Bootstrap specification tests for linear covariance stationary processes

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Journal of Econometrics 133 (2006) 807–839 www.elsevier.com/locate/jeconom

Bootstrap specification tests for linear covariance stationary processes J. Hidalgoa, J.-P. Kreiss,b a

London School of Economics, Houghton Street, London WC2A 2AE, UK Technical University of Braunschweig, Institut Fu¨r Mathematische Stochastik, Technische Universita¨t Braunschweig, Pockelstrasse 14, D-38106, Braunschweig, Germany

b

Available online 10 August 2005

Abstract This paper discusses goodness-of-fit tests for linear covariance stationary processes based on the empirical spectral distribution function. We can show that the limiting distribution of the e tests are functionals of a Gaussian process, say, BðWÞ with W 2 ½0; 1
. Since in general it is not e easy, if at all possible, to find a time deformation gðWÞ such that BðgðWÞÞ is a Brownian (bridge) e process, tests based on BðWÞ will have limited value for the purpose of statistical inference. To circumvent the problem, we propose to bootstrap the test showing its validity. We also provide a Monte-Carlo experiment to examine the finite sample behaviour of the bootstrap. r 2005 Elsevier B.V. All rights reserved. JEL classification: C12; C22 Keywords: Goodness-of-fit; Linear processes; Spectral distribution; Gaussian processes; Bootstrap tests

1. Introduction This paper is concerned with goodness-of-fit tests for linear covariance stationary processes. The tests are based on continuous functionals of the integrated relative error between the periodogram and the spectral density function obtained under the Corresponding author.

E-mail addresses: [email protected] (J. Hidalgo), [email protected] (J.-P. Kreiss). 0304-4076/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jeconom.2005.06.015

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null specification. Goodness-of-fit tests are of long tradition in statistical/ econometric literature. They date back to the work of Kolmogorov (1933) for independent identically distributed (iid) data. In a time series framework, Grenander and Rosenblatt (1957) used Kolmogorov–Smirnov statistics to test for a specific weakly dependent process, say a white-noise process, and later extended by Ibragimov (1963) under the assumption of square integrable spectral density function. More recently, Kokoszka and Mikosch (1997) have allowed for, possibly, infinite variance. See also Dahlhaus (1985) or the review paper by Anderson (1993). The hypothesis in those papers is simple, and the tests are based on functionals of Brownian bridge. In frameworks useful in econometrics they have been extended by Andrews (1997) or Stute (1997) among others, although the null hypothesis depends on a set of unknown parameters to be estimated. The hypothesis testing in the latter two papers is composite. However, as noted by Durbin (1973), when the hypothesis testing is of the latter type, the limit process, on which the tests are based, is no longer a Brownian motion or Brownian bridge. In this context, to implement the test is not straightforward. The first solution was given in Durbin et al. (1975), who employed a principal component decomposition of the limiting Gaussian process as an infinite weighted sum of independent normal random variables. However, for its practical implementation, the infinite sum needs to be truncated and requires, for example, Imhof’s methods to compute the critical values. A similar approach was adopted by Anderson (1997) or Stute (1997) among others but quite apart from the need to choose the level of truncation of the infinite sum, there are some other potential drawbacks. Firstly, this approach is tailor-made, because new critical values need to be computed as the model or the data changes. A potential second drawback is that its implementation requires the estimation of the covariance structure of the limiting process, which may not be an easy task. Thirdly, the computed critical values are in three aspects an approximation of the ‘‘finite-sample’’: (i) the finite sample distribution is replaced by that obtained as the sample size becomes very large, (ii) the asymptotic distribution is not used but rather an approximation to it, i.e. the infinite sum is truncated and (iii) estimates of the covariance structure are used instead of the true one. In other words, the algorithm entails three levels of approximations when implemented to real data. Some alternative solutions have been given in the literature, such as those based on smooth estimates of the spectral density function, see for instance, Hong (1996), Prewitt (1998) and Paparoditis (2000) among others. However, their approach shows a degree of sensitivity, as they need to choose the bandwidth parameter to estimate the spectral density function. Furthermore, their application to general specifications is far from being easily implemented, although the asymptotic relative efficiency, compared to ours, is zero. One possible theoretical difficulty in their approach comes from the fact that the spectral density estimator is not consistent for frequencies near a singularity as is the case with long-memory models. It is for these reasons that we are suggesting in this paper, a bootstrap test that is easy to implement and valid for any choice of a linear covariance model without resorting to the choice of a bandwidth or tuning parameter for its implementation.

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To that end, we intend to bootstrap the test in the frequency domain. For an overview of bootstrapping in the frequency domain, see Paparoditis (2002). Specifically, the bootstrap is based on the discrete Fourier transform of the bootstrap sample obtained by Efron’s (1979) resampling scheme from the original data. Our approach is an obvious one and is quite independent of the model class under consideration. This is in contrast to similar work by Chen and Romano (1999) who use a time domain bootstrap. Given the sample, the critical values are estimated by the conditional quantiles of a bootstrap statistic. Such a statistic is the bootstrap analogue of the original one. This approach has the advantage over those mentioned above that no smoothing or tuning parameter is needed to obtain valid (asymptotic) tests. The outline of the paper is as follows. In the next section we introduce the test, examining in Section 3 its limiting distribution and power against local alternatives converging to the null at the rate n1=2 . Since the covariance structure of the limiting process cannot be transformed, except in a few exceptional cases, into that of a Brownian (bridge) process, Section 4 proposes a bootstrap algorithm to compute the quantiles of the distribution of the test, showing its consistency. In Section 5, a Monte-Carlo experiment is given to illustrate the finite sample performance of the bootstrap test. Section 6 gives the proofs of our results, which make use of five technical lemmas given in Section 7. Finally, Section 8 concludes.

2. The test Consider a zero mean linear covariance stationary process xt which is observed at times t ¼ 1; 2; . . . ; n, with spectral density function, f ðlÞ, defined from the relation Z p gj ¼ Eðxj x0 Þ ¼ f ðlÞeijl dl; j ¼ 0; 1; 2; . . . . p

Under the condition

Rp

p

logðf ðlÞÞ dl4  1, f ðlÞ admits a factorization such that

s2 jAðlÞj2 , 2p P ijz where AðzÞ ¼ 1 j¼0 aj e , aj being the coefficients in the Wold decomposition of xt , that is f ðlÞ ¼

xt ¼

1 X j¼0

aj
tj ;

1 X

a2j o1; a0 ¼ 1,

j¼0

where the innovation sequence
t is a zero mean white noise process with variance s2 . The main objective of the paper is to test the null hypothesis H0 : 8l 2 ½0; p
and for some c0 2 C;

jAðlÞj2 ¼ jAðl; c0 Þj2 ,

(1)

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where C  Rp , a compact set. The alternative is the negation of H0 , which can be written as H1 : For all c 2 C; 9PðcÞ  ½0; p
such that jAðlÞj2 ajAðl; cÞj2 , for l 2 PðcÞ and where PðcÞ has Lebesgue measure greater than zero. Observe that under H0 , we have that the spectral density function of xt is f ðl; c0 Þ¼:

s2 jAðl; c0 Þj2 . 2p

We shall assume throughout the remainder of the paper that the function jAðl; cÞj2 satisfies Z p log jAðl; cÞj2 dl ¼ 0 8c 2 C. (2) p

Condition (2) indicates that for a process with spectral density ðs2 =2pÞjAðl; cÞj2 , the free parameter s2 (functionally independent of c) is the variance of the one-stepahead best linear predictor (see Hannan, 1970, pp. 157–163). Alternatively in the time domain, the test can be formulated as whether or not xt follows the model xt ¼

1 X

aj ðc0 Þ
tj ;

j¼0

1 X

a2j ðc0 Þo1; a0 ðc0 Þ ¼ 1.

(3)

j¼0

(Observe that the normalization a0 ðc0 Þ ¼ 1 is consistent with (2).) We now describe an alternative formulation of the hypothesis testing given in (1) and the test. Denoting Z pW f ðlÞ F ðW; cÞ ¼ 2 dl, jAðl; cÞj2 0 we have that, under H0 , the function SðW; c0 Þ ¼

F ðW; c0 Þ W F ð1; c0 Þ

is zero for all W 2 ½0; 1
, whereas under H1 , SðW; cÞa0 in PðcÞ for all c 2 C. Hence, (1) can equivalently be written as H0 : 9c0 2 C such that SðW; c0 Þ ¼ 0 H1 : 8c 2 C;

SðW; cÞa0 in PðcÞ.

8W 2 ½0; 1
(4)

So, the test can be computed as follows. First, if c0 were known then we could obtain the sample analogue of SðW; c0 Þ as Sn ðW; c0 Þ ¼

F n ðW; c0 Þ  W; F n ð1; c0 Þ

W 2 ½0; 1
,

(5)

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where abbreviating gðlj Þ by gj for a generic function gðÞ, F n ðW; cÞ ¼

½nW=2
Ij 2p X n~ j¼1 jAj ðcÞj2

with
2

n 1

X itlj
Ij ¼ xt e


2pn
t¼1 denoting the periodogram of xt , with lj ¼ 2pj=n, j ¼ 1; . . . ; n~ ¼ ½n=2
. Observe that S n ðW; c0 Þ is a weighted empirical process in D½0; 1
. Also, it is worth mentioning that P½nW=2
~ j¼1 ðI j =jAj ðcÞj2 Þ. However, we prefer our we could have defined F n ðW; cÞ as ðp=nÞ definition of F n ðW; cÞ because F n ð1; c0 Þ gives a consistent estimate of s2 . In empirical examples, c0 is unlikely to be known, so we need to replace c0 in (5) b defined in (10) below, obtaining by a suitable estimate, say the Whittle estimator c b b ¼ F n ðW; cÞ  W; Sn ðW; cÞ b F n ð1; cÞ

W 2 ½0; 1
.

(6)

Hence, the test for H0 given in (1) can be based on whether or not the left-hand side of (6) is significantly different from zero for all W 2 ½0; 1
. More specifically, we propose to test for H0 using b b Zn ¼ jðn~ 1=2 S n ðW; cÞÞ,

(7)

where jðÞ is some continuous functional such that jð0Þ ¼ 0 and jðÞX0. Two such functionals are the Kolmogorov–Smirnov and the normalized Crame´r–von Mises ostatistic given by Bn ¼

sup ~ f‘:‘¼1;...;ng

b jn~ 1=2 S n ðl‘ ; cÞj

and

Cn ¼

n~ 1X b 2, ðn~ 1=2 S n ðl‘ ; cÞÞ n~ ‘¼1

(8)

respectively. We finish this section by discussing the motivation for employing the relative difference between the periodogram and the spectral density function instead of their difference to test for (1). To this end, consider the weighted empirical spectral distribution Z Wp G n ðW; gðlÞÞ ¼ gðlÞIðlÞ dl; W 2 ½0; 1
. 0

Assuming that c0 is known in (3) and similar to the empirical process theory, G n ðW; gðlÞÞ can form the basis for R Wpgoodness-of-fit tests by considering the difference between G n ðW; 1Þ and G c0 ðWÞ ¼ 0 f ðl; c0 Þ dl, W 2 ½0; 1
, that is G n ðW; 1Þ  G c0 ðWÞ.

