Asymptotic Expansions of the QMLEs in theEGARCH(1,1) Model�

Antonis Demos (Athens University of Economics & Business)and

Dimitra Kyriakopoulou (University of Piraeus)

Current Version: March 2009

AbstractIn this paper we develop the Edgeworth expansions of the Quasi Maxi-

mum Likelihood Estimators (QMLEs) of the EGARCH(1; 1) parameters andwe derive their asymptotic properties, in terms of bias and mean squared er-ror. The notions of geometric ergodicity and stationarity are also discussedin shedding light on the asymptotic theory of the EGARCH models. We alsoexamine the effect on the QMLEs of the variance parameters by including anintercept in the mean equation. We check our theoretical results by simula-tions. In doing this, we employ either analytic or numeric derivatives and theconvergence properties of the methods will be also discussed.Keywords: Edgeworth expansion; quasi-maximum likelihood; EGARCH

model; bias approximations; asymptotic properties.JEL classi�cation numbers: C13; C22

1 Introduction

The last years there has been a substantial interest in approximating the exact distrib-utions of econometric estimators in time series models and deriving their asymptoticproperties. Although there is an important and growing literature that deals with theasymptotics of the Generalized Autoregressive Conditional Heteroskedastic (GARCH)models, either in terms of consistency and asymptotic normality of the estimators orin terms of the �nite-sample theory, the asymptotic properties of the estimators in theExponential GARCH (EGARCH) process of Nelson [22] have not been fully explored.

� This is a preliminary version, please do not quote. Comments are welcome. Address all corre-spondence to Dimitra Kyriakopoulou, University of Piraeus, Dept. of Financial Management and Bank-ing, 80 Karaoli & Dimitriou Str., Piraeus, 185 34, Greece. E-mail: dimitra.kyriakopoulou@gmail.com.Tel. no. +302108203125.

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Comparing to the GARCH process, the advantages of the EGARCH model are well-known, with the main one being the fact that the model captures the negative dynamicasymmetries noticed in many �nancial series, i.e. the so-called leverage effects.The asymptotic aspects of the conditionally heteroskedastic models have been dis-

cussed under many different considerations, in order to analyze the statistical propertiesof these estimators. Since the important work of Engle [11] and that of Bollerslev [2],who introduced the Autoregressive Conditional Heteroskedasticity (ARCH) and Gen-eralized ARCH models, respectively, a huge amount of literature on the asymptoticshas appeared in short time. Weiss [28] proved Consistency and Asymptotic Normal-ity (CAN) of the maximum likelihood estimators in ARCH models, assuming normaldistribution of the errors and imposing a rather restrictive condition that the data havebounded fourth moments, excluding in that way from the proof many other interestingconditionally heteroskedastic models. Quite parallel, Lee and Hansen [16] and Lums-daine [18] relaxed the condition which Weiss imposed and they looked at the conse-quences of the possible failure of the normality assumption on the errors, providingconditions under which CAN exist in the GARCH(1; 1) speci�cation (for multivariateframeworks see e.g. Jeantheau [15]; Comte and Lieberman [6]).The problem of proving stationarity in the GARCH context has not been solved

until Nelson [21] provided conditions for the existence and uniqueness of a stationarysolution in the GARCH(1; 1) case. Mixing and moment properties have been inves-tigated for various GARCH and stochastic volatility models, see Carrasco and Chen[5]. The small sample properties of the estimators in the �rst order GARCH model areinvestigated through an asymptotic expansion of the Edgeworth type, as Linton [17]developed1. Nowadays, many researchers work on the asymptotic behaviour of theseestimators, with unceasing interest.Until the in�uential work of Nelson [22], the conditional heteroskedastic models

that had been developed could not explain the asymmetry effects, indicating that al-ternative models might be suitable for �nancial applications. Turning our attentionto asymmetric GARCH models, and more speci�cally to the EGARCH model whichhas become a popular model in applied work, very little is known about its statisti-cal properties. Although we are endowed with the moment structure investigated byHe, Terasvirta and Malmsten [13], the limiting properties of the maximum likelihoodestimators do not exist in the literature. The interest in consistency and asymptoticnormality results of EGARCH has been growing and the problem of the theoreticalproperties not yet been explored await for an answer; see, for example, Straumann andMikosch [26].In this paper we develop the Edgeworth expansions of the Quasi Maximum Like-

lihood Estimators (QMLEs) of the EGARCH(1; 1) parameters and we derive their as-ymptotic properties, in terms of bias and mean squared error. The notions of geometric

2

ergodicity and stationarity are also discussed in shedding light on the asymptotic the-ory of the EGARCHmodels. We also examine the effect on the QMLEs of the varianceparameters by including an intercept in the mean equation. We check our theoreticalresults by simulations. In doing this, we employ either analytic or numeric derivativesand the convergence properties of the methods will be also discussed.

1.1 Literature Review on the EGARCH modelThe literature on GARCH models is really vast, so that readers are refered for exampleto the survey papers of Bera and Higgins [1] and Bollerslev, Chou and Kroner [3],as well as Bollerslev, Engle and Nelson [4]. These papers review the properties andthe estimation techniques, as well as the statistical tests that have been developed andthey comprise the origin that all researchers in this �eld should begin with. Here, wereview some papers that mostly deal with the features and the statistical properties ofthe EGARCH model.In his seminal paper, Nelson [22] pointed out that the simple structure of the GARCH

and related processes, i.e. symmetric non-linear models, is not the most importantcriterion in modelling the �nancial time series. He underlined that there are someunexceptionable drawbacks of the GARCH theory indicating that a new approach isnecessary to be established. The Exponential GARCH model of Nelson meet theselimitations, which can be summarized thereupon. First, volatility tends to rise in re-sponse to "bad news" and fall in response to "good news", so that a sign effect shouldbe introduced in the model. Second, the strict assumption of nonnegative coef�cientsis not imposed as the logarithm of the conditional variance ensures that there is alwayspositivity. Third, interpretation of the persistence of shocks to conditional variance ispossible through the EGARCH context. Nelson also derived a necessary and suf�cientcondition for strict stationarity of the EGARCH process, when lnh2t has an in�nitemoving average representation (see Theorem 2.1, p. 351).Moreover, in the paper of He, Terasvirta and Malmsten [13] we are given the mo-

ment structure of the standard EGARCH(1; 1) model and without assuming normalityof the errors. The existence of unconditional moments and the kurtosis, as well asthe properties of the autocorrelation function of positive powers of absolute observa-tions are examined. The results presented there are important for comparisons with theGARCH model. For an analysis which assumes normality and presents an alternativetechnique for the expression of the autocorrelation function of squared observations,the reader is refered to the paper of Demos [10].The �nite sample properties of the maximum likelihood and quasi-maximum likeli-

hood estimators of the EGARCH(1; 1) process using Monte Carlo methods have beenexamined in the paper of Deb [9]. He used, however, response surface methodology inorder to examine the �nite sample bias and other properties in interest.

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Recently, Dahl and Iglesias [8] analysed the limiting properties, in terms of consis-tency and asymptotic normality, of the estimated parameters in a related but in manyaspects different to the traditional EGARCH model of Nelson, [22]. More speci�cally,they allow for a variant speci�cation of the conditional variance in which the stan-dardised residuals (zt�1), which are introduced in the next section, are replaced by theobserved process and as a consequence the recursive nature of zt�1 in the conditionalvariance function no longer exists. Automatically, this alteration makes the wholeproofs easier, but the investigation of the asymptotic properties of Nelson's model stillremains unsolved.Another related paper is that of Perez and Zaffaroni [23] who derived and com-

pared the �nite sample properties of maximum likelihood and Whittle estimators inEGARCH models. This paper considers also the Fractionally Integrated EGARCH(FIEGARCH) model to represent the long-memory behaviour observed in the corre-lations of squared returns. The results of the paper are illustrated by a Monte Carloexperiment. The outcome of such an analysis is that if someone performs Whittleestimation, its performance is comparable to maximum likelihood, especially whenconsidering the bias of the estimates, while the maximum likelihood is often superiorin terms of mean squared error.To the best of our knowledge, in terms of consistency and asymptotic normality, the

theoretical properties of the QMLEs in the EGARCH model have not been studied inthe literature. Straumann and Mikosch [26] established the asymptotic properties ofthis model but of a lower order, with stochastic recurrence equations methods. Theyencountered dif�culties of establishing invertibility of the model, which is critical inorder to make sure that the likelihood function is well-behaved asymptotically.The organisation of the paper is as follows: Section 2 presents the model and esti-

mators. Section 3 deals with the main results of our analysis. First, analytic derivativesand their expected values are presented. Second, conditions for stationarity of the log-variance derivatives are investigated. In the sequel, the Edgeworth expansions of theQuasi Maximum Likelihood Estimators are derived and theoretical bias approxima-tions of the estimators are calculated. Finally, Section 4 concludes. All proofs, ratherlengthy, are collected in the Appendix at the end. Let us now turn our attention to thede�nition of the EGARCH(1; 1) model and its statistical properties.

2 The Model and Estimators

Let us consider the following model, where the observed data fytgTt=1 are generatedby the EGARCH(1; 1) process, see Nelson [22], in which the conditional variance, ht,depends on both the size and the sign of the lagged residual:

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yt = �+ ut; t = 1; :::T; where (1)ut = zt

pht;

utj At�1 � N (0; ht) ; zt � iid (0; 1)

ln (ht) = �+ �zt�1 + g (zt�1) + � ln (ht�1) ; where (2)g (zt) = jztj � E jztj

futg is a real-valued discrete time stochastic process (the error process) and ht is a pos-itive with probability one At�1-measurable function (the conditional variance), whereAt�1 is the sigma-algebra generated by the past values of zt, i.e. fzt�1; zt�2; zt�3; :::g.The function g (zt) is a well-de�ned function of zt. The process ht is not observedand thus is constructed via recursion using the estimating values of the parameters anda proper initial value for the conditional variance. The only distributional assumptionmade about the innovations z0ts is that they are independently and identically distributed(iid) with zero mean and unit variance. We do not impose any symmetric distributionalproperty, however the proofs automatically become very tedious, but also very chal-lenging. The conditional variance is constrained to be non-negative by the assumptionthat the logarithm of ht is a function of past z0ts. Comparing to relative research, theNelson's paper was the �rst which models the conditional variance as a function ofvariables which are not solely squares of the observations.It is useful to rewrite our model in the multiplicative form; from de�nition (2) it

follows that

ht = exp f�+ �zt�1 + g (zt�1) + � ln (ht�1)g == exp f�g exp f�zt�1g exp f g (zt�1)gh�t�1: (3)

The equations of the EGARCH process create a complicated probabilistic structurewhich is not easily understood. Note from eq. (2) that ln (ht) constitutes a causal AR(1)process with mean �= (1� �) and error sequence [�zt�1 + (jzt�1j � E jzt�1j)]. Theunique stationary solution to (2), provided that j�j < 1, is given by its almost sure(a.s.) representation:

ln (ht) = � (1� �)�1 +1Xk=0

�k (�zt�1�k + g (zt�1�k)))

ln (ht) � � (1� �)�1 a:s:

The conditional variance responds asymmetrically to rises and falls in stock price,which is believed to be important for example in modelling the behaviour of stockreturns. It is an important stylized fact for many assets. The coef�cients (� + ) and

5

(� � ) (if zt � 0 and zt < 0, respectively) show the asymmetry in response to positiveand negative yt. The parameter � is referred to as the leverage parameter, which showsthe effect of the sign of yt. The term [jztj � Ejztj] represents a magnitude effect.Formulae for the higher order moments of ut are given in Nelson [22]. The parameter� can be made a function of time (�t) to accommodate the effect of any non-tradingperiods of forecastable effects.The unconditional mean and variance of yt is:

E (yt) = �;

and

V ar (yt) = exp

��

1� �

� 1Yi=0

E�exp

��i (�z0 + g (z0))

��;

which, under normality of the errors, becomes the following result:

V ar (yt) = exp

0@�� q

2�

1� �

1A 1Yi=0

"exp

�2i ( �)2

2

!���i �

�+ exp

��2i�2

2

����i��#;

where � = + �, � = � � and � (k) is the value of the cumulative standard Normalevaluated at k, i.e. � (k) =

R k�1

1p2�exp

��x2

2

�dx.

