Testing for a unit root in noncausal autoregressive models

Transcription

1 Teting for a unit root in noncaual autoregreive model Saikkonen, Pentti Saikkonen, P & Sandberg, R 216, ' Teting for a unit root in noncaual autoregreive model ' Journal of Time Serie Analyi, vol. 37, no. 1, pp Downloaded from Helda, Univerity of Helinki intitutional repoitory. Thi i an electronic reprint of the original article. Thi reprint may differ from the original in pagination and typographic detail. Pleae cite the original verion.

2 Teting for a Unit Root in Noncaual Autoregreive Model * by Pentti Saikkonen y Univerity of Helinki and Bank of Finland Rickard Sandberg z Stockholm School of Economic May 215 Abtract Thi work develop likelihood-baed unit root tet in the noncaual autoregreive (NCAR) model formulated by Lanne and Saikkonen (211, Journal of Time Serie Econometric 3, I. 3, Article 2). The poible unit root i aumed to appear in the caual autoregreive polynomial and for reaon of identication the error term of the model i uppoed to be non-gauian. In order to derive the tet, aymptotic propertie of the maximum likelihood etimator are etablihed under the unit root hypothei. When the error term of the model i ymmetric the limiting ditribution of the propoed tet depend on a ingle nuiance parameter, and a imple procedure to handle thi diculty in application i propoed. In the cae of kewed error a boottrap procedure to the nuiance parameter problem i dicued. Finite ample propertie of the tet are examined by mean of Monte Carlo imulation. The reult how that the ize propertie of the tet are atifactory and that clear power gain againt correctly pecied tationary NCAR alternative can be achieved in comparion with conventional Dickey-Fuller tet, the M-tet of Luca (1995, Econometric Theory 11, ), and the tet of Rothenberg and Stock (1997, Journal of Econometric 8, ). In an empirical application to a Finnih interet rate erie evidence in favour of a tationary NCAR model with leptokurtic error i found. Key word: Maximum likelihood etimation; Noncaual autoregreive model; Non-Gauian time erie; Unit root; Boottrap. * The author thank the Editor, a Co-Editor, and two anonymou referee for ueful comment. The rt author acknowledge nancial upport from the Academy of Finland, the OP-Pohjola Group Reearch Foundation, and the Finnih Cultural Foundation. Part of thi reearch wa done during hi reearch viit to the Monetary Policy and Reearch Department of the Bank of Finland whoe hopitality i gratefully acknowledged. The econd author acknowledge nancial upport from Jan Wallander' and Tom Hedeliu' Foundation, Grant No. P212-85:1. y Department of Mathematic and Statitic, Univerity of Helinki, P.O. Box 68 (Gutaf Halltromin katu 2b), FIN-14 Univerity of Helinki, pentti.aikkonen@helinki.. z Department of Economic, Center for Economic Statitic, Stockholm School of Economic, P.O. Box 651 (Sveavagen 65), Stockholm, rickard.andberg@hh.e. 1

3 1 Introduction Teting for the unit root hypothei i an important part in the analyi of economic time erie, and ha attracted an enormou amount of interet during the pat decade. In thi context, the mot widely ued model i the conventional (caual) autoregreive (AR) model where the current obervation i expreed a a weighted average of pat obervation and an error term. An eential aumption of the conventional AR model i that the error term i unpredictable by the pat of the conidered time erie. However, in (ay) economic application thi aumption may break down becaue the impact of omitted variable, interrelated with the conidered (univariate) time erie, i ignored. More pecically, if relevant variable are omitted their impact goe (at leat partly) to the error term of the model and, a the conidered time erie may help to predict the omitted variable, the aumed unpredictability condition may break down. A economic variable are typically interrelated, thi point appear particularly pertinent in economic application. In cae like thi the noncaual AR (NCAR) model may provide a viable alternative, for it explicitly allow for the predictability of the error term by the pat of the conidered erie. Early tudie of NCAR model and their extenion, noncaual and (potentially) noninvertible autoregreive moving average (ARMA) model, were mainly motivated by application to natural cience and engineering (ee, e.g., Breidt et al. (1991), Lii and Roenblatt (1996), Huang and Pawitan (2), Roenblatt (2), Breidt et al. (21), Wu and Davi (21), and the reference therein). More recently, a lightly dierent formulation of the NCAR model wa conidered by Lanne and Saikkonen (211) (hereafter L&S), and further tudied by Lanne et al. (212a), Lanne et al. (212b), Lanne et al. (212c), Lanne and Saikkonen (213), and Gourieroux and Zakoian (213). Thee paper demontrate that the NCAR model can uccefully decribe and forecat many economic time erie, and it often outperform it conventional caual alternative in term of model t and forecating accuracy. Even though the propertie of the tationary NCAR model are by now well undertood and aymptotic ditribution theory for variou parameter etimator (typically maximum likelihood etimator) have been developed, the nontationary cae and tet for a unit root have not yet been tudied in the literature. A unit root type nontationarity appear quite common (particularly) in economic time erie, and hence potential application of the NCAR model, thi work aim at propoing unit root tet in the context of the NCAR model of L&S. We develop Wald type unit root tet by auming that the poible unit root appear in the caual autoregreive polynomial of the model, and to thi end we rt derive aymptotic propertie of a (local) maximum likelihood (ML) etimator of the parameter of the model under the unit root hypothei. A in the tationary cae, a non-gauian error term i required to achieve identication (ee, e.g., Brockwell and Davi (1987, pp ) and Roenblatt (2, pp. 1-11)). Thi render the etimation problem nonlinear which, in turn, make the derivation of limiting ditribution le traightforward than in the context of conventional unit root tet, where etimation i carried out by linear leat quare (LS) technique. To addre thi iue, we ue idea imilar to thoe ued in tatitical model whoe likelihood ratio atify the o-called locally aymptotically mixed normal (LAMN) condition (ee Baawa and Scott (1983), Ch. 2). It turn out that the limiting ditribution of our tet are not ditribution free and appear, in general, very complicated depending on a number of nuiance parameter. To obtain tet with manageable limiting ditribution we aume that the error 2