(9)

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Under suitable regularity conditions, see for example Grenander and Rosenblatt (1957), Ibragimov (1963) and more recently Anderson (1993) and references therein, weakly

n1=2 ðG n ðW; 1Þ  G c0 ðWÞÞ ¼) G 1 ðWÞ;

W 2 ½0; 1
,

where G 1 ðWÞ is a Gaussian process in ½0; p
. However, results in Hannan (1976) suggest that one necessary condition for the last result to hold true is the squared integrability of f ðl; c0 Þ. This can be expected from the equality n1 1 1X sinðpjWÞ 1=2 2X sinðjWÞ 1=2 n ðbgj  gj Þ  n gj , p j¼0 j p j¼n j P 1 sinðpjWÞ ¼ pW where gj ¼ Eðxj x0 Þ, bgj ¼ n1 nj t¼1 xt xtþj and with the convention j for j ¼ 0. Thus, one necessary condition for the weak convergence of n1=2 ðG n ðW; 1Þ  Gc0 ðWÞÞ to a Gaussian process is that n1=2 ðbgj  gj Þ converges in distribution to a normal random variable. But, following Hannan (1976), a necessary condition for the latter property is that f ðlÞ 2 L2 ½0; p
. However, since in this paper, we want to allow for all covariance stationary processes, e.g. processes for which f ðlÞ 2 L1 ½0; p
, the previous arguments illustrate the reason to consider the T p -Bartlett statistic in (6) instead of the U p type statistic given in (9).

n1=2 ðG n ðW; 1Þ  G c0 ðWÞÞ ¼

3. Asymptotic properties of the test Given the parametric model (3) and condition (2), we estimate the parameters c, due to its computational convenience, by Whittle’s method, that is, b ¼ arg min F n ð1; cÞ; c c2C

b b s2 ¼ F n ð1; cÞ.

(10)

b given in (6) with c b as in Before we examine the asymptotic properties of S n ðW; cÞ (10), let us introduce the following regularity conditions: C.1: fxt g is a covariance stationary linear process defined as xt ¼

1 X j¼0

aj ðcÞ
tj ;

1 X

a2j ðcÞo1; a0 ðcÞ ¼ 1,

j¼0

where the innovation sequence f
t g is an ergodic process which satisfies Eð
t jFt1 Þ ¼ 0, Eð
2t jFt1 Þ ¼ Eð
2t Þ ¼ s2 a.s., Eðj
t j‘ jFt1 Þ ¼ m‘ o1, ‘ ¼ 3; 4, where Ft is the s-algebra of events generated by
s ; spt, and with the joint fourth cumulant of
t1 ,
t2 ,
t3 and
t4 satisfying  k4 ; t1 ¼ t2 ¼ t3 ¼ t4 ; cumð
t1 ;
t2 ;
t3 ;
t4 Þ ¼ 0 otherwise: C.2: The function jAðl; cÞj2 is even in l and bounded away from zero, and the derivatives rc jAðl; cÞj2 , rl jAðl; cÞj2 , rc rl jAðl; cÞj2 , rc r0c jAðl; cÞj2 are

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continuous in any open set outside the origin and jðq=qlÞAðl; cÞj ¼ OðjAðl; cÞjl1 Þ as l ! 0þ. C.3: c0 is an interior point of a compact set C  Rp . C.4: The set fl : jAðl; c1 Þj2 ajAðl; c2 Þj2 g;

c1 ac2

has positive Lebesgue measure. Also, the matrix Z p rc log jAðl; c0 Þj2 r0c log jAðl; c0 Þj2 dl40. 0

Conditions C.1–C.4 are very much the same as those in Giraitis et al. (2001), so their comments apply in the present paper and no further discussion is needed. b be given in (10). Assuming C.1–C.4, under H0 in (4), Theorem 3.1. Let c weakly

b ¼) S 1 ðWÞ n~ 1=2 S n ðW; cÞ in D½0; 1
endowed with the Skorohod’s metric and where S 1 ðWÞ is a Gaussian process centered at zero with covariance structure (11) KðW1 ; W2 Þ ¼ ðminðW1 ; W2 Þ  W1 W2 Þ  ð1=pÞG0 ðW1 ÞA1 GðW2 Þ, R pW Rp 0 where GðWÞ ¼ 0 fðl; c0 Þ dl, A ¼ 0 fðl; c0 Þf ðl; c0 Þ dl with fðl; c0 Þ ¼ q log jAðl; c0 Þj2 =qc. Proof. The proof of this theorem or any other result will be deferred to Section 6. b given in (11) is different We observe that the limiting covariance of n~ 1=2 S n ðW; cÞ than that obtained when c0 is known. This is expected since n~ 1=2 Sn ðW; Þ is evaluated b instead of c , in which case the second term on the right of (11) would not at c 0 b forms the basis for the construction of appear. The limiting behaviour of n~ 1=2 S n ðW; cÞ a test of (1), as we illustrate in: Corollary 3.2. Let jðÞ be a continuous functional and b Zn defined in (7). Under H0 and d the same conditions of Theorem 3.1, b Zn ! jðS 1 ðWÞÞ. Proof. The proof is an immediate consequence ofTheorem 3.1 and the continuous mapping theorem. & Theorem 3.1/Corollary 3.2 gives the asymptotic distribution of the test under H0 . As in related hypothesis testing problems, one feature required for any test is its consistency. In addition, to shed some light on the structure of the test it is desirable b to describe its power under local alternatives. Specifically, we will show that S n ðW; cÞ can detect departures converging to H0 at the parametric rate n1=2 . To this end, consider the local alternatives   1 Ha : f ðl; cÞ 1 þ 1=2 gðlÞ for some c 2 c, (12) n

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where gðlÞ is some symmetric, nonconstant continuous function in ½0; p
such that ð1=n1=2 Þ gðlÞ4  1 for all nX1. Remark 3.1. We should point out that Ha could have been written as Ha : f ðl; cÞ þ

1 n1=2

geðlÞ for some c 2 c

where geðlÞ is a positive integrable function. However, since we are concerned with the relative error of I j compared to f j ðcÞ ðjAj ðcÞj2 Þ, we found it notationally more convenient to write the alternative hypothesis Ha as given in (12). Theorem 3.3. Let jðÞ be a continuous functional. Assuming C.1–C.4, under Ha given d in (12), b Zn ! jðS 1 ðWÞ þ RðWÞÞ, where Z p Z p RðWÞ ¼ ðIðlppWÞ  WÞgðlÞ dl  GðWÞA1 fðl; cÞgðlÞ dl, 0

0

where IðBÞ denotes the indicator function of the event B. b is a The previous theorem indicates that the limit, under Ha , of n~ 1=2 S n ðW; cÞ noncentered Gaussian process, with ‘‘noncentrality function’’ RðWÞ. So, the test will have nontrivial power under Ha if RðWÞa0, which is the case under condition (2) as we now show. One possibility for RðWÞ ¼ 0 is when gðlÞ is a constant which is ruled out since it would imply that Ha  H0 . Next, suppose that there exists a nonconstant function gðlÞ such that RðWÞ ¼ 0 holds true. However, following the discussion in Stute’s (1997) Section 3 or Durbin et al. (1975), we have that RðWÞ ¼ 0 if and only if gðlÞ belongs to the space spanned by ffk ðl; c0 Þ; k ¼ 1; . . . ; pg where P fk ðl; c0 Þ denotes the kth component of fðl; c0 Þ. That is, we have that gðlÞ ¼ pk¼1 d k fk ðl; c0 Þ. The latter can easily be seen by looking at the principal component decomposition of the Rp 1 process S ðWÞ þ RðWÞ. But, condition (2) implies that logðð1 þ n1=2 gðlÞÞÞ dl ¼ 0 1 0 Rp 2 since 0 jAðl; c0 Þj dl ¼R 0. On the other hand, since g is a linear combination of p fk ðl; c0 Þ we have that 0 n1=2 gðlÞ dl ¼ 0. So, we obtain that   Z p 1 1 gðlÞ  log 1 þ gðlÞ dl ¼ 0. n1=2 n1=2 0 But since the function z  1  log zX0 with equality if z ¼ 1, it implies that for the last displayed equality to hold true gðlÞ ¼ 0, e.g. Ha  H0 . Thus, we conclude that P gðlÞa p1 k¼1 d k fk ðl; c0 Þ so that under Ha , RðWÞa0. Once the statistical properties of b Zn in (7) are known, the question is how to implement the test in practice. Because of the complicated nature of KðW1 ; W2 Þ in (11), to find a (time) transformation that would lead to a Brownian (bridge) process appears to be a difficult task. Although, in principle, the asymptotic distribution could be simulated, since it is a model-dependent random variable, it would imply that a practitioner would need to compute critical values every time a new model is under consideration. Thus, to circumvent the problems mentioned above, we describe and examine a bootstrap approach in the next section.

