A simple test Of normality for time series

Abstract

This paper considers testing for normality for correlated data. The proposed test procedure employs the skewness-kurtosis test statistic, but studentized by standard error estimators that are consistent under serial dependence of the observations. The standard error estimators are sample versions of the asymptotic quantities that do not incorporate any downweighting, and, hence, no smoothing parameter is needed. Therefore, the main feature of our proposed test is its simplicity, because it does not require the selection of any user-chosen parameter such as a smoothing number or the order of an approximating model.We are very grateful to Don Andrews and two referees for useful comments and suggestions. We are especially thankful to a referee who provided a FORTRAN code. Lobato acknowledges financial support from Asociaci n Mexicana de Cultura and from Consejo Nacional de Ciencia y Tecnolog a (CONACYT) under project grant 41893-S. Velasco acknowledges financial support from Spanish Direcci n General de Ense anza Superior, BEC 2001-1270.

Full text

A SIMPLE TEST OF NORMALITY
FOR TIME SERIES
I
G
G
GN
N
NA
A
AC
C
CI
I
IO
O
O
N. L
O
O
OB
B
BA
A
AT
T
TO
O
O
Instituto Tecnológico Autónomo de México (ITAM)
C
A
A
AR
R
RL
L
LO
O
OS
S
S
V
E
E
EL
L
LA
A
AS
S
SC
C
CO
O
O
Institució Catalana de Recerca i Estudis Avançats
and
Universitat Autònoma de Barcelona
This paper considers testing for normality for correlated data+The proposed test
procedure employs the skewness-kurtosis test statistic,but studentized by stan-
dard error estimators that are consistent under serial dependence of the observa-
tions+The standard error estimators are sample versions of the asymptotic quantities
that do not incorporate any downweighting,and,hence,no smoothing parameter
is needed+Therefore,the main feature of our proposed test is its simplicity,because
it does not require the selection of any user-chosen parameter such as a smooth-
ing number or the order of an approximating model+
1. INTRODUCTION
There has been recent interest in testing for normality for economic and finan-
cial data+For instance,Bai and Ng ~2001!test for normality in a set of macro-
economic series,whereas Bontemps and Meddahi ~2002!emphasize financial
applications+Kilian and Demiroglu ~2001!present a variety of cases where test-
ing for normality is of interest for econometricians+These applications include
financial and economic ones where,for instance,assessing whether abnormal
financial profits or economic growth rates are normal is important for the spec-
ification of financial and economic models+They also present methodological
applications where testing for normality is a previous step for the design of
some tests,such as tests for structural stability or tests of forecast encompassing+
In econometrics,testing for normality is customarily performed by means of
the skewness-kurtosis test+The main reasons for its widespread use are its
We are very grateful to Don Andrews and two referees for useful comments and suggestions+We are especially
thankful to a referee who provided a FORTRAN code+Lobato acknowledges financial support from Asociación
Mexicana de Cultura and from Consejo Nacional de Ciencia y Tecnologìa ~CONACYT!under project grant
41893-S+Velasco acknowledges financial support from Spanish Dirección General de Enseñanza Superior,BEC
2001-1270+Address correspondence to:Ignacio N+Lobato,Centro de Investigación Económica,Instituto Tec-
nológico Autónomo de México ~ITAM!,Av+Camino Sta+Teresa 930,México D+F+10700,Mexico;e-mail:
ilobato@itam+mx;and Carlos Velasco,Dep+d’Economia i d’Història Econòmica,UniversitatAutònoma de Bar-
celona,08193 Cerdanyola dep Valles,Spain;e-mail:Carlos+Velasco@uab+es+
Econometric Theory,20,2004,671–689+Printed in the United States of America+
DOI:10+10170S0266466604204030
© 2004 Cambridge University Press 0266-4666004 $12+00 671

straightforward implementation and interpretation+The skewness-kurtosis test
statistic is the sum of the square of the sample skewness and the excess kurto-
sis coefficients properly standardized by their asymptotic variances in the white
noise case,6 and 24,respectively+Implementing the skewness-kurtosis test is
very simple because it compares the skewness-kurtosis test statistic against upper
critical values of a chi-squared distribution with two degrees of freedom ~x
2
2
!+
This test is typically applied to the residual series of dynamic econometric mod-
els ~see,e+g+, Lütkepohl,1991,Sect+4+5!+
In many empirical studies with time series data,the application of the
skewness-kurtosis test is questionable,though+The reason is that the previous
asymptotic variances are correct under the assumption that the model is cor-
rectly specified,implying that the sequence under examination is uncorre-
lated+However,on many occasions either the researcher might specify the model
incorrectly or might not even be interested in modeling the serial correlation+
In both cases,when the considered data are correlated,the asymptotic vari-
ances are no longer 6 and 24 but some functions of all the autocorrelations+In
this situation the skewness-kurtosis test is invalid because it does not control
asymptotically the type I error+
In this paper we propose to employ the standard test statistic based on the
sample skewness and sample kurtosis,but studentized by standard error esti-
mators that are consistent under serial dependence of the observations+The
standard error estimators are sample versions of the asymptotic quantities that
do not incorporate any downweighting,and,hence,no smoothing parameter is
needed+These standard error estimators are consistent even though the asymp-
totic standard errors involve infinite sums of terms that depend on all autocor-
relations+The reason is that in the expression of the asymptotic standard errors,
the autocorrelations enter raised to the cubic or fourth powers+Hence,the pow-
ers of the sample autocorrelations provide stochastic dampening factors,sim-
ilar to the nonstochastic dampening factors that appear in the standard
nonparametric approach+By contrast,Bai and Ng ~2001!and Bontemps and
Meddahi ~2002!rely on smoothing with kernel methods+
Our test can employ either frequency or time domain estimators of the asymp-
totic variances of the sample skewness and the sample excess kurtosis+Although
the proposed test is based on a time domain estimator,in the technical part of
the paper in the Appendixes we stress a frequency domain estimator because it
is relatively easier to handle theoretically+In addition,for conciseness of expo-
sition,we only analyze the univariate case+
The plan of the paper is the following+Section 2 presents the framework+
Section 3 introduces the proposed test statistic and studies its asymptotic theory+
Section 4 discusses the proposed variance estimators+Section 5 examines the
case where the considered series are the residuals of regression and time series
models+Section 6 considers the finite sample performance of the proposed test
in a brief Monte Carlo exercise+The technical material is included in the
Appendixes+
672 IGNACIO N. LOBATO AND CARLOS VELASCO

