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Distribution Free Goodness-of-Fit Tests for Linear Processes

Author(s): Miguel A. Delgado, Javier Hidalgo and Carlos VelascoSource: The Annals of Statistics, Vol. 33, No. 6 (Dec., 2005), pp. 2568-2609Published by: Institute of Mathematical Statistics

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TheAnnals of Statistics

2005,Vol. 33,No. 6, 2568-2609DOI: 10.1214/009053605000000606? Institute fMathematicalStatistics,2005

DISTRIBUTION FREE GOODNESS-OF-FIT TESTS FOR

LINEAR PROCESSES1

By Miguel A. Delgado, Javier Hidalgo and Carlos Velasco

Universidad Carlos III, London School of Economics and Universidad Carlos III

This article proposes a class of goodness-of-fit tests for the autocorrela

tion function of a time series process, including those exhibiting long-range

dependence. Test statistics for composite hypotheses are functionals of a

(approximated) martingale transformation of the BartlettTp-process

with es

timated parameters, which converges in distribution to the standard Brownian

motion under the null hypothesis. We discuss tests of different natures such

as omnibus, directional and Portmanteau-type tests. A Monte Carlo study il

lustrates the performance of the different tests in practice.

1. Introduction and statement of the problem. Let / be the spectral densityfunction of a second-order stationary time series process {X(t)}tez with mean ?i

and covariance function

Cov(X(j),X(0))= r f(X)cos(Xj)dX, j= 0,?l,?2,....

J-n

We shall assume that {X(t)}tez admits theWold representationoo oo

(1) X(t) = ? + J2 fl0X*-

j) witha(0)= 1andJ^ fl20')< ???

7=0 7=0

for some sequence {s(t)}tez satisfying E(s(t))=

0, and E(s(0)e(t))= a2ift = 0

and = 0 for all t^ 0. Under (1), the spectral density function of {X(t)}tez can be

factorized as

a2

f(X)

=?h(X), Ae[0,;r],

withA(?):=|E7?lo?OV7?l2Let

(2)M=lh0: r\ogh0(X)dk=

0,0e?\,

Received May 2002; revised December 2004.

1Supported in part by the Spanish Direcci?n General de Ense?anza Superior (DGES) reference

numbers BEC2001-1270 and SEJ2004-04583/ECON andby

the Economic and Social Research

Council (ESRC) reference number R000239936.

AMS 2000 subject classifications. Primary 62G10, 62M10; secondary 62F17, 62M15.

Key words and phrases. Nonparametric model checking, spectral distribution, linear processes,

martingale decomposition, local alternatives, omnibus, smooth and directional tests, long-range al

ternatives.

2568

This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM

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GOODNESS-OF-FIT FOR LINEAR PROCESSES 2569

where 0 c F is a compact parameter space. Much of the existing time se

ries literature is concerned with parametric estimation and testing, assuming

that h belongs to M, that is, h?

h?0 for some Oo e &, because the parame

ter #o and the functional form of ho summarize the autocorrelation structure of

{X(t)}tez- Notice that h eM in (2) guarantees that a(0) = 1 in (1) and a2 =

min#G0 2/?r f(X)/heCk)dX. For our purposes, a2 can be considered a nuisance

parameter, as well as the mean \x.

Classical parameterizations that accommodate alternative models are the

ARMA, ARFIMA, fractional noise and Bloomfield [4] exponential models

(see [35] for definitions). For instance, in an ARFIMA specification, M consists

of all functions indexed by a parameter vector 0 ?(d,r?',8f)\ where 0 e ? C

(-1/2, 1/2) x RPi x RP\ of the form

(3) he(k) =1

l-e iX\2dZ,(elX)

?G[0,7T],\$>8(eiX)\

such thatEr?

and 0? are the moving average and autoregressive polynomials of

orders p\ and /?2, respectively, with no common zeros, all lying outside the unit

circle.

Before statistical inference on the true value Oo is made, one needs to test the

hypothesis Ho :h e M, which can be equivalently stated as

(4) Ho : ^? = - for all X e [0,n] and some 00 e 0,Go0(tt) n

where

rk f(X) -

Ge(X):=2 ^?^dX, ag[0,tt]../o h?(X)

Under Ho, Gq0is the spectral distribution function of the innovation process

{?(t)}tGz and Go0(n)= a2.

Given a record {X(t)}J=l and a consistent estimator 0t ofOo under Ho, a naturalestimator of G#0 is defined as

Gqtj(X), where

(5)?":=T

g ??w'Ae[?'"1

Here T =[T/2], [z] being the integer part of z, and for a generic time series

process {V(t)}te%,

T 2

t=\

Iv(Xj):=2nT

t=i

denotes the periodogram of{V(t)}J=l

evaluated at the Fourier frequency Xj=

2nj/ T for positive integers j.








2570 M. A. DELGADO, J.HIDALGO AND C. VELASCO

The formulation of Ho in (4) suggests use of Bartlett's7),-process

as a basis for

testing Hq. TheTp -process is defined as

?0,7 (A.)._

fi/2?Gej&) _

k_iGejin) il Xe[0,7z].

Notice that oiqj is scale invariant and that, for j ^ Omod(r), Iy{Xj)is mean

invariant, so omission of j= 0 in the definition of Go j entails mean correction.

That is, (xqj is independent of both ?i and a2.

Under short-range dependence and Ho, we have that

max E Ixfrj)

ho0(Xj)

hfrj)=

o(iy,

see [7], Theorem 10.3.1, page 346. So, it is expected that ciqqJ will be asymptoti

cally equivalent to Bartlett's[/^-process

for {s(t)}tez,

a^(X):=T1/2 G?T(X)

X

G?T(7T)71

Xe[0,n],

with

2n[TX/iz]

7= 1

Xe[0,7T].

In fact, under suitable regularity conditions, we shall show below that the afore

mentioned equivalence also holds under long-range dependence. Observe that the

Up -process a,j and theTp -process a$0iT are identical when {X(t)}tez is a white

noise process.

TheUp -process a? is useful for testing simple hypotheses when the innovations

{?(Ol/lican be

easily computed,as is the case when

{X(t)}tezis an AR model.

However, there are many other models of interest whose innovations{?(t)}J=l

cannot be directly computed, for example, Bloomfield's exponential model, or dif

ficult to obtain, as in models exhibiting long-range dependence, such as ARFIMA

models. In those cases, it appears computationally much simpler to use oto0j for

testing simple hypotheses.

The empirical processes a? and otoj, with fixed 9, are random elements

in D[0, 7t], the space of right continuous functions on [0, n] with left-hand

side limits, the c?dl?g space. The functional space D[0, n] is endowed with the

Skorohod metric (see, e.g., [3]) and convergence in distribution in the correspond

ing topology will be denoted by "=^".

Under suitable regularity conditions on {?(t)}tez, it is well known that

(6) ??T=*Bl









whereB^

is the standardized tied down Brownian motion at n. In terms of the

standard Brownian motion B on [0, 1], B\can be represented as

B^(X)=b(^-^B(\),

??[0,7T].

Grenander and Rosenblatt [18] proved (6) assuming that {e(t)}tez is a se

quence of independent and identically distributed (i.i.d.) random variables with

eight bounded moments. The i.i.d. condition was relaxed by Dahlhaus [10], who

assumed that [s(t)}t?z behaves as amartingale difference, but still assuming eight

bounded moments. Recently Kliippelberg and Mikosch [27] proved (6) under i.i.d.

{?(t)}teZ, but assuming only four bounded moments. The i.i.d. requirement is re

laxed by the following assumption:

Al. The innovation process {s(t)}tez satisfies E(s(t)r\!Ft-i)=

?ir with /xr con

stant (??i= 0 and ?jL2 cr2) for r =

1,..., 4 and all t =0, ?1,..., where

Ft is the sigma algebra generated by {s(s), s <t}.

Assumption Al appears to be a minimal requirement to establish a functional

central limit theorem for oij, due to the quadratic nature of the periodogram.

To establish the asymptotic equivalence between a$0j and otj,we introduce

the

followingsmoothness

assumptions

on h :

A2. (a) h is a positive and continuously differentiable function on (0, n}\

(b) |dlogA(X)/3A.|=

0(X~] ) as X -> 0+.

This condition is very general and allows for a possible singularity of h at X= 0.

It holds for models exhibiting long-range dependence, like ARFIMA(/?2, d, p\)

models with d / 0, as can easily be checked using (3) and that |1?

elk\?

|2sin(?/2)|.

THEOREM 1. Assuming Al and A2, under Ho, (6) holds and

sup \aeo,T(X)-a^(X)\=op(\).A. [0,7T]

We can relax the location of the possible singularity in h at any other frequencyX ^ 0, as in [23] or, more recently, [14], or even allow for more than one singu

larity. However, for notational simplicity we have taken the singularity, if any, at

X = 0. If the location of the singularity is at ?0 ^ 0, then A2 is modified to the

following:

A2;. (a) ft is a positive and continuously differentiable function on [0, ?0) U

(Xo, it]',

(b) \dlogh(X)/dX\= 0(\X

-X?rl) as X -+ Xo.









We now comment on the results of Theorem 1. This theorem indicates that

aoQj is asymptotically pivotal. One consequence is that critical regions of tests

based on a continuous functional (p :D[0, n] h> R can be easily obtained. Dif

ferent functionals cp lead to tests with different power properties. Among themare omnibus, directional and/or Portmanteau-type tests. For example, classical

functionals which lead to omnibus tests are the Kolmogorov-Smirnov (cp(g)=

suP?g[0,7t] g(^)l) an<3e Cram?r-von Mises ((p(g)= n~l f? g(X)2dX), whereas

Portmanteau tests, defined as weighted sums of squared estimated autocorrelations

of the innovations, and directional tests are obtained by choosing an appropriate

functional <p-, ee Section 3 for details.

