IGARCH models

ARTICLE IN PRESS

Journal of Econometrics 140 (2007) 849–873 www.elsevier.com/locate/jeconom

Self-weighted and local quasi-maximum likelihood estimators for ARMA-GARCH/IGARCH models Shiqing Ling Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong Available online 1 September 2006

Abstract The limit distribution of the quasi-maximum likelihood estimator (QMLE) for parameters in the ARMA-GARCH model remains an open problem when the process has infinite 4th moment. We propose a self-weighted QMLE and show that it is consistent and asymptotically normal under only a fractional moment condition. Based on this estimator, the asymptotic normality of the local QMLE is established for the ARMA model with GARCH (finite variance) and IGARCH errors. Using the self-weighted and the local QMLEs, we construct Wald statistics for testing linear restrictions on the parameters, and their limiting distributions are given. In addition, we show that the tail index of the IGARCH process is always 2, which is independently of interest. r 2006 Elsevier B.V. All rights reserved. JEL classification: C12; C22 Keywords: Asymptotic normality; ARMA-GARCH model; GARCH model; Quasi-maximum likelihood estimation; Self-weighted estimation

1. Introduction (G)ARCH-type models have been extensively used in economics and finance since Engle (1982) and Bollerslev (1986). Weiss (1986) first presented the asymptotic theory for the ARMA-ARCH model assuming pffiffiffiffifinite fourth moment (see also Tsay, 1987). The ARCH (1) model is defined as et ¼ Zt ht and ht ¼ a0 þ ae2t1 , where a0 40, a40 and Zt  i.i.d. Nð0; 1Þ. It is well known that fet g is strictly stationary when a 2 ð0; 3:5620   Þ, and et has a Tel.: +852 23587459; fax: +852 23591016.

E-mail address: [email protected]. 0304-4076/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jeconom.2006.07.016

ARTICLE IN PRESS 850

S. Ling / Journal of Econometrics 140 (2007) 849–873

finite second moment when a 2 ð0; 1Þ, a finite fourth moment when a 2 ð0; 0:57   Þ and a finite eighth moment when a 2 ð0; 0:3   Þ. It is clear that the moment conditions of et link directly to the restriction on the parameters and that Ee4t o1 is a very strong condition. The problem on the asymptotic theory for (G)ARCH-type models under weak moment conditions has attracted a lot of attention in econometrics and statistics. For the GARCH(1,1) model, including the case when Ee2t ¼ 1, Lee and Hansen (1994) and Lumsdaine (1996) showed that the quasi-maximum likelihood estimator (QMLE) of the parameters is consistent and asymptotically normal. For the GARCHðr; sÞ model as given below in Section 2, Berkes et al. (2003) proved strong consistency and asymptotic normality of the QMLE. The same results were proved by Hall and Yao (2003) and Francq and Zakoı¨ an (2004). But Hall and Yao (2003) assumed that Ee2t o1 and discussed the case when EZ4t ¼ 1. As far as we know, the weakest condition is presented by Francq and Zakoı¨ an (2004). Consistency of the QMLE was also discussed by Jeantheau (1998) and Ling and McAleer (2003a). We also refer to recent works by Peng and Yao (2003) and Berkes and Horvath (2004) in this area. Basically, the asymptotic theory of the GARCH model is known. However, we know less about the asymptotic theory of the ARMA-GARCH model. When Ee4t o1, the consistency and asymptotic normality of its local QMLE were given by Ling and Li (1997), while the strong consistency and asymptotic normality of its global QMLE were proved by Francq and Zakoı¨ an (2004). It is challenging to establish the limiting distribution of the QMLE when Ee4t ¼ 1. Li et al. (2002) identified this as an open problem. This difficulty was also recognized by Francq and Zakoı¨ an (2004). To understand this, we begin with the AR(1) model: yt ¼ fyt1 þ et with et in (2.2) and let ^ be the LSE of f. Then, pffiffinffiðf ^  fÞ ¼ ðPn y2 =nÞ1 ðPn y et =pffiffinffiÞ. The central f n n t¼1 t1 t¼1 t1 limit theorem requires Eðyt1 et Þ2 ¼ Eðy2t1 ht Þo1 and hence the condition Ee4t o1 should ^ . This is similar to the AR model with i.i.d. be minimal for asymptotic normality of f n errors. The minimal condition for asymptotic normality of the LSE is Ey2t o1. When Ey2t ¼ 1, the asymptotic distributions of existing estimators, such as LSE, LAD and Mestimators, do not have a closed form (see Davis et al., 1992). Ling, 2005 introduced a selfweighted LAD estimator. The purpose of the weighting in Ling (2005) is to downweight the covariance matrix such that asymptotic normality can be recovered. For the ARCH model, a similar weighted LAD and Lp -estimator are used by Horvath and Liese (2004) and Chan and Peng (2005). This paper first proposes a self-weighted QMLE for the ARMA-GARCH model. The basic idea is to use a weight to control the quasi-information matrix as in (3.4) below. We show that the self-weighted QMLE is consistent and asymptotically normal under only a fractional moment condition, i.e., Ejet ji o1 for some i40. For the QMLE, ht ðyÞ itself is one sort of weight in the quasi-log-likelihood function (3.3). Ling (2004) shows that the QMLE is asymptotically normal even if Ee2t ¼ 1 for a double AR model and, hence, it is reasonable to expect that asymptotic normality of the QMLE holds when Ee4t ¼ 1. We next establish asymptotic normality of the local QMLE for ARMA-GARCH (finite variance) and -IGARCH models. Based on the two estimators, Wald statistics are constructed for testing linear restrictions on the parameters. When Ee2t ¼ 1, our result is entirely different from that in Mikosch et al. (1995) for the infinite-variance ARMA with i.i.d errors. This paper is organized as follows. Section 2 presents the model and assumptions. Section 3 studies the self-weighted QMLE. Section 4 studies the local QMLE. Concluding remarks are offered in Section 5. All proofs are given in the Appendix.

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

851

2. Model and assumptions Assume that fyt : t ¼ 0; 1; 2; . . .g are generated by the ARMA-GARCH model: yt ¼ m þ

p X

fi yti þ

i¼1

e t ¼ Zt

q X

ci eti þ et ,

(2.1)

i¼1

pffiffiffiffi ht and

ht ¼ a0 þ

r X

ai e2ti þ

s X

i¼1

bi hti ,

(2.2)

i¼1

where a0 40, ai X0 ði ¼ 1; . . . ; rÞ, bj X0 ðj ¼ 1; . . . ; sÞ, and Zt is a sequence of i.i.d. random variables with zero mean and variance 1. Denote g ¼ ðm; f1 ; . . . ; fp ; c1 ; . . . ; cq Þ0 , d ¼ ða0 ; a1 ; . . . ; ar ; b1 ; . . . ; bs Þ0 , and y ¼ ðg0 ; d0 Þ0 . The parameter subspaces, Yg  Rpþqþ1 and Yd  R0rþsþ1 , are compact, where R ¼ ð1; 1Þ and R0 ¼ ½0; 1Þ. Let Y ¼P Yg  Yd and m ¼ pP þ q þ r þ s þ 2 andPy0 be the true value ofP y. Denote aðzÞ ¼ ri¼1 ai zi , bðzÞ ¼ 1  si¼1 bi zi ,fðzÞ ¼ 1  pi¼1 fi zi and cðzÞ ¼ 1 þ qi¼1 ci zi . We introduce the following conditions: Assumption 2.1. y0 is an interior point in Y and for each y 2 Y, fðzÞa0 and cðzÞa0 when jzjp1, and fðzÞ and cðzÞ have no common root with fp a0 or cq a0. P Assumption 2.2. aðzÞ and bðzÞ have no common root, að1Þa0, ar þ bs a0, and si¼1 bi o1 for each y 2 Y. Assumption 2.3. Z2t has a nondegenerate distribution with EZ2t ¼ 1. Assumption 2.4. Ejet j2i o1 for some i40. Assumption 2.1 implies the stationarity, invertibility and identifiability of model (2.1), under which it follows that c1 ðzÞ ¼

1 X

ac ðiÞzi

and

fðzÞc1 ðzÞ ¼

i¼0

1 X

ag ðiÞzi ,

(2.3)

i¼0

where supYg ac ðiÞ ¼ Oðri Þ and supYg ag ðiÞ ¼ Oðri Þ with r 2 ð0; 1Þ. Assumption 2.2 is the identifiability condition for model (2.2) and, by Lemma 2.1 in Ling (1999), the condition P s i¼1 bi o1 is equivalent to ! b1    bs 0prðGÞo1 where G ¼ , (2.4) I s1 O I k is the k  k identity matrix and rðBÞ is the spectral radius of matrix B. Under this condition, we have b1 ðzÞ ¼

1 X i¼0

ab ðiÞzi

and

aðzÞb1 ðzÞ ¼

1 X

ad ðiÞzi ,

(2.5)

i¼1

where supYd ab ðiÞ ¼ Oðri Þ, supYd ad ðiÞ ¼ Oðri Þ and r ¼ rðGÞ. Assumption 2.3 is necessary to ensure that ht is not almost surely (a.s.) a constant. When i ¼ 1, the necessary and