Proof. The proof of the unconditional variance is given in the Appendix.The last expressions above may be dif�cult to compute. We have to approximate

the in�nite products by a product with a �nite number of terms. Such expressions willarise in many different points below in our analysis.To estimate the parameters of the model in eq. (1) and eq. (2), we employ the quasi-

maximum likelihood estimation. Maximum likelihood is the procedure which is mostoften used in estimating the parameters in time series models, but for most applicationsit is very dif�cult to justify the conditional normality assumption. Therefore, the log-likelihood function may be misspeci�ed. However, we can still obtain estimates bymaximizing a Gaussian quasi-log-likelihood function and under the auxiliary assump-tion of an iid distribution for the standardized innovations z0ts. The estimators whichare derived by this maximization problem are the so-called Quasi Maximum Likeli-hood Estimators (QMLEs). The fact that the we maximize a quasi-log-likelihood isjusti�ed by the evidence that distributions of asset returns are often thick tailed andas a consequence the normality assumption is violated and hence the likelihood is notwell speci�ed.An important and really interesting feature of our model is that the assumption of

the block diagonality of the information matrix no longer holds. This is also the casefor the ARCH-Mmodel and the asymmetric model of the Augmented ARCH (see Bera

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and Higgins [1], p. 349; also Bollerslev, Engle and Nelson [4], p. 2981). This impliesthat the off-diagonal blocks involving partial derivatives with respect to both meanand variance parameters are not null matrices, while this is the case in other GARCH-type models. Below we present analytic proofs of this argument in the context of theEGARCH(1; 1) model and these results disaccord with Malmsten [19], even if thedistribution of the innovations is symmetric, which implies that Ez3 = 0.Nelson [22] proposes to estimate the EGARCH model by maximum likelihood as-

suming that the innovations have a Generalized Error Distribution (GED). The appealof this distribution comes from the fact that it nests the normal distribution as a specialcase, and can also display both thinner and thicker tails than the normal, depending onthe value achieved by the tail thickness parameter2.In the EGARCH(1; 1) model, there is no explicit expression of the probability den-

sity of the vector (y1; :::; yT )0 since the distribution of (h1; :::; hT )0 is not known. Toovercome this dif�culty, we consider an approximate conditional log-likelihood in-stead. Some assumptions are also required for the initial values of the conditionalvariance ht, which should be drawn from the stationary distribution, and the squaredstandardized residuals z2t . Assuming that z0 = 0 and ln (h0) = �

1�� , we obtain a goodapproximation to the conditional Gaussian log-likelihood, as follows:

` (�; �; �; �; j z0; h0) = �T2ln (2�)� 1

2

TXt=1

ln (ht)�TXt=1

(yt � �)2

2ht=

= �T2ln (2�)� 1

2

TXt=1

ln (ht)�1

2

TXt=1

z2t : (4)

Notice that ht and zt are both functions of ! and �, where ! = (�; �; �; )=, i.e. thevector of unknown log-variance parameters, so that both are functions of'=

�!=; �

�=,which represents the vector of all unknown parameters. The �rst order conditions arerecursive and consequently do not have explicit solutions.The likelihood function is derived as though the errors are conditionally normal and

is still maximized at the true parameters. Having speci�ed the log-likelihood function,the quasi maximum likelihood estimator is then de�ned as

c�T = argmax�2�

1

T

TXt=1

` (�) : (5)

Because the likelihood function is misspeci�ed, the form of the covariance matrix ofthe QMLEs is more complex than the usual inverse of the information matrix, seeWhite [29] and [30]. More speci�cally, if we assume that the random functions ht (')

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are almost surely twice continuously differentiable then the following matricesG0 andF0 are well-de�ned:

F0 = � 2

E (z40 � 1)G0;

G0 =E (z40 � 1)

4E

�1

h20

�@h0 ('0)

@'

�0�@h0 ('0)

@'

��:

The asymptotic covariance matrix is given by F�10 G0F�10 . If z0 is actually standard

Gaussian, then G0 = �F0 and thus the covariance matrix asymptotically is given bythe well-known result of the inverse of the information matrix. There are stronglyconsistent estimators of the matricesG0 and F0; indeed:

cAn =1

n(Li)0 (Li)

a:s:! G0; cBn = 1

nLij

a:s:! F0;

where Li = @`('0)@'

and Lij = @2`('0)@'@'0 ; for i; j 2 f�; �; �; �; g.

In summary, the steps in the estimation method are the following. If the type of thedistribution of zt is not speci�ed, it is impossible to determine a likelihood function.A common approach in such a situation is to suppose that zt � iidN (0; 1). Now itis possible to determine a so-called Gaussian quasi log-likelihood and solving a maxi-mization problem to derive the quasi maximum likelihood estimator. Instead of usingthe exact log-likelihood, we concentrate on �nding an approximation to a conditionallikelihood. We then make use of proper initial values in order to obtain a good ap-proximation to the conditional likelihood. In that way, we derive an estimator which isasymptotically equivalent to the QMLE that we obtained as the solution of the maxi-mization problem.Let us proceed with the main results of our analysis, beginning with the analytic

derivatives of the log-likelihood function and their expected values.

3 The Main Results

3.1 Analytic derivatives and their expected valuesIn this section we present the derivatives of the log-likelihood function and their ex-pected values, which are needed in the sequel to calculate the cumulants of the Edge-worth distribution and to evaluate the asymptotic bias of the QMLEs. It is of greatimportance to mention that there are no such analytic results in the literature, and it isespecially this feature that makes this analysis to differ from the previous one, that ofLinton [17], who studied the case of the GARCH(1; 1) model.

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Before presenting our �rst results, it is worthwhile to discuss the usefulness of theanalytic derivatives and their expected values, in terms of the asymptotic theory. Thelimit behaviour of the sequences of functions of �rst and second order derivatives ofthe log-likelihood function are essential to be studied for establishing the asymptoticnormality of the QMLEs. By means of a Taylor expansion of the score function of thelog-likelihood, the asymptotic normality is established. Consequently, one has to studythe differentiability properties of the conditional variance, which is a priliminary butcrucial step in large-sample theory. Moreover, the expected values of the derivativesof the log-likelihood are needed in order to calculate the cumulants of the approximatedistribution and these results are then used to calculate the bias expressions of theestimators. Let us �rst proceed with the derivatives of the log-likelihood function andtheir analytic representation.

Following the notation employed in Linton (1997), i.e. ht;� = @ ln(ht)@� , the derivatives

of the log-likelihood function with respect to all the parameters are:

L� =1

2

TXt=1

�z2t � 1

�ht;� +

TXt=1

ztpht;

L�� =1

2

TXt=1

�z2t � 1

�ht;�;� �

TXt=1

�1

ht+ 2

ztphtht;� +

1

2z2t h

2t;�

�;

L��� =1

2

TXt=1

�z2t � 1

�ht;�;�;� + 3

TXt=1

1

htht;�

�3TXt=1

ztpht

�ht;�;� � h2t;�

�� 12

TXt=1

z2t�3ht;�ht;�;� � h3t;�

�while for i; j; k 2 f�; �; ; �g the derivatives are:

Li =1

2

TXt=1

�z2t � 1

�ht;i;

Lij =1

2

TXt=1

�z2t � 1

�ht;i;j �

1

2

TXt=1

z2t h2t;i;

Lijk =1

2

TXt=1

�z2t � 1

�ht;i;j;k �

1

2

TXt=1

z2t�3ht;iht;j;k � h3t;i

�:

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The cross derivatives are given by the following expressions:

Li� =1

2

TXt=1

�z2t � 1

�ht;i;� �

1

2

TXt=1

z2t ht;iht;� �TXt=1

ztphtht;i;

Li�� =1

2

TXt=1

�z2t � 1

�ht;i;�;� � 2

TXt=1

ztpht(ht;i;� � ht;iht;�)

�12

TXt=1

z2t�2ht;�ht;i;� � ht;ih

2t;� + ht;iht;�;�

�+

TXt=1

1

htht;i;

Lij� =1

2

TXt=1

�z2t � 1

�ht;i;j;� �

TXt=1

ztpht(ht;i;j � ht;iht;j)

�12

TXt=1

z2t (ht;jht;i;� � ht;jht;iht;� + ht;i;jht;� + ht;iht;j;�) :

Note that the log-likelihood derivatives are expressions of the log-variance deriva-tives, ht;�, where the latter are given in the Appendix. The expected values of thelog-likelihood derivatives are also given in the Appendix.The cross-products of the log-likelihood derivatives are:

For i; j 2 f�; �; ; �g,

LiLij =1

2

TXt=1

�z2t � 1

�ht;i

1

2

TXt=1

�z2t � 1

�ht;i;j �

1

2

TXt=1

z2t h2t;i

!;

LiLj� =1

2

TXt=1

�z2t � 1

�ht;i

1

2

TXt=1

�z2t � 1

�ht;j;� �

1

2

TXt=1

z2t ht;jht;� �TXt=1

ztphtht;i

!;

LiL�� =1

2

TXt=1

�z2t � 1

�ht;i

"1

2

TXt=1

�z2t � 1

�ht;�;� �

TXt=1

�1

ht+ 2

ztphtht;� +

1

2z2t h

2t;�

�#;

L�Lij =

1

2

TXt=1

�z2t � 1

�ht;� +

TXt=1

ztpht

! 1

2

TXt=1

�z2t � 1

�ht;i;j �

1

2

TXt=1

z2t h2t;i

!;

L�Lj� =

1

2

TXt=1

�z2t � 1

�ht;� +

TXt=1

ztpht

! 12

PTt=1 (z

2t � 1)ht;j;� � 1

2

PTt=1 z

2t ht;jht;�

�PT

t=1ztphtht;i

!;

L�L�� =

1

2

TXt=1

�z2t � 1

�ht;� +

TXt=1

ztpht

!"12

PTt=1 (z

2t � 1)ht;�;�

�PT

t=1

�1ht+ 2 ztp

htht;� +

12z2t h

2t;�

� # :10

The expectations of the cross-products are given in the Appendix.Let us turn our attention to the conditions for stationarity of the log-variance deriv-

atives.