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4. Bootstrap test for H0 Since Efron’s (1979) seminal paper on the bootstrap, an immense effort has been devoted to its development. The primary motivation for this effort is that it has proved to be a very useful statistical tool. We can cite two main examples/reasons. First, bootstrap methods are capable of approximating the finite sample distribution of statistics better than those based on their asymptotic counterparts. And secondly, and perhaps the most important, it allows computing valid asymptotic quantiles of the limiting distribution in situations where (1) the limiting distribution is unknown or (2) even if known, the practitioner is unable to compute its quantiles. In the present paper we face the latter situation. Following our comments at the end of the previous section, the aim of this section is thus to propose a bootstrap b given in (6). The resampling method must be such that the procedure for n~ 1=2 Sn ðW; cÞ conditional distribution, given x ¼ ðx1 ; . . . ; xn Þ0 , of the bootstrap statistic, say b Zn ,   consistently estimates the distribution of jðS1 Þ under H0 . That is, b Zn !d  jðS1 Þ in probability under H0 , where ‘‘!d  ’’ denotes p

lim Pr½b Zn pzj x
! GðzÞ,

n!1



at each continuity point z of GðzÞ ¼ PrðjðS 1 ÞpzÞ as defined in Gine´ and Zinn (1990). Moreover, under local alternatives Ha , b Zn must also converge, in bootstrap distribution to jðS 1 Þ, whereas under the alternative H1 , b Zn should be bounded in probability to have good power properties. b in (6) We first describe and motivate the bootstrap approach. Looking at S n ðW; cÞ 2 b b and regarding 2pI j =ðF n ð1; cÞjAj ðcÞj Þ  1 as if they were residuals of some regression model, one could be tempted to use the following wild bootstrap: ! n~ 2pI j 1 X ðIðlj ppWÞ  WÞ  1 mj (13) b j ðcÞj b 2 n~ 1=2 j¼1 F n ð1; cÞjA as in Stute et al. (1998), where mj are zero mean iid random variables with unit variance. However, this bootstrap is inconsistent, as we now illustrate. Let E ðzÞ denote the bootstrap expectation of a random variable z. Then, using arguments from the proof of Theorem 3.1 and denoting JðW; jÞ ¼ Iðlj ppWÞ  W, ! !2 2pI j E JðW; jÞ  1 mj b j ðcÞj b 2 n~ 1=2 j¼1 F n ð1; cÞjA 8 ! !2 9 n~ < = X 2pI j 1 b  c Þ0 f ðc Þ þ op ð1Þ ¼ J2 ðW; jÞ  1 þ ð c 0 j 0 2 : ; n~ j¼1 s2 jAj ðc0 Þj !2 n~ p 2pI j 1X 2 ¼ J ðW; jÞ 2  1 þ op ð1Þ !ðW  W2 Þ, 2 n~ j¼1 s jAj ðc0 Þj 

n~ 1 X

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b  c ¼ Op ðn1=2 Þ and F n ð1; cÞ b is a n1=2 -consistent estimator as is well known that c 0 1 2 for s (see the proof of Theorem 3.1), and where f j ðc0 ÞI j has been approximated by a w21 random variable. Thus, in the limiting covariance structure of (13), the b  c Þ does not appear. contribution in (11) due to the limiting distribution of n1=2 ðc 0 Alternatively, we could consider ! n~ n~ X 2pI j 2pI p 1 1X JðW; jÞ m  , (14) b n~ 1=2 j¼1 b 2 j n~ p¼1 jAp ðcÞj b 2 jAj ðcÞj F n ð1; cÞ where now mj are unit mean and variance iid random variables. However, similar to the bootstrap given in (13), (14) would be also inconsistent. The intuition for its b ¼ 2p Pn~ jAp ðcÞj b 2 I p =n~ and inconsistency is as follows. Recalling that F n ð1; cÞ p¼1 arguing as above, (14) is ! n~ Ij 2p X JðW; jÞ mj  1 jAj ðc0 Þj2 s2 n~ 1=2 j¼1 b  c Þ0 þ n1=2 ðc 0

n~ 2p X JðW; jÞfj ðc0 Þmj þ op ð1Þ. s2 n~ 1=2 j¼1

Proceeding as in the proof of Proposition 6.1 below, the first term is   n~ 2pI
;j 1 X JðW; jÞ m  1 þ op ð1Þ s2 j n~ 1=2 j¼1   n~ n~ 2pI
;j 1X 1 X JðW; jÞ  1 m þ JðW; jÞðmj  1Þ þ op ð1Þ. ¼ j s2 n~ j¼1 n~ 1=2 j¼1 From here, it is clear that the covariance structure of the limiting distribution will have an extra term, added to the right of (11), due to the second term on the right of the last displayed equation. Therefore, in this paper we propose the following bootstrap. P et ¼ ðxt  xÞ=b Step 1: For t ¼ 1; . . . ; n, let x sx where x ¼ n1 nt¼1 xt and 2 1 Pn 2 b sx ¼ ðn  1Þ t¼1 ðxt  xÞ , and a random sample of size n with replacement from et . Denote that sample as x ¼ ðx1 ; . . . ; xn Þ0 . the empirical distribution of x  ~ compute the bootstrap periodogram Step 2: For j ¼ 1; . . . ; n, b  , I j ¼ f j ðcÞI x;j where b b ¼ F n ð1; cÞ jAj ðcÞj b 2; f j ðcÞ 2p

I x;j


2

n 1

X  itlj
¼
xt e

n
t¼1

b by and the bootstrap analogue of c b  ¼ arg min F  ð1; cÞ; c n c2C



b Þ, b s2 ¼ F n ð1; c

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where F n ðW; cÞ ¼

~ ½nW
I j 2p X . n~ j¼1 jAj ðcÞj2

(15)

Step 3: Compute the bootstrap statistic bÞ Sn ðW; c

bÞ F n ðW; c ¼  W. bÞ F  ð1; c

(16)

n

Some other procedures are possible. Similar to that in Franke and Ha¨rdle (1992), for the bootstrap of the spectral density estimator, we could replace Steps 1 and 2 by Step 10 : We draw independent exponential variables w1 ; . . . ; wn~ with parameter 1 and b ~ compute the bootstrap periodogram I  j ¼ f j ðcÞwj for j ¼ 1; . . . ; n. 0 b Step 2 : Define the bootstrap analogue of c by b  ¼ arg min F  ð1; cÞ; c n c2C

but with

I j

b  Þ, b s2 ¼ F n ð1; c

in (15) replaced by I  j .

A related bootstrap was considered in Delgado and Hidalgo (2000) to test for a FARIMA model. The bootstrap is based on the Cholewsky’s decomposition of the b estimated covariance matrix of the data, gðjk  jj; cÞ k;j¼1;...;n , to obtain the innovations
t of the model. Although the algorithm appears feasible in the context of FARIMA models, it seems difficult to implement it in the context of general linear models such as Bloomfield’s (1973) (fractionally integrated) exponential, FEXP, model, see Robinson (1994) for definitions or the fractional Gaussian noise (fgn) introduced by Mandelbrot and Van Ness (1968) model. The main reason is that for the latter two models, contrary to the FARIMA model, to obtain an explicit function for the autocovariance in terms of the parameters of the model does not seem easy if at all possible. In this context, one possible solution could be to replace gðjÞ by its sample analogue, but to show the validity of this approach remains an open question. Theorem 4.1. Assuming C.1–C.4, under H0 [ Ha , 

d b  ÞÞ ! jðS1 ðWÞÞ in probability. jðn~ 1=2 Sn ðW; c

Thus, Theorem 4.1 indicates that the bootstrap statistic given in (16) is consistent. That is, let jðÞ be a continuous functional designed to test for H0 , and let cfn;ð1aÞ and cað1aÞ be such that f b Prfjjðn~ 1=2 S n ðW; cÞÞj4c n;ð1aÞ g ¼ a

and a b lim Prfjjðn~ 1=2 S n ðW; cÞÞj4c ð1aÞ g ¼ a,

n!1

ARTICLE IN PRESS 818

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respectively. So Theorems 3.1 and 4.1 indicate that cfn;ð1aÞ ! cað1aÞ and cð1aÞ ! cað1aÞ , respectively, where cð1aÞ is defined as 

b ÞÞj4c g ¼ a. Prfjjðn~ 1=2 S n ðW; c ð1aÞ 

b ÞÞ is not available, Typically, the finite sample distribution of jðn~ 1=2 Sn ðW; c  although the critical values cð1aÞ can be approximated, as accurately as desired, by standard Monte-Carlo simulation. To that end, consider the bootstrap samples ð‘Þ 0  b I ð‘Þ ¼ ðI ð‘Þ 1 ; . . . ; I n~ Þ for ‘ ¼ 1; . . . ; B, and compute S n ðW; c Þ as in (16) for each ‘.  B Then, cð1aÞ is approximated by the value cð1aÞ that satisfies B 1 X b  ÞÞ4cB Þ ¼ a. Iðjðn~ 1=2 Sn ðW; c ð1aÞ B ‘¼1

Next we study the behaviour of the bootstrap tests under the alternative H1 . Corollary 4.2. Assuming C.1–C.4, under H1 , 

d b  ÞÞ ! jðn~ 1=2 Sn ðW; c jðSe1 ðWÞÞ in probability,

where Se1 ðWÞ is a centred Gaussian process with covariance structure KðW1 ; W2 Þ but with b c0 replaced by c1 ¼ p lim c. Proof. The proof is exactly the same as that of Theorem 4.1 with the only difference b  c ¼ op ð1Þ we write c b  c ¼ op ð1Þ and c instead of that instead of writing c 0 1 1 c0 . &

5. Monte-Carlo experiment In this section, we illustrate by a small Monte-Carlo experiment the performance of the bootstrap in small samples. The models considered were xt ¼ rxt1 þ
t

(17)

with r ¼ 0:3, 0.5 and 0.9, and ð1  LÞd xt ¼
t

(18)

for d ¼ 0:2, 0.3 and 0.4, and where in (17) and (18)
t is a sequence of iid Nð0; 1Þ random variables. In all the experiments, we have generated 10 000 Monte-Carlo samples and we have used B ¼ 299 bootstrap resamples. We have considered sample sizes of n ¼ 64; 128 and 256 and the Kolmogorov–Smirnov and Crame´r–von Mises functionals given in (8), and denoted by Bn and C n in Tables 1–4 below. Table 1 studies the performance of the level of the bootstrap test when the null hypothesis is that the process is an ARð1Þ model given in (17) with parameter r ¼ 0:3, 0.5 and 0.9. In Table 2, we examined the performance of the test under H0 when the model follows a FARIMAð0; d; 0Þ process with d ¼ 0:2, 0.3, 0.4 given in (18).

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Table 1 Proportion of rejections, in 10 000 Monte-Carlo experiments, under H0 when testing that the process is an ARð1Þ a ð%Þ

r ¼ 0:3 Cn

r ¼ 0:5 Bn

r ¼ 0:9

Cn

Bn

Cn

Bn

n ¼ 64

1 5 10

0.52 3.99 8.85

0.51 4.17 9.21

0.93 5.18 10.61

0.82 5.30 10.60

1.04 5.81 11.52

1.14 5.29 10.65

n ¼ 128

1 5 10

1.08 5.10 10.26

0.91 4.92 10.21

1.11 5.51 10.66

0.86 5.41 10.39

1.16 5.53 10.95

1.16 5.28 10.74

n ¼ 256

1 5 10

1.13 5.38 10.14

1.15 5.46 10.51

1.20 5.37 10.52

1.17 5.60 10.50

1.14 5.69 10.95

1.19 5.49 10.83

Observations generated as xt ¼ rxt1 þ
t ,
t iid Nð0; 1Þ. Bootstrap critical values are computed based on 299 bootstrap samples.

