Weighted Least Squares Approximate Restricted Likelihood. Estimation for Vector Autoregressive Processes

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1 Wightd Last Squars Approximat Rstrictd Liklihood Estimation for Vctor Autorgrssiv Procsss BY WILLA W. CHEN Dpartmnt of Statistics, Txas A&M Univrsity Collg Station, Txas 77843, U.S.A. wchn@stat.tamu.du AND ROHIT S. DEO Nw York Univrsity, 44 Wst 4th Strt, Nw York, Nw York 10012, U.S.A. rdo@strn.nyu.du SUMMARY W driv a wightd last squars approximat rstrictd liklihood stimator for a k- dimnsional pth ordr autorgrssiv modl with intrcpt, for which xact liklihood optimization is gnrally infasibl du to th paramtr spac which is complicatd and highdimnsional, involving pk 2 paramtrs. Th wightd last squars stimator has significantly rducd bias and man squard rror than th ordinary last squars stimator for both stationary and non-stationary procsss. Furthrmor, at th unit root, th limiting distribution of th wightd last squars approximat rstrictd liklihood stimator is shown to b th zrointrcpt Dicky Fullr distribution, unlik th ordinary last squars with intrcpt stimator which has a diffrnt distribution with significantly highr bias. Som ky words: Autorgrssiv; Bias; Rstrictd maximum liklihood; Unbiasd stimating quations

2 W. W. CHEN AND R. S. DEO 1. INTRODUCTION Th pth ordr autorgrssiv modls, ARp), ar commonly usd in multipl tim sris as thy ar asy to stimat via last squars and to us in forcasting. Including an intrcpt in univariat autorgrssiv modls incrass th finit sampl bias of th ordinary last squars and maximum liklihood stimators Shaman & Stin, 1988, Chang & Rinsl, 2000). For xampl, in th AR1) with cofficint α and no intrcpt, th bias of both th maximum liklihood and ordinary last squars stimators basd on n obsrvations is 2αn 1, but bcoms 1 + 3α) n 1 whn an intrcpt is includd. Thus, th bias is almost doubld whn th autorgrssiv cofficint is clos to unity. At th unit root th intrcpt has an impact that prsists vn asymptotically, with diffrnt limiting distributions for th ordinary last squars stimator with and without intrcpt. Chang & Rinsl 2000) found that rstrictd maximum liklihood stimation of autorgrssiv modls with intrcpt yilds stimats with rducd bias which is idntical up to O n 1) to that of th maximum liklihood and ordinary last squars stimats for zro-intrcpt autorgrssiv modls. S also Kang t al 2003). Hnc, including an intrcpt has no impact on th rstrictd maximum liklihood stimator and its bias dos not incras, unlik that of th maximum liklihood and ordinary last squars stimators. Othr advantags of th rstrictd maximum liklihood stimator in tim sris applications includ bttr modl slction proprtis Tunnicliff-Wilson, 1989) and wll-bhavd liklihood ratio tsts Chn & Do, 2009). Howvr, th rstrictd liklihood optimisation is infasibl in vctor autorgrssiv modls, whr th paramtr spac is vry complicatd and dfind implicitly through th roots of a charactristic quation. W obtain a wightd last squars approximat rstrictd maximum liklihood stimator which is asy to comput and has th sam suprior bias proprtis of th ordinary last squars stimator without intrcpt. At th unit root th limiting distribution of th proposd stimator for th AR modl with intrcpt is idntical to th Dicky Fullr distribution