2. FRAMEWORK
Notation+Let x
t
be an ergodic strictly stationary process with mean mand
centered moments denoted by m
k
5E~x
t
2m!
k
for knatural,with [m
k
5
n
21
(
t51
n
~x
t
2Sx!
k
being the corresponding sample moments where Sxis the
sample mean and nis the sample size+In addition,g~ j!denotes the population
autocovariance of order j,g~ j!5E@~ x
1
2m!~ x
11j
2m!#,and [g~ j!is the
corresponding sample autocovariance,[g~ j!5n
21
(
t51
n26j6
~x
t
2Sx!~x
t16j6
2Sx!+
Notice that m
2
5g~0!+Let f~l! be the spectral density function of x
t
,defined by
g~ j!5
E
P
f~l!exp~ijl! dlj50,1,2,+++, (1)
where P5@2p,p#,and let I~l! denote the periodogram I~l! 56w~l!6
2
where w~l! 5~2pn!
2102
(
t51
n
x
t
exp~itl!+In addition,k
q
~j
1
,+++,j
q21
!denotes
the qth-order cumulant of x
1
,x
11j
1
,+++,x
11j
q21
,and the marginal cumulant of
order qis k
q
5k
q
~0,+++,0!+
Null and alternative hypotheses. The null hypothesis of interest is that the
marginal distribution of x
t
is normal+For the independent case,omnibus tests
for this null hypothesis such as the Shapiro–Wilk test ~Shapiro and Wilk,1965!,
which is based on order statistics,or tests based on the distance between the
empirical distribution function and the normal cumulative distribution function
such as the Kolmogorov–Smirnov,the Cramér–von Mises,or the Anderson–
Darling test have been proposed+A test based on L
2
distance between Gaussian
and empirical characteristic functions has been introduced by Epps and Pulley
~1983!and developed by Henze and others+For more details see Mardia ~1980!,
Henze ~1997!,Epps ~1999!,and references therein+For the independent case,
the omnibus tests are consistent,but it has been shown that their finite sample
performance can be very poor ~see,e+g+, Shapiro,Wilk,and Chen,1968!+For
the weak dependent case,no such analysis exists because inference with these
omnibus test statistics is problematic as a result of the fact that their asymptotic
distributions are nonstandard and case dependent+Hence,the standard applica-
tion of these tests to weak dependent time series sequences is invalid ~see Gleser
and Moore,1983!+The only developed test of which we are aware is the one
by Epps ~1987!that is based on the characteristic function+However,Epps’s
procedure is hard to implement because sensing functions g~x
t
,v!have to be
selected,a joint spectral density of sensing functions has to be found,a matrix
has to be inverted,and a quadratic form has to be minimized to estimate the
marginal mean and variance+In addition,there is the disadvantage of having to
choose the parameters vthat enter g~{,v!+
In practice,instead of the previous omnibus tests,the common procedure
just tests whether the third and fourth marginal moments coincide with those
of the normal distribution+Equivalently,in terms of the cumulants,it is tested
that the third and fourth marginal cumulants are zero instead of testing that all
SIMPLE TEST OF NORMALITY FOR TIME SERIES 673

higher order marginal cumulants are zero+We follow this practice,and in this
paper we test that the marginal distribution is normal by testing that m
3
50
and m
4
53m
2
2
+Of course,the derived tests are not consistent because they
cannot detect deviations from normality that are not reflected in the third or
fourth moments+
The skewness-kurtosis test. This test compares the skewness-kurtosis test
statistic
SK 5n[m
3
2
6[m
2
3
1n~[m
4
23[m
2
!
2
24 [m
2
4
against upper critical values of a x
2
2
distribution ~see Bowman and Shenton,
1975!+Apart from the fact that Jarque and Bera ~1987!have shown the opti-
mality of this test within the Pearson family of distributions,the popularity of
this approach resides in its simplicity as we mentioned previously+In fact,now-
adays most econometrics packages customarily report the SK test,which is called
the Jarque–Bera test+
The SK test procedure is justified on the following grounds+When the con-
sidered series x
t
is an uncorrelated Gaussian process,the following limiting
result holds:
M
nS[m
3
[m
4
23[m
2
2
Dr
d
NS6m
2
3
0
024m
2
4
D
,(2)
where r
d
denotes convergence in distribution+However,when x
t
is a Gauss-
ian process satisfying the weak dependent condition
(
j50
`
6g~ j!6,`, (3)
the result ~2!is replaced by
M
nS[m
3
[m
4
23[m
2
2
Dr
d
NS6F
~3!
0
024F
~4!
D
,(4)
where
F
~k!
5(
i52`
`
g~i!
k
,(5)
for k53,4~see Lomnicki,1961;Gasser,1975!+Notice that condition ~3!guar-
antees that all F
~k!
are well defined because it entails that (6g~ j!6
r
,`, for
all natural r+
Hence,when the series exhibits serial correlation,the SK test is invalid because
the denominators of its components do not estimate consistently the true asymp-
674 IGNACIO N. LOBATO AND CARLOS VELASCO

totic variances in ~4!,implying that asymptotically its rejection probabilities do
not coincide with the desired nominal levels under the null hypothesis+
3. THE GENERALIZED SKEWNESS-KURTOSIS TEST
In the previous section we have seen that the SK test is invalid when the con-
sidered process x
t
exhibits serial correlation+One strategy to overcome this prob-
lem is to carry out a two-step test where the SK procedure is applied after testing
that the considered series is uncorrelated+However,this solution is not simple
because there is an obvious pretest problem in such a sequential procedure and,
furthermore,testing for uncorrelatedness for non-Gaussian series is rather chal-
lenging ~see Lobato,Nankervis,and Savin,2002!+
Looking at ~4!two natural solutions appear+The first one consists of modi-
fying the SK test statistic by including consistent estimators of F
~3!
and F
~4!
in
the denominators of its components+This solution is proposed by Gasser ~1975,
Sect+6!,who suggested truncating the infinite sums that appear in the asymp-
totic variances+However,he did not provide any formal analysis or any recom-
mendation about the selection of the truncation number+As we will see,our
proposed procedure overcomes these difficulties because it does not require the
selection of any truncation number+The second solution estimates the unknown
asymptotic variances with the bootstrap;that is,it employs the SK test statistic
with bootstrap-based critical values+Implementing the bootstrap in a time series
context is problematic because generally valid bootstrap procedures require the
introduction of an arbitrary user-chosen number,typically a block length ~see,
e+g+, Davison and Hinkley,1997,Ch+8!+Therefore in this paper we follow the
first approach+Furthermore,in our case the bootstrap does not present a clear
theoretical advantage because the SK statistic is not asymptotically pivotal+
Before introducing our test statistic,let us consider the following estimator
of F
~k!
,which is the sample analog of ~5!:
ZF
~k!
5(
j512n
n21
[g~ j!
k
+(6)
In the next section we consider alternative versions of this estimator and study
their large sample properties;in particular,Lemma 1 establishes the consis-
tency of ZF
~k!
for F
~k!
for Gaussian processes that satisfy condition ~3!+Then,
our proposed test statistic,the generalized SK statistic,is
G5n[m
3
2
6ZF
~3!
1n~[m
4
23[m
2
!
2
24 ZF
~4!
+
The Gstatistic does not require the introduction of any user chosen number,
and,in view of ~4!and Lemma 1 in the next section,the proposed test con-
sists of comparing the Gtest statistic against upper critical values from a x
2
2
distribution+
SIMPLE TEST OF NORMALITY FOR TIME SERIES 675

