Order Determination for Multivariate Autoregressive Processes Using Resampling Methods

Transcription

1 joural of mulivariae aalysis 57, (1996) aricle o Order Deermiaio for Mulivariae Auoregressive Processes Usig Resamplig Mehods Chaghua Che ad Richard A. Davis* Colorado Sae Uiversiy ad Peer J. Brockwell* Royal Melboure Isiue of Techology, Melboure, Ausralia Le X 1,..., X be observaios from a mulivariae AR( p) model wih ukow order p. A resamplig procedure is proposed for esimaig he order p. The classical crieria, such as AIC ad BIC, esimae he order p as he miimizer of he fucio $(k)=l ( 7 k )+C k 2 where is he sample size, k is he order of he fied model, 7 k is a esimae of he whie oise covariace marix, ad C is a sequece of specified cosas (for AIC, C =2m 2, for Haa ad Qui's modificaio of BIC, C = 2m 2 (l l ), where m is he dimesio of he daa vecor). A resamplig scheme is proposed o esimae a improved pealy facor C. Codiioal o he daa, his procedure produces a cosise esimae of p. Simulaio resuls suppor he effeciveess of his procedure whe compared wih some of he radiioal order selecio crieria. Commes are also made o he use of YuleWalker as opposed o codiioal leas squares esimaios for order selecio Academic Press, Ic. 1. Iroducio I his paper, we exed he resuls of Che e al. (1993) for AR model selecio usig resamplig mehods o he mulivariae problem. Le Received Jue 23, 1993; revised July AMS 1980 subjec classificaio: 62M10. Key words ad phrases: mulivariae auoregressive processes, order deermiaio, AIC, YuleWalker esimaio, resamplig. * Research suppored by NSF Gras DMS ad X Copyrigh 1996 by Academic Press, Ic. All righs of reproducio i ay form reserved.

2 176 CHEN, DAVIS, AND BROCKWELL [X 1,..., X ] be cosecuive observaios from he m-variae AR( p) model, X &8 1 X &1 &}}}&8 p X &p =Z (1.1) where 8 1,..., 8 p are real m_m marices, de(i m &8 1 z&}}}&8 p z p ){0 for z 1, ad [Z ]IID(0, 7) (i.e. [Z ] is a idepede ad ideically disribued sequece of radom m-vecors wih mea vecor 0 ad covariace marix 7) wih 7 osigular. Mos order selecio crieria esimae he order of he model by miimizig a objecive fucio of he form: $(k)=l( 7 k )+C k, (1.2) where C is a sequece of prespecified cosas ad 7 k is a esimae of 7 based o fiig a m-dimesioal AR(k) model o he daa ( A deoes he deermia of he marix A). The esimaed order, p^, is he defied as he miimizer of $(k) over some suiable rage of values. Typical values for he pealy facor are: C(, k)= 2km2, AIC (Akaike (1973)) C(, k)= 2(km2 +m(m+1)2), AIC &(km+m+1) C (Hurvich ad Tsai (1989), (1991)) C(, k)= 2c(log log ) km2, (Haa ad Qui (1979)). As i he uivariae case, Haa ad Qui's pealy facor produces a srogly cosise esimae of he rue order p (see Haa ad Deisler, 1988). If he rue order of he model were kow o be p, he we show (see Proposiio 2.2) ha here is a rage of values of C for which $(k) has is miimum a k=p. We call such pealy facors correc. I his paper we use a resamplig scheme o approximae he rage of correc pealy facors. The appeal of his mehod is ha i produces a rage of pealy facors which are correc for he simulaed daa. These es realizaios are geeraed from cadidae AR modelsmodels which have bee fied o he origial daa. Sice hese pealy fucios are correc for he simulaed daa from he cadidae models ad sice he origial model is believed o belog o he class of cadidae models, i is reasoable o suppose he pealy facors will be early correc for he origial daa. This is bore ou i Secio 2 where we show ha his procedure produces a weakly cosise esimae of he rue order of he model ad i Secio 3 where we

