Pseudo Maximum Likelihood

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Pseudo-Maximum Likelihood Estimation of ARCH() Models

Author(s): Peter M. Robinson and Paolo ZaffaroniSource: The Annals of Statistics, Vol. 34, No. 3 (Jun., 2006), pp. 1049-1074Published by: Institute of Mathematical StatisticsStable URL: http://www.jstor.org/stable/25463451

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1050 P.M. ROBINSON AND P. ZAFFARONIfor some unknown /xo.

ARCH(oo) processes, extending the ARCH(m), m < oo, process of Engle [11]and the GARCH(n,m) process of Bollerslev [4], were considered by Robinson[29] as a class of parametric alternatives in testing for serial independence of yt.

Empirical evidence of Whistler [35] andDing, Granger and Engle [10] has suggested the possibility of long memory autocorrelation in the squares of financialdata. Taking [contrary to the first requirement in (3)] coo= 0, such long memory inxf driven by (1) and (2)was considered by Robinson [29], the \//ojbeing the autoregressiveweights of a fractionally integratedprocess, implyingYlJLi ^0j = 1;see also Ding and Granger [9]. For such x//oj, and the same objective function aswas employed to generate the tests of Robinson [29], Koulikov [20] establishedasymptotic statistical properties of estimates of fo- On the other hand, under ourassumption coo > 0, Giraitis, Kokoszka and Leipus [13] found that such \/soj areinconsistent with covariance stationarity of xt, which holds when Y,JLi i^Oj < 1Finite variance of xt implies summability of coefficients of a linear moving average in martingale differences representation of xf; see [37]. In this paper we donot assume finite variance of xt, but rather that xt has a finite fractional momentof degree less than 2. The first requirement in (3) was shown by Kazakevicius andLeipus [18] to be necessary for existence of an xt satisfying (1) and (2). The intermediate requirement in (3) is sufficient but not necessary for a.s. positivity of a},and is imposed here to facilitate a clearer focus on the xf/oj,which decay, possiblyslowly, but never vanish.

We wish to estimate the (r + 2) x 1 vector #o = (coo, Mo, fo)' on the basis of observations yt, t= 1,..., T, the prime denoting transposition. The case when /xo isknown, for example, /xo= 0, is covered by a simplified version of our treatment. Ifthe yt were instead unobserved regression errors, we have /xo= 0, but would thenneed to replace xt by residuals in what follows; the details of this extension wouldbe relatively straightforward. Another relatively straightforward extension wouldcover simultaneous estimation of the regression parameters coo and fo, after replacing /xo by amore general parametric function; as in (1), (2) and (5), efficiencygain is afforded by simultaneous estimation.Under stronger restrictions than YlJLi i^Oj < 1,Giraitis and Robinson [14] considered discrete-frequency Whittle estimation of fo, based on the squared observations yf (with /xo known to be zero), this being asymptotically equivalent toconstrained least squares regression of y} on the yf_s, s > 0, amethod employedin special cases of (2) by Engle [11] and Bollerslev [4]. In these the spectral density of yf, when it exists, has a convenient closed form. This property, along withavailability of the fast Fourier transform, makes discrete-frequency Whittle esti

mation based on the y} a computationally attractive option for point estimation,even in very long financial time series. However, it has a number of disadvantages,as discussed by Giraitis and Robinson [14]: it is not only asymptotically inefficient under Gaussian st, but never asymptotically efficient; it requires finiteness


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ESTIMATION OF ARCH(oo) MODELS 1051of fourth moments of yt for consistency and of eighth moments for asymptoticnormality, which are sometimes considered unacceptable for financial data; itslimit covariance matrix is relatively complicated to estimate; it is less well motivated inARCH models than in stochastic volatility and nonlinear moving average

models, such as those of Taylor [33], Robinson and Zaffaroni [30, 31], Harvey[15], Breidt, Crato and de Lima [5] and Zaffaroni [36], where the actual likelihoodis computationally relatively intractable, while Whittle estimation also plays a lessspecial role in the short-memory-in-jr2 ARCH models of Giraitis and Robinson[14] than in the long-memory-in-^2 models of the previous five references, whereit entails automatic "compensation" for possible lack of square-integrability of thespectrum of yf. Mikosch and Straumann [26] have shown that a finite fourth mo

