A Model Specification Test For GARCH(1,1 ... - Stochastik · A Model Speciﬁcation Test For...

Scandinavian Journal of Statistics, Vol. 42: 1167 –1193, 2015

doi: 10.1111/sjos.12158© 2015 Board of the Foundation of the Scandinavian Journal of Statistics. Published by Wiley Publishing Ltd.

A Model Specification Test ForGARCH(1,1) ProcessesANNE LEUCHT and JENS-PETER KREISSInstitut für Mathematische Stochastik, Technische Universität Braunschweig

MICHAEL H. NEUMANNInstitut für Mathematik, Friedrich-Schiller-Universität Jena

ABSTRACT. We provide a consistent specification test for generalized autoregressive conditionalheteroscedastic (GARCH (1,1)) models based on a test statistic of Cramér-von Mises type. Becausethe limit distribution of the test statistic under the null hypothesis depends on unknown quantitiesin a complicated manner, we propose a model-based (semiparametric) bootstrap method to approx-imate critical values of the test and to verify its asymptotic validity. Finally, we illuminate the finitesample behaviour of the test by some simulations.

Key words: bootstrap, Cramér-von Mises test, generalized autoregressive conditional het-eroscedacity processes, V -statistic

1. Introduction

Conditionally, heteroscedastic time series are frequently used in the finance literature to modelthe evolution of stock prizes, exchange rates and interest rates. Starting with the papers by Engle(1982) on autoregressive conditional heteroscedastic models (ARCH) and Bollerslev (1986) ongeneralized ARCH (GARCH) models, numerous variants of these models have been proposedfor financial time series modelling, see, for example Francq & Zakoïan (2010) for a detailedoverview. The question of parameter estimation in these models has been studied intensively.Exemplarily, we refer the reader to Straumann (2005), Francq & Zakoïan (2010), Tinkl (2013)and references therein.

There is also an overwhelming amount of model specification tests in the econometricliterature. However, these methods typically rely on the assumption that the informationvariables and the response variables are observable. This condition is violated in the caseof GARCH models, where unobserved volatilities enter the information variable. Hence,standard tests cannot be applied, and certain additional approximation procedures haveto be invoked. It turns out that the literature on specification tests for conditionally het-eroscedastic time series is comparatively rare. Berkes et al. (2004) proposed a portmanteaugoodness-of-fit test for GARCH(1,1) models. Their test statistic is a quadratic form ofweighted autocorrelations of the squared residuals of a GARCH(1,1) process fitted to thedata, whose dimension increases with the sample size. They showed that its limit distri-bution is an (infinite) weighted sum of independent �2

1-distributed random variables under

the null hypothesis but did not consider the behaviour under alternatives. In the moregeneral context of conditional mean and variance time series models, Escanciano (2010)proposed a specification test that, contrary to our test, does not require bootstrap. He men-tioned in connection with his Example 1 that this test however is not consistent againstall alternatives.

In the present paper, we propose a specification test of Cramér-von Mises type for aGARCH(1,1) hypothesis against general alternatives. We face the particular problem that some

1168 A. Leucht et al. Scand J Statist 42

of the explanatory variables are not observed and have to be approximated. It turns out thatour test statistic can be approximated by a Cramér-von Mises (V -) statistic, and it follows fromresults of Leucht & Neumann (2013a) that the latter converges to a weighted sum of indepen-dent �2

1variables. In contrast to Berkes et al. (2004), where the weights in the limit correspond

to the weights in the test statistic itself, here, these quantities depend on the properties of theunderlying process in a complicated way. Therefore, the asymptotic result cannot be used fordetermining an appropriate critical value. We propose to apply a model-based (semiparamet-ric) bootstrap method to approximate the null distribution of the test statistic that eventuallyyields an appropriate critical value for the test. Bootstrap consistency for statistics of L2-typehas already been shown in several previous papers. Escanciano (2007a, 2008) showed consis-tency of the wild bootstrap in the context of tests with an underlying martingale structureunder the null hypothesis. Leucht & Neumann (2013a, b) proved consistency of model-basedbootstrap and a variant of the dependent wild bootstrap, respectively, for statistics that can beapproximated by a degenerate V -statistic. In contrast to the method of proof used in Leucht& Neumann (2013a), we take this opportunity and present a different approach of provingbootstrap consistency: rather than imitating the derivation of the limit distribution of the teststatistic on the bootstrap side, we use coupling arguments to show consistency. This approachwas successfully applied to U -statistics and V -statistics of independent random variables byDehling & Mikosch (1994) and Leucht & Neumann (2009); however, it seems to be new in thecontext of dependent data. Finally, we would like to mention that our theory can perhaps begeneralized to GARCH models of higher order. To present the main ideas in as a transparentas possible manner, we restrict ourselves to the simple GARCH(1,1) case.

2. Assumptions and some preliminaries on GARCH(1,1) processes

Suppose that we observe Y0; : : : ; Yn, where .Yt /t2Z is a (strictly) stationary process satisfyingthe model equation

Yt D �t "t ;

where �t WD E�Y 2t j Yt�1; Yt�2; : : :

�and "t are stochastically independent and ."t /t is a

sequence of independent and identically distributed (i.i.d.) random variables. Moreover, weassume E"2

0D 1. In the GARCH literature, it is sometimes also assumed that E"0 D 0,

which would imply that �2t D var .Yt j Yt�1; Yt�2; : : : /, see, for example Straumann &Mikosch (2006). However, our theory can also be applied in the non-centred case similarlyto Nelson (1990) and Berkes et al. (2004). Therefore, we proceed without the assumptionE"0 D 0.

A GARCH(1,1) process .Yt /t2Z satisfies the equations

�2t D ! C ˛Y 2t�1 C ˇ�2t�1 (1)

and

Yt D �t "t : (2)

We consider the test problem

H0 W .Yt /t2Z 2M0 against H1 W .Yt /t2Z 2MnM0

© 2015 Board of the Foundation of the Scandinavian Journal of Statistics.

Scand J Statist 42 A test for GARCH(1,1) models 1169

with

M0 D°.Yt /t2Z j Yt D �t "t with E

�Y 2t j Yt�1; Yt�2; : : :

�D �2t D ! C ˛Y 2t�1

Cˇ�2t�1; � D .!; ˛; ˇ/0 2 ‚

±;

M D°.Yt /t2Z j Yt D �t "t with E

�Y 2t j Yt�1; Yt�2; : : :

�D �2t D f

�Yt�1; �

2t�1

�±and ‚ D ¹� D .!; ˛; ˇ/0 j ! > 0; ˛; ˇ � 0º.

Typical asymmetric alternatives contained in M are generalized quadratic ARCH(GQARCH(1,1)) processes introduced by Sentana (1995), where

�2t D ! C ˛.Yt�1 � ı/2 C ˇ �2t�1

or Glosten–Jagannathan–Runkle (GJR)-GARCH processes with

�2t D ! C ˛ Y 2t�1 C ˇ �2t�1 C ı Y 2t�1 Yt�1<0;

introduced by Glosten et al. (1993), which are frequently used in finance. If .Yt /t describes asequence of log returns of an asset and if ı > 0, then negative shocks have a larger impacton the conditional volatilities than positive ones. If we had only one of these two particularalternatives in mind, we could simply test whether or not ı D 0. However, other deviationsfrom the null are of a more complicated structure, for example, the model equation for thevolatilities of exponential GARCH(1,1) processes is given by

ln �2t D ! C ˛

²�Yt�1

�t�1C �

�ˇYt�1

�t�1

ˇ�E

ˇYt�1

�t�1

ˇ�³C ˇ ln �2t�1; !; ˛; ˇ; �; � 2 R;

Nelson (1991). Therefore, we strive for a more general test that is also consistent against unspec-ified deviations from a GARCH(1,1) model. It can be expected that our test procedure can begeneralized to a test for GARCH(p; q) specification, but this extension would be very technicaland is therefore not carried out here.Under H0, we denote by �0 D .!0; ˛0; ˇ0/0 the true parameter and assume

(A1)(i) !0 > 0, ˛0; ˇ0 � 0 and

(ii) ."t /t2Z i.i.d., E"20D 1, and E

�ln�ˇ0 C ˛0"

20

�< 0.