3 97 98 Wightd Last Squars Approximat Rstrictd Liklihood 3 for th zro-intrcpt modl, unlik that of th ordinary last squars with intrcpt stimator. Our rsults hav consquncs for forcasting, unit root tsts and cointgration tsts THE APPROXIMATE RESTRICTED LIKELIHOOD ESTIMATOR 2 1. Motivation Assum that Y =Y T 1,...,Y T n )T follows th k dimnsional ARp) procss, givn by Y t = µ+ỹt, p Ỹt = A iỹt i + t 1) i=1 whr t is an indpndnt N 0,Σ ) sris and th roots of I k z p p i=1 A iz p i = 0 ar at most on in absolut valu. Th initial valus of Ỹt ar assumd to b Ỹt = 0 for t = 0, 1,..., p + 1. Ltting Σ = var Y ), th log rstrictd liklihood for Y is RL = 1 2 log Σ 1 2 log XT Σ 1 X 1 Q, 2) 2 whr Q = Y T Σ 1 Y Y T Σ 1 X X T Σ 1 X ) 1 XT Σ 1 Y, and X is th nk k matrix I k,...,i k ) T. For simplicity, w illustrat th wightd last squars approximat rstrictd maximum liklihood stimator through th univariat AR1) modl, whr Ỹt = αỹt 1 + t in 1). From 2), th rstrictd maximum liklihood stimator, ˆα REML is th minimisr of RLα,σ 2 1) ) = n log σ log w α) 1 Q, 3) 2 whr wα) = {1 + n 1)1 α) 2 } 1. Th ordinary last squars with intrcpt stimator, ˆα OLS, minimiss th objctiv function Q OLS α) = 1 σ 2 n t=2 { )} 2 Y t Ȳ1) α Y t 1 Ȳ0) whr Ȳ1) = n 1) 1 n t=2 Y t and Ȳ0) = n 1) 1 n t=2 Y t 1. Th log rstrictd liklihood in 3) has an unbiasd stimating quation, i.., E RL/ α) = 0 Chang & Rinsl, 2000). Howvr, Q OLS α) dos not satisfy this condition du to th man corrction, which incrass

4 W. W. CHEN AND R. S. DEO th bias of ˆα OLS. Lmma 1 blow shows that th xponnt trm Q in 3) can b writtn as Qα) = Q OLS α) + w α) Q C α), 4) whr Q C α) = n 1) 1 σ 2 {Ȳ1) Y 1 αȳ0) Y 1 )} 2. Sinc E α Q C ) = w 1 α), w gt { } { RLα) QOLS α) 0 = E α = E α α α + w α) Q α) } C. α Thus, th objctiv function, Q I α) = Q OLS α) + w α 0 )Q C α), whr α 0 is th tru valu of α, also has an unbiasd stimating quation. Furthrmor, Q I α) RLα) in th sns that both 153 first drivativs ar O p n) and thir diffrnc, w 1 α) w α) Q C w α) = O p 1). Thrfor, th objctiv functions RL and Q I hav similar stimating quations and th minimisr, ˆα IWLS = arg min α Q I α) can b thought of as an infasibl wightd last squars approximation to ˆα REML and should hav lss bias than ˆα OLS, just lik ˆα REML. 157 Th wightd last squars approximat rstrictd maximum liklihood stimator that w considr uss a consistnt initial stimat of α, say ˆα, in th wight function w ) in Q I to yild a fasibl stimator, ˆα WLS = arg min α {Q OLS α) + w ˆα) Q C α)}. Sinc gnrally ˆα α 0 ) = O p n 1/2 ) in th stationary cas and ˆα α 0 ) = O p n 1 ) in th unit root cas, rplacing wα 0 ) by wˆα) rsults in an rror of O p n 3/2 ) in th stationary cas and O p n 1 ) in th unit root cas, suggsting that th diffrnc btwn ˆα IWLS and ˆα WLS is small. Indd, both our thortical rsults in Thorms 1 and 2 as wll as our simulations show that th fasibl wightd last squars stimator approximats th rstrictd maximum liklihood stimator Th vctor autorgrssiv cas Lt H = H 1,...,H p ) whr H 1 = p i=1 A i and H i = p j=i A j for i = 2,...,p ar th cofficint matrics of th ARp) modl xprssd in Dicky Fullr form. W first giv th wightd last squars objctiv function for th vctor autorgrssiv procss. Lt n p = n p, Ȳ 1) = n 1 p nt=p+1 Y t, Ȳ 0) = n 1 nt=p+1 p Y t 1 and Z s = n 1 nt=p+1 p Y t s