3 MULTIVARIATE AUTOREGRESSIVE PROCESSES 177 show ha perform exremely well i a simulaio sudy. I Secio 4, we discuss he effec of usig YuleWalker as opposed o codiioal leas squares esimaes for order selecio. 2. The Use of Resamplig o Choose a Pealy Facor I his secio, we develop he ecessary heory o esablish weak cosisecy of a resamplig procedure for model selecio. May of he resuls i his secio parallel hose for he uivariae case give i Secio 2 of Che e al. (1993), however, he argumes used o derive hem are subsaially more difficul i wo impora respecs. The firs is he eed for he sric moooiciy propery of he deermia of he predicio error covariace marices esablished i Proposiio 2.1. The secod is he delicae proof of he covergece of he boosrapped mulivariae Yule-Walker esimae codiioal o he daa. Le [X, =0, \1,...] be he m-variae AR( p) process defied by (1.1). The esimaed order, p^, based o observaios X 1,..., X, is he miimizer of $(k) where $(k)=l( 7 k )+C k, (2.1) C is a sequece of as ye uspecified cosas ad 7 k is he YuleWalker esimae of 7 based o fiig a m-dimesioal AR(k) model o he daa. For k=1, 2,..., le X k+1=8 k1 X k +}}}+8 kk X 1 deoe he bes liear predicor of X k+1 i erms of X k,..., X 1. The coefficie marices 8 k1,..., 8 kk ad he predicio error covariace marix 7 k =E(X k+1 &X k+1) (X k+1 &X k+1)$ are give by he YuleWalker equaios, 1( j)=8 k1 1(j&1)+ } }} +8 kk 1( j&k), j=1,..., k, k 7 k =1(0)& : i=1 8 ki 1(k&i), (2.2) where 1(h) :=E[X h X$ 0 ] is he marix covariace fucio. Sice 8 p { 0 m_m i (1.1) (he rue order is p), we have for kp 8 ki = {8 i, 0 m_m, ip, i>p, (2.3) ad 7 k =7, (2.4)

4 178 CHEN, DAVIS, AND BROCKWELL ad for k<p 7 k 7, (2.5) where AB meas he marix A&B is oegaive defiie ad 0 m_m is he m_m zero marix. The YuleWalker esimaes 8 k1,..., 8 kk ad 7 k are obaied by replacig 1(h) by he sample marix covariace fucio i equaios (2.2). &h 1 (h)={1 : X j+h X$ j, if h0, j=1 1 $(&h), if h<0, Proposiio 2.1. For a m-variae AR( p) process we have 7 = 7 p < 7 j for j=0,..., p&1. Proof. I follows from he defiiio of X k+1 ad 7 k ha 7 k 7 k+1 ad hece 7 k 7 k+1. So i suffices o show 7 p < 7 p&1. From he calculaios o p. 422 of Brockwell ad Davis (1991), i follows easily ha 7 p =7 p&1 &8 pp 7 p&18$ pp where 7 p&1 is he oe-sep error-covariace marix of he bes liear predicor of X 0 i erms of X 1,..., X p&1. Wriig 7 p&1 =AA$, where A is osigular, we have from he above relaio ha 0< 7 p 7 &1 p&1 = I&A$&1 8 pp 7 p&18$ pp A &1 m = ` i=1 (1&* i ) (2.6) where * 1,..., * m are he eigevalues of A$ &1 8 pp 7 p&18$ pp A &1. Sice 8 pp = 8 p {0 ad 7 p&1 is posiive defiie, his marix mus have a leas oe ozero eigevalue. Cosequely, he boud i (2.6) is sricly less ha oe. This complees he proof of he proposiio. K