ment is necessary for consistency of Whittle estimates, and that convergence ratesare slowed by fat tails in et.For Gaussian et, a widely-used approximate maximum likelihood estimate isdefined as follows. Denote by 9 = (co, /x, ?')' any admissible value of #o and define

xt(v) = yt -M,oo

7= 1for (eZ, and

t-\

j=ifor f > 1,where 1() denotes the indicator function. Define also

* = -T^+lna^6^ 9t(0) = -^+\nd}(9), 1< t< T,erf(6) af(9)T TQt(9) = T~l J^ltiO), QT(9) = T~x j>(0),t=\ t=\

?t = zrgmin Qt(9), 9j ? argmin Qt(9),0e? 0e?where @ is a prescribed compact subset of Er+2. The quantities with over-bar areintroduced due to yt being unobservable for t < 0; 9j is uncomputable. Becausewe do not assume Gaussianity in the asymptotic theory, we refer to ?t as a pseudomaximum likelihood estimate (PMLE).We establish strong consistency of 9j and asymptotic normality of Txl2(9j ?#o), as T ? oo, for a class of ^(f) sequences. In the case of the first property this is accomplished by first showing strong consistency of 9j and thenthat 9t ?9t ? 0, a.s. In the case of the second we likewise first show it for


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1052 P.M. ROBINSON AND P. ZAFFARONI

TXI2(9T - do) and then show that 9T-9T= op(T~x/2), but the latter property, andthus the asymptotic normality of Tx/2(9j ? #o)> is achieved only under a restrictedset of possible fo values, and this seems of practical concern in relation to somepopular choices of the x//j(t;)- These results are presented in the following section,along with a description of regularity conditions and partial proof details. Thestructure of the proof is similar in several respects to earlier ones for the GARCHcase of (2), especially that of Berkes, Horvath and Kokoszka [3]. Sections 3 and 4apply the results to particular models.

2. Assumptions and main results. Our assumptions are as follows.

Assumption A(q), q > 2. The et are i.i.d. random variables with Eso = 0,Es^ = I, E\so\q < oo and probability density function f(s) satisfying

f(s) = 0(L(\8\-x)\8\b) as?^0,for b > ? 1 and a function L that is slowly varying at the origin.

Assumption B. There exist coi,cou, lil, l^u such that 0 < col < cou < oo,?oo < /xl < /X(/ < oo, and a compact set T e Rr such that 0 = [col, 1,(6) inf^(f)>0;

(7) sup \jrj(f) < Kj ~l~x for some d > 0;(8) foj


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ESTIMATION OF ARCH(oo) MODELS 1053Assumption F(/). For all j > 1, Vy(f) has continuous kth derivative on Tsuch that, with ?7 denoting the /th element of f,

(IDr^-s^K)1-'for all r\> 0 and all //*= 1,..., r, h = 1,..., &, k < I.

Assumption G. For each f Y, there exist integers jt = jt (?),/ = 1,..., r,such that 1< ./l (?)< < jr(C) < ?? andrank{*Ul,...Jr)(f)}=r,

where

*,;.*)tt) -l


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1054 P.M. ROBINSON AND P. ZAFFARONIProof. The proof follows as in, for example, [17], Theorem 6, from uniforma.s. convergence over 0 of Qt(9) to Q(9) ? Eqo(0) established in Lemma 7, thefact thatQT(9T) < Qt(0), andLemma 10.Theorem 1. For some 8>0, let Assumptions A(2 + 8), B, C, D, E, F(l) and

G hold. Then(15) 9j ?> 9o a.s. as T -* oo.

Proof. From Lemmas 7 and 8, Qt(0) converges uniformly to Q(9) a.s.,whence the proof is as indicated for Proposition 1.