On the other hand, in practical situations the parameters 9o are not known and,

thus,they

have to be

replaced by

some estimate 9j. In this situation, as Theorem 2

below shows, theTp-process

is no longer asymptotically pivotal and, hence, the

aforementioned tests are not useful for practical purposes. The unknown critical

values of functionals of theTp-process

with estimated parameters can be approxi

mated with the assistance of bootstrap methods. This approach has been proposed

by Chen and Romano [9] and Hainz and Dahlhaus [19] for short-range models us

ing theUp -process and by Delgado and Hidalgo [11], who allow also long-range

dependence models using the7^-process. Alternatively, asymptotically distribu

tion free tests can be obtained by introducing a tuning parameter that must behave

in some required way as the sample size increases. Among them, the most popularone is the Portmanteau test, although it has only been justified for testing short

range models. Box and Pierce [5] showed that the partial sum of the squared resid

ual autocorrelations of a stationary ARMA process is approximately chi-squared

distributed assuming that the number of autocorrelations considered diverges to in

finity with the sample size at an appropriate rate. A different approach, in the spirit

of Durbin, Knott and Taylor [12] for the classical empirical process, is that in

Anderson [2], who proposed to approximate the critical values of the Cram?r-von

Mises tests for astationary

AR model. The method considers a truncated version

of the spectral representation of olqtj with estimated orthogonal components. The

number of estimated orthogonal components must suitably increase with the sam

ple size. A similar idea was proposed by Velilla [46] for ARMA models. Finally,

another alternative uses the distance between a smooth estimator of the spectral

density function and its parametric estimator under i/o- This approach provides

asymptotically distribution free tests for short-range models assuming a suitable

behavior of the smoothing parameter as the sample size diverges; see, for exam

ple, Prewitt [34] and Paparoditis [33]. However, the final outcome of all these tests

depends on the arbitrary choice of the tuning/smoothing parameters, for which no

relevant theory is available.

This article solves some limitations of existing asymptotically pivotal tests, only

justified under short-range dependence, by considering an asymptotically pivotal

transformation of olqtj related to the cusum of recursive residuals proposed by









Brown, Durbin and Evans [8]. We show that our testing procedure is valid un

der long-range specifications. In the next section we provide regularity condi

tions for the weak convergence of aoTj and its asymptotically distribution free

transformation. In Section 3 we discuss the behavior of tests of avery

different

nature?omnibus, directional and smooth/Portmanteau?under local alternatives

converging to the null at the rate T~1^2. Section 4 reports the results of a small

Monte Carlo experiment. Some final remarks are placed in Section 5. Section 6

provides lemmata with some auxiliary results, which are employed to prove, in

Section 7, the main results of the paper.

2. Tests based on a martingale transformation of theTp -process with esti

mated parameters. A popular estimator of Oo is theWhittle estimator

(7) Oj :=argminG^,r(7r),OeS

with Got defined in (5).Let us define

3 1 ^(t>e(X) =?logh0(X), ST := ~

?^0(^0^0(^)7= 1

and introduce the following assumptions:

A3, (a)(j>o0

is a continuously differentiable function on (0, 7r]; (b)||30#O(?)/

3A.||=

0(1/A.) as X -* 0+; and for some 0 < 8 < 1 and all X e (0, n], there

exists aK < oo such that (c) sup{#: \\o-OoWs8}o(^)\\ < ^ I og? |;(d)

1sup

{6: \\e-$o\\<S/2} l|0?

0o||'

hon(X) ,

\+(/)'0o(X)(0-Oo)

K , 2

<^log2?;e(X)

and (e) X?#0 :=n~l f? (/)o0(X)(f)f0o(X)dXis positive definite.

These assumptions are standard when analyzing the asymptotic distribution of

theWhittle estimator 6j and they are satisfied for all parametric linear processes

used in practice. Standard ARMA models satisfy a stronger condition, replacingthe upper bounds inA3(c) and (d) by a constant independent of X. It can easily be

shown that A3 is satisfied for ARFIMA models. Note that A3(e) and Lemma 1 in

Section 6 imply that St is positive definite for T large enough.

A4. The estimator in (7) satisfies the asymptotic linearization

(8) TXI2(0T-

Oo)= S~l r (t>o0(X)a0o,T(dX) op(\).jo

Theexpansion (8),

inassumption A4,

is satisfied under A1-A3 and additional

standard identification conditions; see [15, 20] or [45] for a later reference.

Define

cxooW =Bl(X)-(ljo (po.Wd'X^1 j*

c/>o0(X)Bl(dX).









THEOREM 2. Under Ho and assuming A1-A4, uniformly in X e[0, tc\.

11 [fx/n] \ r*(a) aw(?)

=

a?rW-L ? ^e^j))ST ?^ 4%W^{dX)+

op{\)\(b) aoTJ^<XoQ.

7= 1

Theorem 2 indicates that the asymptotic critical values of tests based on olqtjcannot be tabulated. However, we can use a transformation of olqtj that converges

in distribution to the standard Brownian motion. To this end, it is of interest to re

alize that Theorem 2(a) provides an asymptotic representation of aeTj as a scaled

cumulative sum (cusum) of the least squares residuals in an artificial regression

model. For that purpose, observe that by (2), and using the fact that 4>o0 is inte

grable [A3(c)L

(9)/*o

<t>eJX)dX= 0.

Now, because Lemma 1 in Section 6 with ?(X)=

(/>o0(X) and (9) imply that

II2k=i 00b(^*)Il=

OQ?? ^)? ^e uniform asymptotic expansion inTheorem 2(a)indicates that

supA. [0,7T]

2tt 1 [TX/Tt]

a?Tj{X)-^)T^ gUT(J) op(l),

where

UT?)=

Isfrj)-Y?0frj) J2y?o(xk)Y?0&k).k=\

-1 j

J2yOo(Xk)Ie&k),k=\

j = l,...9T,

are the least squares residuals in an artificial regression model with dependent vari

ableI?(Xj)

and a vector of explanatory variablesye0(Xj)

:= (1,(/>o0(Xj)Y.

This fact

suggests employing the cusum of recursive residuals for constructing asymptoti

cally pivotal tests, as were proposed by Brown, Durbin and Evans [8]; see also [39].

Let us define

1 TAo,tU)'.= ~ J2 Yo(^k)Y?(^k)

k=j+l

and assume the following:

A5. Aqqj(T)is nonsingular for T = T

?p

?1.









The (scaled) cusum of forward recursive least squares residuals is defined as

l7i j [TX/n]

where

eT(j) :=/e(Xy)

-

y?0(^)Mj),;

=1,..., 7\

are the forward least squares residuals and

f

TT(j):=A-QlT(j)l ? YOo(^)Ie(h)k=j+\

It is worth observing that the motivation to employ only the first T Fourier fre

quencies to compute the recursive residuals is due to the singularity of Aoj(j) for

all j>f.

The empirical process ?jcan be written as a linear transformation of a?,

??T(X)=

?oQjc4(X), Xe[0,7tl

where, for any function g e D[0,7r],

?e9Tg(V=g(1Fk)-1F ? Yo^MotU) Ye(Vg(dX).

\i / ij=l Jxj+i

The transformation ?oqJ has the limiting version dC?,defined as

1 ?1

where

X?g(k)= g(k)- -

f Ye0(X)A^(k) ye0(i)g(di)dk,I J \) JA.

Ae0W:= i yeoCk)y?0Ck)dLa.

Notice that Xoa is the martingale innovation of a^; see [25].

This type of martingale transformation has been used by Khmaladze [25]

and Aki [1] in the standard goodness-of-fit testing problem, by Nikabadze and

Stute [32] for goodness-of-fit of distribution functions under random censor

ship, by Stute, Thies and Zhu [42], Koul and Stute [28, 29] and Khmaladze and

Koul [26] for dynamic regression models, and by Stute and Zhu [43] for generalized linear models.

Henceforth, Bn(X) := B(X/n) for ? e [0, n}.

THEOREM 3. Under Ho and assuming A1-A5,

?T^Bn.









Because?j

cannot be computed in practice, as it depends on Oo, it is suggestedone use

?oTj, where

?ejiX)'.=

?e,Tae,T{X)

[TX/7T]2n 1~

Gej(n)W/2 jzYand

J2 eej(j), Xe[0,n],

ho(Xj)

are the forward recursive residuals in the linear projection ofIxi^j)/h$(Xj)

on

yo(Xj), and where

f

be,T(j) A??Tu?? y^-?aTk^+l

he(Xk)

In order to establish the asymptotic equivalence between?j

and ?oTj,we also

need some extra smoothness assumptions on the model under the null.

A6. For some 0 < S < 1 and all X e (0, n], there exists a constant K < oo such

that

SUP ,m\ .Jl^W-^oW-^oW^-^li< ^llog^l,

{0:\\0-Oo\\<8} WUU0\\

and <posatisfies A3(a)-(c).

This assumption holds for all models used in practice, such as ARFIMA in (3),

Bloomfield's exponential model and the fractional noise models mentioned before.

In fact, they satisfy even the stronger condition with jST|log ? |replaced by K.

THEOREM 4. Under Ho and assuming A1-A6,

sup \?eTj{X)-?^X)\=op{\).ke[0,7t]

Theorem 4 holds true, mutatis mutandis, with 9j replaced by any T^-consis

tent estimator. Also, from a computational point of view, it is worth observing that

A$,t?)

=

A9J{j

+

1)-y w-w- , n n ,T+ Ye(xj)AejO + Vre&j)

and

bej{j)= b9j{j + D + A^lT(j)ye(Xj)

[%^j

~Y^J^tU + 1)









see [8] for similar arguments.

Alternatively to ?oTj,we could have considered the cusum of backward recur

sive residuals, that is,

ljz 1[TX/7T]

?oTJ^)'=r-TT7?72 Y. ?orjU), Xe[0,nl

GoTj(n)T''/j=p+l

where

-eej(j) :=^4

"Yo^j)?>ej(j), j = p + l,...,f,

he(Xj)

boj(j) :=?qXt(j)-~ V Ye(Xk) y* , and A^rO") := ~

7] y&(**)y0(**).

In this case, we can take advantage of the computational formulae

A0T(j + 1)=

A?/r0)-

y -,7-W., ,. ,r + Yo^j+\)Ao tO)Y6(Xj+i)

and

??.rO"+ 1)= ?>ej(j) + A~lT(j + l)yo(Xj+i)/jf(^+i)

M\/+i)-y#(^/+i)^,rO')

This formulation may be useful in small samples when we suspect that themain

discrepancy between the null and the alternative is near n. However, from Theo

rems 3 and 4, it is easily seen that the empirical processes ?oTj and ?oTjhave

the same asymptotic behavior.

Let <p:D[0, n] -? R be a continuous functional. Under Ho and the conditions

in Theorem 4,

<p{?oT,T)^(p(Bn),

as a consequence of the continuous mapping theorem. For instance,

Kt?

sup

j=U...T??TT(C)\S sup \Bn(X)\= sup |?(a>)|,

w / 1 ?g[0,7t] ?>e[0,l]

The above limiting distributions are tabulated; see, for example, [40], pages

34 and 748.









3. Local alternatives: omnibus, directional and Portmanteau tests. In this

section we shall show that tests based on?eTj

are able to detect local alternatives

of the type

HlT:h(X)=h0o(X)(l

+rJ^KX)

+isriX)^,

X e [0, tt] and for some 9o e 0,

wheref? l(X)dX

=0, l(X) satisfies the same properties as

0#oin A3(a)-(c), r is

a constant, possibly unknown, and for some finite To, \st(-)\ is integrable for all

T > To. Let us consider some examples.

Example 1. If we wish to study departures from the white noise hypothesis

in the direction of fractional alternatives, we have

h(X) 1- =

-^777, Xe[0, TT],

h0o(X) |2sin(?/2)|2^1/2

for some d ^ 0. By a simple Taylor expansion up to the second term,

l(X)= -2log |2sin(?/2)| and r = d,

respectively, with the remainder function sj being such that, for some 0 < e < 1,

\stM\<

K\X\~ for all large T and some K < oo.

Example 2. Ifwe consider departures in the direction of MA( 1) alternatives,

we obtain

(?) =i_^J_2cos(A)+ ~r?2, Xe[0,n].

ho0(X) 'TW

Thus, r =rj, l(X)

= ?2cos(?) and sj(X)

=if2.

Example 3. Ifwe consider departures in the direction of AR( 1) alternatives,

then

h(X) |\ ? 1 i ^"11

8~^2cos(X)+

??2j

, X e [0, tt].ho0(X)

Thus, r = S and l(X)=

2cos(?) with \sT(X)\< K, for all large T and some

K < oo.

For X [0, tt], let us define

(10) L(X):=- ? \l(X)-y?0(X)A?(X)- [_*ye0Cx)l(X)dx\dXand

M(X) := Bn(X) + r L(X), X e [0, tt].

We have the following theorem.









THEOREM 5. Assuming the same assumptions as in Theorem 4, under H\t,

?oTj^M.

Using the fact that M and Bn are identically distributed, except for the de

terministic shift r L, and taking into account that 21/2sin((y?

1/2)?) and

1/0*?

l/2)27r2are the eigenfunctions and eigenvalues in the Kac-Siegert rep

resentation of Bn [24], the orthogonal components of M,

m (j) :=2"2{j-

i) rSin((;-

\)k)M{X)dk, j= 1,2,..

J u

are independently distributed normal random variables with mean r ?(j) and

variance 1,where

?(j)=

2^2(j -\)jfs'm((j

-\)X)L(X)dX, j= l,2,....

Using the (asymptotically) orthogonal components of ?oT,T,

mT(j)=

21/2(y-

\)j71

sin((y-

\)X)?oTj(X) dX, 7= 1,2.

we obtain the spectral representation,

?eTj(X)= 21/z >-??-, X e [0, n].

By Theorem 5 and the continuous mapping theorem, finitely many of the ra^Oys

converge in distribution to the corresponding m(j)'s under H\t. Using Parseval's

theorem,

oo 2/ -\

j=l(j-

1/2)

Using similar arguments to those in [13] in the context of the standard empirical

process with estimated parameters, tests based on

n

W?s:=J2 2tU),7=1

with a reasonable choice of n > 1, will lead to gains in power, compared to ?Y,

in the direction of alternatives with significant autocorrelations at high lags. These

Portmanteau tests are related to Neyman's [31] smooth tests, a compromise between omnibus and directional tests, and for each n > 1, under H\t we have that

Wn,T^X2n{r2t?2U))









That is, tests based on Wnj are asymptotically pivotal under i/o (r=

0) for each

choice of n, and more importantly, they are able to detect local alternatives con

verging to the null at the parametric rate T~1/2, provided that ?(j) ^ 0 for some

j=

1,..., n. The latter is in contrast with the classical Portmanteau tests based on

nj

(H)QnTj'=t^Tl'2pTU))\7= 1

where pr(j) is some estimate of the jth autocorrelation of the residuals. It has

been shown that Qnjj is approximately distributed as aXnT-p

under i/o specify

ing a short-range model and assuming that nj diverges as T?

oo. On the other

hand, the resulting test is able to detect alternatives converging to the null at the

rate rij T~x?2 (see, e.g., [21]), which is slower than T~x/2.

In practice, it is recommended that one use the discrete version

n

Wn,r :=?>40')

7=1

oiWnj, With

^-^-?)tS*(0--?)tMt)On the other hand, optimal tests of i/o in the direction H\j can be constructed

applying results in [16] (see also [17] and references therein), as was suggested

by Stute [41] in the context of goodness-of-fit testing of a regression function.

Asymptotically, testing for i/o in the direction of H\j is equivalent to testing

Ho :E(m(j))= 0 for all j > 1, againstHi :E(m(j))

= x ?(j) for all j > 1with

L known, but maybe with unknown r. Under i/o, the distribution of {m(j)}j>\is

completely specified, as it is also under H\ when the parameter r is known. Then

the likelihood-ratio for a finite-dimensional set (ra(l),..., m(n)) is

(12) A?=expir??(j) (m(j)-

^Y^))'

Grenander [16] showed that An -+p A as n -> oo, and that the most power

ful test at significance level a has a critical region of the form {Aoo > k], with

Po{Aoo > k)= a if

YlJLi?2U)

< ??- The latter condition is satisfied in our con

text by Parseval's theorem and A3(c) because / is a square integrable function.

Define

t:~a:r=i?2?))i/2'









Then under Ho, \/f= N (0,1), and in view of (12), \jr forms a basis to obtain op

timal critical regions. When the sign of r is known, the critical region of the uni

formly most powerful test at significance level a is {x/r> z\-a} when r > 0 and

{xjf< ?zi-a) when r < 0, where zv is the v quantile of the standard normal. Also,

when the sign of r is unknown, the most powerful unbiased test at significance

level a has critical region given by {\\//\ >z\-a/2}

These arguments suggest an (asymptotically) optimal Neyman-Pearson test in

the direction of H\t based on the first n orthogonal components of ?oTj, using

the test statistic

1 Enj=it?)"hT?)fnJ~

(E^i^O'))1/2

'

Schoenfeld [38] proposes the same type of statistic in the standard goodness-of-fit

testing context. Under Ho and the assumptions in previous sections, we have that

d

tynj-

N(0, 1) as T -> oo for each fixed n.

Also, arguing as in Schoenfeld's [38] Theorem 3, the convergence in distri

bution of \\fnTj when h 7 increases with T can be shown. Approximately op

timal tests for Ho in the direction of H\t reject Ho at significance level a

when\\?fnTs\

>zi-a/2 if T has unknown sign, ^rnTj

> z\-a when r > 0 and

^nT,T< ?z\-a when r < 0.

4. Some Monte Carlo experiments. A small Monte Carlo study was carried

out to investigate the finite sample performance of the different tests. To that end,

we considered theAR(1), MA(1) andARFIMA(0, d0, 0) models

(13) \-80L)X(t) = 8(t),

(14)X(t) = (\-r]oL)8(t),

(15) l-L)d?X(t)= e(t),

respectively, where the parameter Oo equals 8o, r?o and do for the different models

and L is the lag operator. The innovations{s(t)}J=l

are i.i.d. <M(0, 1), and the sam

ple sizes used are T = 200 and 500 with different values of the parameters So, r)o

and d0. For models (13) and (14), we considered <50, ?o=

-0.8, -0.5, 0.0,0.5, 0.8,whereas for model (15) do

=0.0, 0.2,0.4. The ARFIMA model was simulated us

ing an algorithm by Hosking [22].

For the three models and all values of Oo, we computed the proportion of re

jections in 50,000 generated samples for both sample sizes. Whittle estimates are








2582 M. A. DELGADO, J. HIDALGO AND C. VELASCO

obtained according to (7). For each of the models considered (f>ois given by

AR(1), 9 = 8:folk) =? log |1 8eik\~2

= -2-8~ ??SX

d8?l

l-2?cosA + ?2'

^ i?n ik\2 ^ rl~ cos^

MA(1), ? =ij:^(A)

=?log|l-^,A|z

= 2

9r? 1?

2r?cosA. + r?z

ARFIMA(0,rf, 0), 9 =d:(?)d(X)

= ?log|1

-^p^

=-2log |2sin(?/2)|.

od

We also report, as a benchmark, the proportion of rejections using

J= \ J

which is suitable for testing simple hypotheses. In addition, for the sake of com

parison, we provide the results for the Box and Pierce [5] test statistic (11) using

several values of nj increasing with T, where PtU), j> 1, are the sample auto

correlations of the residuals{?(t)}J=l. Specifically, for the AR(1) model,

8(t) = (l-8TL)X(t),

with X(t) = 0fovt<0; for theMA(1) model,

m=

X(t)-r?THt-l),

with ?(0)= 0; and for theARFIMA(0, d, 0) model,

?-i

Ht) =J2?U,dT)X(t-j),

7=0

where ?(j, d) are the coefficients in the formal expansion

oo

j=o

with

?{j,d) =-r?~t/)-, T(a)= [??xa-le-xdx.

The standardized values of QnT,T, (QnT,T-

/ir)/V2wrare compared with the

5% critical value of the standard normal (see Hong [21]) instead of the usual

X(n _i) approximation correcting by the loss of degrees of freedom due to pa

rameter estimation, which is justified under Gaussianity. The two approximations

provide a similar proportion of rejections. We also tried the weighting suggested

by Ljung and Box [30], which produced very similar results.