3.2 Conditions for stationarity of the log-variance derivativesIn this section we investigate under which conditions there is a stationary solution tothe log-variance derivatives, needed for the existence and the evaluation of the log-likelihood derivatives, and hence in order to calculate the bias expressions of the QM-LEs. The existence, stationarity and ergodicity of the second order derivatives of theconditional variance are necessary so that the Taylor expansion of the �rst order deriv-atives of the log-likelihood is validated.Let the following:

ht;�ht;�� =1

4(�zt�1 + jzt�1j)h2t�1;� +

1

4(�zt�1 + jzt�1j)

�� � 1

2�zt�1 �

1

2 jzt�1j

�h3t�1;�

+

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1;�;�

+

�� � 1

2�zt�1 �

1

2 jzt�1j

�2ht�1;�ht�1;�;�: (6)

In order to calculate the expected value of the above expression, we �rst assume thatE�h2t;��; E�h3t;��and E (ht;�;�) exist. Next, de�ne:

A (zt�1) =1

4(�zt�1 + jzt�1j)h2t�1;�

+1

4(�zt�1 + jzt�1j)

�� � 1

2�zt�1 �

1

2 jzt�1j

�h3t�1;�

+

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1;�;�;

and

B2 (zt�1) =

�� � 1

2�zt�1 �

1

2 jzt�1j

�2:

Then,

ht;�ht;�� = A (zt�1) +B2 (zt�1)ht�1;�ht�1;�;� =

= A (zt�1) +

1Xk=1

k�1Yi=0

B2 (zt�1�i)A (zt�1�k) :

11

The in�nite sum converges almost surely. To see this, let:

Sn = A (zt�1) +nXk=1

k�1Yi=0

B2 (zt�1�i)A (zt�1�k) :

Then we have:

E (Sn) = E [A (zt�1)] +

nXk=1

E

"k�1Yi=0

B2 (zt�1�i)

#E [A (zt�1�k)] =

= E [A (zt�1)]

"nXk=0

�E�B2 (zt�1�i)

�k#:

Thus, E (limn!1 Sn) = E [A (zt�1)] f1� E [B2 (zt�1�i)]g�1 < 1, providing thatE [A (zt�1)] < 1. In order to ensure the existence of a stationary solution to the eq.(6), we should impose the condition that���B2 (zt�1�i)

��� < 1:In a similar manner, the rest stationarity conditions of all log-variance derivatives

and the products of them follow. Let us now proceed with the asymptotic expansionsof the QMLEs.

3.3 Asymptotic Expansions of the QMLEsThere are relatively few papers that theoretically investigate the small sample proper-ties of estimators in non-linear time series models due to the dif�culty of the analy-sis and the awkward expressions that one can end up with (e.g. Iglesias and Phillips[14]). In particular, analytical work on bias and mean squared error of the estimatorsin the general context of GARCH models has not received much attention in the liter-ature, until Linton [17] calculated the formal Edgeworth distribution function for theGARCH(1; 1) process.Due to the dif�cult expressions for the bias of the estimators, Linton used numerical

integration in order to evaluate their magnitude. We make one step further and presentanalytic results of the expected values of the log-likelihood derivatives needed in thesequel in order to calculate the bias approximations. These expressions are presentedin such a form so that one may choose a speci�c distribution for the errors and endup with the desired results. These theoretical outcomes can then be checked throughsimulations in terms of accuracy and be compared with numeric derivatives.The Edgeworth expansions have a long history in econometrics3. Taniguchi, [27],

developed the expansions for many time series models. The Edgeworth expansion in

12

the likelihood context is derived through a speci�c method, which expands the scorefunctions of the log-likelihood in a power series in the likelihood derivatives aboutthe true parameter value and then inverts this expansion to yield an expansion for thestandardized estimator. Let b� be an estimator of � and

� =pn�b� � �

�The Edgeworth expansion of �, with an error of O (T�1), is given by

P�� � m

�= �

�m!

�� �

�m!

�� 0 + 1

�m!

�+ 2

�m!

�2�;

where � (z) = 1p2�exp

�� z2

2

�and � (�) =

R ��1 � (z) dz.

That is, as we expresspT�b� � �

�as a function of the �rst and second order deriv-

atives of the log-likelihood, standardised appropriately and evaluated at the true para-meter values, we are then able to calculate the approximate Edgeworth distribution.Let ' =

��1; �2; �3; �4; �5

�==�p

T (b�� �) ;pT (b�� �) ;

pT�b� � �

�;pT (b � ) ;

pT�b� � �

�;�=. Consider

the Taylor expansion for 1pT

@`(b')@', where b' = �b�; b�;b�; b ; b��= around the true value

' = (�; �; �; ; �)= as:

0 =1pT

@`

@�j+1

T

5Xk=1

@2` (')

@�j@�k�k +

1

2T 3=2

5Xk;l=1

@3` ('�)

@�j@�l@�k�k�l;

where j = 1; :::; 5 and 1pT@`@�j, @2`@�j@�l

are evaluated at the true values and @3`@�j@�l@�k

isevaluated at an intermediate point, say '�, between the estimates and the true values.The above expression can be written as

0 =1pT

@`

@�j+

5Xi=1

�Aji +

wjipT

��i +

1

2pT

5Xk;i=1

Kjik�i�k +Op�T�1

�� fj ('; v) +Op

�T�1

�; j = 1; :::; 5

where Aij = 1TE�@2`(')@�j@�i

�, Kjik =

1TE�

@3`(')@�j@�i@�k

�,

wij =1pT

�@2` (')

@�j@�i� TAij

�;

13

for i; j; k = 1; :::; 5. Let us de�ne a vector v containing the non-zero elements of1pT@`@�iand wij , for i; j = 1; :::; 5. Solving for �j , and j = 1; :::; 5, as continuously

differentiable functions of v, gives:

�j (v) =

qXa=1

@�j (0)

@vava +

1

2

qXa;b=1

@2�j (0)

@va@vbvavb +Op

�T�1

��

qXa=1

e(j)a va +1

2pT

qXa;b=1

e(j)ab vavb +Op

�T�1

�where e(j)a =

@�j(0)

@va, e(j)ab =

pT@2�j(0)

@va@vb, and q is the dimension of the vector v.

Now the derivatives can be found by solving the following system of equations, forj; k = 1; :::; 5 and a; b; c = 1; :::; q:

0 =5Xk=1

Ajke(k)a +

@fj (0; 0)

@va;

0 =5Xk=1

1pT

5Xl=1

Kjkle(l)b +

@2fj (0; 0)

@vb@e�k!e(k)a +

5Xk=1

@2fj (0; 0)

@va@e�k e(k)b

+1pT

5Xk=1

Ajke(k)ab :

The expected values of all the elements of the vector v; together with the derivativesea, eab, are then used in order to calculate appropriately the polynomial coef�cients iin the Edgeworth expansion. Let us turn our attention to the theoretical bias approxi-mations of the QMLEs.

3.4 Bias ApproximationsIn this section we develop the bias approximations for the QMLEs, under two dif-ferent considerations. We distinguish between the case when the mean parameter isknown, hence not estimated, and the case when this parameter is unknown and shouldbe estimated with the other parameters of the model. One of the main advantages ofdeveloping the bias expressions is to use them as a bias correction mechanism. This isone of the practical applications of the bias approximations. A bias reduction is onlypossible in the case we are familiar with the exact bias expressions. Moreover, theseresults help to analyse the consequences of introducing restrictions in the log-varianceparameters. With these expressions, one can compute the Edgeworth approximate dis-tribution. It is important to explore the theoretical properties of the estimators so thatthe statistical inference is possible.

14

We use a McCullagh [20] result for the standardized estimator having a stochasticexpansion, see in [20] p.209, and taking expectations we end up with the asymptoticbias of the QML estimator. Our next step is to check our bias approximations throughsimulation. Note that McCullagh's expansion has already been applied in the literatureto retrieve the bias in many nonlinear models, such as Linton [17]. When dealing withnonlinear models, it is very common to have the bias expressions in terms of expecta-tions and applying these expressions for bias correction. At this point, it is importantto state brie�y the main differences between our analysis and that of Linton. Firstof all, we generalize the �nite-sample analysis of heteroskedastic time series modelsconsidering a non-symmetric distribution of the errors. Furthermore, we show that theblock-diagonality of the information matrix does not hold in our case, which meansthat there are new terms in the bias expressions of the estimators.

Assumption 3.4.1 We assume that the errors have third and fourth order cumu-lants, denoted by �3 and �4, respectively, where the latter is given by:

�4 = E�z4 � 3

�:

We further assume that the errors have bounded J th moments, for some J > 6,so that the bias expressions exist.

Under the above assumptions, we are now able to present our Theorem which is use-ful for the evaluation of the bias approximations of all estimators and also to constructthe Edgeworth expansions in this setting.

Theorem 3.4.1 Given that zt � iid (0; 1) and non-symmetric, and for i; j; k 2f�; �; �; ; �g unless the parameter � is used seperately to underline the differ-ence, the following moments of the log-likelihood derivatives converge to �nitlimits as T !1:cij =

1TE (Lij) = �1

2� i;j;

cijk =1TE (Lijk) = �1

2(� ij;k + � ik;j + � jk;i � � i;j;k) ;

cij;k =1TE (LijLk) = �1

4

�� zzk;i;j � (�4 + 2) (� ij;k � � i;j;k)

�;

c�� =1TE (L��) = �

�� + ��;�

2

�;

ci�� =1TE (Li��) = �i � 1

2(� i;�� + 2��i;� � ��;i;�) ;

c��� =1TE (L���) = �1

2

�3���;� � � 3�

�+ 3��;

ci�;� =1TE (Li�L�) = �1

4

�4�i � (�4 + 2) (� i�;� � � i;�;�)

+� zz�;i� + 2�zhi;� + 2�3

�2�hi;� � �hi�

� � ;ci�;j =

1TE (Li�Lj) = �1

4

�� (�4 + 2) (� i�;j � � i;j;�) + � zz�;i� + 2�3�

hij

;

15

c��;i =1TE (L��Li) = �1

4

�� (�4 + 2) (���;i � � i;�;�) + � zzi;�� + 4�3�

hi;�

;

cij;� =1TE (LijL�) = �1

4

�� (�4 + 2) (� ij;� � � i;j;�) + � zz�;ij + 2�

zhi;j

+2�3�2�hi;j � �hij

� �;

c��;� =1TE (L��L�) = �1

4

�8�� � (�4 + 2) (���;� � ��;�;�)

+� zz�;�� + 2�zh�;� + 2�3

�3�h�;� � �h��

� � ;where � i = 1

T

PTt=1E (ht;i) ; � i;j =

1T

PTt=1E (ht;iht;j) ; � ij;k =

1T

PTt=1E (ht;ijht;k)

and � i;j;k = 1T

PTt=1E (ht;iht;jht;k).

