Journal of Econometrics. Properties and estimation of asymmetric exponential power distribution

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1 Journal of Econometrics Contents lists available at ScienceDirect Journal of Econometrics journal homeage: Proerties and estimation of asymmetric exonential ower distribution Dongming Zhu a Victoria Zinde-Walsh bc a HSBC School of Business Peking University China b Deartment of Economics McGill University Canada c CIREQ Canada a r t i c l e i n f o a b s t r a c t Article history: Received 3 October 7 Received in revised form Setember 8 Acceted 4 Setember 8 Available online 5 October 8 JEL classification: C3 C6 Keywords: Asymmetric distributions Maximum likelihood estimation The new distribution class Asymmetric Exonential Power Distribution AEPD roosed in this aer generalizes the class of Skewed Exonential Power Distributions SEPD in a way that in addition to skewness introduces different decay rates of density in the left and right tails. Our arametrization rovides an interretable role for each arameter. We derive moments and moment-based measures: skewness kurtosis exected shortfall. It is demonstrated that a maximum entroy roerty holds for the AEPD distributions. We establish consistency asymtotic normality and efficiency of the maximum likelihood estimators over a large art of the arameter sace by dealing with the roblems created by non-smooth likelihood function and derive exlicit analytical exressions of the asymtotic covariance matrix; where the results aly to the SEPD class they enlarge on the current literature. Also we give a convenient stochastic reresentation of the distribution; our Monte Carlo study illustrates the theoretical results. We also rovide some emirical evidence for the usefulness of emloying AEPD errors in GARCH tye models for redicting downside market risk of financial assets. 8 Elsevier B.V. All rights reserved.. Introduction Observed characteristics of many financial data series have motivated exloration of classes of distributions that can accommodate roerties such as fat-tailedness and skewness while nesting distributions tyically used in estimation such as the normal and skew-normal. An imortant desired roerty of any such class is that it ermits maximum likelihood estimation of all arameters. Obtaining closed-form exressions for the moments of interest such as the mean variance skewness and kurtosis as well as comonents of the information matrix rovides useful interretable features of the distributions in the class. For alications in risk management one may in addition be interested in closed-form exressions for value-at-risk and exected shortfall of asset/ortfolio returns. Classes of non-symmetric distributions that nest the skewnormal were constructed by Azzalini 986. Other classes of distributions with the desired roerties of accommodating heavy tails and skewness the Skewed Exonential Power Distribution SEPD classes were roosed in Fernandez et al. 995 Theodossiou and Komunjer 7; they all generalize the generalized error distribution GED class. Many financial alications of the GED as well as its skew extensions have been considered in Hsieh 989 Nelson 99 Duan 999 Rachev and Mittinik Theodossiou Ayebo and Kozubowski 4 Komunjer 7 Christoffersen et al. 5 and others. Esecially in alications to otion ricing the GED and its skew extensions are referred to Student-t distributions because it is found that Student-t distributions are not suited to model continuously comounded returns see Duan 999 and Theodossiou. Since all moments of the GED exist the moments of exonential transformations of GED random variables needed to rice otions can be evaluated. Ayebo and Kozubowski 4 resented basic roerties of the SEPD of Fernandez et al. 995 derived maximum likelihood estimators of scale and skewness arameters given other arameters and discussed its alications in finance. Komunjer 7 exlored moments also see Theodossiou as well as measures such as value at risk and exected shortfall useful in financial alications. DiCiccio and Monti 4 studied roerties of MLEs of the Azzalini s 986 SEPD and obtained results for the information matrix not in closed form and for inferential roerties of MLE. Corresonding address: Deartment of Economics McGill University 855 Sherbrooke Street West Montreal PQ H3A T7 Canada. Tel.: address: victoria.zinde-walsh@mcgill.ca V. Zinde-Walsh. The GED class was roosed first by Subbotin 93 and Box and Tiao 973 called such a distribution the Exonential Power Distribution EPD. It is also called the Generalized Power Distribution or the Generalized Lalace Distribution /$ see front matter 8 Elsevier B.V. All rights reserved. doi:.6/j.jeconom