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

852

sufficient condition for Assumption 2.4 is r X

a0i þ

i¼1

s X

b0i o1,

(2.6)

i¼1

under which the GARCH model (2.2) has a finite variance. Furthermore, we give the necessary and sufficient condition for Assumption 2.4 when i 2 ð0; 1 in the following theorem. Theorem 2.1. Let i 2 ð0; 1 and b~ ¼ minfa0i ; b0j : i ¼ 0; 1; . . . ; r; j ¼ 1; . . . ; sg. Suppose fet g is generated by model (2.2). (i) If i   iY 0 1   there exists an integer i0 such that E Ak  o1, (2.7)  k¼0  ~ then fet g is strictly stationary and ergodic with Ejet j2i o1; (ii) If b40 and fet g is strictly ~ stationary with Ejet j2i o1, then (2.7) holds; (iii) if b40, r X i¼1

a0i þ

s X

b0j ¼ 1,

(2.8)

j¼1

and Zt has a positive density on R such that EjZt jt o1 for all tot0 and EjZt jt0 ¼ 1 for some t0 2 ð0; 1, then limx!1 x2 Pðjet j4xÞ exists and is positive, where 1 0 a01 Z2t    a0r Z2t b01 Z2t    b0s Z2t C B C B I r1 O O C, At ¼ B B a01  a0r b01  b0s C A @ O I s1 O pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and kBk ¼ trðBB0 Þ for a vector or matrix B. We can show that (2.6) and (2.7) are equivalent when i ¼ 1. Under (2.8), model (2.2) is the IGARCH model with an infinite variance. Theorem 2.1 (iii) implies that the tail index of the IGARCHðr; sÞ process is always 2. When r ¼ s ¼ 1, this tail index was also obtained by Basrak et al. (2002). 3. Self-weighted QMLE for ARMA-GARCH Given the observations fyn ; . . . ; y1 g and the initial values fy0 ; y1 ; y2 ; . . .g which are generated by models (2.1)–(2.2), we can write the parametric model as et ðgÞ ¼ yt  m 

p X

fi yti 

i¼1

Zt ðyÞ ¼ et ðgÞ=

pffiffiffiffiffiffiffiffiffiffi ht ðyÞ and

q X

ci eti ðgÞ,

(3.1)

i¼1

ht ðyÞ ¼ a0 þ

r X i¼1

ai e2ti ðgÞ þ

s X i¼1

bi hti ðyÞ.

(3.2)

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

853

Here, Zt ðy0 Þ ¼ Zt , et ðg0 Þ ¼ et and ht ðy0 Þ ¼ ht . The log-quasi-likelihood function based on fet ðgÞ : t ¼ 1; . . . ; ng is Ln ðyÞ ¼

n 1X l t ðyÞ and n t¼1

1 e2 ðgÞ . l t ðyÞ ¼  log ht ðyÞ  t 2 2ht ðyÞ

(3.3)

The QMLE of y0 is defined as the maximizer of Ln ðyÞ on Y. The quasi-score function and the quasi-information matrix are given in the Appendix. Eq. (A.18) in the Appendix shows that  2  q l t ðyÞ 4 2  (3.4) sup  qg qg0 pCxrt1 ð1 þ Zt Þ, Y P i where xrt ¼ 1 þ 1 i¼0 r jyti j, and r 2 ð0; 1Þ and C are some constants not depending on t. This is why we generally need Ee4t o1 for asymptotic normality of the QMLE. It is possible to obtain an estimator such that it is asymptotically normal if we can downweight x4rt1 . Thus, we introduce the weighted log-quasi-likelihood function: Lsn ðyÞ ¼

n 1X wt l t ðyÞ, n t¼1

(3.5)

where l t ðyÞ is defined as in (3.3) and wt satisfies the following assumption: Assumption 3.1. wt ¼ wðyt1 ; yt2 ; . . .Þ and w is a measurable, positive and bounded function on RZ0 with Eðwt x4rt1 Þo1 and Z0 ¼ f0; 1; 2; . . .g for any r 2 ð0; 1Þ. In practice, we do not have the initial values yi when ip0 and hence they have to be replaced by some constants. Denote et ðgÞ, ht ðyÞ and wt as e~t ðgÞ, h~t ðyÞ and w~ t , respectively, when yi is a constant not depending on parameters when ip0. The weighted log-quasilikelihood function (3.5) is modified as follows: 1 L~ sn ðyÞ ¼ n

n X t¼1

w~ t l~t ðyÞ

and

1 e~ 2 ðgÞ l~t ðyÞ ¼  log h~t ðyÞ  t . 2 h~t ðyÞ

(3.6)

The following assumption makes the initial value yi ignorable when ip0: Assumption 3.2. Ejwt  w~ t ji0 =4 ¼ Oðt2 Þ, where i0 ¼ minfi; 1g. Since the weight wt is determined by fyt g itself, the maximizer of L~ sn ðyÞ on Y is called the self-weighted QMLE, denoted as y^ sn . Let !p and !L , respectively, denote convergence in probability and in distribution as n ! 1. Our result for y^ sn is as follows. Theorem 3.1. Suppose that Assumptions 2.1–2.4 and 3.1–3.2 hold. Then, ðiÞ

y^ sn !p y0 , pffiffiffi ^ 1 nðysn  y0 Þ!L Nð0; S1 0 O0 S0 Þ

if EZ4t o1 and J40,   pffiffiffi where S0 ¼ E½wt U t ðy0 ÞU 0t ðy0 Þ, O0 ¼ E½w2t U t ðy0 ÞJU 0t ðy0 Þ, J ¼ k13 kk3 , k3 ¼ EZ3t = 2, k ¼ pffiffiffi 1=2 ðEZ4t  1Þ=2 and U t ðyÞ ¼ ½ht qet ðgÞ=qy; ð 2ht Þ1 qht ðyÞ=qy. ðiiÞ

It is readily shown that J40 if and only if PðZ2t  cZt  1 ¼ 0Þo1 for any c 2 R. A simple condition for this is that Zt has a positive density on some interval. Since y0 is an interior point, it excludes the ARCH as a special case of model (2.2). However, our

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

854

framework allows us to deal with an ARCH model when it has been correctly identified. From its proof, Theorem 3.1 holds for the ARMA-ARCH model by removing the components corresponding to bj ’s, and it holds similarly for the AR-ARCH and ARCH models. The matrices S0 and O0 can be consistently estimated by their sample averages, i.e., 1 S^ 0n ¼ n

n X

0

w~ t U^ st U^ st

and

t¼1

^ 0n ¼ 1 O n

n X

0

w~ 2t U^ st J^ sn U^ st ,

(3.7)

t¼1

where U^ st ¼ U~ t ðy^ sn Þ and J^ sn is defined as J with k3 and k replaced by 2 33 n n ^ X 1 1 X e~4 ðy^ sn Þ 1 6 e~ t ðysn Þ 7  , w~ t 4qffiffiffiffiffiffiffiffiffiffiffiffiffi5 and k^ sn ¼ w~ t t2 k^ 3sn ¼ pffiffiffi 2w^ 1 t¼1 h~ ðy^ Þ 2 2w^ 1 t¼1 h~t ðy^ sn Þ t sn P respectively, and w^ 1 ¼ nt¼1 w~ t . Based on these, we can undertake statistical inference for the ARMA-GARCH model, such as the goodness-of-fit test in Li and Mak (1994). Here, we only consider the Wald test statistic, denoted by W sn , for the p1 linear hypothesis of the form: H0 : Gy ¼ y10 , in the usual notation. Using a similar method as for (A.17) and ^ 0n ¼ O0 þ op ð1Þ. Thus, by Theorem 3.1, ^ 0n ¼ S0 þ op ð1Þ and O (A.19), we can show that S we have a corollary as follows. Corollary 3.1. If Assumptions 2.1–2.4 and 3.1–3.2 hold, J40 and EZ4t o1, then it follows that 1

1

^ 1n S ^ G0 Þ1 ðGy^ sn  y10 Þ!L w2 , W sn ¼ nðGy^ sn  y10 Þ0 ðGS^ 1n O p1 1n ^ 0n are defined as in (3.7). ^ 0n and O under H0 , where S We should mention that W sn cannot be used for testing if the coefficients in the GARCH part are zero since y0 is an interior point in Y under H0 . To use the result in this section, we need to select a weight wt . Obviously, there are a lot of weights that satisfy Assumptions 3.1–3.2. When i ¼ 12 (i.e., Ejet jo1), as in Ling (2005), one natural weight is ( )!4 1 X 1 1 wt ¼ max 1; C jy jIfjytk j4Cg , (3.8) 9 tk k¼1 k for some C40. This weight satisfies Assumptions 3.1–3.2 and has a connection to Huber’s robust estimator for the regression model. It downweights the quasi-information matrices with large points (in absolute value) such that the magnitudes of its elements are not larger than C 4 , but takes full advantage of all matrices without these points. When q ¼ s ¼ 0 (AR-ARCH model), for any i40, the weight can be selected as ( )!4 pþr X 1 1 wt ¼ max 1; C jy jIfjytk j4Cg . (3.9) 9 tk k¼1 k   When i 2 0; 12 and q40 or s40, the weight wt may not be well defined and needs to be modified as follows: ( )!4 1 X 1 wt ¼ max 1; C 1 jy jIfjytk j4Cg . (3.10) 1þ8=i tk k¼1 k