4 term of the model ha a ymmetric ditribution. Then the limiting ditribution of our tet only depend on a ingle nuiance parameter determined by the ditribution of the error term, and thi problem can be rather eaily circumvented by uing etimated critical value (decribed in Section 5.1). Extending thi approach to kewed error appear infeaible o that a boottrap procedure (decribed in Section 5.2) i dicued in order to relax the ymmetry aumption. We examine the practical relevance of our aymptotic tet by mean of Monte Carlo imulation. The reult how that our tet perform atifactorily in term of ize and their power againt correctly pecied tationary NCAR alternative i very good in comparion with conventional Dickey-Fuller (DF ) tet, the M-tet of Luca (1995), and the likelihood-baed unit root tet of Rothenberg and Stock (1997). We alo demontrate that our boottrap procedure work very well in cae where the error ditribution i kewed. To illutrate the practical implementation of our tet we preent an application to a Finnih interet rate erie for which a tationary NCAR model with Student' t-ditributed error (ymmetric or kewed) i found to provide a good decription. The plan of the paper i a follow. Section 2 dene the conidered NCAR model and dicue the teting problem. Parameter etimation and related aymptotic reult are preented in Section 3 and ued in Section 4 to obtain our unit root tet. Section 5 report the reult of the Monte Carlo imulation and Section 6 preent the empirical application. Section 7 conclude. Three appendice contain mathematical proof and ome technical detail. Finally, the following notation i ued throughout the paper. The notation! p ignie convergence in probability and! d i ued for convergence in ditribution and alo for weak convergence in a function pace. We write B (u) BM () for a Brownian motion B (u) with indicated variance or covariance matrix. Unle otherwie tated, all vector will be treated a column vector and, for notational convenience, we hall write x = (x 1 ; :::; x n ) for the (column) vector x where the component x i may be either calar or vector (or both). 2 Model and teting problem Following L&S we conider the NCAR model (B) ' B 1 y t = t ; t = 1; 2; :::; (1) where t i a equence of independent and identically ditributed (IID) random variable with mean and nite variance 2 >, B i the uual backward hift operator (By t = y t k for k = ; 1; :::), and (B) = 1 1 B r B r and ' B 1 = 1 ' 1 B 1 ' B. L&S aume that the polynomial (z) and ' (z) (z 2 C) have their root outide the unit circle in which cae the dierence equation (1) ha a tationary olution. In thi paper, we allow for the poibility that, due to a unit root in the caual autoregreive polynomial (z), the proce y t i a nontationary integrated proce. Thu, we aume that r > and proceed in the conventional way by writing the lag polynomial (B) a (B) = B 1 B r 1 B r 1 ; (2) where = 1 B i the dierence operator. Our focu i in teting for the unit root null hypothei H : = againt the tationary alternative H 1 : <. At thi point we abtract from any 3

5 determinitic term uch a a contant term or linear time trend in the proce. Thee extenion will be dicued in Section 4.2. Unle otherwie tated we aume throughout the paper that the null hypothei H hold and that the root of the polynomial (z) = 1 1 z r 1 z r 1 and ' (z) lie outide the unit circle or, formally, that (z) 6= for jzj 1 and ' (z) 6= for jzj 1: (3) Uing equation (2) we can write equation (1) a y t = y t y t r 1 y t r+1 + v t ; t = 1; 2; :::; (4) where the proce v t = (B) y t = ' B 1 1 t ha the forward moving average repreentation v t = 1X j t+j ; = 1: (5) j= Here j i the coecient of z j in the Laurent erie expanion of ' z 1 1. By the latter condition in (3) thi expanion i well dened for jzj b ' with ome b ' < 1 and with the coecient j decaying to zero at a geometric rate a j! 1. Equation (4) how that our teting problem can be thought of a teting for a unit root in an AR(r) proce with tationary error following the purely noncaual AR(; ) proce ' B 1 v t = t (a in L&S we ue the acronym AR(r; ) for the model dened in equation (1)). When r = 1 the lagged dierence vanih from the right hand ide of equation (4) which become a pecial cae of a rt-order autoregreion with general tationary (or hort-memory) error. Teting for a unit root in uch context ha been conidered in a number of paper ince the work of Phillip (1987) and Phillip and Perron (1988). That the error in (4) are generated by a purely noncaual AR(; ) proce ditinguihe our formulation from it previou counterpart. For later ue we alo introduce the (caual) AR(r) proce u t = ' B 1 y t or (B) u t = t (t = 1; 2; :::). Under the null hypothei, (B) u t = t and the former condition in (3) yield the conventional backward moving average repreentation u t = 1X j t j ; = 1; (6) j= where the coecient j of the power erie repreentation of (z) 1 decay to zero at a geometric rate a j! 1 for jzj b and ome b > 1. Thu, u t i a nontationary I(1) proce. Finally, note that equation (1) and the condition in (3) imply that there exit initial value uch that the dierenced proce y t ha the two-ided moving average repreentation y t = 1X j= 1 j t j ; (7) where j i the coecient of z j in the Laurent erie expanion of (z) 1 ' z 1 1 def = (z) o that (z) = P 1 j= 1 jz j exit for b ' jzj b with b ' < 1 < b dened above and with j decaying to zero at a geometric rate a jjj! 1. The repreentation (7) implie that y t i a tationary and ergodic proce with nite econd moment. Hence, the invariance principle and 4

6 weak convergence reult of ample covariance matrice given in Phillip (1988) apply to y t for any (random or nonrandom) initial value y. Thi implie that the uual aymptotic reult needed to develop limit theory for unit root tet are available. To implify preentation we aume that, under the null hypothei, the procee y t and u t are tationary and not only aymptotically tationary. We derive a unit root tet in a likelihood framework imilar to that in L&S (for the employed aumption, ee alo Andrew et al. (26)). Thu, we impoe the following aumption on the error term in (1). Aumption 1. The zero mean error term t i a equence of non-gauian IID random variable with a (Lebegue) denity 1 f 1 x; which depend on the (nite and poitive) error variance 2 and (poibly) on the parameter vector (d 1) taking value in an open et R d. A dicued in Breidt et al. (1991), Roenblatt (2, pp. 1-11), L&S, and other, caual and noncaual autoregreion are tatitically inditinguihable if the error term (and hence the oberved proce) i Gauian. Thi explain why Aumption 1 include the requirement of non- Gauian error. Further aumption on the denity function f (x; ) will be made later. We cloe thi ection with a remark on the conceivable poibility of teting for a unit root in the noncaual polynomial ' (). A equation (4) and the ubequent dicuion indicate a poible unit root in the caual polynomial () make the teting problem conceptually very imilar to it previou counterpart, where the exitence of a unit root mean that y t, the value of the conidered proce at time t, can be expreed a a um of the current and pat value of a tationary proce and an initial value y. If a unit root were in the noncaual polynomial ' () the counterpart of thi would (preumably) be that y t hould be expreed a a um of the current and future value of a tationary proce. However, without truncation uch a um doe not converge and, therefore, cannot be ued to dene a proce for all t >. For purpoe of unit root teting one could truncate the um at the lat value of the conidered erie, y T ay, although uch an approach may not lend itelf a natural interpretation. A potential technical diculty i that conventional invariance principle are not directly applicable to the reulting proce and it function, uch a the component of the core and Heian of the log-likelihood function involving the unit root parameter, implying that the problem of teting for a unit root in the noncaual polynomial may lead to a rather involved aymptotic ditribution theory. In thi paper we therefore conne ourelve to the cae where a unit root appear in the caual autoregreive polynomial. 3 Parameter etimation 3.1 Approximate likelihood function To obtain our tet we rt dicu the likelihood function baed on the oberved time erie fy 1 ; :::; y T g generated by the AR(r; ) proce (1). Proceeding in the ame way a in Section 3.1 of L&S ugget approximating the log-likelihood function by l T () = g t () ; (8) 5