2 D. Zhu V. Zinde-Walsh / Journal of Econometrics However for some alications in finance and risk management the skew extensions may not be rich enough to cature all the asymmetry of distributions of asset returns articularly asymmetry in the tails. For examle it is found esecially for ortfolios such as S&P5 and NASDAQ that ex ost innovations from estimated GARCH models even with a leverage effect are not normally distributed the QQ lot of ex ost innovations tyically shows that the fit in the uer tail is good but the lower tail is heavier than that of the normal distribution see Figure 6 of Bradley and Taqqu 3 for NASDAQ Figure 4. of Christoffersen 3 for S&P5. To cature the asymmetry in the tails this aer extends the SEPD to a fully asymmetric exonential ower distribution AEPD where heavy-tailedness itself may be asymmetric with different tail exonents on different sides of the distribution. We demonstrate that the AEPD class has desired roerties: interretable arameters to reresent location scale and shae closed-form exressions for the moments as well as for value at risk and exected shortfall. A maximum entroy roerty is shown to hold and a stochastic reresentation of the AEPD is given. We develo asymtotic roerties of the MLE consistency and asymtotic normality and obtain fully closed-form exressions for the information matrix for all arameters. Thus we also rovide new theoretical results such as closed-form exressions for the asymtotic covariance matrix and consistency and asymtotic normality of MLE for some SEPD classes exanding on results currently available in the literature. Comaring the AEPD with Azzalini s 986 SEPD class both classes have continuous but non-differentiable densities; the latter density however involves an integral normal cdf. Also the AEPD has more flexible tail behavior and analytical exressions for mode and moments; while for Azzalini s 986 SEPD the left tail is always thinner than the right one its odd moments involve infinite series exansions and it is not ossible to find an analytic exression for the mode. In addition note that DiCiccio and Monti 4 were not able to rovide closed form exressions for the information matrix nor comlete roofs of asymtotics for MLE for Azzalini s 986 SEPD. In addition we rovide some emirical evidence for the usefulness of emloying AEPD errors in GARCH tye models for redicting the value at risk of financial assets. The aer is organized as follows. Section exlains the relation between EPD SEPD and AEPD classes highlighting the main features of the new AEPD class. The interretation of arameters is rovided in Section 3. Section 4 gives basic roerties of the AEPD such as analytical exressions of cdf quantiles moments and exected shortfall. In Section 5 we establish consistency and asymtotic normality of the MLE and Section 6 rovides some finite samle Monte Carlo results. We also rovide an alication of the AEPD in Section 7. Technical results and roofs are collected in the Aendices A C; more detailed roofs are in Zhu and Zinde- Walsh 7. Aendix D shows grahs of AEPD densities for different arameter combinations.. The relation between EPD SEPD and AEPD The density function of the EPD or GED is usually defined as: f EP x µ σ σ K EP ex x µ σ where µ R and σ > are the location and scale arameters resectively > is the shae arameter and K EP is the normalizing constant K EP / / Ɣ + /]. If X is a random variable with the EPD density then the location arameter µ EX medx the median of X; the scale arameter σ E X µ / which is the L -norm deviation has an interretation similar to that of the standard deviation of the normal distribution. When the shae arameter gets smaller and smaller the EPD becomes more and more heavy-tailed and letokurtic. With and + the EPD reduces to the normal Lalace and uniform distributions resectively. So far there are two different methods to extend the EPD to a skewed exonential ower distribution SEPD. Azzalini 986 first roosed a family of SEPD based on the fact that if f is a density symmetric about and Π an absolutely continuous distribution function such that its df Π is symmetric about then Πλxf x is a density for any real λ. Taking f f EP and Π normal cdf or EPD s cdf we get Azzalini s SEPD class. Fernandez et al. 995 extended the EPD class to another family of SEPD by using a two-iece method in which an additional skew arameter γ is introduced also see Kotz et al. 7. By a method similar to that of Fernandez et