Table 1

Empirical size of omnibus and Portmanteau tests at 5% significance

T = 200 T = 500

Ct Cj ?3,r ?6,r ?io,r ?20,r Cr Cr ?3,r ?6,r ?i5,r ?35,r

?0,H0:AR(1)-0.8 4.92 4.69 3.34 3.72 3.91 3.61 5.07 5.17 3.56 3.87 4.35 3.97

-0.5 4.38 4.96 2.80 3.38 3.60 3.41 4.96 5.16 3.12 3.75 4.17 3.82

0.0 4.07 4.96 2.66 3.35 3.45 3.37 4.62 5.10 3.00 3.63 4.11 3.82

0.5 3.59 4.95 2.67 3.33 3.57 3.40 4.50 5.04 2.97 3.82 4.17 3.80

0.8 3.08 4.92 2.89 3.44 3.73 3.54 4.27 5.11 3.33 3.77 4.32 3.88

?K),#o:MA(l)

-0.84.25

8.37 4.324.54 4.42

3.95 4.89 6.674.13

4.39 4.56 4.07-0.5 4.16 5.06 2.83 3.41 3.65 3.38 4.89 5.18 3.13 3.76 4.15 3.83

0.0 4.08 4.96 2.51 3.26 3.46 3.32 4.62 5.10 2.94 3.61 4.05 3.82

0.5 3.60 5.08 2.65 3.30 3.55 3.41 4.49 5.15 2.96 3.77 4.13 3.82

0.8 3.89 7.72 15.33 15.30 15.33 15.05 4.63 6.42 8.03 8.44 8.68 8.17

do,H0:l(d)0.0 3.53 4.96 2.76 3.40 3.68 3.47 4.48 5.10 3.13 3.90 4.29 3.83

0.2 3.54 4.95 2.76 3.39 3.63 3.46 4.54 5.15 3.14 3.89 4.27 3.81

0.4 3.58 5.21 2.79 3.39 3.59 3.44 4.58 5.37 3.14 3.88 4.27 3.80

First we analyze the size accuracy of the Cramer-von Mises test based on?oTj

The empirical sizes of the tests based on ?V, reported in Table 1, are reasonablyclose to the nominal ones. The asymptotic approximation improves noticeably

when the sample size increases from T = 200 to T =500, this improvement being

uniform for all the models, although the empirical size is smaller than the nominal

level. Tests based onQnTj have serious size distortions for the smaller sample

size and large values of |?7| in the MA(1) model, since Whittle estimates can be

quite biased in these cases. The empirical size of tests based on Qnjj dependssubstantially on the number of autocorrelations used. In addition, for the largerchoices of nT implemented, Qnjj over-rejects Ho. The usual recommendation

ut =o(Jxl2) also seems reasonable here, in terms of size accuracy.

Next we study the power performance of the tests. To this end, we report first, in

Table 2, the proportion of rejections under the alternative hypothesis for different

nonnested specifications with the model specified under the null. We cannot con

clude that one test is clearly superior to the others in any of the four cases analyzed.As expected, the power of the Portmanteau test decreases as n^ increases. In view

of Tables 1 and 2, we can conclude that a choice of large nj, around T~1/2, produces reasonable size accuracy, but such a choice is not the best possible one in

order tomaximize the power. The test based on Cj is fairly powerful compared to

the Portmanteau test for all cases considered, and itworks remarkably well when

testing an AR(1) in the direction of anMA(1) alternative.









Table 2

Empirical power of omnibus and Portmanteau tests at 5% significance

T = 200 T = 500

Ct 03,7 06,7 010,7 020,7 ^7 03,7 06,7 ?l5,7 035,7

r?, /0:AR(1), /fi:MA(l)-0.8 100.00 99.97 99.95 99.25 92.34 100.00 100.00 100.00 100.00 100.00

-0.5 80.82 70.16 55.53 44.38 31.25 99.84 99.23 97.54 88.65 68.72

0.2 7.12 5.04 4.98 4.86 4.34 12.16 8.31 7.35 6.27 5.21

0.5 70.82 72.03 57.50 46.06 32.15 98.59 99.32 97.83 89.19 69.29

0.8 99.56 99.99 99.95 99.30 92.76 100.00 100.00 100.00 100.00 100.00

8,H0:MA(l)9Hi:AR(l)-0.8 100.00 100.00 100.00 100.00 99.99 100.00 100.00 100.00 100.00 100.00

-0.5 84.36 77.15 66.51 57.37 44.02 99.73 99.47 98.45 94.26 82.89

0.2 7.16 3.71 3.99 3.94 3.63 12.04 6.65 6.42 5.73 4.80

0.5 77.08 74.86 64.04 54.79 31.78 99.19 99.41 98.35 93.77 82.04

0.8 100.00 100.00 100.00 100.00 99.97 100.00 100.00 100.00 100.00 100.00

8,H0:l(d),H{:AR(\)0.2 11.34 12.84 13.00 11.27 13.13 34.92 33.35 33.01 23.98 15.71

0.5 26.81 34.11 41.17 35.55 24.94 75.29 81.36 87.81 80.73 58.52

0.8 9.82 12.86 21.01 21.32 15.41 33.21 38.74 57.53 61.63 39.15

d,H0:AR(l),Hi:l(d)

0.1 8.22 4.98 5.66 5.11 4.83 16.79 12.07 14.09 12.34 9.100.2 19.90 13.74 16.20 15.23 11.81 51.77 45.04 53.29 47.54 36.11

0.3 36.03 25.92 32.00 30.50 24.35 82.80 74.84 85.12 81.44 69.62

0.4 48.83 34.86 43.78 43.31 35.48 94.40 87.30 95.56 94.31 87.38

Finally, we analyze the power of the different tests when testing an AR(1)

specification in the direction of local ARFIMA(l,d, 0) with d =x/Tx?2, and

in the direction of local ARMA(1, 1) alternatives with moving average parame

ter ?]? r/T1/2, for different values of r. The proportion of rejections for these

designs is reported in Tables 3 and 4. We also consider tests based on the test sta

tistics Wnj and yjfnj (one-sided and two-sided, \?r+Tand \\?rnj\ resp.), choosing

n = 3 and 6, which has been recommended by Stute, Thies and Zhu [42] for a

different goodness-of-fit test problem. Of course, tests based on the first n (as

ymptotic) orthogonal components of ?oTjare sensitive to the choice of n, as also

happens with tests based on the n (asymptotic) orthogonal components of otoTj

(the estimated autocorrelations of the innovations) in Portmanteau tests. The om

nibus test based on Cj still works fairly well comparedto the

others, includingthe optimal and smooth tests. The directional tests are the most powerful in the

directions for which they are designed, and the tests based on Wnj and Qnjj

work very similarly, though Wnj exhibits a better size precision for the choices

of n considered.









Table 3

Empirical size and power under local alternatives at 5% significance

i/0:AR(l), Jfir1:ARFIMA(l,i/=

r/r1/2,0)

t p CT WXT ^6,r \h,T\ \+6,t\ f^T +t9T ?3,r Q^t

7 = 200

0 0.0 4.07 3.19 2.59 4.70 4.81 4.48 5.12 2.66 3.35

0.5 3.59 2.98 2.32 3.79 4.24 3.62 3.99 2.67 3.33

0.8 3.08 2.52 1.94 3.94 3.10 3.75 4.02 2.89 3.44

1 0.0 6.26 5.40 4.37 8.39 11.13 13.44 16.63 3.68 4.25

0.5 3.57 2.90 2.26 3.45 4.19 4.19 5.64 2.73 3.37

0.8 3.01 2.25 1.66 4.10 4.52 7.80 8.53 3.87 4.41

2 0.0 12.19 12.04 10.53 19.93 26.15 28.94 35.10 7.80 9.13

0.5 3.44 2.91 2.36 3.47 4.15 4.25 6.27 2.91 3.58

0.8 4.84 3.16 2.19 9.17 10.33 16.59 17.98 8.45 7.58

3 0.0 21.92 23.63 21.27 35.77 44.37 47.20 54.61 15.17 18.02

0.5 3.26 2.74 2.39 3.65 4.43 4.99 6.48 3.27 3.92

0.8 9.13 6.61 4.10 20.13 22.90 31.95 35.14 21.18 16.12

4 0.0 33.38 27.13 24.15 50.40 59.39 62.18 69.12 23.88 29.88

0.5 3.41 2.47 2.38 4.09 4.75 6.80 7.61 4.32 4.67

0.8 17.48 14.65 9.09 38.10 43.37 53.13 57.56 46.00 33.97

T = 5000 0.0 4.62 4.22 3.66 4.81 4.78 4.57 5.06 3.00 3.63

0.5 4.50 3.99 3.40 4.26 4.58 4.27 4.43 2.97 3.82

0.8 4.27 3.56 3.09 3.90 3.85 4.63 3.63 3.33 3.77

1 0.0 6.93 7.03 6.29 9.35 11.62 14.63 17.54 4.37 5.13

0.5 4.58 4.42 4.08 4.85 5.35 58.30 7.43 3.02 3.93

0.8 4.74 4.13 3.47 5.72 5.90 9.61 9.83 4.12 4.64

2 0.0 14.22 15.51 14.23 23.43 29.37 33.47 39.37 10.03 11.60

0.5 4.69 4.72 4.67 4.83 6.49 6.37 10.18 3.08 4.21

0.8 7.36 6.13 4.73 11.57 12.08 19.11 19.817.27 7.38

3 0.0 26.86 31.03 29.55 44.70 53.35 56.44 63.59 21.28 24.91

0.5 4.65 5.04 5.48 4.71 7.14 5.44 11.31 3.30 4.60

0.8 13.56 11.62 8.18 23.46 24.65 34.56 35.78 15.23 13.51

4 0.0 43.62 51.19 49.81 66.34 74.28 75.93 81.84 37.13 43.93

0.5 4.65 5.18 6.35 5.05 7.03 5.09 10.80 3.81 5.09

0.8 24.44 23.10 16.17 42.07 44.05 54.86 56.23 31.28 25.74

\irnj\ denotes two-sided tests, whereas\?r^T

are one-sided (right-hand side) tests.

5. Final remarks. Our results can be extended to goodness-of-fit tests of

models that can accommodate simultaneously stationary and nonstationary time

series. For instance, if the increments Y(t) := (1?

