2014 Abadir Distaso Giraitis Koul Asymptotic Normality for Weighted Sums of Linear Processes

8/13/2019 2014 Abadir Distaso Giraitis Koul Asymptotic Normality for Weighted Sums of Linear Processes

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Econometric Theory,30, 2014, 252284.doi:10.1017/S0266466613000182

ASYMPTOTIC NORMALITY FOR

WEIGHTED SUMS OF LINEAR

PROCESSES

KARIM M. ABADIR AND WALTER DISTASO

Imperial College London

LIUDAS GIRAITIS

Queen Mary, University of London

HIRA

L. KOUL

Michigan State University

We establish asymptotic normality of weighted sums of linear processes with gen-

eral triangular array weights and when the innovations in the linear process are

martingale differences. The results are obtained under minimal conditions on the

weights and innovations. We also obtain weak convergence of weighted partial sum

processes. The results are applicable to linear processes that have short or long mem-

ory or exhibit seasonal long memory behavior. In particular, they are applicable to

GARCH and ARCH() models and to their squares. They are also useful in de-riving asymptotic normality of kernel-type estimators of a nonparametric regression

function with short or long memory moving average errors.

1. INTRODUCTION

Numerous inference procedures in statistics and econometrics are based on the

sums

Sn=n

j=1

Xj , Wn=n

j=1

zn jXj

of a linear process{Xj }, where{zn j , 1 j n} is an array of known real num-bers. In this paper we focus on deriving asymptotic distributions ofSn and Wn .

In addition, the weak convergence property of the corresponding partial sum

The authors would like to thank Donatas Surgailis for providing part (iii) of Theorem 2.2, Lemma 3.1, and other

useful comments, and to the Editor and the three reviewers for their constructive suggestions. Research is in part

supported by ESRC grant RES062230790 and USA NSF DMS Grant 0704130. Address correspondence to Liudas

Giraitis, School of Ecnomics and Finance, Queen Mary, University of London, Mile End Rd., London E14NS, United

Kingdom; e-mail: [email protected]

252 c Cambridge University Press 2013


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CLT FOR WEIGHTED SUMS OF LINEAR PROCESS 253

processes is also discussed. The linear process is assumed to be a moving average

with martingale differences innovations and may exhibit short- or long-range de-

pendence. It will be shown that{Var(Tn)}1/2(Tn ETn ), with Tn= Sn orWn ,converges weakly to a normal distribution under easily verifiable minimal as-sumptions on weights and innovations. In particular, these assumptions are valid

for the nonlinear squared autoregressive conditional heteroskedasticity (ARCH)

process where innovations are conditionally heteroskedasticity martingale differ-

ences. The proofs use the central limit theorem (CLT) for martingale differences.

Numerous testing procedures, e.g., testing for unit root, Cumulative Sum

(CUSUM) change-point detection, and Kwiatkowski, Phillips, Schmidt and Shin

(1992) (KPSS) test for stationarity, are based on the weak convergence of the par-

tial sum process S[nt],0 t 1 to a Gaussian process, and, in particular, requireverification of the CLT for Sn . The CLT for weighted sums Wn , in turn, is used inkernel estimation and in spectral analysis, e.g., obtaining asymptotic normality of

the discrete Fourier transforms of{Xj }. It is thus of interest to provide easy-to-useCLTs and invariance principles for the above statistics.

The work of Ibragimov and Linnik (1971) contains a number of useful results

on classical asymptotic theory of weakly dependent random variables. Davydov

(1970) obtained a weak convergence result for the partial sum process of lin-

ear processes with independent and identically distributed (i.i.d.) innovations,

whereas Phillips and Solo (1992) developed CLT and invariance principles for

sums of linear processes based on Beveridge-Nelson decomposition. Peligrad andUtev (2006) extended Ibragimov and Linnik (Thm. 18.6.5) for linear processes

with innovations following a more general dependence framework. Gordin (1969)

introduced a general method for proving CLTs for stationary processes using a

martingale approximation. In the case of short memory, his method gives the same

result as the Beveridge-Nelson decomposition. Wu and Woodroofe (2004) ob-

tained a CLT for the sums of stationary and ergodic sequences using a martingale

approximation method. Merlevede, Peligrad, and Utev (2006) provide a further

survey of some recent results on CLT and its weak invariance principle for sta-

tionary processes.Section 2 deals with asymptotic normality ofSn and Wn , whereas in Section 3

we discuss weak convergence of the corresponding partial sum processes.

Section 4 contains examples and applications, while Section 5 includes simula-

tions. In the sequel, all limits are taken as n , unless specified otherwise; pand D , respectively, denote the convergence in probability and in distribution,Z = {0,1,2, . . .}; and Nk(,) denotes the k-dimensional normal distribu-tion with mean vectorand covariance matrix,k 1. We write N for N1. Forany two sequences of real numbers,an, bn ,an bn meansan /bn 1.