Table 2 Proportion of rejections, in 10 000 Monte-Carlo experiments, under H0 when testing that the process is a FARIMAð0; d; 0Þ process with parameter d and the innovations
t are Nð0; 1Þ a ð%Þ

d ¼ 0:2

d ¼ 0:3

d ¼ 0:4

Cn

Bn

Cn

Bn

Cn

Bn

n ¼ 64

1 5 10

0.55 3.68 7.88

0.70 3.99 8.39

0.76 4.14 8.76

0.73 4.60 9.10

0.80 4.55 9.62

0.88 4.87 9.79

n ¼ 128

1 5 10

0.70 4.10 9.09

0.67 4.28 9.03

0.79 4.33 9.43

0.71 4.44 9.20

0.72 4.22 9.00

0.70 4.26 8.92

n ¼ 256

1 5 10

1.03 4.97 9.99

0.93 4.95 9.79

0.97 4.86 9.99

0.93 4.93 9.91

0.79 4.45 9.44

0.75 4.65 9.46

Bootstrap critical values are computed based on 299 bootstrap samples.

Tables 1 and 2 illustrate that the results for both models (17) and (18) are qualitatively similar, showing that the test gives good empirical sizes even for sample sizes as small as n ¼ 64. Also, as it may be expected when the parameters r and/or d become closer to the nonstationarity region, the results are qualitatively worse,

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Table 3 Proportion of rejections, in 10 000 Monte-Carlo experiments, under H1 when testing that the process is a FARIMAð0; d; 0Þ and the observations are generated according to an ARð1Þ with parameter r and the innovations
t are Nð0; 1Þ a ð%Þ

r ¼ 0:3

r ¼ 0:5

r ¼ 0:9

Cn

Bn

Cn

Bn

Cn

Bn

n ¼ 64

1 5 10

4.20 16.89 28.87

3.77 15.19 26.19

7.50 25.75 40.10

6.64 22.77 35.80

85.67 94.68 96.76

80.91 92.38 95.34

n ¼ 128

1 5 10

13.25 34.62 58.71

10.23 29.30 43.25

18.90 47.41 63.63

15.66 40.24 56.59

99.26 99.88 99.97

98.75 99.72 99.91

n ¼ 256

1 5 10

35.47 62.55 74.91

26.52 52.28 67.73

53.22 81.53 90.40

41.56 72.57 84.63

100.0 100.0 100.0

100.0 100.0 100.0

Bootstrap critical values are computed based on 299 bootstrap samples.

Table 4 Proportion of rejections, in 10 000 Monte-Carlo experiments under H1 , when testing that the process is an ARð1Þ and the observations are generated according to a FARIMAð0; d; 0Þ process with parameter d and the innovations
t are Nð0; 1Þ a ð%Þ

d ¼ 0:2 Cn

d ¼ 0:3 Bn

d ¼ 0:4

Cn

Bn

Cn

Bn

n ¼ 64

1 5 10

0.60 3.53 7.76

0.47 3.27 7.88

1.27 5.99 11.62

0.91 5.14 10.43

2.03 7.99 14.50

1.64 6.77 12.59

n ¼ 128

1 5 10

2.77 10.25 17.89

1.65 8.48 15.49

6.48 18.83 29.63

3.85 13.89 23.65

10.05 26.87 39.00

6.84 20.41 31.56

n ¼ 256

1 5 10

10.32 26.81 38.99

6.16 20.81 33.35

24.71 48.97 61.94

13.91 36.44 51.54

41.18 66.95 77.13

27.65 53.39 66.79

Bootstrap critical values are computed based on 299 bootstrap samples.

indicating that larger sample sizes are needed. Although, the results at the 1% level are a bit disappointing, we notice that as the sample size increases, the results become better. In particular, for n ¼ 256 the results are very satisfactory.

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Tables 3 and 4 examine the power of the test. More specifically, Table 3 describes the empirical power when testing that the model is an ARð1Þ process, but the true model is a FARIMAð0; d; 0Þ process with parameter d ¼ 0:2, 0.3, or 0.4, whereas in Table 4, we report the power of the tests when testing that the data follow a FARIMAð0; d; 0Þ process but the true model is an ARð1Þ with parameter 0.5. Glancing at these two tables we observe, not surprisingly, that the power of the test increases with the sample size. Of course, this is what one can expect as, in finite samples, the power depends very much on how far away the true model is from the hypothetical one. Moreover, as r and/or d increases the power increases as Tables 3 and 4 illustrate. Again, this is according to what we would expect, as the ARð1Þ model becomes a poorer approximation to the FARIMAð0; d; 0Þ model for bigger d or vice versa. On the other hand, the power in Table 3 is bigger than that in Table 4. This can be explained as the dependence for an ARð1Þ is far smaller than for a FARIMAð0; d; 0Þ. Another feature which we can draw from Table 4 is that the power for small sample sizes, e.g. n ¼ 64, is not much different than the size. This is somehow expected because for that small sample size, the ARð1Þ can be a good and accurate approximation to the FARIMAð0; d; 0Þ, specially for small values of d. Fortunately, we observe that as the sample size increases, so does power. In particular, for sample sizes of n ¼ 256 the power seems to indicate that even for d ¼ 0:2, the test is able to reject the ARð1Þ hypothesis.

6. Proofs of the results For the next three propositions, the function z : ð0; p
! Rp satisfies that kz ðlÞkpCj log lj‘ , for some ‘X1, and kqzðkÞ ðlÞ=qlkpCl1 j log lj‘1 for all l40 and k ¼ 1; . . . ; p, and where zðkÞ denoted the kth element of z. Finally, henceforth C will denote a generic finite positive constant. ðkÞ

Proposition 6.1. Assuming C.1–C.4, for all k ¼ 1; . . . ; p,
!
~ ½nW


1 X Ij

ðkÞ sup
1=2 zj  I
¼ op ð1Þ.
;j 2
jA ðc Þj ~ W2½0;1

n j 0 j¼1

(19)

Proof. Let uj ¼ A1 j ðc0 Þwx;j , vj ¼ w
;j and Aj ðc0 Þ ¼ Aðlj ; c0 Þ, where wx;j ¼ ð2pnÞ1=2

n X

xt eitlj

and

w
;j ¼ ð2pnÞ1=2

t¼1

n X


t eitlj .

t¼1

Then, the left-hand side of (19) is bounded by

~ ~ ½nW
½nW


1 X 1 X ðkÞ

ðkÞ 2 C sup 1=2 zj juj  vj j þ C sup
1=2 zj vj ðuj  vj Þ
,

~ ~ W2½0;1
n W2½0;1
n j¼1 j¼1 where c denotes the conjugate of the complex number c.

(20)

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The first term of (20) is op ð1Þ by Markov’s inequality, because its expectation is, in absolute value, bounded by       n~ 1 X s2 s2 s2 ðkÞ 2 jz j Eju j  v Þ  v Þ   Eðu  Eðu j j j j j j 2p 2p 2p n~ 1=2 j¼1 !   n~ s2 1 X log j jzðkÞ þ Ejvj j2  ¼ O 1=2 j j j 2p n~ j¼1 because Ejvj j2 ¼ s2 ð2pÞ1 and by Theorems 1 and 2 of Robinson (1995a) and its extension to all jpn~ given in Lemma 4.4 of Giraitis et al. (2001). Next, the second term of (20) is bounded by





~ ½n~ b
½nW


1 X X

1

ðkÞ ðkÞ
(21) C
1=2 zj vj ðuj  vj Þ
þ C sup
1=2 zj vj ðuj  vj Þ

,

n~ ~ j¼½n~b
þ1 W2ð0;1

n
j¼1 P where 0obo1=4, and the convention that dc ¼ 0 if doc. Proceeding as with the proof of (4.8) in Robinson (1995b) and that jzðkÞ ðlÞjpCj log lj‘ , the second moment of the first term is Oðnð2b3Þ=3 log2‘þ2 nÞ. So, Markov’s inequality implies that the first term of (21) is op ð1Þ. To complete the proof we need to show that the second term of (21) is also op ð1Þ. To that end, we will show that the finite dimensional distributions of the expression inside the absolute value converge to zero in probability and tightness. First,
2
~ 2
½nW

1 X

ðkÞ E
1=2 zj vj ðuj  vj Þ
¼ a1 þ a2 þ b1 þ b2 ,

n~ j¼½nW ~
þ1 1

where a1 ¼

1 n~

~ 2
½nW X

zðkÞ2 fEjuj j2 Ejvj j2 þ jEðvj uj Þj2 þ jEðvj uj Þj2 þ 2ðEjvj j2 Þ2 þ ðEv2j Þ2 j

~ 1
þ1 j¼½nW

 2Ejvj j2 Eðvj uj Þ  jEv2j jEðuj vj Þ  2Ejvj j2 Eðuj vj Þ  Eðuj vj ÞEðv2j Þg,

a2 ¼

1 n~

~ 2
½nW X

zðkÞ2 fcumðvj ; vj ; uj ; uj Þ þ cumðvj ; vj ; vj ; vj Þ  cumðvj ; vj ; vj ; uj Þ j

~ 1
þ1 j¼½nW

 cumðvj ; vj ; vj ; uj Þg,    s2 s2 Eðvj vk ÞEðuj uk Þ þ Eðvj uj Þ  Eðvk uk Þ  2p 2p ~ 1
ojokp½nW ~ 2
½nW     s2 s2 þ Eðvj uj Þ  þ Eðvk uk Þ  þ Eðvj uk ÞEðuj vk Þ þ Eðvj vk ÞEðvj vk Þ 2p 2p

1 b1 ¼ n~

X

ðkÞ zðkÞ j zk



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823

    s2 s2 2 þ Ejvk j  þ Eðvj vk ÞEðvj vk Þ  Eðvj vk ÞEðuj vk Þ  Eðvj uj Þ  2p 2p    s2 Eðvj vk ÞEðuj vk Þ  Eðvj uk ÞEðvj uk Þ  Eðvk uk Þ   Eðvj uk ÞEðvk vj Þ , 2p and b2 ¼

1 n~

X

ðkÞ zðkÞ j zk fcumðvj ; vk ; uj ; uk Þ þ cumðvj ; vj ; vk ; vk Þ

~ 1
ojokp½nW ~ 2
½nW

 cumðvj ; vk ; uj ; vk Þ  cumðvj ; vk ; vj ; uk Þg. By routine extension of the proof of the term (4.8) in Robinson (1995b) to ½0; p
but using Lemma 4.4 of Giraitis et al. (2001) when needed, it follows that a1 þ a2 ¼ p