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

855

In this case, we need to determine i such that Assumption 2.4 holds. This is not an easy task, but there are two ways to do this. One way is to use a simulation method to verify the condition (2.7) for a possible i. This is the same as the method used for verifying the stationarity condition of model (2.2) suggested by Bougerol and Picard (1992) and Basrak et al. (2002). But this method needs some information on the distribution of Zt . The test statistic in Koul and Ling (2006) can be used to test possible distributions. Another way is to use the Hill estimator to estimate the tail index of fyt g and its estimator may provide some useful guidelines for the choice of i. The constant i can be any value less than the tail index of fyt g. In general, it is difficult to compare the efficiency of estimators based on different 1 weights. However, when EkU t ðy0 Þk2 o1, S1 0 O0 S0 , based on the weights in (3.8)–(3.10), converges to the asymptotic covariance matrix of the QMLE in Section 4 as C ! 1. When EkU t ðy0 Þk2 ¼ 1 (see the example on the AR-ARCH model in Francq and Zakoı¨ an, 2004), for the weight in (3.9) or (3.10), we have 1 1 kS1 0 O0 S0 kplkS0 k ! 0

as C ! 1,

for some l40. That is, the asymptotic variance of the self-weighted QMLE can be as small as we want only if C is large enough. For the self-weighted LAD estimator for the infinitevariance AR model with i.i.d. errors in Ling (2005), simulation results show that it works well when C is the 90% or 95%-quantile of data fy1 ; . . . ; yn g. 4. Local QMLE for ARMA-GARCH/IGARCH When studying the quasi-information matrix in detail, we find that EH gt ðyÞo1 only if 0 Ee2t o1, where H gt ðyÞ ¼ h2 t ðyÞ½qht ðyÞ=qg½qht ðyÞ=qg  (see the proof in the Appendix). Another second-moment condition is required by the factor e2t ðgÞ=ht ðyÞ in (A.3). Note that e2t ðg0 Þ=ht ðy0 Þ ¼ Z2t is independent of Ft1 . If we restrict the estimator on the subspace pffiffiffi Yn ¼ fy : ky  y0 kpM= ng for any fixed M40, e2t ðgÞ=ht ðyÞ is expected to be sufficiently close to Z2t . Thus, Ee2t o1 may be sufficient for asymptotic normality of the local QMLE. In addition, since the tail index of the IGARCH process is 2, ht ðyÞ may be able to reduce a little bit of the required moment of et in H gt ðyÞ such that the asymptotic normality of the local QMLE holds for the ARMA-IGARCH model. Thus, this section focuses on the local QMLE. By using y^ sn in Theorem 3.1 as an initial estimator of y0 , we obtain the local MLE through the following one-step iteration: " #1 n n 2~ ^ X X l q ð y Þ ql~t ðy^ sn Þ t sn y^ n ¼ y^ sn  . (4.1) 0 qy qy qy t¼1 t¼1 For this local QMLE, we have the following result: Theorem 4.1. Suppose that Assumptions 2.1–2.3 and 3.1–3.2 hold and that (2.6) or the condition of Theorem 2.1 (iii) is satisfied. If EZ4t o1, J40 and y^ n is obtained through (4.1), then pffiffiffi ^ nðyn  y0 Þ!L Nð0; S1 OS1 Þ, where S ¼ E½U t ðy0 ÞU 0t ðy0 Þ and O ¼ E½U t ðy0 ÞJU 0t ðy0 Þ.

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

856

For the ARMA-IGARCH model, the condition b01 a0 is critical in the proof (see Lemmas A.5–A.6) and hence Theorem 4.1 cannot be applied to the ARMA-IARCH model. However, when Ee2t o1, we can take ~i ¼ 0 in Lemma A.5 such that all the proofs follow. Thus, Theorem 4.1 in this case holds for ARMA-ARCH, GARCH and ARCH models only if we remove the reductant parameters and the corresponding components in the covariance matrix. In general, it is not easy to compare the efficiency of the QMLE and the self-weighted QMLE in Theorem 3.1. However, when J ¼ diagf1; 1g, i.e., Zt has the same moments as those of Nð0; 1Þ up to fourth-order, we can show that the QMLE is more efficient than the self-weighted QMLE. In fact, in this case, S ¼ O ¼ EðX 1t X 01t þ X 2t X 02t Þ, S0 ¼ Eðwt X 1t X 01t þ wt X 2t X 02t Þ

and

O0 ¼ Eðw2t X 1t X 01t þ w2t X 2t X 02t Þ,

1=2

where X 1t ¼ ht qet ðgÞ=qy and X 2t ¼ ð2ht Þ1 qht ðyÞ=qy. Let b and c be two any m-dimensional constant vectors. Then, c0 S0 bb0 S0 c ¼ fE½ðc0 X 1t wt ÞðX 01t bÞ þ ðc0 X 2t wt ÞðX 02t bÞg2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p Eðc0 X 1t wt Þ2 EðX 01t bÞ2 þ Eðc0 X 2t wt Þ2 EðX 02t bÞ2 p½Eðc0 X 1t wt Þ2 þ Eðc0 X 2t wt Þ2 ½EðX 01t bÞ2 þ EðX 02t bÞ2  ¼ c0 O0 cb0 Sb ¼ c0 O0 ðb0 SbÞc. Thus, O0 b0 Sb  S0 bb0 S0 X0 (a positive semi-definite matrix) and hence b0 S0 O1 0 S0 b ¼ 1=2 1=2 1 1 trðO0 S0 bb0 S0 O0 Þptrðb0 SbÞ ¼ b0 Sb. Thus, we have S1 O S XS . 0 0 0 For the pure (G)ARCH model (2.2), the asymptotic covariance matrices of the QMLE y^ n and the self-weighted QMLE y^ sn are S1 OS1 ¼ kE1 ðX 2t X 02t Þ, 1 1 0 2 0 1 0 S1 0 O0 S0 ¼ kE ðwt X 2t X 2t ÞEðwt X 2t X 2t ÞE ðwt X 2t X 2t Þ,

respectively. Using the same method as for the previous case, we can show that 1 1 1 S1 0 O0 S0 XS OS . Thus, the QMLE is always more efficient than the self-weighted QMLE. For the pure ARCH model, the asymptotic covariance of the weighted L2 1 0 estimator in Horvath and Liese (2004) is S1 1 O1 S1 , where S1 ¼ Eðwt Z t1 Z t1 Þ and O1 ¼ 0 0 2 2 2 2 kEðwt ht Z t1 Zt1 Þ with Z t ¼ ð1; et ; . . . ; etr Þ . Note that X 2t ¼ Zt =ht in this special case. In general, the QMLE is more efficient than the weighted L2 -estimator. In fact, for any m  1 nonzero constant vectors b and c, 

 2 1 0 c0 S1 bb0 S1 c ¼ E ðwt ht c0 Z t1 Þ b Z t1 ht

2 1 0 2 0 pEðwt ht c Z t1 Þ E b Z t1 ¼ c0 O1 cb0 Sb=k. ht 1=2

1=2

Thus, S1 bb0 S1 pO1 b0 Sb=k and hence b0 S1 O1 S1 bb0 S1 O1 Þpb0 Sb=k. 1 S1 b ¼ trðO1 1 1 1 1 1 Thus, S1 O1 S1 XkS . Based on the weight (3.9), S0 O0 S0 ! kS1 as C ! 1, while 1 2 4 S1 1 O1 S1 ! 1 if EkZ t k o1 but EkZ t k ¼ 1 as C ! 1. Using kZ t1 k=ht p a constant a.s., we can show that the self-weighted QMLE based on the weight (3.9) with a large C is

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

more efficient than the weighted-L2 estimator based kZ t1 k2 Þsuggested by Horvath and Liese (2004). The covariance matrices, S and O, can be estimated by ^n ¼ 1 S n

n X

0 U^ t U^ t

t¼1

and

^n ¼ 1 O n

n X

on

857

the

weight

0 U^ t J^ n U^ t ,

1=ð1þ

(4.2)

t¼1

where U^ t ¼ U~ t ðy^ n Þ and J^ n is defined as J with k3 and k replaced by k^ 3n and k^ n , 2 33 n n ^ X 1 1 X e~4t ðy^ n Þ 1 6 e~t ðyn Þ 7  . k^ 3n ¼ pffiffiffi 4qffiffiffiffiffiffiffiffiffiffiffiffi5 and k^ n ¼ 2n t¼1 h~2 ðy^ Þ 2 2n t¼1 h~t ðy^ n Þ t n Using Lemmas A.5–A.6, Lemmas 6 as for Lemma 7, we can show that J^ n ¼ J þ op ð1Þ, ^ n ¼ O þ op ð1Þ. Thus, by Theorem 4.1, we have the following ^ n ¼ S þ op ð1Þ and O S corollary for testing the null hypothesis H 0 , which is defined as in Corollary 3.1: Corollary 4.1. Under the assumption of Theorem 4.1, it follows that  1 1 ^ n S^ 1 G0 W n ¼ nðGy^ n  y10 Þ0 GS^ n O ðGy^ n  y10 Þ!L w2p1 , n ^ n are defined as in (4.2). ^ n and O under H 0 , where S 5. Concluding remarks Given a data set, different estimators may give different results in practice. To see if the QMLE should be used, it will be helpful to estimate the tail index of the data. If it is greater than 4, then the global QMLE can be used. If it is in ½2; 4, a two-step estimator should be considered, i.e., first obtaining a self-weighted QMLE and then using it to obtain the QMLE via a one-step iteration. When Zt Nð0; 1Þ, the QMLE is the MLE and hence it is efficient. Theorem 3 with Theorem 3.1 provide an approach to obtain an efficient estimator for the ARMA-GARCH (finite variance)/IGARCH models. Such an approach is novel and has never appeared in the literature before. To make sure if Zt Nð0; 1Þ, we can perform a test by using the statistic in Koul and Ling (2006). If there is strong evidence that Zt is not normal, we can further perform the adaptive estimator along the lines as in Drost et al. (1997) and Ling and McAleer (2003b), subject to some regular conditions on the density of Zt . When the tail index is in (0, 2), we should consider only the self-weighted QMLE. But the results will depend on the choice of weights or the constant C in (3.8)–(3.10). As the Co-Editor has noted, we should acknowledge that using other weights may produce different finite sample results and a different asymptotic variance. We do not have a theory to support the choice of the weight or C yet. Which result is more reliable should depend on the features of the data and their source. It remains a difficult problem to set a sense for comparing different estimators and to select the C or other weights such that the estimator is optimal under this sense. In summary, this paper has proposed a self-weighted QMLE for parameters in the ARMA-GARCH model and showed that it is consistent and asymptotically normal under only a fractional moment condition of GARCH errors. Asymptotic normality of the local QMLE has been established for ARMA model with GARCH and IGARCH errors.