L)X(t), t =0, ?1,..., are









Table 4

Empirical size and power under local alternatives at 5% significance

H0 :AR(1), Hx :ARMA(1,1), r? r/T1/2

r p CT W3yT w6iT \f3,r\ \t6,Tl ft,T Kt ?3,r ?6,r

r = 200

0 0.0 4.13 3.09 3.58 3.98 4.39 4.18 4.39 2.65 3.36

0.5 3.62 2.80 2.22 3.68 4.04 3.93 4.14 2.67 3.31

0.8 3.06 2.38 1.86 3.00 3.21 3.45 3.64 2.93 3.46

1 0.0 4.22 3.10 2.58 3.88 4.23 3.74 3.93 2.76 3.40

0.5 5.52 4.08 2.90 5.51 5.76 8.86 9.20 3.08 3.61

0.8 7.81 5.63 3.66 7.77 7.98 13.13 13.62 5.47 5.05

2 0.0 5.01 3.50 2.79 3.77 4.06 3.36 3.46 3.45 3.82

0.5 8.53 6.10 4.02 8.58 9.06 14.33 14.61 4.51 4.56

0.8 18.07 13.73 8.53 20.63 21.26 30.93 31.41 12.52 10.66

3 0.0 7.79 5.04 3.76 4.62 4.92 6.00 6.06 5.60 5.32

0.5 10.64 7.80 5.16 10.84 11.25 17.39 17.87 5.76 5.41

0.8 32.10 27.17 17.65 37.68 38.18 50.25 50.49 23.84 20.09

4 0.0 14.60 9.51 6.65 10.86 11.01 16.70 16.78 11.03 8.99

0.5 10.67 8.16 5.42 10.65 11.01 17.11 17.57 5.93 5.56

0.8 45.29 42.62 29.55 52.48 52.79 64.96 64.97 36.18 31.63

T = 500

0 0.0 4.70 4.43 3.86 4.66 5.68 4.52 4.62 2.99 3.64

0.5 4.50 4.23 3.70 4.53 4.55 4.50 4.52 2.99 3.80

0.8 4.39 3.94 3.40 4.22 4.26 4.37 4.38 3.34 3.78

1 o.O 4.74 4.37 3.83 4.70 4.75 4.31 4.35 3.02 3.70

0.5 6.68 5.72 4.73 6.71 6.61 10.25 10.36 3.75 4.32

0.8 9.56 8.06 6.00 10.03 10.08 16.20 16.28 6.26 5.82

2 0.0 5.00 4.47 3.90 4.76 4.87 3.61 3.62 3.34 3.90

0.5 11.06 8.94 6.81 11.48 11.43 18.23 18.17 6.06 5.88

0.8 23.21 19.66 13.89 26.87 26.88 38.01 37.99 15.66 13.35

3 0.0 6.31 5.17 4.38 4.95 5.03 3.19 3.18 4.25 4.55

0.5 16.44 13.17 9.58 17.26 17.24 26.26 26.03 9.45 8.39

0.8 42.78 38.92 28.30 50.11 49.91 62.36 62.42 32.23 27.37

4 0.0 9.48 6.98 5.57 5.09 5.16 4.09 4.07 6.40 5.98

0.5 21.08 17.22 12.42 22.10 21.95 32.15 31.99 12.84 10.89

0.8 62.44 60.69 47.41 70.99 70.86 80.69 80.67 52.01 46.42

\\?rnj\denotes two-sided tests, whereas

\?r?Tare one-sided (right-hand side) tests.

second order stationary with zero mean and spectral density g such that

lim \X\2(d~l)g(X)= G > 0 for some d e [0.5,1.5),

?-*0+









we can define the pseudo-spectral density function of {X(t)}t z, f, as

1

11~?

<*>=

71?5??2*W

Thus, when d^\, g has a singularity at X=0, as happens with many long-range

dependent time series (cf. A2). If {X(t)}tez is stationary, / becomes the standard

spectral density function.

If either {Y(t)}tez or {X(t)}tsz satisfies Wold's decomposition, / admits the

factorization

a2

f(X.)= ?A (A.),

2n

where h satisfies A2. Thus, given a parametric family 3i, for example, the

ARFIMA specification given in (3), aTp-process

for testing that h e 3i is

<rW:=?1/2r^jW kII

Xe[0,n],

G%tT(7T) IT]

where G T is analogous to Goj, but using the tapered periodogram, for example,

\Ej=]w(t)X(t)eia\2/?(*):=2nYj=\^2(t)

Here 0t=

argmin^e GT(n)

is the Whittle estimator proposed by Velasco and

Robinson [45], which admits a similar asymptotic first order expansion as in (8),

and where w is a taper function, for example, the full cosine taper

u;(i)=

-(l-cos(-^)),

t= l,...,T.

If the full cosine taper is used, because of its desirable asymptotic properties

(see [44]), it is recommended in practice to base our tests on the empirical

process ? T, where

/P?\1/2 2nm

>os(n)

with

IW(X )

eZrUV-=i^-Y?>Q.j)b%T(j),

Tk=j+\h^Xk)









and

fl?r-lim r?-'?4?-354* 7-oo(?j=1 w2(t))2 18*

Under appropriate regularity conditions, it can be proved using tools in [44]

and[45]thatj8?fr=??,r.Finally, the methodology can be extended to test the correlation structure of

the innovations of regression models (e.g., distributed-lags models) using the mar

tingale part of thet/^-process

based on the residuals. When E(z(t)u(s))= 0 for

all t, s, where{z(t)}J=l

are the regressors and[u(t)}J=l

the error term, the resid

ualUp -process is asymptotically equivalent to the

Up -process based on the true

innovations, and there is no need to use tests based on the martingale part of the

Up -process. When E(z(t)u(t

?

s))^0 for some s > 0, the first-order expansion ofthe residual

Up -process depends on the cross-spectrum of the innovations and re

gressors. However, it seems possible to apply the results in this paper to implement

tests based on the (approximate) martingale part of thisUp -process with estimated

parameters.

6. Lemmas. This section provides a series of lemmas which will be used in

the proofs of the main results. Some of them can be of independent interest. Hence

forth, z^ denotes the kth element of a p x 1 vector z and K a finite positive

constant. Also,we

shall abbreviate g(Xj) by gj fora

generic function g(X).

LEMMA 1. Let ? : (0, tt] -> Rp be a function such that U(X)\\<

K\ \ogXf,

I > 1, and \\d?(X)/dX\\<

KX~l\ log X\?~l for all X > 0. Then, as T -> oo,

II [fxM 1 ck I(16) sup ~

Y. O"- / Kix)dx\K7T]\\T

~TTJoAe[0,7r]|| 7= 1

<?^.

Proof. The left-hand side of (16) is bounded by

ll rx II II[TkM i rk I(17) sup -/?(x)dx\

+ sup ~Yl O"-/ ttx)dx\

? [0,7T/f)l|7r0 " ?G[7T/f,7r]|| j= \ n J0I

The first term of (17) isbounded by

-U(x)\\dx<K \iogxfdx<KK-^~.TJO JO

Next, by the triangle inequality, the second term of (17) is bounded by

supke[7T/T,n]

(18)

i i rtT-CW-- / S(x)dxT TTJO

I[Tl.M-1 ?(j+\)n/TL

i^J nj+w/i

+ sup-

Y, /~ Uj-K(x)\\dx.

k?[7T/T,7T]n

j= \ JJnlT









The first term of (18) is bounded by Kf-^logf)1 since ||f(jc)|| <K\logxf.

Next, by the mean value theorem, the second term of (18) is bounded by

f-i

; ""_. _ f-idxj 1 l-rr IT

t,1 /o'+l

*?/ ,fj=ljj*/t

{j+\)iz/T 1

f logx| dx <y=1

J Jjn/T

K(\ogf)1D

The next lemma corresponds to Giraitis, Hidalgo and Robinson's [14]

Lemma 4.4, which we state without proof for easy reference. For this pur

pose, let uj:=

hJl/2(2nT)-l/2Y:L\X(t)eia', vj

:=(2nT)-x'2Y,J=\ s(t)eiaJ

andRxeW

be thespectral coherency ([6], pages 256-257)

between X and s.

Also, herewith c will denote the conjugate of the complex number c.

LEMMA 2. Assuming Al and A2, then, as T ?> oo, the following relations

hold uniformly over 1<j < k <T\

E(ujVj)=

Rxsj + 0(j~1 logt/ + 1));

E(ujVj)=

0{j~l\og(j + \)y,

max(|E(W^)|, \E(ukvj)\)=

0(j~{ log(*));

max(|E(u^)|, |E(i;*ii;)|)=

0(j~l log(k)).

The next lemma corresponds to the proof of expression (4.8) of [37], pages

1648-1651, using the orders of magnitude of the terms a\, ai, b\ and bi in [37]and Lemma 3 there, but using our Lemma 2 instead of Robinson's [36] Theorems

1 and 2 when appropriate.

LEMMA 3. Let ? :[0, n] ?> Rp satisfy the same conditions on4>o0

in

A3(a)-(c). Then, assumingAl

and A2,as

T?>

oo, for l<r<s<T,h=

1, P

EJ2q 'vjiUj-Vj)

j=r

j=rK\og2(T)J2\r^og(T)

+ J2U~2^g2(T) +rlk-^2).

Lemma 4. Let C:[0,tt]

A3(a)-(c) and write

k=r )

satisfy the same conditions on (?>o0 in

[TX/n]

?7= 1








2590

and

M. A. DELGADO, J.HIDALGO AND C. VELASCO

[TX/7Z]IXJ

-?).

Then, under the conditions of Theorem I,for some 0 < 5 < 1/6,

(19) ? i J tE sup \\a'T(X)-a'T(X)\\=

0(T-d).Xe[0,n]

PROOF. It suffices to show that (19) holds for each element of the vector

:r(X)-

a\bounded by

oi\(X)?

dj{X). Then, by the triangle inequality the left-hand side of (19) is

i [TX/n]1 " -.(?I

(20)

A. [0,7T]I '

/==i

+ 2E sup?G[0,7T]

Y ?^VjQ?j-Vj)i/27= 1

The first termof (20) is bounded by

.^yv^ii/W-i2--27= 1

-K^-?)+H2-?)l

fl/27

= 1

2_ /">?\-l?2 .(*),by Lemma 2, because E| v,-1

?(2n) a ,and by assumption \?j\<K log 7.

Next, to show that the second term of (20) is 0(T~S), it suffices to show that

(21) E max

?=i.f sq/?E^j vMj~vj)7-1/2

7= 10(r_?).

By the triangle inequality the left-hand side of (21) is bounded by

(22) E maxs=l,...,[f?]

1

7-1/2Y,?jVj(uj-Vj)7= 1

(23) + E maxs=[f?]+l,...,f

+ E

1 [f?]

7-1/2?fj }Vj(.Uj-Vj)7= 1

7-1/2 E fj 'Vjiuj-Vj):=[T?]+\









where\

< ? <\- Using the inequality

(24) (suplcpl) =sup|cp|2 <Y^\cp\2'V P / P v

by the Cauchy-Schwarz inequality the square of (22) is bounded by

{f?]

S=\Y,?jk)Vj(uj-Vj)

7= 1

=0(f2?~x log4T) =

0(T~28)

using Lemma 3.

To complete the proof, we need to show that (23)=

0(T~8). To that end, let

q=0,..., [f?]?