1 n~

~ 2
½nW X

zðkÞ2 j

~ 1
þ1 j¼½nW

log j log2þ2‘ n pC j n~ 1þb

~ 2
½nW X

1

~ 1
þ1 j¼½nW

C ðW2  W1 Þlog2þ2‘ n, n~ b

ð22Þ

and b1 þ b2 is bounded by 1=2

1=2

C n~ b log2þ2‘ ðnÞððW2  W1 Þ þ ðW2  W1 Þ2 ÞpC n~ b ðW2  W1 Þlog2þ2‘ n, 1=2

1=2

because ðW2  W1 ÞpCðW2  W1 Þ1=2 and where we have employed the fact that ~ 1
. So, the finite dimensional distributions of the second term of (21) ½n~ b
p½nW converge to zero in probability by Markov’s inequality and the fact that b40. To complete the proof we need to show tightness. By Billingsley’s (1968) Theorem 15.6, it suffices to show that, for some d; g40 and all 0pW1 oWoW2 p1,

g

g ! ~ 1
~ 2
½nW ½nW
1

1
X X



ðkÞ ðkÞ E
1=2 zj vj ðuj  vj Þ

1=2 zj vj ðuj  vj Þ
pCðW2  W1 Þ1þd .
n~ j¼½nW
þ1


n~ j¼½nW ~ ~ 1
þ1 (23) First, observe that we can take n~ pW2  W1 since otherwise (23) holds trivially. Because ðW  W1 ÞðW2  WÞpðW2  W1 Þ2 , by Cauchy–Schwarz inequality it suffices to show that
2g
~ 2
½nW

1 X

1þd E
1=2 zðkÞ . j vj ðuj  vj Þ
pCðW2  W1 Þ

n~ j¼½nW ~ 1
þ1 1

But this is the case since, choosing g ¼ 1, we have already shown that the left side of the last displayed inequality is C ðW2  W1 Þlog2þ2‘ npCðW2  W1 Þ1þd n~ b because n~ 1 pW2  W1 and b40.

&

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824

Proposition 6.2. Assuming C.1, the finite dimensional distributions of   ~ ½nW
2p X s2 zj I
;j  Rn ðWÞ ¼ 1=2 2p n~ j¼1 converge to those of a Gaussian process with covariance structure X1 ðW1 ; W2 Þþ s4 X2 ðW1 ; W2 Þ, where the ðk1 ; k2 Þth element of the matrices X1 ðW1 ; W2 Þ and X2 ðW1 ; W2 Þ are  Z  Z pW2  ð2 þ k4 Þs4 1 pW1 ðk1 Þ 1 ðk2 Þ z ðxÞ dx z ðxÞ dx , 2 p 0 p 0  Z pW1  Z pW2  Z pðW1 ^W2 Þ 1 1 1 zðk1 Þ ðxÞzðk2 Þ ðxÞ dx  zðk1 Þ ðxÞ dx zðk2 Þ ðxÞ dx , 4p 0 2p 0 2p 0 respectively. Proof. Fix W1 ; . . . ; Wq and constants a1 ; . . . ; aq . Observing that ! ! ~ ½nW
n n t1 X X 1 X 1 X 2 2 z ð
 s Þ þ

s cts ðWÞ, Rn ðWÞ ¼ t j t n1=2 t¼1 21=2 n~ j¼1 t¼2 s¼1

(24)

where the kth element of the p 1 vector cs ðWÞ is 1 3=2 ~ cðkÞ s ðWÞ ¼ 2 n

~ ½nW
X

zðkÞ j cosðslj Þ,

(25)

j¼1

we have that by Crame´r–Wold device, it suffices to investigate the limit distribution of ! ~ p
½nW q q n n X X 1 X 1X 1 X 2 2 ap Rn ðWp Þ ¼ 1=2 ap zj ð
 s Þ þ zt , (26) t n~ j¼1 n1=2 t¼1 2 p¼1 p¼1 t¼2 where zt ¼
t

t1 X s¼1


s

q X

! ap cts ðWp Þ ,

p¼1

suppressing any reference to n in zt and cts ðWp Þ, for p ¼ 1; . . . ; q. The first and second terms on the right of (26) are uncorrelated since, by C.1, for all tos, Eð
t
s ð
2r  1ÞÞ ¼ 0. Next, the first term on the right of (26) converges to R pWp R pWp P Nð0; ½ðð2 þ k4 Þs4 Þ=2p2
qp1 ;p2 ¼1 ap1 ð 0 1 zðxÞ dx 0 2 z0 ðxÞ dxÞap2 Þ because by stanP dard CLT for martingale differences, n1=2 nt¼1 ð
2t  s2 Þ converges in distribution R P½nW ~
1 pWp ðkÞ to Nð0; ð2 þ k4 Þs4 Þ and by Lemma 7.1, jn~ 1 j¼1p zðkÞ z ðxÞ dxj ¼ j p 0 Oðn1 log‘ nÞ for all k ¼ 1; . . . ; p. So, it remains to examine the behaviour of the second term on the right of (26). Because zðkÞ t , the kth element of the vector zt , forms a martingale difference triangular

ARTICLE IN PRESS J. Hidalgo, J.-P. Kreiss / Journal of Econometrics 133 (2006) 807–839

825

array sequence, Hall and Heyde (1980) imply that it suffices to check that, for k1 and k2 ¼ 1; . . . ; p, n X

(a)

4 1 Þ ðk 2 Þ Eðzðk t zt jFt1 Þ  s

t¼2 n X

(b)

q X q X

P

1 ;k 2 Þ ap1 Xðk ðWp1 ; Wp2 Þap2 ! 0 2

p1 ¼1 p2 ¼1

P

ðkÞ EðzðkÞ2 t Iðjzt j4dÞÞ ! 0 for all d40 and k ¼ 1; . . . ; p.

t¼2

We begin with (a), whose left side is 2

s

n X t1 X t¼2 s¼1

 s4

q X


2s

q X

p1 ;p2 ¼1 1 ;k 2 Þ ap1 Xðk ðWp1 ; Wp2 Þap2 2

p1 ;p2 ¼1

þs

2

!

ðk2 Þ 1Þ ap1 cðk ts ðWp1 Þcts ðWp2 Þap2

(

n t1 X X


s1
s2

t¼2 1¼s1 as2

q X

! ðk1 Þ ap cts ðWp Þ 1

p¼1

q X

s2

ð
2t  s2 Þ

nt X

q X

s¼1

p1 ;p2 ¼1

t¼2

þs

4

 s4

n1 X nt X

q X

t¼2 s¼1

p1 ;p2 ¼1

q X q X

.

ð27Þ

p¼1

First, we examine the first two terms of (27) which is n1 X

!) 2Þ ap cðk ts2 ðWp Þ

!

ap1 csðk1 Þ ðWp1 Þcsðk2 Þ ðWp2 Þap2 !

ap1 csðk1 Þ ðWp1 Þcsðk2 Þ ðWp2 Þap2

1 ;k 2 Þ ap1 Xðk ðWp1 ; Wp2 Þap2 . 2

ð28Þ

p1 ¼1 p2 ¼1

By Lemma 7.2, the last two terms of (28) converge to zero whereas C.1 implies that the first term has zero mean and, using standard inequalities, variance bounded by ! q n1 X nt nt X X X ðk1 Þ2 ðk2 Þ2 C jap1 ap2 j cs ðWp1 Þ cs ðWp2 Þ ¼ oð1Þ (29) p1 ;p2 ¼1

t¼2

s¼1

s¼1

as we now show. First, for W 2 ½0; 1
, 1=2 jcðkÞ s ðWÞjpn

and by summation by parts, it is also Oðn=sÞ, because by Zygmund (1977)

j
X
 n

ðkÞ z‘ cosðsl‘ Þ
¼ O

‘¼1
s

(30)

(31)

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826

~ whereas for npspn ~ ~ ‘ Þ and if 1pspn,  1, because cosðsl‘ Þ ¼ ð1Þ‘ cosððs  nÞl ðkÞ z ðuÞ is an integrable function. So, we can restrict ourselves to the sum for those ~ But, spn. n~ X

~ ½X n=m


cðkÞ2 s ðWp Þ ¼

s¼1

n~ X

cðkÞ2 s ðWp Þ þ

s¼1

cðkÞ2 s ðWp Þ

~ s¼½n=m
þ1

0

1   X 1 n 1 1 m 2 A @ þ þ ¼O ¼O s n m n s¼½n=m
m n2 ~

ð32Þ

Pq P ðkÞ because qp¼1 cðkÞ s ðWp Þ ¼ p¼1 cns ðWp Þ, and where for the first and second terms on the right of (32) we have used (30) and (31), respectively, and the definition of 2 3 2 ~ x
with cðkÞ s ðWp Þ. So, the left side of (29) is Oðm n þ n m Þ ¼ oð1Þ choosing m ¼ ½n 1=2oxo1. To complete the proof of part (a), we need to show that the last term of (27) converges to zero in probability, for which it suffices to show that its second moment converges to zero. But a typical element of the second moment is s2

n t1 X X

ðk2 Þ2 1 Þ2 cðk ts1 ðWp Þcts2 ðWp Þ

t¼2 s1 as2 ¼1

þ 2s2

n X t1 u1 X X t¼3 u¼2 s1 as2 ¼1

ðk1 Þ ðk2 Þ ðk1 Þ ðk2 Þ cts ðWp Þcus ðWp Þcts ðWp Þcus ðWp Þ. 1 2 1 2

The first term of the last displayed expression is o(1) proceeding as in the proof of (29) and, by the Cauchy–Schwarz inequality, the second term is bounded by ! n X t1 u1 u1 X X X ðk1 Þ2 ðk2 Þ2 cts ðWp Þ cus ðWp Þ C t¼3 u¼2

pC

n X t¼1

s¼1

ctðk1 Þ2 ðWp Þ

!

s¼1 n X t1 X

t1 X

! csðk2 Þ2 ðWp Þ

.

ð33Þ

t¼3 u¼2 s¼tuþ1

Proceeding as with (32), the second bracketed term on the right of (33) is  2 n2 n~  n  X X nn sðn  s  1Þcsðk2 Þ2 ðWp Þp2n scsðk2 Þ2 ðWp Þ ¼ O þ log , m nm2 s¼1 s¼1 using (30) and (31) as in (32). Also, by (30) and (31), the first bracketed term on the right of (33) is Oðm1 þ mn2 Þ. So, the right-hand side of (33) is  2  n log n O þ ¼ oð1Þ, m m3 choosing m ¼ ½n~ x
with x4 23. (Observe that this choice of m is also valid for (32) as long as xo1.) Now Markov’s inequality implies that the last term of (27) ¼ op ð1Þ, which concludes the proof of part (a).