5 MULTIVARIATE AUTOREGRESSIVE PROCESSES 179 The followig proposiio deermies he rage of correc pealy facors (i.e., pealy facors such ha he miimizer of $( j, C )isp) whe he rue order of he model is kow. Proposiio 2.2. Le X 1,..., X be observaios from he AR( p) model (1.1) ad le Kp be a fixed ieger. Defie, for p=k, : ={0, max p+1lk{ l( 7 p )&l( 7 l ), for p<k, l&p = for p=0, ; ={+, mi 0lp&1{ l( 7 l )&l( 7 p ), for p>0, p&l = (2.7) (2.8) where 7 0=1 (0). The as, (a) (b) : 0 ad ; b>0 a.s. Pu If : ;, he for ay C #(:,; ), $(k, C)=l( 7 (k) )+kc. $(p, C )= mi [$(l, C )]. (2.9) 0lK I oher words, wih his choice of pealy facor, he order of model (1.1) will be correcly esimaed by he miimizer of (2.1). Proof. (a) From Theorem of Haa ad Deisler (1988), he YuleWalker esimaes are srogly cosise. Thus by (2.3)(2.6), we have, for p<k : w a.s. ad from Proposiio 2.1, max p< jk l( 7 p )&l( 7 j ) =0 j&p ; w a.s. mi 0j<p l( 7 j )&l( 7 p ) =b>0, p&j p>0. The proof for he cases p=k ad p=0 are immediae.

6 180 CHEN, DAVIS, AND BROCKWELL (b) whece Sice C :, we have for j>p, l( 7 p )&l( 7 j )+C (p&j) <l( 7 p )&l( 7 j )+ l( 7 p )&l( 7 j ) j&p =0, (p&j) $( j, C )=l( 7 ( j) )+jc >l( 7 ( p) )+pc =$( p, C ). (2.10) O he oher had, sice C ;, we have for j<p $( p, C )&$( j, C )=l( 7 p )&l( 7 j )+C (p&j)<0. (2.11) Combiig (2.10) a (2.11), we ge (2.9). Whe he rue order p of he model is kow, he rage of pealy facors C which leads o he correc model ideificaio usig (2.1) is oempy, a leas for large. Whe p is ukow, we use a resamplig procedure o geerae es sequeces from a AR(k) model for k=0,..., K 1, from which he above proposiio may be applied o compue a ierval of suiable values for C for each k. The iersecio of hese K 1 iervals he gives us a rage from which o choose a pealy facor o be used i (2.1) applied o he origial daa. The fac ha he iersecio of hese K 1 iervals is asympoically oempy is he coe of he followig proposiio. Proposiio 2.3. I addiio o he assumpios i Proposiio 2.2, assume he oise vecor has fiie fourh momes. Defie he residual sequece K Z =X &8 K1X &1 &}}}&8 KKX &K for =K+1,...,. For ay fixed ieger K 1 (0K 1 p), le [Y (k) 1,..., Y(k) ] be observaios from he AR(k) model Y (k) &8 k1y (k) &}}}&8 &1 kky (k) =Z &k *, =k+1,..., ; k=0, 1,..., K 1, where [Z *] is a iid sequece geeraed from he empirical disribuio fucio (correced o have mea zero) based o [Z ]. (For k=0, Y (0) = Z *). For each k=0, 1,..., K 1, le I (k) =[:(k), ;(k) ] deoe he ierval