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

858

Wald statistics have been investigated for testing linear restrictions on the parameters in the model. In all results, we assume that EZ4t o1. This assumption can be relaxed by using the LAD and the self-weighted method in this paper. The self-weighted principle can be applied to other estimators, such as the M- and the quantile-estimators, to other ARCHtype models, such as threshold AR-ARCH models, and to multivariate time series models. Acknowledgments The author greatly thanks Co-Editor Peter M. Robinson, the Associate Editor, two referees, Professors V.A. Unkefer and Hao Yu for their invaluable comments and thanks Professors C. Francq and L. Horvath for sending their reprints during the revision of this paper and the Hong Kong RGC (CERG #HKUST6273/03H, #HKUST6428/06H and #HKUST602205/05P) for financial support. This paper is the revised version of Ling (2003), which included the self-weighted LSE for the ARMA-GARCH model and was completed when the author visited the Department of Statistics, Stanford University. The author thanks the department for its hospitality. This early version was presented in the 2004 International Symposium on Forecasting, Sydney, and the 2004 International Chinese Statistical Association, Singapore. Appendix A. Proofs ProofQ of Theorem 2.1. By (2.7) and Jensen’s inequality, we can show that 0 1 Elnk ii¼0 Ai ko0. Using the same method as in Bougerol and Picard (1992), we can show that the following representation holds: " # j1 1 Y X pffiffiffiffi 0 et ¼ Zt ht and ht ¼ a00 1 þ urþ1 Ati ztj a:s:; j¼1

i¼0

and hence fet g is strictly stationary and ergodic, zt ¼ ðZ2t ; 0; . . . ; 0; 1; . . . ; 0Þ0ðrþsÞ1 with the first component Z2t and the ðr þ 1Þth component 1, and ui ¼ ð0; . . . ; 0; 1; . . . ; 0Þ0ðrþsÞ1 with the ith component P 01.1 Qj1 Qi0 1 ~ Let Bt ¼ zt þ ij¼1 i¼0 Ati . We rewrite ht as i¼0 Ati ztj and At ¼ " # 1 k 1 Y X u0 ht ¼ a00 u0 Bt þ A~ ti i Btki , (A.1) rþ1

rþ1

k¼1

0 1

0

i1 ¼0

where u0rþ1 zt ¼ 1 is used. By (A.1), it follows that Ehit pOð1Þ þ Oð1Þ

1 X

ðEkA~ t ki Þk o1.

k¼1

Thus, Ejet j o1 and hence (i) holds. Note that model (2.2) has only one strictly stationary solution representation P withQthe j1 (A.1) (see Bougerol and Picard, 1992). Denote hJt ¼ a00 ð1 þ Jj¼1 u0rþ1 i¼0 Ati xtj Þ. Then ðhJt  hJ1;t Þi ! 0 a.s. when J ! 1. fhiJt g is an increasing sequence in terms of J and EQsupJ hiJt pEhit o1. Thus, by the dominated convergence theorem, we have i i i ~ Eðu0rþ1 J1 i¼0 Ati ztj Þ ¼ EðhJt  hJ1;t Þ =a00 ! 0Qas J ! 1. Since b40, Q 2 1 there exist J 1 J 1 1 0 0 and J 2 such that all the elements of d 1t ðurþ1 i¼0 Ati Þ and d 2t Ji¼0 Ati ztj are a.s. 2i

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

859

positive, and 0oEd i1it o1 and 0oEd i2it o1, where djit is the ith element of d jt as j ¼ 1; 2. Note that !i " ! #i J 1 þiY J1Y þi0 1 0 þJ 2 1 0 0 E urþ1 Ati ztj ¼ E d 1t Ati d 2;J 1 þi0 ! 0, i¼J 1

i¼0

QJ 1 þi0 1

as i0 ! 1. Let aijt be the ði; jÞ-element of i¼J 1 Ati . From the preceding equation, we know that Eðd 1it aijt d 2j;J 1 þi0 Þi ! 0 as i0 ! 1. Since d 1it , aijt , and d 2j;J 1 þi0 areQindependent, J 1 þi0 1 i we know Ati ki ijt ! 0 as i0 ! 1. Thus, there exists i0 such that Ek i¼J 1 Qi0 1that Ea i ¼ Ek i¼0 Ati k o1, i.e., (ii) holds. 2 (iii) Since (2.6) is the necessary and Qi0 1sufficient condition for Eet o1, (2.8) implies 2 Eet ¼ 1. By (ii) of this theorem, Ek k¼0 Ak kX1 for any i0 X1. Thus, lnEk

n Y

Ak kX0,

k¼1

Q Q for all nX1. Note that u0i nk¼1 Ak uj is the ði; jÞth element of nk¼1 Ak . We have 2   !2 31=2  Y n rþs X rþs n X Y   E Ak  ¼ E4 u0i A k uj 5 k¼1  i¼1 j¼1 k¼1 " !# rþs X rþs n rþs X rþs Y X X 0 pE ui Ak uj ðu0i An uj Þpðr þ sÞ2 , ¼ i¼1 j¼1

k¼1

i¼1 j¼1

where A ¼ EAt . By the previous two inequalities, it follows that 1 ln EkA1 . . . An k ¼ 0. n!1 n By Theorems 2.4 and 3.1 (B) in Basrak et al. (2002), the conclusion (iii) holds. This completes the proof. & lim

We next give some basic formulas as follows: q et ðgÞ ¼  c1 ð1Þ, qm q et ðgÞ ¼  c1 ðBÞytj ; 1pjpp, q fj q et ðgÞ ¼  c1 ðBÞetj ðyÞ; 1pjpq, qcj a0 þ b1 ðBÞaðBÞe2t ðgÞ, ht ðyÞ ¼ bð1Þ q ht ðyÞ ¼ b1 ðBÞ~et ðyÞ, qd

qht ðyÞ qet ðgÞ 1 ¼ 2b ðBÞaðBÞ et ðgÞ , qg qg where e~ t ðyÞ ¼ ½1; e2t1 ðgÞ; . . . ; e2tr ðgÞ; ht1 ðyÞ; . . . ; hts ðyÞ0 and B is the back-shift operator. Similarly, we can write down the formula for q2 et ðgÞ=qgi1 qgi2 and q2 ht ðyÞ =qyj 1 qyj 2 , where i1 ; i2 ¼ 1; . . . ; p þ q þ 1 and j 1 ; j 2 ¼ 1; . . . ; m. We further give the quasi-score function and

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

860

the quasi-information matrix as follows.

2  ql t ðyÞ et ðgÞ qet ðgÞ 1 et ðgÞ qht ðyÞ ¼ þ , 1 qy ht ðyÞ qy 2ht ðyÞ ht ðyÞ qy

(A.2)

q2 l t ðyÞ 1 q et ðgÞ q et ðgÞ 1 qht ðyÞ qht ðyÞ e2t ðgÞ ¼   qg qg0 ht ðyÞ qg qg0 qg0 ht ðyÞ 2h2t ðyÞ qg " #

1 e2t ðgÞ 1 qht ðyÞ qht ðyÞ 1 q2 ht ðyÞ 1  2 þ þ 2 ht ðyÞ qg0 ht ðyÞ qg qg0 ht ðyÞ qg 

2et ðgÞ q et ðgÞ q ht ðyÞ et ðgÞ q2 et ðgÞ  , ht ðyÞ qg qg0 ht ðyÞ qg qg0

ðA:3Þ

q2 l t ðyÞ 1 qht ðyÞ qht ðyÞ e2t ðgÞ 0 ¼  0 qd qd 2h2t ðyÞ qd qd ht ðyÞ #

" 1 e2t ðgÞ 1 qht ðyÞ qht ðyÞ 1 q2 ht ðyÞ 1  2 þ . 0 þ 2 ht ðyÞ ht ðyÞ qd qd0 ht ðyÞ qd qd

ðA:4Þ

Similarly, we can write down q2 l t ðyÞ=ðqg qd0 Þ. We now give three basic Lemmas. They are commonly used in the proof. Lemma A.1. Let xrt be defined as in (3.4). If Assumptions 2.1–2.2 hold, then there exist constants C and r 2 ð0; 1Þ such that the following holds uniformly in Y:    2  qet ðgÞ q et ðgÞ     are bounded a:s: by Cxrt1 , and  ðiÞ et1 ðgÞ;  qg  qg qg0  ðiiÞ ht ðyÞ is bounded a:s: by Cx2rt1 . Proof. It directly comes from (2.3) and (2.5). This completes the proof.

&

Lemma A.2. Let xrt be defined as in (3.4). Under Assumptions 2.1–2.2, there exists a neighborhood Y0 of y0 and a constant r 2 ð0; 1Þ such that    1 qht ðyÞ  pCxi1 , ðiÞ sup rt1  Y0 ht ðyÞ qd    1 q2 ht ðyÞ i1  ðiiÞ sup 0 pCxrt1 ,  Y0 ht ðyÞ qd qd for any i1 2 ð0; 1Þ, where C is a constant independent of i1 and t. Proof. Let G be defined as in (2.4). It is not difficult to show that ht ðyÞ ¼ C y þ

r X 1 X i¼1 j¼1

ai u0 G j ue2tij ðgÞ,

(A.5)

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

where C y ¼ a0 =ð1 

Ps

i¼1 bs Þ

861

and u0 ¼ ð1; 0; . . . ; 0Þs1 . Similarly, we can show that

1 X qht ðyÞ ¼ u0 Gj uhtkj ðyÞ qbk j¼1

¼ C1 þ

r X 1 X 1 X

ai u0 G j uu0 G j1 ue2tkijj 1 ðgÞ,

ðA:6Þ

i¼1 j¼1 j 1 ¼1

where C 1 is some constant. Note that xð1 þ xÞ1 oxi1 as x40 for any i1 2 ð0; 1Þ. We have, for any i1 2 ð0; 1Þ, " Cðk; i; j; j 1 Þ max Y

#

ai u0 G kþjþj 1 ue2tkijj1 ðgÞ C y þ ai u0 Gkþjþj 1 ue2tkijj 1 ðgÞ

p max½ai u0 G kþjþj1 ue2tkijj1 ðgÞ=C y i1 Y

pC 2 rjþj1 jetkijj1 ðgÞji1 ,

ðA:7Þ

where C 2 is a constant. Since y0 is an interior point in Y, there exists a neighborhood Y0 of y0 such that b ¼ minfb1 : y 2 Y0 g40. Furthermore, because each element of G is nonnegative, it is easy to see that, for any constant vector c with all elements being nonnegative, c0 uu0 cpc0 Gc= b, and hence u0 G j uu0 G j1 upu0 G j GG j1 u= b ¼ u0 Gjþj 1 þ1 u= b. Again, since all the elements of G are nonnegative, it follows that u0 G kþjþj1 u ¼ u0 GG kþjþj 1 1 u ¼ ðb1 ; . . . ; bs ÞG kþjþj1 1 u X b u0 G kþjþj 1 1 u X    Xbk1 u0 G jþj 1 þ1 uXbk u0 G j uu0 G j1 u.