al. 995 Theodossiou and Komunjer 7 resectively constructed seemingly different classes of SEPD which are actually rearametrizations of that of Fernandez et al However Komunjer s 7 asymmetry arameter α is interestingly interreted as the robability such that the location arameter is exactly the α- quantile of the SEPD. Noting the interretable nature of the arameters this aer follows a similar method to construct the AEPD. The AEPD density has the following form: f AEP x β α α σ K EP ex x µ α σ if x µ; α α σ K EP ex x µ α σ if x > µ where β α µ σ T is the arameter vector µ R and σ > still reresent location and scale resectively α is the skewness arameter > and > are the left and right tail arameters resectively K EP is the same as in and α is defined as α αk EP /αk EP + αk EP ]. 3 Note that α α K EP α α K EP αk EP + αk EP B. 4 The AEPD density function is still continuous at every oint and unimodal with mode at µ. The arameter α in the AEPD density rovides scale adjustments resectively to the left and right arts of the density so as to ensure continuity of the density under changes of shae arameters α. If imlying α α the AEPD reduce to a new version of SEPD: f SEP x β σ K EP ex x µ ασ if x µ; σ K EP ex x µ ασ if x > µ which is equivalent to those of Fernandez et al. 995 Theodossiou and Komunjer 7. This new version of SEPD rovides new interesting interretations for scale and skewness in terms of L distances. The skewness arameter α lays the same role as the arameter γ of Fernandez et al By rearametrization α γ / + γ and σ / / γ + /γ σ / the SEPD 5 will become that of Fernandez et al. 995; A referee ointed out various other references e.g. Arellano-Valle et al. 5 and Salinas et al. 7 that generalize the class of asymmetric models beyond ower distribution classes however none of those classes considers asymmetry in the tails of the distribution. 5

3 88 D. Zhu V. Zinde-Walsh / Journal of Econometrics a re-scaling of the density leads to Komunjer s 7; letting α + λ/ σ θσ / and µ µ δσ the density will be that i.e. f y µ σ λ in Equation of Theodossiou where θ and δ are given in Equations and 3 of Theodossiou. With α / the SEPD 5 reduces to the EPD. The skewed Lalace distribution and skewed normal distribution are secial cases of the SEPD resectively with and. According to the skewness measure of Arnold and Groeneveld 995 -CDFmode the SEPD density is skewed to the right for α < / and to the left for α > /. A convenient rearametrization of is obtained by rescaling f AEP x θ σ ex x µ ασ K EP if x µ; σ ex x µ ασ K EP if x > µ where θ α µ σ T. From the rescaled AEPD density 6 we can clearly observe the effects of the shae arameters on the distribution. The density in the form 6 is used in deriving a closed form exression for the information matrix of the maximum likelihood estimator MLE. 3. Interretation of arameters of AEPD The main tools that are used for interretation are various L r sace related distance measures. Define for r > d L r E X µ r X µ] /r d R r E X µ r X > µ] /r resectively called the L r -norm deviation or distance conditional on X µ and the L r -norm deviation conditional on X > µ. The total conditional deviation or distance is dr d L r + d R r; the L r -norm deviation X µ r E X µ r /r. Suose now that random variable X has the AEPD density defined in with shae arameters α location µ and scale σ. Proosition. The following relations hold: a PX µ α; also d L α σ and d R α σ where α is defined in 3; b σ d L + d R ]; c there is a ositive function r c deending on arameter and increasing in its argument c such that α d L r c d L r c + d R r c ; c > max{lb lb } where lb Ɣ + /] ex{ Ψ /} and Ψ x Ɣ x/ɣx is a digamma function. d d L r α σ M r ασ ξ r/b and d R r α σ M r ασ ξ r/b where M r / {Ɣr + //Ɣ/} /r ; ξ r K EP M r is strictly increasing in r and decreasing in B and K EP are defined above. Proof. See Aendix A. From art a the location µ is the α-quantile of the r.v. X and the scale σ is related to the left and right L conditional deviations by the arameter α for the SEPD α α. Part b reresents the scale σ via an average of the left and right conditional deviations. It follows from a that the ratio of the left conditional deviation to the total is α for SEPD just α. Part c gives an interretation of α with two adjusted order functions r c i i ; in the SEPD case the left and right conditional deviations enter with a same order thus then α d L r/dr for any r >. 