7 MULTIVARIATE AUTOREGRESSIVE PROCESSES 181 obaied whe Proposiio 2.2 is applied o he series [Y (k) ] wih p=k. The for almos all sample sequeces of [X ], we have (a) (b) : :=max 0kK1 [: (k) ; :=mi 0kK1 [; (k) ] w P ] w P 0, b 0, where w P deoes covergece i probabiliy codiioal o X 1,..., X. I paricular, I = K 1 k=0 I(k) (=[:, ; ] if : ; ) coverges i codiioal probabiliy o a oempy se. Proof. Sice he argume is early ideical o he proof give for Proposiio 2.2 i Che, e al. (1993), we oly give a skech of he argume. I suffices o show ha : (k) w P 0 ad ; (k) w P b k 0. (2.12) Le 1 *(h) :=(1) j=1 Y(k) +h Y(k)$ deoe he sample covariace marix fucio of he Y (k) 's. We begi by showig ha he codiioal covariace marix of 1*(h), deoed by Cov (1 *(h)), covergeces o 0, a.s. By Remark 2 o p. 424 of Brockwell ad Davis (1991), he YuleWalker esimaes produce a causal model so ha [Y (k) ] has he represeaio = : j=0 Y (k) 9 jz* &j where he 9 j's are he coefficie marices i he power series expasio of he marix fucio (1&8 k1z& }}}&8^ kkz k ) &1 o z 1. The codiioal mea of 1 *(h) is give by 1*(h):=E (Y (k) +h Y(k)$ )= : j=0 9 j+h7*9 $ j (2.13) where E ( } ) deoes expecaio relaive o P ad 7* is he sample covariace marix of [Z^ ]. The srog cosisecy of he YuleWalker esimaes implies ha for almos all sample pahs, 8 k 8 k ad 7 k 7 k. I follows ha 9 j 9 j as (9 j are he coefficie marices i he expasio of he marix fucio (1&8 k1 z&}}}&8 kk z k ) &1 ). Now for l=0, 1,..., wrie 9 l=[9 l(i, j)] m i, j=1. I is easy o see ha here exis cosas C>0 ad {<1, depedig o he sample pah, such ha 9 l(i, j) Cr l (2.14)

8 182 CHEN, DAVIS, AND BROCKWELL for l=0, 1,... ad large. Similarly, oe ca show ha 7* 7 k which combied wih (2.13) ad (2.14) yields 1*(h) : j=0 9 j+h 7 k 9$ j (2.15) as. Usig hese resuls ogeher wih he relaios esablished i he proof of Theorem 6, p. 210 of (Haa (1970)), we have lim sup Cov (1 *(h))< a.s. This implies Cov (1 *(h))0 a.s. which ogeher wih (2.15) yields 1 *(h) w P : j =0 9 j+h 7 k 9$ j. The weak cosisecy of he sample ac implies ha he YuleWalker esimaes, i fiig a AR( j) model o he daa [Y (k) ], are also weakly cosise relaive o P. The limis i (2.12) ow follow usig he argume give for Proposiio 2.1. K Theorem 2.4. Uder he assumpios of Proposiio 2.3, le I =[:, ; ] ad suppose 8 jj {0 for j=1,..., K 1. Defie c; l() +, C ={: c; l, if : <;, oherwise, where c>0 is ay cosa such ha C # I. If p^ is he miimizer of $(k, C ) for 0kK, he for almos all sample sequeces of [X ], as. p^ w P p (2.17) Proof. Sice for almos all sample pahs, C w P 0 ad C log log w P,