ðA:8Þ

Thus, by (A.5) and (A.8), we have " max

# ai u0 G j uu0 G j1 ue2tkijj1 ðgÞ ht ðyÞ

Y0

" p max Y0

ai u0 G j uu0 G j1 ue2tkijj 1 ðgÞ Cy þ

ai u0 G kþjþj1 ue2tkijj 1 ðyÞ

# p

1 bk

Cðk; i; j; j 1 Þ.

ðA:9Þ

By (A.6), (A.7) and (A.9), for any i1 2 ð0; 1Þ, it follows that !   1 X  1 qht ðyÞ C 4 j i1   sup p 1þ r jetkj ðgÞj ; h ðyÞ qb  bk Y0

t

k

k ¼ 1; . . . ; s,

(A.10)

j¼1

where r 2 ð0; 1Þ and C40 are some constants. Similarly, we can show that  i1 !      1 qh ðyÞ   1 X    t j r etkj ðgÞ ; sup pC 1 þ     h qa ðyÞ t k Y0 j¼1

k ¼ 1; . . . ; r.

(A.11)

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

862

Let r be the maximizer of the r in (A.11) and Lemma A.1(i). By Lemma A.1 (i), 1 X

rj jetkj ðgÞji1 pC

j¼1

1 X

rj jxrtkj ji1

j¼1

pC 1 þ

1 X

!i1 " j

r jxrtkj j

j¼1

pC 1 þ

1 X

!i1 rj jxrtkj j

j¼1

pC 1 þ

1 X j¼1

1 X j¼1

1 X j¼1

j

r

1 X

l

jx j i1 P1rtkj r j ð1 þ j¼1 r jxrtkj jÞi1

#

j

rð1i1 Þj ! i1

r jytkjl j

1 pCxirtk1 , ~

ðA:12Þ

l¼0

for some r~ 2 ð0; 1Þ, where the constant C may be different in different inequalities. Thus, using (A.10)–(A.12), we can claim that (i) holds. Similarly, we can show that (ii) holds. This completes the proof. & Lemma A.3. Let xrt be defined as in (3.4). Under Assumptions 2.1–2.2, there exist constants C and r 2 ð0; 1Þ such that    1 qh ðyÞ   t ðiÞ suppffiffiffiffiffiffiffiffiffiffi pCxrt1 , Y  ht ðyÞ qg     1 q2 h ðyÞ   t ðiiÞ suppffiffiffiffiffiffiffiffiffiffi pCxrt1 , 0 Y  ht ðyÞ qg qg     1 q2 h ðyÞ   t ðiiiÞ suppffiffiffiffiffiffiffiffiffiffi pCxrt1 . 0   qd qg ht ðyÞ Y Proof. By (2.5), we have the following expansions: ht ðyÞ ¼ a0 b1 ð1Þ þ

1 X

ad ðiÞe2ti ðgÞ,

i¼1



1 X qht ðyÞ qeti ðgÞ ¼2 ad ðiÞ eti ðgÞ . qg qg i¼1 Thus, ht ðyÞXad ðiÞe2ti ðgÞ. By Lemma A.1(i), it follows that      1 qh ðyÞ X 1 ad ðiÞeti ðgÞ qeti ðgÞ     t pffiffiffiffiffiffiffiffiffiffi suppffiffiffiffiffiffiffiffiffiffi  ¼ 2 sup      qg qg ht ðyÞ ht ðyÞ Y Y i¼1   X 1 pffiffiffiffiffiffiffiffiffi qeti ðgÞ   p2 sup ad ðiÞ , px qg  rt1 Y  i¼1 i.e., (i) holds. Similarly, we can show that (ii)–(iii) hold. This completes the proof.

&

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

863

Lemma A.4. Under Assumptions 2.1–2.2, for any i1 2 ð0; 1Þ, it follows that ðiÞ ðiiÞ

ðiiiÞ

1 , sup jl t ðyÞ  l~t ðyÞjpOðrt ÞR1þi t Y   ql ðyÞ ql~ ðyÞ   t t 1  , sup pOðrt ÞR2þi t qy  Y  qy   q2 l ðyÞ q2 l~ ðyÞ   t t t 3þi1 , sup 0  0 pOðr ÞRt   qy qy qy qy Y

where Rt ¼ 1 þ

Pt

2 i¼0 xrti

and xrt is defined as in (3.4) for some r 2 ð0; 1Þ.

Proof. It is straightforward to show that there exists a r 2 ð0; 1Þ such that sup jet ðgÞ  e~t ðgÞj ¼ Oðrt Þxr0 , Yg

sup je2t ðgÞ  e~2t ðgÞj ¼ Oðrt Þxrt xr0 , Yg

sup jht ðyÞ  h~t ðyÞjpOð1Þ Y

t1 X

ri sup je2ti ðgÞ  e~2ti ðgÞj þ Oðrt Þx2r0 Yg

i¼1

pOðrt Þxr0

t1 X

xrti þ Oðrt Þx2r0 pOðrt ÞRt ,

i¼1

 i1    1  1 1  1    sup k  k pOð1Þ sup  k  k  Y ht ðyÞ Y ht ðyÞ h~ ðyÞ h~ ðyÞ t

t

pOð1Þ sup jht ðyÞ  h~t ðyÞji1 ¼ Oðrt ÞRit1 , Y

for any i1 2 ð0; 1Þ and kX1. Note that sup j~e2t ðgÞjpOð1Þx2rt . There is a constant C40 Y such that jl t ðyÞ  l~t ðyÞjp log½1 þ Cjht ðyÞ  h~t ðyÞj

  1 1 2 1  2 2  þ jet ðgÞ  e~ t ðgÞj þ e~ t ðgÞ  . ht ht h~t

By the preceding inequalities, we can show that (i) holds. Furthermore, we have   qet ðgÞ q~et ðgÞ  ¼ Oðrt Þxr0 , sup   qg qg  Yg    qh ðyÞ qh~ ðyÞ t1 X  qeti ðgÞ   t t  sup ri sup pOð1Þ eti ðgÞ qg qg  Y  qg Y g i¼1  q~eti ðgÞ  þ Oðrt Þx2 ~eti ðgÞ r0 qg  t1 X pOðrt Þxr0 ðxrti þ xrti1 Þ þ Oðrt Þx2r0 pOðrt ÞRt , i¼1

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

864

  qh ðyÞ qh~ ðyÞ t1 X     t t  sup ri supe2ti ðgÞ  e~2ti ðgÞ þ Oðrt Þx2r0 pOð1Þ   qd qd Y Yg i¼1 pOðrt Þxr0

t1 X

xrti þ Oðrt Þx2r0 pOðrt ÞRt .

i¼1

Using these inequalities, we can show that (ii)–(iii) hold. This completes the proof.

&

Proof of Theorem 3.1(i). First, the space Y is compact and y0 is an interior point in Y. Second, L~ sn ðyÞ is continuous in y 2 Y and is a measurable function of fyi , i ¼ t; t  1; . . .g for all y 2 Y. Third, by Lemma A.1, it follows that   qet ðgÞ   sup jet ðgÞjpjet ðg0 Þj þ sup kg  g0 k sup qg  Y Y Y pffiffiffiffiffiffiffiffiffiffiffiffi pjZt j ht ðy0 Þ þ Oð1Þxrt1 pOð1ÞjZt jð1 þ xrt1 Þ,

1p sup Y

ðA:13Þ

ht ðyÞ pOð1Þx2rt1 , a0

(A.14)

for some r 2 ð0; 1Þ, where a0 ¼ inf Y fa0 : y 2 Yg. By Assumption 3.1 and (A.13)–(A.14), E supY ½wt e2t ðgÞ=ht ðyÞo1. Since w is a bounded function, by Jensen’s inequality, i=2 Assumption 2.4 and (A.14), E supY jwt log ht ðyÞjpOð1ÞE supY j log ht ðyÞjpOð1Þ i=2 ½log E supY ðht ðyÞ=a0 Þ þ j log a0 jo1. Thus, we can claim that E supY jwt l t ðyÞjo1. By the ergodic theorem, Lsn ðyÞ ! E½wt l t ðyÞ a.s. for each y 2 Y. Furthermore, by Theorem 3.1 in Ling and McAleer (2003a), it follows that sup jLsn ðyÞ  E½wt l t ðyÞj!p 0.

(A.15)

Y

It is straightforward to show that supY ½jl t ðyÞj þ jl~t ðyÞjpOð1Þx2rt . By Lemma A.4 (i), sup jwt l t ðyÞ  w~ t l~t ðyÞjpwt sup jl t ðyÞ  l~t ðyÞj þ jwt  w~ t j sup l~t ðyÞj Y

Y

Y

pOð1Þr

t

Eðjwt 

w~ t jx2rt Þi0 =8 pfEjwt

1 R1þi t

þ Oð1Þjwt 

w~ t jx2rt ,

i =2

 w~ t ji0 =4 Exrt0 g1=2 ¼ Oðt1 Þ.