7 where g t () = log f 1 (u t (') u t 1 (') 1 u t 1 (') r 1 u t r+1 (')) ; log = log f 1 (v t (; ) ' 1 v t+1 (; ) ' v t+ (; )) ; log : Here u t (') and v t (; ) ignify the erie u t = ' B 1 y t and v t = (B) y t, repectively, treated a function of the parameter ' = (' 1 ; :::; ' ) and (; ) = (; 1 ; :::; r 1 ), and the parameter vector = (; ; '; ; ) ((r d) 1) contain the parameter of the model. Maximizing l T () over permiible value of give an (approximate) ML etimator of. In what follow, we drop the word \approximate" from the ML etimator and related quantitie. Above we aumed unrealitically that the order of the model, r and, are known. A in Breidt et al. (1991) and L&S we pecify thee order in practice a follow. Firt, we t a conventional caual AR model by LS and determine it order by uing conventional procedure uch a model election criteria and reidual diagnotic. We deem a caual model adequate when it reidual how no ign of autocorrelation. Due to the aforementioned identiability iue we alo need to check for the non-gauianity of the reidual becaue otherwie there i no point to conider noncaual model. If non-gauianity i upported by the data a non-gauian error ditribution i adopted and all caual and noncaual model of the elected order are etimated. Of thee model the one that maximize the likelihood function i elected and it adequacy i evaluated by conventional diagnotic tool. In practice a purely noncaual model (r = ; > ) may turn out to be the mot appropriate choice but, due to the aumption r >, it i not in accordance with the aumed formulation. If one want to perform a formal tet in a cae like thi one may augment the model with a rt-order caual polynomial and bae the tet on the AR(1; ) model. 3.2 Score vector and Heian matrix A our goal i to derive a Wald type tet for the unit root hypothei, we have to aume that the likelihood function atie conventional dierentiability condition imilar to thoe ued in the related previou work of Andrew et al. (26) and L&S. Thu, we impoe the following aumption. Aumption 2. For all (x; ) 2 (R; ), f (x; ) > and f (x; ) i twice continuouly dierentiable with repect to (x; ) and an even function of x, that i, f (x; ) = f ( x; ). Unlike the aforementioned previou author we require that the function f (; ) i even. A will be dicued in Section 4.1, thi aumption i impoed to implify the limiting ditribution of the obtained unit root tet. However, in Appendix B we derive the aymptotic ditribution of our unit root tet when thi aumption i relaxed. Thee derivation make evident that thi limiting ditribution i of no or only little practical ue. For cae where a kewed error ditribution i expected to be plauible, a boottrap procedure i uggeted to obtain an approximation to the aymptotic ditribution of our tet. An example of uch a boottrap procedure i outlined in Section 5.2. For the derivation of the Wald type tet we need to etimate the unretricted model and derive the limiting ditribution of the ML etimator of under the null hypothei. Becaue the data are aumed to be generated by a nontationary I(1) proce the derivation of the limiting 6

8 ditribution of the ML etimator involve feature dierent from thoe in the previou literature on tationary NCAR model. Moreover, a the etimation problem i nonlinear, the preence of an I(1) proce implie that method ued in the context of conventional unit root tet baed on linear LS etimation are not directly applicable. Therefore, we ue idea imilar to thoe developed for likelihood-baed tatitical model whoe etimation theory i nontandard in the ene that the information matrix i random even aymptotically. Such nonergodic model are dicued in Baawa and Scott (1983) and Jeganathan (1995) amongt other, and to facilitate their treatment we introduce the notation for the true value of and imilarly for it component. A the null hypothei i aumed to hold, the true value of i zero. We hall now derive weak limit of (appropriately tandardized verion of) the core vector and Heian matrix aociated with the log-likelihood function evaluated at the true parameter value. We ue a ubcript to ignify a partial derivative indicated by the ubcript; for intance g ;t () t () =@, f x (x; ) (x; ) =@x, and f (x; ) (x; ) =@. Denote V t+1 = (v t+1 ; :::; v t+ ) and U t 1 = (u t 1 ; :::; u t r+1 ) where v t are u t have the repreentation (5) and (6) with the coecient replaced by their true value ;j and ;j o that the latter, for example, i obtained from (z) 1 = P 1 j= ;jz j. The rt and econd partial derivative of g t (), the log-likelihood function baed on a ingle obervation, are preented in Appendix A. When evaluated at the true parameter value, the vector of rt partial derivative i g ;t ( ) 1 e 3 x;tu t 1 g ;t ( ) 1 e x;tu t 1 g ;t ( ) = g ';t ( ) = e x;tv t+1 4 g ;t ( ) (e x;t t + ) 5 g ;t ( ) e ;t where e x;t = f x 1 t; =f 1 t; and e;t = f 1 t; =f 1 t;. To obtain the weak limit of the core, we have to aume that the error denity f (x; ) ati- e regularity condition uch a thoe employed by Andrew et al. (26) and L&S. Rather than preenting the needed condition explicitly we implify the preentation by uing uitable \high level" aumption that can be veried by uing the regularity condition given in the aforementioned paper. To thi end, it i convenient to write = (; #) = (; # 1 ; # 2 ) where # 1 = (; ') and # 2 = (; ). The core of # (evaluated at ) i clearly a tationary and ergodic proce imilar to the core in L&S. We make the following aumption. Aumption 3. (i) E [e x;t ] = and E e 2 x;t = J, where J = R (fx (x; ) 2 =f (x; ))dx > 1 i nite. Moreover, Cov [ t ; e x;t ] =. (ii) The core vector g #;t ( ) = (g #1 ;t ( ) ; g #2 ;t ( )) ha zero expectation and nite poitive denite covariance matrix = diag( 1 ; 2 ) where i = Cov [g #i ;t ( )] (i = 1; 2) and the partition i conformable to that of g #;t ( ). Part (i) of thi aumption can be veried by uing the denition of e x;t, the regularity condition in Andrew et al. (26) and L&S, and direct calculation. Specically, the expreion of Cov [ t ; e x;t ] i obtained from the denition of e x;t and condition (A2) of thee paper, wherea condition (A5) implie that the inequality J > 1 hold if and only if the ditribution of t i non-gauian. Thi inequality and the explicit expreion of the matrice 1 and 2 obtained from L&S can further be 7

