View
217
Download
0
Embed Size (px)
Citation preview
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 1/43
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
Stable URL: http://www.jstor.org/stable/25463388 .
Accessed: 27/06/2013 03:38
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp
.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].
.
Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve and extend access to The
Annals of Statistics.
http://www.jstor.org
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AMAll use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 2/43
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
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 3/43
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.
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 4/43
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
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 5/43
GOODNESS-OF-FIT FOR LINEAR PROCESSES 2571
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.
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 6/43
2572 M. A. DELGADO, J.HIDALGO AND C. VELASCO
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
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 7/43
GOODNESS-OF-FIT FOR LINEAR PROCESSES 2573
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).
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 8/43
2574 M. A. DELGADO, J.HIDALGO AND C. VELASCO
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.
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 9/43
GOODNESS-OF-FIT FOR LINEAR PROCESSES 2575
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.
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 10/43
2576 M. A. DELGADO, J.HIDALGO AND C. VELASCO
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)
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 11/43
GOODNESS-OF-FIT FOR LINEAR PROCESSES 2577
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.
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 12/43
2578 M. A. DELGADO, J.HIDALGO AND C. VELASCO
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.
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 13/43
GOODNESS-OF-FIT FOR LINEAR PROCESSES 2579
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))
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 14/43
2580 M. A. DELGADO, J.HIDALGO AND C. VELASCO
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'
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 15/43
GOODNESS-OF-FIT FOR LINEAR PROCESSES 2581
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
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 16/43
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.
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 17/43
GOODNESS-OF-FIT FOR LINEAR PROCESSES 2583
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.
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 18/43
2584 M. A. DELGADO, J.HIDALGO AND C. VELASCO
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.
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 19/43
GOODNESS-OF-FIT FOR LINEAR PROCESSES 2585
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
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 20/43
2586 M. A. DELGADO, J.HIDALGO AND C. VELASCO
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+
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 21/43
GOODNESS-OF-FIT FOR LINEAR PROCESSES 2587
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)
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 22/43
2588 M. A. DELGADO, J.HIDALGO AND C. VELASCO
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
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 23/43
GOODNESS-OF-FIT FOR LINEAR PROCESSES 2589
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
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 24/43
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?]+\
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 25/43
GOODNESS-OF-FIT FOR LINEAR PROCESSES 2591
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)
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 26/43
2592 M. A. DELGADO, J.HIDALGO AND C. VELASCO
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).
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 27/43
GOODNESS-OF-FIT FOR LINEAR PROCESSES 2593
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(^i-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/^Aif^/TT] \
iLzlLlL v f(*iM*2)\j2 \f L^ V V /
^ P= [T?/7T]+\'
9 [ i/tt] [f?2/7T]
JJL^ ^P\ L^i >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 ?(?
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 28/43
2594 M. A. DELGADO, J.HIDALGO AND C. VELASCO
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
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 29/43
GOODNESS-OF-FIT FOR LINEAR PROCESSES 2595
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 .
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 30/43
2596 M. A. DELGADO, J.HIDALGO AND C. VELASCO
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.
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 31/43
GOODNESS-OF-FIT FOR LINEAR PROCESSES 2597
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
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 32/43
2598 M. A. DELGADO, J.HIDALGO AND C. VELASCO
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.
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 33/43
GOODNESS-OF-FIT FOR LINEAR PROCESSES 2599
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
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 34/43
2600 M. A. DELGADO, J.HIDALGO AND C. VELASCO
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.
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 35/43
GOODNESS-OF-FIT FOR LINEAR PROCESSES 2601
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),
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 36/43
2602 M. A. DELGADO, J.HIDALGO AND C. VELASCO
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)
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 37/43
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).
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 38/43
2604 M. A. DELGADO, J.HIDALGO AND C. VELASCO
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
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 39/43
GOODNESS-OF-FIT FOR LINEAR PROCESSES 2605
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)^E ^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 ll^o^llll^o?rWll^ II0,7 II*=1 j=k+\
1[ /*]
i-i
rW(l-f)[^A./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.
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 40/43
2606 M. A. DELGADO, J.HIDALGO AND C. VELASCO
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
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 41/43
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.