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To complete the proof of the proposition, Pwe need to prove part (b). To that end, it suffices to show the sufficient condition nt¼2 EðzðkÞ4 t Þ ! 0, whose proof is similar to that in Robinson (1995b), so it is omitted. & Proposition 6.3. Assuming C.1, the process Rn ðWÞ as defined in Proposition 6.2, is tight. P½nW ~ 2
Proof. The first term on the right of (24) is tight since jn~ 1 j¼½ ~ 1
þ1 zj jpCjW1  nW P n ð1þdÞ=2 1=2 2 2 2 W2 j for some d40 by the definition of zðlÞ and Eðn t¼2 ð
t  s ÞÞ oC by C.1. So, it suffices to examine the tightness condition for the second term of (24). To that end, write EðkÞ n ðWÞ ¼

n X t¼2


t

t1 X


s cðkÞ ts ðWÞ.

s¼1

ðkÞ First, by definition of zðkÞ j ðWÞ, En ðWÞ is a process which belongs to D½0; 1
, so by Billingsley’s (1968) Theorem 15.6, it suffices to show the moment condition 1þd ðkÞ 4 EjEðkÞ n ðW2 Þ  En ðW1 Þj pCðW2  W1 Þ

for some d40. But, the left side of the last displayed inequality is 2 0 13 tj 1 n 4 Y X X A5
ðtj Þ@
ðsj Þe cðkÞ E4 tj sj ðW; W2 Þ 2¼t1 pt2 pt3 pt4 j¼1

pC

4 Y

0 @

j¼1

X 1psj ptj pn

sj ¼1

11=2 A e cðkÞ2 tj sj ðW; W2 Þ

!2

X

¼C

1psptpn

e cðkÞ2 ts ðW; W2 Þ

,

proceeding as in the proof of Lemma 5.4 of Giraitis et al. (2001), where ðkÞ ðkÞ e cðkÞ t ðW1 ; W2 Þ ¼ ct ðW1 Þ  ct ðW2 Þ. But by an obvious extension of Lemma 7.2, the right side of the last displayed equation is bounded by 1 C 4p

Z

pW2 pW



1 zðkÞ2 ðuÞ du  2p

Z

pW2

zðkÞ ðuÞ du

2 !2

pCðW2  WÞ1þd .

pW

R pW R pW This concludes the proof since 0 2 zðkÞ2 ðuÞ du and 0 2 zðkÞ ðuÞ du are Lipschitz’s continuous with parameter ð1 þ dÞ=2. & 6.1. Proof of Theorem 3.1 ~  W. By definition Write JðW; jÞ ¼ Iðjp½nW
Þ b  WF n ð1; cÞ b ¼ F n ðW; cÞ

n~ Ij 2p X bn ðWÞ, JðW; jÞ þP n~ j¼1 jAj ðc0 Þj2

(34)

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where

! n~ 2 X I jA ðc Þj 2p j j 0 bn ðWÞ ¼ JðW; jÞ 1 . P b 2 n~ j¼1 jAj ðc0 Þj2 jAj ðcÞj

Next, under C.1–C.4, see for instance Hannan (1973), Hosoya (1997) or Velasco and Robinson (2000), we have that b ¼ c þ ðA=pÞ1 ð2pÞbn þ oðn1=2 Þ a:s:; c 0 ~ uniformly in W 2 ½0; 1
, where bn ¼ F 1 n ð1; c0 Þn other hand, Proposition 6.1 implies that bn ¼

(35) 1 Pn~

j¼1

fj ðc0 ÞjAj ðc0 Þj2 I j . On the

n~ X 1 f ðc ÞI
;j þ op ðn1=2 Þ ¼ Op ðn1=2 Þ F n ð1; c0 Þn~ j¼1 j 0

by Proposition 6.2 with zðlÞ ¼ fðl; c0 Þ there. So, using standard linearization of b 2 around jAj ðc Þj2 , and recalling that fðl; c Þ ¼ q log jAj ðc Þj2 =qc, we obtain jAj ðcÞj 0 0 0 bn ðWÞ is that P bn ðWÞ ¼  2p P n~

n~ X

JðW; jÞf0j ðc0 Þ

j¼1

Ij b  c Þð1 þ Op ðn1=2 ÞÞ. ðc 0 jAj ðc0 Þj2

(36)

Because by C.2, fðl; c0 Þ satisfies the same conditions of zðlÞ, we have that ~ ½nW
n~ Ij 2p X s2 X JðW; jÞfj ðc0 Þ ¼ f ðc Þ þ op ðn1=2 Þ 2 n~ j¼1 j 0 n~ j¼1 jAj ðc0 Þj

¼

s2 GðWÞ þ op ðn1=2 Þ p

Pn~ by Propositions 6.1 and 6.2 R p and that by Lemma 7.1 j j¼1 fj ðc0 Þj ¼ Oðlog nÞ, recallPthat (2) implies that 0 fðl; c0 Þ dl ¼ 0, and because again by Lemma 7.1, ~ nW
jn~ 1 ½j¼1 fj ðc0 Þ  p1 GðWÞj ¼ Oðn1 log nÞ with zðlÞ ¼ fðl; cÞ there. Moreover, by Propositions 6.1 and 6.2 and using the linearization (35), we have that b ¼ F n ð1; c Þ þ Op ðn1=2 Þ ¼ s2 þ Op ðn1=2 Þ. F n ð1; cÞ 0 So, gathering the previous identity, (34)–(36), we conclude that ! n~ n~ Ij 2p X 2p X 0 1 b JðW; jÞ ðWÞA fj ðc0 Þ S n ðW; cÞ ¼  G jAj ðc0 Þj2 s2 n~ 1=2 j¼1 s2 n~ 1=2 j¼1

Ij þ op ð1Þ. jAj ðc0 Þj2

But by Proposition 6.1, we have that the right side of the last displayed equality is n~ I
;j 2p X ðJðW; jÞ  G0 ðWÞA1 fj ðc0 ÞÞ 2 þ op ð1Þ. s ~n1=2 j¼1

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From here the conclusion is straightforward by Propositions 6.2 and 6.3, taking zðlÞ ¼ JðW; lÞ  G0 ðWÞA1 fðl; c0 Þ there and observing that the function inside the brackets of the last displayed expression integrates to zero for all W, so that the component due to X1 is zero. & 6.2. Proof of Theorem 3.3 b is Using the expansion given in (35), under Ha , Sn ðW; cÞ ! n~ n~ Ij 1X 1 X JðW; jÞ  1 þ 3=2 JðW; jÞgj fj n~ j¼1 2n~ j¼1

! n~ n~ X Ij I 1X 1 j bc Þþ ðc  JðW; jÞfj ðc0 Þ JðW; jÞgj 1 , 0 f j ðc0 Þ fj n~ j¼1 2n~ 3=2 j¼1

ð37Þ

which is !

n~ 1 X JðW; jÞgj 2n~ 3=2 j¼1   n~ 1X 1 b  c Þ þ op ðn1=2 Þ,  JðW; jÞfj ðc0 Þ 1 þ 1=2 gj ðc 0 n~ j¼1 n

n~ Ij 1X JðW; jÞ 1 fj n~ j¼1

þ

ð38Þ

as we know show. Consider, for example, last term on the right of (37), which is !   n~ n~ I j 2p 1 X 1 X 2p JðW; jÞgj  I
;j þ 3=2 JðW; jÞgj 2 I
;j  1 ¼ op ðn1=2 Þ f j s2 s 2n~ 3=2 j¼1 2n~ j¼1 because by Propositions 6.2 and 6.3, the second term on the left of the last displayed equality is Op ðn1 Þ, whereas by Proposition 6.1, the first term on the left of the last displayed equation is op ðn1 Þ with zðlÞ ¼ JðW; lÞgðlÞ there. So (37) is indeed (38). Now, proceeding as in the proof of Theorem 3.1, ! Z n~ p X I 1 j 1=2 b ¼ JðW; jÞ  1 þ ðIðlppWÞ  WÞgðlÞ dl n~ Sn ðW; cÞ fj n~ 1=2 j¼1 0 b  c Þ þ op ð1Þ.  p1 G0 ðWÞn~ 1=2 ðc 0 R d b  c Þ ! NðA1 p fðlÞgðlÞ dl; 2pA1 Þ. From here, the But, under Ha , n~ 1=2 ðc 0 0 conclusion of the theorem is standard. & As was done with the proof of Theorem 3.1, we first show three propositions from which the proof of Theorem 4.1 will easily follow. Also, the function zðl; cÞ is continuously differentiable in c and uniformly in a neighbourhood of c ¼ c0 , it satisfies the same conditions as the function zðlÞ in Propositions 6.1–6.3.

ARTICLE IN PRESS J. Hidalgo, J.-P. Kreiss / Journal of Econometrics 133 (2006) 807–839

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Proposition 6.4. Under the same conditions of Proposition 6.1, for all k1 ; k2 ¼ 1; . . . ; p, P

E ðRn ðW1 ÞRn ðW2 ÞÞ !

2 þ kx;4 Xðk1 ;k2 Þ ðW1 ; W2 Þ þ X2ðk1 ;k2 Þ ðW1 ; W2 Þ, ð2 þ k4 Þs4 1

(39)

where s4x kx;4 ¼ cumðxt ; xt ; xt ; xt Þ and Rn ðWÞ ¼

~ ½nW
1 X

n~ 1=2

b   1Þ. zj ðcÞðI x;j

j¼1

Proof. By definition, Rn ðWÞ ¼

1

~ ½nW
X

21=2 n~

j¼1

! b zj ðcÞ

n

! n n t1 X X 1 X 2  b ðx  1Þ þ x xs cts ðW; cÞ, t t 1=2 t¼1

t¼2

s¼1

where 1 3=2 b ~ cðkÞ s ðW; cÞ ¼ 2 n

~ ½nW
X

b zðkÞ j ðcÞ cosðslj Þ.

(40)

j¼1

Because for all tos, E ðxt xs ðx2 r  1ÞÞ, (39) is shown if ! ! !2 ~
~
½nW ½nW n 1 1 X1 ðk1 Þ b 1 X2 ðk2 Þ b 1 X  2 z ðcÞ z ðcÞ E ðx  1Þ 2 n~ j¼1 j n~ j¼1 j n1=2 t¼1 t P

!