9 ued to verify the poitive denitene of the covariance matrix 1 in part (ii), wherea, due to the generality of the error ditribution, the poitive denitene of 2 ha to be aumed. The other condition in part (ii) can be veried by uing the regularity condition impoed on the denity function f (x; ) in the aforementioned paper. Aumption 3(i) and a tandard functional central limit theorem for IID equence yield [T u] X T 1=2 t=1 (e x;t ; t ) d! (B ex (u) ; B (u)) BM " J 2 #! ; (9) where the covariance matrix i poitive denite when t i non-gauian. Uing Aumption 1-3 we can further derive the limiting ditribution of the core vector of. The reult i preented in the following lemma. Lemma 1. Suppoe that Aumption 1-3 hold. Then, and T 1 g ;t ( ) d! Z 1 = T 1=2 1 (1) Z 1 B (u) db ex (u) (1) g #;t ( ) d! Z 2 N (; ) : (11) Moreover, joint weak convergence applie with Z 1 and Z 2 independent. The proof of thi lemma i preented in Appendix B. A dicued therein, the requirement that the function f (; ) i even i needed to etablih the independence tatement (further dicuion on thi iue will be given at the end of Section 4.1). Next conider the Heian matrix aociated with the log-likelihood function l T (). Expreion for the required econd partial derivative are obtained from Appendix A. Similarly to the rt partial derivative we ue notation uch a g ;t () 2 g t () =@@, f xx (x; ) 2 f (x; ) =@x 2, and f x (x; ) 2 f (x; ) =@x@. We alo dene and and make the following aumption. e xx;t = f xx 1 t; f 1 t; e 2 x;t e x;t = f x 1 t; f 1 t; f 1 t; f 1 e x;t ; t; Aumption 4. E [e xx;t ] = E e 2 x;t and E [g##;t ( )] = with given in Aumption 3(ii). Moreover, E [e xx;t t ] = and E [e x;t ] =. Similarly to Aumption 3, thi aumption can be veried by uing the regularity condition in Andrew et al. (26) and L&S. The rt moment equality i obtained from Aumption (A3) of thee paper, wherea the econd one tate that the negative of the Heian matrix of the log-likelihood function with repect to the hort-run parameter # equal the covariance matrix of the core of #, a fact that can be etablihed by direct calculation (ee L&S). A for the lat two 8

10 moment condition, both e xx;t t and e x;t are odd function of t o that, given Aumption 2, only nitene of the expectation i required. Thi in turn can be obtained from condition (A7) of Andrew et al. (26) and L&S. Now we can prove the following lemma. Lemma 2. Suppoe that Aumption 1-4 hold. Then, and T 2 g ;t ( ) d! J 2 (1) 2 T 1 T 3=2 Z 1 B 2 (u) d (u) def = g ( ) ; (12) g ##;t ( ) p! ; (13) g #;t ( ) p! : (14) Moreover, the weak convergence in (12) and in Lemma 1 hold jointly, and g ( ) and Z 2 are independent. Uing the limit obtained in Lemma 1 and 2 we dene Z = (Z 1 ; Z 2 ) and G ( ) = diag(g ( ) ; ), and we alo introduce the matrix D T = diag T; T 1=2 I r++d. The following propoition i an immediate conequence of Lemma 1 and 2. Propoition 1. Suppoe that Aumption 1-4 hold. Then, and S T ( ) def = D 1 G T ( ) def = D 1 T T g ;t ( )! d Z (15) g ;t ( ) D 1 T d! G ( ) ; (16) where the weak convergence in (15) and (16) hold jointly with (Z 1 ; G ( )) and Z 2 independent. In the next ection we derive the limiting ditribution of the ML etimator of the parameter by uing Propoition 1 and argument imilar to thoe ued by Baawa and Scott (1983, Ch. 2.4) in the context of tatitical model whoe likelihood ratio atify the LAMN condition. 3.3 Limiting ditribution of the ML etimator To obtain the limiting ditribution of the ML etimator of the parameter we have to upplement the aumption made o far by condition on the tandardized Heian matrix G T () def = D 1 P T T g ;t () D 1 T. A ucient \high level" condition, ued by Baawa and Scott (1983, pp ) in a more general form, require that, for all c >, where N T;c = f : D T k up kg T () G T ( )k! p ; (17) 2N T;c k cg. A dicued in Appendix C, thi condition can be veried by uing aumption imilar to thoe ued by Lii and Roenblatt (1996) in the context of (tationary) 9

11 noncaual and noninvertible ARMA model and by Meitz and Saikkonen (213) in the context of a (tationary) noninvertible ARMA model with conditionally heterokedatic error. Propoition 1 combined with condition (17) enable u to etablih the limiting ditribution of the ML etimator of under the unit root hypothei. Propoition 2. Suppoe that Aumption 1-4 and condition (17) hold. Then, with probability approaching one, there exit a equence of local maximizer of the log-likelihood function ^ T = (^ T ; ^# T ) uch that Moreover, G T (^ T ) G T ( ) p!. d! D T (^ T ); G T ( ) G ( ) 1 Z; G ( ) : Propoition 2 can be proved along the ame line a Theorem 1 and 2 of Baawa and Scott (1983, pp ). An outline of the needed argument i provided in Appendix B. Now all ingredient for the derivation of our unit root tet are available. 4 Tet procedure 4.1 Tet tatitic With Propoition 2 at hand it i traightforward to derive Wald type unit root tet. A we are intereted in one-ided (tationary) alternative we ue a \t-ratio" type tet tatitic dened a T def = q T ^ T ; G 1;1 T (^ T ) where G 1;1 T (^ T ) abbreviate the (1,1)-element of G T (^ T ) 1. The following propoition preent the aymptotic ditribution of T. Propoition 3. Suppoe that Aumption 1-4 and condition (17) hold. Then T d! J Z 1 1=2 Z 1 W 2 (u) d (u) W (u) dw (u) (J 1) 1=2 Z 1 def W (u) dw (u) = (J ) ; where W (u) = 1 B (u) BM (1), and W (u) BM (1) i independent of W (u). To ee how thi reult can be obtained, note that Propoition 2 and the continuou mapping theorem yield T d! J Z 1 1=2 Z 1 B 2 (u) d (u) B (u) db ex (u) : The tated reult i obtained by replacing the Brownian motion B ex (u) on the right hand ide by the expreion B ex (u) = 1 B (u) + (J 1) 1=2 W (u) = W (u) + (J 1) 1=2 W (u) ; obtained via a Choleky decompoition of the covariance matrix in (9). (18) 1