6 Part d allows us to investigate the effect of shae arameters a. These shae arameters have a common effect on both d L r and d R r through B αk EP + αk EP which reresents a scale adjustment effect. Ignoring the common effect α has the same effect on the AEPD as it does on the SEPD but the left and right tail arameters and resectively control the left and right L r -deviation d L r and d R r. Since ξ r is a strictly decreasing function of for any given r a smaller or leads to a larger left or right L r -deviation thus AEPD with a smaller or has a heavier left or right tail. The effect of or on the left or right tail can be measured by a generalized kurtosis index kur L r or kur R r for r > called the left or right generalized kurtosis similar to Mineo 989 who defined generalized kurtosis as E X µ E X µ and showed that for EPD it is +. The left and right generalized kurtoses are defined as kur L r d L r/d L r] r kur R r d R r/d R r] r. With r we get the usual definition of kurtosis. Proosition. For the AEPD the left and right generalized kurtosis can be exressed as follows: r + kur L r Ɣ Ɣ kur R r Ɣ Ɣ r + / Ɣ r + 7 / r + Ɣ ; 8 they are strictly decreasing resectively in and for any given r > and strictly increasing in r given >. Proof. See Aendix A. From the exressions for kur L r and kur R r of the AEPD the heaviness of the left or right tail is controlled by only or. If < then kur L r > kur R r imlying that the left tail is heavier than the right. When i < i the AEPD is more heavy-tailed than the normal distribution. The left or right tail arameter or is directly related to the left or right generalized kurtosis by the relation: kur L + or kur R +. Further results about kurtosis via moments are in the next section. Fig. in Aendix D lots the AEPD densities of the form 6 with µ and σ for combinations of shae arameters α. The first lot shows that for given and the density curve shifts to the right with α decreasing but its mode does not change; the second lot shows how controls only the right tail heavier and heavier for smaller and smaller. The effect of skewness arameter and tail arameters on tails is comared in the last lot. Although a smaller α leads to a fatter right tail this influence eventually is dominated by the effect of a smaller. 4. Basic roerties of the AEPD 4.. Cumulative distribution quantile function and moments All the formulae in this section follow straightforwardly from results for the classical EPD summarized in III in Aendix A. Suose that X is a random variable with the standard AEPD density µ σ. Denote a b min{a b} a b max{a b} by Gx; γ the gamma cdf: x Gx; γ Ɣγ z γ ex zdz 9

4 D. Zhu V. Zinde-Walsh / Journal of Econometrics and by G x; γ the inverse function of Gx; γ. Then for the standard AEPD the cdf can be exressed via G ; : F AEP x α x α G ; ] α if x x α + αg ; α if x > and the quantile function is exressed via G ; F υ α AEP α G υ α ; ] / if υ α α G υ α ; ] / if υ > α where υ. Note that for any measurable function hx of the standard AEPD random variable X we have EhX] αehx X ] + αehx X > ] imlying that all unconditional moments can be exressed as a weighted sum of two conditional moments. Therefore we first give the conditional moments of the standard AEPD r.v. X. From exression for the absolute moment of EPD 4 we get for any real r > E X r X < α ] r E Z r B r α r H r E X r X > α ] r E Z r B r α r H r 3 where Z is a random variable that has the standard EPD density µ σ in with ower index B is defined in 4 H r r Ɣ +r /Ɣ+r. For any non-negative integer k the kth right-conditional moment EX k X > has the same exression as in 3 while the k th left-conditional moment EX k X < has an exression slightly different from 3 : EX k X < α ] k E Z k B k α k H k. Thus the kth moment of the standard AEPD r.v. X equals EX k B k k α +k H k + α +k H k ] k and its r-absolute moment is exressed as E X r B r α +r H r + α +r H r ] r >. 5 In articular the mean and variance of the standard AEPD r.v. X are given as follows: EX B α Ɣ/ Ɣ / α Ɣ/ Ɣ / ] 6 { VarX α 3 Ɣ3/ B Ɣ 3 / + α3 Ɣ3/ Ɣ 3 / α Ɣ/ Ɣ / ] } Ɣ/ α. 7 Ɣ / 3 Note that these moments reresent lower artial moments which are used in finance literature as risk measures see Bawa 975. We see that all moments can be exressed simly and conveniently in terms of gamma function. In the case of the SEPD and we get simlified exressions for moments: EX k / k k α +k + α +k ] Ɣ + k//ɣ/ 8 E X r / r α +r + α +r ] Ɣ + r//ɣ/ 9 where k... and r >. These rovide an advantage over Azzalini s 986 SEPD class where the exressions for the odd moments involve infinite series exansions; 8 is a rearametrization of formulae of Fernandez et al. 995 and Komunjer Value at risk and exected shortfall Value at risk VaR for return on a