2. CLT FOR WEIGHTED SUMS

In this section we consider asymptotic normality of the sums Sn and Wn when

{Xj } is a moving average process,


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254 KARIM M. ABADIR ET AL.

Xj=

k=0akjk=

j

k=

ajkk, j Z,

k=0a2k< . (2.1)

Here,(j

,Fj

), j

Z is a martingale difference sequence (m.d.s.) with constant

variance, where Fj := -field{i , i j}, j Z, i.e., E(j |Fj1) = 0, E2j= 2 K) 0, as K .First, we discuss the asymptotic normality ofSn . Theorem 18.6.5 of Ibragimov

and Linnik (1971) gives a CLT forSnin the case of i.i.d. innovations under general

conditions. Theorem 2.1 below extends it to m.d. innovations and shows that any

rate of divergence Var(Sn) of the variance guarantees the CLT. Peligradand Utev (2006, Prop. 4) proved this result when

j

is a stationary and ergodic

m.d.s. For clarity, we provide a brief proof based on the ideas of these works.

Note that if{Xj } is a zero-mean Gaussian process, then (Var(Sn))1/2 Sn=DN(0, 1) for all n 1, and the question about the asymptotic distribution of Snreduces to finding the asymptotics of the Var(Sn).

THEOREM 2.1. Suppose {Xj } is a linear process (2.1) where {j } is either astationary and ergodic m.d.s., or satisfies Assumption 2.1. Then

2n := Var(Sn) implies

1n SnD N(0, 1). (2.2)Proof. For simplicity of notation, set ak= 0, k= 1,2, . . . in (2.1), and let

cn j= 1

n

n

k=max(j,1) akj= 1

n

n

k=1 akj , j Z. Without loss of generality,assume 2= 1. Then,

1n Sn =n

j=

cn j j , 2n Var(Sn ) =

n

j=

c2n j= 1, n 1. (2.3)


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Next we show

n

j=cn j cn,j1

2

=o(1), cn :

=max

jn |cn j

| =o(1), (2.4)

which together with Lemma 2.1 below implies (2.2).

To prove (2.4), note that for any s n,t= 1, 2, . . . ,s

j=st

c2n j=s

j=st

cn,j1 +

cn j cn,j1

2,

c2n,s = c2n,st1 + 2s

j=

s

t

cn,j1

cn j cn,j1

+

s

j=

s

tcn j cn,j1

2

.

Because jn c2n,j= 1, then for every s Z and n 1, limt c2n,st1= 0.Sincetis arbitrary, take the limit t and use the Cauchy-Schwarz inequalityto obtain

c2n,s = 2s

j=

cn,j1

cn j cn,j1+ s

j=

cn j cn,j1

2

2

n

j=

c2n j1/2

Bn

+B2n

2Bn

+B2n , (2.5)

where Bn :=

nj=(cn j cn,j1)2

1/2does not depend ons . Since by defini-

tioncn j cn,j1 = 1n (a1j anj+1),

B2n 4 2n

j=0a2j= o(1),

because 2n

and

j

=0 a

2j 0. (2.12)

Letqn := nj=Md2n j Vj denote the left-hand side (l.h.s.) of (2.11). We shall showthat

Eqn 1, Var(qn) 0,


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which yields E(qn 1)2 0 and, together with Chebyshevs inequality, will im-ply (2.11). The first claim follows from EVj=E2j= 1, (2.9) and nj= d2n j= 1.To prove the second claim, let 0 :

=maxj

ZEV

2j K

[. . .] +n

j,k=M: |jk|K

[. . .]

K

n

j=

d2n j

2+ d2n

n

j=M

d2n j k: |jk|K

|V(j,j )V(k, k)|1/2 K + 0(2K+ 1)d2n 0,

by Assumptions 2.1(a,b) and (2.6). This completes the proof of (2.11).

To prove (2.12), note that the expected value of the l.h.s. of (2.14) is bounded

above by

n

j=M

d2n jE

2j I|j | d1n max

jE

2j I|j | d1n 0,

by Assumptions 2.1(c) and (2.6), which together with the Markov inequality

implies (2.12), and also completes the proof of part (i) of the lemma.

Case (ii). The proof of (2.10) combines arguments used in the proofs of

Peligrad and Utev (1997, Thm. 2.1; 2006, Prop. 4). Let Xn j := dn j j ,Mj n. By Hall and Heyde (1980, Thm. 3.2) to prove (2.10), it suffices to verify(a) maxMjn |Xn j | p 0, (b) nj=MX2n jp1, and (c) EmaxMjnX2n j=O(1).

First, we show that (2.8) implies (2.6). To see this, use the bound analogous

to (2.5), which does not depend on any particular form ofcn j , to obtain d2n,s

2Bn +B2n ,where now Bn :=

nj=

dn j dn,j1

21/2. Hence, (2.8) implies

dn 0.Now, claims (a) and (c) follow because for any >0,

E n

j=M

dn j j

2I(|dn j j | )

E21I(|1| d1n ) nj=

d2n j 0.