Denote by Kj the j th cumulant of st and introduceG0 = (2+ k4)M - 2k3(N + N') + P, Ho = M + \P,

whereM = E(tot& N = E(a-X r0)e'2, P = E(a^2)e2ef2,

for ro= ro(#o)> *t(0) = (9/90) log0/(0), and e2 the second column of the (r+2) x (r + 2) identity matrix. In case /xo is known (e.g., to be zero), we omit thesecond row and column from M, and have instead Go = (2 + k^)M, Ho = M. Incase st isGaussian, ic$= k4 = 0, so Go = 2Ho = 2M + P.

PROPOSITION 2. Let Assumptions A(4), B, C, D, E, F(3) and G hold. Then

Tx/2(9T-90) -i N(0, H-xG0HqX) asT^oo.Proof. Write


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ESTIMATION OF ARCH(oo) MODELS 1055we define ||A|| = {tr(A'A)}1/2 for any real matrix A. Now ut(00) = rt(90)(l -s2) ? 2e2st/ot is, by Lemmas 2, 3 and 7, a stationary ergodic martingale difference vector with finite variance, so from Brown [6] and the Cramer-Wold device, T1/2Q{j}(Oo) -*d N(0, Go) as T -> oo. Finally, by Lemma 7 and Theorem 1,Ht -+p Ho, whence the proof is completed in standard fashion.

Define- m\ 9^(g) - m\ um\ d24t(0)ut(0) = ??, gt(9) = ut(9)ut(9), ht(9) = ,

T ^ TGT(9) = T~x J>(0), HT(9) = T~x?ft,(0).t=\ t=\

THEOREM 2. Let Assumptions A(4), B, C, D, E, F(3), G and H hold. Then(17) Tl/2(9T-90) 4> N(0, H^1GoHq1) as T^ oo,

_ 1 _ 1 ? _ 1/V ? /V? _ 1 ^am/ //0 GoH0 is strongly consistently estimated byHT (9t)Gj (9t)HT (9j).Proof. We have

O=Q{t\0t) = Qt\Oo) + Ht(9t - Oo),where Qj\0) = (d/dO)QT(0) andHT has as its ith row the ir/z row of HT(9)evaluated at 0 = of, for \\of - #0|| < ||0r- 0O||.Thus, from (16),

0T-0t = (H~x - H-x)Q^(Oo) - H~x\Qf(Oo) - Q(Tl)(90)},where the inverses exist a.s. for all sufficiently large T by Lemma 9. In view ofProposition 2 and Lemma 8, (17) follows on showing that

Q{t1\Oo)-Q(t)(9o)=op(T-x?2).The left-hand side can be written (B\j + B2? + B^t)/T', where

T T tBXT = ?>2fci? B2T = -?>2 - l)b2t, B3T = -2e2J2?tb3t,t=\ t=\ t=\

with-2(1)/ 2 -2\ 2(1) -2(1) 2 -2

otK\of-af) V or/' a,2-

erfo\t=-?a-, b2t = ?5-ry-, fc3, = -?r-,erferf ofcrfcrtfor of = of(9o), a2(1) = a2(>0), a2(1) = a2(1)(c90), with o^(9) =(d/d9)of(9), a2(1)(0) = (d/d9)df(9). We show thatBiT = op(Tx'2), i= 1,2,3.For the remainder of this proof, we drop the zero subscript in \Jsoj.


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1056 P.M. ROBINSON AND P. ZAFFARONIConsider first B\t. We have

\ 7=1 7= 1 Iwhere ^j = V*,- (fo)- FromAssumption F(l),

|a,2(I) | < 1+ 2? *, |*,_,-1+ a:E ^'"xlj,7=1 7=1for all ?7> 0. Now

t-\lt-\ \'/2/oo \1/2

E^7^-7l(oi> 0,

r-l

7= 1

From (8),

7= 1It follows that(19) \\a^\\/a^K^\

On the other hand, by the cr-inequality ([23], page 157) and (10),oo oo(20) E(af - a}Y f ^ < Ktx-p^+m~^,j=t j=tchoosing n < 1? l/{p(do + 1)}, which (14) enables. Applying the cr-inequalityagain,

TE\\BlTr


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ESTIMATION OF ARCH(oo) MODELS 1057

Applying (21), this is0(1) when p > 2/(do + 1),while when p < 2/(do + 1),wemay choose r\ so small to bound it by