According to theorem 2 in Nelson (1990), (A1) ensures that there exists a unique strictlystationary and ergodic solution to (1) and (2) that can be rewritten ((10) in Nelson (1990)) as

�2t D !0

"1 C

1XkD1

kYiD1

�ˇ0 C ˛0"

2t�i

�#: (3)

Making repeatedly use of the models (1) and (2), we get

�2t D !0 C ˛0Y2t�1 C ˇ0�

2t�1

D�!0 C ˛0Y

2t�1

�C ˇ0

�!0 C ˛0Y

2t�2

�C � � � C ˇK�10

�!0 C ˛0Y

2t�K

�C ˇK0 �

2t�K :

(4)



Because��2t�t2Z

is stationary and because E�ln�ˇ0 C ˛0"

20

�< 0 implies ˇ0 < 1, we see that

the last summand on the right hand side of (4) tends to zero as K !1, that is, we obtain thealternative representation

�2t D

1XkD1

ˇk�10

�!0 C ˛0Y

2t�k

�D

!0

1 � ˇ0C ˛0

1XkD1

ˇk�10 Y 2t�k : (5)

For any parameter � D .!; ˛; ˇ/0, we define a stationary sequence of approximations of thevolatilities that are based on the Yt s but correspond to the model with parameter � as

�2t .�/ D!

1 � ˇC ˛

1XkD1

ˇk�1Y 2t�k : (6)

We have the obvious equality �2t .�0/ D �2t but, more importantly, for � close to �0;

��2t .�/

�t2Z

is strictly stationary and ergodic. This implies that for any fixed value of an estimator b�n of �0,�2t .

b�n/ serves as a suitable approximation to the (non-stationary) estimated volatilitiesb�2t thatwill be specified in the succeeding texts. The following lemma shows that the aforementioneddefinition is correct if � is sufficiently close to �0 and that �2t .�/ converges to �2t as � ! �0.Here and in the sequel, we use the notation a _ b WD max¹a; bº.

Lemma 1.(i) For � D .!; ˛; ˇ/0 2 ‚, satisfying EŒln..ˇ0 _ ˇ/C .˛0 _ ˛/"20/� < 0, .�2t .�//t2Z is the

unique stationary solution to

�2t .�/ D ! C ˛Y 2t�1 C ˇ�2t�1.�/; t 2 Z: (7)

�2t .�/ is finite with probability 1.(ii) sup�2‚W k��0k�ı

ˇ�2t .�/ � �

2t

ˇ�!ı!0

0 with probability 1.

We intend to test a composite hypothesis, that is, the GARCH(1,1) parameters are unknownand have to be estimated. In accordance with Francq & Zakoïan (2004), we use the quasi-maximum likelihood estimator (QMLE) with a normal reference distribution, which isdefined as

b�n D �b!n; bn; bn�0 D arg min�2‚0

NLn.�/:

Here,

‚0 D ¹� D .!; ˛; ˇ/0 j ˇ � �0; u1 � min¹!; ˛; ˇº � max¹!; ˛; ˇº � u2º

with some 0 < u1 < u2 < 1 and �0 2 .0; 1/, and �n NLn denotes the logarithmic quasi-likelihood function (constant terms are ignored here), given by

NLn.�/ D1

n

nXtD1

log N�2t .�/ C

Y 2t

N�2t .�/

!;

where

N�2t .�/ D ! C ˛Yt�1 C ˇ N�2t�1.�/; t � 1:

In principle, the initial value N�20.�/ can be chosen arbitrarily. For the sake of definiteness, we

follow the suggestion (2.7) in Francq & Zakoïan (2004) and set N�20.�/ D Y 2

0.



We assume

(A2)(i) Ej"0j

4 <1 and var."20/ > 0 and

(ii) �0 is in the interior of ‚0.

Francq & Zakoïan (2004) proved strong consistency and asymptotic normality of the QMLEin the framework of GARCH(p; q) processes. Under the aforementioned conditions, we obtainfrom their results, in the special case of a GARCH(1,1) process considered here, the Bahadurlinearization

b�n � �0 D 1

n

nXtD1

Lt C oP

�n�1=2

�with Lt D ."2t � 1/.EŒ

RW0.�0/�/�1 P�

2t .�0/

�2t .�0/;

(8)

where P�2t .�/ D .@�2t .�/=@!; @�

2t .�/=@˛; @�

2t .�/=@ˇ/

0,W0.�/ D log �20.�/CY 2

0=�20.�/ and RW0

denotes its Hessian with respect to � , cf. Francq & Zakoïan (2004, page 618).

Remark 1. The practical derivation of the QMLE is based on a non-linear optimization prob-lem and therefore is computationally intensive. For that reason, Kristensen & Linton (2006)proposed a moment-based approach to estimate the GARCH parameters. They provide explicitexpressions for their estimators; however, their method is only reliable for very large samplesizes. Therefore, and for sake of definiteness, we stick to the QMLE in the sequel.

3. The test statistic and its asymptotics

We propose a test of Cramér-von Mises type. At first glance, the statistic

NTn D n

ZR2

´1

n

nXtD1

�Y 2t �b�2t � w.´1 � Yt�1; ´2 �b�2t�1/

μ2Q.d´1; d´2/

seems natural, whereb�20D Y 2

0andb�2t D b!nCbnY 2t�1Cbnb�2t�1 (t D 1; : : : ; n) is a model-based

approximation of the unobserved volatility. Here, w is a weight function and Q a probabilitymeasure. Because E.Y 2t j Yt�1; �

2t�1

/ � �2t D 0 under H0, one would reject the null hypoth-esis if the value of the test statistic is large. However, the test statistic is of a very complicatedstructure, and critical values cannot be determined directly. A bootstrap-aided testing proce-dure will be proposed in the succeeding texts to circumvent these difficulties. In order to showits asymptotic validity, we would have to impose certain moment constraints such as finitefourth moments of Yt . The latter assumption would be rather restrictive and would rule out,for example, IGARCH processes (˛Cˇ D 1) that are frequently applied in financial mathemat-ics, see Lee & Hansen B. E (1994). In contrast, moment assumptions on the innovations are farless restrictive and have already been presumed by Berkes et al. (2003) and Francq & Zakoïan(2004) to obtain asymptotic normality of the QMLE for the GARCH(1,1) parameter vector.It turns out that moment conditions on the innovations suffice to derive the asymptotics of theslightly modified test statistic

bTn D n

ZR2

´1

n

nXtD1

Y 2tb�2t � 1

!w.´1 � Yt�1; ´2 �b�2t�1/

μ2Q.d´1; d´2/:

We will show that a bootstrap-aided test based on this statistic is consistent and asymptoticallylevel-� .



We first show that the test statistic can be approximated by a von Mises (V -) statistic notdepending on the estimators .b�2t /t andb�n but on the true quantities .�2t /t and �0, respectively.The limit distribution of this approximating statistic can then easily be obtained from recentresults by Leucht & Neumann (2013a). To this end, we need the kernel of the V -statistic beingcontinuous that is ensured by the next assumption. Furthermore, in order to keep the effect ofapproximating the unobserved volatilities in the weight function negligible, we require an extracondition on the smoothness of w. We make the following assumptions regarding the weightfunction and the measure Q:

(A3) Q is a probability measure on .R2;B.R2//. The weight function w is non-negative,bounded and measurable. Moreover, there is some Cw <1 such thatZ

R2

jw.´ � .y1; s1/0/ � w.´ � .y2; s2/

0/j2Q.d´/ � Cw k.y1; s1/0 � .y2; s2/

0k2: (9)

Remark 2. Equation (9) is obviously satisfied if w is Lipschitz continuous. It is also satisfiedif w.´ � .y; s/0/ D ..y; s/0 � ´/ and Q has bounded marginal densities q1 and q2. Here andbelow .a1; a2/

0 � .b1; b2/0 means that a1 � b1 and a2 � b2.

Subsequently, we will abbreviate the information variable�Yt�1; �

2t�1

.�/�0

by It�1.�/. In

particular, we have It�1.�0/ D�Yt�1; �

2t�1

�0.