ortfolio or an asset is defined as the υ-quantile of the distribution of returns with a negative value corresonding to a loss. Here the quantile function F υ AEP α of rovides VaR at υ for the historical distribution of returns in the AEPD class i.e. VaR AEP υ F υ α AEP. The Exected Shortfall ES of a standard AEPD random variable X ES AEP q E X X < q also called Conditional Value at Risk CVaR reresents the negative exected return or loss conditional on it being below the threshold q. It can be exressed in terms of the gamma CDFs with arameters / / / and / : ES AEP q α C G q α ; / q ; G q α ; / αα C α α q C G α ; / q α + αg α ; / q > where C / Ɣ//Ɣ/ Gx; γ is the gamma cdf given in 9. Recall that G x; γ is the inverse function of Gx; γ. For q VaR AEP υ the ES as a function of confidence level υ denoted by ES AEP υ can be exressed as follows: ES υ AEP { α υ υ αα C G G α ; υ α { αα C α α C υ υ α G G α ; ; / ]} ; / ]} υ > α. In ractice ES is often used in the following form: Eq X X < q q + E X X < q which is the average loss when an asset return falls below q; the exression follows from ES AEP q or ES AEP υ Maximum entroy roerty In a distribution class maximum entroy is achieved by a distribution that encodes information in the least biased way without giving any referential measure weight to any art of the distribution other than what is required by the distribution class itself. Here we consider a class of absolutely continuous

5 9 D. Zhu V. Zinde-Walsh / Journal of Econometrics distributions with secific shae moment constraints on the left and right deviations and show that the AEPD as defined in 6 has the maximum entroy roerty in that class. Secifically consider for arameters θ α µ σ an absolute deviation function of x R scaled differently on two sides of µ : yx Lx; θ + Rx; θ with Lx; θ Ɣ + / x µ x < µ ασ Rx; θ Ɣ + / x µ x > µ. ασ Define a class Ωα µ σ of absolutely continuous distributions having densities x with suort + that satisfy the following moment constraints on the left and right deviations of yx : / Lx; θ yx α / x < µxdx ; / Rx; θ yx α / x > µxdx. This class allows for the location µ scale σ and three shae arameters α that roduce different effects: when arameter α alone governs which of the sides gets a larger weight when differ the smaller imarts a heavier tail to its side regardless of a. Thus such a class for fixed values of the arameters gives rise to distributions that could fit required roerties for shae in terms of the left/right conditional deviations. Proosition 3. The AEPD distribution in 6 has maximum entroy in the class Ωα µ σ. Proof. See Aendix A. 5. Asymtotic roerties of the maximum likelihood estimator Since AEPD generalizes the EPD and SEPD classes we note the asymtotic results available for the latter two classes. The MLE for the EPD arameters and its roerties are investigated in Agrò 995 where the information matrix Iβ and the covariance matrix are derived; for > consistency asymtotic normality and efficiency of MLE are roved; other theoretical results for the MLE are available when is known. Ayebo and Kozubowski 4 focused on estimators of scale σ and skewness arameter α in the SEPD by assuming that location µ and tail arameter are known; they gave the exressions for the MLEs of σ and α showed that they are consistent asymtotically normal and efficient and rovided the asymtotic covariance matrix for this subset of arameters. DiCiccio and Monti 4 investigated roerties of the MLE of all arameters for the Azzalini s SEPD class but they did not give a closed-form exression for information matrix and did not rovide a rigorous roof of asymtotics for the MLEs which is needed due to the non-smoothness of the log-likelihood function. Here we establish consistency asymtotic normality and efficiency for MLE of all arameters in the AEPD class which nests EPD and SEPD with > and > and rovide a closed-form asymtotic covariance matrix of the MLE. Suose that the true density f y θ with θ α µ σ belongs to the AEPD class given in 6 with arameter vector θ in a arameter sace Θ Ξ {θ θ α µ σ σ > α µ R} where Θ is assumed to be a comact set and θ to be an interior oint of Θ. For a random samle y y y... y T the log-likelihood function l T θ y T t ln f y t θ is given as follows: T Ɣ + / µ y t l T θ y T ln σ y t µ ασ t T Ɣ + / y t µ y t > µ. ασ t Note that the AEPD does not satisfy the regularity conditions under which the ML estimator has the usual T -asymtotics because of a non-differentiable likelihood function. However we nonetheless establish consistency of the MLE by using Theorem.5 in Newey and