To show (b), we need to verify

n

j=M

d2n j 2

jp1. (2.13)

We do this by truncation. For ak 1, writen

j=M

d2n j 2j=n

j=M

d2n j +n

j=M

d2n j2j k1 kl=1

2j+l+

n

j=M

d2n j

k1

k

l=1

2j+l 1

=:qn1 + qn2 + qn3.


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In view of (2.9),qn1 = nj= d2n j + O(1/ log n) 1. Since E2j= 1,

E|qn2| k1E

n

j=Mkd2

n j

2

jk

l=1

n+l

j=M+l d2

n,jl 2

j k1

n

j=

|kd2n j d2n,j1 . . . d2n,jk|E

2j

+ dnE

2n+1 + . . . + 2n+k+ 2M

kn

j= |d2n j d

2n,j1|+ (k+ 1)d

2n 0, k 1,

because nj= |d2n j d2n,j1|

nj=

dn j dn,j1

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nj= d

2n j

1/2 0, by (2.8).

Finally, since {j } is stationary and ergodic, by the ergodic theorem, see, e.g.,Stout (1974, Cor. 3.5.2), E

k1 kl=1 2j+l E21 = Ek1 kl=1 2l1 0,k, and thus E|qn3| o(1)nj= d2n j = o(1), which implies (2.13) and completesthe proof. n

Remark 2.2. Note that the above lemma clearly includes the case of{j } IID(0, 1). Furthermore, it gives a generalization of Theorem V.1.2.1 of Hajek and

Sidak (1967) wheredn j= 0, j 0 and {j } IID(0, 1). Ifj IID(0, 1), assump-tion (2.6) implies the weak Lindeberg condition, see Hall and Heyde (1980, p. 53),

and is minimal. Indeed, let Sn= nj=1 dn j j , dn j=

1 + cnj n , c> 0. If >0, then max1jn |dn j | = o(1) and the CLT holds, as also shown in Phillipsand Magdalinos (2007). However, if = 0, then max1jn |dn j | =(1 + c)1,(2.6) does not hold and SnD j=1(1 + c)j j , which is Gaussian only if{j }is Gaussian, as pointed out in Anderson (1959). Peligrad and Utev (1997, Thm.2.1) derived Lemma 2.1, requiring instead of Assumption 2.1 that the m.d.s. {j }be a pairwise mixing sequence. In their Example 2.1 they showed that (2.6) and

stationarity and ergodicity of m.d.s. alone are not sufficient for the CLT.

The next corollary provides a straightforward generalization of Lemma 2.1 to a

two-sided moving average of m.d.s., while the result for weighted sums of a sta-

tionary ergodic process stated in Proposition 2.1 is useful in various applications.

COROLLARY 2.1. Suppose Sn= j= dn j j, n 1, where{j } is anm.d.s., and{dn j } are such thatjZ d2n j= 1.

(i) If{j } satisfies Assumption 2.1 and maxjZ |dn j | = o(1), then SnDN(0, 2).

(ii) In addition, if{j } is stationary ergodic andjZ(dn j dn,j1)2 = o(1),


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then SnD N(0, 2), and

jZ

d2n j 2

j= E21+op(1). (2.14)

PROPOSITION 2.1. Suppose {j } is a stationary ergodic sequence, E|1| 0, |u| u0, for some c > 0, and u 0> 0, and (2.20) holds, thenthe conditions of Theorem 2.2(iii)are satisfied, and CLT (2.18) holds.(c) Suppose Assumption 2.1 is satisfied, there exists c> 0such that f(u) c>0, u , and

max1jn

|zn j | = o n

j=1

z2n j

1/2. (2.22)

Then the conditions of Theorem 2.2(i)are satisfied, and CLT (2.18) holds.

Proof.(a) Let

G(

u):=

n

j=1e

ijuzn j ,

u . SinceX(

k) = eiku f(u)d u,

2n=n

j,k=1

zn j X(j k)znk=

f(u)|G(u)|2du

= 2f(0)n

j=1

z2n j + in, in := 2n2f(0)n

j=1

z2n j . (2.23)

It remains to show that

in= 2n2f(0) n

j=1z2n j

= o nj=1

z2n j

, (2.24)

which proves (2.21) and together with (2.20) verifies condition (iii) of Theorem

2.2 and thus (2.18).