5 Wightd Last Squars Approximat Rstrictd Liklihood 5 for s = 2,...,p. St L T t = {Y t 1 Ȳ0)) T, Y t 1 Z 1 ) T,..., Y t p+1 Z p 1 ) T }. Dfin R = R 1,...,R p ), whr R i = Y i+1 Y 1, i = 1,...,p 1, R p = n 1/2 nt=p+1 p Y t Y 1 ) and U = U 1,...,U p ), whr Us T = {Y s Y 1 ) T, Ys T,..., Y T n n Up T = np 1/2 Y t 1 Y 1 ) T, Yt 1,..., T t=p+1 t=p+1 2,0T p s) 1 n t=p+1 Also, lt D = {I k H 1 H 2 ) T,...,I k H 1 H p ) T,n 1/2 p I k H 1 ) T }Σ 1/2. } for s = 1,...,p 1, Yt p+1 T. LEMMA 1. Th log rstrictd liklihood in 2) can b writtn as RL = 1/2{n 1)log Σ log W + Q}, whr W = I pk + D T Σ D) 1 is positiv dfinit and Q = n t=p+1 Y t Ȳ1) HL t ) T Σ 1 Y t Ȳ1) HL t ) + vc R HU) T I p Σ 1/2 )WI p Σ 1/2 vc R HU). Th xprssion for th log rstrictd liklihood in th gnral cas mirrors that for th simpl univariat AR1) statd in 3) and 4). Th first trm in Q is th sum of squars for th ordinary last squars stimator, whil th scond trm can b intrprtd as a corrction trm. Though Σ and W ar unknown, thy can b stimatd using any consistnt stimator of H, such as th ordinary last squars stimator. Lt ˆΣ and Ŵ =Ŵij) p i,j=1 b any such consistnt stimats of Σ and W. Th wightd last squars approximat rstrictd maximum liklihood stimator is th minimisr of Q valuatd at ˆΣ and Ŵ, givn by n vcĥwls ) = L t L T t t=p+1 n 1 vc 2ˆΣ t=p+1 2ˆΣ 1 + p p i=1 j=1 p Y t )L Ȳ1) T t + ) U i Uj T + U jui T p i=1 j=1 ˆΣ 1/2 ˆΣ 1/2 Ŵ ˆΣ 1/2 ij Ŵ ij ˆΣ 1/2 1 5) ) R i Uj T + R jui T. Th finit sampl bias of th ordinary last squars stimator of vctor AR modls both with and without intrcpt has bn obtaind by Yamamoto & Kunitomo 1984). Th nxt thorm shows that th bias of ĤWLS is idntical up to o n 1) to that of th ordinary last squars stimator in th modl without intrcpt, Ĥ OLS,

6 241 6 W. W. CHEN AND R. S. DEO 242 a) b) Bias Ratio Bias Ratio nd root 2nd root Fig. 1. Bias ratios of ĤOLS squar), Ĥ WLS, circl), Ĥ WLS2 triangl) and ĤOLS,0 cross) against Ĥ REML basd on 10,000 rplications of bivariat AR1) with n = 100. a) Th first root is 0.8. b) th first root is 1. THEOREM 1. Lt Y t follow 1). Thn EĤWLS H) = EĤOLS,0 H) + on 1 ), whr th bias of ĤOLS,0, which is On 1 ), is givn in Yamamoto & Kunitomo 1984). Th nxt thorm shows that th bias of th xact rstrictd liklihood stimator from quation 2) obtaind undr th assumption that Σ is known, Ĥ REML,Σ, is also idntical up to o n 1) to that of ĤOLS,0. THEOREM 2. Lt Y t follow 1) and assum that Σ is known. Thn EĤREML,Σ H) = EĤOLS,0 H) + on 1 ), whr th bias of ĤOLS,0, which is On 1 ), is givn in Yamamoto & Kunitomo 1984). Th bias of th rstrictd maximum liklihood stimator whn Σ is unknown, Ĥ REML, is xpctd to b no bttr than that of ĤREML,Σ, which is th sam as that of ĤOLS,0. Indd, Figur 1 shows that th bias of ĤREML is almost idntical to that of ĤOLS,0. Thus, Thorms 1 and 2 togthr show that ĤWLS attains th bias of both ĤREML and ĤOLS,