McFadden 994 and under certain arameter restrictions establish the usual asymtotic normality for the AEPD s MLE by using Theorem 3 as well as its corollary in Huber 967. Proosition 4. Consistency of MLE. The MLE θ of θ is consistent i.e. θ θ. Proof. See Aendix C. Proosition 5. The information matrix equality Iθ Hθ holds for > / > /. The elements of the Fisher information matrix φ ij φ ij E ln f y t ; θ / θ i ] ln f y t ; θ / θ j ] with φ ij φ ji and θ j the jth element of arameter vector θ α µ σ T are as follows 4 : φ α α φ φ 3 φ 4 σ α + α φ 5 φ α + / σ 3 Ψ + / φ 3 φ 5 α σ φ 4 σ Ψ Ψ + / ] φ 35 α σ φ 55 α + α σ φ 33 α + / 3 Ψ + / φ 34 Ψ Ψ + / ] σ φ 44 Ɣ/ Ɣ / ασ + Ɣ/ Ɣ / ασ φ 45 σ 3 where all the φ ij are evaluated at the true values α µ σ. Proof. See Aendix B. The information matrix for the MLE of the SEPD is given below; to our knowledge these results were not available in the literature so far. 4 By using ƔxƔ x π/ sinπx see Artin or Farrell and Ross the element of φ 44 can also be exressed as φ 44 π / σ α sinπ/ + /. α sinπ/

6 D. Zhu V. Zinde-Walsh / Journal of Econometrics Corollary 6. For the SEPD ln f ln f + ln f from 48; the terms φ ij involving and become φ + φ 3 φ + φ 33 φ 4 + φ 34 and φ 5 + φ 35. The information matrix for the MLE of the SEPD arameters α µ σ is: Iθ Hθ + α α σ α α σ α α + Ψ + 4 σ Ɣ/Ɣ /. σ α α σ Proosition 7. Asymtotic Normality of MLE Suose that > and >. Then the MLE θ of θ is asymtotically normal i.e. T θ θ d N I θ where Iθ is the Fisher information matrix: Iθ E θ ln f Y t θ θ ln f Y t θ ] rovided by 3; it can be consistently estimated by I θ. Proof. See Aendix C. The information matrix equality Iθ Hθ holds only for > / and > / because E ln f µ ] as an element of Iθ may not exist or may be negative for some oints of / and or /. Since Iθ is continuous for all θ Ξ satisfying > / and > / it follows from the consistency of θ that I θ is a consistent estimator of Iθ. The restriction > and > ensures that the exected score vector converges uniformly in the neighborhood of the location arameter and is required for the estimation of the location arameter. The restriction is not an imediment in most alications. Even for the GARCH otion ricing model with GED conditional distribution in Duan 999 this restriction is imosed in order to ensure the existence of the exected simle return. If µ is known then the usual T -asymtotics hold for the MLEs of other arameters α σ without any restrictions. When location arameter µ can be consistently estimated by a nonarametric method see Andrews et al. 97 and Bickel the MLEs of other arameters are still consistent asymtotically normal but may not be efficient. 6. Performance of MLE in simulation A stochastic reresentation of a distribution is imortant to simulation studies. For given values of arameters and α < α < i > i we can generate standard AEPD random numbers by the following method: first generate three random numbers U W and W where U is drawn from standard uniform distribution U and W i i is from the gamma distribution with shae arameter / i and df f Wi w Ɣ/ i w /i ex w; second define a random variable Y : signu α Y αw / + α W / Ɣ + / signu α + Ɣ + / σ ] 4 where signx + if x > if x < and if x. It is straightforward to show that random variable Y has the density 6 of standard AEPD location µ scale σ. An alternative method is the inverse method i.e. using Y F AEP U to generate standard AEPD random numbers where U is a standard uniform random variable and F AEP is the standard AEPD cdf. However this method is very time-consuming while the method given in 4 allows us to generate AEPD random numbers more quickly in Matlab. To assess the asymtotic roerties of the MLE in finite samles following Agrò 995 a numerical investigation of bias and variance of MLEs was made using samle sizes of T We choose µ σ and various different true values of α : α.3.5 and i i. To save sace here we only reort the cases of α.3 and.5. For each set of true values of arameters and every samle size N relications are drawn from the AEPD with the set of arameter values and then N ML estimates θ i i... N are obtained using these samles. So we can estimate the means and standard deviations of the MLEs of arameters denoted resectively by M θ and STD θ M θ N θ i N i N STD θ θ i M θ ] / N i and comare these estimated standard deviations with their theoretical values which are taken from the square root of the diagonal elements of Cramer Rao bound i.e. I θ/t. Simulation results are resented in Table. All entries labeled Mean of MLEs reort M θ and those in STD Ratio rows are the ratios of