR(p)) with GARCH (,) errors. It is defied by the followig formula: { X t = φ X t + φ X t + + φ p X t p + u t, u t = σ t ε t, σt = ω + αu t + βσ t, t Z (.) +, where φ p 0,ω>0, α 0, β 0, ad α + β<, ad the scaled coditioal error {ε t } is assumed to be a sequece of i.i.d radom variables. Received: 006--3. MR Subject Classificatio: 6F05, 60F05. Keywords: uit root, AR (p)-garch (,), self-ormalized, Dickey-Fuller test statistic. Digital Object Idetifier(DOI): 0.007/s766-008-009-x. Supported by the Natioal Natural Sciece Foudatio of Chia (0476; 06776).
98 Appl. Math. J. Chiese Uiv. Vol. 3, No. Like the most popular volatility models, ARCH ad GARCH models are discussed i the literature frequetly. For GARCH model, whe α = β =0,{u t } defied i model reduce to i.i.d white oises, ad i such a case the limitig distributio of uit root test statistic has bee ivestigated for a log time. Whe p =ad{u t } are GARCH type errors satisfyig Eu 4 t <, the asymptotic distributios of test statistics are the same as those obtaied i the i.i.d case, see for example [7-0]. These papers are at the forefrot of research i this area, they gave the asymptotic iferece for uit root processes with GARCH errors uder weak coditios. Also, Wag [3] has abolished the restrictio of Eu 4 t < ad obtaied the asymptotic distributio of test statistic for AR ()-GARCH (,) model. I the higher order cases, a represetatio of (.) which is used for testig purpose ca be writte as follows: p X t = ρx t + a k ΔX t k + u t, (.) k= where ΔX t k := X t k X t k, ρ = p i= φ i ad a k = p j=k+ φ j. The uit root problem for this model focuses o testig H 0 : ρ = agaist H : ρ <. Let ˆΘ =(â,, â p ) be the estimator of Θ =(a,,a p ). ˆΘ ca be take as the least-square estimator, maximum likelihood estimator or M-estimator. A basic test statistic of the ull hypothesis is give by ˆT := (ˆρ ), (.3) where ˆρ is the least-square estimator (LSE) of ρ. If the determiistic drift term β is icluded i the regressio, that is, p X t = β + ρx t + a k ΔX t k + u t, (.4) k= the correspodig LSE of ρ deotes ˆρ,c, ad the test statistic becomes ˆT,c := (ˆρ,c ). (.5) Elighteed by the former researches, this paper focuses its attetio o AR (p)-garch (,) model without Eu 4 t <. We establish correspodig fuctios of the self-ormalized partial sums for the Dickey-Fuller test statistics, which provides a coveiet way to derive asymptotic distributio. The results i our paper are more geeral tha those cited above. This paper is orgaized as follows. I, we state the assumptios ad give the mai results. roofs are collected i 3. Assumptios ad the mai results At the begiig of this sectio we pay attetio to the followig assumptios eeded i this paper. Assumptio.. Eε t =adα + β<. Remark.. Uder the Assumptio., E ε t +δ = is allowable for all δ>0. Ad whe α + β<, oly the existece of the secod momet of u t is required. I the literature o uit root with GARCH errors, this is the weakest coditio for the uit root distributio to exist.
YANG Xiao-rog, et al. A ote o self-ormalized Dickey-Fuller test for uit root i autoregressive 99 Assumptio.. The polyomial a(z) := a z a z a p z p satisfies a(z) 0 for z. Ad for all k p, we have a k â k = O p ( ). (.) Remark.. The foregoig assumptio is imposed to esure that if ρ =,theφ(z) has oly oe root o the uit disc, the root z =. Moreover, the coditio (.) is a mild oe. We see that uder the ull hypothesis, (.) becomes a statioary AR (p )-GARCH (,) process based o the differece series, the -cosistece of the estimator (â,, â p )isaatural assumptio. Let S = t= u t, V = t= u t.here[x] deotes the largest iteger less tha or equal to x, sigifies the weak covergece of the space D[0, ] edowed with the Skorohod topology, ad W (r) stads for a stadard Browia motio. The mai results of this paper are stated as follows. Theorem.. If ρ =, the observatios follow the regressive model (.) (i.e. β = 0), with iitial value (X 0,X,,X p ), the uder the Assumptios.,. we have S [r] = W (r), 0 r, (.) V W (t)dw (t) ad ˆT = ( p k= a 0 k)ξ, where ξ =. W (t)dt 0 Theorem.. If ρ =, the observatios follow the regressive model (.4)( i.e. β 0), with iitial value (X 0,X,,X p ), the uder the Assumptios.,. we have that (.) holds ad ˆT,c = ( p k= a k)ζ, where ζ = (W () ) W () W (r)dr 0 3 roofs 0 W (r)dr ( 0 W (r)dr). roof of Theorem.. We first rewrite the test statistic as the self-ormalized Dickey-Fuller test statistic. ut S = t= u t, V = t= u t. S /V is called self-ormalized sums. Sice ˆρ is the LSE of ρ, it ca be described as follows: { ˆρ =argmi ρ Ω (X t ρx t } p k= âkδx t k ), where Ω is the parameter space. We get by simple algebra that ˆρ = X t (X t p k= âkδx t k ). By (.) ad the fact that 3 X t ΔX t i ˆT = (ˆρ ) = X t 0, we immediately get X t u t + o p (). X t Defie ψ (z) = j=0 ψ jz j =( p k= âk), ad usig the polyomial decompositio give by hillips ad Solo [,p97],wegetx t = ψ () t k= u k + η t η 0, where ψ () ψ() = p k= a, η t = k α j u t j, j=0 α j = ψ i. i=j+
00 Appl. Math. J. Chiese Uiv. Vol. 3, No. Notice that X t u t = ψ () Q = ψ () ( + j= t u i ) u j j= u t + = ψ () u j j= (η t η 0 )u t =: Q + Q. + ( ( S ) ) V. Ad uder Assumptio. we get EQ =0adEQ = O( ). Sice Xt = ψ () t u j + o p () = ψ () ( ) S + o p (), j= so ˆT = ψ () (S V ) + o p () = ψ () ((S /V ) ) S t (S t/v ) + o p(). (3.) (3.) demostrates that ˆT ca be represeted as the fuctio of the self-ormalized sums S /V. So we call it the self-ormalized Dickey-Fuller test statistic. If (.) holds, the asymptotic distributio of ˆT is derived from (.) ad the cotiuous mappig theorem. Thus, it remais to prove (.). For this purpose we eed the followig two lemmas. Lemma 3.. Set s = ES = EV, the uder Assumptios. ad., Ad for all x>0, s t= V s a.s.. E[u t I ( ut >xs )] 0 as. roof. See [3] Lemma ad Lemma. Lemma 3.. Let s be defied the same as i Lemma 3.. If the Lideberg coditio holds, amely for all x>0, Ad if the s t= roof. See [6, p99] Theorem 4.. E[u t I ( ut >xs )] 0 as. V s λ > 0 S [r] V a.s., = W (r). Hece, (.) holds immediately from Lemma 3. ad Lemma 3., ad we fulfill the proof of Theorem.. roof of Theorem.. It is aalogous to Theorem., thus we just coclude the result with a brief proof. ˆT,c = (ˆρ,c ) = (X t X)u t (X t X) + o p().
YANG Xiao-rog, et al. A ote o self-ormalized Dickey-Fuller test for uit root i autoregressive 0 where X = X t. Similar to the proof of Theorem., we obtai ˆT,c = ψ () [(S /V ) ] (S /V ) S t/v [ (S t/v ) ( S + o p (). t/v ) ] Thus asymptotic distributio of ˆT,c follows from (.) ad the cotiuous mappig theorem. Refereces Billigsley. Covergece of robability Measure, New York: Wiley, 968. Cha N H, Wei C Z. Limitig distributio of least squares estimates of ustable autoregressive process, A Statist, 988, 6: 367-40. 3 Cha N H, Tra L T. O the first-order autoregressive process with ifiite variace, Ecoometric Theory, 989, 5: 354-36. 4 Dickey D A, Fuller W A. Distributio of the estimators for autoregressive time series with a uit root, J Amer Statist Assoc, 979, 74: 47-43. 5 Fuller W A. Itroductio to Statistical Time Series, New York: Wiley, 976. 6 Hall, Heyde C C. Martigale Limit Theory ad Its Applicatio, New York: Academic ress, 980. 7 Lig S Q, Li W K. Limit distributios of maximum likelihood estimators for ustable autoregressive movig-average time series with geeral autoregressive heteroscedastic errors, A Statist, 998, 6: 84-5. 8 Lig S Q, Li W K. Asymptotic iferece for uit root processes with GARCH (,) errors, Ecoometric Theory, 003, 9: 54 564. 9 Lig S Q, Li W K, McAleer M. Estimatio ad testig for uit processes with GARCH (,) errors: theory ad Mote Carlo evidece, Ecoometric Review, 003, : 79-0. 0 atula S G. Estimatio of autoregressive models with ARCH errors, Sakhyā (Ser B), 988, 50: 9-38. hillips C B, erro. Testig for a uit root i time series regressio, Biometrika, 988, 75: 334-346. hillips C B, Solo V. Asymptotics for liear processes, A Statist, 99, 0: 97-00. 3 Wag G W. A ote o uit root tests with heavy-tailed GARCH errors, Statist robab Lett, 006, 76: 075-079. Dept. of Math., Zhejiag Uiv., Hagzhou 3007, Chia.