7 Wightd Last Squars Approximat Rstrictd Liklihood 7 For a bivariat AR1) modl with n = 100 obsrvations, Figur 1 compars th bias of th ordinary last squars with and without intrcpt stimators, dnotd by ĤOLS and ĤOLS,0 rspctivly, th wightd last squars stimator, Ĥ WLS, and th itratd wightd last squars 292 stimator, Ĥ WLS2, computd using ĤWLS in Ŵ, to that of th rstrictd maximum liklihood stimator, Ĥ REML. Th cofficint matrix H was configurd as 6) in 3 with on root fixd at 0.8 in plot a) and at 1 in plot b), whil th othr root varid from 0.4 up. Thus, th two plots ncompass full-stationarity, on unit root and two unit roots. Each curv compars th squar root of th sum of squard bias of an stimator ovr th four lmnts of Ĥ to that of ĤREML. As xpctd, Ĥ OLS,0 has bias that is almost idntical to that of ĤREML, whil ĤOLS has uniformly significantly largr bias. Th bias of ĤWLS is uniformly significantly smallr than that of ĤOLS and approachs that of ĤREML and ĤOLS,0, whil th itratd wightd last squars ĤWLS 2 has uniformly smallr bias than ĤWLS. Th nxt thorm shows that th limiting distribution of ĤWLS at th unit root is unaffctd by th intrcpt, unlik that of th ordinary last squars stimator. THEOREM 3. Lt H 1 = I and Ĥ b any initial stimator such that nĥ1 I k ) = O p 1) and n 1/2 Ĥi H i ) = O p 1) for i = 2,...,p. Lt M = diag{ni k,n 1/2 I p 1)k }, thn ĤWLS H)M Ψ in distribution, whr Ψ is th asymptotic distribution of th ordinary last squars stimator assuming µ = 0 dscribd in Thorm of Fullr 1996). Both ĤOLS and thus ĤWLS satisfy th assumptions of Thorm SIMULATION STUDY W compar through simulations th finit sampl prformanc of th itratd wightd last squars stimator, Ĥ WLS2, computd using ĤWLS in Ŵ, to that of th ordinary last squars stimator. Only rsults for th bivariat AR1) ar rportd hr, whil rsults for highr ordr modls can b found at wchn/wlsrlsupp.pdf. Howvr, our con

8 337 8 W. W. CHEN AND R. S. DEO Tabl 1. Simulation bias and root man squar rror for sampl siz n = Paramtrs n Bias n RMSE z 11 z 22 Est. â 11 â 21 â 12 â 22 â 11 â 21 â 12 â WLS REML OLS WLS REML OLS WLS REML OLS RMSE, root man squar rror; WLS 2, th itratd wightd last squars stimator; REML, th rstrictd maximum liklihood stimator; OLS,th ordinary last squars stimator. clusions for th AR1) modl rportd hr ar rprsntativ of th othr modls w studid. Hr, w also rport rsults for th rstrictd maximum liklihood stimator, sinc it is fasibl to comput it for th bivariat AR1) modl. Tabls 1 and 2 rport rsults from 10,000 rplications of a bivariat AR1) modl with sampl sizs n = 100 and n = 200 rspctivly. Lt z 11 and z 22 b charactristic roots of H, and st z 11 0 Z = 0.4 z 22, B = 2 0.5, Σ = ) Th matrix H is dfind to b BZB 1. Th itratd wightd last squars stimator almost always has smallr bias and root man squar rror than th ordinary last squars stimator and th bias rduction can b substantial, vn far from th unit root. Nar th unit root, this bias rduction is attaind without an xcssiv incras in varianc as sn by th root man squar

9 385 Wightd Last Squars Approximat Rstrictd Liklihood 9 Tabl 2. Simulation bias and root man squar rror for sampl siz n = Paramtrs n Bias n RMSE z 11 z 22 Est. â 11 â 21 â 12 â 22 â 11 â 21 â 12 â WLS REML OLS WLS REML OLS WLS REML OLS RMSE, root man squar rror; WLS 2, th itratd wightd last squars stimator; REML, th rstrictd maximum liklihood stimator; OLS,th ordinary last squars stimator. rror. Thr ar som instancs of an incras in varianc for th itratd wightd last squars stimator whn far from th unit root. Th rstrictd maximum liklihood stimator rducs bias at all paramtr valus without inflating th varianc. As prdictd by Thorms 1 and 2, th bias of th wightd last squars stimator approachs that of th rstrictd liklihood stimator as th sampl siz incrass. This convrgnc is mor apparnt in th unit root cas, sinc thn th stimators ar known to convrg to thir limit distribution at a fastr rat than in th stationary cas. This phnomnon is also obsrvd in Figur ACKNOWLEDGEMENT Chn s rsarch was supportd by a grant from th U.S. National Scinc foundation