KT2-p(dQ+\)(\-r)) < ^rp/2-{l+2(db+l)(l-//)}[p/2-2/{l+2(db+l)(l-i;)}] = 0(Tp/2^using (12) [which requires (13)] and arbitrariness of n. Thus, B\j = op(Tx/2) byMarkov's inequality.Consider Z^r- By independence of st and ?>2r,by the cr-inequality when p \,T TE\\B2T\\2p < K?(?|?0|4p + \)E\\b2t\\2p < KY,(E\\b4t\\2p + E\\b5t\\2p),t=\ t=\

whereu - ?t ~Gt u - Gt {?t ~?t )04t ? -o > &5t ? To 9

?t Gt?tThus, from Assumptions F(l) and H,

/ OO oo \1164,11


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1058 P.M. ROBINSON AND P. ZAFFARONIand thence, B2T = op(Tx/2).Next,

T|2p

TE\\B3T\\2e < KE J2?tb3t < KÊôÊb]',t=i t=i

applying the cr -inequality when p < \ and von Bahr and Esseen [34] when p > ^.Now b3t < (erf - of)xl2&-2, so from (20),T oo

E\\B3T\\2p 2/(do+ 1))+ (InT)l(p = 2/(do+ 1))

+ T2-P{d?+Xh(p< 2/(db + D)}= o(Tp),

much as before. Thence, B3j = op(Tx/2).It remains to consider the last statement of the theorem, which follows on standard application of Propositions 1 and 2, Theorem 1 and Lemmas 7 and 8.

In earlier versions of this paper we checked the conditions in the case ofGARCH(h, m) models in which the x/fj(^) decay exponentially and we allow thepossibility that the GARCH coefficients lie in a subspace of dimension less thanm + n\ the details are available from the authors on request. However, the literatureon asymptotic theory for estimates of GARCH models is now extensive, recent references including [3, 7, 12, 16, 22, 32], along with investigations of the propertiesof the models themselves; see recently [2, 18, 25]. We focus instead on alternativemodels which have received less attention, and for which our theoretical framework is primarily intended.We introduce the generating function

oo(22) iAU;0=?^W, kl


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In these references coo= 0 was assumed in (2), but we assume coo> 0 and generalize (23) as follows. Introduce the functions aj = fl/(f)> bj = &/(?) and, for m > 1,n > 0, n + m > r,

m n(25) a(z', f) = X>7V, Mz; C)= 1- J^bjz'Hn > 1);

y=i 7=iand for all f GY,(26) 0/> 0, j = \,...,m; bj>0, j = l,...,n;(27) ft(z;?)#0, |z|l,n>0, and

let d and the aj, bj be continuously dijferentiable. For some 8 > 0, letAssumptionsA(2 + 8), B, C and E /*6>/d,vWf/ialls eT satisfying (26)-(28), (30) and\ 9 1mnk\-?(a\,...,am,b],...,bn,d) >=r.

Then (15) is true. Let also d and the aj, bj be thrice continuously dijferentiableand do > \. Then (17) is true.

PROOF. Denoting by cj (j > 1) and dj (j > 0) the coefficients of z7 in theexpansions of a(z\ $)/b(z\ ?)> z-1{l ? (1 ? z)d], respectively, we have V'y(C) ?Yl{ZocJ-kdk, .7> 1- From [3], the Cj are bounded above and below by positive, exponentially decaying sequences when n > 1, and are all nonnegative whenn = 0. Since the dj are all positive, it follows that (6) holds. Also, Stirling's approximation indicates that j~d~x/K < dj < Kj~d~x, so the V^/(f) satisfy the sameinequalities. Compactness of Y, smoothness of d, and (30), imply d(t;) > d, tocheck (7). The above argument indicates that tyoj < Kj~d?~x < Kk~d?~x < Kxjrokfor j > k > 1, so (8) holds, and thus Assumption D. With regard to (11), notethat (d/dd)^(z\ O = -{a(z; 0/b(z; 0}z-1(l - z/ln(l - z)9where the coef

ficient of zJ in -z_1(l - z)d\n(l - z) is EJk=ik~xdj-k < K(lnj)j-d-x


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Kj-(d+\)(i-r)) < Kfj^tt) for any rj> 0. Derivatives with respect to the aj,bjare dominated, and higher derivatives can be dealt with similarly, to complete thechecking of Assumption F(/). To check Assumption G, suppress reference to ? ina, b, ty and