In the next assumption we introduce the class of processes under the alter-
native hypothesis for which both EF
~k!
and ZF
~k!
converge to bounded positive
constants,and hence whenever m
3
Þ0orm
4
Þ3m
2
2
,the Gtest rejects with
probability tending to 1 as ntends to infinity+Notice that the conditions of
Gasser ~1975!that involve summability conditions of cumulants of all orders
are relaxed to cumulants up to order 16 using an extension of Theorem 3 in
Rosenblatt ~1985,p+58!+
Assumption A+The process x
t
satisfies Ex
t16
,`, and,for q52,3,+++,16,
(
j
1
52`
`
+++ (
j
q21
52`
`
6k
q
~j
1
,+++,j
q21
!6,`, (7)
and,for k53,4,
(
j51
`
@E6~E~x
0
2m!
k
6I
2j
!2m
k
6
2
#
102
,`, (8)
where I
2j
denotes the s-field generated by x
t
,t#2j,and,for k53,4,
E@~ x
0
2m!
k
2m
k
#
2
12(
j51
`
E~@~x
0
2m!
k
2m
k
#@~x
j
2m!
k
2m
k
#! .0+(9)
Assumption A is a weak dependent assumption that implies that the higher
order spectral densities up to the sixteenth order are bounded and continuous+
For the case q52,expression ~7!implies that condition ~3!holds+We require
finite moments up to the sixteenth order because we need to evaluate the vari-
ance of the fourth power of the sample autocovariances+Notice that condition
~9!assures that the asymptotic variances of estimates are positive+
The following theorem establishes the asymptotic properties of the Gtest+
THEOREM 1+
(i) Under the null hypothesis and for Gaussian processes that satisfy con-
dition (3), G r
d
x
2
2
+
(ii) Under Assumption A, the test statistic G diverges to infinity whenever
m
3
Þ0or m
4
Þ3m
2
2
.
The asymptotic null distribution is straightforward to derive given the con-
sistency of ZF
~k!
for F
~k!
that is proved in Lemma 1 in the next section+The
proof of ~ii!is omitted because it follows easily using that under the alternative
hypothesis ZF
~k!
converges to a bounded positive constant ~by ~7!and ~9!!,
whereas the numerator of Gdiverges+
676 IGNACIO N. LOBATO AND CARLOS VELASCO

4. CONSISTENT VARIANCE ESTIMATORS
Following the literature on nonparametric estimation of asymptotic covariance
matrices,the standard approach to estimate F
~k!
consistently employs a smoothed
estimator such as
(
j512n
n21
w
j
[g~ j!
k
+(10)
In ~10!the weights $w
j
%are usually obtained through a lag window $w
j
5
w~j0M!% such that the weight function w~{! verifies some regularity proper-
ties and Mis a smoothing number that grows slowly with n+Note that the
introduction of the smoothing number leads to estimators whose rate of con-
vergence is usually slower than the parametric rate+We stress that in this
approach the weights $w
j
%provide a nonstochastic dampening on the [g~ j!
k
for large j+Because of this dampening,the estimator in ~10!is consistent for
~5!as it happens in the case k51,where f~0!is consistently estimated by
autocorrelation robust estimators ~see,e+g+, Robinson and Velasco,1997!+
As mentioned in the introduction,the main problem with the smoothing
approach is that statistical inference can be very sensitive to the selection of
the user-chosen weights;in our context,the discussion in Section I in Robin-
son ~1998!is especially relevant+In the absence of a clear and rigorously jus-
tified procedure to select the smoothing number in our testing framework,we
prefer to analyze estimators that do not require any smoothing+
Our first estimator ZF
~k!
,introduced in equation ~6!,also admits a frequency
domain version ~see Appendix A!+For technical reasons,in this paper we con-
sider a second estimator that can be motivated by writing F
~k!
in terms of the
spectral density function of the x
t
process using ~1!:
F
~k!
5(
j52`
`
g~ j!
k
5(
j52`
`
)
h51
k
H
E
P
f~v
h
!exp~ijv
h
!dv
h
J
52p
E
P
k21
f~v
1
1{{{ 1v
k21
!)
h51
k21
$f~v
h
!dv
h
%+(11)
The sample analog of the previous equation renders the following alternative
estimator for F
~k!
:
EF
~k!
5~2p!
k
n
k21
(
j
1
51
n21
+++ (
j
k21
51
n21
I~l
j
1
!+++I~l
j
k21
!I~l
j
1
1{{{ 1l
j
k21
!,(12)
where l
j
52pj0n+The estimator EF
~k!
can also be written in the time domain
by plugging
I~l
j
!51
2p(
t512n
n21
exp~itl
j
![g~t!,jÞ0,mod n,(13)
SIMPLE TEST OF NORMALITY FOR TIME SERIES 677

into equation ~12!+After some algebra,in Appendix A it is shown that
EF
~k!
5(
t512n
n21
[g~t!$ [g~t!1[g~n26t6!%
k21
+(14)
Notice that both expressions for EF
~k!
are numerically identical,but in the Appen-
dixes,for technical reasons,we stress the frequency domain version ~12!+Expres-
sion ~12!guarantees that EF
~k!
is positive in finite samples+
The next lemma states the consistency of EF
~k!
and ZF
~k!
for F
~k!
+This lemma
is the substantive technical contribution of the paper+Its proof is inAppendix B+
LEMMA 1+Under the null hypothesis, for Gaussian time series that satisfy
condition (3),
(i) EF
~k!
5F
~k!
1o
p
~1!and
(ii) ZF
~k!
2EF
~k!
5o
p
~1!for k 53,4.
At first look,consistency of ZF
~k!
and EF
~k!
could be surprising because no
smoothing parameter has been introduced+Robinson ~1998!analyzes a special
regression model where smoothing is not necessary for establishing consis-
tency of asymptotic covariance matrix estimators+The reason is that the spe-
cific form of the covariance matrix that he considers ~see his equation ~1+2!!
allows for a stochastic dampening of some sample autocovariances by other
sample autocovariances+The time domain versions ~6!and ~14!provide a sim-
ilar intuition where the powers of the sample autocovariances provide the sto-
chastic dampening factors+
In the frequency domain,~11!provides a complementary explanation+Recall
that in time series the standard problem is that the relevant asymptotic variance
depends on the spectral density function evaluated at a unique point,typically
the zero frequency,f~0!+However,in our case ~11!shows that the asymptotic
variance,F
~k!
,is a convolution of the spectral density function,instead of a
single value+Intuitively,in the first case a user-chosen smoothing number is
required to estimate the local quantity,f~0!,whereas in our case no such num-
ber is needed because we are estimating a global quantity+
5. RESIDUAL TESTING
The previous sections analyze the case where raw data are under examination+
However,in practice the test is commonly applied to the residuals of regres-
sion or time series models+Again,two approaches can be used:first,the Gtest
that we propose and,second,employing the SK statistic with bootstrap-based
critical values+The bootstrap has been employed by Kilian and Demiroglu
~2000!+However,as mentioned in Section 3,application of the bootstrap is not
an obvious task in a time series context+Kilian and Demiroglu perform a para-
metric bootstrap that could be justified if the model were correctly specified,
although in this case the SK test would also be asymptotically valid+However,
678 IGNACIO N. LOBATO AND CARLOS VELASCO