By the preceding inequality, and Assumptions 2.4 and 3.2, we can show that n 1X sup jwt l t ðyÞ  w~ t l~t ðyÞj ¼ op ð1Þ. n t¼1 Y

(A.16)

By (A.15)–(A.16), it follows that sup jL~ sn ðyÞ  E½wt l t ðyÞj ¼ op ð1Þ. Y

(A.17)

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

865

Fourth, by (2.3), it follows that et ðgÞ  et ðg0 Þ ¼

1 X ½ag ðiÞ  ag0 ðiÞyti i¼1

¼ ðg  g0 Þ0

1 X

ag ðiÞyti ¼ ðg  g0 Þ0

i¼1

qet ðg Þ , qg

where g is between y and y0 . Thus, wt ½et ðg0 Þ þ ðg  g0 Þ0 qet ðy Þ=qy2 E½wt l t ðyÞ ¼  Ewt log ht ðyÞ  E ht ðyÞ  wt ht ðy0 Þ ¼ Ewt log ht ðyÞ  E ht ðyÞ

w qe ðy Þ qet ðy Þ t t 0  ðg  g0 Þ E ðg  g0 Þ L1 ðyÞ þ L2 ðyÞ. qg0 ht ðyÞ qg L2 ðyÞ obtains its maximum at zero if and only if g ¼ g0 , and

ht ðy0 Þ ht ðy0 Þ  L1 ðyÞ ¼ Ewt log  Ewt log ht ðy0 Þ. ht ðyÞ ht ðyÞ Note that, for any M40, gðMÞ log M  Mp  1 with equality only if M ¼ 1. Let M t ¼ ht ðy0 Þ=ht ðyÞ. When M t ¼ 1 a.s., we have E½wt gðM t Þ ¼ Ewt . If PðM t ¼ 1Þa1, then PðgðM t Þogð1ÞÞa0, so that E½wt gðM t ÞoE½wt gð1Þ ¼ Ewt . Thus, L1 ðyÞ reaches at its maximum 1  E½wt log ht ðy0 Þ, and this occurs if and only if ht ðyÞ ¼ ht ðy0 Þ. Since maxY E½wt l t ðyÞpmaxY L1 ðyÞ þ maxY L2 ðyÞ, maxY E½wt l t ðyÞ ¼ 1  E½wt log ht ðy0 Þ if and only if maxY L2 ðyÞ ¼ 0 and maxY L1 ðyÞ ¼ 1  E½wt log ht ðy0 Þ, which occurs if and only if g ¼ g0 and ht ðyÞ ¼ ht ðy0 Þ a.s. (see e.g., Francq and Zakoı¨ an, 2004). Since ht ðyÞjg¼g0 ¼ ht ðy0 Þ a.s. if and only if d ¼ d0 , we can claim that L1 ðyÞ reaches its maximum 1  E log ht ðy0 Þ if and only if y ¼ y0 . Thus, E½wt l t ðyÞ is uniquely maximized at y0 . Thus, we have established all the conditions for consistency in Theorem 4.1.1 in Amemiya (1985) and hence (i) holds. This completes the proof. & Proof of Theorem 3.1(ii). First, y^ sn !p y0 as n ! 1. Second, q2 l t ðyÞ=qy qy0 exists and is continuous in Y. Third, by (A.3), (A.13) and Lemmas A.1 and A.3, 8 2    2   < q et ðgÞ2  q l t ðyÞ 1 qh ðyÞ  2  t  þ pffiffiffiffiffiffiffiffiffiffi pOð1Þ sup  sup  e ðgÞ  qg   qg qg0   ht ðyÞ qg  t Y Y : 0 2  1   1 q2 h ðyÞ 2   1 qh ðyÞ  A   t t þ et ðgÞ þ 1 @pffiffiffiffiffiffiffiffiffiffi  þ pffiffiffiffiffiffiffiffiffiffi   ht ðyÞ qg   ht ðyÞ qg qg0      2 )  q et ðgÞ q et ðgÞ 1 q h ðyÞ   t  pffiffiffiffiffiffiffiffiffiffi  þjet ðgÞj  þ jet ðgÞj  qg   qg qg0   ht ðyÞ qg0 

ARTICLE IN PRESS 866

S. Ling / Journal of Econometrics 140 (2007) 849–873

n pOð1Þ x2rt1 þ x4rt1 ð1 þ jZt jÞ2 þ ½x2rt1 ð1 þ jZt jÞ2 þ 1 o ðx2rt1 þ xrt1 Þ þ xrt1 ð1 þ jZt jÞðx2rt1 þ xrt1 Þ pOð1Þx4rt1 ð1 þ Z2t Þ,

ðA:18Þ

where Oð1Þ holds uniformly in t. By (A.4), (A.13) and Lemma A.2, there exists a neighborhood Y0 of y0 such that (   2   1 qht ðyÞ2 2 q l t ðyÞ   e ðgÞ   sup 0 pOð1Þ sup  ht ðyÞ qd  t Y0 qd qd Y0 "   #)  1 qht ðyÞ2  1 q2 ht ðyÞ 2    þðet ðgÞ þ 1Þ  h ðyÞ qd  þ h ðyÞ qd qd0  t t pOð1Þfx4rt1 ð1 þ jZt jÞ2 þ ½x2rt1 ð1 þ jZt jÞ2 þ 1ðxrt1 þ x2rt1 Þg, where Oð1Þ holds uniformly in t. Now, by Assumption 3.1, it is readily seen that E supY kwt q2 l t ðyÞ=qg qg0 ko1 and E supY0 k wt q2 l t ðyÞ=qd qd0 ko1. Similarly, we can show that E supY0 kwt q2 l t ðyÞ=qg qd0 ko1. Thus, we can claim that    q2 l t ðyÞ  w E sup  t qy qy0 o1. Y0 By the ergodic theorem and Theorem 3.1 in Ling and McAleer (2003a), we can show that q2 Lsn ðyÞ=qy qy0 converges to E½wt q2 l t ðyÞqy qy0  uniformly in Y0 in probability. Similar to (A.16), using Lemma A.4(iii), we can show that supY0 kq2 Lsn ðyn Þ=qy qy0  q2 L~ sn ðyn Þ=qy qy0 k ¼ op ð1Þ. Since E½wt q2 l t ðyÞqy qy0  is continuous in terms of y, for any sequence yn such that yn ! y0 in probability, we can show that q2 L~ sn ðyn Þ 1 ¼  S0 þ op ð1Þ. 0 qyqy 2

(A.19)

Fourth, for the previous neighborhood Y0 , by (A.4), (A.13) and Lemmas A.1–A.3,        ql t ðyÞ qet ðgÞ  1 qht ðyÞ 2       pOð1Þ sup jet ðgÞj þ ½e ðgÞ þ 1 sup qy  qy  ht ðyÞ qy  t Y0 Y0 n o pOð1Þ ð1 þ jZt jÞx2rt1 þ ðxrt1 þ xrt1 Þ½ð1 þ jZt jÞ2 x2rt1 þ 1 pOð1Þx3rt1 ð1 þ Z2t Þ, where Oð1Þ holds uniformly in t. Thus,

2 ql t ðy0 Þ ql t ðy0 Þ O0 ¼ 4E wt o1. qy qy0 Similar to the proof of Lemma 4.2 in Ling and McAleer (2003a), we can show that S0 and O0 are positive definite (see also Francq and Zakoı¨ an, 2004). By the central limiting theorem, we have qL 0 =4Þ. Similar to (A.16), using Lemma A.4(ii), we pffiffisnffi ðy0 Þ=qy!L Nð0; O can show that nkqLsn ðy0 Þ=qy  qL~ sn ðy0 Þ=qyk ¼ op ð1Þ and hence qL~ sn ðy0 Þ=qy!L Nð0; O0 =4Þ. Thus, pwe in Theorem 4.1.3 in Amemiya ffiffiffi have established all the conditions 1 (1985) and hence nðy^ sn  y0 Þ!L Nð0; S1 0 O0 S0 Þ. This completes the proof. &

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

867

The following lemma is a key result for the ARMA-IGARCH model. P i Lemma A.5. Let xrt be defined as in (3.4) and x0Rt ¼ 1 þ 1 i¼0 R jeti j. If Assumptions 2.1–2.2 hold, then for any r, R 2 ð0; 1Þ, there exist constants R1 ; ~i 2 ð0; 1Þ and C not depending on t such that ðiÞ ðiiÞ

xrt pCx0R1 t a:s:; x0Rt1 i pffiffiffiffiffiffiffiffiffiffiffiffi pCx1~ 0R1 t1 ht ðy0 Þ

a:s::

Proof. By Assumption 2.1, yt has the following expansion: yt ¼ Oð1Þ

1 X

ri eti ,

i¼0

where r 2 ð0; 1Þ. By this, it is straightforward to show that (i) holds. Let G 0 be defined as G in (2.4) with bi ¼ b0i . Since all b0i 40, it is not difficult to see that u0 G j0 uXbj01 , where u is i j=2 defined as in (A.5). Let ~i 2 ð0; 1Þ such that b~i01 4R. Note that b011 o1. Then, by (A.5), it follows that 2 !i !~i 3 1 i 2 X x0Rt1 R b e 01 ti 5 pffiffiffiffiffiffiffiffiffiffiffiffi pOð1Þ41 þ jeti j12~i ~i C þ bi01 e2ti b ht ðy0 Þ y0 01 i¼0 " !i # 1 X R 12~i j 2 ~i=2 pOð1Þ 1 þ jeti j ðb01 eti Þ b~i01 i¼0 " !i # 1 X R 1~i pOð1Þ 1 þ jeti j1~i px0R , 1 t1 b~i01 i¼0 for some R1 2 ð0; 1Þ, where the last step holds using the same method as for (A.12). Thus, (ii) holds. This completes the proof. & pffiffiffi Lemma A.6. If the assumptions of Theorem 4.1 hold and nkyn  y0 kpM, then it follows that pffiffiffiffiffiffiffiffiffiffiffiffi ðiÞ et ðgn Þ ¼ et ðg0 Þ þ op ð1Þ ht ðy0 Þ, ðiiÞ ht ðyn Þ ¼ ht ðy0 Þ þ op ð1Þht ðy0 Þ, where op ð1Þ holds uniformly in t ¼ 1; . . . ; n. Proof. First, it is straightforward to show that et ðgn Þ ¼ et ðg0 Þ þ Oðn1=2 Þxrt1 . Furthermore, by Lemma A.5(i), we have a constant R 2 ð0; 1Þ such that et ðgn Þ ¼ et ðg0 Þ þ Oðn1=2 Þx0Rt1 . By Theorem 2.1 (iii), for any ~i; R 2 ð0; 1Þ, we have pffiffiffi i max x1~ 0Rt1 = n ¼ op ð1Þ, 1ptpn

(A.20) i 2 Eðx1~ 0Rt Þ o1.