Lemma 2. Suppose that H0 holds true and that (A1)–(A3) are satisfied. Then,

bTn � Tn D oP .1/;

where

Tn D

ZR2

´1pn

nXtD1

�"2t � 1

�w.´� It�1.�0//�E�0

"P�21 .�0/

�21 .�0/w.´� I0.�0//

#0Lt

!μ2Q.d´/:

Note that Tn is a V -statistic that is degenerated under H0, that is, it can be represented asTn D n

�1Pns;tD1 h.Xs ; Xt / with Eh.X0; x/ D 0 8x, where Xt D

�"2t ; Yt�1; �

2t�1

; L0t�0

and

h.x; Nx/ D

ZR2

´.x1 � 1/w

�´ � .x2; x3/

0��E�0

"P�21.�0/

�21.�0/

w .´ � I0.�0//

#0x4

μ

�

´. Nx1 � 1/w

�´ � . Nx2; Nx3/

0��E�0

"P�21.�0/

�21.�0/

w.´ � I0.�0//

#0Nx4

μQ.d´/:

Thus, its asymptotics can be immediately deduced from a recent result on degenerate V -statistics of ergodic data by Leucht & Neumann (2013a). In conjunction with the previouslemma, we obtain the limit distribution of bTn.

Proposition 1. Suppose that H0 holds true and that (A1)–(A3) are satisfied. Then,

bTn d�! Z D

1XkD1

kZ2k :

Here, .Zk/k is a sequence of independent standard normal random variables, and .k/k denotesthe (finite or countably infinite) sequence of non-zero eigenvalues of the equation ˆ.x/ DRh.x; Nx/ˆ. Nx/PX

�0.d Nx/, enumerated according to their multiplicity.



Now, we consider the behaviour of the test statistic under fixed alternatives. To this end, weassume the parameter estimator that is obtained by the quasi-maximum likelihood approachdescribed in Section 2 to be consistent for some pseudo-true parameter N�0.

(A4) b�n P�! N�0 2 ‚0.

Additionally, we impose some regularity conditions on the model under the alternative.

(A5) .Yt /t2Z is strictly stationary and ergodic. Moreover, EjY0js < 1 for some s > 0 andEjY0=�0. N�0/j

4Cı 2 .0;1/.

Note that the first moment condition ensures almost sure finiteness of��2t .�/

�t2Z

for � in aneighbourhood of N�0. As expected, the test statistic turns out to be asymptotically unboundedunder H1.

Proposition 2. Suppose that (A1) and (A3)–(A5) hold true, with �0 replaced by N�0. Then,

(i) n�1 bTn P�!

RR2

®E��Y 21=�21

�N�0�� 1

�w�´ � I0

�N�0��¯2

Q.d´/:

(ii) If additionally the relation E��Y 21=�21

�N�0�� 1

�w�´ � I0

�N�0��¤ 0 for all ´ 2 ˘ and

some ˘ with Q.˘/ > 0 holds true, then

bTn P�!1:

Remark 3

(i) Provided that Q has an everywhere positive density, a weight function that satisfies theadditional condition in Proposition 2 for all H1 scenarios is w.´ � It / D It�´ that isfrequently used in Cramér-von Mises type tests, cf. Lemma 1(d) in Escanciano (2006).

(ii) We conjecture that our test can also detect Pitman alternatives at ratepn. Under slightly

stronger moment and dependence conditions, corresponding results for V -statistics havebeen derived by Leucht (2012), see Proposition 4.1 therein.

4. A bootstrap-based test

We see from the previous section that the null distribution of the test statistic bTn and also itslimit distribution depend on the unknown parameter �0 in a complicated way. In particular, theeigenvalues .k/k appearing in the limit are unknown, and it is not clear at all how they canbe computed in an efficient manner. Therefore, (asymptotic) critical values of a test based onthis statistic cannot be derived directly. The bootstrap offers a convenient tool to circumventthese difficulties. In the present context, a model-based bootstrap is probably the first choicebecause it can be expected to be more precise than alternative model-free methods. We proposethe following algorithm:

(1) Compute the residuals

et D Yt=b�t ; t D 1; : : : ; n:

(2) Calculate standardized versions

b"t D et=

vuutn�1nXsD1

e2s :



(3) Draw independent bootstrap innovations

"�t � Fn;b"; where Fn;b".x/ D n�1nXtD1

.b"t � x/ :(4) Compute ��t

2, Y �t recursively:

��02D b!n "1 C 1X

kD1

kYiD1

�bn C bn"�2�i�

#;

Y �0 D ��0 "�0 ;

and then, for t D 1; : : : ; n,

��t2D b!n C bnY �2t�1 C b

n��2t�1;

Y �t D ��t "�t :

(In our simulations, we replaced the infinite sum in the definition of ��20

by a finite one.Alternatively, we could generate a reasonable number of presample values not used inthe bootstrap statistics to reach an approximate stationary regime.)

(5) Compute the bootstrap statistic

bT �n D n

ZR2

´1

n

nXtD1

Y �2tb��2t � 1

!w.´1 � Y

�t�1; ´2 �b��2t�1/

μ2Q.d´1; d´2/;

withb��20D Y �2

0andb��2t D b!�n Cb�n Y �2t�1C b�nb��2t�1. Here,b��n D �b!�n ; b�n ; b�n�0 is the

QMLE based on the bootstrap sample.(6) Repeat steps (3) to (5) B times and, for a nominal size of � 2 .0; 1/, choose t�� as any

.1 � �/-quantile of the empirical distribution of bT �n;1; : : : ; bT �

n;B.

(7) Reject the null hypothesis if bTn > t�� .

Remark 4. Instead of using the QMLE approach and re-estimating the (known) volatiliteson the bootstrap side which is computationally costly, one can probably apply a wildbootstrap method to the approximating statistic Tn since the summands there form a mar-tingale difference array. In similar settings, validity of this method has been verified e.g. inEscanciano (2008). In the present context, this would require to approximate the unknown

quantity E�0

P�21 .�0/

�21 .�0/w .´ � I0.�0//

�which can in general only be represented by an infinite

sum, and to invert an estimate of E�RW0 .�0/

which might become numerically intense again.

In order to validate asymptotic correctness of the algorithm previously, we do not imitateall the proofs of Section 3. Instead, an appropriate coupling of Xt D

�"t ; Yt�1; �

2t�1

; L0t�0

and

X�t D�"�t2; Y �t�1

; ��2t�1

; L�t0�0

(with ��2t�1D ��2

t�1

�b�n� and ��2t .�/ being the bootstrap ana-

logue to �2t .�/) directly results in a coupling of the corresponding test statistics on the original

and on the bootstrap side. Here, L�t D�"�2t � 1

� �E�hRW �0

�b�n�i��1 P��2t .b�n/=��2t �b�n�.

To express distributional convergence in conjunction with the additional qualification‘almost surely’ properly and to describe closeness of two distributions both depending on n, weuse the Lévy metric dL that is defined, for distribution functions G and H on R, as

dL.G;H/ D inf°" W G.x � "/ � " � H.x/ � G.x C "/C " 8x 2 R

±:



Applied to random variables U and V with c.d.f. FU and FV , we also use the notationdL.U; V /. A first step towards our proof of bootstrap consistency is performed by the followinglemma.

Lemma 3. Assume that H0 holds true and that (A1)–(A3) are fulfilled. Then,

dL�Fn;b"; F"� a:s:�! 0:

Now, we are in the position to construct a coupling of the random variables Xt D�"2t ; Yt�1; �

2t�1

; L0t�0

appearing in the approximating V -statistic Tn with the random variables

X�t D�"�t2; Y �t�1

; ��2t�1

; L�t0�0

.

Lemma 4. Assume that H0 holds true and that (A1)–(A3) are fulfilled. On a sufficiently rich

probability space . Q ; eA; QP /, there exists independent random vectors��Q"t ; Q"�t

�0�t2Z

such that

Q"tdD "t

Q"�tdD "�t

and, with QLt ; QYt ; Q�2t and QL�t ; QY�t ; Q�

�2t being versions based on the Q"s and Q"�s , respectively,

E QP

h.Q"�t � Q"t /

2 C k QL�t �QLtk

22 C j

QY �t�1 �QYt�1j ^ 1 C jQ�

�2t�1 � Q�

2t�1j ^ 1

ia:s:�! 0:

As a consequence of the aforementioned coupling, the following assertion provides a usefulapproximation for the two hypothetical volatility processes.

Corollary 1. Assume that H0 holds true and that (A1)–(A3) are fulfilled. On the probabilityspace . Q ; QA; QP / from Lemma 4,

E QP

"sup�2‚0

ˇQ��2t .�/ � Q�2t .�/

ˇ^ 1

#QP�! 0;

where Q��2t .�/ and Q�2t .�/ are versions of the ��2t .�/ and �2t .�/, respectively, based on the QY �t andQYt .