Hz) = b(zrx{l - (1- z)dh Y(z) = b(z)~xa(z),and note that

9^(z) j-Uc^ 1?-=zJ 0(z), j = l,...,m,daj??=ZJ y(z)c/)(z), j = l,...,n,dbj

?f(z) Y(z) d?ZT~ =-(1 - z)a log(l - z).ad zChoose ji (f) = i for i= 1,..., m + n, ? e T, leaving jm+n+\ (t;) to be determinedsubsequently. Fix ? and write U = x^(ji,...jr)(0^ partitioning it in the ratio m + n:land calling its (i, j)th submatrix [/,-y.We first show that the (m + n) x (m + n)matrix U\\ is nonsingular. Write R for the n x (m+ n) matrix with (/, j)th elementYj-i, and S for the (m + n) x (m + a) matrix with (/, y)th element 07_/+i, where0y

=y^r

= 0 for j< 0, and for j

> 0, 0y and Yjare

respectively given byOO 004>(z)= ^jzj, y(z) = J2 yjzJ>

j=\ 7 = 1

these series converging absolutely for \z\ < 1 in view of (30). Noting that \fris given by (d/d^)%l/(z) = J2JL\ Wj zJ, we find that the first m rows of U\\ canbe written (Im, 0)S, where Im is the m-rowed identity matrix, O is the m x nmatrix of zeroes and, when n > 1 the last n rows of U\\ can be written RS. Now5 is upper-triangular with nonzero diagonal elements. Thus, for n = 0, U\ \= S isnonsingular. For n>l,U\\is nonsingular if and only if the n x n matrix R2 having(/, j)th element Ym+j-i and consisting of the last n columns of R is nonsingular.This is not so if and only if the y/, j = m,..., m + n ? 1, are generated by a

homogeneous linear difference equation of degree n ? 1, that is, if there existscalars A.o, M,..., Xn-\, not all zero, such that

n-\X0Yj - ^2^iYj-i =0' J =m,...,m+n- 1.1=1

But it follows from (25) and (27) that they are generated by the linear differenceequation

n-\Yj - J2 biyj-i =7lh j =m,...,m+n-l,i=\


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ESTIMATION OF ARCH(oo) MODELS 1061where nm = am + bnym-n, iij = bnyj-n for j = m + 1,..., m + n

? 1. Sincebn ^ 0, the Ttj are all zero if and only if ym-n = ?am/bn and yj = 0 for j =m + \?n,...,m ? 1. But this implies ym = 0 also, and thence, yj =0, ally > raft + 1. For ra < ft, this is inconsistent with the requirement aj > 0, j = I,... ,m,and for ra > ft, it implies a has a factor b, which is inconsistent with (28). Thus,U\i is nonsingular when n > 1.Nonsingularity of [/ follows if U22 ? U2\ U^XU\2.For large enough jm+n+\ = Jm+n+\ (?), this must be true because f/22 decays like(\njm+n+\)j~^\.^ whereas the elements of U\2 are 0(^m+n+]) for some /J e(0, 1). Thus Assumption G is true, and thence (15). Clearly (13) is true, so underthe additional conditions so is Assumption H, and thence (17).

For ra = 1, ft = 0, (29) reduces to (23) when a\ = 1, while when a\ e (0, 1),it gives model (4.24) of Ding and Granger [9]. The important difference betweenthese two cases is that the covariance stationarity condition xj/(l; t;o) < 1 is satisfied in the second but not in the first. In general with (29), as with the GARCH

model, xt is covariance stationary when a(l; ?0) < 6(1; fo) but not otherwise. Wecompare (29)with

(31)V(z; ?) = 1-77?7?0 ~ z) 'b(z\0with d again satisfying (30) and a and b again given as in (25), though we nowallow ra = 0, meaning a(z\ f) = 0. Thus, with ra = ft = 0, (31) reduces to (23).