REFERENCES
[1] Aki, S. (1986). Some test statistics based on the martingale term of the empirical distribution
function. A??. Inst. Statist. Math. 38 1-21. MR0837233
[2] Anderson, T. W. (1997). Goodness-of-fit tests for autoregressive processes. /. Time Ser. Anal.
18 321-339. MR1466880
[3] BILLINGSLEY, P. (1968). Convergence of Probability Measures. Wiley, New York.
MR0233396
[4] Bloomfield, P. (1973). An exponential model for the spectrum of a scalar time series. Bio
metrika 60 217-226. MR0323048
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 42/43
2608 M. A. DELGADO, J.HIDALGO AND C. VELASCO
[5] Box, G. E. P. and Pierce, D. A. (1970). Distribution of residual autocorrelations in
autoregressive-integrated moving average time series models. /. Amer. Statist. Assoc. 65
1509-1526. MR0273762
[6] Brillinger, D. R. (1981). Time Series, Data Analysis and Theory, 2nd ed. Holden-Day, San
Francisco. MR0595684
[7] BROCKWELL, P. J. and DAVIS, R. A. (1991). Time Series: Theory and Methods, 2nd ed.
Springer,New York.MR1093459
[8] Brown, R. L., Durbin, J. and Evans, J. M. (1975). Techniques for testing constancy of re
gression relationships over time (with discussion). J. Roy. Statist. Soc. Ser. B 37 149-192.
MR0378310
[9] Chen, H. and ROMANO J. P. (1999). Bootstrap-assisted goodness-of-fit tests in the frequency
domain. J. Time Ser. Anal. 20 619-654. MR1749578
[10] DAHLHAUS, R. (1985). On the asymptotic distribution of Bartlett'sUp-statistic.
/. Time Ser.
Anal. 6 213-227.
[11] Delgado, M. A. and Hidalgo, J. (2000). Bootstrap goodness-of-fit test for linear processes.
Preprint, Universidad Carlos III de Madrid.
[12] Durbin, J., Knott, M. and Taylor, C. C. (1975). Components of Cram?r-von Mises sta
tistics. II. J. Roy. Statist. Soc. Ser. B 37 216-237. MR0386136
[13] Eubank, R. L. and LaRiccia, V. N. (1992). Asymptotic comparison of Cram?r-von Mises
and nonparametric function estimation techniques for testing goodness-of-fit. Ann. Statist.
20 2071-2086. MR1193326
[14] Giraitis, L., Hidalgo, J. and Robinson, P. M. (2001). Gaussian estimation of parametric
spectral density with unknown pole. Ann. Statist. 29 987-1023. MR1869236
[15] Giraitis, L. and Surgailis, D. (1990). A central limit theorem for quadratic forms
instrongly dependent
linear variables and itsapplication
toasymptotic normality
of
Whittle's estimate. Probab. Theory Related Fields 86 87-104. MR1061950
[16] Grenander, U. (1950). Stochastic processes and statistical inference. Ark. Mat. 1 195-277.
MR0039202
[17] Grenander, U. (1981). Abstract Inference. Wiley, New York. MR0599175
[18] Grenander, U. and Rosenblatt, M. (1957). Statistical Analysis of Stationary Time Series.
Wiley, New York.MR0084975
[19] Hainz, G. and Dahlhaus, R. (2000). Spectral domain bootstrap tests for stationary time
series. Preprint.
[20] HANNAN, E. J. (1973). The asymptotic theory of linear time-series models. J. Appl. Probability
10 130-145. MR0365960
[21] HONG, Y. (1996). Consistent testing for serial correlation of unknown form. Econometrica 64
837-864. MR1399220
[22] HOSKING, J. R. M. (1984). Modeling persistence in hydrological time series using fractional
differencing. Water Resources Research 20 1898-1908.