The asymptotics of the test statistic bTn heavily relies on the linearization of the QMLE b�n. Wenow establish this property for the bootstrap QMLE.


b��n �b�n D 1

n

nXtD1

L�t C oP�

�1pn

�;

where we write R�n D oP�.an/ if P �.jR�nj=janj > "/P�! 0; 8 " > 0.

These results enable us to derive a bootstrap analogue to Lemma 2.


bT �n D T �n C oP�.1/;



where T �n D n�1Pns;tD1 h

��X�s ; X

�t

�with

h�.x; Nx/ D

ZR2

8<:.x1 � 1/w .´� .x2; x3/0/ � E�b�n

24 P��21�b�n�

��21

�b�n� w�´� I�0

�b�n��350 x4

9>=>;�

8<:. Nx1 � 1/w �

´� . Nx2; Nx3/0��E�b�n

24 P��21�b�n�

��21

�b�n�w�´� I�0

�b�n��350 Nx4

9>=>;Q.d´/:Hence, bootstrap validity under the null hypothesis can be deduced from the following result.

Theorem 1. Assume that H0 holds true and that (A1)–(A3) are fulfilled. Then,

dL�T �n ; Tn

� P�! 0:

To show that our bootstrap method is also valid under fixed alternatives, we additionallyassume

(A6)

(i) b��n �b�n D oP�.1/ and(ii) E

�ln�N C NY 2

0=�2. N�/

�< 0.

Lemma 7. Suppose that (A3) to (A6) are fulfilled. Then,

E�hn�1bT �n i P

�! 0:

Thus, the aforementioned algorithm leads to a consistent, asymptotic level-� test.

Corollary 2(i) Assume that H0 holds true, that (A1) - (A3) are fulfilled and that additionally

EŒh.X1; X1/� > 0. Then,

P�bTn > t��

��!n!1

�:

(ii) Under H1 and if additionally the prerequisites of Proposition 2 and (A6) are satisfied, then

P�bTn > t��

��!n!1

1:

Remark 5. Our test has similarities to the methodology proposed by Escanciano (2008) inthe general context of mean and variance specification testing. While our test is based on themarked empirical process n�1=2

PntD1

�Y 2t =b�2t � 1� w �´1 � Yt�1; ´2 �b�2t �, Escanciano’s test

is based on the processes .n � j C 1/�1=2PntDj

�Y 2t �b�2t � w.Yt�j ; ´/, for j D 1; 2; : : : , that

is, he uses only observable random variables in the weight function. However, in a previousversion of that paper, Escanciano (2007b), he allows for models with infinitely many explana-tory variables. In principle, using representation (5) previously, it seems that these results couldbe applied in our case, too. In order to apply his result to our test problem, we would have toassume that certain moments of the observed process exist; see his assumption A1(b). However,this typically is not guaranteed in financial time series as we already discussed at the beginningof Section 3.



5. Numerical examples

5.1. Simulation study

We illustrate the finite sample behaviour of the proposed test by some simulations. We usethe indicator function w.´ � It / D It�´ as a weight function that is admissible in view ofRemark 3 and chooseQ D N .0; 25/˝N .0; 25/. Straightforward calculations show that in thiscase, the test statistic simplifies to

bTn D 1

n

nXs;tD1

Y 2sb�2s � 1

! Y 2tb�2t � 1

!.1 �ˆ0;25 .max ¹Ys�1; Yt�1º//

��1 �ˆ0;25

�max

°b�2s�1;b�2t�1±�� ;where ˆ0;25.�/ D ˆ.�=5/ and ˆ denote the cumulative distribution function of N .0; 1/. Tostudy the performance of our test under the null hypothesis and certain alternative scenarios,we choose the innovations to be standard normal, draw samples of size n D 500 and n D 1000and run a Monte-Carlo simulation N D 500 times, each with B D 500 bootstrap replications.In order to meet our assumption of stationarity, we discarded 500 presample data values ofthe corresponding processes. The implementation was carried out with the aid of the statisticalsoftware package R, see R Core Team (2012). To estimate the GARCH parameters, we use theroutine of Tinkl (2013).

The rejection frequencies of our test under three null scenarios and for nominal significancelevels � D 0:05 and � D 0:1 are summarized in Table 1. Here, we choose two scenarios,where the parameter estimation procedure performs very well, � D .0:20; 0:15; 0:25/0 and� D .0:20; 0:25; 0:35/0. The parameter � D .0:04; 0:08; 0:90/ is motivated by financial appli-cations; it was estimated from the German stock index (DAX), see Kreiss & Neuhaus (2006,Section 14). Tables 2 and 3 report the finite sample behaviour of our procedure under five H1scenarios. We consider GQARCH and GJR-GARCH alternatives for the first two choices of � .We also considered the GJR-GARCH scenario for higher values of ˇ that typically appearin finance. However, to keep stationarity, we could not take ˇ D 0:9 here. Instead, we usedGJR-GARCH processes with parameters .!; ˛; ˇ; ı/ equal to .0:04; 0:08; 0:9�ı; ı/ for ı D 0:25and ı D 0:5.

Table 1. Empirical size

Table 2. Power: GQARCH



Table 3. Power: GJR-GARCH

Table 4. Analysis of the Standard & Poor’s (S&P) 500

It can be seen that the prescribed size is kept well for samples of size 1000 that one can assumein financial applications. The power behaviour is convincing for all of our alternatives. Havinga particular alternative in mind, the power can even be increased by a tailor-made choice ofthe weights w and Q, cf. Anderson & Darling (1954) in the case of generalized Cramér-vonMises statistics.

5.2. Application to financial data

GARCH models are frequently used to model economic phenomena, for example, log returnsof financial assets. Here, we consider the Standard & Poor’s 500 stock market index. We applyour test with the weights of the previous subsection to the daily log returns of three consecutiveperiods, 2 January 1990 to 31 December 1993, 3 January 1994 to 31 December 1997 and 2January 1998 to 28 August 2002. These data have already been investigated by Escanciano(2010). He also compared the results of his test to another specification test by Wooldridge(1990). For the first period, the latter test accepts the null hypothesis of a GARCH(1,1) model.The decision of the type of test proposed by Escanciano (2010) is not definite and depends onthe choice of the weight function. His Table 4 shows that the test rejects the null hypothesis forfour out of five different choices of the weight functions. Our test accepts the null hypothesis. Atest of Escanciano & Velasco (2006) rejects the hypothesis of a martingale difference array forthe second period. In line with these findings, our test rejects the GARCH(1,1) specification.Finally, in the last period, the tests by Escanciano (2010), Wooldridge (1990) and our test acceptthe GARCH(1,1) model. The results of our test are summarized in Table 4.

6. Proofs

Throughout this section, we shortly writeR

instead ofRR2

. Moreover, C denotes a generic,finite constant that may change its value from one line to another.

Proof of Lemma 1. First of all, finiteness of �2t .�/ follows from a simple coupling argu-ment. According to (5), �2t .�/ consists only of non-negative summands. Hence, the seriesP1kD1 ˇ

k�1Y 2t�k

converges possibly to infinity. We show next that this series is actually finite



with probability 1 under the assumption E�ln�.ˇ0 _ ˇ/C .˛0 _ ˛/"

20

�< 0. To this end, we

compare �2t .�/ with

L�2t D .!0 _ !/

"1 C

1XkD1

kYiD1

�.ˇ0 _ ˇ/ C .˛0 _ ˛/"

2t�i

�#: (10)

In contrast to �2t .�/, L�2t is the solution to a system of GARCH(1,1) equations,

L�2t D .!0 _ !/ C .˛0 _ ˛/ LY2t�1 C .ˇ0 _ ˇ/ L�

2t�1;

LYt D L�t "t :

In view of our assumption of EŒln..ˇ0 _ ˇ/ C .˛0 _ ˛/"20/� < 0, we can use available theory,and we obtain from theorem 2 of Nelson (1990) that L�2t is finite with probability 1. Comparing(3) with (10), we see that �2t � L�

2t almost surely. Hence, Y 2t � LY

2t , and because L�2t can be

rewritten as

L�2t D

1XkD1

.ˇ0 _ ˇ/k�1

�.!0 _ !/ C .˛0 _ ˛/ LY

2t�k

�;

we see that �2t .�/ � L�2t . Hence, �2t .�/ is also finite with probability 1. Now we obtain that

�2t .�/ D ! C ˛ Y 2t�1 C ˇ

1XkD1

ˇk�1�! C ˛Y 2t�1�k

�D ! C ˛ Y 2t�1 C ˇ �2t�1.�/;

that is, .�2t .�//t2Z solves the system of ( 7). As for uniqueness, assume that . Q�2t /t2Z is anyarbitrary stationary solution to (7). Then, we obtain from a repeated application of thisequation thatˇ

�2t .�/ � Q�2t

ˇ� ˇK

ˇ�2t�K.�/ � Q�

2t�K

ˇ:

Because our assumption EŒln..ˇ0_ˇ/C .˛0_˛/"20/� < 0 implies that ˇ < 1, we conclude thatQ�2t D �

2t .�/ a.s. for all t 2 Z.