12 Propoition 3 implie that the limiting ditribution of tet tatitic T i free of nuiance parameter except for the parameter J. For ubequent analyi and dicuion we notice that for Student' t-ditributed error with > 2 degree of freedom ( + 1) J = ( 2) ( + 3) : (19) Of coure, the obtained limiting ditribution i of limited practical ue becaue it depend on the the nuiance parameter J. Fortunately, thi problem i rather eaily circumvented and i further dicued in Section 5.1. The ditribution of the limiting variable (J ) i a weighted average of a tandard normal ditribution and a Dickey-Fuller type of ditribution. More pecically, letting J! 1 in (18) a tandard normal ditribution i obtained, a Z 1 1=2 Z 1 lim (J ) = W 2 (u) d (u) W (u) dw (u) = N(; 1); J!1 where the econd equality hold true becaue R 1 W (u) dw (u) i a cale mixture of normal ditribution and can be written a R 1 R W 1 1=2 (u) dw (u) = W 2 (u) d (u). On the other hand, letting J! 1 in (18) the Dickey-Fuller type of ditribution i obtained, a Z 1 1=2 Z 1 lim (J ) = W 2 (u) d (u) W (u) dw (u) : J!1 That the limiting ditribution of T i relatively imple, depending only on the nuiance parameter J, i achieved by auming that the function f (; ) i even. Thi aumption i ued to etablih the independence of g ( ) and Z 2 in Lemma 2, and further the independence of (Z 1 ; G ( )) and Z 2 in Propoition 1, and it i alo ued to jutify the block diagonality of G ( ) (ee the proof of Lemma 2 for ome detail). If thee reult do not hold the limiting ditribution of T will be a coniderably more complicated function of the hort-run parameter of the model (ee Appendix B), making the implementation of the reulting tet very dicult. 4.2 Tet allowing for determinitic term The reult of Propoition 3 only applie to mean-zero data. To accommodate erie with trend component we conider the model x t = + t + y t ; t = 1; 2; :::; where x t i the oberved time erie and y t i a noncaual AR(r; ) proce. The trend coecient and are etimated by LS to obtain the etimate ^ and ^ after which the tet tatitic T introduced in the preceding ection i formed by uing y t = x t ^ in the cae of demeaned data and y t = x t ^ ^t in the cae of detrended data. A in other unit root tet, the ditribution of the reulting tet tatitic depend on the trend component choen, and therefore we denote the tet tatitic by T (m), where m =, m = 1, and m = 2 refer to mean-zero, demeaned, and detrended data, repectively. The reult of Propoition 3 applie even for T (1) and T (2) a long a the Brownian motion W (u) i replaced by correponding detrended Brownian motion (ee, e.g., Park and Phillip (1988)). 11

13 5 Simulation tudie 5.1 Etimated critical value The problem of the nuiance parameter J (2 (; 1)) appearing in the limiting ditribution of tet tatitic T (m) i addreed next. We hall rt illutrate how the value of the parameter J aect the ditribution of (J ) (ee (18)). It turn out to be convenient to tudy thi eect by uing the correlation between the two Brownian motion B (u) and B ex (u), that i, = J 1=2 2 (; 1) (ee (9)). The following gure diplay the 1% (dotted line), 5% (dahed line), and 1% (dahed-dotted line) percentile of the ditribution of (J ) a a function of. Figure 1 Percentile of the ditribution of (J ) a a function of = J 1=2 Note: 1t percentile (dotted line), 5th percentile (dahed line), and 1th percentile (dah-dotted line) for the aymptotic ditribution of the T (m) tatitic. The Brownian motion appearing in the limiting ditribution of tet tatitic T (m) are approximated uing (appropriately caled) um of normal IID(; 1) variable with T = 5; and 5; replication. In Figure 1 a monotonically decreaing relationhip between the percentile and i een. A already mentioned, the Dickey-Fuller ditribution and the tandard normal ditribution are obtained a limiting cae by letting J! 1 (! 1) and J! 1 (! ), repectively. Thu, in Figure 1 the 1%, 5%, and 1% critical value for the DF -tatitic and a tandard normal variate are found at the very left and very right, repectively. 12

14 Due to the monotonicity of the percentile in it i obviou that if the value of J were known Figure 1 could be ued to determine (conventional) critical value. Taking a more rigorou approach we proceed intead with curve etimation of the percentile by tting a econd-order polynomial cv ;m () = b + b 1 + b 2 2 for 2 f:1; :5; :1g and m 2 f; 1; 2g. The curve etimate, obtained by LS, yield the coecient in Table 1 that can be ued to compute aymptotic critical value. To exemplify how Table 1 can be ued, aume that we wih to tet the unit root hypothei in the NCAR model at a 1% ignicance level in the cae of demeaned data with J = 2 ( = 1= p 2). Then, the etimated aymptotic critical value equal cv :1;1 (2) = 1:276 1:584 (1= p 2) + :289 (1= p 2) 2 = 2:252. To thi end, the value of J i in practice obviouly not known and mut be etimated. In the cae of Student' t-ditributed error we can ue equation (19) with the etimator ^ ued in place of. More generally, in cae where the ditribution of the error term comprie le traightforward calculation of J we may, by virtue of Aumption 3(i), ue the etimator bj = 1 T r " f x (^ 1^ t ; ^) # 2 f(^ 1^ t ; ^) ; (2) where ^ t = ^u t ^^ut 1 ^ 1 ^u t 1 ^ r 1 ^u t r+1 with ^u t = ^'(B 1 )y t. Table 1 Coecient to compute aymptotic critical value cv ;m () of tet tatitic T (m) Cae Signicance level () b b 1 b 2 R 2 mean-zero data 1% 2:321 :492 :251 :998 (m = ) 5% 1:639 :495 :187 :999 1% 1:276 :48 :131 :999 demeaned data 1% 2:322 1:578 :474 1: (m = 1) 5% 1:639 1:591 :367 1: 1% 1:276 1:584 :289 1: detrended data 1% 2:324 2:21 :575 1: (m = 2) 5% 1:64 2:23 :462 1: 1% 1:276 2:231 :381 1: Note: For each ignicance level and each trend pecication the coecient b, b 1, and b 2 are obtained from the regreion of cv ;m() on (1; ; 2 ) (uing LS). R 2 i the regreion coecient of determination. 13