1with 5< ? < ?. By the triangle inequality (23) is bounded by

r s [f?]+q(s)f/[f^]

?-

?ij=[T?]+\ j=[T?]+\

J

E1

max

r1/2 i=[f^]+i,...,fkr^^y-^')

(25)

E1

max

Tx?2 s=[T?]+\,...T

|[f^]+^(5)f/[fq

7=[^l+l

where g(s) denotes the value of q=

0,..., [f?]?

1 such that [T^] + q(s)T/[T?]

is thelargest integer

smaller than or

equal

to s, and

using

the convention

Y1?

= 0

if d < c.

By the definition of q(s) and the Cauchy-Schwarz inequality, the square of the

second term of (25) is bounded by

E? max

T0=o,...,[fq--l

[T*

[T?]+qT/[TS] |2

E Sjk)vjQ*j-vj)j=[f?]+l

i y e4=0

[7^]+<5r77[r<n2

f/ Vj(Uj-Vj)j=[TP]+\

by (24). But, using Lemma 3, we have that the right-hand side of the last displayed

inequality is bounded by

K>^LIT?'(l+̂ ?'-s

q=0T?

+\q\]?2T1'2?-^

< K log4 T(fg~? + f?~l/2)<

KT~28,

where \q\+ = max{l, \q\}. To complete the proof, we need to show that the first

term in (25) is 0(T~8). To that end, we note that this term is bounded by

E ~ max max

T{'z 0=o,...,[fq-is

j= \+[f?]+qT/[T<;]

?j Vj(uj-Vj)









where^the max5 runs for ail values s = 1+ [f?] + qf/[f?],..., [f?] + (q +

l)T/[T?]. By the Cauchy-Schwarz inequality and (24), the square of the last dis

played expression is bounded by

1 [f?]-l [T?]+(q+DT/[TS]

'fL-^i ??J Y K) VjQ*j-vj)q=0 s=l+[f?]+qT/[TS] lj=l+[T?]+qT/[TS]

Af [TSy-1 [f^]+to+l)f/[fff]

f j f(l-?)/2

T?2~

2

<?- >; >; \ +E E I" / ^ / jIII 3/

9=0 s=\+[f?]+qf/[fs],|<?l+ M +

<

A:l5?I(f1-ffiogT-+ f3(1-?)/2)

<^f(1-3^2log47

< tf f~2a,

where in the first inequality we have used Lemma 3 and that, for q> 1 and ^

> 0,

5^ / [T?]+(q+DT/[TS] \

y r*<-~_^o_i y n~^ ~ ~

~(T? +aTl-?)^ \ ~^ ~ ~ j

j=i+[T?]+qT/[T<;]v ^H }

v=i+[r^]+^r/[rq /

-^

This completes the proof. D

Remark 1. Lemma 4 holds for aT (X)anda\ (X) replaced by

??^(A.):=

c?t(tt)?

aj(X), o?t(X):=

?^T(n)?

?j(X),

respectively. This is so because the triangle inequality implies that

E sup \?^T(X)-a^T(X)\<2E sup \a^T(X)-

&T(X)\.?e[0,7t] Xe[0,7z]

Define, for ?xand & e [0, tt],

2[T?/7T]

(26) Cs(?,u)=Y~Yj2 E ?pCos(sXp),

p=[Tfi/7i]+l

where ? is as in Lemma 1 and ?i < ?.

Lemma 5. For0<?? <u\,U2<n,asT -* oo,

T-lT-t

(27>E E C^ #l)c's(V? #2) = gift, #1, #2>(l + o(l)),t=\ 5= 1

where gbJL,?i,?2)= n~l

f?lA*2 ?(u)?'(u)du-

(tt'1 f?1 ?(u)du)(n-1 x

ff?'{u)du).









PROOF. A typical component of the matrix on the left-hand side of (27) is

A [f?i/n] IT?2M T-\T-t

4fE

t?0E

ffEE^K)^)pi=[f?/7t]+\ p2=[T?/jT]+\ t=\ s=\

A [f?l/7T]A[f?2/7t] T-\T-t

(28)

T2Tp={T?/7i}+\

'=1 a=\

2[T?\/n\

/7l=[fM/7T]+l

[f#2/*] T-lT-t

xE <S2) E(cos(5?pi+^)+cos(î-p2)}

/72=

[r/x/7T]+lt= \ 5=1

Because cos2 ? =(1 + cos(2?))/2, then using formulae in [6], page 13, we have

thatEL"/ HTsZ\ ?(sXp)

= (T- l)2/4 and, for px / p2,

T-\ T-t

E JHC0S(sXp\+P2)+ cos(sXP]-P2)}

= -T

t=\ 5=1

and, hence, we conclude that the right-hand side of (28) is, recalling that

f =[T/2],

(T 1\2 / 1 [f?i/Âif^/TT] \

iLzlLlL v f(*iM*2)\j2 \f L^ V V /

^ P= [T?/7T]+\'

9 [ i/tt] [f?2/7T]

JJL^ ^P\ Lî >P2

Pl=[T?/7T]+\ P2 = [Tp/7T]+\

P2?P\

=g{k^k2)(?,ul,u2){l+o(l)),

by Lemma 1 and where g{kuk2\/?, ?\, ?2) denotes the (k\, k2)th element of the

matrix g(?ji, ?\, ?2).

We now introduce the following notation. For 0 < v\ < v2 < n,

[fv2/Jt]

_(s2(t)-a2)),P

=[rui/7T]+1

T t~\

(30) S2J(vx,v2) :=J2s(t)1}2s(s)ct-S(vl, v2),

t=2 s=\

I 1 [fv2/n] \/fl/2

T

(29) 8hT(vuv2):=lf ? f,W ?(?









where ct(-, ) is given in (26) and ? is as in Lemma 1.

LEMMA 6. Let 0 < v\ < v < v2 < tt. Then assuming Al, for k =\,..., p

andfor

some?

> 0 and 0 < 8 <1,

(31)E(\S$(vi,v)\?\S$(v, v2)\?)

< K(v2-

v{)2-8, j= 1, 2,

(k) (k)where S\ j(v\, v) and g^ fi^i? v) are the kth components of (29) and (30), re

spectively.

Proof. We begin with j= 1. By Lemma 1,

[7W*1

after we notice that we can take T~l <(t>2

? f i), since otherwise (31) holds triv

ially. On the other hand, Al implies thatE(?f=1(?2(i)

-a2))2

< KT. So, using

the inequality (v2?

v)(v?

v\) < (v2?

v\)2 and the Cauchy-Schwarz inequality,

we have thatE(\s{k)T(vi, v)\\s[k)T(v, v2)\) < K(v2

-vi)2~*.

To complete the proof, it suffices to examine that the inequality in (31) holds

for j= 2. Now

4

E{8^T(vx,v2)f= 16fj E c^lSj(vi9v2)E(e(ti)e(si).. .e(t4)e(s4)).

j=

\\<Sj<tj<T

Since the number of equal indices in the set [t\, s\,..., t4, s4] does not exceed 4,

by assumption Al it follows that \E(e(t\)s(s\).. .s(t4)s(s4))\< K. Moreover,

by Al the inequality \E(e{t\)e(s\)... s(t4)s(s4))\ / 0 can hold only if any tj,Sjare repeated in {t\,s\,..., t4, s4} at least twice. Hence, by the Cauchy-Schwarz

inequality, we obtain that

4 / \l/2

E(^!,,2))4<*n E ($lj(VuV2))2)=l\l<Sj<tj<T

I

\l<s<t<T I

But by Lemma 5 theright-hand side of the lastdisplayed equation is bounded by

2N2

k(- r{^k\u)f du (- P ?;{k\u)du\7t Jv\ \7T Jv\

<K{v2-vx)2-s

because\f^(?{k)(x))pdx\

<K\v2

-i>i|1_,5/2 for p

=1,2. This concludes the

proof choosing ?= 2 by the Cauchy-Schwarz inequality. D









LEMMA 7. Denote r?p\? Iep?

a2/(2n) and

i 2tt y?^ 2 2tt ^?-v

RT(v)=fy?L. SPVP and RT(V)

=

fiJ2

?^ SpVpiP=\ p=[Tv/n]+\

(0<V<7t)

with ? as in Lemma 1. Let 0 <v\ < v < v2 <n. Then assuming Al, for some

?>0and0<8<l:

(a) E{\\RJT(v2)-

RJT(v)f\\RJT(v)-

R^f)< K(v2

-v\)2~8,

(32).7

=1,2.

(b) RJT(v) -i eV(0,4tt2V(^(i;)), 7= 1,2,

where V(l)(v)=

a4$ r(u)rf(u)du/n + o4k$ r(u)du$ ?\u)du/n2 and

V(2)(v)= a4 f* ?(u)?f(u)du/7t+cr4K f* ?(u)du f* ?f(u)du/n2, with k denot

ing the fourth cumulant of{?(t)/a}tez.

PROOF. We begin with (a).We shall considerRj(v) only, R\(v) being simi

larly handled. From the definition of r\p, and

2n [Tv2/n]

R2(V)-R2(v2)=

j?n Y, hip*p=[Tv/jz]+\

we have that

R2T(v)-

R2T(v2)=

8hT(v, v2) + 82J(v, v2),

where ?\j(v,v2) and S2j(v,v2) are given in (29) and (30), respectively.Now (32) follows immediately from Lemma 6 and standard inequalities.

Part (b). We will examine RxT(v) S Jsi(0,4n2V(i)(v)), the proof for j=

2 being handled identically. But this follows by an obvious extension of Theorem 4.2

of [14] because ?(u) satisfies the same conditions on hn(u) there. D

LEMMA 8. Assume A1-A4. Then we have that, for some 0 < 8 < 1/6,

2n(IXj a2\_

2n / a2\<a>

J??2L, Sj\j?--2?)-T?? ?- ?JVe'J-2?)? 1 ,JJ=l

/ 2 [TV*]\

(33)-\j

?W^jjfWVr-eo)

+ Op -7 .









7=[n./jr]+l

r^V7T]+l

E^o,y)fl/2^-0o)

+oP(?),

2tt

7*1/2y=[7-?/7r]+i

V 7

\ j=[Tk/n]+\

where theOp(l/Ts)

terms are uniform in X e [0, 7t], and where ?(u) and ||f (u)\\

are as in Lemma 1.

Proof. We examine (a), part (b) being handled similarly. The difference be

tween the left-hand side of (33) and the first term on its right-hand side is

0 [TX/7T] j.