(ii)

Because E"20< 1, the function .˛; ˇ/ 7! EŒln.ˇ C ˛"2

0/� is continuous. Hence, there exists a

sufficiently small ı0 > 0 such thatEŒln..ˇ0Cı0/C.˛0Cı0/"20/� < 0. If � 2 ‚ and k��0k � ı0,then there exists a stationary solution .�2t .�//t2Z to (7). We have the representations

�2t .�/ D

1XkD1

ˇk�1�! C ˛Y 2t�k

�(11)

and

�2t D

1XkD1

ˇk�10

�!0 C ˛0Y

2t�k

�: (12)

If � ! �0, then all summands on the right hand side of (11) converge to their counterparts in(12). Moreover, they are majorized by .ˇ0 _ ˇ/k�1

�.!0 _ !/C .˛0 _ ˛/ Y

2t�k

�. Because

1XkD1

.ˇ0 _ ˇ/k�1

�.!0 _ !/C .˛0 _ ˛/ Y

2t�k

�D L�2t < 1;



we obtain by majorized convergence that

sup�2‚W k��0k�ı

ˇ�2t .�/ � �

2t

ˇ�!ı!0

0:

Proof of Lemma 2. The proof of this lemma consists of a series of approximations that can bemostly derived by standard arguments. To save space, we sketch the main steps only. Detailsare provided in the online supplementary material at the publisher’s web site.

In a first step, we show that the substitution of our estimated volatilitiesb�2t by their station-

ary counterparts �2t�b�n� is asymptotically negligible. (To be precise, for any non-random � 2

‚,��2t .�/

�t

forms a stationary process.) Let QTn be the statistic based on �2t�b�n� instead ofb�2t ,

that is,

QTn D

Z 8<: 1pn

nXtD1

0@ Y 2t

�2t

�b�n� � 11A w

�´ � It�1

�b�n��9=;2

Q.d´/: (13)

Becauseb�2t � �2t �b�n� D btn

�b�20� �2

0

�b�n�� and bn P�! ˇ < 1, we can easily show that

bTn D QTn C oP .1/:

In order to show that

QTn D Tn C oP .1/; (14)

we decompose the square root of the integrand in QTn as

1pn

nXtD1

0@ Y 2t

�2t

�b�n� � 11A w

�´ � It�1

�b�n��

D1pn

nXtD1

Y 2t

�2t .�0/� 1

!w.´ � It�1.�0// �E�0

"P�21.�0/

�21.�0/

w.´ � I0.�0//

#0Lt

C1pn

nXtD1

Y 2t

�2t .�0/� 1

!�w�´ � It�1

�b�n�� w.´ � It�1.�0//�

C1pn

nXtD1

0@ Y 2t

�2t

�b�n� �Y 2t

�2t .�0/

1A�w �´ � It�1 �b�n�� w.´ � It�1.�0//�

�1pn

nXtD1

"2tP�2t .�0/

�2t .�0/w.´ � It�1.�0// � E�0

"P�21.�0/

�21.�0/

w.´ � I0.�0//

#0!1

n

nXsD1

Ls

C1pn

nXtD1

0@ Y 2t

�2t

�b�n� �Y 2t

�2t .�0/C "2t

"P�2t .�0/

�2t .�0/

#0 �b�n � �0�1A w.´ � It�1.�0//

�1pn

nXtD1

"2t

"P�2t .�0/

�2t .�0/

#0 b�n � �0 � 1

n

nXsD1

Ls

!w.´ � It�1.�0//

DW Rn;0.´/ C Rn;1.´/ C Rn;2.´/ � Rn;3.´/ C Rn;4.´/ � Rn;5.´/

(15)



say. The first term on the right hand side, Rn;0.´/, is just the square root of the inte-grand of Tn, and it follows by (A2), (A3) and (4.29) in Francq & Zakoïan (2004) thatE�0

�RR2n;0.´/Q.d´/

<1, which impliesZ

R2n;0.´/Q.d´/ D OP .1/: (16)

It remains to show thatZR2n;i .´/Q.d´/ D oP .1/; for i D 1; : : : ; 5; (17)

which can actually be achieved by lengthy but tedious calculations. Equations (16) and (17)yield (14), which completes the proof.

Proof of Proposition 2

(i) The overall structure of the proof is similar to the one of Lemma 2.Within the proof of Lemma 2 (in the online supplement), we repeatedly apply relations

(4.26) and (4.29) of Francq & Zakoïan (2004). Under the alternative, the observations donot arise from a GARCH(1,1) process. Therefore, these results are not applicable directly.Still, replacing �0 by N�0, these relations remain valid because the proofs of Francq &Zakoïan (2004) only rely on the definition of �2t .�/; t 2 Z; � 2 ‚0; and the momentconditions stated in (A5). As under H0, the effect of choosing an arbitrary initial volatil-ityb�0 is asymptotically negligible, that is, n�1bTn D n�1 QTnCoP .1/, where eTn is definedas in (14).

Now, we decompose the square root of the integrand of n�1eTn as

1

n

nXtD1

0@ Y 2t

�2t

�b�n� � 11A w

�´ � It�1

�b�n��

D1

n

nXtD1

Y 2t

�2t .N�0/� 1

!w�´ � It�1. N�0/

�C1

n

nXtD1

Y 2t

�2t .N�0/� 1

! �w�´ � It�1

�b�n�� w �´ � It�1 � N�0��

C1

n

nXtD1

0@ Y 2t

�2t

�b�n� �Y 2t

�2t�N�0�1A w

�´ � It�1

�b�n��DW Un;1.´/ C Un;2.´/ C Un;3.´/:

Firstly, it can be shown thatZU 2n;1.´/Q.d´/

P�!

Z �Eh�Y 21 =�

21

�N�0�� 1

�w�´ � I0

�N�0��i�2

Q.d´/:

Moreover, under (A3) and (A5), we obtainZU 2n;2.´/Q.d´/ D OP .1/

1

n

nXtD1

°1 ^ j�2t

�b�n� � �2t � N�0� j± :In analogy to the proof of Lemma 1(ii), asymptotic negligibility of the second factor onthe right-hand side can be verified. Finally, we get

jUn;3.´/j D oP .1/kwk21n

nXtD1

Y 2t

�2t�N�0� k P�2t � N�n;t � k

�2t

�b�n�© 2015 Board of the Foundation of the Scandinavian Journal of Statistics.


for some random N�n;t betweenb�n and N�0. Thus,RU 2n;3.´/Q.d´/ is of order oP .1/ under

(A5) and finally n�1bTn D R E��Y 21 =�21 � N�0� � 1�w �´ � I0 � N�0��2Q.d´/C oP .1/.(ii) The assertion follows immediately from part (i), and the extra condition presumed under

(ii). �

Proof of Lemma 3. Recall that et and b"t denote the raw and the standardized residuals,respectively. We define

F".x/ D P."0 � x/; Fn;".x/ D1

n

nXtD1

."t � x/;

Fn;e.x/ D1

n

nXtD1

.et � x/; Fn;b".x/ D 1

n

nXtD1

.b"t � x/ :First of all, we obtain from the Glivenko–Cantelli theorem that

dL .Fn;"; F"/a:s�! 0: (18)

Next, we will prove that

1

n

nXtD1

jet � "t ja:s:�! 0; (19)

which then implies

dL .Fn;e; Fn;"/a:s:�! 0: (20)