10 W. W. CHEN AND R. S. DEO APPENDIX Proofs Lt λ max B) and λ min B) b th largst and smallst ignvalu of a matrix B rspctivly and lt B = λmax 1/2 B T B) b its matrix norm. As ĤWLS is man invariant, w assum that µ = 0. Proofs of Lmma 1, Thorms 2 and 3 ar at wchn/wlsrlsupp.pdf. LEMMA 2. Lt Z t b a stationary Gaussian vctor tim sris and S = n t=1 Zt Z ) Z t Z ) T. Thn, for any positiv intgr h, E S 1 h = O n h) and E ns 1 E 1 n 1 S) h = On h/2 ). Proof of Lmma 2. For a positiv intgr m to b dfind latr, n 1 S = n 1 n/m s=1 B s, whr B s = s+1)m t=1+sm Zt Z ) Z t Z ) T. W hav λ min n 1 S ) n 1 n/m s=1 λ min B s ). Hnc, by inquality of harmonic and arithmtic mans of positiv numbrs and Jnsn s inquality λ h min n 1 S ) h h n/m n h λ min B s ) 1 n/m λ 1 min n B s) 1 n s=1 For any 1 s n/m, th vctors { Z t Z } s+1)m t=1+sm s=1 n/m s=1 λ h min B s). hav a Gaussian distribution with a nonsingular varianc matrix uniformly in s. Th proof of Lmma 8 of Chn & Hurvich 2006) yilds sup n sup 1 s n/m E { λ h min B s) } < C 0, whr C 0 is th hth momnt of λ max of an invrs Wishart matrix with m dgrs of frdom. Choosing m > 2h + k 1 nsurs that this hth momnt is finit von Rosn, 1988), giving C 0 < and proving th first part. Th scond assrtion follows by th fact that A 1 B 1 = A 1 B A)B 1 and Höldr s inquality. LEMMA 3. Lt h > 0, thn i) for Ĥ = ĤOLS,0 or ĤOLS, E h ˆΣ Σ ) = O n h/2, E Ĥ H 0 h = O n h/2) ; 1 ii) for th hth momnts of ˆΣ, w hav E ˆΣ 1 h = O1), E ˆΣ 1 Σ 1 h = O n h/2) ; and 1/2 1/2 iii) for th hth momnts of ˆΣ Ŵ ˆΣ, E ˆΣ 1/2 1/2 Ŵ ˆΣ Σ 1/2 WΣ 1/2 h = O n 3h/2). Proof of Lmma 3. Höldr s inquality and Lmma 2 yild i). For ii), lt Ψ = n 1) 1 n t=2 η tη T t, whr η t = { Y t 1 Ȳ0)) T, t ē 1) ) T } T, thn ˆΣ is th Schur complmnt of Ψ rlativ to n 1)S 1 Y. Hnc, E ˆΣ h = E{ Ψ h n 1 S Y h } E 1/2 Ψ 2h E 1/2 n 1 S Y 2h = O 1) by Lmma