Also, � = 1T

PTt=1E

�1ht

�, and �i = 1

T

PTt=1E

�1htht;i

�,

while � zzk;i;j = 1T

PPs<t

E [(z2s � 1)hs;kht;iht;j] ; � zhi;j = 1T

PPs<t

E�zs

1phtht;iht;j

�;

�hi;� =1T

PTt=1E

�1phtht;iht;�

�and �hi� = 1

T

PTt=1E

�1phtht;i;�

�.

Proof. Given in the Appendix.The basic approach to �nding bias approximations requires that we �nd expres-

sions for the cij; cijk and cjk;l. Let us �rst condider the case when the mean parameteris known and not estimated. With techniques of McCullagh [20], the standardized es-timators, derived from choosing � to solve Li (�; �) = 0, for i 2 f�; �; ; �g, have thefollowing stochastic expansions4:

pTnb�i � �i

o� �cijZj +

1pT

�cijcklZjkZl � cijcklcmncj lnZkZm=2

+OP

�1

T

�;

(7)where

Zj = T�1=2Ljand

Zjk = T�1=2 fLjk � E (Ljk)gare evaluated at the true parameters and are jointly asymptotically normal. Raisingpairs of indices signi�es inversion, i.e. cij = (cij)�1.Taking expectations of the right-hand side in eq. (7), we get:

Ehp

T'0nb� (�)� �

oi� 1p

T'ic

ijckl fcjk;l + cjkl (�4 + 2) =4g ;

where ' is the 5�1 parameter vector. If �4 = 0, QML equalsML and then the aboveformula equals the one of Cox and Snell (1968), i.e.:

16

Ehp

T'0nb� (�)� �

oi� 1p

T'ic

ijckl�cjk;l +

1

2cjkl

�:

Let us now consider the other case, when the mean parameter is unknown and esti-mated with the other parameters. Hence, if we incorporate the effects of estimating �,the stochastic expansions take the following form:

pTnb�i (b�)� �i

o�pTnb�i (�)� �i

o� 1p

T

�cijcklZjkZl � cijcklcmncj lnZkZm=2

;

where now i; j; k; l 2 f�; �; ; �; �g. Taking expectations of the right-hand side, we�nd the asymptotic bias of the estimators in this case.In terms of the mean squared error, from eq. (7) we have up to OP

�1T

�:

Ehp

T'0nb� (�)� �

oi2� �'icij (�4 + 2) =2; (8)

which is the asymptotic variance. If we let the remainder to be of O�T�3=2

�, then the

mean squared error is again evaluated by eq. (8), with the difference now that therewould be added terms of O (T�1). Of course, as T ! 1, the mean squared errorapproaches the asymptotic variance.

4 ConclusionsGaussian quasi-maximum likelihood estimation is a popular method which is widelyused for inference in time series models. In this paper we study the asymptotic proper-ties of the QMLEs in the EGARCH(1; 1)model of Nelson. The interest of establishingthese theoretical properties has been growing but it remains a problem that awaits foran answer. In the current context, we present analytic derivatives both of the log-likelihood and the log-variance functions and also their expected values. We furtherdevelop theoretical bias approximations for the QMLEs of the parameters and we �ndconditions for stationarity of the log-variance derivatives. The theoretical results in thispaper can be used to bias-correct the QMLEs in practice directly. In small or moderate-sized samples, a bias correction could be appreciable and it is helpful to have a roughestimate of its size.The next steps in our research are to check the whole theoretical results through

simulations and then examine the sensitivity of the bias approximations to different as-sumptions on the variance parameters. An interesting topic would be the investigationof necessary and suf�cient conditions for the existence and validity of the Edgeworthapproximations in this context. These issues are ongoing research.

17

Notes1The validity of the Edgeworth expansions in the GARCH model is established in the paper of

Corradi and Iglesias, [7].2In a recent paper, Zaffaroni [31] estimates the EGARCH parameters with Whittle methods and the

asymptotic distribution theory of these estimators is established.3For an excellent review on the asymptotic expansion of the Edgeworth type, the reader is refered to

Rothenberg, [24].4We make use of the summation convention, that is: cijZj =

Xj

cijZj .

5 References

[1] Bera, A. K. and Higgins, M. L. (1993) ARCH models: properties, estimation andtesting, Journal of Economic Surveys, 7, 305-362.

[2] Bollerslev, T. (1986) A generalized autoregressive conditional heteroskedasticity,Journal of Econometrics, 31, 307-327.

[3] Bollerslev, T., Chou, R. Y. and Kroner, K. F. (1992) ARCH modelling in �nance:a review of the theory and empirical evidence, Journal of Econometrics, 52, 5-59.

[4] Bollerslev, T., Engle, R. F. and Nelson, D. B. (1994) ARCH models, in The Hand-book of Econometrics, Vol. 4, edited by R. F. Engle and D. L. McFadden.

[5] Carrasco, M. and Chen, X. (2002)Mixing andmoment properties of various GARCHand stochastic volatility models, Econometric Theory, 18, 17-39.

[6] Comte, F. and Lieberman, O. (2003) Asymptotic theory for multivariate GARCHprocesses, Journal of Multivariate Analysis, 84, 61-84.

[7] Corradi, V. and Iglesias, E. M. (2008) Bootstrap re�nements for QML estimatorsof the GARCH(1; 1) parameters, Journal of Econometrics, 144, 500-510.

[8] Dahl, C. M. and Iglesias, E. M. (2008) The limiting properties of the QMLE in ageneral class of asymmetric volatility models, Working paper.

[9] Deb, P. (1996) Finite sample properties of maximum likelihood and quasi-maximum

18

likelihood estimators of EGARCH models, Econometric Reviews, 15, 51-68.

[10]Demos, A. (2000) The autocorrelation function of conditionally heteroskedastic inmean models, Working paper, Athens University of Economics and Business.

[11]Engle, R. F. (1982) Autoregressive conditional heteroskedasticity with estimatesof the variance of U.K. in�ation, Econometrica, 50, 987-1008.

[12]Fiorentini, G., Calzolari, G. and Panattoni, L. (1996) Analytic derivatives and thecomputation of GARCH estimates, Journal of Applied Econometrics, 11, 399-417.

[13]He, C., Terasvirta, T. and Malmsten, H. (2002) Moment structure of a family of�rst-order exponential GARCH models, Econometric Theory, 18, 868-885.

[14]Iglesias, E. M. and Phillips, G. D. A. (2002) Small sample estimation bias inARCH models, Cardiff University.

[15]Jeantheau, T. (1998) Strong consistency of estimators of multivariate ARCH mod-els, Econometric Theory, 14, 70-86.

[16]Lee, S-W. and Hansen, B. E. (1994) Asymptotic theory for the GARCH(1,1) quasimaximum likelihood estimator, Econometric Theory, 10, 29-52.

[17]Linton, O. (1997) An asymptotic expansion in the GARCH(1,1) model, Econo-metric theory, 13, 558-581.

[18]Lumsdaine, R. L. (1996) Consistency and asymptotic normality of the quasi-maximumlikelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) mod-els, Econometrica, 64, 575-596.

[19]Malmsten, H. (2004) Evaluating exponential GARCH models, SSE/EFI WorkingPaper Series in Economics and Finance, no. 564, Stockholm school of economics.

[20]McCullagh, P. Tensor methods in statistics, Monographs on Statistics and AppliedProbability, 1986, Chapman and Hall.

[21]Nelson, D. B. (1990) Stationarity and persistence in the GARCH(1,1) model,Econometric Theory, 6, 318-334.

[22]Nelson, D. B. (1991) Conditional heteroskedasticity in asset returns: a new ap-

19

proach, Econometrica, 59, 347-370.

[23]Perez, A. and Zaffaroni, P. (2008) Finite-sample properties of maximum likelihoodand Whittle estimators in EGARCH and FIEGARCH models, Quantitative andQualitative Analysis in Social Sciences, 2, 78-97.

[24]Rothenberg, T. J. (1986) Approximating the distributions of econometric estima-tors and test statistics, in The Handbook of Econometrics, vol. 2, Amsterdam:North-Holland.

[25]Straumann, D. Estimation in conditionally heteroskedastic time series models,Lecture Notes in Statistics, no. 181, 2005, Berlin: Springer.

[26]Straumann, D. and Mikosch, T. (2006) Quasi-maximum likelihood estimation inconditionally heteroskedastic time series: a stochastic recurrence equations ap-proach, Annals of Statistics, 34, 2449-2495.

[27]Taniguchi, M. (1991) Higher order asymptotic theory for time series analysis,Berlin: Springer-Verlag.

[28]Weiss, A. A. (1986) Asymptotic theory for ARCH models: estimation and testing,Econometric Theory, 2, 107-131.

[29]White, H. (1982) Maximum likelihood estimation of misspeci�ed models, Econo-metrica, 50, 1-25.

[30]White, H. Estimation, inference and speci�cation analysis, Cambridge UniversityPress, 1994, Cambridge.

[31]Zaffaroni, P. (2008) Whittle estimation of EGARCH and other exponential volatil-ity models, Journal of Econometrics, forthcoming.