15 5.2 Boottrapped p-value and critical value A already mentioned, if the ymmetry condition in Aumption 2 i relaxed the limiting ditribution of our unit root tet depend on everal nuiance parameter in a very complex way (for detail, ee Appendix B). In thi ection, we dicu a boottrap procedure that can be ued to obtain approximation to the critical value and p-value of our tet that do not rely on the ymmetry condition of Aumption 2. Our approach cloely follow the boottrap procedure decribed in Caner and Hanen (21). tep: The boottrap ditribution of tet tatitic T (= T (1)) i obtained by the following imple (i) Ue the oberved time erie fy 1 ; :::; y T g and the aumed ditribution for the error term t to compute ^ = (^; ^; ^'; ^; ^), the unretricted ML etimate of, and furthermore the value of the unit root tet tatitic T. (ii) Generate T b random draw f b 1 ; b 2 ; :::; b g from the etimated error ditribution with denity T b ^ 1 f ^ 1 t ; ^, and inert thee draw and the etimate (; ^; ^') into the NCAR pecication (1) to yield ^ (B) ^' B 1 y b t = b t; t = 1; 2; :::; T b ; (21) where ^ (B) = ^ 1 B ^ r 1 B r 1 = 1 ^1 B ^r B r, and the lat equation dene the coecient ^ 1 ; :::; ^ r, and ^' B 1 = 1 ^' 1 B 1 ^' B. The reaon for dening the lag polynomial ^ (B) in thi way i to enure that the boottrap ample obey the null hypothei of a unit root. A boottrap ample fy b 1 ; yb 2 ; :::; yb T b g i obtained via equation (21) by generating rt the \noncaual" part a v b t = ^' 1 v b t ^' v b t+ + b t; t = T b ; T b 1; :::; 1; where v b T b +1 = = vb =, and thereafter the \caual" part a T b + where y b r+1 = = yb =. y b t = ^ 1 y b t ^ r y b t r + v b t ; t = 1; 2; :::; T b ; (iii) Ue the boottrap ample fy b 1 ; yb 2 ; :::; yb T b g to compute the value of our unit root tet tatitic denoted by b T b. (iv) Repeat the reampling cheme in (ii) and (iii) BR time to yield the boottrap ditribution of the tet tatitic T, from which, e.g., approximate boottrap p-value can be computed a the average number of time b T b i maller than T. In practice the number of boottrap replication BR i et relatively large in order to get reaonable approximation. The number of boottrap draw T b may be et equal to the (eective) ample ize. However, to eliminate the eect of the terminal and tarting value one may generate 2 extra obervation (ay) and dicard the rt and lat 1 obervation at the end and beginning of each realization. The propertie of thi boottrap procedure in the cae of ymmetric and kewed error are examined in the next ection. 14

16 5.3 Empirical ize and power imulation In thi ection, we examine nite ample propertie of the T (m)-tet for m 2 f; 1; 2g by mean of imulation experiment. The nominal ignicance level employed i 5%, and the benchmark proce i a noncaual autoregreive proce a dened in (1) with r = = 1, and with the independent and identically ditributed error term t having Student' t-ditribution with degree of freedom equal to 3 and tandard deviation equal to :1. Realization fy 1 ; :::; y T g from thi proce are generated a decribed in tep (ii) of the boottrap cheme (ee the preceding ection). To eliminate eect of the terminal and tarting value, 1 obervation at the end and beginning of each realization are dicarded. Finally, in all experiment the true order of the proce i aumed known (i.e. r = = 1), and the etimation of the parameter ^ = (^; ^' 1 ; ^; ^) i carried out in GAUSS 12 uing the BHHH algorithm in the CML library. In the rt experiment the empirical ize of the T (m)-tet i examined in the cae of Student' t-ditributed error when the parameter ' 1 i varied and etimated (aymptotic) critical value baed on dierent etimate of J are ued. The parameter value and ample ize conidered are 1 = 1 ( = ), ' 1 2 f:1; :5; :9g and T 2 f1; 25g, repectively. Moreover, all the reult in thi experiment are baed on 1; realization of the fy 1 ; :::; y T g proce, and for each realization 5% critical value are obtained by the econd-order polynomial in Table 2 uing (19) with ^ ( J b 1 ), the etimate in (2) ( J b 2 ), and J = 2 (the true value) a etimate. The outcome of thi experiment are reported in Table 2a. Table 2a Empirical ize of the T (m)-tet in the cae of a ymmetric error ditribution mean-zero data demeaned data detrended data Sample (m = ) (m = 1) (m = 2) Size ' 1 ' 1 ' 1 T :1 :5 :9 :1 :5 :9 :1 :5 :9 J = 2 :53 :53 :83 :59 :59 :86 :58 :56 :98 1 J1 b 2:127 :52 :52 :83 :54 :54 :8 :56 :59 :97 bj 2 2:122 :52 :52 :83 :54 :53 :8 :55 :59 :97 J = 2 :54 :47 :45 :63 :61 :55 :57 :52 :56 25 J1 b 2:183 :53 :46 :44 :58 :58 :54 :56 :52 :57 bj 2 2:181 :53 :46 :44 :58 :58 :54 :56 :52 :57 Note: The reult are baed on 1, replication, and the nominal ize of the tet i 5%. Reported etimated value for J are baed on the average value over the number of replication for each ample ize in the cae of demeaned data with ' 1 = :5. 15