[23] HOSOYA, Y (1997). A limit theory for long-range dependence and statistical inference on
related models. Ann. Statist. 25 105-137. MR1429919
[24] Kac, M. and Siegert, A. J. F. (1947). An explicit representation of a stationary Gaussian
process A??. Math. Statist. 18 438-442. MR0021672
[25] Khmaladze, E. V. (1981). A martingale approach in the theory of goodness-of-fit tests. The
ory Probab. Appl. 26 240-257. MR0616619
[26] Khmaladze, E. V. and Koul, H. (2004). Martingale transforms goodness-of-fit tests in
regression models. Ann. Statist. 32 995-1034. MR2065196
[27] KL?PPELBERG, C. and MlKOSCH, T. (1996). The integrated periodogram for stable processes.
Ann. Statist. 24 1855-1879. MR1421152
[28] KOUL, H. and Stute, W. (1998). Regression model fitting with long memory errors. J. Statist.
Plann. Inference 71 35-56. MR1651851
This content downloaded from 201.230.67.71 on Thu, 27 Jun 2013 03:38:29 AM
All use subject to JSTOR Terms and Conditions
7/28/2019 Miguel Delgado, Javier Hidalgo y Carlos Velasco - Distribution Free Goodness-Of-Fit Tests for Linear Processes
http://slidepdf.com/reader/full/miguel-delgado-javier-hidalgo-y-carlos-velasco-distribution-free-goodness-of-fit 43/43
GOODNESS-OF-FIT FOR LINEAR PROCESSES 2609
[29] KOUL, H. and STUTE, W. (1999). Nonparametric model checks for time series. Ann. Statist.
27 204-236. MR1701108
[30] Ljung, G. M. and Box, G. E. P. (1978). On a measure of lack of fit in time series models.
Biometrika 65 297-303.
[31] NEYMAN, J. (1937). "Smooth" test for goodness of fit. Skand. Aktuarietidskr. 20 150-199.
[32] Nik ab adze, A. and Stute, W. (1997). Model checks under random censorship. Statist.
Probab. Lett. 32 249-259. MR 1440835
[33] Paparoditis, E. (2000). Spectral density based goodness-of-fit tests for time series models.
Scand. J. Statist. 27 143-176. MR1774049
[34] Prewitt, K. (1998). Goodness-of-fit test in parametric time series models. J. Time Ser. Anal.
19 549-574. MR 1646250
[35] ROBINSON, P. M. (1994). Time series with strong dependence. In Advances in Econometrics:
Sixth World Congress 1 (C. A. Sims, ed.) 47-95. Cambridge Univ. Press. MR1278267
[36] Robinson, P. M. (1995). Log-periodogram regression of time series with long range depen
dence. Ann. Statist. 23 1048-1072. MR1345214
[37] ROBINSON, P. M. (1995). Gaussian semiparametric estimation of long-range dependence. Ann.
Statist. 23 1630-1661. MR1370301
[38] SCHOENFELD, D. A. (1977). Asymptotic properties of tests based on linear combinations of
the orthogonal components of the Cramer-von Mises statistic. Ann. Statist. 5 1017-1026.
MR0448698
[39] Sen, P. K. (1982). Invariance principles for recursive residuals. Ann. Statist. 10 307-312.
MR0642743
[40] SHORACK, G. R. and WELLNER, J. A. (1986). Empirical Processes with Applications to Sta
tistics. Wiley, New York. MR0838963
[41] Stute, W. (1997). Nonparametric model checks for regression. Ann. Statist. 25 613-641.
MR 1439316
[42] Stute, W., Thies, S. and Zhu, L. (1998). Model checks for regression: An innovation
process approach. Ann. Statist. 26 1916-1934. MR 1673284
[43] STUTE, W. and Zhu, L. (2002). Model checks for generalized linear models. Scand. J. Statist.
29 535-545. MR1925573
[44] VELASCO, C. (1999). Non-stationary log-periodogram regression J. Econometrics 91
325-371. MR1703950
[45] Velasco, C. and Robinson, P. M. (2000). Whittle pseudo-maximum likelihood estimation
for nonstationary time series. J. Amer. Statist. Assoc. 95 1229-1243. MR 1804246
[46] Velilla, S. (1994). A goodness-of-fit test for autoregressive-moving-average models based
on the standardized sample spectral distribution of the residuals /. Time Ser. Anal. 15
637-647. MR1312327
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]
J. Hidalgo
Department of Economics
London School of Economics
Houghton Street
London W2A 2AE
United Kingdom
E-mail: [email protected]