Verification of (2.24) is based on two facts, which will be proved later:

|G(u)|2du = 2n

j=1z2n j , sup|u| |uG(u)| = o

n

j=1z2n j. (2.25)

We proceed as follows. Let >0. Choose > 0, such that sup0u|f(u) f(0)| . Then by (2.25),


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in |u|

|f(u) f(0)||G(u)|2du

|u||G(u)|2du +


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Remark 2.5. Proposition 2.2 verifies the CLT for smooth weights zn j and

stationary and ergodic m.d.s.{j }; e.g., in Corollary 4.4 below, it is shown that(2.20) holds for kernel weights zn j= K((nx j )/(nb)) in nonparametric re-gression setups. As long as fis continuous at zero, (2.20) yields the asymptoticbehavior of the variance (2.21) and the asymptotic normality ofWn .

If (2.20) does not hold, Theorem 2.2(i) or Theorem 2.2(ii) may be applied.

For instance, alternating weights zn j= (1)j = eij do not satisfy (2.20), butTheorem 2.2(i) is applicable. Indeed, maxj |zn j | = 1 and nj=1z2n j = n. In addition,if f(u) f() >0, u , then, letting Dn(u):= nl=1 eilu ,

Var(Wn) =n

j,k

=1

zn j X(j k)znk=

|Dn(u + )|2 f(u)du

=

0f(u )|Dn (u)|2du +

0

f(u + )|Dn(u)|2du

= f()

|Dn(u)|2du + o(n) = 2f( )n + o(n),

using f( ) = f(), as in the proof of (2.21). This shows the applicability ofTheorem 2.2(i) and hence the CLT for Wn . Note that here (2.21) for Var(Wn)does

not hold.

The following proposition, which is valid for any short memory covariance-stationary process{Xj }, provides an upper bound for Var(Wn) and analyzes itsasymptotic behavior.

PROPOSITION 2.3. Let{Xj } be a covariance-stationary process with zeromean, finite variance, and covariance function such that

kZ

| (k)| < . (2.27)

Let Wn be as in (2.17) with zn j satisfy conditions (2.20) and (2.22). Then, n

j=1z2n j

1Var(Wn)

kZ| (k)|,

n

j=1

z2n j

1Var(Wn) 2 :=

kZ (k). (2.28)

Proof.Under (2.27),

f(u)= (2 )1 kZ

eiku (k) (2 )1 kZ

| (k)| < , u ,

f(u) f(0) = (2)1 kZ

(k) = (2 )1 2, u 0.


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Thus, by (2.23) and (2.25),

Var(Wn) =

f(u)|G(u)|2du supu

f(u)

|G(u)|2du

kZ

| (k)|n

j=1

z2n j ,

which proves the first bound of (2.28). The proof of the second bound is the same

as that of (2.21). This completes the proof of the proposition. n

The following result is a multivariate generalization of Theorem 2.2.

THEOREM 2.3. Let z(i )n,j, i

=1, . . . , k be k arrays of real weights, and

{Xj

}is

a linear process (2.1) with m.d.s. {j }. Assume that sums W(i)n :=nj=1z(i )n,jXj and(

(i )n )

2 := Var(W(i)n ), i= 1, . . . , k satisfy one of the conditions (a) or (b).(a){j } is stationary and ergodic, and each sum W(i)n , i= 1, . . . , k satisfies

condition (iii) of Theorem 2.2.

(b){j } satisfies Assumption 2.1, and each sum W(i )n ,i= 1, . . . , k, satisfies oneof the conditions of (i)(iii) of Theorem 2.2.

Let for some (positive definite) matrix,CovW(i )n / (i )n , W(j )n / (j)n i,j=1,...,k

. (2.29)

Then,W(1)n /

(1)n , . . . , W

(k)n /

(k)n

D Nk(0,). (2.30)

Proof.Similarly, as in (2.19), write

S

(i )

n := W(i )

n /(i )

n =n

j=d(i )

n jj , i= 1, . . . , k.To prove (2.30), in view of the Cramer-Wold device, it suffices to show that for

everya = (a1, . . . , ak) Rk andk 1,Sn := a1 S(1)n + + akS(k)n D N(0, aaT). (2.31)Write Sn= nj= dn j j where dn j= a1d(1)n j + + akd(k)n j . Condition (2.29)implies that

Var(Sn) = E21n

j=

d2n j aaT.

Assume that m.d.s. satisfies Assumption 2.1. As seen in the proof of Theorem

2.2, any one of the conditions (i), (ii), or (iii) assures that maxjn |d(i )n j | 0,


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i= 1, . . . , k, which yields maxjn |dn j | 0.Hence, coefficientsdn j ofSn satisfyassumptions of Lemma 2.1, which implies (2.31).

Assume that m.d.s. is stationary ergodic. In the proof of Theorem 2.2, it was

shown that condition (iii) yields jn d(i )n j d(i)n,j12 0,i= 1, . . . , k. Conse-quently, jn(dn j dn,j1)2 0, and (2.31) follows by Lemma 2.1. n

3. WEAK CONVERGENCE OF PARTIAL SUM PROCESSES

A number of econometric applications require the weak convergence of a suitably

standardized partial sums process Sn ( ) = [n ]j=1Xj , >0 to some limit processS(),0 1. Observe that for each n , Sn(),0 1,is a step function in , belonging to the Skorokhod functional space D[0, 1].