To this end, we split up

1

n

nXtD1

jet � "t j �1

n

nXtD1

j"t jˇ�t .�0/=�t

�b�n� � 1ˇ C 1

n

nXtD1

j"t j�t .�0/

ˇˇ 1b�t � 1

�t

�b�n�ˇˇ :

It follows from (A2) and (4.26) of Francq & Zakoïan (2004) thatE�j"0j sup� W k��0k�ı j�0.�0/=�0.�/ � 1j

< 1 for some ı > 0. Therefore, we obtain from

Lemma 1 that E�j"0j sup� W k��0k�ı j�0.�0/=�0.�/ � 1j

�!ı!0 0. This implies, in con-

junction with b�n a:s:�! �0 and by the ergodic theorem (see, e.g. Theorem 2.3 on page 48 in

Bradley (2007)) that

P

supn�n0

1

n

nXtD1

j"t jj�t .�0/=�t

�b�n� � 1j > "! �!n0!1

0

holds for all " > 0. Therefore, we obtain that

1

n

nXtD1

j"t jˇ�t .�0/=�t

�b�n� � 1ˇ a:s:�! 0: (21)

Because j�2t�b�n� �b�2t j � btnj�20 �b�n� �b�20 j andb�2t � b!n (for t � 1), we obtain

1

n

nXtD1

j"t j�t .�0/

ˇˇ 1b�t � 1

�t

�b�n�ˇˇ � 1

n

nXtD1

j"t j�t .�0/

�t

�b�n�btnj�

20

�b�n� �b�20 jb!n :



Furthermore, because

j�20

�b�n� �b�20 jb!n a:s:�!

j�20.�0/ �b�20 j!0

and

bn

a:s:�! ˇ0 < 1;

we conclude that

1

n

nXtD1

j"t j�t .�0/

ˇˇ 1b�t � 1

�t

�b�n�ˇˇ a:s:�! 0: (22)

Next, (21) and (22) yield (19) and therefore also (20). Moreover, we obtain by the strong law

of large numbers that n�1PntD1 "

2t

a:s:�! E

�"20

D 1. In analogy to the proof of (19), it can be

shown that

1

n

nXtD1

je2t � "2t ja:s:�! 0:

Therefore, we end up with

1

n

nXtD1

e2ta:s:�! 1: (23)

Hence,

dL�Fn;b"; Fn;e� a:s:�! 0: (24)

From (18), (20) and (24), we conclude that

dL�Fn;b"; F"� a:s:�! 0; (25)

as required.

Proof of Lemma 4. Recall that "�t has the distribution function Fn;b". Because E�h"�t2iD

E�"2tD 1, it follows from our Lemma 3 and Lemma 8.3 in Bickel & Freedman (1981) that

d2�"�t ; "t

� a:s:�! 0; (26)

where d2.U; V / D inf°E. QU � QV /2 W QU

dD U; QV

dD V

±denotes Mallows’ distance between the

random variables U and V . Because ."t /t2Z and ."�t /t2Z are both sequences of i.i.d. randomvariables, we can construct, on an appropriate probability space . Q ; QA; QP /, a sequence of i.i.d.random vectors ..Q"�t ; Q"t /

0/t2Z such that

Q"tdD "t

Q"�tdD "�t

and

E QP

h�Q"�t � Q"t

�2iD d2

�"�t ; "t

� a:s:�! 0: (27)



The proof of

E QP

hj QY �t�1 �

QYt�1j ^ 1 C jQ��2t�1 � Q�

2t�1j ^ 1

ia:s:�! 0 (28)

is more delicate because we have to deal here with infinite series. Let " > 0 be arbitrary. Wedefine approximations

Q�2t;K D !0

"1 C

KXkD1

kYiD1

�ˇ0 C ˛0 Q"

2t�i

�#; Q��2t;K D b!n

"1 C

KXkD1

kYiD1

�bn C bn Q"�2t�i�

#:

It follows from (27) and b�n a:s:�! �0 that

QP�j Q�2t;K � Q�

�2t;K j > "

�a:s:�! 0

holds for all K 2 N. Because the infinite series defining Q�2t converges, we have

QP�j Q�2t � Q�

2t;K j > "

�� "

ifKDK."/ is sufficiently large. Furthermore, becauseE�hln�bn C bn"�t 2�i a:s:�! E

hln�ˇ0 C

˛0"20

�i< 0, we have that

QP�j Q��2t � Q��2t;K� j > "

�� "; a:s:

for all n � N , where K� is sufficiently large and non-random and N is random but finite. ForNK D max¹K;K�º, we obtain that

QP�j Q�2t � Q�

�2t j > 3 "

�� 3 "; a:s:

for all n larger than some random but finite value. This, however, implies that

E QP

hj Q��2t�1 � Q�

2t�1j ^ 1

ia:s:�! 0:

The proof of the fact

E QP�j QY �t�1 �

QYt�1j ^ 1 a:s:�! 0 (29)

is similar and therefore omitted.It remains to establish a coupling of . QL�t /t and . QLt /t . To this end, we first show that

E�hRW �0

�b�n�i a:s:�! E�0�RW0.�0/

with W �

0.�/ WD log ��2

0.�/ C Y �2

0=��20.�/ and ��2t .�/ D

!=.1 � ˇ/ C ˛P1kD1 ˇ

k�1Y �2t�k

. In accordance with (4.13) in Francq & Zakoïan (2004),we get

RW �t .�/ WD

1 �

Y �2t

��2t .�/

!1

��2t .�/

@2��2t .�/

@�@� 0C

2Y �2t

��2t .�/� 1

!@��2t .�/

@�

@��2t@� 0

:

We obtain explicit formulas for the derivatives appearing in the formula previously by substi-tuting the original random variables by their bootstrap counterparts in (4.15), (4.16) and (4.20)to (4.22) of that paper. They depend on the parameters and lagged Y �s in a smooth manner, andwe get the desired convergence by dominated convergence theorem from almost sure conver-gence of b�n to � , (29) and from the bootstrap analogues to (4.25) in Francq & Zakoïan (2004)and to Lemma 2.3 in Berkes et al. (2003).



Similarly, we obtain

E QP

�� PQ�t�2 �b�n� = Q��2t �b�n� � PQ�t 2.�0/= Q�2t .�0/��2 a:s:�! 0;

which in turn leads to

E QP kQL�t �

QLtk2 a:s:�! 0: (30)

This finally completes the proof.

Proof of Corollary 1. We have

Q�2t .�/ D

!

1 � ˇC ˛

KXkD1

ˇk�1 QY 2t�k

!C ˛

1XkDKC1

ˇk�1 QY 2t�k DW Q�2t;K.�/ C Rt;K.�/;

and, analogously,

Q��2t .�/ D

!

1 � ˇC ˛

KXkD1

ˇk�1 QY �2t�k

!C ˛

1XkDKC1

ˇk�1 QY �2t�k DW Q��2t;K.�/CR

�t;K.�/:

Because ˇ � �0 < 1 and ˛ � u2 <1 for all .!; ˛; ˇ/0 2 ‚0, we obtain from Lemma 4, forarbitrary K <1,

E QP

"sup�2‚0

ˇQ��2t;K.�/ � Q�

2t;K.�/

ˇ^ 1=3

#� u2

KXkD1

E QP

hˇQY �2t�k �

QY 2t�k

ˇ^ 1=3

ia:s:�! 0:

(31)

To estimate the remainder terms Rt;K.�/ and R�t;K

.�/, we choose any � 2 .1; 1=�0/. Then,

1XkDKC1

QP . QY 2t�k > �k/ D

1XkD1

P.Y 20 > �KCk/

D

1XkD1

P

�2 logY0

log �� K > k

�

� E

�2 logY0

log �� K

�C

�;

which tends to zero as K !1. On the other hand, if QY 2t�k� �k for all k > K, we obtain that

sup�2‚0

Rt;K.�/ � u2

1XkDKC1

�k�10 �k D .�0�/K u2�

1 � �0�:

Because this upper estimate also tends to zero as K !1, we conclude

E QP

"sup�2‚0

Rt;K ^ 1=3

#�!K!1

0: (32)

Finally, because

E��2 logY �

0

log �� K

�C

�P�! E

�2 logY0

log �� K

�C

�;



we get in analogy to (32) that

E QP

"sup�2‚0

R�t;K ^ 1=3

#QP�! 0: (33)

The assertion follows now from (31) to (33).