Thus, (A.21)

for any ~i; R 2 ð0; 1Þ. Thus, by (A.20) and Lemma A.5 (ii), we can see that (i) holds.

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

868

Define G n and G 0 as G in (2.4) with y ¼ yn and y0 , respectively. By (A.5), " # r X 1 X 0 j 0 j 2 ht ðyn Þ  ht ðy0 Þ ¼ C yn  C y0 þ ðani u Gn u  a0i u G 0 uÞetij ðg0 Þ i¼1 j¼1

þ

r X 1 X

h i ani u0 G jn u e2tij ðgn Þ  e2tij ðg0 Þ

i¼1 j¼1

B1n þ B2n ,

ðA:22Þ

where upffiffiffiis defined as p inffiffiffi (A.5). There is a constant c40 such that G0 ð1  c= nÞpG n pG0 ð1 þ c= nÞ, where ‘‘BpC’’ for matrices B ¼ ðbij Þ pand ffiffiffi j C ¼ ðcij Þ 0 j 0 j means that b pc for all i and j. Thus, ju G n u  u G uÞjp maxfjð1  c= nÞ  1j; jð1þ ij ij 0 pffiffiffi pffiffiffi pffiffiffi c= nÞj  1jgu0 Gj0 upOðj= nÞð1 þ c= nÞj u0 G j0 u. Furthermore, since C yn  C y0 ¼ Oðn1=2 Þ and a0i and ani are bounded, it follows that



X r X 1 1 1 B1n pO pffiffiffi þ O pffiffiffi u0 G jn ue2tij ðg0 Þ n n i¼1 j¼1   r X 1 X   þ a0i u0 G jn u  u0 G j0 uÞe2tij ðg0 Þ i¼1 j¼1



X  r X 1

1 1 c j 0 j 2 pO pffiffiffi þ O pffiffiffi 1 þ pffiffiffi ju G 0 uetij . n n i¼1 j¼1 n Using (A.5) and (A.7), as n is large enough, we have



 j a0i u0 G j0 ue2tij c j a0i u0 G0 ue2tij c j j 1 þ pffiffiffi pj 1 þ pffiffiffi ht ðy0 Þ n n C y0 þ a0i u0 G j0 ue2tij

 c j pj 1 þ pffiffiffi ða0i u0 G j0 ue2tij Þi1 pOðrj Þjetij j2i1 , n for some r 2 ð0; 1Þ and any i1 2 ð0; 1Þ. Thus, as for (A.12), we can show that



X r X 1 1 1 B1n pO pffiffiffi þ ht ðy0 ÞO pffiffiffi rj jetij j2i1 n n i¼1 j¼1 #

" r X 1 2i1 pO pffiffiffi 1 þ ht ðy0 Þ x0Rtij ¼ ht ðy0 Þop ð1Þ, n i¼1

ðA:23Þ

by (A.21), where i1 2 ð0; 12Þ. By (A.5), ht ðy0 Þ=a0i Xu0 G j0 ue2tij . By (A.20), " # r X 1 r X 1 X 1 X 1 B2n ¼ Oð1Þ pffiffiffi u0 G jn ujetij jx0Rtij þ u0 G jn ux20Rtij n i¼1 j¼1 n i¼1 j¼1 " #  r X 1

r X 1 1 X c j 0 j 1X j 2 1 þ pffiffiffi u G0 ujetij jx0Rtij þ r x0Rtij pOð1Þ pffiffiffi n i¼1 j¼1 n n i¼1 j¼1

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

869

"pffiffiffiffiffiffiffiffiffiffiffiffi #  r X 1

r X 1 ht ðy0 Þ X c j 0 j 1=2 1X pffiffiffi pOð1Þ 1 þ pffiffiffi ðu G 0 uÞ x0Rtij þ rj x20Rtij n i¼1 j¼1 n n i¼1 j¼1 !2



 X r X 1 r X 1 1 pffiffiffiffiffiffiffiffiffiffiffiffi X 1 j j pO pffiffiffi r x0Rtij þ O r x0Rtij , ht ðy0 Þ n n i¼1 j¼1 i¼1 j¼1 P P P1 j k for some r; R 2 ð0; 1Þ. Reordering ri¼1 1 j¼1 k¼0 r R jetijk j, we can show that there exists R~ 2 ð0; 1Þ such that "pffiffiffiffiffiffiffiffiffiffiffiffi # ht ðy0 Þ 1 2 pffiffiffi x0~Rt1 þ x0~Rt1 ¼ op ð1Þht ðy0 Þ, B2n ¼ Oð1Þ (A.24) n n by (A.21) and Lemma A.5 (ii), where op ð1Þ holds uniformly in t ¼ 1; . . . ; n. By (A.22)–(A.24), (ii) holds. This completes the proof. & pffiffiffi Lemma A.7. If the assumptions of Theorem 3 hold and nkyn  y0 kpM, then it follows that n n 1X q2 l t ðyn Þ 1 X q2 l t ðy0 Þ þ op ð1Þ 0 ¼ n t¼1 qyqy n t¼1 qy qy0

for any fixed constant M.

Proof. We first show that n n 1X q2 l t ðyn Þ 1 X q2 l t ðy0 Þ ¼ þ op ð1Þ. n t¼1 qg qg0 n t¼1 qg qg0

(A.25)

q2 l t ðyÞ=qg qg0 in (A.3) includes five terms. We only provide the proof of the following equation, while other terms in (A.3) can be proved either easily or similarly. n n 1X qht ðyn Þ qht ðyn Þ e2t ðgn Þ 1 X qht ðy0 Þ qht ðy0 Þ e2t ðg0 Þ ¼ þ op ð1Þ. n t¼1 qg qg0 h3t ðyn Þ n t¼1 qg qg0 h3t ðy0 Þ

By Lemma A.6(ii), we have       1 1   ht ðyn Þ  ht ðy0 Þ   pffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi   ht ðyn Þ  ht ðy0 Þ ht ðyn Þht ðy0 Þð ht ðyn Þ þ ht ðy0 Þ op ð1Þ op ð1Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffi , ht ðy0 Þ½1 þ op ð1Þ½ 1 þ op ð1Þ þ 1 ht ðy0 Þ where op ð1Þ holds uniformly in t ¼ 1; . . . ; n. By Lemma A.6, jet ðgn Þ  et ðg0 Þj op ð1Þ pffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ op ð1Þ, 1 þ op ð1Þ ht ðyn Þ where op ð1Þ holds uniformly in t ¼ 1; . . . ; n. By the preceding two inequalities,      e ðg Þ  1 et ðg0 Þ  jet ðgn Þ  et ðg0 Þj 1   t n  pffiffiffiffiffiffiffiffiffiffiffiffi þ jet ðg0 Þjpffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffip  ht ðyn Þ  ht ðyn Þ ht ðy0 Þ ht ðyn Þ ht ðy0 Þ jet ðg Þjop ð1Þ pop ð1Þ þ p0ffiffiffiffiffiffiffiffiffiffiffiffi ¼ op ð1Þ þ jZt jop ð1Þ, ht ðy0 Þ

(A.26)

ARTICLE IN PRESS 870

S. Ling / Journal of Econometrics 140 (2007) 849–873

where op ð1Þ holds uniformly in t ¼ 1; . . . ; n. Thus,    2  2     e ðg Þ   et ðgn Þ e2t ðg0 Þ   et ðg0 Þ  et ðgn Þ e ðg Þ e ðg Þ    t t t 0 n 0   ffipffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffi h ðy Þ  h ðy Þp2pffiffiffiffiffiffiffiffiffiffiffi     ht ðy0 Þ ht ðyn Þ ht ðy0 Þ ht ðyn Þ ht ðy0 Þ t n t 0 ¼ op ð1Þ þ Z2t op ð1Þ.