in the absence of the knowledge of the true data generating process,a paramet-
ric bootstrap is invalid;that is,there is no guarantee that the type I error is
controlled properly asymptotically+As mentioned previously,bootstrap proce-
dures valid for time series require the introduction of a user-chosen number,
typically a block number,complicating statistical inference in finite samples+
Next,we introduce a general assumption that validates the use of the Gsta-
tistic applied to the residuals of many dynamic econometric models where the
correlation structure is not correctly specified or it is not specified at all+In this
section,[x
t
denotes the residuals of the regression or time series model,and x
t
denotes the true disturbances+
Assumption B+Let the Gaussian process x
t
satisfy ~3!and let e
t
5x
t
2[x
t
satisfy
(
t51
n
e
t2
5O
p
~1!and (
t51
n
e
t4
5o
p
~n
2104
!+(15)
The first condition in ~15!guarantees the consistency of the estimates of F
~k!
based on residuals,whereas the second guarantees that the residual SK test has
the same asymptotic distribution as the original SK test+Assumption B is very
general and covers many interesting cases such as linear regressions with pos-
sible trending stochastic and deterministic regressors that satisfy Grenander’s
conditions and weakly dependent errors+In this case e
t
5~Zb2b!
'
Z
t
,where
Z
t
isap-dimensional sequence of regressors,so ~15!implies that (
t51
n
e
t2
5
~Zb2b!
'
Z
'
Z~Zb2b! 5O
p
~1!,allowing for the components of Zbto have
different convergence rates+A leading example with stochastic Z
t
is a regres-
sion between cointegrated variables+For stationary Z
t
,another interesting
application is when [x
t
are the residuals obtained through possibly misspecified
AR~p!regressions;that is,[x
t
5y
t
2Zb
'
Z
t
with Z
t
5~y
t21
,+++,y
t2p
!
'
,
and
M
n~Zb2b! 5O
p
~1!for some vector bsuch that the polynomial b~v! 5
12(
j51
p
b
j
v
j
has no roots on or inside the unit circle+For this case,if
Assumption B holds for y
t
,the limit process x
t
5y
t
2b
'
Z
t
5b~L!y
t
inherits
the weak dependence properties of y
t
,but notice that x
t
is autocorrelated unless
y
t
follows an AR~q!process with q#p+
In Appendix C we prove the following lemma,which shows that the use of
residuals does not affect the consistent studentization that we propose in this
paper+
LEMMA 2+Under the null hypothesis and Assumption B, for k 53,4,
(
12n
n21
[g
[x
~j!
k
5(
12n
n21
[g
x
~j!
k
1o
p
~1!+
Finally,using the previous lemma and Hölder’s inequality,it is straight-
forward to prove the next theorem,which establishes that the asymptotic null
distribution of the Gtest statistic applied to the residuals of many dynamic
SIMPLE TEST OF NORMALITY FOR TIME SERIES 679

econometric models whose correlation structure is ignored or misspecified is
still x
2
2
and that whenever m
3
Þ0orm
4
Þ3m
2
2
the Gtest rejects with prob-
ability tending to 1 as ntends to infinity+
THEOREM 2+Let ZG be the test statistic G calculated from residuals [x
t
+
(i) Under the null hypothesis and Assumption B, ZGr
d
x
2
2
+
(ii) If m
3
Þ0or m
4
Þ3m
2
2
and Assumptions A and B hold,then the test
statistic ZG diverges to infinity.
6. FINITE SAMPLE PERFORMANCE
This section compares briefly the finite sample behavior of the previous tests
with the Epps ~1987!test+Under the null hypothesis we generate data from an
AR~1!process x
t
5fx
t21
1«
t
,where «
t
is independent and identically distrib-
uted N~0,1!and the autoregressive parameter ftakes eight values:20+9,20+5,
0,0+5,0+6,0+7,0+8,and 0+9+We report the results for a detailed grid of positive
values of fbecause positive autocorrelation is particularly relevant for many
empirical applications+
Along with the null hypothesis,we consider also testing the null that the
skewness is zero by using the first components of the SK and Gstatistics+
Namely,we compute the skewness test statistic S5n[m
3
2
06[m
2
3
and the gener-
alized skewness test statistic GS 5n[m
3
2
06ZF
~3!
and compare them with upper
critical values from a x
1
2
+We have not reported the results of a kurtosis test
because of the well-known slow convergence of the sample kurtosis to the
normal asymptotic distribution even in the white noise case ~see,e+g+, Bow-
man and Shenton,1975,p+243!+In Tables 1A and 1B we report the empirical
rejection probabilities for the tests for three sample sizes,n5100,500,and
1,000,and three nominal levels,a50+10,0+05,and 0+01+In these experi-
ments 5,000 replications are carried out+
The main conclusions derived from Table 1A are the following+For the case
of testing symmetry,the Stest is not reliable since it severely underrejects for
the cases when f,0 and substantially overrejects for the cases when f.0+
This result could be expected because when fis negative,(
j51
`
g
j3
is negative,
leading to overestimation of the asymptotic variance and then to underrejection
of the Stest,whereas when fis positive the opposite effect occurs+The most
interesting evidence is the magnitude of these distortions,which are very large
for negative values of fand all sample sizes,whereas for positive fthe dis-
tortions are increasing steadily with the sample size+On the contrary,for the
GS test the empirical rejection probabilities are very close to the nominal lev-
els for all the parameter values and all sample sizes ~the only exception is when
n5100 and f50+9!+
Table 1B reports the results for testing normality for the three tests,SK,G,
and Epps test+The SK test,which is the sum of the skewness test and the kur-
680 IGNACIO N. LOBATO AND CARLOS VELASCO