ARCH(oo) models with yjrgiven by (31) were proposed by Baillie, Bollerslev andMikkelsen [1] and called FIGARCH(ft, do, ra). In general, though (31) also giveshyperbolically decaying x/roj, it differs in some notable respects. Application of(26)-(28) again ensures positivity of \jrj (f) in case of FGARCH and facilitates theabove proof, but sufficient conditions in FIGARCH are less apparent in general,though Baillie, Bollerslev and Mikkelsen [1] indicated that they can be obtained.

Also, unlike FGARCH, FIGARCH xt never has finite variance.The requirement do > \ for the central limit theorem in Corollary 1would alsobe imposed in a corresponding result for FIGARCH. This is automatically satisfiedinGARCH models but if only do e (0, ^] in (13) is possible in the general settingof Section 3, it appears that the asymptotic bias in 9j is of order at least T~xl2,

whereas that for 9j is always o(T~xl2). Assumption H copes with the replacementof cr2(r^) by of (9), the truncation error varying inversely with do- Inspection of theproof of Theorem 2 indicates that this bias problem is due to the term H~1B\t.The factor of ? of in b\t is nonnegative, and if j~do~x is an exact rate for ifroj,of ? of exceeds t~d?/K as t -> 00 with probability approaching one. So far asthe factor ot /of mb\t is concerned, the second element of o2^ [see (18)] haszero mean, but the first is positive, and though the \/f- can have elements of eithersign, whenever do < \ it seems unlikely that the last r elements of B\j can be


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op(Tx/1). Nor is there scope for relaxing (12) by strengthening other conditions.With regard to implications for choice of p, when do > 2d + \, (14) entails norestriction over (9).Though results of Giraitis, Kokoszka and Leipus [13] indicate existence of a

stationary solution of (l)-(3) when t/t(1;^0)< 1,Kazakevicius and Leipus [19]have questioned the existence of strictly stationary FIGARCH processes, and thusthe relevance of Assumption E here. The same reservations can be expressedabout FGARCH when a(l; f0) > b(\\ Co), andmore generally aboutARCH(oo)processes with t/Kl; ?o) > 1. A sufficient condition for (10) can be deduced asfollows. Recursive substitution gives

oo / oo oo\

aj < K + K?( ? ? Vo;, ihj,slj1eljl-h ~$-h-...-X/=1 Vl = l 7/= l /so by the cr -inequality,

00 / oo oo

'=1 Vi = l 7/=lv Ip \2p \p -\2p\X \bt-J\-J2\ \kt-J\-Ji\ I

Thus, from Lemma 2,00 / oo \ '

E\xt\2(,


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1064 P.M. ROBINSON AND P. ZAFFARONInot be very close and the "integrated" case is less easy to distinguish than inGARCH and FGARCH models (though itwould be possible to alternatively scalethe weights by infinite sums to achieve equality).

The following corollary covers (34) and (35) simultaneously, and implies thespecial case when the f are specified a priori, for example, to be nonnegativeintegers; strictly speaking, when the true value of f\ is unknown, Assumption Cprevents it from being zero.

COROLLARY2. Let^r(z\t;)be given by (22) and (34) or (35)with ra > 1andlet d and the et, f be continuously differentiable. For some 8 > 0, let AssumptionsA(2 + 8), B, C and E hold, with all f e Y satisfying (36)-(38) andrank ?(


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ESTIMATION OF ARCH(oo) MODELS 10655. Technical lemmas. Define

00 ooaf(9) = a>+J2 irj(0xf_j, of = cou+ YJ supV; (?)*,-,7=1

'7=1?6T

Lemma 1. Under Assumptions B and D,for all 9 e?, t e Z,K~xof(9) < a2(9) < Kaf(9) a.s.