20

Appendix A. Proof of the unconditional varianceWe write the variance equation as follows:

ln (ht) = �� + �

1Xi=0

�izt�1�i +

1Xi=0

�i (jzt�1�ij � E jzt�1�ij) ;

where �� = �1�� . Taking the expectation of the exponential of ln (ht) we have:

E exp (lnht) = exp (��)E exp

�P1i=0

���izt�1�i + �i (jzt�1�ij � E jzt�1�ij)

��=

= exp (��)E1Yi=0

exp���izt�1�i + �i (jzt�1�ij � E jzt�1�ij)

�=

= exp (��)1Yi=0

E exp���izt�1�i + �i (jzt�1�ij � E jzt�1�ij)

�Now,1Yi=0

E exp���izt�1�i + �i (jzt�1�ij � E jzt�1�ij)

�=

=1Yi=0

exp�� E jzt�1�ij �i

�E exp

���izt�1�i + �i jzt�1�ij

�=

= exp�� Ejzj

1��

� 1Yi=0

E exp���izt�1�i + �i jzt�1�ij

�

E exp���iz + �i jzj

� �1 = ��i

�2 = �i

= E exp [�1z + �2 jzj] = 1p2�

R1�1 exp

��1z + �2 jzj � 1

2z2�dz =

= 1p2�

R 0�1 exp

��12

��2 (�1 � �2) z + z2 � (�1 � �2)

2�� dz+ 1p

2�

R10exp

��12

��2 (�1 + �2) z + z2 � (�1 + �2)

2�� dz == 1p

2�

R 0�1 exp

��12

��2 (�1 � �2) z + z2 � (�1 � �2)

2�� dz+ 1p

2�

R10exp

��12

��2 (�1 + �2) z + z2 � (�1 + �2)

2�� dz == exp

�(�1��2)2

2

�1p2�

R 0�1 exp

��12(z � (�1 � �2))

2� dz+exp

�(�1+�2)

2

2

�1p2�

R 0�1 exp

��12(z � (�1 + �2))

2� dz == exp

�(�1��2)2

2

�1p2�

R �(�1��2)�1 exp

��12u2�du+exp

�(�1+�2)

2

2

�1p2�

R �(�1+�2)�1 exp

��12u2�du =

= exp�(�1��2)2

2

�� (� (�1 � �2)) + exp

�(�1+�2)

2

2

�(1� � (� (�1 + �2))) =

21

= exp�(�1��2)2

2

�� (� (�1 � �2)) + exp

�(�1+�2)

2

2

�� (�1 + �2) =

= exp��2i( ��)2

2

����i ( � �)

�+ exp

��2i( +�)2

2

����i ( + �)

�= exp (�)� (�B) + exp (�)� (A) ;where � = �2i( +�)2

2, � = �2i( ��)2

2, A = �i ( + �) and B = �i ( � �), and � (�)

is the cumulative distribution function of the standard normal distribution.Therefore,

E exp (q lnht) = exp

��� E jzj1� �

� 1Yi=0

0@ exp��2i( ��)2

2

����i ( � �)

�+exp

��2i( +�)2

2

����i ( + �)

�1A

= exp ()1Yi=0

(exp (�)� (�B) + exp (�)� (A))

= b�;

where = �� Ejzj1�� .

Appendix B. Expected values of the log-likelihood derivativesThe expected values of all �rst order derivatives are equal to zero.Second order derivatives:For i; j 2 f�; �; ; �g ;

E (Lij) = �T2E (ht;iht;j) ;

E (L�j) = �T2E (ht;�ht;j) ;

E (L��) = �TE�1

ht

�� T

2E�h2t;��:

Third order derivatives:For i 2 f�; �; ; �g ;

E (Liii) = �T

2E�3ht;iht;i;i � h3t;i

�;

for i 2 f�; �; ; �g ; j 2 f�; �; ; �; �g ;

E (Liij) = �T

2E�ht;jht;i;i � h2t;iht;j + 2ht;iht;i;j

�;

22

for i; j 2 f�; �; ; �g ; k 2 f�; �; ; �; �g ;

E (Lijk) = �T

2E (ht;jht;i;k + ht;kht;i;j + ht;iht;j;k � ht;jht;iht;k; ) ;

for i 2 f�; �; ; �g ; j 2 f�g ;

E (Lijj) = �T

2E�ht;iht;j;j + 2ht;jht;i;j � ht;i (ht;j)

2�+ TE

�1

htht;i

�;

for j 2 f�g ;

E (Ljjj) = �T

2E�3ht;jht;j;j � h3t;j

�+ TE

�31

htht;j

�:

In this Appendix we make a list of the results that are needed for the bias approx-imations. Please note that the last Appendix should be studied �rst in order to befamiliarized with the symbols used.First, provided that

���2 + 14�2 + 1

4 2 � �E jzj+ 1

2� E (z jzj)

�� < 1, the expectedvalues of all second order derivatives are:1. E (L��) = �T

2

1+2(�� 12 Ejzj)E;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2� E(zjzj))

2. E (L��) = �T2

E(ln(ht�1))+(�� 12 Ejzj)LE;�+(�� 1

2 Ejzj)E;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2� E(zjzj))

3. E (L� ) = �T2

� 12 [E(zjzj)+ (1�E2jzj)]E;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

4. E (L��) = �T2

� 12[�+ E(zjzj)]E;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

5. E (L��) = �T2

�(�+ EI)E�

1ph

�+(�� 1

2 Ejzj)E;�+[�(�� Ejzj)+ �EI]E� 1

2E;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

6. E (L��) = �T2

E(ln2(ht�1))+2(�� 12 jzt�1j)LE;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

7. E (L� ) = �T2

(�� 12 Ejzj)LE; � 1

2 [�E(zjzj)+ (1�E2jzj)]E;�1�(�2+ 1

4�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

8. E (L��) = �T2

� 12[�+ E(zjzj)]E;�+(�� 1

2 Ejzj)LE;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

9. E (L��) = �T2

�(�+ EI)LE� 12+[�( Ejzj��)�� EI]E� 1

2E;�+(�� 1

2 Ejzj)LE;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

10.E (L ) = �T2

1�E2jzj1�(�2+ 1

4�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

23

11.E (L� ) = �T2

E(zjzj)1�(�2+ 1

4�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

12.E (L� ) = �T2

� EIg(z)E�

1ph

�� 12 [�E(zjzj)+ (1�E2jzj)]E;��(�(�� Ejzj)+ �EI)E� 1

2E;

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

13.E (L��) = �T2

1

1�(�22+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

14.E (L��) = �T2

� 12(�+ E(zjzj))E(ht;�)� EjzjE

�1ph

�+[�( Ejzj��)�� EI]E� 1

2E;�

1�(�22+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

15.E (L��) = �TE�1ht

�� T

2

(�2+ 2+2 �EI)E�

1pht�1

��2(�(�� Ejzj)+ �EI)E� 1

2E;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

:

Second, the expected values of the third order derivatives are:1. E (L���) = �T

2E�3 (ht;�ht;�;�)� h3t;�

�2. E (L���) = �T

2E�ht;�ht;�;� � h2t;�ht;� + 2ht;�ht;�;�

�3. E (L�� ) = �T

2E�ht; ht;�;� � h2t;�ht; + 2ht;�ht;�;

�4. E (L���) = �T

2E�ht;�ht;�;� � h2t;�ht;� + 2ht;�ht;�;�

�5. E (L���) = �T

2E�ht;�;�ht;� + 2 (ht;�ht;�;�)� h2t;�ht;�

�6. E (L���) = �T

2E�ht;�ht;�;� + 2ht;�ht;�;� � ht;�h

2t;�

�7. E (L�� ) = �T

2E (ht;�ht;�; + ht; ht;�;� + ht;�ht;�; � ht;�ht;�ht; )

8. E (L���) = �T2E (ht;�ht;�;� + ht;�ht;�;� + ht;�ht;�;� � ht;�ht;�ht;�)

9. E (L���) = �T2E (ht;�ht;�;� + ht;�;�ht;� + ht;�ht;�;� � ht;�ht;�ht;�)

10.E (L���) = �T2E (ht;�ht;�;� + ht;�;�ht;� + ht;�ht;�;� � ht;�ht;�ht;�)

11.E (L� ) = �T2E�ht;�ht; ; + 2ht; ht;�; � ht;�h

2t;

�12.E (L� �) = �T

2E (ht;�ht;�; + ht; ht;�;� + ht;�ht; ;� � ht;�ht; ht;�)

13.E (L� �) = �T2E (ht;�ht; ;� � ht;�ht; ht;� + ht; ;�ht;� + ht; ht;�;�)

14.E (L���) = �T2E�ht;�ht;�;� + 2ht;�ht;�;� � ht;�h

2t;�

�15.E (L���) = �T

2E (ht;�ht;�;� � ht;�ht;�ht;� + ht;�;�ht;� + ht;�ht;�;�)

16.E (L���) = �T2E�ht;�ht;�;� + 2ht;�ht;�;� � ht;� (ht;�)

2�+ TE�1htht;�

�17.E (L���) = �T

2E�3ht;�ht;�;� � Eh3t;�

�18.E (L�� ) = �T

2E�2ht;�ht; ;� + ht; ht;�;� � h2t;�ht;

�19.E (L���) = �T

2E�2ht;�ht;�;� + ht;�ht;�;� � h2t;�ht;�

�24

20.E (L���) = �T2

�2E (ht;�ht;�;�) + E (ht;�;�ht;�)� E

�h2t;�ht;�

��21.E (L� ) = �T

2E�ht;�ht; ; + 2ht; ht;�; � ht;�h

2t;

�22.E (L� �) = �T

2E (ht;�ht;�; + ht; ht;�;� + ht;�ht; ;� � ht;�ht; ht;�)

23.E (L� �) = �T2E (ht;�ht; ;� � ht;�ht; ht;� + ht; ;�ht;� + ht; ht;�;�)

24.E (L���) = �T2E�ht;�ht;�;� + 2ht;�ht;�;� � ht;�h

2t;�

�25.E (L���) = �T

2E (ht;�ht;�;� � ht;�ht;�ht;� + ht;�;�ht;� + ht;�ht;�;�)

26.E�L���

�= �T

2E�ht;�ht;�;� + 2ht;�ht;�;� � ht;� (ht;�)

2�+ TE�1htht;�

�27.E (L ) = �T

2E�3 (ht; ht; ; )�

�h3t; ��

28.E (L �) = �T2E�ht;�ht; ; + 2ht; ht; ;� � h2t; ht;�

�29.E (L �) = �T

2E�2ht; ht; ;� � h2t; ht;� + ht; ; ht;�

�30.E (L���) = �T

2E�ht; ht;�;� + 2ht;�ht; ;� � ht; h

2t;�

�31.E (L ��) = �T

2E (ht; ht;�;� � ht; ht;�ht;� + ht;�; ht;� + ht;�ht; ;�)

32.E (L ��) = �T2E�ht; ht;�;� + 2ht;�ht; ;� � ht; h

2t;�

�+ TE

�1htht;

�33.E (L���) = �T

2E�2ht;�ht;�;� � h2t;�ht;� + ht;�;�ht;�

�34.E (L���) = �T

2E�ht;�ht;�;� + 2ht;�ht;�;� � ht;�h

2t;�

�+ TE

�1htht;�

�35.E (L���) = �T

2E�3ht;�ht;�;� � h3t;�

�+ TE

�3 1htht;�

�:

Appendix C. Expected values of the log-variance derivativesIn the current Appendix, we present some of the results for the expected values ofthe log-variance derivatives and more speci�cally those that are needed for the evalua-tion of some of the expected values of the third order log-likelihood derivatives of theprevious Appendix, that is:Assuming �rst

���2 + 14�2 + 1

4 2 � �E jzj+ 1

2 �E (z jzj)

�� < 1 and���3 + 34��2 � 1

8�3Ez3 � 3

2 ��2E jzj � ��E (z jzj) + 1

4�2E jzj3

�+ 3

4 2�� � 1

2�Ez3

�� 1

8 3E jzj3

�� <1, we have:

1. E (ht;�ht;�;�) =14 EjzjE

(;�)2+( 14� Ejzj�

18 2� 1

8�2� 1

4� E(zjzj))E(;�)3+(��

12 Ejzj)E;�;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

2. E�h3t;��=

1+3(�� 12 Ejzj)E;�+3(�2+ 1

4�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))E(;�)2

1�[�3+ 34��2� 1

8�3Ez3� 3

2 (�2Ejzj���E(zjzj)+ 1

4�2Ejzj3)+ 3

4 2(�� 1

2�Ez3)� 1

8 3Ejzj3]

25

3. E (ht;�ht;�;�) =14 EjzjLE

(;�)2+(�2+ 1

4�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))E(;�)2;�+(��

12 Ejzj)LE;�;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

4. E (ht;�ht;�;�) =E;�+(�� 1

2 Ejzj)E(;�)2+

14 EjzjE;�;�+( 14� Ejzj�

18 2� 1

8�2� 1

4� E(zjzj))E(;�)2;�+(��

12 Ejzj)E;�;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

5. E (ht; ht;�;�) =14[�E(zjzj)+ (1�Ejzj)]E

(;�)2� 12[�E(zjzj)+ (1�Ejzj)]E;�;�+( 14� Ejzj�

18 2� 1

8�2� 1

4� E(zjzj))E(;�)2;

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

6. E (ht;�ht;�;�) =14[�+ E(zjzj)]E

(;�)2� 12[�+ E(zjzj)]E;�;�+( 14� Ejzj�

18 2� 1

8�2� 1

4� E(zjzj))E(;�)2;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

7. E�h2t;�ht;�

�=

�[�+ E(zjzj)]E;�+( 12 �Ejzj3���+ 1

4(�2+ 2)Ez3�� E(zjzj))E(;�)2+2(�

2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))E;�;�

1�[�3+ 34��2� 1

8�3Ez3� 3

2 (�2Ejzj���E(zjzj)+ 1

4�2Ejzj3)+ 3

4 2(�� 1

2�Ez3)� 1

8 3Ejzj3]

8. E (ht;�ht;�;�) =14 EjzjE;�;�+(�� 1

2 Ejzj)E;�;�+ 1

4[�+ E(zjzj)]E

(;�)2+( 14� Ejzj�

18 2� 1

8�2� 1

4� E(zjzj))E(;�)2;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

9. E (ht;�ht;�;�) =14 EjzjE

(;�)2+2E;�+(�� 1

2 Ejzj)E;�;�+2(�� 1

2 Ejzj)E;�;�+( 14� jzt�1j�

18 2� 1

8�2� 1

4� E(zjzj))E;�(;�)2

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

10.E (ht;�ht;�;�) =LE;�+(�� 1

2 Ejzj)E;�;�+ 1

4 EjzjLE;�;�+( 14� jzt�1j�

18 2� 1

8�2� 1

4� E(zjzj))E;�(;�)2+(��

12 Ejzj)LE;�;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

11.E (ht;�ht;�; ) =� 12EjzjLE;�+ 1

4 EjzjLE;�; +(�� 1

2 Ejzj)LE;�; � 1

2(�Ejzj�12�E(zjzj)� 1

2 )E;�E;�+ 1

4(� Ejzj�12 2� 1

2�2�� E(zjzj))E;�;�;

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

12.E (ht;�ht;�;�) =14 EjzjLE;�;�+(�� 1

2 Ejzj)LE;�;�+ 1

4(�+ E(zjzj))E;�;�+ 1

4(� Ejzj�12 2� 1

2�2� �E(zjzj))E;�;�;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

13.E (ht;�ht;�;�) =14 EjzjE;�;�+(�� 1

2 Ejzj)E;�;�+(�� 1

2 Ejzj)E;�;�+ 1

4(�+ E(zjzj))E;�;�+ 1

4(� Ejzj�12 2� 1

2�2� �E(zjzj))E;�;�;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

14.E (ht;�ht;�;�) =14(�+ E(zjzj))E;�;�� 1

2(�+ E(zjzj))E;�;�+(�� 1

2 Ejzj)E;�;�+( 14� Ejzj�

18�2� 1

8 2� 1

4 �E(zjzj))E;�;�;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

15.E (ht;�ht; ; ) =14 EjzjE

(; )2+(�� 1

2 Ejzj)E; ; �(�Ejzj� 1

2 � 1

2�E(zjzj))E;�; + 1

4(� Ejzj�12 2� 1

2�2� �E(zjzj))E;�(; )2

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

16.E (ht;�ht; ;�) =� 12(�Ejzj�

12�E(zjzj)� 1

2 )E;�;�+ 1

4(�+ E(zjzj))E;�; + 1

4(� Ejzj�12 2� 1

2�2� �E(zjzj))E;�; ;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

17.E (ht;�ht;�;�) =14 EjzjE

(;�)2+(�� 1

2 Ejzj)E;�;�+ 1

2(�+ E(zjzj))E;�;�+ 1

4(� Ejzj�12 2� 1

2�2� �E(zjzj))E;�(;�)2

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

18.E (ht;�ht;�;�) =� 12E;�+

14(�+ E(zjzj))E;�;�� 1

2(�+ E(zjzj))E;�;�+ 1

4(�+ E(zjzj))E;�;�+ 1

4(� Ejzj�12 2� 1

2�2� �E(zjzj))E;�(;�)2

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

19.E (ht;�ht;�;�) =14 EjzjLE

(;�)2+2LE;�+(�� 1

2 Ejzj)LE;�;�+2(�� 1

2 Ejzj)E(;�)2+

14(� Ejzj�

12 2� 1

2�2� �E(zjzj))E(;�)3

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

20.E�h3t;��=

L3+3(�� 12 Ejzj)L2E;�+3(�2+ 1

4�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))LE(;�)2

1�[�3+ 34��2� 1

8�3Ez3� 3

2 (�2Ejzj���E(zjzj)+ 1

4�2Ejzj3)+ 3

4 2(�� 1

2�Ez3)� 1

8 3Ejzj3]

26

21.E (ht;�ht;�;�) =14(�+ E(zjzj))E

(;�)2� 12(�+ E(zjzj))E;�;�+ 1

4(� Ejzj�12 2� 1

2�2� �E(zjzj))E(;�)2;�+2(��

12 Ejzj)E;�;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

22.E (ht;�ht;�;�) =14 EjzjLE

(;�)2+(�� 1

2 Ejzj)LE;�;�+ 1

2(�+ E(zjzj))E;�;�+ 1

4(� Ejzj�12 2� 1

2�2� �E(zjzj))E;�(;�)2

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

23.E�h3t; �=(Ejzj3�3Ejzj+2E3jzj)+3[( 14 �

2+ 14 2)(Ejzj3�Ejzj)� �(1�E2jzj)���E(zjzj)+ 1

2 �(Ez3�EjzjE(zjzj))]E(; )2

1�[�3+ 34��2� 1

8�3Ez3� 3

2 (�2Ejzj���E(zjzj)+ 1

4�2Ejzj3)+ 3

4 2(�� 1

2�Ez3)� 1

8 3Ejzj3]

24.E (ht;�ht; ; ) =14(�+ E(zjzj))E

(; )2� 12(�+ E(zjzj))E; ; + 1

4(� Ejzj�12 2� 1

2�2� �E(zjzj))E(; )2;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

25.E (ht; ht; ;�) =14(�+ E(zjzj))E

(; )2+ 14(� Ejzj�

12 2� 1

2�2� �E(zjzj))E(; )2;�

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

26.E�h2t; ht;�

�=

Ez3�2EjzjE(zjzj)+[�( 12 Ejzj3��)+ 1

4(�2+ 2)Ez3�� E(zjzj)]E(; )2

1�[�3+ 34��2� 1

8�3Ez3� 3

2 (�2Ejzj���E(zjzj)+ 1

4�2Ejzj3)+ 3

4 2(�� 1

2�Ez3)� 1

8 3Ejzj3]

27.E (ht; ht;�;�) =14 [�E(zjzj)+ (1�E2jzj)]E(;�)2�

12 [�E(zjzj)+ (1�E2jzj)]E;�;�+

14(� Ejzj�

12 2� 1

2�2� �E(zjzj))E; (;�)2

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

28.E (ht;�ht; ;�) =� 12(�Ejzj�

12�E(zjzj)� 1

2 )E(;�)2+

14(� Ejzj�

12 2� 1

2�2� �E(zjzj))E; (;�)2

1�(�2+ 14�2+ 1

4 2� �Ejzj+ 1

2 �E(zjzj))

29.E�ht; h

2t;�

�=

Ejzj3�Ejzj+[( 14 �2+ 1

4 2)(Ejzj3�Ejzj)� �(1�E2jzj)���E(zjzj)+ 1

2 �(Ez3�EjzjE(zjzj))]E(;�)2

1�[�3+ 34��2� 1

8�3Ez3� 3

2 (�2Ejzj���E(zjzj)+ 1

4�2Ejzj3)+ 3

4 2(�� 1

2�Ez3)� 1

8 3Ejzj3]

:

The whole results are available on demand from the corresponding author.