Tl/2 .t? "hoojlhorj(34)

+ -72jttT^M / j.. . x -,? [ /*]

7-1/2

First we notice that

(35) 9T-eo=

Op(T-V2),

which follows by (8) in assumption A4, and because

(recall that under Ho, hj=

ho0j), by Lemma 4 and Markov's inequality, and

T

s)

k=\

(37)

2tt d ( 1 fn

2fi/2 E^b?*/g?*~*

^l0'~

y (/)e0(u)^0o(u)du

=

/ (t>o0(u)Bn(du)Jo

by Lemma 7 with f(w)=

0??(m). Notice also that??=i00b,*= O (log7) by

Lemma 1 because (9) and A3 part (c) implies that (/>o0(X) satisfies the same condi

tions on ?(X) in Lemma 1.









Next, A3 part (d) implies that, uniformly in X e [0, n], the norm of the first term

of (34) isbounded by

[TX/n]

Ixj(38) KTV2\\eT-9o\\2~ J2 \^g2Xj\Uj\\-^ = Op(T-^2),Tp h0oJ

because (35) implies that we can take S? KT~l/2 inA3 part (d) so thatXj8

< K

when 8 < KT~1/2 and j> 1, and also because by Markov's inequality and Lem

mas 4 and 7,

[TX/7T]

sup?g[0,tt]

=Op(T-x'2),

and because by Lemma 1with ||f (m)||| log {u)\ there,

[TX/n]j x

V I1022A,-Ill?-;IIT

upke[0,n]

~J] | og2Xj| HO I-- / |log2(M)|||C(?)||^

/ -. 7? JO7= 1

=o(T-1'2)

The second term of (34) isOp(T 8) by Lemma 4 and Markov's inequality. Next,

proceeding similarly as in (38), since ? (X)(pre (X) satisfies the same conditions

as r(X)\\ogX\, the third term of (34) is

T-{o2Y}J^n\j<t>oQjTl,2(?T

-%) +

Op(T~8),which concludes the proof. D

LEMMA 9. Assuming A\,for any 0 < v < (1?

8)/4, with 8 as in Lemma 1,we have that, for all k = 1,..., p,

-S^lfruTT) 8{^T(X2,n)^2 a)2-8-2v

(39) (a) E J - J<K(X2-XX

\ (n -X])v (n -X2)v )

/8^k)T(X\,iz) 8fUx2,n)\4

(40) (b) E( lT

K

?J

-

?'T ?J) SK(X2-XX)2-8-4"\ (t? -X\)v (tt -X2)v J

?(*).for all 0 < X\ < X2 < n, and where

8\Kj(X\, X2) and8^t(X\, X2) are given in

(29) and (30), respectively.

PROOF. We begin with (b). By standard inequalities the left-hand side of (40)is bounded by

K^O^?)^ ^^

-

^)V??a, *>)4.

By Lemma 6, for any 0 < 8 < 1,we have that the last displayed expression is

bounded by

(?2-Al)2_S / 1 1 \4 2(41) ,2-5









Consider the case X2?

X\< 2 1(tt

?X2) first. By the mean value theorem

(41) is

K(X2-Xx)2'8

(n-Xrfv

+K v4(X2-Xx)A

{TT-

XX)4"(tT-

X2)S+*"-2 (?(TT-

XX) + (1-

?)(TT-

?2))4"4"

<K(X2

-Xx)2~8~4v + K(tt

-X2)-8-4v~2(X2

-XX)\

where ?=

?(Xx,X2) e (0, 1), and then because tt?

Xx > X2?

Xx and tt ?Xx

>

tt?

X2 > 0. But the right-hand side of the last displayed inequality is bounded by

K(X2

-

Xx)2-8~4v since X2-Xx <2'1(tt

-

X2).Next, consider the case for which 2~l (tt

?X2) < X2

?Xx. Using the inequality

a??

h? <(a

?b)? for any 0 < ? < 1 and a>b,we have that (41) is bounded by

\4u/ x2-<5

K(X2-XX)+K~t-, ,4vi-, .av <K(X2-XX)

(tt?

X\rv(TT?

X2yv

where we have used 0 <X2?

Xx <tt?

Xx and tt ?X2 < 2(X2

?Xx). This completes

the proof of part (b).

Next part (a).By

definition and Al, the left-hand side of (39) is bounded by

K[Tk2M

T ?-" ^J

(k)

NJ=[TX2/tt]+\

<K(X2-Xx)2-8-2u

by Lemma 1, and then proceed as in part (b). D

In what follows we shall abbreviateYo,qA~^j(q) by Hej(q).

LEMMA 10. Assuming A1-A5, for all 6 > 0,

lim lim supPr

(42)

sup ? [T^]H9o,T(k)

f _2^i fl/2k=[TXoM+l

j=k+l \h0T,j 2*J> e = 0.









Proof. Abbreviateh^ -Ixj

-hj by xj and take ?o > n/2 without loss

of generality. Noting thath^ -Ixj

-a2/(2n)

=xj + r]j, where r?j

=Isj

-

a2/(2tt),we have

supXq<X<7T

1 [TX/n] rj (h. f

f _L fx/2 L Y<k>,j{Kjrij)k=[TX0/7T]+\ j=k+l

K T/ k \8/2

<f _E llH*.7-(*)?(i-i)(43)

^=[7-X0/3r]+l

sup

[[fX0/7T]<k<7

(l-k/T)

-8/2

+ sup[TXo/x]<k<T

y 1/2

(l-)t/f)-?/2

E KW*/

71/2j=k+\

for any 0 < 8 < 1.The first factor on the right-hand side of (43) is bounded by

K1

X- II I

k=[TX0/it]+l (i-iy/2"'<^(7-[y?/7r])

8/2

using

^)|-H)~'

because || o0(X)\\> K (n?

X) by assumption A5 and because Lemma 1 implies

that^V{TX,in]<k<T II^otW

-Ao0([kn/f])\\

=0(T~l log2T).

Next, by Lemma 9 the second term inside the braces on the right-hand side

of (43) isOp(\)

for 8 > 0 small enough, whereas Lemma 8 and (35) imply that

the first term is bounded by

(l-k/T)-8/2

sup

[Tk0/jT]<k<T

+ Op\ sup

\TXo/n]<k<T

J2 YOojt'oojJ=k+\

Op(\)

d-k/T)-8/2

f8

=Op(\n-X0\8/2),

because of T~l < f~x <inf[7?o/7r]<^<^(l

-k/f), 0 < 5 < 1, and an obvious

extension of Lemma 1 but with ?(k)=

yo0(X)(j)f0o(X)there. So, (43) is

Op(\n-

Xq\8), which implies that (42) holds because 8 > 0. D









LEMMA 11. Assuming A1-A6,

f

(44) sup

Ae[0,jr]

1

7-1/2

= n.(^T

?o\^)= [T\/7Z]+\

Proof. The expression inside the norm on the left-hand side of (44) is

1T

fl/2?-j

j=

[Tk/7T]+\ ooj[^-Iej)(OT-9o)

(45)1 a

+f?j2 E KjUj-^)(0t-00)j=[TK/it]+l

T 9

j=[TX/k]+\"T'J

By A6 and then noting that \a?

b\ < (a?

b) + 2b for a > 0 and b > 0, thenorm of the third termof (45) isbounded by

Wot-M2 T

K^-^ 7-1/2 ?ll0g(?7)l

<K

7= 1

||?r-?o||2

'*.,

Vj

a

2n

= 0

fl/2

/logT

pVri/2

?f>s<^-?+vf>^'

)by (35) and then using Lemmas 8 and 7 with f (A.)

=|log A.|, and Lemma 1, respec

tively. So, uniformly in X, the third term of (45) is op(\). Likewise, the first term

of (45) isOp(T~1/2) uniformly inX using Lemma 8with f (X)= <po0(X)nd (35).Observe that <po0(X) satisfies the same conditions as ?(X) in Lemma 8 by A6. Fi

nally, the second term of (45) isOp(T~xl2) by Lemma 7 with ?(X)

=<po0(X).

D

LEMMA 12. Assuming A1-A6, for all > 0,

[TXM

Eim lim supPr j supX0->7t T-+00 [x0<X<tt

(46)

HeTj(q)

q=[TX0/7T]+l

T\/2

> \= 0.









Proof. Notice that (35) implies that it suffices to show (46) in the set {\\6t?

001| <KT~l/2mj1},

where mT +mjXT~x/2

-> 0. On the other hand, Lemma 11

and then Lemma 8 imply that, uniformly in q,

(47)

T / 1 T \

\? E YeT,jXj(?T E ro0j<l>o0j)fl/2(Oo-OT)

+ Op(T-s),j=q+\ \ j=q+\ I

1 f 1 f

T72 Y<hJrlj TFx?E Y0oJlj+ Op(T 1/2),7-1/2

7=9+17-1/2

.

proceeding as in the proof of (44) but with xj + x]3 replaced by r\j there. Observe

that we can take A,o> n/2. Next, uniformly in q, A6 implies that

sup _\\AeT,T(q)-Aeoj(q)\\=(7T-Xo)Op(\\eT-9o\\),[Tk0/7t]<q<T

which will imply that, with probability approaching one, as T -+ oo,

\\A?T(q)\\<

|A?rte)|(l+ KT-Xl2m^)

<K(\-Vj,

because ||A#0(?)||>

?^-1(7r-

X) and Lemma 1 implies that

sup _\\Ao0,T(q)-Ao0([q7T/T])\\ = 0(T-1 log2T).[TX0/n]<q<T

So, we have that, for 0 < 8 < 1/2,

supf _2?i fl/2

q=[TXa/n}+\

K supXo<X<7T

[TX/jt]

(48)

^ _E n^o<?=[T?0/^]+1

E ^,7 7-X

ii(i qYl+s'

x { sup

\[TkoM<q<T 0-f)

-a/2

T-l/27=9+1

E ^0,^

+ O?(|^-?0|a/2)

by (47) and because T~l < f~l <^[Tx0/7z]<q<T(l ~ l/?)- But Lemma 9 implies that

sup

[TX0/7t]<q<T 7=9+1

=On(l),









and A3 implies that

1%n] ? ?A qVl+8/2̂ (T-YTXq/tt^8'2

sup -s L WyooA1_7 ^M-s-?