Table 1. Empirical rejection probabilities for 3 sample sizes and 3 nominal
levels
n100 500 1,000
f0+10 0+05 0+01 0+10 0+05 0+01 0+10 0+05 0+01
A+Sand GS tests
20+9S0+001 0+000 0+000 0+000 0+000 0+000 0+000 0+000 0+000
GS 0+083 0+038 0+008 0+097 0+047 0+012 0+091 0+045 0+010
20+5S0+056 0+025 0+005 0+063 0+027 0+005 0+064 0+026 0+005
GS 0+093 0+047 0+011 0+103 0+052 0+010 0+101 0+051 0+009
0S0+092 0+047 0+011 0+105 0+056 0+009 0+104 0+053 0+012
GS 0+097 0+051 0+012 0+105 0+056 0+010 0+104 0+054 0+013
0+5S0+117 0+064 0+019 0+146 0+080 0+023 0+152 0+090 0+025
GS 0+095 0+048 0+012 0+100 0+052 0+011 0+105 0+054 0+012
0+6S0+140 0+081 0+026 0+175 0+107 0+037 0+181 0+109 0+039
GS 0+094 0+046 0+011 0+097 0+050 0+009 0+098 0+052 0+009
0+7S0+176 0+109 0+039 0+233 0+158 0+061 0+231 0+157 0+069
GS 0+091 0+046 0+010 0+099 0+049 0+011 0+097 0+049 0+009
0+8S0+228 0+153 0+065 0+315 0+231 0+116 0+321 0+238 0+123
GS 0+089 0+045 0+009 0+098 0+049 0+011 0+092 0+049 0+012
0+9S0+278 0+196 0+090 0+442 0+361 0+238 0+467 0+389 0+261
GS 0+067 0+029 0+005 0+090 0+047 0+013 0+096 0+049 0+010
B+SK,G,and Epps tests
20+9SK 0+078 0+034 0+009 0+229 0+157 0+072 0+261 0+195 0+106
G0+031 0+014 0+004 0+062 0+038 0+013 0+067 0+038 0+012
E0+172 0+109 0+046 0+121 0+072 0+018 0+126 0+068 0+015
20+5SK 0+051 0+032 0+015 0+079 0+043 0+016 0+082 0+044 0+010
G0+065 0+039 0+014 0+090 0+047 0+014 0+095 0+047 0+011
E0+118 0+063 0+020 0+109 0+059 0+013 0+104 0+055 0+011
0SK 0+069 0+045 0+021 0+094 0+048 0+014 0+095 0+047 0+014
G0+070 0+045 0+021 0+094 0+048 0+014 0+095 0+048 0+014
E0+123 0+067 0+020 0+099 0+055 0+013 0+106 0+055 0+010
0+5SK 0+080 0+050 0+023 0+120 0+071 0+025 0+138 0+082 0+026
G0+063 0+040 0+015 0+084 0+045 0+014 0+094 0+053 0+014
E0+130 0+069 0+021 0+113 0+061 0+017 0+101 0+054 0+014
0+6SK 0+093 0+058 0+025 0+157 0+095 0+036 0+170 0+104 0+039
G0+056 0+035 0+015 0+079 0+045 0+017 0+088 0+047 0+013
E0+139 0+079 0+022 0+128 0+070 0+017 0+110 0+060 0+014
0+7SK 0+114 0+075 0+036 0+221 0+142 0+063 0+238 0+158 0+067
G0+054 0+033 0+012 0+076 0+043 0+017 0+081 0+046 0+015
E0+158 0+087 0+026 0+134 0+078 0+019 0+121 0+064 0+011
0+8SK 0+168 0+103 0+047 0+329 0+236 0+114 0+356 0+266 0+132
G0+045 0+026 0+009 0+064 0+042 0+018 0+075 0+041 0+016
E0+185 0+115 0+036 0+141 0+081 0+024 0+120 0+060 0+016
0+9SK 0+267 0+154 0+062 0+549 0+440 0+265 0+585 0+489 0+323
G0+027 0+015 0+006 0+056 0+036 0+013 0+067 0+043 0+013
E0+236 0+155 0+064 0+180 0+110 0+033 0+149 0+084 0+026
Note: Data follow a Gaussian AR~1!process with parameter f+Sample size is denoted by n+
SIMPLE TEST OF NORMALITY FOR TIME SERIES 681

tosis test,inherits their characteristics+Notice that for the cases where f,0,
there is a fair amount of compensation between the skewness and kurtosis,mak-
ing the distortions of the SK test much smaller than those of its components+
The Gtest inherits the slow convergence from the kurtosis,but using the white
noise case as benchmark,it appears to be robust to the presence of moderate
serial correlation+When 6f650+9,the Gtest is severely affected by its kurto-
sis component+In fact,even for n51,000 the Gtest appears to be very con-
servative+For the cases f50+7 and f50+8,a similar pattern can be observed+
Similar to the Gtest,the Epps test is also insensitive to moderate serial corre-
lation+However,for the case f520+9,and also for the most interesting cases
where f$0+7,the Epps test appears to be too liberal+
We also conducted power experiments for data generated by the previous
AR~1!model for six different distributions:standard log-normal,student’s t
with 10 degrees of freedom,x
1
2
,x
10
2
,beta with parameters ~1,1!,and beta with
parameters ~2,1!+Although distributions with bounded support are not that pop-
ular in econometrics,it is well known that in the independent and identically
distributed ~i+i+d!setting the SK test performs very poorly against such alterna-
tives+Hence,it is of interest to examine the performance of the Gtest in these
difficult cases+Table 2 reports the power results for the Gand the Epps tests
for three sample sizes,n5100,500,and 1,000,respectively,and for a 5%
nominal level+In these experiments 2,000 replications are carried out+The main
conclusions from these tables are the following+For both tests it appears that
the sign of the autocorrelation has little relevance in terms of power ~although
generally the empirical power is slightly greater for positive f!+Using the white
noise as the reference case,higher values for 6f6lead to a decrease in the empir-
ical power that in some cases is very exacerbated+The empirical rejection prob-
abilities for the Gtest are particularly high for heavily skewed distributions
such as the lognormal or the x
1
2
+For these cases the Gtest is clearly preferable
to the Epps test+When the distribution is symmetric or slightly skewed,both
tests are comparable+For the t
10
and the x
10
2
distributions,the Gtest presents
higher empirical power,especially for a moderate degree of serial correlation+
For these cases and when 6f650+9,the tests present very low empirical power
even for n51,000+Notice that when n51,000 and f50+7or0+8,for the x
10
2
case,the empirical powers of both the Epps test and,especially,the Gtest are
moderately high,but the power deteriorates suddenly for f50+9+For the beta
distributions,both tests ~and especially the Gtest!appear to be very sensitive
to a high degree of serial correlation+In fact,when 6f650+9,the power of
both tests is very low even when n51,000+Here again,there is a sudden
decrease in the empirical power when fincreases from 0+6to0+7 for n5500
and when fincreases from 0+7to0+8 for n51,000+
We end with a suggestion on further research+In this section we have seen
that for small sample sizes,because of the slow convergence of the sample
kurtosis coefficient,the Gtest presents significant size distortions even in the
white noise case+One potential way of improving the finite sample perfor-
682 IGNACIO N. LOBATO AND CARLOS VELASCO