Appendix D. Expected values of cross products of thelog-likelihood derivativesIn this Appendix, we present the expected values of cross-products of the log-likelihoodderivatives. To conserve space, we present only some indicative. That is,

1. 1TE (L�L��) = �1

4

�PPs<t

E [(z2s � 1)hs;�ht;�ht;�]� (�4 + 2)PT

t=1E�ht;�ht;�;� � h3t;�

��

2. 1TE (L�L��) = �1

4

24 PPs<t

E [(z2s � 1)hs;�ht;�ht;�]� (�4 + 2)PT

t=1E�ht;�ht;�;� � ht;�h

2t;�

�+2�3

PTt=1E

�1phth2t;�

�35

3. 1TE (L�L��) = �1

4

24 PPs<t

E�(z2s � 1)hs;�h2t;�

�� (�4 + 2)

PTt=1E

�ht;�ht;�;� � ht;�h

2t;�

�+4�3

PTt=1E

�1phtht;�ht;�

�35

4. 1TE (L�L��) = �1

4

264PPs<t

E�(z2s � 1)hs;�h2t;�

�� (�4 + 2)

PTt=1E

�ht;�ht;�;� � ht;�h

2t;�

�+2PPs<t

E�zs

1phth2t;�

�+ 2�3

PTt=1E

�1phth2t;� � 1p

htht;�;�

�375

27

5. 1TE (L�L��) = �1

4

266664PPs<t

E [(z2s � 1)hs;�ht;�ht;�]� (�4 + 2)PT

t=1E�ht;�ht;�;� � ht;�h

2t;�

�+2PPs<t

E�zs

1phtht;�ht;�

�+ 2�3

PTt=1E

�2phtht;�ht;� � 1p

htht;�;�

�+4PT

t=1E�1htht;�

�377775

6. 1TE (L�L��) = �1

4

266664PPs<t

E�(z2s � 1)hs;�h2t;�

�� (�4 + 2)

PTt=1E

�ht;�ht;�;� � h3t;�

�+2PPs<t

E�zs

1phth2t;�

�+ 2�3

PTt=1E

�3phth2t;� � 1p

htht;�;�

�+8PT

t=1E�1htht;�

�377775

At this point, we should note that these results differ from those in the paper ofLinton [17], due to the fact that we assume non-symmetric distribution of the errors andalso none of these expressions are zero, since the block-diagonality of the informationmatrix in our case that we study the EGARCH(1; 1) model does not hold.Analytic proof of the �rst result is given as follows:

L�L�� =1

2

TXt=1

�z2t � 1

�ht;�

1

2

TXt=1

�z2t � 1

�ht;�;� �

1

2

TXt=1

z2t (ht;�)2

!

=1

4

TXt=1

�z2t � 1

�ht;�

TXt=1

�z2t � 1

�ht;�;� �

1

4

TXt=1

�z2t � 1

�ht;�

TXt=1

z2t h2t;�

=1

4

TXt=1

�z2t � 1

�2ht;�ht;�;� +

1

4

TXt=1

�z2t � 1

�ht;�

TXs=t+1

�z2s � 1

�hs;�;�

+1

4

TXt=1

�z2t � 1

�ht;�

t�1Xs=1

�z2s � 1

�hs;�;�

�14

TXt=1

�z2t � 1

�z2t h

3t;� �

1

4

TXt=1

�z2t � 1

�ht;�

TXs=t+1

z2sh2s;� �

1

4

TXt=1

�z2t � 1

�ht;�

t�1Xs=1

z2sh2s;�

Hence

E (L�L��) =T (�4 + 2)

4

�E (ht;�ht;�;�)� E

�h3t;����14EXXs<t

�z2s � 1

�z2t h

2t;�hs;�;

whereht;� = 1+

�� � 1

2�zt�1 � 1

2 jzt�1j

�ht�1;� and h2t;� = 1+2

�� � 1

2�zt�1 � 1

2 jzt�1j

�ht�1;�+�

� � 12�zt�1 � 1

2 jzt�1j

�2h2t�1;�:

28

Letht+k;� = 1+

�� � 1

2�zt+k�1 � 1

2 jzt+k�1j

�ht+k�1;� and h2t+k;� = 1+2

�� � 1

2�zt+k�1 � 1

2 jzt+k�1j

�ht+k�1;�+�

� � 12�zt+k�1 � 1

2 jzt+k�1j

�2h2t+k�1;�:

Hence,E�(z2t � 1)h2t+k;�ht;�

�=

= Eh(z2t � 1)

hht;� + 2

�� � 1

2�zt+k�1 � 1

2 jzt+k�1j

�ht+k�1;�ht;� +

�� � 1

2�zt+k�1 � 1

2 jzt+k�1j

�2h2t+k�1;�ht;�

ii=

k>1= 2

�� � 1

2 E jzj

�E (z2t � 1)ht+k�1;�ht;�+

��2 + 1

4

��2 + 2

�� � E jzj+ 1

2� E (z jzj)

�E (z2t � 1)h2t+k�1;�ht;�:

k = 1 : E�(z2t � 1)h2t+1;�ht;�

�= E

h(z2t � 1)

hht;� + 2

�� � 1

2�zt � 1

2 jztj

�h2t;� +

�� � 1

2�zt � 1

2 jztj

�2h3t;�

ii=

= 2E�(z2t � 1)

�� � 1

2�zt � 1

2 jztj

��Eh2t;�+E

h(z2t � 1)

�� � 1

2�zt � 1

2 jztj

�2iEh3t;�:

Hence,

E��z2t � 1

�h2t+k;�ht;�

� k>1= 2

�� � 1

2 E jzj

���12�Ez3 � 1

2 �E jzj3 � E jzj

���� � 1

2 E jzj

�k�2Eh2t;�

+

��2 +

1

4

��2 + 2

�� � E jzj+ 1

2� E (z jzj)

�E�z2t � 1

�h2t+k�1;�ht;�:

Set: A = 2�� � 1

2 E jzj

� ��12�Ez3 � 1

2 �E jzj3 � E jzj

��Eh2t;� and C = �2 +

14

��2 + 2

�� � E jzj+ 1

2� E (z jzj) :

We have that: E�(z2t � 1)h2t+k;�ht;�

� k>1= A

�� � 1

2 E jzj

�k�2+CE (z2t � 1)h2t+k�1;�ht;�:

By repeating substitution,E�(z2t � 1)h2t+k;�ht;�

� k>1= A

h�� � 1

2 E jzj

�k�2+ C

�� � 1

2 E jzj

�k�3+ :::+ Ck�2

i+Ck�1E (z2t � 1)h2t+1;�ht;�.This formula can be written as:

E�(z2t � 1)h2t+k;�ht;�

� k>1= A

Ck�1�(�� 12 Ejzj)

k�1

C�(�� 12 Ejzj)

+ Ck�1E (z2t � 1)h2t+1;�ht;�:

Consequently,

E�(z2t � 1)h2t+k;�ht;�

� k�1= A

Ck�1�(�� 12 Ejzj)

k�1

C�(�� 12 Ejzj)

+Ck�1h2E�(z2t � 1)

�� � 1

2�zt � 1

2 jztj

��Eh2t;� + E

h(z2t � 1)

�� � 1

2�zt � 1

2 jztj

�2iEh3t;�

i:

Appendix E. Proof of the Theorem

The proof comes immediately from the results of Appendix B and Appendix D.

29

Appendix F. The log-variance derivativesIn this Appendix we present the expressions of the log-variance derivatives, in a formuseful to explore their properties.

ht;� = 1 +

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1;�

ht;�;� =

�1

4�zt�1 +

1

4 jzt�1j

�h2t�1;� +

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1;�;�

ht;�;� = ht�1;�+

�1

4�zt�1 +

1

4 jzt�1j

�ht�1;�ht�1;�+

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1;�;�

ht;�; = �1

2jzt�1jht�1;�+

�1

4�zt�1 +

1

4 jzt�1j

�ht�1;�ht�1; +

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1;�;

ht;�;� = �1

2zt�1ht�1;�+

�1

4�zt�1 +

1

4 jzt�1j

�ht�1;�ht�1;�+

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1;�;�

ht;�;� =1

2(� + [I (zt�1�0)� I (zt�1<0)])

1pht�1

ht�1;� +1

4(�zt�1 + jzt�1j)ht�1;�ht�1;�

+

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1;�;�

ht;� = ln (ht�1) +

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1;�

ht;�;� =

�1

4�zt�1 +

1

4 jzt�1j

�h2t�1;� +2ht�1;� +

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1;�;�

ht;�; = ht�1; �1

2jzt�1jht�1;�+

1

4(�zt�1 + jzt�1j)ht�1;�ht�1; +

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1;�;

ht;�;� = ht�1;��1

2zt�1ht�1;�+

1

4(�zt�1 + jzt�1j)ht�1;�ht�1;�+

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1;�;�

ht;�;� = ht�1;� +1

2(� + [I (zt�1�0)� I (zt�1<0)])

1pht�1

ht�1;� +1

4(�zt�1 + jzt�1j)ht�1;�ht�1;�

+

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1;�;�

ht; = g (zt�1) +

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1;

30

ht; ; = � jzt�1jht�1; +1

4(�zt�1 + jzt�1j)h2t�1; +

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1; ;

ht; ;� = �1

2jzt�1jht�1;��

1

2zt�1ht�1; +

1

4(�zt�1 + jzt�1j)ht�1; ht�1;�+

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1; ;�

ht; ;� = � [I (zt�1�0)� I (zt�1<0)]1pht�1

+1

2(� + [I (zt�1�0)� I (zt�1<0)])

1pht�1

ht�1;

�12jzt�1jht�1;� +

1

4(�zt�1 + jzt�1j)ht�1; ht�1;� +

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1; ;�

ht;� = zt�1 +

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1;�

ht;�;� = �zt�1ht�1;�+1

4(�zt�1 + jzt�1j)h2t�1;�+

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1;�;�

ht;�;� = � 1pht�1

+1

2(� + [I (zt�1�0)� I (zt�1<0)])

1pht�1

ht�1;� �1

2zt�1ht�1;�

+1

4(�zt�1 + jzt�1j)ht�1;�ht�1;� +

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1;�;�

ht;� = � (� + [I (zt�1�0)� I (zt�1<0)])1pht�1

+

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1;�

ht;�;� = (� + [I (zt�1 � 0)� I (zt�1 < 0)])1pht�1

ht�1;� +1

4(�zt�1 + jzt�1j)h2t�1;�

+

�� � 1

2�zt�1 �

1

2 jzt�1j

�ht�1;�;�

Appendix G. Some additional calculationsHow do we evaluate E

�exp [� lnht]h

2t;�

:

First, note thath2t;� = 1 + 2

�� � 1

2�zt�1 � 1

2 jzt�1j

�ht�1;� +

�� � 1

2�zt�1 � 1

2 jzt�1j

�2h2t�1;�

=P1

i=0

�1 + 2

�� � 1

2�zt�1�i � 1

2 jzt�1�ij

�ht�1�i;�

��i�1Yj=0

�� � 1

2�zt�1�j � 1

2 jzt�1�jj

�2Hence,

31

E�exp [� lnht]h

2t;�

=P1

i=0

i�1Yj=0

En�� � 1

2�z � 1

2 jzj

�2exp

���j (�+ �z + g (z))

�o��

E��i + 2E��� � 1

2�z � 1

2 jzj

�exp

���i (�+ �z + g (z))

�E��iE;�

= E�E(;�)2 :

Additionally, we need:

E (exp (k lnht) ln (ht)) = E

exp (k lnht)

�

1� �+ �

1Xs=0

�szt�1�s + 1Xs=0

�sg (zt�1�s)

!!