Proof of Lemma 5. The assertion can be proved along the lines of the proof of Theorem 2.2 inFrancq & Zakoïan (2004) if b��n �b�n D oP�.1/. We proceed in two steps.

(i) Weak consistency of b��n .

Let > 0 be arbitrary. By strong consistency of b�n for �0, it suffices to show that

P ��b��n 2 U�� P

�! 0; (34)

where U� D ¹� W k� � �0k � º \‚0.Before we deduce this conclusion via coupling arguments, some preliminary considerations

involving the behaviour of the log-likelihood process . NLn.�//�2‚0 on the original side are inorder. It can be seen from the proof of Theorem 2.1 in Francq & Zakoïan (2004) that for all� ¤ �0, there exist sufficiently small �.�/ > 0, ı.�/ > 0 and a sufficiently largeM.�/ <1 suchthat, for U.�/ D

®N� W k N� � �k < �.�/

¯\‚0,

E�0

infN�2U.�/

�ln �2t . N�/ C Y 2t =�

2t .N�/�^M.�/

�� E�0

hln �20 .�0/ C 1

iC ı.�/: (35)

Because the set U� is a compact subset of R3 and is covered by the open sets®

N� W k N� � �k < �.�/¯, � 2 U� , we can extract a finite subcover, that is, there exists

� .1/; : : : ; � .N/ 2 U� such that

U�

N[iD1

U.� .i//:

Let Ln;M .�/ D n�1PntD1

®�ln �2t .�/C Y

2t =�

2t .�/

�^M.�/

¯. Because the underlying pro-

cess is strictly stationary and ergodic, we obtain by the ergodic theorem, for M D

max¹M.� .1//; : : : ;M.� .N//º,

P�0

inf�2U�

Ln;M .�/ �1

n

nXtD1

�ln �2t .�0/ C Y 2t =�

2t .�0/

�!

�

NXiD1

P�0

1

n

nXtD1

infN�2U.�.i//

°ln �2t . N�/ C Y 2t =�

2t .N�/±^M

�1

n

nXtD1

�ln �2t .�0/ C Y 2t =�

2t .�0/

�Cı.� .i//

2

!�!n!1

0:

(36)

We now show that the log-likelihood process NL�n, given by

NL�n.�/ D1

n

nXtD1

ln N��2t .�/ CY �2t

N��2t .�/;



behaves in a similar manner as NLn. Here, N��2t .�/ denotes the bootstrap analogue to N�2t .�/.Because we have to deal with a triangular scheme on the bootstrap side, we do not have toolssuch as the ergodic theorem at hand, and a direct imitation of the consistency proof from theoriginal side would be presumably rather cumbersome. Fortunately, some coupling argumentscan be employed to complete the proof in a simple manner.

Recall that b��n is defined as a minimizer of NL�n. In the following, we show that

P ��

inf�2U�

NL�n.�/ > NL�n�b�n�� P

�! 1; (37)

which then implies (34).Let L�n.�/ D n�1

PntD1 ln ��2t .�/ C Y �2t =��2t .�/ be the analogue to NL�n.�/, where only

N��2t .�/ is replaced by the stationary approximation ��2t .�/. We can prove in complete analogyto (i) in the proof of Theorem 2.1 in Francq & Zakoïan (2004) that

sup�2‚0

ˇNL�n.�/ � L�n.�/

ˇD oP�.1/: (38)

Next, we prove that with QL�n�b�n� D n�1Pn

tD1 ln Q��2t�b�n�C QY �2t = Q��2t

�b�n�,

ˇQL�n�b�n� � QLn.�0/ˇ QP

�! 0: (39)

We split up

ˇQL�n�b�n� � QLn.�0/ˇ � 1

n

nXtD1

ˇQ"�2t � Q"

2t

ˇ

C1

n

nXtD1

ˇ�ln Q��2t

�b�n� ^M��

ln Q�2t�b�n� ^M�ˇ

C1

n

nXtD1

�ln Q�2t

�b�n� � M�C

C1

n

nXtD1

�ln Q��2t

�b�n� � M�C

C1

n

nXtD1

ˇln Q�2t

�b�n� � ln Q�2t .�0/ˇ

D Tn;1 C � � � C Tn;5:

It follows from Lemma 4 that

Tn;1 C Tn;2QP�! 0:

We obtain from monotone convergence that

E QP

"sup

� W k��0k�ı

.ln Q�2t .�/ �M/C

#�!M!1

0 (40)

for some ı > 0. Because b�n a:s:�! �0, we conclude that

Tn;3QP�! 0:



Furthermore, it can be deduced similarly to Theorem 3 in Nelson (1991) thatE� Q��s0

�b�n� � Cwith probability tending to one for some s > 0 and C <1. Hence, using Lemma 4 and (40),

Tn;4QP�! 0:

Finally, it follows from E�sup� W k��0k�ı j ln �

20.�/ � ln �2

0.�0/j

!ı!0 0 and b�n a:s:�! �0 that

Tn;5QP�! 0;

which completes the proof of (39).Define QL�

n;M.�/ D n�1

PntD1

�ln Q��2t .�/ C QY �2t = Q��2t .�/

�^M . We obtain from Corollary 1

that

sup�2‚0

ˇQL�n;M .�/ � QLn;M .�/

ˇ QP�! 0: (41)

From (35), (36) and (38) to (41), we obtain (37) and therefore (34).

(ii) Linearization of b��n .

With the same arguments as in the proof of Theorem 2.2 of Francq & Zakoïan (2004), weobtain

�E�hRW �0

�b�n�i��1 1n

nXtD1

�RW �t�N�n;i;j

��i;jD1;2;3

�b��n �b�n� D 1

n

nXtD1

L�t C oP��n�1=2

�

for some N�n;i;j between b�n and b��n , and it remains to prove that

1

n

nXtD1


��i;jD1;2;3

�E�hRW �0

�b�n�i D oP�.1/:

By part (i) of this proof and the bootstrap analogue to (iii) of the proof of Theorem 2.2 inFrancq & Zakoïan (2004), we get

1

n

nXtD1


��i;jD1;2;3

�E�hRW �0

�b�n�i D 1

n

nXtD1

RW �t

�b�n��E�h RW �0 �b�n�i C oP�.1/:

Asymptotic negligibility of the right hand side follows from n�1PntD1

RWt .�0/�E�RW0.�0/

D

oP .1/ (see (vi) of the proof of Francq & Zakoïan (2004)), and E�hRW �0

�b�n�i a:s:�! E�RW0.�0/

if additionally

1

n

nXtD1

RW �t

�b�n� � RWt .�0/ D oP�.1/:This in turn can be deduced from Lemma 4 in conjunction with (4.20) to (4.22) in Francq &Zakoïan (2004).



Proof of Lemma 6. The proof can be carried out in complete analogy to the verification ofLemma 2 if the bootstrap counterparts of the assumptions (A1) and (A2) are satisfied becausethe Bahadur linearization of the estimator (8) is valid by Lemma 5.

Obviously, (A1)(i) is satisfied with b�n instead of �0. The bootstrap innovations ."�t /t are

i.i.d. and have a unit second moment. Moreover, E�hln�bn C bn"�12�i < 0 with probability

tending to one, which then yields the bootstrap version of (A1)(ii).Next, we consider the bootstrap analogue of (A2). Clearly, by Lemma 3, we obtain a non-

degenerate distribution of the "�t2s with probability tending to one. Finally, we have to show

that E��j"�0j4� C with probability tending to one for a finite constant C . In view of (23),

we get

E�hj"�t j

4iD j1C oP .1/jn

�1

nXnD1

e4t D j1C oP .1/jn�1

nXnD1

"4t C oP .1/;

where the latter relation can be verified similarly to (19). Now, the strong law of large numbersimplies the desired boundedness of moments.