9 MULTIVARIATE AUTOREGRESSIVE PROCESSES 183 he heorem follows a oce from Theorem of Haa ad Deisler (1988). K Remark 1. I order o impleme our resamplig procedure for deermiig C usig Theorem 2.4, we mus firs specify K 1 p such ha 8 jj {0 for j=1,..., K 1. Ideally, we would like o opimize over as large a class of es models as possible i order o esure ha a model close o he rue model is icluded i our es se. This requires a large K 1. If our iiial choice for K 1 happes o be bigger ha p, he i is likely ha he se I will be empy. I his case, he value of K 1 is reduced i seps of size 1 uil a oempy I is achieved. Remark 2. While we have assumed i Proposiio 2.3 ha he es series [Y (k) ] has bee geeraed from he cadidae AR(k) model, his is o ecessary. If isead, [Y (k) 1,..., Y(k) ] were geeraed from he model Y (k) &A k1 Y (k) &}}}&A &1 kky (k) =Z* &k where A k1,..., A kk are prespecified coefficie marices, he he coclusios of Proposiio 2.3 ad Theorem 2.4 would remai uchaged. Two poeial advaages of geeraig he es sequeces i his way are ha he codiio K 1 p is o loger required ad ha a coefficie marix A kk very differe from he zero marix icreases he likelihood ha he se I =[:, ; ]is oempy. 3. Implemeaio ad Simulaio The resamplig procedure for order selecio of mulivariae AR models is basically he same as for he uivariae case. Assume ha X 1,..., X are observaios from a mulivariae AR( p) process defied as i (1.1). The order selecio procedure is implemeed as follows: Sep 1. Choose a fixed ieger K which is believed o be greaer ha he rue order p ad compue he YuleWalker esimaes 8 K1,..., 8 KK, 7 K from he observed daa, [X ] =1. The residual sequece is give by Z =X &8 K1X &1 &}}}&8 KKX &K for =K+1,...,. Ceer he residuals by subracig off he sample mea (1&K) =K+1 Z. For simpliciy, we use he same oaio [Z ] K+1 for he ceered residuals. Sep 2. Compue he YuleWalker esimaes 8 k, 7 k, k=0, 1,..., K from he observed daa, [X ]. =1

10 184 CHEN, DAVIS, AND BROCKWELL Sep 3. Choose a posiive ieger K 1 K. For k=0,..., K 1 geerae observaios Y (k) 1,..., Y(k) from he model Y (k) &8 k1y (k) &}}}&8 &1 kky (k) =Z* &k where [Z *] is a IID sequece sampled from he ceered residuals, [Z ] K+1. The case k=0 correspods o Y (0) =Z *. Sep 4. For k=0,..., K 1, compue he YuleWalker esimae of he iovaio covariace marix i fiig a AR( j) model o [Y (k) ] for =1 j=0,..., K. Deoe his esimae by 7 j(k). Sep 5. For k=0,..., K 1 compue ad : (k) if k=k, ={0, log 7 k(k) &log 7 j(k) max, if k<k, k< jk j&k ; (k) ={, if k=0, log 7 j(k) &log 7 k(k) mi, if k>0. 0j<k k&j Sep 6. Compue : = max [: (k) ] 0lK 1 ad ; = mi 0kK 1 [; (k) ]. Sep 7. If : <; se C =: + c(log ) ; where c>0 is such ha C ;.If: ;, he reduce he value of K 1 by 1 ad reur o Sep 6. Sep 8. The esimaed order p^ is defied o be he miimizer of $(k, C )=log 7 k +kc for 0kK.

11 MULTIVARIATE AUTOREGRESSIVE PROCESSES 185 To reduce samplig variabiliy, i is ofe beeficial o geerae may replicaes of he es series [Y (k) ] (k) =1 i Sep 4. The compued values of : ad ; (k) i Sep 5 are he replaced by heir respecive averages over he replicaios. We also cosidered a modificaio of his procedure meioed i Remark 2 of he precedig secio. The modificaio occurs i Sep 4 where he es series [Y (k) ] is geeraed from a AR(k) model wih a prespecified sequece of parameer vecors (A k1,..., A kk )$. I our simulaio sudy, we compared our proposed procedure ad is modificaio wih he followig four well kow order selecio crieria: AIC log 7 k + 2km2 AICC log 7 k + 2(km2 +1) &(km 2 +2) H 6 Q BIC (2log log ) km2 log 7 k + log[ 7 k +m]+(m 2 k+m(m+1)2) log where 8 ad 7 k are he YuleWalker esimaes of he coefficies ad iovaio covariace marix i fiig a m-variae AR(k) model o he daa. Noe ha he defiiio of AICC here (see Brockwell ad Davis (1991), p. 432) differs from he AIC C defied i Secio 1. However, i he simulaio resuls of Hurvich ad Tsai (1991), AICC ouperformed AIC C i erms of he frequecy of correc ideificaios of he rue order of he model, ad hece we have used he former i our sudy. We geeraed 100 sample pahs of various leghs from each of he followig AR models: X = \&1.0 & X &1+Z (3.1) X = \ & X &1+ \& &0.4+ X &2+Z (3.2) X = \&0.17 & &0.1+ X &1+ \& &0.25+ X &2 + \ &0.37+ X &3+Z (3.3)