-q=[TX0/7T]+l

and, hence, the left-hand side of (48) isOp(\tt

?Xo\8/2). From here we conclude

that (46) holds because S> 0. D

7. Proofs. This section provides the proofs of the main results which are

based on the series of lemmas given in the previous section.

Proof of Theorem 1. Part (a) follows by Lemma 4 with f (X)= 1 there.

The proof of part (b) follows immediately from part (a) and Lemma 7 with

?(X)= l there. D

Proof of Theorem 2. Part (a).By Lemma 8 with ? (X)= 1 there and the

definitions of Gqt andGj,

we have that

T^2(Got,t(X)-G?t(X))

r2 [TX/7T]

(49)HyE ^oJ\fl/2(9T-eo)

+ op(l)

(a2[TX/n] \

2n=

-{TE

^,;j^-1G?o>r(7r)fl/2E^ 'he?<k

+ Op(l),

by (8) and (9), and where theop{\)

term is uniform in X e [0,7r]. Likewise,

(50)1/2(Ger,r(7r)-G^(7r))=

op(l)

because of (36) and (37) and, by Lemma 1with ?(X)=

<f>e0(X)nd (9),we have

||f_1 ?j=1 faoj II 0(T_1 logT). So, (50) holds. Also, it isworth noticing that

Lemma 1with ?(k)=

<?>00(A)<^o(A)mplies that \\ST Se0ll= 0(T~l log2T).

On the other hand, noting that (50) and Al imply that

(51)GT{7i)= o2 + Op{T-x'2),

and that

\Ge0,T(n)-

G0T(n)\=op(T-V2)








GOODNESS-OF-FIT FOR LINEAR PROCESSES

by Lemma 4, thenby (49), (50) and (36), uniformly inX,we obtain that

m on^^1/2(^jW-G?r(?))aeTj(X)

=aT-(X)-\

2603

(52)

G?T(jt)

+ G0t T(k)f x/2(-?\GeTj{Ti) G?T(n)

1 [TX/7T]

=4(A)- i

?; _i 27T L

Y?Y^^kh^ + op(l),

which concludes the proof of part (a).

Next part (b). Taking into account part (a), part (b) follows because Lemma 7

guarantees the fidi's convergence of a? and its tightness. Tightness of the second

term on the right-hand side of (52) follows by (37) and Lemma 1 and then be

causeJo (pe0(u)du is Holder's continuous of order greater than 1/2 by A3. This

concludes the proof of the theorem. D

Proof of Theorem 3. Using (51) and recalling thatHqj(j)=

y'e jAglT(j),we obtain that

[TX/n]

(53)

where theop(\)

term is uniform in X e [0, tt].

Suppose, to be shown later, that the convergence in [0, Xq] holds for any0 < Xo < tt. Then, because Bn and the limit of the process T~xl2

YS\=X\ (hj?

a2/2tt) are continuous in [0, tt], Billingsley's [3] Theorem 4.2 implies that it suf

fices to show that, for all e > 0,

lim lim supPr| sup

X0^n t^ooI X0<X<n

[TX/n]

Ej= [TXn/7T]+l

%r(i)fl/2

? Ve0,k(^Ie,k-yj=j+\

> 6 =0,

which follows by Lemma 10; compare the second term on the right-hand side

of (43).









So, to complete the proof, we need to show that, for any 0 < Xo < n,

(54)J

-Ho0j(j)~ ? Yeo,ky/^h,k-l)\=?^y2BM,

in [0, Xo]. Fidi's convergence follows by Lemma 7, part (b) after we note that the

second term on the right-hand side of (53) is

f I , kA[TX/7i] \2n

1

1 Z/l** $2nfTTjElf E

Heo,T(j)jY9o,k[^2kk

and(T~lJ2j=\ Ho0tT(j))yo0,k satisfies the same conditions of Lemma 7

for ?(X), for example, those of hn(X) in [14], Theorem 4.2. Then, it suffices to

prove tightness. Since a? is tight, we only need to show the tightness condition of

1UA/7rJ / 1 T( a2\\

(55) ArW=~ E %rO')l^ E

YOoty**-^))By Billingsley's [3] Theorem 15.6, it suffices to show that

E(|Ar(0)-

AT(?)\\AT(X)-

AT&)$< K\X

-?\28

for all 0 < ?jl< ? < X < n and some 8 > 1/2. Observe that we can take T~l <

\X?

/x|, since otherwise the last inequality is trivial. Because (X?

?)(??

?x) <

(X?

?jl)2, by the Cauchy-Schwarz inequality, it suffices to show the last displayed

inequality holds for E| At(X)?

A^(/x)|2, which is

[TX/n] f f

f3j,k=[T?/7t]+l ^1=7+ 1?2=*+l

xE[{l--"-?(,^-?]H^(k)K

[TX/n

? \\He0,T(j)\\\\H9o,T(k)\T2

j,k=[Tti,/n]+\

<K(\H(X)

-H(?)\2 + f-2log2 f ),

where

H(X):=tt-1 HeJx)dx and \\HT(k)-

H(k)\\ < KT~X logT./n









and

[TX/n]

HT(X)'.=f-x

EI|tfflb.r0')||

by Lemma 1. From here we conclude by Billingsley's [3] Theorem 15.6, because

H(X) is a monotonie, continuous and nondecreasing function such that \H(X)?

H(/i)\ < K\X-

?\8, 8 > 1/2 and f~x <\X- p\. D

Proof of Theorem 4. By definition of ?ej and?j,

it suffices to show that

(56)

and

.[TX/nj7 j1 x?v / lx,k

T\/2

1 /1 [7^] u _ 1 ? //xj G0tJ(tt)\\

(57)1

Gqtj(tt)

[TX/n

XU E Her.Hk)Ê ^H?T^^X,j Gqtj(tt)

2tt

converge to zero uniformly in X e [0, tt]. Expression (56) isop(\), uniformly in

X [0, tt], because the contribution due to the term in brackets in the last line

of (52), that is,-0^ j2tt(G?t(tt))-xSj

1f_1/2y!=x <t>e0,kh,k, is easily seen to be

zero. Next, because

x [TX/n] jf

isE llô^llllô?rWll^ II0,7 II*=1 j=k+\

1[ /*]

i-i

rW(l-f)[Â./7T]

K? E l^ll<**=1

by integrability of yo0 and || eQj(k)(\-

k/f)~x ||> 0 by A3 andA5, it impliesthat the contribution to (56) due to the term

op(\)on the right-hand side of (52) is

negligible.









Next we examine (57). Because of (50) and (51), it suffices to show that

1

(58)

[TX/7T]

E

in)

V- { *XJ a2^ YOoJ

[I-^J=k+i ^?tJ *>

?r,r(fc)v^ {III _ ?fl/2 ?- Y9T'J\hn 2:

j=k+i KneTj ^

He0j(k)

fl/2

HeTj{k)

2tt)

converges to zero uniformly in X e [0, n], after observing that

f

supk

[0,n]

[TX/tt] T [TX/jz]

J2 n0Tj{k)J2 YotJ- E HOo.r(k) Ye0Jk=\ j=k+\ k=\ j=k+\

= 0.

First, we observe that Lemmas 10 and 12 imply that it suffices to show the

uniform convergence in X e [0, ?o] for any Xo <tt. But (58) is equal to

1 [TX/7T] 1 f (I2\(59) f ? /fcr.rtfW E ^J-YeTA^--)=k+l

[TX/7T]

(60) +

1^/-"J i ^ / r ^z \

r Jk=l r 7=^1 U^7 2jr/

So, the theorem follows if (59) and (60) areop(l) uniformly in ? g [0, ?o].

To that end, we first show that

(61)

(62)

(63)

1 [Tk/n]

sup ~?

?G[0,7T]T j= lj-4>otj\\ =?p(l)>

sup\\A??}T(X)A. [0,A.0] 3,^ AZl(X)\\=o(l),

sup \\A0tJ(X)1 '^

AzK(X)\\=op(l).T** [(U0]

(61) follows proceeding as with the proof of (44) in Lemma 11, but without the

factorh?l ??xj

?cr2/(2n)', (62) follows because assumption A5 implies that

AoQ(Xo)> 0 and because, by assumption A3,

||0ob(A.)0^(A.)||satisfies the same

conditions on ? (X) in Lemma 1, so that

sup ||A0o(X)-

A0o,t(X)\\ = 0(T~l log2T)\A. [(Uo]

and (63) follows proceeding aswith theproof of (61) and (62).Now we show that (59) is

op(l) uniformly in ? g [0, ?o], which follows by

Lemma 11 and (61)-(63), noting that(y?oj

-

y?TJ)=

(0,0^,

-

4>eTj)>so








GOODNESS-OF-FIT FOR LINEAR PROCESSES

does (60) by (61) and (63) and noting that

1

2607

supA.

[0,ji\

7-1/2T\ /tt-1-LI

UT,J

=Op(l)

j=[TX/n]+\

by Lemmas 7 and 8 with ?(k)=

yo0(X) there and observing (35) and that by

Lemma 1,f~xETj=[fx/n]+] YOoj^j

-+fx Ye0W<i>e0(x)dx.

D

Proof of Theorem 5. Under HXt, we have that, by definition,

2n[f^]IXja2r ,

7= 1 ' 7= 1

+2lZT

J?J?

[TX/jt] j

7=1 VHj

Ixj ^2\[TX/jt]

2n)+T2 ^

*^ ?,

By Lemmas 1, 4 and 7 with f (?)=

rl(X), and because |5r| is integrable, we have

Ge0,T(k) T ? h.j +j^j

Ku)du+ op{T-xl2).7= 1

So, using (51)because

f? l(u)du =

0,we have

that, uniformlyin X e

[0,7r],

fl/2/G,0,r(A)AX

fl/2/2*

tf^r]_A.

_r_^\

+ op(\)

=ctr(X)+

-/ l(u)du + op(\).

tt Jo

From here the conclusion is straightforward. D

Acknowledgments. We thank an Associate Editor, two referees and Hira Koul

for their constructive comments on previous versions of this article which have led

to substantial improvement of the paper. Of course, all remaining errors are our

sole responsibility.

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M. A. Delgado

C. Velasco

Departamento de Econom?a

Universidad Carlos III de Madrid

C./Madrid 126-128

Getafe, 28903 Madrid

Spain

E-mail: [email protected]

[email protected]

J. Hidalgo

Department of Economics

London School of Economics

Houghton Street

London W2A 2AE

United Kingdom

E-mail: [email protected]

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