mance is by using the bootstrap+Because the Gtest statistic is asymptotically
pivotal,it can be expected that application of the bootstrap will deliver an asymp-
totic refinement+Hence,it would be interesting to study the implementation of
the Gstatistic with bootstrap-based critical values+
Table 2. Empirical rejection probabilities at the 0+05 nominal levels for the G
and Epps ~E!tests for 3 sample sizes
fLog Nt
10
x
1
2
x
10
2
Beta~1,1!Beta~2,1!
n5100
20+9G0+291 0+045 0+175 0+048 0+006 0+035
E0+041 0+115 0+052 0+115 0+120 0+128
20+5G0+999 0+187 0+998 0+437 0+002 0+121
E0+673 0+043 0+843 0+200 0+557 0+484
0G10+299 1 0+798 0+511 0+740
E0+971 0+079 0+996 0+537 0+993 0+978
0+5G10+177 1 0+435 0+001 0+114
E0+865 0+059 0+953 0+215 0+532 0+544
0+6G0+992 0+122 0+985 0+312 0+001 0+064
E0+612 0+057 0+782 0+121 0+257 0+270
0+7G0+936 0+080 0+899 0+188 0+004 0+040
E0+318 0+062 0+392 0+060 0+099 0+080
0+8G0+742 0+045 0+591 0+104 0+007 0+034
E0+146 0+083 0+120 0+017 0+025 0+008
0+9G0+371 0+042 0+187 0+055 0+013 0+028
E0+111 0+144 0+054 0+006 0+010 0+003
n5500
20+9G0+959 0+080 0+734 0+126 0+023 0+078
E0+400 0+064 0+226 0+075 0+101 0+097
20+5G10+484 1 0+995 0+971 0+995
E10+132 1 0+756 0+992 0+987
0G10+773 1 1 1 1
E10+320 1 0+996 1 1
0+5G10+471 1 0+998 0+964 1
E10+139 1 0+857 0+992 0+998
0+6G10+323 1 0+960 0+465 0+914
E0+999 0+113 1 0+701 0+785 0+901
0+7G10+194 1 0+773 0+059 0+430
E0+999 0+085 1 0+473 0+326 0+551
0+8G10+105 1 0+403 0+037 0+116
E0+980 0+077 0+950 0+249 0+105 0+217
0+9G0+947 0+075 0+737 0+133 0+030 0+062
E0+572 0+090 0+382 0+059 0+032 0+027
continued
SIMPLE TEST OF NORMALITY FOR TIME SERIES 683

REFERENCES
Bai,J+&S+Ng ~2001!Tests for Skewness,Kurtosis,and Normality for Time Series Data+Preprint,
Boston College+
Bontemps,C+&N+Meddahi ~2002!Testing Normality:A GMM Approach+Preprint,Université de
Montréal+
Bowman,K+O+&L+R+Shenton ~1975!Omnibus test contours for departures from normality based
on
M
b
1
and b2+Biometrika 62,243–250+
Brillinger,D+R+~1981!Time Series: Data Analysis and Theory+Holden Day+
Davison,A+C+&D+V+Hinkley ~1997!Bootstrap Methods and Their Application+Cambridge Uni-
versity Press+
Epps,T+W+~1987!Testing that a stationary time series is Gaussian+Annals of Statistics 15,1683–1698+
Epps,T+W+~1999!Limiting behavior of the integrated characteristic function test for normality
under Gram-Charlier alternatives+Statistics and Probability Letters 42,175–184+
Epps,T+W+&L+B+Pulley ~1983!A test for normality based on the empirical characteristic func-
tion+Biometrika 70,723–726+
Gasser,T+~1975!Goodness-of-fit tests for correlated data+Biometrika 62,563–570+
Gleser,L+J+&D+S+Moore ~1983!The effect of dependence on chi-squared and empiric distribu-
tion tests of fit+Annals of Statistics 11,1100–1108+
Henze,N+~1997!A new approach to the BEHP tests for multivariate normality+Journal of Multi-
variate Analysis 62,1–23+
Jarque,C+M+&A+K+Bera ~1987!A test for normality of observations and regression residuals+
International Statistical Review 55,163–172+
Kilian,L+&U+Demiroglu ~2000!Residual-based tests for normality in autoregressions:Asymp-
totic theory and simulation evidence+Journal of Business and Economic Statistics 18,40–50+
Lobato,I+N+, J+C+Nankervis,&N+E+Savin ~2002!Testing for zero autocorrelation in the presence
of statistical dependence+Econometric Theory 18,730–743+
Table 2. Continued
fLog Nt
10
x
1
2
x
10
2
Beta~1,1!Beta~2,1!
n51,000
20+9G10+091 0+978 0+245 0+058 0+116
E0+881 0+064 0+484 0+086 0+097 0+101
20+5G10+700 1 1 1 1
E10+243 1 0+971 1 1
0G10+953 1 1 1 1
E10+583 1 1 1 1
0+5G10+710 1 1 1 1
E10+250 1 0+991 1 1
0+6G10+489 1 1 0+959 1
E10+176 1 0+946 0+974 0+997
0+7G10+298 1 0+980 0+352 0+907
E10+113 1 0+766 0+570 0+845
0+8G10+148 1 0+688 0+068 0+331
E10+076 1 0+439 0+164 0+403
0+9G10+094 0+980 0+219 0+061 0+111
E0+962 0+086 0+724 0+168 0+075 0+118
Note: Data follow an AR~1!process with parameter f+
684 IGNACIO N. LOBATO AND CARLOS VELASCO