12 186 CHEN, DAVIS, AND BROCKWELL where [Z ] is a IID sequece of N(0, 7) radom variables wih 7= \ 1.0 &0.08 & The AR(1) ad AR(2) models were used by Hurvich ad Tsai (1991) i heir simulaio sudy. I all of our simulaios we ook K=10, K 1 =2, ad c=5.0. The bouds : (k) ad ; (k) were compued as a average based o 50 replicaes of he es series. For he modified procedure (MDC), he parameer vecors were ad A 21 = \ A 11 = \& , , A 22= \ The frequecies of he esimaed orders for each of he 6 crieria are summarized i Tables IV (he mehod described above ad is modificaio are lised as DC ad MDC, respecively). As i he uivariae case, he modified procedure (MDC) geerally ouperformed he oher procedures, alhough he margi was o as grea as i he uivariae case. I a sese, order selecio for mulivariae AR's is easier ha i he uivariae case sice a icrease i he order of he model from k o k+1 adds m 2 more parameers o he model. A more delicae model ideificaio problem is o ideify boh he order of he model ad possible cosrais o he parameers. TABLE I Frequecies of Esimaed Order i 100 Replicaios from he AR(1) Model Give by (3.1) wih Sample Sizes 50 ad 100 (i Pareheses) Esimaed order Crierio AIC 0(0) 96(88) 0(8) 4(0) 0(2) 0(0) AICC 0 (0) 98 (96) 0 (4) 2 (0) 0 (0) 0 (0) H 6 Q 0 (0) 98 (98) 0 (2) 2 (0) 0 (0) 0 (0) BIC 0 (0) 99 (100) 1 (0) 0 (0) 0 (0) 0 (0) DC 0(0) 100(97) 0(1) 0(1) 0(1) 0(0) MDC 0 (0) 100 (100) 0 (0) 0 (0) 0 (0) 0 (0)

13 MULTIVARIATE AUTOREGRESSIVE PROCESSES 187 TABLE II Frequecies of Esimaed Order i 100 Replicaios from he AR(2) Model Give by (3.2) wih Sample Sizes 50 ad 100 (i Pareheses) Esimaed order Crierio AIC 0 (0) 0 (0) 89 (88) 7 (11) 2 (0) 2 (1) AICC 0 (0) 2 (0) 98 (95) 0 (5) 0 (0) 0 (0) H 6 Q 0 (0) 2 (0) 96 (99) 2 (1) 0 (0) 0 (0) BIC 0 (0) 4 (0) 96 (100) 0 (0) 0 (0) 0 (0) DC 0(0) 1(0) 97(99) 2(1) 0(0) 0(0) MDC 0 (0) 6 (0) 94 (100) 0 (0) 0 (0) 0 (0) TABLE III Frequecies of Esimaed Order i 100 Replicaios from he AR(3) Model Give by (3.3) wih Samples Sizes 50, 100 (i Pareheses), ad 200 (i Square Brackes) Esimaed order Crierio AIC 2 (0) [0] 2 (0) [0] 6 (0) [0] 62 (88) [92] 16 (8 [8] 12 (4) [0] AICC 16 (0) [0] 8 (2) [0] 14 (0) [0] 62 (96) [96] 0 (2) [4] 0 (0) [0] H 6 Q 16 (0) [0] 4 (2) [0] 12 (0) [0] 62 (98) [99] 4 (0) [1] 2 (0) [0] BIC 54 (8) [0] 6 (4) [0] 8 (4) [0] 32 (84) [100] 0 (0) [0] 0 (0) [0] DC 12 (2) [0] 0 (2) [0] 10 (0) [0] 66 (88) [99] 8 (6) [1] 4 (2) [0] MDC 12 (0) [0] 2 (2) [0] 12 (0) [0] 63 (92) [100] 7 (6) [0] 4 (0) [0] TABLE IV Frequecies of Correc Order Selecio i 100 Replicaios from he Models (3.1)(3.4) Usig DC Sample size Model AR(1) AR(2) AR(3)