=�

1� �E (exp (k lnht)) + E

exp (k lnht)

1Xs=0

�s (�zt�1�s + g (zt�1�s))

!=

�

1� �E (exp (k lnht))

+1Xs=0

�sE

exp

�k�1�� + k�

P1j=0 �

jzt�1�j + k P1

j=0 �jg (zt�1�j)

�(�zt�1�s + g (zt�1�s))

!

=�

1� �E (exp (k lnht)) + exp

�k�

1� �

��

1Xs=0

�sE (exp (k��szt�1�s + k �sg (zt�1�s)) (�zt�1�s + g (zt�1�s)))

�1Y

j=0 6=s

E�exp

�k��jzt�1�j + k �jg (zt�1�j)

��=

�

1� �E (exp (k lnht)) + exp

�k�

1� �

� 1Yj=0

E�exp

�k��jz + k �jg (z)

���

1Xs=0

�sE (exp (k��sz + k �sg (z)) [�z + g (z)])

E [exp (k��sz + k �sg (z))]

= EkL

Appendix H. Expected values of the �rst & second orderlog-variance derivativesWe assume

��� � 12 E jzj

�� < 1.First order derivatives:

32

1. E (ht;�) = 11��+ 1

2 E(jzj)

2. E (ht;�) = �

(1��+ 2Ejzj)(1��)

3. E (ht; ) = 04. E (ht;�) = 0

5. E (ht;�) = ��E� 1

2

1�(�� 12 Ejzj)

:

Second order derivatives:1. E (ht;�;�) =

14 EjzjE

(;�)2

1�(�� 12 Ejzj)

2. E (ht;�;�) =E;�+

14 EjzjE;�;�

1�(�� 12 Ejzj)

3. E (ht;�; ) =� 12EjzjE;�+ 1

4 EjzjE;�;

1�(�� 12 Ejzj)

4. E (ht;�;�) =14 EjzjE;�;�

1�(�� 12 Ejzj)

5. E (ht;�;�) =12(�+ EI)E� 1

2E;�+

14 EjzjE;�;�

1�(�� 12 Ejzj)

6. E (ht;�;�) =14 EjzjE

(;�)2+2E;�

1�(�� 12 Ejzj)

7. E (ht;�; ) =� 12EjzjE;�+ 1

4 EjzjE;�;

1�(�� 12 Ejzj)

8. E (ht;�;�) =14 EjzjE;�;�

1�(�� 12 Ejzj)

9. E (ht;�;�) =E;�+

12(�+ EI)E� 1

2E;�+

14 EjzjE;�;�

1�(�� 12 Ejzj)

10.E (ht; ; ) =14 EjzjE

(; )2

1�(�� 12 Ejzj)

11.E (ht; ;�) = 0

12.E (ht; ;�) =�EIE� 1

2+ 12(�+ EI)E� 1

2E; � 1

2EjzjE;�+ 1

4 EjzjE; ;�

1�(�� 12 Ejzj)

13.E (ht;�;�) =14 EjzjE

(;�)2

1�(�� 12 Ejzj)

14.E (ht;�;�) =�E� 1

2+ 12(�+ EI)E� 1

2E;�+

14 EjzjE;�;�

1�(�� 12 Ejzj)

15.E (ht;�;�) =(�+ EI)E� 1

2E;�+

14 EjzjE

(;�)2

1�(�� 12 Ejzj)

:

33

Appendix I. Useful formulae

Let I denote the Indicator function, then:

E [(� + [I (zt�1�0)� I (zt�1<0)]) (�zt�1 + g (zt�1))] = �E jzj+ 2EIg(z)

E [(� + [I (zt�1 � 0)� I (zt�1 < 0)]) (�zt�1 + jzt�1j)] = 2� E [jzj]

E [(�zt�1 + g (zt�1)) (�zt�1 + jzt�1j)] =��2 + 2

�1� E2 jzj

�+ 2� E (z jzj)

�E

�(�zt�1 + g (zt�1))

�� � 1

2�zt�1 �

1

2 jzt�1j

��= �1

2�2 � 1

2 2�1� E2 jzj

�� � E (z jzj)

E

�zt�1 (�zt�1 + g (zt�1))

�� � 1

2�zt�1 �

1

2 jzt�1j

��=

��� � �E jzj3 + 1

2 �E jzj � 1

2�2Ez3

+ �E (z jzj)� 12 2 (Ez3 � E jzjE (z jzj))

�

E

"(�zt�1 + g (zt�1))

�� � 1

2�zt�1 �

1

2 jzt�1j

�2#=

26666414���2 + 3 2

�E (z3)

+14 � 2 + 3�2

�E jzj3

���2 � � 2 � 2� �E (z jzj)�14 ��2 + 2

�E jzj

+� 2E2 jzj � 12 2�E (z jzj)E jzj

377775

E�jzt�1j2 g (zt�1)

�= E

�z2t�1g (zt�1)

�= E jzj3 � E jzj

E�g2 (zt�1)

�= 1� E2 jzj

E�g3 (zt�1)

�= E

�jzt�1j3 � 3 jzt�1j2E jzj+ 3 jzt�1jE2 jzj � E3 jzj

�=

�E jzj3 � 3E jzj+ 2E3 jzj

�E�g2 (zt�1) jzt�1j

�=

�E jzj3 � 2E jzj+ E3 jzj

�34

E

�(� + [I (zt�1�0)� I (zt�1<0)])

�� � 1

2�zt�1 �

1

2 jzt�1j

��= � (� � E jzj) + �EI

E

�zt�1 (� + [I (zt�1�0)� I (zt�1<0)])

�� � 1

2�zt�1 �

1

2 jzt�1j

��= �1

2�2 � 1

2 2

� �E (z jzj) + � E jzj

E

�g (zt�1) (� + [I (zt�1�0)� I (zt�1<0)])

�� � 1

2�zt�1 �

1

2 jzt�1j

��= �1

2�2E (z jzj) + �E2 jzj

� � � 12 2EIz2 � �E jzjEI

E

"(� + [I (zt�1�0)� I (zt�1<0)])

�� � 1

2�zt�1 �

1

2 jzt�1j

�2#=

24 �2� + 14�3 + 3

4 2�

�2� �E jzj+ �2 EI+14 � 2 + 3�2

�E (z jzj)

35E

�g (zt�1) zt�1

�� � 1

2�zt�1 �

1

2 jzt�1j

��=

��12��E jzj3 � E jzj

�+ �E (z jzj)

�12 (Ez3 � E jzjE (z jzj))

�E

"g (zt�1)

�� � 1

2�zt�1 �

1

2 jzt�1j

�2#=

�14

��2 + 2

� �E jzj3 � E jzj

�� � (1� E2 jzj)

���E (z jzj) + 12� (Ez3 � E jzjE (z jzj))

�E

�g2 (zt�1)

�� � 1

2�zt�1 �

1

2 jzt�1j

��=

�� (1� E2 jzj)� 1

2 �E jzj3 � 2E jzj+ E3 jzj

��12� (Ez3 � 2E jzjE (z jzj))

�E

"zt�1

�� � 1

2�zt�1 �

1

2 jzt�1j

�2#= �

�1

2 E jzj3 � �

�+1

4

��2 + 2

�Ez3 � � E (z jzj)

E

�(�zt�1 + jzt�1j)

�� � 1

2�zt�1 �

1

2 jzt�1j

��= � E jzj � 1

2 2 � 1

2�2 � � E (z jzj)

�� � 1

2�zt�1 �

1

2 jzt�1j

�2=

��2 + 1

4�2z2t�1 +

14 2 jzt�1j2

���zt�1 � � jzt�1j+ 12 �zt�1 jzt�1j

�E

�� � 1

2�zt�1 �

1

2 jzt�1j

�2=

��2 +

1

4�2 +

1

4 2 � �E jzj+ 1

2 �E (z jzj)

��� � 1

2�zt�1 �

1

2 jzt�1j

�3=

��3 � 3

2�2�zt�1 +

34��2z2t�1 � 1

8�3z3t�1 � 3

2�2 jzt�1j+ 3

2� �zt�1 jzt�1j

�38 �2 jzt�1j3 + 3

4� 2 jzt�1j2 � 3

8 2�z3t�1 � 1

8 3 jzt�1j3

�E

�� � 1

2�zt�1 �

1

2 jzt�1j

�3=

��3 + 3

4��2 + 3

4� 2 � 1

8���2 + 3 2

�E (z3)� 3

2�2 E jzj

+32� �E (z jzj)� 1

8 � 2 + 3�2

�E jzj3

�

35

Appendix J. SymbolsThe next symbols are used in the paper and more speci�cally in the expressions of theexpected values of all the derivatives.

E (exp (k lnht) ln (ht)) = EkL E (exp (k lnht) ln (ht�1)) = EkL(�1)

E (exp [� ln (ht)] exp [� ln (ht�1)]) = E�E� (�1)E�exp [� ln (ht)] exp [� ln (ht�1)] Izt�1

�= E�EI� (�1)

E (exp (� lnht) lnht exp (� lnht�1)) = E�LE� (�1)E�exp [� ln (ht)] lnht exp [� ln (ht�1)] Izt�1

�= E�LEI� (�1)

E�ln2 (ht)

�= L2 E (ln (ht))

3 = L3 etc:

E (ht;�) = E;� E (ht;�) = E;� E (ht;�)2 = E(;�)2 E (ht;�)

3 = E(;�)3 etc:

E (ln (ht)ht;�) = LE;� E (ln (ht)ht;�) = LE;� E (ht;� ln (ht)) = LE;� etc

E�ht;� (ln (ht))

2� = L2E;� E�ln (ht)h

2t;�

�= LE(;�)2

E (exp (� lnht)ht;�) = EkE;� E (exp (k lnht)ht;�) = EkE;�

E (exp (k lnht)ht;�) = EkE;� etc:

E�exp (� lnht)h

2t;�

�= E�E(;�)2 E

�exp (� lnht)h

3t;�

�= E�E(;�)3 etc:

E�ht;� (ht;�)

2� = E;�(;�)2

E (ht�1;�ht�1;�) = E;�;� E (ht;�ht;�) = E;�;� E (ht;�ht; ) = E;�; etc:

E (exp [� ln (ht)]ht;�ht;�) = E�E;�;� etc

E (ht;�;�) = E;�;� E (ht;�;�) = E;�;� etc:

E (ht;�ht;�;�) = E;�;�;�

E (ln (ht)ht;�;�) = LE;�;�

E (exp (� lnht)ht;�;�) = E�E;�;�E (ht;�ht;�;�) = E;�;�;�

E (exp (� lnht) ln (ht)ht;�) = E�LE;�

E (ht;�ht;�;�) = E;�;�;� E (ht;�ht;�;�) = E;�;�;�

E (exp (� lnht)ht;�;�) = E�E;�;�

36