Proof. A simple extension of [21], Lemma 1.LEMMA 2. Under Assumptions A(2), B, C, D and E,for all t eZ,

(39) E\xt\2p < Ea2p < E mpa2p(9) < KEofp < KE\xt\2p < K,(40) inf a2(9) > 0, sup a2(9) < Kaf < oo a.s.,

(41) ?sup|lnar2(^)| < K.PROOF. With respect to (39), the first inequality follows from Jensen's in

equality, the second is obvious, the third follows from Lemma 1, the fourth follows from the cr-inequality, (7) and (9), while the last one is due to (10). The

proof of (40) uses Lemma 1, o2(9) > col, (10) and [23], page 121. To prove (41),| lnx| < x + x~x for x > 0 and Lemma 2 give

E sup |lna2(9)\ < p~xE supa2p(9) + e\ inf


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1066 P.M. ROBINSON AND P. ZAFFARONIIt suffices to show that the last integral is bounded. For all 8 > 0, there exists r\> 0such that L(s~l) < ?~s, e e (0, rj), so

/OO2 rT] 1 1e~te /(e) ds2pq/[(b+l)(q-2p)]. D

LEMMA 6. Under Assumptions A(2), B, C, D, E and F(l),for all p > 0 andk


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Proof. Take i'i< 12< < h- First assume i\ > 3, whence, for given fc and

90,-, 90,-t ^ ?-;^'where ?,-(?) = 9*^(?)/9??,-2 90,-2. Now

I OO I OO

'7=1 I 7=1so using Lemma 1,

It suffices to take p > 1. By Holder's inequality,oo r oo\ P/P ( oo \ 1-P/P

E it;(f)ix2_,-< E i?y(f)ip/p^(f),-w"^ E wk2.,7=117= 1 J 1.7 = 1JSO

By Assumption F(/), for all n > 0sup (Ol'iM?)'"'' ^ * SUPWS)p~r,p < Kj-v+wo-wKC T 1,we may choose 77such that (d + l)(p ? pn) > 1. Thus,

E sup {?-=- } < oo.oee\ o*2{6) IThe above proof implies that also

{001p

E 1^(011


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1068 P.M. ROBINSON AND P.ZAFFARONIwhere now f;(?) = 3*-V;(?)/9?i2-2 9^-2, while if i2= 2, i3 > 2,

where now ?,(?) = dk~2x//j(^)/d^3-2 3^-2- In the first of these cases theproof is seen to be very similar to that above after noting that, by the Cauchyinequality, (46) isbounded by

Ioooo 1 1/2 ooE woixLj E i*7(?)i + *E i*y(?)i,7= 1 7= 1 J7= 1

while in the second it ismore immediate; we thus omit the details. We are left withthe cases i\ ? 12= 13= 2 and i\ = 1, both of which are trivial. The details for(45) are very similar (the truncations in numerator and denominator match), andare thus omitted.

DefineT

gt(9) = ut(9)uft(9), GT(9) = T~x J>(0).t=\LEMMA 7. For some 8 > 0, under Assumptions A(2 +8), B, C, D,E and F(l),

(47) sup \QT(9) - Q(9)\ -* 0 0.5. as T - 00,6>e0am/ (?(#) is continuous in 9. If also Assumption F(2) holds,(48) sup ||Gr(0) - G(9)\\ -+ 0 as. as T -+ 00, 0 as. as r -> 00,6> 0awd //(0) w continuous in 9.

Proof. To prove (47), note first that, by Lemmas 1, 2, 3 and 5,supE\q0(9)\ < sup?|logoro2(0)| + sup?xo(#) < oc.00 0

Thus, by ergodicityQT(9)^Q(9) a.s.,

for all 9 0. Then uniform convergence follows on establishing the equicontinuity property

sup \Qt(9)-Qt(9)\^0 a.s.,0:\\0-e\\


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ESTIMATION OF ARCH(oo) MODELS 1069as s -> 0, and continuity of Q(9). By the mean value theorem it suffices to showthat

\\dQT(9)\\, |3G(0)||SUP?^? + SUP~^~ < ?? a*s-'0 II SO II e II 39 ||which, by Loeve ([23], page 121) and identity of distribution, is implied by?sup@ ||wo(#)|| < oo. By the definition of ut(9), and x2(/x) < K(x2 + 1),\\vt(0)\\< 2(|*,| + 1),we have

\\ut(9)\\


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1070 P.M. ROBINSON AND P. ZAFFARONIBecause

ooa2(9) = of (9)+Y, ^MOxlj(^),

7=0ln(l + x) < x for x > 0 and a2(9) >coL>0, it follows that

T\AT(9)\


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ESTIMATION OF ARCH(oo) MODELS 1071as s ?> 0. This completes the proof of (50). We omit the proofs of (51) and (52) asthey involve the same kind of arguments.