14 188 CHEN, DAVIS, AND BROCKWELL TABLE V Frequecies of Correc Order Selecio i 100 Replicaios from he Models (3.1)(3.4) Usig MDC Sample size Model AR(1) AR(2) AR(3) YuleWalker vs Codiioal Leas-Squares I he simulaio resuls of he precedig secio, he parameers were all esimaed usig he YuleWalker equaios. We also compared hese resuls wih esimaes based o codiioal leas squares (see Hurvich 6 Tsai (1991) ad Lu kepohl (1991)). For small o moderae sample sizes, he order selecio crieria performed cosisely beer wih he YuleWalker esimaes. This differece is eve more proouced i he uivariae case. (See Che e al. (1993) where YuleWalker esimaio was compared wih boh Burg ad maximum likelihood esimaio.) To demosrae he differece i performace bewee he wo esimaes, we geeraed 500 ime series of legh 30 from he model (3.1), X = \&1.0 & X &1+Z, where [Z ] is a IID sequece of N(0, 7) radom vecors. A AR(6) model was fied o each series usig boh YuleWalker ad codiioal leas squares esimaio. The 24_1 parameer vecor for he model is give by a$=(&1.0,.96, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, &1.5, 1.4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0). Figures 1 ad 2 show he boxplos of he YuleWalker ad leas squares esimaes respecively of he compoes of he vecor a. Oe ca clearly see ha he error bars of he YuleWalker esimaes are sysemaically shorer ha hose of he leas squares esimaes. The YuleWalker esimaors appear o be beer behaved ha codiioal leas squares esimaors for fiig over-parameerized auoregressive models, givig geerally smaller values for he coefficies a lags greaer ha he rue order of he model. Accurae fiig of over-parameerized models is a impora igredie i model selecio.

15 MULTIVARIATE AUTOREGRESSIVE PROCESSES 189 Fig. 1. AR(6) coefficies fied by YuleWalker. The rue model is he AR(1) (3.1). Box plos are based o 500 replicaes. Fig. 2. AR(6) coefficies fied by leas squares. The rue model is he AR(1) (3.1). Box plos are based o 500 replicaes.

16 190 CHEN, DAVIS, AND BROCKWELL Refereces Akaike, H. (1969). Fiig auoregressive models for predicio. A. Is. Sais. Mah Brockwell, P. J., ad Davis, R. A. (1991). Time Series: Theory ad Mehods. Spriger- Verlag, New York. Che, C., Davis, R. A., Brockwell, P. J., ad Bai, Z. D. (1993). Order deermiaio for auoregressive processes usig resamplig mehods. Saisica Siica Haa, E. J. (1970). Muliple Time Series. J. Wiley, New York. Haa, E. J., ad Deisler, M. (1988). The Saisical Theory of Liear Sysems. Wiley, New York. Haa, E. J., ad Qui, B. G. (1979). The deermiaio of he order of a auoregressio. J. R. Sais. Soc. B. 41 (2) Hurvich, C. M., ad Tsai, C. L. (1989). Regressio ad ime series model selecio i small samples. Biomerika 76 (2) Hurvich, C. M., ad Tsai, C. L. (1991). A correced Akaike iformaio crierio for vecor auoregresive model selecio. J. Time Series Aal Lu kepohl, H. (1991). Iroducio o Muliple Time Series Aalysis. Spriger-Verlag, Berli.