11 Wightd Last Squars Approximat Rstrictd Liklihood 11 Lmma 2 and th fact that ˆΣ 1 ˆΣ 1 ˆΣ k 1 yilds ii). By i), ii) and Lmma 2, E Ŵ h = On h ) and E Ŵ W h ) = O n 3h/2, thus iii) follows by Höldr s inquality. Proof of Thorm 1. As notd by Yamamoto & Kunitomo 1984) & Nicholls and Pop 1988) it suffics to obtain th bias for an AR1) procss sinc an ARp) can always b r-xprssd as a suitabl AR1). Sinc QĤOLS,0 ) = QĤWLS ) QĤOLS,0 ), w gt 1 QĤOLS,0 ) = 2ˆΣ ĤWLS ĤOLS,0 ) n t=2 ) Yt 1 Ȳ0) Yt 1 Ȳ0)) T +2n 1)ˆΣ 1/2 1/2 Ŵ ˆΣ ĤWLS ĤOLS,0 ) ) ) T Ȳ 0) Y 1 Ȳ0) Y 1. Thus vcĥwls ĤOLS,0) = G + J) 1 vc{ QĤOLS,0)}, whr J = n 1)Γ Γ = Ȳ 0) Y 1 ) Ȳ0) Y 1 ) and G = SY ˆΣ 1, S Y = n t=2 Yt 1 Ȳ0)) Yt 1 Ȳ0)). 1/2 1/2 ˆΣ Ŵ ˆΣ, Lt Q I b th objctiv function of Ĥ IWLS, it is sufficint to show i) E G + J) 1 = n 1) 1 {Σ 1 Y Σ } + on 1 ), ii) E G + J) 1 2 = On 2 ), iii) E{ Q I ĤOLS,0 )} = o1), iv) E Q I ĤOLS,0) 2 = O1), and v) E Q I ĤOLS,0) QĤOLS,0) 2 = on 2 ). Not that J is rankd on, thus G + J) 1 = G 1 {1 + trac JG 1) }G 1 JG 1. Furthrmor, E G 1 Σ 1 Y ˆΣ n 1) h E {n 1)S 1 Y Σ 1 Y } ˆΣ h n 1) h + Σ h Y E ˆΣ Σ h n 1) h. By Lmmas 2, 3, both trms on th right hand sid ar On 3 ). Sinc {1 + trac JG 1) } 1 1 and E G 1 JG 1 2 = On 4 ) by Lmma 3, i) and ii) follow. Noting that n t=2 Y ty T t 1 = Ĥ OLS,0 n t=2 Y t 1Y t 1, w gt Q I ĤOLS,0) = 2n 1){ΩW 0, Σ ) + RW 0, Σ )}, whr ΩW 0, Σ ) = Σ 1/2 I W 0 )Σ 1/2 I H 0 )Ȳ0)Ȳ T 0) Σ 1/2 W 0 Σ 1/2 { RW 0, Σ ) = Σ 1/2 I W 0 )Σ 1/2 H 0 ĤOLS,0) Ȳ0) + n 1) 1 T Y n Y 1 ) }Ȳ 0) Σ 1/2 W 0 Σ 1/2 { H 0 ĤOLS,0 I H 0 )Y 1 Y T 1, A1) ) Y 1 I ĤOLS,0) Ȳ0) n 1) 1 Y n Y 1 ) Sinc EY 1 Y T 1 ) = Σ, En 1)Ȳ0)Ȳ T 0) ) = I H 0) 1 Σ I H T 0 ) 1 + o1), W 0 = On 1 ) and I W 0 = n 1)W 0 Σ 1/2 I H 0 )Σ I H 0 ) T Σ 1/2, w hav EΩ) = on 1 ) from A1). Noting } Y T

12 W. W. CHEN AND R. S. DEO that I W 0 = O1), it is asy to vrify that E RW 0, Σ ) 2 = On 3 ) by Lmma 3. Finally, v) also follows from Lmma REFERENCES CHEANG, W. K. & REINSEL, G. 2000). Bias Rduction of Autorgrssiv Estimats in Tim Sris Rgrssion Modl through Rstrictd Maximum Liklihood, J. Am. Statist. Assoc. 95, CHEN, W. W. & HURVICH, C. M. 2006). Smiparamtric stimation of fractional cointgrating subspacs, Ann. Statist. 34, CHEN, W. W. & DEO, R. S. 2009). Bias Rduction and Liklihood Basd Almost-Exactly Sizd Hypothsis Tsting in Prdictiv Rgrssions using th Rstrictd Liklihood, forthcoming Economtric Thory FULLER, W. 1996). Introduction to Statistical Tim Sris, Scond Edition, John Wily & Sons, Inc. KANG, W., SHIN, D. & LEE, Y. 2003). Biass of th Rstrictd Maximum Liklihood Estimators for ARMA Procsss with Polynomial Tim Trnd, J. Statist. Plan. Infr. 116, NICHOLLS, D. F. & POPE, A. I.1988). Bias in th Dstimation of Multivariat Autorgrssions. J. Mult. Anal. 30A, SHAMAN, P. & STINE, R. 1988). Th Bias of Autorgrssiv Cofficint Estimators, J. Am. Statist. Assoc. 83, TUNNICLIFFE-WILSON, G. 1989). On th Us of Marginal Liklihood in Tim Sris Modl Estimation. J. R. Statist. Soc. B 51,15-27 VON ROSEN, D. 1988). Momnts for th invrtd Wishart distribution. Scand. J. Statist. 15, YAMAMOTO, T.& KUNITOMO, N1984). Asymptotic bias of th last squars stimator for multivariat autorgrssiv modls. Ann, Inst. Statist. Math. 36,