From Billingsley (1968), we recall that a sequence of stochastic processes

{Zn()}, n 1 in D[0, 1] is said to converge weakly to a stochastic processZ() C[0, 1], and we write Zn Z, if every finite dimensional distribution of{Zn()} converges to that of Z() and if{Zn ()} is tight with respect to the uni-form metric; see also Pollard (1984). The uniform topology is stronger than the

Skorokhod J1-topology, and the verification of tightness in the uniform metric is

relatively easier.

Using the arguments in Section 12 and Theorem 15.5, p. 127 of Billingsley

(1968), see also Pollard (1984, Ch. V.1, Thm. 3), one can show that a sufficient

condition for tightness of{Zn} is the following: There exists a sequence of nonde-creasing right continuous functions Fn on [0, 1] that are uniformly bounded and

converge uniformly to a continuous function Fsuch that for some >1, >0,

and 0 s< t 1,E|Zn(t) Zn (s)| C[Fn (t) Fn(s)], n 1, (3.1)

whereCmay depend on , but not ons,t, andn.

Now, let Sn= Sn(1)and 2n := Var(Sn). Our goal here is to establish the weakconvergence of

{

1

n S

n()}

. For this purpose, we need to establish that its finite di-

mensional distributions converge to those of the limit process, denoted by f d d,and that the process is tight in uniform metric. We first focus on the finite dimen-

sional convergence.

According to the Lamperti theorem (1962), if 1n Sn( )

f d d S( ), (3.2)then for some H (0, 1)and a positive slowly varying function L, 2n

=Var(Sn)

=n2HL(n). (3.3)

In most applications

2n= Var(Sn) s2n2H , for some 0< H


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In the case of a linear process{Xj } of (2.1), with m.d. stationary and ergodicinnovations {j }, (3.4) is also sufficient for (3.2).

The limits in this section will be described by the fractional Brownian motion

(fBm),BH(),0 1, with parameter 0 0

f d d

BH( )

>0, (3.6)

where BHis the fBm with parameter H.

Proof. LetTn ( ) := 1n Sn( ). Assumption (3.4) implies that for any 0 < < 1,

Var(

Tn ()) =

Var(Sn ())

Var(Sn) =(n )2H(1

+o(1))

n2H(1 + o(1)) 2H

,and hence,

Cov(Tn(t), Tn (s)) = (1/2){Var(Tn (t)) + Var(Tn(s)) Var

Tn(t) Tn(s)}

rH(t, s), 00, and k 1, Sn= a1 Sn ([nt1]) + + akSn ([ntk])satisfy

1

n SnD S:= a1BH(t1) + + akBH(tk).By (3.7), Var(Sn) , and the rest of the proof repeats the lines of proof ofTheorem 2.1. n

The following proposition is a simplified version of the result obtained by

Taqqu (1975). It shows that long memory of the summands{Xj } a priori guar-antees the tightness of the normalized partial sum process. It does not require

{Xj } to be a linear process.PROPOSITION 3.2. Let

{Xj

}be a second-order stationary process satisfying

(3.4), with1/2< H


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Proof. To check tightness, we shall verify (3.1) for the process Tn(t) := 1n Sn (t). Let Fn (t):= [nt]/n, F(t):= t,0 t 1. Observe that supt |Fn(t) F(t)| 0 and Fis continuous on [0, 1]. By covariance stationarity of the incre-ments,

ETn (t) Tn(s)2 =E 1n [nt][ns ]

j=1Xj

2

=

[nt] [ns ]n

2H(1 + o(1))(1 + o(1)) C

[nt] [ns]

n

2H= C

Fn(t) Fn(s)

2H

, (3.9)

for some >0. Since := 2H >1, this verifies (3.1) for the Tn process with= 2 and completes the proof. nThe following lemma, whereSn= Sn(1), gives a useful bound for moments of

the sums of a linear process with martingal difference (m.d.) innovations and is

useful in proving the tightness of the process {Tn(t)} in the cases of short and neg-ative memory processes{Xj }. It extends the well-known Burkholder-Rosenthalinequality for martingales and some other inequalities involving m.d.s. of Dhar-

madhikari, Fabian, and Jogdeo (1968, Theorem in Section 1) and Borovskikh and

Korolyuk (1997, Chap. 3).

LEMMA 3.1. Let{Xj } be a linear process (2.1) with m.d. innovations{j },such thatE2j= 2 for all j, and:= maxj E|j |p < , for some p 2. Then

E|Sn|p cES2n

p/2, n 1, (3.10)

where c>0 depends only on p and.