LEMMA 9. For some 8 > 0, under Assumptions A(2 + 8), B, C, D, E, F(l)and G, M(9) isfinite and positive definite for all 9 e 0.

Proof. Fix 9 e Q. Finiteness of M(9) follows from Lemma 6. Positivedefiniteness follows (by an argument similar to that of Lumsdaine [24] inthe GARCH(1, 1) case) if, for all nonnull (r + 2) x 1 vectors A,, \'M(9)\ =?{A/r0(#)}2 > 0, that is, that

(55) X'to(9)o$(9)^0 a.s.,since 0 < ofi(9) < oo a.s. Define

Tta)(9)= ^-lnof(9) = ot-2(9),ocoM0) = ? lna2(0) = -2at-2(6) T ^;(?)*,-;(/*),

o CO

^(9) = --lnof(9) = o-2(9) J2 IrfhOxf-ifa),a? 7=1so that rt(9) = (Ttco(9), rtfl(9), t^(9))'. Write k = (X\,X2, Xf3)\ where A.i and X2are scalar and A.3 is r x 1. Consider first the case k\ = X2 = 0, A.3 0. Suppose(55) does not hold. Then we must have

00J^X.'3iff\Oxf_j(fi) = 0 a.s.;=i

If A.^j(1)(?) i- 0, it follows that00

(56) (


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1072 P.M. ROBINSON AND P. ZAFFARONILet k be the smallest integer such that (f) ^ 0- Then (57) implies

r oo?t-k = cr~_\(9) M-/xo-^_1(?) E fj(P)xt-j(ti)j=k+\

But the left-hand side is nondegenerate and independent of the right-hand side, so(57) cannot hold. Next consider the case X\ = 0, X2 ?" 0> ^3 ?" 0- If (55) is not true,then, taking X2 = 1,we must have

00(58) Y{X^f\Oxt-j(^)-2fj(^)}xt-j(^) = 0 a.s.7=1

Let k be the smallest integer such that either^V^CO 7 0 or V*(f) ?" 0; thepreceding argument indicates that there exists such k. Then we have{2^(0 - X'3\l/(kX)(t;)(crt-k8t-k + /xo - li)}{cft-k?t-k + Mo - M}00= E {^?\Oxt-j(v) -2^j(0}xt-j(li) a.s.7=^+1

The left-hand side is a.s. nonzero and involves the nondegenerate random variable8t-k, which is independent of the right-hand side, so (58) cannot hold. We are leftwith the cases where X\ ^ 0. Taking X\ = ? 1 and noting that cr2(9)rta)(9) = 1,the preceding arguments indicate that there exist no X2 and A3 such that

^2cr2(9)rt^(9) + Xf3a2(9)rH(9) = 1 a.s.LEMMA 10. For some 8 > 0, under Assumptions A(2 + 8), B, C, D, E, F(l)

andH,inf Q(9) > Q(90).

Proof. We have

Q(0) - Q(9o) = E -^-lnj-^?[-1 +(/z-/z0)2? ?,? .^v ) mko) [cup) \a2(9)\ J ^ } lcr2(9).The second term on the right-hand side is zero only when \i = /xo and is positiveotherwise. Because x ? hut ? 1> 0 for x > 0, with equality only when x ? 1, itremains to show that

(59) lncr^(9)= lna02 a.s., some 9 #


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Acknowledgments. We thank an Associate Editor and referees for a numberof helpful comments that have led to a considerable improvement in the paper, andFabrizio Iacone for help with the numerical calculations referred to in Section 3.REFERENCES

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Department of EconomicsLondon School of EconomicsHoughton StreetLondon WC2A 2AEUnited KingdomE-MAIL: [email protected]

Tanaka Business SchoolImperial College LondonSouth Kensington CampusLondon SW7 2AZUnited KingdomE-MAIL: [email protected]

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