2 þ kx;4 ðk1 ;k2 Þ X ðW1 ; W2 Þ ð2 þ k4 Þs4 1 n X



E

xt

t¼2

t1 X

ð41Þ !

ðk1 Þ b xs cts ðW1 ; cÞ

s¼1

n X t¼2

xt

t1 X

!! 2Þ b xs cðk ts ðW2 ; cÞ

P

1 ;k 2 Þ ! Xðk ðW1 ; W2 Þ. 2

s¼1

(42) Since, conditional on E

n

x, xt  !2

n 1 X ðx2 t  1Þ 1=2 t¼1

is an iidð0; 1Þ sequence of random variables, ¼

n p 1 X ðx4t  b s4x Þ ! 2 þ kx;4 4 b sx n t¼1

(43)

because by a well-known argument, see Stout’s (1974, Theorem 3.5.8), xt is ergodic P by C.1, which implies that n1 nt¼1 x4t , say, converges in probability to ð3 þ kx;4 Þs4x . On the other hand, for all k ¼ 1; . . . ; p ~
~
½nW ½nW 1 X1 ðkÞ b 1 X1 ðkÞ zj ðcÞ ¼ z ðc0 Þ þ op ð1Þ, n~ j¼1 n~ j¼1 j

from here the convergence in (41) follows by Lemma 7.1 and arguments in Lemma 7.4.

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Thus, we are left to show (42). To this end, conditional on x, 

t1 X

 zðkÞ t ðWÞ ¼ xt

b xs cðkÞ ts ðW; cÞ,

(44)

s¼1

b in z ðWÞ forms a martingale difference triangular array sequence, where reference to c t  has been suppressed for notational convenience. Let Ft be the smallest sigma algebra generated by fxs ; sptg conditional on x. Since E ðxt Þ ¼ 0 and E ðx2 t Þ ¼ 1, 
!
n n X X
1Þ 2Þ E zðk ðW1 Þ zðk ðW2 Þ
Ft1 t t
t¼2 t¼2 ¼

n X t1 X

ðk1 Þ b ðk2 Þ b x2 s cts ðW1 ; cÞcts ðW2 ; cÞ

t¼2 s¼1

þ

n t1 X X t¼2 1¼s1 as2

ðk1 Þ ðk2 Þ b ts b xs1 xs2 cts ðW1 ; cÞc ðW2 ; cÞ. 1 2

ð45Þ

1 ;k 2 Þ The first term on the right of (45) converges in probability to Xðk ðW1 ; W2 Þ as we 2 now show. First, that term is

n1 nt n1 X nt X X X ðk1 Þ ðk2 Þ b b b ðk2 Þ ðW2 ; cÞ. b ðx2  1Þ c ðW ; cÞc ðW ; cÞ þ csðk1 Þ ðW1 ; cÞc 1 2 t s s s t¼1

s¼1

(46)

t¼1 s¼1

1 ;k 2 Þ By Lemma 7.3, the second term converges in probability to Xðk ðW1 ; W2 Þ, whereas 2 the first term has, conditional on x, mean zero and variance which is



ð2 þ kx;4 Þ

n1 X nt X t¼1

!2 b ðk2 Þ ðW2 ; cÞ b csðk1 Þ ðW1 ; cÞc s

ð1 þ op ð1ÞÞ,

s¼1

using the arguments in (43). b for all W 2 ½0; 1
and that Next, by the continuous differentiability of zj ðcÞ b  c Þ ¼ op ð1Þ, ðc 0 1=2 b jcðkÞ ð1 þ op ð1ÞÞ, s ðW; cÞj ¼ n

(47)

b  c Þ ¼ op ð1Þ and and by summation by parts, it is also op ðn=sÞ, because by ðc 0 Zygmund (1977),

j

X  n

b zðkÞ , (48)
j ðcÞ cosðsl‘ Þ
¼ Op

‘¼1 s ~ whereas for npspn ~ ~ ‘ Þ and if 1pspn,  1 because cosðsl‘ Þ ¼ ð1Þ‘ cosððs  nÞl zðuÞ ¼ zðu; c0 Þ and its derivative with respect to c are both integrable functions. So,

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~ But, we can restrict ourselves to the sum on spn. n~ X

1Þ b ðk2 Þ b cðk s ðW1 ; cÞcs ðW2 ; cÞ ¼

~ ½X n=m


b ðk2 Þ ðW2 ; cÞ b csðk1 Þ ðW1 ; cÞc s

s¼1

s¼1

n~ X

þ

1Þ b ðk2 Þ b cðk s ðW1 ; cÞcs ðW2 ; cÞ

~ s¼½n=m
þ1

0

X

1 n 1 þ ¼ Op @ nm n

1 s2 A ¼ Op



1 m þ m n2

~ s¼½n=m
þ1

 ð49Þ

ðkÞ b b because cðkÞ s ðW; cÞ ¼ cns ðW; cÞ and by (47) and (48). Thus, 0 !2 !2 1 n1 X nt n1 X n~ X X ðk1 Þ ðk Þ ðk Þ ðk Þ b 2 ðW2 ; cÞ b b 2 ðW2 ; cÞ b A c ðW1 ; cÞc ¼ O@ c 1 ðW1 ; cÞc s

t¼1

s

s

s¼1

t¼1

 ¼ Op

s

s¼1

n m2 þ m 2 n3

 ¼ op ð1Þ,

choosing m ¼ ½n~ x
with 12 oxo1. That completes the proof that the first term on the 1 ;k 2 Þ right of (45) converges in probability to Xðk ðW1 ; W2 Þ. 2 To complete the proof of the proposition, we are left to prove that the second term on the right of (45) is op ð1Þ. But, conditional on x, its first moment is 0, whereas its  second moment is n X

minðt1;u1Þ X s1 ;s2 ¼1

t;u¼2

¼

ðk1 Þ ðk2 Þ b ðk1 Þ ðW; cÞc b ts b ðk2 Þ ðW; cÞ b cts ðW; cÞc ðW; cÞc us1 us2 1 2

n X

t1 X

ðk1 Þ2 ðk2 Þ2 b ts b cts ðW; cÞc ðW; cÞ 1 2

t¼2 s1 as2 ¼1

þ

n X t1 u1 X X t¼3 u¼2 s1 as2 ¼1

1Þ b ðk1 Þ b ðk2 Þ b ðk2 Þ b cðk ts1 ðW; cÞcus1 ðW; cÞcts2 ðW; cÞcus2 ðW; cÞ.

Now, proceeding as we did in Proposition 6.2 and using argument employed for (46), we have that the right side of the last displayed equality is op ð1Þ. That concludes the proof of the proposition. & Proposition 6.5. Under the same conditions of Proposition 6.4, the finite dimensional distributions of Rn ðWÞ converge in bootstrap law to those of a centred Gaussian process. Proof. Fix W ¼ ðW1 ; . . . ; Wp Þ0 and constants a1 ; . . . ; aq . By the Crame´r–Wold device  and definition of the Rn ðWÞ, it suffices to examine the limit distribution of ! ~
½nW q q n n X X X 1 Xp 1 X 2 b ap 1=2 zj ðcÞ ðx  1Þ þ a zt ðWp Þ, (50) p t 1=2 n ~ 2 n j¼1 t¼1 t¼2 p¼1 p¼1

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where zt ðWÞ is given in (44). We already know, see arguments in Proposition 6.4,  that the first and second terms of (50) are uncorrelated. On the other hand,  since xt are iid, it is easy to show that the first term converges in (bootstrap) distribution to 0 1 ! q X 2 þ k x;4 ðk ;k Þ A. ap1 X 1 2 ðWp1 ; Wp2 Þap2 N@0; ð2 þ k4 Þs4 1 p ;p ¼1 1

k1 ;k2 þ1;...;p

2

So, we are left to examine the second term of (50). To simplify the proof, let us assume that q ¼ 1. Then, proceeding as in the proof of Proposition 6.4, cf. (45), n~ X

P

1Þ 2Þ E ðzðk ðW1 Þzðk ðW2 ÞjFt1 Þ ! X2ðk1 ;k2 Þ ðWp1 ; Wp2 Þ, t t

j¼1

and so, it remains to verify the Lindeberg’s condition, that is 8d40, n X

P

E ½zðkÞ2 ðWÞIðjzðkÞ t t ðWÞj4dÞ
! 0, 

t¼2



P P or the sufficient condition nt¼2 E ½zðkÞ4 ðWÞ
! 0, whose proof is essentially that of t  Robinson (1995b) proceeding as in Proposition 6.2, and thus it is omitted. & Proposition 6.6. Under the same conditions of Proposition 6.4, conditional on x, Rn ðWÞ  is tight. Proof. First, observe that 1

~ ½nW
X

!

¼

b zðkÞ j ðcÞ

n 1 X ðx2 t  1Þ 1=2

!

b þ EðkÞ (51) n ðW; cÞ, n 21=2 n~ j¼1 t¼1 Pn Pt1  ðkÞ  b b where EðkÞ n ðW; cÞ ¼ t¼2 xt s¼1 xs ct ðW2 ; cÞ. The first termP on the right of (51) is tight from (43) and the fact that by the triangle ~ 2
½nW ðkÞ b inequality, jn~ 1 j¼½ ~ 1
þ1 zj ðcÞj is bounded by nW



~ 2
~ 2
½nW ½nW

1 X

1 X



ðkÞ ðkÞ b ðkÞ zj ðc0 Þ
þ
ðzj ðcÞ  zj ðc0 ÞÞ




n~ j¼½nW
~ n ~ 1
þ1 ~ 1
þ1 j¼½nW
~ 2

½nW X
q ðkÞ
b  c j1 e
,
z ðcÞ pCjW1  W2 jð1þdÞ=2 þ Cjc ð52Þ 0
qc j
n~ j¼½nW ~
þ1 RðkÞ n ðWÞ

1

e is an intermediate point between c and c. b But, jc b  c j ¼ Op ðn1=2 Þ, so we and c 0 0 conclude that the second moment of the first term on the right of (51) is bounded by Hn ðW1 ; W2 ÞjW1  W2 j1þd , where Hn ð; Þ is bounded in probability, because ðq=qcÞzðkÞ j ðcÞ satisfies the same conditions of z in Lemma 7.1 in a neighbourhood as c0 .