Proof. Setting dn j= nk=max(j,1) akj , write Sn= nj= dn j j . BecauseE2

j= 2

, ES2

n= 2

n

j=d2

n j

2, (3.11)with a constant Cp>0 depending only on p.

The inequality (3.11) remains valid also for n = .


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Proof. Let n< . For p 2, by the Burkholder-Rosenthals inequality (seeHall and Heyde, 1980, p. 24),

E nj=1

Yj p Cp nj=1

E|Yj |p +E nj=1

E[Y2j|Fj1]p/2. (3.12)Recall the fact that for any real numbers a,b, and for 0 < 1,|a + b| |a| +|b| . Apply this fact with = 2/p 1 to obtain

n

j=1

E|Yj |p2/p

n

j=1

E|Yj |p

2/p. (3.13)

To bound the second term on the r.h.s. of (3.12), first use the CauchySchwarz

inequality for the conditional expectation with p/2 1 to obtain E[Y2j|Fj1] (E[|Yj |p|Fj1])2/p, 1 j n. Next, by Minkowski inequality for any r.v.s XandY,

(E|X+ Y|r)1/r (E|X|r)1/r + (E|Y|r)1/r, r 1.These facts, in turn, imply

E

nj=1E

Y2j|Fj1

p/2 Enj=1(E|Yj |p|Fj1])2/pp/2 nj=1 E|Yj |p2/pp/2,

which together with (3.13) and (3.12) proves (3.11).

Consider now the case of an infinite sum. Let p 2. We claim

E

j=1

Yj

p Cpj=1

(E|Yj |p)2/pp/2

. (3.14)

This inequality is trivially true if the r.h.s. of (3.14) is infinite. Assume it is finite.

By (3.11), for any 1


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Claim (3.14) now is proved upon taking limit as n in this bound. nWe are now ready to state and prove the following weak convergence result.

THEOREM 3.1. Assume that the m.d.s. {j } in the linear process {Xj } is eitherstationary and ergodic or satisfies Assumption 2.1. Suppose that Sn satisfies (3.4)

with0< H1/H.Then,

1n Sn () BH(), in D[0, 1]and uniform metric, (3.15)where BHis a fBm.

Proof. Proposition 3.1 implies the finite dimensional convergence. To prove

tightness, we shall use (3.10). For H

1/2, by (3.10) and (3.9),

E|Tn (t) Tn(s)|p cE(Tn(t) Tn(s))2p/2 cFn(t) Fn(s)H p.This verifies (3.1) for the Tn process with = p, = H p>1. For H >1/2,(3.15) follows from Proposition 3.2. This completes the proof. n

Remark 3.1. A functional CLT for the partial sums process of a short memory

linear process {Xj } of (2.1) with m.d. innovations andaj s such that j=0 j |aj | 0, bk 0, k 1,

k=1bk


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4.1.2. Squared ARCH Process. A centered squared ARCH process

Xj= r2jEr2j,

withrj as in (4.1) and satisfying E1/2

[4

0 ]k=1 bk


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Use this bound, C-S, and BE1/240m, Cov( 2mt, 2m j ) = 0. We will show that for |t j | >m,max

jE( 2j 2m j )2 m 0, m . (4.7)

The latter with the bound |Cov(X, Y)| (EX2EY2)1/2 and (4.5) implies

|Cov( 2t,

2j)

| = Cov( 2mt, 2m j ) + Cov( 2t 2mt, 2m j )+ Cov( 2mt, 2j 2m j ) + Cov( 2t 2mt, 2j 2m j )

C(1/2m + m ) 0, m , (4.8)

which verifies Assumption 2.1(b) with K= m. To verify (4.7), notice thatbj 0implies hk,j hm,k,j . Hence, with Bm := ms=1 bs andbs= bsI(1 s m), asin (4.4),

E(hk,j

hm,k,j )

2

=

s1,...,sk=1(bs1 . . . bsk

bs1 . . . bsk)

2j

s1

. . . 2j

s1

sk

2(4.9)

s1,...,sk=1

{bs1 . . . bsk bs1 . . . bsk}2

(E40)k

= (BkBkm )2(E40)k.Next, since 0< Bm B, by the mean value theorem, Bk Bkm k Bk1(B Bm ) k Bkm ,wherem := (B Bm )/B 0, asm . Hence, as in (4.5),

E( 2j

2m j )

2

=b20E

k=1{hk,j

hm,k,j

}2

b20

k=1kBE1/2[40 ]

k2 2m= C2m 0, m , (4.10)which proves (4.7).


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Assumption 2.1(c) follows by Markovs inequality, noting that Er4j =E41 E

41 < .