Lomnicki,Z+A+~1961!Tests for departure from normality in the case of linear stochastic pro-
cesses+Metrika 4,37–62+
Lütkepohl,H+~1991!Introduction to Multiple Time Series Analysis+Springer Verlag+
Mardia,K+V+~1980!Tests of univariate and multivariate normality+In P+R+Krishnaiah ~ed+!,Hand-
book of Statistics: Robust Inference,vol+1,pp+279–320+North-Holland+
Robinson,P+M+~1998!Inference-without-smoothing in the presence of nonparametric autocorrela-
tion+Econometrica 66,1163–1182+
Robinson,P+M+&C+Velasco ~1997!Autocorrelation robust inference+In G+S+Maddala & C+R+
Rao ~eds+!,Handbook of Statistics: Robust Inference,vol+15,pp+267–298+North-Holland+
Rosenblatt,M+~1985!Stationary Sequences and Random Fields+Birkhäuser+
Shapiro,S+S+&M+B+Wilk,~1965!An analysis of variance test for normality ~complete samples!+
Biometrika 52,591–611+
Shapiro,S+S+, M+B+Wilk,&H+J+Chen ~1968!A comparative study of various tests for normality+
Journal of the American Statistical Association 63,1343–1372+
Zygmund,A+~1977!Trigonometric Series+Cambridge University Press+
APPENDIX A
This Appendix provides the alternative versions of ZF
~k!
and EF
~k!
+First,the ZF
~k!
estima-
tor can be written in the frequency domain as follows:
ZF
~k!
5(
j512n
n21
[g~ j!
k
5(
j512n
n21
)
h51
k
H
E
P
I~v
h
!exp~ijv
h
!dv
h
J
5)
h51
k
H
E
P
I~v
h
!dv
h
J
(
j512n
n21
exp$ij~v
1
1{{{ 1v
k
!%
5
E
P
k
I
x2Sx
~v
1
!+++I
x2Sx
~v
k
!D
n
~v
1
1{{{ 1v
k
!dv
1
+++dv
k
,
where D
n
~v!5(
j512n
n21
exp~ijv!satisfies *
P
D
n
~v!dv52pand D
n
~v!r2pd~v50!as
nr`, where drepresents the Dirac’s delta function+Hence,for large nwe obtain the
following approximate expression for ZF
~k!
in the frequency domain:
ZF
~k!
'2p
E
P
k21
I
x2Sx
~l
1
!+++I
x2Sx
~l
k21
!I
x2Sx
~l
1
1{{{ 1l
k21
!dl
1
+++dl
k21
+(A.1)
Equation ~12!is the natural discrete approximation of ~A+1!+
Second,to obtain the time domain expression of EF
~k!
we just plug ~13!into equation
~12!to get
EF
~k!
51
n
k21
(
t
1
512n
n21
[g~t
1
!+++ (
t
k21
512n
n21
[g~t
k21
!(
t
k
512n
n21
[g~t
k
!
3(
j
1
51
n
+++ (
j
k21
51
n
exp$i~t
1
l
j
1
1{{{ 1t
k21
l
j
k21
1t
k
~l
j
1
1{{{ 1l
j
k21
!!%
51
n
k21
(
t
1
512n
n21
[g~t
1
!+++ (
t
k21
512n
n21
[g~t
k21
!(
t
k
512n
n21
[g~t
k
!f
n
~l
t
1
1l
t
k
!+++f
n
~l
t
k21
1l
t
k
!,
SIMPLE TEST OF NORMALITY FOR TIME SERIES 685

where f
n
~l! 5(
t51
n
exp~itl!+Finally,using that f
n
~l
j
!50ifl
j
52pj0n,jÞ0 mod n,
and f
n
~0!5n,and denoting the indicator function by 1,we obtain for j51,+++,k21,
1
n(
t
j
512n
n21
[g~t
j
!f
n
~l
t
j
1l
t
k
!5[g~2t
k
!1[g~n2t
k
!1
$t
k
.0%
1[g~2n2t
k
!1
$t
k
,0%
5[g~t
k
!1[g~n26t
k
6!,
where we have used that [gis even+Then ~14!follows immediately+
APPENDIX B
Proof of Lemma 1(i). We just report the analysis for EF
~3!
because the analysis for
EF
~4!
is similar but notationally more involved+We prove consistency by checking the
sufficient conditions that EF
~3!
is asymptotically unbiased and that its variance goes to
zero as nr`+
First,we consider the expectation of EF
~3!
,
E@EF
~3!
#5~2p!
3
n
2
(
j
1
51
n21
(
j
2
51
n21
E@I~l
j
1
!I~l
j
2
!I~l
j
1
1l
j
2
!# +
Using the definition of I~l!,
E@I~l
j
1
!I~l
j
2
!I~l
j
1
1l
j
2
!#
5E@w~l
j
1
!w~2l
j
1
!w~l
j
2
!w~2l
j
2
!w~l
j
1
1l
j
2
!w~2l
j
1
2l
j
2
!# (B.1)
5(
n
cum~n
1
!{{{cum~n
q
!,
where the summation in nruns for all possible partitions n5n
1
ø{{{øn
q
,q51,2,3of
the 6-tuple
$j
1
,2j
1
,j
2
,2j
2
,j
1
1j
2
,2j
1
2j
2
%(B.2)
such that n
i
5$n
i
~1!,+++,n
i
~p
i
!% and (
i51
q
p
i
56 and where cum~n
i
!stands for
cum~w~l
n
i
~1!
!,+++,w~l
n
i
~p
i
!
!! ~See Brillinger,1981,pp+20–21!+
To evaluate the expectation ~B+1!,by Gaussianity the only cumulants different from
zero are second-order cumulants,k
2
,with q53+Hence
E@EF
~3!
#51
n
5
(
j
1
51
n21
(
j
2
51
n21
H
(
k
2
3
E
P
3
)
i51,2,3
@f~m
i
!f
n
~m
i
1l
n
i
~1!
!f
n
~l
n
i
~2!
2m
i
!dm
i
#
J
,
(B.3)
where the sum in k
2
3
is for all the different 3-tuples n
1
øn
2
øn
3
of pairs n
i
5
~n
i
~1!,n
i
~2!! formed with all the permutations of the coefficients in ~B+2!+In fact,fol-
lowing Brillinger ~1981,Theorem 4+3+1!,the only relevant combinations in the sum in
k
2
3
are those for which n
i
~1!1n
i
~2!50 mod n,i51,2,3+Therefore,using that
686 IGNACIO N. LOBATO AND CARLOS VELASCO