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ðkÞ b b Next, the second term on the right of (51). Denoting e cðkÞ t ðcÞ ¼ ct ðW2 ; cÞ and proceeding as with the proof of Proposition 6.3, !2 X ðkÞ2  ðkÞ ðkÞ 4 b  E ðW1 ; cÞj b pC b e , E jE ðW2 ; cÞ c ðcÞ

b cðkÞ t ðW1 ; cÞ,

n

n

ts

1psptpn

which, by the Cauchy–Schwarz inequality, is bounded by !2 !2 X X ðkÞ2 ðkÞ b ðkÞ 2 e ðe cts ðcÞ  e cts ðc0 ÞÞ . cts ðc0 Þ þ C C 1psptpn

1psptpn

From here, proceeding as in Proposition 6.3, but using (52) and Lemma 7.3 instead of Lemma 7.2 when appropriate, it is straightforward to see that the last displayed expression is also bounded by Hn ðW1 ; W2 ÞjW1  W2 j1þd . Hence we have shown that ðkÞ g ðkÞ ðkÞ g 1þd E ðjRðkÞ n ðW2 Þ  Rn ðWÞj jRn ðWÞ  Rn ðW1 Þj ÞpHn ðW1 ; W2 ÞjW1  W2 j

for all 0pWoW1 oW2 p1, because jW2  WjjW  W1 jpjW2  W1 j2 , which implies that  RðkÞ n ðWÞ is tight for all k ¼ 1; . . . ; p, so it is Rn ðWÞ. & 6.3. Proof of Theorem 4.1 The arguments are similar to those of Theorem 3.1, using Propositions 6.4–6.6 instead of Propositions 6.1–6.3. First, by Lemma 7.5, and continuity of fðl; cÞ in c, !1 n~ n~ X  1 2p X 0 b ¼c b b ðcÞÞ b b   1Þð1 þ op ð1ÞÞ c ðfj ðcÞf f0 ðcÞðI j x;j n j¼1 n~ j¼1 j n~ X b   1Þð1 þ op ð1ÞÞ, b  ðA=pÞ1 2p f0 ðcÞðI ¼c x;j n~ j¼1 j

ð53Þ

b the definition of A where for the second equality, we have used the consistency of c, and then Lemmas 7.4 and 7.1. Then, proceeding as in the proof of Theorem 3.1, we have that ~ 1
½nW n~ n~ X X 1X 0 b 1 2p  b ¼ 2p b   1Þ. Sn ðW; cÞ JðW; jÞI  z ð cÞðA=pÞ fj ðcÞðI j x;j x;j n~ j¼1 n~ 1=2 j¼1 n~ 1=2 j¼1

But proceeding as in the proof of Proposition 6.4, we have that ~
½nW 1 X1 0 b 1 z ðcÞ ¼ GðWÞ þ op ð1Þ, p n~ j¼1 j

and hence b ¼ Sn ðW; cÞ

n~ 2p X  b ðJðW; jÞ  G0 ðWÞA1 fj ðcÞÞðI x;j  1Þ. n~ 1=2 j¼1

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835

From here the proof follows as in Theorem 3.1 but using Propositions 6.5 and 6.6 b instead of Propositions 6.2 and 6.3 when appropriate, and observing again that fj ðcÞ can be replaced by fj ðc0 Þ. &

7. Technical lemmas Lemma 7.1. Let z be as in Proposition 6.1. As n ! 1, for all k ¼ 1; . . . ; p

Z ~ ½nW


1 X 1 Wp ðkÞ log‘ n

ðkÞ . sup
zj  z ðxÞ dx
pC
~ j¼1 p 0 n W2½0;1

n

(54)

Proof. The left side of (54) is bounded by


Z Wp
Z ~ ½nW


1 X
1
1 Wp ðkÞ

ðkÞ ðkÞ

sup
z ðxÞ dx
þ sup
zj  z ðxÞ dx
.

~ j¼1 p 0 W2½0;n~ 1 Þ p 0 W2½n~ 1 ;1
n

(55)

The first term of (55) is bounded by 1 p

Z

p=n~

jzðkÞ ðxÞj dxpC 0

Z

p=n~

j log xj‘ dxpC

0

log‘ n . n

Next, by the triangle inequality, the second term of (55) is bounded by

Z Z pðjþ1Þ=n~ ~ ½nW
1
1
1 p=n~ ðkÞ 1 X
ðkÞ
ðkÞ sup
z ðpWÞ  z ðxÞ dx
þ sup jzðkÞ j  z ðxÞj dx.

~ n p p 1 1 ~ 0 pj=n W2½n~ ;1
W2½n~ ;1
j¼1

(56) The first term of (56) is obviously bounded by Cn1 ðlog nÞ‘ since jzðkÞ ðxÞjpCj log xj‘ . Next, by the mean value theorem, the second term of (56) is bounded by C

Z ~ n1 X j¼1

pðjþ1Þ=n~

pj=n~



Z ~ n1 X
1

jp 1 pðjþ1Þ=n~
j log xj‘1 dxpC j log xj‘1 dx  x
lj
n~ j ~ pj= n j¼1 p

C log‘ n : n

&

Lemma 7.2. Let cs ðWÞ be as in (25) and X2 ðW1 ; W2 Þ as defined in Proposition 6.2. Then, n X nt X t¼2 s¼1

ðk1 ;k2 Þ ðk2 Þ 1Þ cðk ðW1 ; W2 Þð1 þ oð1ÞÞ. s ðW1 Þcs ðW2 Þ ¼ X2

(57)

ARTICLE IN PRESS J. Hidalgo, J.-P. Kreiss / Journal of Econometrics 133 (2006) 807–839

836

Proof. The left side of (57) is ~
~ 2
½nW ½nW n1 X nt X 1 X1 ðk1 Þ X z zkðk2 Þ cosðslj Þ cosðslk Þ j 3 4n~ j¼1 t¼1 s¼1 k¼1

1 4n~ 3

¼

þ

~ 1X ~ 2
½nW
^½nW

1 8n~ 3

zjðk1 Þ zjðk2 Þ

cos2 ðslj Þ

t¼1 s¼1

j¼1 ~ 1
½nW X

n X nt X

~ 2
½nW X

1Þ zðk j

j¼1

zkðk2 Þ

n X nt X

cosðsðlj þ lk ÞÞ þ cosðsðlj  lk ÞÞ.

ð58Þ

t¼1 s¼1

jak¼1

Because, see for instance Robinson (1995b), n X nt X

cos2 ðslj Þ ¼

t¼1 s¼1

ðn  1Þ2 4

(59)

and for jak n X nt X

cosðsðlj þ lk ÞÞ þ cosðsðlj  lk ÞÞ ¼ n,

(60)

t¼1 s¼1

and recalling that n~ ¼ n=2, the right side of (58) is ðn  1Þ2 16n~ 2 ¼

1 n~

~ 2
½n~ WX 1
^½nW j¼1

1 ;k 2 Þ Xðk ðW1 ; W2 Þð1 2

by Lemma 7.1.

! zjðk1 Þ zjðk2 Þ



~
~ 2
½nW ½nW 1 X1 ðk1 Þ X z zkðk2 Þ j 4n~ 2 j¼1 jak¼1

þ oð1ÞÞ

&

b be as defined in (40). Then, for all W1 , W2 2 ½0; 1
, Lemma 7.3. Let cs ðW; cÞ n1 X nt X

ðk1 ;k2 Þ 1Þ b ðk2 Þ b cðk ðW1 ; W2 Þð1 þ op ð1ÞÞ. s ðW1 ; cÞcs ðW2 ; cÞ ¼ X2

t¼1 s¼1

Proof. As in Lemma 7.2 and using (59) and (60), the first term on the left side of the last displayed expression is ! ~
^½nW ~
~
~
½nW ½nW ½nW ðn  1Þ2 1 1X 2 ðk1 Þ b ðk2 Þ b 1 X1 ðk1 Þ b X2 ðk2 Þ b z ð cÞz ð cÞ  z ð cÞ zk ðcÞ. (61) j j n~ j¼1 16n~ 2 4n~ 2 j¼1 j jak¼1 Because zj ðcÞ is continuously differentiable in c and C.2 implies that it is integrable, by the mean value theorem, continuity of the derivative and that C.1–C.4 implies

ARTICLE IN PRESS J. Hidalgo, J.-P. Kreiss / Journal of Econometrics 133 (2006) 807–839

837

b  c Þ ¼ Op ðn1=2 Þ as was shown in Theorem 3.1, we have that that ðc 0 ! ~
^½nW ~
½nW ðn  1Þ2 1 1X 2 ðk1 Þ ðk2 Þ ð61Þ  zj ðc0 Þzj ðc0 Þ n~ j¼1 16n~ 2 

~
~ 2
½nW ½nW X 1 X1 ðk1 Þ z ðc Þ zkðk2 Þ ðc0 Þ 0 j 4n~ 2 j¼1 jak¼1

is op ð1Þ by standard arguments. However Lemma 7.2 implies that the last two terms on the left of the last displayed expression converge to Xðk1 ;k2 Þ ðW1 ; W2 Þ, which concludes the proof of the lemma. & Lemma 7.4. Assuming C.1–C.4, n~ 1

n~ X

P

b ! p1 GðWÞ. JðW; jÞfj ðcÞ

j¼1

Proof. The left side is n~ 1

n~ X

b  f ðc ÞÞ þ n1 JðW; jÞðfj ðcÞ j 0

j¼1

n~ X

JðW; jÞfj ðc0 Þ.

j¼1

R Wp The second term converges to p1 0 fðl; cÞ dl by Lemma 7.1. So, the proof of the lemma is completed if the first term converges to zero in probability. But this is the b  c ¼ Op ðn1=2 Þ and the convergence of the case by standard arguments because c 0 sum to the integral is uniform in c. & b be such that it converges almost surely to c 2 C. Then Lemma 7.5. Let c 1 b ¼ op ð1Þ. c  c Proof. Since conditional on the sample x, xt is, by construction, an iid zero mean  sequence of random variables with variance 1, then it is an ergodic sequence in a quadratic mean sense. Then, proceeding as with the proof of Hannan’s (1973) Lemma 1, uniformly in c 2 C, Z p n~ b I j 1 X f ðl; cÞ P dl ! 0.  n j¼1n~ f j ðcÞ f ðl; cÞ p Now, proceeding as in the proof of Theorem 1 of Hannan (1973), we have that b ¼ op ð1Þ because c  c b ¼ arg min c c2C

Z

p

p

b f ðl; cÞ dl. f ðl; cÞ

This concludes the proof.

&

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8. Conclusion The paper has described and examined a test for linear covariance stationary processes where the alternative is left unspecified. Because the test is not pivotal and computing the critical values is an extremely difficult task, if at all possible, we have bootstrapped the test and have shown its validity. The bootstrap is obtained in the frequency domain, in contrast to alternative ones in the time domain. The former algorithm seems preferable than the latter, specially when the covariance function of the data cannot explicitly be written in terms of the underlying parameters of the model.

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