17 In Table 2a, the reported etimate for J are (for each ample ize) baed on the average number of replication in the cae of demeaned data with ' 1 = :5. It i een that thee etimate are cloe to the true value even for moderate ample ize. For the other cae the etimate of J are imilar and therefore omitted. It i further noticed that the empirical ize i cloe to the nominal ize for mot of the cae conidered, and the inuence of the parameter ' 1 appear to be modet. One exception, though, i for T = 1 and ' 1 = :9, where the tet i omewhat over-ized o that ome cautioune i required. Taken the reult in Table 2a together, it appear that the aymptotic ditribution of the T (m)-tet, alo with the value of J etimated, yield reaonable approximation to the nite ample ditribution even for relatively mall ample ize, variou trend component, and a wide range of parameter value for ' 1. In the econd experiment the empirical ize of the T (m)-tet i examined in the cae where the error term ha a kewed Student' t-ditribution but the regular Student' t-ditribution i (incorrectly) aumed in the tet. In thi and the ubequent experiment critical value are baed on the etimate b J 2 in (2). The kewed t-ditribution employed i the one of Azzalini and Genton (28) which, in addition to the parameter and, alo include a kewne parameter. 1 The kewne parameter i aumed to take on the value = (ymmetric error), = :66 (error with kewne 1:33), and = 2 (error with kewne 3). The etup for thi experiment i the ame a in the rt experiment except that, to conerve pace, the reult with zero mean data are excluded (thee reult are available upon requet from the author). A before we let = :1, but chooe = 4 to make the conventional kewne meaure well dened. Finally, we alo report the empirical ize of the boottrap verion of our tet baed on the correctly pecied kewed error ditribution. The boottrap verion of our tet, denoted by b T (m), i baed on 5 boottrap replication and on 1; Monte Carlo replication. The outcome of thi experiment are reported in Table 2b. 1 The denity of the kewed t-ditribution of Azzalini and Genton (28) parameterized to have mean zero and variance 2 take the form: f z t; m ( ; ) 1 ( ; ) ; 1 ( ; ) ; ; = 2 ( ; ) 1 t (z t; ) T q ( + 1) = + 2 ; + 1 ; where m ( ; ) = (1 + 2 ) 1=2 (=) 1=2 (( 1)=2)= (=2), 2 ( ; ) = (=2) (( 2)=2)= (=2)-m 2 ( ; ), and z t = ( ; ) 1 t + m ( ; ) 1 ( ; ). Furthermore, t and T denote the Student t denity and ditribution function, repectively. 16

18 Table 2b Empirical ize of the T (m)-tet and it boottrap verion in the cae of kewed error demeaned data (m = 1) Sample = = :66 = 2 Size Tet ' 1 ' 1 ' 1 T :1 :5 :9 :1 :5 :9 :1 :5 :9 1 T (1) :51 :53 :75 :52 :57 :79 :47 :56 :19 b T (1) :5 :48 :51 :53 :47 :51 :51 :55 :53 25 T (1) :54 :46 :46 :59 :43 :5 :56 :47 :6 b T (1) :51 :53 :49 :48 :49 :52 :49 :49 :54 detrended data (m = 2) Sample = = :66 = 2 Size Tet ' 1 ' 1 ' 1 T :1 :5 :9 :1 :5 :9 :1 :5 :9 1 T (2) :45 :53 :87 :49 :56 :71 :42 :51 :75 b T (2) :47 :49 :53 :52 :51 :5 :46 :48 :51 25 T (2) :52 :51 :56 :48 :5 :48 :51 :49 :59 b T (2) :45 :53 :5 :47 :53 :49 :54 :55 :51 Note: b T ignie the boottrap verion of our tet in the cae of kewed t-ditributed error. All reult are baed on 1, replication, and the number of boottrap replication for the b T tet i 5. Nominal ize of the tet are 5%. Etimated critical value for the T -tet are baed on the etimate J b 2 in (2). The reult in Table 2b indicate that, except for the cae T = 1 and ' 1 = :9, the T (m) tet i not very enitive to violation of the ymmetry condition in Aumption 2. From Table 2b it i alo een that the performance of the b T (m) tet i very atifactory for all ample ize and all value of conidered. Thu, one could conider uing it alway in combination with a ditribution allowing for kewed error. However, limited imulation experiment (reult available upon requet from the author) indicate that in the cae of ymmetric error thi lead to a light lo of power compared to uing the T tet that aume ymmetric error. In our third Monte Carlo experiment the power of the T (m)-tet i examined. The data are generated a decribed in our rt experiment above with ' 1 = :5 and 1 2 [:6; 1:] ( 2 [ :4; ]). The ample ize conidered are T 2 f1; 25g. For comparion we we alo report the outcome of the conventional Dickey-Fuller unit root t-tet baed on an AR(2) proce, the t-type unit root tet of Luca (1995) baed M-etimation in an AR(1) model and an aumption of trictly tationary trong-mixing error, and the unit root tet of Rothenberg and Stock (1997) baed on the ML etimation of an AR(2) model and an aumption of Student' t-ditributed error. Thee tet are denoted by DF (m), M(m), and RS(m), repectively. 2 The DF (m)-tet i a natural alternative to our tet in that it i widely ued among practitioner, and it ha alo been hown to be rather robut 2 Following Luca (1995) we ue the Huber -function (x) = min fc; max( c; x)g with c = 1:345 to obtain the 17

19 againt variou mipecication. The M(m)-tet can alo be viewed a a natural alternative, for it i deigned to be robut againt innovation outlier (fat-tailed ditribution). Finally, the RS(m) i a natural alternative in the ene that it explicitly aume nonnormal error. The reult of thi experiment are ummarized in Figure 2. Figure 2 Empirical power of the tet T (m), DF (m), M(m) and RS(m) T = 1 T = 25 Note: T (m)-tet olid line, DF (m)-tet dotted line, M(m)-tet dahed-dotted line, RS(m)-tet dahed line, and nominal ize hort-dahed line. The reult are baed on 1, replication and the nominal ize of the tet i 5%. Etimated critical value for the T -tet are baed on the etimate b J 2 in (2). M-etimator. Furthermore, to operationalize the M(m)-tet, nuiance i parameter are etimated by the Newey-Wet etimator with the lag-truncation parameter et at h4(t=1) 2=9. Finally, in the computation of the M-etimator a cale free verion i ued (ee Luca, 1995, p. 337), and an iterative weighted LS algorithm (imilar to the one decribed in Van Dijk, Frane, and Luca, 1999, p. 219) i applied. 18