ðA:27Þ

By Lemmas A.3, A.5(ii) and A.6(ii), there exists a neighborhood Y0 of y0 such that   qht ðyÞ qht ðyÞ 1  1 Oð1Þx2r;t1 i  sup p ¼ Op ð1Þx2~ 0R;t1 ,  qg qg0 ht ðyÞ ht ðyn Þ ht ðy0 Þ½1 þ op ð1Þ Y0

(A.28)

where Op ð1Þ holds uniformly in t ¼ 1; . . . ; n. By (A.27)–(A.28), it follows that   n 1X qht ðyn Þ qht ðyn Þ 1  e2t ðgn Þ e2t ðg0 Þ   n t¼1 qg qg0 h2t ðyn Þ ht ðyn Þ ht ðy0 Þ n   op ð1Þ X i 2~i 2 p x2~ 0R;t1 þ x0R;t1 Zt ¼ op ð1Þ, n t¼1 i since Ex2~ 0R;t1 o1. Thus, by Lemma A.6(ii) and (A.28), it is not hard to see that " # n 1X qht ðyn Þ qht ðyn Þ Z2t qht ðyn Þ qht ðyn Þ Z2t  n t¼1 qg qg0 h2t ðyn Þ qg qg0 ht ðyn Þht ðy0 Þ 

n 1X qht ðyn Þ qht ðyn Þ Z2t 1 1  ¼ n t¼1 qg qg0 ht ðyn Þ ht ðyn Þ ht ðy0 Þ " # n n op ð1Þ X qht ðyn Þ qht ðyn Þ Z2t 1X ð1Þ x2~i Z2 ¼ op ð1Þ, ¼ ¼ o p n t¼1 qg qg0 h2t ðyn Þ n t¼1 0R;t1 t

Pn 2~i 2 since t¼1 x0R;t1 Zt =n ¼ Op ð1Þ, where op ð1Þ holds uniformly in t ¼ 1; . . . ; n. By the two preceding inequalities, n n 1X qht ðyn Þ qht ðyn Þ e2t ðgn Þ 1X qht ðyn Þ qht ðyn Þ Z2t ¼ þ op ð1Þ 3 n t¼1 qg qg0 ht ðyn Þ n t¼1 qg qg0 h2t ðyn Þ

¼

n 1X qht ðyn Þ qht ðyn Þ Z2t þ op ð1Þ. 0 n t¼1 qg qg ht ðyn Þht ðy0 Þ 1=2

By Lemmas A.3 and A.5, E supfkht Y0

2~i ðyÞqht ðyÞ=qgk2 Z2t =ht ðy0 ÞgpOð1ÞEx0R;t1 o1.

Furthermore, by the preceding equation, the dominated convergence theorem and Markov’s theorem, we can show that (A.26) holds. We next show that n n 1X q2 l t ðyn Þ 1 X q2 l t ðy0 Þ þ op ð1Þ. 0 ¼ n t¼1 qd qd n t¼1 qd qd0

(A.29)

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

871

There are two terms in q2 l t ðyÞ=qd qd0 , see (A.4). We only provide the proof of the following equation, while another term can be similarly proved. n n 1X qht ðyn Þ qht ðyn Þ e2t ðgn Þ 1 X qht ðy0 Þ qht ðy0 Þ e2t ðg0 Þ ¼ þ op ð1Þ. 0 n t¼1 qd qd h3t ðyn Þ n t¼1 qd qd0 h3t ðy0 Þ

By Lemma A.2, it follows that   qh ðy Þ qh ðy Þ 1    t n t n 1 . pOð1Þxirt1   qd qd0 h2t ðyn Þ

(A.30)

(A.31)

By (A.27) and (A.31), we have qht ðyn Þ qht ðyn Þ e2t ðgn Þ qht ðyn Þ qht ðyn Þ e2t ðg0 Þ 1 ð1 þ Z2t Þ. ¼ þ op ð1Þxirt1 qd qd0 h3t ðyn Þ qd qd0 h2t ðyn Þht ðy0 Þ 2 2 By Lemma A.2, E ðsupY0 kh1 t ðyÞqht ðyÞ=qdk Zt Þo1. Furthermore, by the preceding two equations, the dominated convergence theorem P and Markov’s theorem,Pwe can show that (A.30) holds. Similarly, we can show that n1 nt¼1 q2 l t ðyn Þ=qd qg0 ¼ n1 nt¼1 q2 l t ðy0 Þ=qd qg0 þop ð1Þ. This completes the proof. &

Proof of Theorem 4.1. We first show that   2 n X q l t ðy0 Þ q2 l t ðy0 Þ 1 o1 and 1  E 0  0 ¼  S þ op ð1Þ. qy qy n t¼1 qy qy 2

(A.32)

By Lemma A.5(ii) and Lemma A.3, we have     qh ðy Þ qh ðy Þ e2 ðg Þ  qht ðy0 Þ 1 2  t 0 t 0 t 0    pE E   qg h ðy Þ o1.  qg qg0 h3t ðy0 Þ t 0 Similarly, we can show that the other terms in E½q2 l t ðy0 Þ=qy qy0  are finite. Thus, the first part of (A.32) holds. The second part of (A.32) holds by the ergodic theorem. By Taylor’s expansion with each component, we have n X ql~t ðy^ sn Þ t¼1

qy

¼

n X ql~t ðy0 Þ t¼1

qy

þ

n X q2 l~t ðy Þ n

t¼1

qy qy0

ðy^ sn  y0 Þ,

(A.33)

where y n lies between y0 and y~ sn . By Lemma A.4(iii), Lemma A.7, and (A.32), we can show that n 1X q2 l~t ðy^ sn Þ 1 ¼  S þ op ð1Þ n t¼1 qy qy0 2

and

n 1X q2 l~t ðy n Þ 1 ¼  S þ op ð1Þ. n t¼1 qy qy0 2

By Lemma A.4(ii), we can show that n n 1 X ql~t ðy^ 0 Þ 1 X ql t ðy0 Þ pffiffiffi ¼ pffiffiffi þ op ð1Þ. n t¼1 qy n t¼1 qy

ARTICLE IN PRESS 872

S. Ling / Journal of Econometrics 140 (2007) 849–873

As in Ling and McAleer (2003a) Francq and Zakoı¨ an (2004), we can show that S40 pffiffiand ffi and O40. Furthermore, since nðy^ sn  y0 Þ ¼ Op ð1Þ, by (A.33), we have )

1 ( X

n 1 1 ql ðy Þ 1 t 0 y^ n ¼ y^ sn   S þ op ð1Þ þ  S þ op ð1Þ ðy^ sn  y0 Þ 2 n t¼1 qy 2

 1 þ op pffiffiffi n

 n 2S1 X ql t ðy0 Þ 1 þ op pffiffiffi . ¼ y0 þ n t¼1 qy n Finally, by the central limiting theorem, we can show that the conclusion holds. This completes the proof. &

References Amemiya, T., 1985. Advanced Econometrics. Cambridge, Harvard University Press, Cambridge, MA. Basrak, B., Davis, R.A., Mikosh, T., 2002. Regular variation of GARCH processes. Stochastic Processes and their Application 99, 95–115. Berkes, I., Horvath, L., 2004. The efficiency of the estimates of the parameters in GARCH processes. Annals of Statistics 32, 633–655. Berkes, I., Horvath, L., Kokoszka, P., 2003. GARCH processes: structure and estimation. Bernoulli 9, 201–207. Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327. Bougerol, P., Picard, N.M., 1992. Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115–127. Chan, N.H., Peng, L., 2005. Weighted least absolute deviations estimation for an AR(1) process with ARCH(1) errors. Biometrika 92, 477–484. Davis, R.A., Knight, K., Liu, J., 1992. M-estimation for autoregressions with infinite variance. Stochastic Processes and their Application 40, 145–180. Drost, F.C., Klaassen, C.A.J., Werker, B.J.M., 1997. Adaptive estimation in time series models. Annals of Statistics 25, 786–817. Engle, R.F., 1982. Autoregressive conditional heteroskedasticity with estimates of variance of U.K. inflation. Econometrica 50, 987–1008. Francq, C., Zakoı¨ an, J.M., 2004. Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10, 605–637. Hall, P., Yao, Q.W., 2003. Inference in ARCH and GARCH models. Eonometrica 71, 285–317. Horvath, L., Liese, F., 2004. Lp-estimators in ARCH models. Journal of Statistics Plann. Inference 119, 277–309. Jeantheau, T., 1998. Strong consistency of estimators for multivariate ARCH models. Econometric Theory 14, 70–86. Koul, H.L. and Ling, S., 2006. Fitting an error distribution in some heteroscedastic time series models. Annals of Statistics 34, 994–1012. Lee, S.-W., Hansen, R.E., 1994. Asymptotic theory for GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory 10, 29–52. Li, W.K., Mak, T.K., 1994. On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity. Journal of Time Series Analysis 15, 627–636. Li, W.K., Ling, S., McAleer, M., 2002. A survey of recent theoretical results for time series models with GARCH errors. Journal of Economic Survey 16, 245–269. Ling, S., 1999. On probability properties of a double threshold ARMA conditional heteroskedasticity model. Journal of Applied Probability 36, 688–705. Ling, S., 2003. Self-weighted LSE and MLE for ARMA-GARCH Models. Unpublished working paper. HKUST.

ARTICLE IN PRESS S. Ling / Journal of Econometrics 140 (2007) 849–873

873

Ling, S., 2004. Estimation and testing of stationarity for double autoregressive models. Journal of the Royal Statistical Society: Series B 66, 63–78. Ling, S., 2005. Self-weighted LAD estimation for infinite variance autoregressive models. Journal of the Royal Statistical Society B 67, 381–393. Ling, S., Li, W.K., 1997. Fractional autoregressive integrated moving-average time series conditional heteroskedasticity. Journal of American Statistical Association 92, 1184–1194. Ling, S., McAleer, M., 2003a. Asymptotic theory for a new vector ARMA-GARCH model. Econometric Theory 19, 280–310. Ling, S., McAleer, M., 2003b. On adaptive estimation in nonstationary ARMA models with GARCH errors. Annals of Statist. 31, 642–674. Lumsdaine, R.L., 1996. Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models. Econometrica 64, 575–596. Mikosch, T., Gadrich, T., Klppelberg, C., Adler, R.J., 1995. Parameter estimation for ARMA models with infinite variance innovations. Annals of Statist. 23, 305–326. Peng, L. and Yao, Q.W., 2003. Least absolute deviations estimation for ARCH and GARCH models. Biometrika 967–975. Tsay, R.S., 1987. Conditional heteroscedastic time series models. Journal of American Statistical Association 7, 590–604. Weiss, A.A., 1986. Asymptotic theory for ARCH models: estimation and testing. Econometrics Theory 2, 107–131.