Proof of Theorem 1. The coupling of the Xt with the X�t constructed in the proof of Lemma 4implies a coupling of Tn and T �n . Denote by QTn and QT �n the versions of Tn and T �n based onthe coupled variables QXt and QX�t , respectively. Firstly, note that the kernels of the V -statisticsare of the form h.x; y/ D

Rg.x; ´/g.y; ´/Q.d´/ and h�.x; y/ D

Rg�.x; ´/g�.y; ´/Q.d´/,

respectively, where

g.x; ´/ D .x1 � 1/w .´1 � x2; ´2 � x3/ �E�0

"P�21.�0/

�21.�0/

w�´1 � Y0; ´2 � �

20 .�0/

�#0x4;

g�.x; ´/ D .x1 � 1/w .´1 � x2; ´2 � x3/ �E�b�n264 P��1 2

�b�n��1

2�b�n�w

�´1 � Y

�0 ; ´2 � �

�0

2�b�n��

3750

x4:

We obtain by Minkowski’s inequality that

ˇqQTn �

qQT �n

ˇ

D

ˇˇ8<:Z

1pn

nXtD1

g�QXt ; ´

�!2Q.d´/

9=;1=2

�

8<:Z

1pn

nXtD1

g��QX�t ; ´

�!2Q.d´/

9=;1=2

ˇˇ

�

8<:Z

1pn

nXtD1

�g�QXt ; ´

�� g�

�QX�t ; ´

�!2Q.d´/

9=;1=2

:

It follows from the independence of the vectors�Q"t ; Q"�t

�0that for t > s,

E QP�g�QXt ; ´

�� g�

�QX�t ; ´

�j QXs ; QX

�s

�D 0 a:s:



This implies

E QP

�qQTn �

qQT �n

�2�1

n

nXs;tD1

E QP

²Z �g�QXs ; ´

�� g�

�QX�s ; ´

� �g�QXt ; ´

�� g�

�QX�t ; ´

�Q.d´/

³D E QP

Z �g�QXt ; ´

�� g�

�QX�t ; ´

�2Q.d´/

� 2 E QP

Z °�Q"21 � 1

�w�´1 � QY0; ´2 � Q�

20

��Q"�21 � 1

�w�´1 � QY

�0 ; ´2 � Q�

�20

�±2Q.d´/

C 2 E QP

Z 8<:E�0"P�21.�0/

�21.�0/

w.´1 � Y0; ´2 � �20 .�0//

#0QL1

�E�b�n264 P��1

2�b�n�

��1

2�b�n�w

�´1 � Y

�0 ; ´2 � �

�0

2�b�n��

3750

QL�1

9>=>;2

Q.d´/:

The assertion of the theorem now follows from Lemma 4 and

Z 0B@E�b�n264 P��1 2

�b�n��1

2�b�n�w

�´1 � Y

�0 ; ´2 � �

�0

2�b�n��

375�E�0"P�21.�0/

�21.�0/

w�´1 � Y0; ´2 � �

20 .�0/

�#1CA2

Q.d´/a:s:�! 0;

where the latter follows from (A3) and the proof of Lemma 4.

Proof of Lemma 7. We split up

n�1bT �n D Z´1

n

nXtD1

Y �t

2�b��t �2 � 1!w�´1 � Y

�t�1; ´2 �b��2t�1�

μ2Q.d´1; d´2/

� 3

Z ´1

n

nXtD1

�"�t2� 1

�w�Z1 � Y

�t�1; ´2 � �

�2t�1

�μ2Q.d´1; d´2/

C 3

Z ´1

n

nXtD1

Y �t

2�b��t �2 � "�t 2!w�´1 � Y

�t�1; ´2 �b��2t�1�

μ2Q.d´1; d´2/

C 3

Z ´1

n

nXtD1

�"�t2� 1

� �w�´1 � Y

�t�1; ´2 �b��2t�1� � w �´1 � Y �t�1; ´2 � ��2t�1��

μ2Q.d´1; d´2/

DW R�n;1 C R�n;2 C R�n;3

and show asymptotic negligibility for each summand separately.The sum in the integrand of R�

n;1is a sum of martingale differences. Therefore, and because

"�t is independent of�Y �t�1

; ��2t�1

�, we have

E�R�n;1 D3

nE�

�"�12� 1

�2 Zw2

�´1 � Y

�t�1; ´2 � �

�2t�1

�Q.d´1; d´2/

�D OP

�n�1

�



if E�"�1

4D OP .1/. To verify this, it suffices to show that n�1

PntD1 e

kt

P�! EjY 2

0=�2jk=2,

for k D 2; 4. We only consider k D 4 here; the other case can be treated similarly. Firstly, weobtainˇ

ˇ 1n

nXtD1

e4t �1

n

nXtD1

ˇˇ Y 2t

�2t

�b�n�ˇˇ2ˇˇ � OP .1/ 1b! n

nXtD1

btn

ˇˇ Y 2t

�2t�N�0� �2t � N�0��2t

�b�n�ˇˇ2

D oP .1/ (42)

because it can be shown in analogy to Francq & Zakoïan (2004, (4.25)) that for suitable openneighbourhoods Uk

�N�0�

of N�0,

E

0@ sup�2Uk. N�0/

�2t�N�0�

�2t .�/

1Ak <1; k 2 N: (43)

Moreover, Markov’s inequality yieldsˇˇ 1n

nXtD1

ˇˇ Y 2t

�2t

�b�n�ˇˇ2

�1

n

nXtD1

ˇˇ Y 2t

�2t�N�0� ˇˇ2

ˇˇ D oP .1/ (44)

because by (43) and (A4),

E

ˇˇ Y 2t

�2t�N�0� ˇˇ2

ˇˇ1 �

0@ �2t�N�0�

�2t

�b�n�1A2ˇˇ �!n!1

0:

Finally, the ergodic theorem implies that

1

n

nXtD1

ˇˇ Y 2t

�2´ .N�0/

ˇˇ2

a:s:�! E

ˇˇ Y 2

0

�20

�N�0� ˇˇ2

;

which then together with (42) and (44) yields E�"�1

4D OP .1/.

In order to verify that R�n;2

is asymptotically negligible, we show that

1

n

nXtD1

ˇY �2tb��2t � "�2t

ˇD oP�.1/: (45)

To this end, we first consider

1

n

nXtD1

ˇˇY �2tb��2t �

Y �2t

��2t

�b��n�ˇˇ D oP�.1/

nXtD1

�t0Y �2t

��2t

�b��n� : (46)

As E�"�1

4D OP .1/, the term on the right hand side of (46) is of order oP�.1/ if

E�

ˇˇ sup�2U . N�0/ �

�2t .�/

sup Q�2U. N�0/ ��2t

�Q��ˇˇ2

D OP .1/

for some neighbourhood U. N�0/ of �0. The latter can again be verified along the lines of(4.25) in the work of Francq & Zakoïan (2004) noting that under (A6) for some ı > 0,P�E��ln�N C N"�

1

�� ı

��!n!1

1. This also implies that there is a unique stationary solution



��t D ��t

�b�n� to the GARCH equation system on the bootstrap side with probability tending

to one. Now, (45) follows from

1

n

nXtD1

ˇˇ Y �2t

��2t

�b��n� � "�2tˇˇ � kb��n �b�nk2 1n

nXtD1

"�2t

sup�2U. N�0/ k P�t�2.�/k2

sup Q�2U. N�0/ ��2t

�Q�� C oP�.1/ D oP� .1/;

which in turn can be deduced similarly to (4.15) to (4.19) in the work of Francq & Zakoïan(2004).

Finally in view of (A3), R�n;3

tends to zero if

1

n

nXtD1

min°kwk21;

ˇb��2t�1 � ��2t�1 ˇ± D oP�.1/:This in turn can be obtained using similar arguments as before.

Proof of Corollary 2. We prove only (i) because (ii) follows directly from Proposition 2 andLemma 7. [Correction added on 27 May 2015, after first online publication: equation deleted.]

Part (i) of the corollary follows from Proposition 1, Lemma 6 and Theorem 1 if the limitdistribution of bTn is continuous, see also Lemma 2.11 of van der Vaart (1998). The limit variableZ D

Pk kZ

2k

has indeed a continuous distribution if at least one of the ks is non-zero,which in turn follows from

EŒh .X1; X1/� > 0: (47)

Acknowledgements

The authors are grateful to Fabian Tinkl, Friedrich-Alexander Universität Erlangen-Nürnberg,for providing an R-code for QMLE based on GARCH(1,1) processes. The authors thank ananonymous referee for his/her valuable comments that led to a significant improvement ofthe manuscript. This research was partly funded by the German Research Foundation DFG,project NE 606/2-2.

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Received November 2013, in final form March 2015

Anne Leucht, Institut für Mathematische Stochastik, Technische Universität Braunschweig, Pockelsstraße11, Braunschweig, 38106, Germany.E-mail: [email protected]

Supporting information

Additional information for this article is available online including Appendix S1. Detailedproof of Lemma 2.


http://www.R-project.org

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