20 Figure 2 how that, in general, the T (m)-tet i more powerful than the three alternative conidered, and in ome cae it uperiority i quite ubtantial. For intance, in the cae of detrended data with T = 25 and 1 = :95, the dierence in power between the T (2)-tet and the DF (2), M(2), and RS(2)-tet are (approximately) a large a :4, :25, and :15 unit, repectively. The good performance of the T (m)-tet i of coure not urpriing becaue, unlike the other tet conidered, the T (m)-tet i baed on the correctly pecied NCAR model. In practice it application require chooing two order, r and, a well a pecifying the error ditribution, which involve preteting, not taken into account, in our power imulation. Thi hould be kept in mind when one compare the power of the T (m)-tet to the conidered alternative, epecially to the Dickey-Fuller tet whoe application only require chooing one AR order. We alo examined the power of the boottrap verion of our tet with the error having both ymmetric and kewed t-ditribution. Reult of thee experiment are available upon requet from the author. Here we only note that, overall, the reult were imilar to thoe obtained in the ymmetric cae in Figure 2. 6 Empirical application In thi ection, we provide an empirical illutration of our tet by analyzing a Finnih interet rate erie (Government bond). Thee data range from 1988:Q1 to 212:Q4 (quarterly obervation) and yield a ample ize of 1 obervation. The erie, obtained from IMF' International Financial Statitic, i hown in Figure 3. For interet rate erie (in general) it i mot natural to ue demeaned data. But, a the Finnih interet rate erie i trending in the ample we will alo conider the cae of detrended data. A a rt tep in our analyi we t an AR(p) model to the data by LS and thereafter check if the reidual erie appear non-gauian. For the cae of demeaned data both AIC and BIC elect an AR(3) model, wherea for the cae of detrended data an AR(2) model i elected by both AIC and Figure 3 Finnih Government bond BIC (the maximum lag conidered wa p max = 4(T=1) 2=9 = 4). Even though the null hypothei of no 4th-order remaining erial correlation i not rejected by the Ljung-Box (LB) tet for the two 19

21 reidual erie (p-value: :492 and :811 for demeaned and detrended reidual erie, repectively), we nd that the normality aumption i trongly rejected by the Lomnicki, Jarque, and Bera (LJB) tet (p-value: < :1 and < :1 for demeaned and detrended reidual erie, repectively), and ome evidence of 4th-order ARCH eect are alo found by the McLeod-Li (McL) tet (p-value: :89 and :25 for demeaned and detrended reidual erie, repectively). 3 In addition, quantilequantile plot of the reidual of the AR(3) and AR(2) model (not hown here) indicate that a normal ditribution i not appropriate becaue exce kurtoi in the data i left unexplained. Taking thee reult together it eem worthwhile to proceed with etimation and unit root teting of NCAR pecication, and to capture the leptokurtic behavior of the reidual erie we will adopt t-ditributed error. More pecically, for demeaned and detrended data we conider an AR(r; ) model with r + = p = 3 and r + = p = 2, repectively, and conduct unit root teting for the AR(1; 2) and AR(2; 1) pecication in the former cae and for the AR(1; 1) pecication in the latter cae. A will be dicued below, thee noncaual model are upported by the pecication trategy dicued in Section 3.1. For comparion we alo employ the DF -tet, the M-tet, and RS(m)-tet baed on AR(3) and AR(2) model in the cae of demeaned and detrended data, repectively. Finally, for the aforementioned NCAR pecication we will alo report the outcome of our unit root tet when the error are aumed to have the kewed verion of Student' t- ditribution dicued in the preceding ection. The outcome of thee unit root tet a well a variou etimation reult and LB, LJB, and McL mipecication tet are reported in Table 3 and 4 below. 4 3 The kewne part of the LJB-tet i ignicant at 7.6% and 8.8 % level for demeaned and detrended data, repectively, indicating that the rejection of Gauian error mainly tem from the kurtoi part of the LJB-tet. 4 A dicued in L&S (ee p. 12), we ue leat abolute deviation etimator to nd tarting value for, and ' ( ~, ~ and ~', ay), and thereafter maximize l T ( ~ ; ~; ~'; ; ) to alo nd tarting value for and. 2

22 Table 3 Unit root teting for demeaned and detrended Finnih interet rate erie T = 97 m = 1 Model Tet Outcome cv :5;1 cv :5;1 (^ 1 ) cv :5;1 (^ 2 ) AR(3) DF 1:41 2:86 AR(1) M :38 3:6 AR(3)-t RS :213 2:75 AR(2; 1)-t T 5:11 2:558 2:577 AR(1; 2)-t T 5:997 2:542 2:531 AR(2; 1)-St T 4:811(< :1) AR(1; 2)-St T 5:845(< :1) T = 98 m = 2 Model Tet Outcome cv :5;2 cv :5;2 (^ 1 ) cv :5;2 (^ 2 ) AR(2) DF 3:213 3:41 AR(1) M 1:435 3:66 AR(2)-t RS 1:62 3:28 AR(1; 1)-t T 5:379 3:72 3:48 AR(1; 1)-St T 6:482(< :1) Note: T i the eective ample ize. AR(r; ) abbreviate an autoregreive model with rth and th-order polynomial (B) and '(B 1 ), repectively. N, t, and St refer to Gauian, t-ditributed, and kewed t-ditributed error, repectively. and denote ignicance at the 1 and the 1 percent level, repectively. cv :5;m (m = 1; 2) i the 5% critical value for the DF tet, the M-tet, and the RS-tet, and cv :5;m(^ 1 ) and cv :5;m(^ 2 ) (m = 1; 2) are the 5% etimated critical value obtained by letting ^ 1 = J b 1=2 1 and ^ 2 = J b 1=2 2 ( J b 1 and J b 2 are etimator of J uing (19) and (2), repectively). In the cae of kewed error, boottrap p-value (uing 5 boottrap replication) are reported in parenthee for the T tet. 21

23 Table 4 NCAR etimation reult for demeaned and detrended Finnih interet rate erie m = 1 AR(r; ) pecication T = 97 AR(3; )-N AR(3; )-t AR(2; 1)-t AR(1; 2)-t AR(; 3)-t AR(3; )-St AR(2; 1)-St AR(1; 2)-St AR(; 3)-St :15 (:13) 1 :568 (:99) 2 :18 (:99) :3 (:13) :59 (:91) :84 (:14) :472 (:93) :56 (:91) ' 1 :941 (:25) :425 (:71) :873 (:68) ' 2 :56 (:67) ' 3 :447 (:17) :457 (:85) 3:585 (1:356) :477 (:126) 3:17 (:987) :498 (:155) 2:831 (:868) LL 59:556 49:111 46:52 45:637 53:437 49:32 45:975 45:564 53:1 LB(4) :492 :324 :674 :348 :522 :317 :712 :38 :432 McL(4) :89 :233 :256 :36 < :1 :228 :26 :3 < :1 LJB < :1 1:55 (:118) :695 (:16) :13 (:93) :461 (:69) 4:129 (1:731) :1 (:14) :58 (:99) :82 (:11) :473 (:111) 3:381 (1:411) :219 (:513) :463 (:96) :58 (:88) :938 (:27) :462 (:11) 3:165 (1:83) :198 (:516) :417 (:71) :874 (:66) :52 (:67) :478 (:132) 2:964 (:971) :18 (:482) 1:576 (:16) :78 (:142) :124 (:85) :486 (:95) 3:737 (1:594) :534 (:53) 22