(iii) Here, Vj

= 4j Var(

20)

=E[2j

|Fj

1]. Thus, verifying Assumption 2.1(a)

is equivalent to proving maxj E 8j < . Toward this goal, first note that by theHolders inequality and by arguing as for (4.4), Eh4k,j B4k(E80)k. Together withassumption BE1/4[80 ]m ,

maxj

E

4j 4m j2 m 0, m , (4.12)

which together with (4.11), as in (4.8), implies that

|Cov 4t, 4j | = Cov 4mt, 4m j+ Cov 4t 4mt, 4m j

+Cov

4mt,

4j

4m j+ Cov 4t 4mt, 4j 4m j C1/2m + m 0, m .

Finally, to prove (4.12), use Holder inequality, as in (4.9) and (4.10), to obtain

E

hk,j hm,k,j4

s1,...,sk=1bs1 . . . bsk

m

s1,...,sk=1

bs1 . . . bsk

4E80

k

BkBkm

4

E80

k

E80

k

k4B4k4m ,

E 2j 2m j4 b40Ek=1

hk,j hm,k,j4

b40

k=1k(BE1/4[80 ])

k

4 4m= C4m 0, m ,

because BE1/4[80 ]


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Assumption 2.1(c) follows by Markovs inequality, noting that E4j CE81 E

81 < , which completes the proof of the proposition. n

4.1.3. Stochastic Volatility m.d. Processes. By a stochastic volatility model

one usually understands a stationary processj ,j Z of the formj= j j , j IID(0, 1), jZ,where the (volatility) process j >0 is a function of the past information up to

time j 1. Let Fj1 be the -field generated by past innovations s , s j 1,and E 2j < . Then E[j |Fj1] = 0, 2j= Var(j |Fj1), and {j } is a white noiseprocess, which by Theorem 3.5.8 of Stout (1974) is also stationary and ergodic.

It is often assumed that the volatility process j= h(j ), j Z is a nonlin-ear function of a stationary Gaussian or linear process{j }, see, e.g., Robinson(2001). The choice of h(j )= exp(j ) includes the exponential generalizedARCH (EGARCH) model, proposed by Nelson (1991). A related class of stochas-

tic volatility models with long memory in{j } was introduced and studied inBreidt, Crato, and de Lima (1998), Harvey (1998), and Surgailis and Viano

(2002). As a rule, the volatility process Vj 2j in these models is a stationaryprocess with autocovariances decaying to zero and thus satisfies Assumption 2.1.

4.1.4. Sample ACF. Let {j } be a stationary and ergodic m.d.s. such that 2=E20 < . Consider the sample autocorrelation k= k/0, k 1, wherek=n1 nj=k+1(j )(jk ) and= n1 nj=1 j . It is well known that in thecase of i.i.d. random variables,

nkD N(0, 1) and (

n1, . . . ,

nk) D

N(0,I), where the limit is a vector ofk independent standard normal variables.

These results are widely used for testing for the absence of correlation, see, e.g.,

Brockwell and Davis (1991). The next proposition shows that, in general, this

asymptotic normality result is true only for i.i.d. sequences.

Letbe m

mmatrix with(j, k)th elementj,k=

E[21

1j

1k

]/ 4

.

PROPOSITION 4.2. Let{j } be a stationary and ergodic m.d.s. with 2 :=E20 < such thatE

21

21k

< , for k= 1, . . . , m, where m is a given positive

integer. Then,n1, . . . ,

nm

D Nm (0,). (4.13)In particular, (4.13) holds in case of ARCH processj= rj of (4.1), such thatE

1/2(40)k=1 bk


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ergodicity of 2j , 0p E20 . Hence,

nk= ( 2)1n1/2 nj=k+1 j jk+O(n1/2) N(0, k,k).

To prove (4.13), let (k)

j

=j j

k , k

=1, . . . , m. Then for any real numbers

c1, . . . , cm ,j := c1(1)j + . . . + cm (m)j is a stationary ergodic m.d.s., with E2j= 2 := c211,1 + . . . + c2m m,m . Let Sn := nj=m+1 j . Then ES2n= 2 n , andby Theorem 2.1, n1/2 SnD N(0, 2 ), which by the Cramer-Wold device im-plies the claim of (4.13).

In case of ARCH processrj , (4.1) and (4.3) implyk,k 1 and show thatisdiagonal if, in addition, E3j= 0. n

Remark 4.1. Proposition 4.2 shows that for the ARCH m.d.s. (4.1), the limit

variance ofnk, as the rule, is greater than one, and so the 95% confidence bandfor k= 0 is wider than in the i.i.d. case. Moreover, the limit matrix may benondiagonal, unless the distribution ofj in (4.1) is symmetric.

4.2. Regression Models

In this section we discuss application of the above results for obtaining asymptotic

normality of least squares (LS) estimators in parametric regression models and

kernel-type estimators in nonparametric regression models.

But first we verify the above conditions for asymptotic normality Sn and weakconvergence of the partial sum process {Sn(), 0 1} of a linear process {Xj }of (2.1) in some typical cases.