6f
n
~m!6#2 min$6m6
21
,n%~see Zygmund,1977,pp+49–51!,and the continuity of f~l!
implied by ~3!,we obtain that ~B+3!is
E@EF
~3!
#5~2p!
3
n
2
(
j
1
51
n21
(
j
2
51
n21
H
(
k
2
3
E
P
3
)
i51,2,3
@f~m
i
!F
n
~2!
~m
i
2l
n
i
!dm
i
#
J
1o~1!
5~2p!
3
n
2
(
j
1
51
n21
(
j
2
51
n21
f~l
j
1
!f~l
j
2
!f~l
j
1
1j
2
!1o~1!
52p
E
P
2
f~l!f~l!f~l 1m! dldm1o~1!
5F
~3!
1o~1!,as nr`, (B.4)
where F
n
~2!
~m! 5~2pn!
21
6f
n
~m!6
2
and *
P
F
n
~2!
~m!dm51+
Second,we study the variance of EF
~3!
,
Var @EF
~3!
#5cum~EF
~3!
,EF
~3!
!5(
n
cum~n
1
!{{{cum~n
q
!+
Now,we need to consider all the indecomposable partitions n5n
1
ø{{{øn
q
,q51,+++,6
of the following array with 12 elements:
j
1
2j
1
j
2
2j
2
j
1
1j
2
2j
1
2j
2
,
j
1
'
2j
1
'
j
2
'
2j
2
'
j
1
'
1j
2
'
2j
1
'
2j
2
'
+(B.5)
By Gaussianity,the relevant partitions only involve six second-order cumulants,that is,
Var @EF
~3!
#51
n
10
(
j
1
51
n21
(
j
2
51
n21
(
j
1
'
51
n21
(
j
2
'
51
n21
H
(
k
2
6
E
P
6
)
i51
6
$f~m
i
!f
n
~m
i
1l
n
i
~1!
!
3f
n
~l
n
i
~2!
2m
i
!dm
i
%
J
(B.6)
where the sum in k
2
6
is for all the different 6-tuples n5n
1
ø{{{øn
6
of pairs n
i
5
~n
i
~1!,n
i
~2!! constructed in such a way that at least one n
i
in nhas elements in each of
the rows of the array ~B+5!to guarantee an indecomposable partition+Following the
same arguments,the only terms that contribute to the leading term of the variance of
EF
~3!
are those in ~B+6!characterized by a restriction n
i
~1!1n
i
~2!50 mod n,for just
one i[$1,+++,6%~e+g+, j
1
52j
1
'
!+Then,taking into account all the possible partitions
~633!and using the continuity of f,the variance of EF
~3!
is
Var@EF
~3!
#5~2p!
6
n
4
18 (
j
1
51
n21
(
j
2
51
n21
(
j
3
51
n21
f
2
~l
j
1
!f~l
j
2
!f~l
j
3
!f~l
j
1
1l
j
2
!f~l
j
1
1l
j
3
!1o~n
21
!
5O~n
21
!5o~1!(B.7)
as nr`+ Hence,from ~B+4!and ~B+7!we conclude that EF
~3!
5F
~3!
1o
p
~1!+n
SIMPLE TEST OF NORMALITY FOR TIME SERIES 687

Proof of Lemma 1(ii). Notice that
ZF
~k!
2EF
~k!
5(
t512n
n21
[g~t!
k21
[g~n26t6!1{{{ 1(
t512n
n21
[g~t![g~n26t6!
k21
52(
t51
n21
[g~t!
k21
[g~n26t6!1{{{ 12(
t51
n21
[g~t![g~n26t6!
k21
,(B.8)
because [g~n!50+Then,setting M5n
102
,the first element in ~B+8!is equal to
2(
t51
M
[g~t!
k21
[g~n2t!12(
t5M11
n21
[g~t!
k21
[g~n2t!+(B.9)
Now,E[g~n2t!
2
5O~M
2
n
22
!for 0 ,t#M,and using the same methods of the proof
of Lemma 1~i!,it is easy to see that for p52,4,6,
E[g~t!
p
5O~g~t!
p
1n
2p02
!+
Hence,we obtain that for k53,4,
E
*
(
t51
M
[g~t!
k21
[g~n2t!
*
#S(
t51
M
E[g~t!
2~k21!
(
t51
M
E[g~n2t!
2
D
102
5OSS(
t51
n
$g~t!
2~k21!
1n
122~k21!
%M
3
n
22
D
102
D
5O~M
302
n
21
!5o~1!+
Next,
E
*
(
t5M11
n21
[g~t!
k21
[g~n2t!
*
#S(
t5M11
n21
E[g~t!
2~k21!
(
t5M11
n21
E[g~n2t!
2
D
102
,
where (
t5M11
n21
E[g~t!
2~k21!
5O~(
t5M11
n21
$g~t!
2~k21!
1n
12k
%! 5o~1!as nr`for k5
3,4 and (
t5M11
n21
E[g~n2t!
2
5O~(
t51
n21
$g~t!
2
1n
21
%! 5O~11(
t50
`
6g~t!6!5O~1!+
Hence,both terms on the right-hand side of ~B+9!are o
p
~1!+Similar reasoning can be
used to show that the remaining terms in ~B+8!are also asymptotically negligible and
conclude that ZF
~k!
2EF
~k!
5o
p
~1!+n
APPENDIX C
Proof of Lemma 2. Write
[g
[x
~j!2[g
x
~j!51
n(
t51
n26j6
e
t
e
t26j6
11
n(
t51
n26j6
e
t
x
t26j6
11
n(
t51
n26j6
e
t26j6
x
t
,
5A~j!1B~j!1C~j!,
688 IGNACIO N. LOBATO AND CARLOS VELASCO

say+Thus,
(
j512n
n21
[g
[x
~j!
4
5(
j512n
n21
$[g
x
~j!
4
14[g
x
~j!
3
A~j!1{{{ 1A
4
~j!1B
4
~j!1C
4
~j!%+
Hence,using from Appendix B that (
12n
n21
[g
x
~j!
4
5O
p
~1!and the Cauchy–Schwartz
inequality,we only need to show that
(
j512n
n21
A
4
~j!1(
j512n
n21
B
4
~j!1(
j512n
n21
C
4
~j!5o
p
~1!+
First,
(
j512n
n21
A
4
~j!51
n
4
(
j512n
n21
S(
t51
n26j6
e
t
e
t26j6
D
4
#2n
23
S(
t51
n
e
t2
D
4
5O
p
~n
23
!5o
p
~1!,
where we have used Assumption B+
Second,
(
12n
n21
B
4
~j!51
n
4
(
j512n
n21
F
(
t51
n26j6
e
t
x
t26j6
G
4
#1
n
4
(
j512n
n21
F
(
t51
n26j6
e
t2
(
t51
n26j6
x
t26j6
2
G
2
#2n
21
F
[g
x
~0!(
t51
n
e
t2
G
2
5O
p
~n
21
!5o
p
~1!,
where we have employed the Cauchy–Schwartz inequality+The analysis of (
12n
n21
C
4
~j!
is omitted because it is similar to that of (
12n
n21
B
4
~j!+n
SIMPLE TEST OF NORMALITY FOR TIME SERIES 689