Let (j ), j= 0, 1, 2, . . . denote the autocovariance function of{Xj } and fits spectral density. Note that (j ) := Cov(Xj ,X0)= 2k=0 akak+j , j=0, 1, 2,. . . .Consider the assumption in terms of f.

f(u) c|u|2d, u 0, |d| 0. (4.14)In terms of (j ), consider the condition

jZ

| (j )| < , 2 := jZ

(j ) >0, ford = 0,

(j ) c |j |1+2d, 0


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where

s2 = 2f(0), d= 0;

s2 = cf R

sin

2

(u/2)(u/2)2

|u|2ddu, 0< |d|


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Now, note that= nj=1zn jXj /nj=1z2n j=:Wn/nj=1z2n j . Define

v2d:= 10 g2(u)dukZ (k), d= 0,

:= c 1

0

10

g(u)g(v)|u v|1+2ddudv, 0K

| (s)| 0, K ,

whereas for any |i | K,n1 nk=1zn,k+iznk (i ) (i )1

0 g2(u)du.Whence,

limK

limn

n

j,k

=1:

|j

k|

K

zn jznk (j k) =

1

0g2(u)du lim

K

|i|

K

(i ) = v20 ,

which proves 2n v20.Next, consider the case 0


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n12d 2n= n12dn

k,j=1

zn jzn,kX(j k)

= n12dcn

k,j=1:k=jg jng kn|j k|1+2d+ o(1)

c 1

0

10

g(u)g(v)|u v|1+2ddudv= v2d.

This completes the proof of (4.22) and the corollary. n

4.2.2. A Non-Stationary Process. As another application of Theorem 2.2,

consider the weighted sum Vn := n

j=1zn j Yj of a nonstationary unit root processYj := js=1Xs , j= 1, 2, . . .. Set (u) =

1u g(v)dv,0 u 1. Also, define

v2d:= 1

02(u)du

kZ (k)

, d= 0,

:= c 1

0

10

(u)(v)|u v|1+2ddudv, 0


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Then,n (x) (x) = Dn (x) + n (x) (x). Typically the bias term n(x) (x) is negligible compared to Dn(x), and asymptotic distribution ofn (x) (x)is determined by that ofDn (x). Fix anx (0, 1)and let

zn j := K1nx Kx n j

nb

. (4.23)

Then, clearly Dn(x)is like aWn . Define

2d,K := K22 kZ

(k), d= 0,

:= c 1

0

10

K(u)K(v)|u v|1+2ddudv, 0


29/33


30/33


FIGURE3. Kernel density ofVar( n (x))1/2( n (x)(x))(solid line),N(0, 1)density(dashed line).5.0.5. ARFIMA-ARCH

We start from the case where Xj in (2.1) is ARFIMA(1, d, 0) model with the

AR parameterr= 0.8 and m.d. innovationsj generated as ARCH(1)process:j= j j , j N(0, 1),

2j= 0 + 12j1, j= 1, . . . , n. (5.1)

We take 0= 0.2, 1= 0.8, so that the E2

j = 1. We simulated the cases d=0.1, 0.2, 0.3, 0.4, which gave qualitatively similar results. For the sake of brevity,we report the results ford= 0.3 only. We analyze the asymptotic normality of thesuitably standardized sums

Wn=n

j=1

zn jXj , Vn=n

j=1

zn j Yj ,

whereYj= js=1Xs , with the weights zn j= (j/n)2 + cos(j/n), j= 1, 2, . . . , n.Var(Wn) and Var(Vn) are estimated by the Monte Carlo variances. We then plota kernel estimate of the densities ofVar(Wn )1/2Wn andVar(Vn)1/2 Vn usinga Gaussian kernel with bandwidth b chosen according Silvermans (1986) rule,

bn = (4/3)1/5n1/5, and superimpose the standard normal density. The results forARFIMA(1, 0.3, 0),r= 0.8 process {Xj } with ARCH(1)innovations in Figure 1


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show a very close resemblance between the solid line representing the kernel

density estimate and the dashed line representing N(0, 1)density.

AR-ARCH

Next we analyze asymptotic normality of Wn and Vn in the case of AR(1)

processXj= Xj1 +j , j Z,with = 0.6 and ARCH(1)errorsj as in (5.1).The results in Figure 2 show that the kernel density estimate ofVar(Wn)1/2Wnseems to have less probability mass close to the mean than the N(0, 1) densityand is slightly positively skewed.

Nonparametric regression

To illustrate the fit of normal approximation in nonparametric estimation, we

use the regression model Yj

=(j/n)

+Xj , j

=1, . . . , n,where Xj s follow an

ARFIMA(1, 0.3, 0)-ARCH(1) process with AR parameter r= 0.8 generated asin (5.1). We set(j/n) = (j/n)2 + cos(j/n), and(x) is estimated at x= 1/4by a kernel-type estimator

n (x) =

nj=1K

nxj

nb

Yj

nj=1K

nxj

nb

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