simulated standard deviations STD θ to the theoretical ones from I θ/t. From our simulation studies we can see that the estimates θ of all arameters seem asymtotically unbiased for all given true values and that their variance seems to be aroaching the Cramer Rao bound for the cases of > and > see the cases in which.5.5 and.5.5. However for the cases of or the behavior of the variance aears to be roblematic. More secifically although the estimates of scale σ aear always efficient in all cases there are significantly large ratios of standard deviation for the other arameters esecially for µ and α; but the larger the values of the tail arameters and the more efficient the estimates of α and µ aear to be. Other observed henomena are that because of fewer observations on the left side estimates of the left tail arameter have slower convergence and lower efficiency than those of the right tail see the cases in which or.5; in general the MLE is more efficient in the cases with larger tail arameter or than for those with smaller tail arameters. Finally we want to oint out that for a small samle say a size less than 5 the likelihood function may not have any maximum oint. This roblem still exists for the GED and is discussed in detail in Agrò Forecasting value at risk: An emirical examination In this section we examine forecast for value at risk VaR with GARCH tye model and comare erformance for error distributions given by GED SEPD and our AEPD. 7.. Model and data GARCH tye models have been widely and successfully used to model financial asset returns. In general a return rocess r {r t }

7 9 D. Zhu V. Zinde-Walsh / Journal of Econometrics Table Simulation results for the MLE of the AEPD. P P.5 alha.3.7 sigma mu alha sigma mu T Mean of MLEs alha.3.7 sigma mu alha sigma mu T STD ratio alha.3 sigma mu alha.3.5 sigma mu T Mean of MLEs alha.3 sigma mu alha.3.5 sigma mu T STD ratio alha.3.5 sigma mu alha sigma mu T Mean of MLEs alha.3.5 sigma mu alha sigma mu T STD ratio alha.3.5 sigma mu alha sigma mu T Mean of MLEs alha.3.5 sigma mu alha sigma mu T STD ratio is modeled as 5 r t m t + σ t z t 5 where following tradition m t and σ t are the conditional mean and variance of r t given the information set available at time t i.e. m t E t r t and σ t E t r t m t z t are the i.i.d. innovations with zero mean and unit variance. To cature the leverage effect we adot the non-linear asymmetric GARCH NGARCH structure of Engle and Ng 993. The conditional distribution of the return rocess is modelled as the AEPD tye distribution. For simlicity we assume m t m for 5 As noted by Andersen et al. 5 this reresentation is not entirely general as there could be higher-order conditional deendence in the innovations. any t; the return series r t is an AEPD-NGARCH rocess r t m + σ t z t z t i.i.d.aepd σ t b + b σ t + b σ t z t c b + b σ t + b r t m cσ t. 6 The arameter c in the NGARCH equation 6 catures the leverage effect; that is a ositive value of c gives rise to a negative correlation between the innovations in the asset return and its conditional volatility. The j-ste-ahead forecast of σ t+j denoted by σ t+j t is defined as σ t+j t E t σ t+j ; it is σ t+ t b + b σ t + b r t m cσ t 7 σ t+j t b + b + b + c ] σ t+j t j. 8

8 D. Zhu V. Zinde-Walsh / Journal of Econometrics Table Parameter estimates for the AEPD-NGARCH models. m b b b c α AEPD AEPD α SEPD GED We consider daily returns 6 on the S&P5 comosite index. Emirical evidence has indicated that high frequency data continue to exhibit conditional tail-fatness even after allowing for the GARCH effect see Bollerslev et al. 99. Our samle covers the eriod from January 99 to December 3 and the samle size is T 38. The data set is from CRSP Center for Research in Security Prices University of Chicago. 7.. Estimation and goodness of fit The maximum likelihood estimate of the arameter vector φ where φ m b b b c α is obtained by maximizing the log-likelihood function T Lφ; r {log δ log σ t + log f Y ω + δ r } t m β 9 σ t t where f Y β is the standard density function of the AEPD with the distributional arameters β α ω ωβ and δ δβ denote the mean and standard deviation of f Y β resectively and as functions of β both are given in 6 and 7. 7 To show the significance of asymmetric behavior in the tails we consider the AEPD and nested distribution classes: the AEPD with α / to reresent asymmetry arising only from different tail behavior the SEPD i.e. AEPD with the GED i.e. AEPD with α / and. The ML estimates of the arameters and their standard deviations are dislayed in Table. Following Mittnik and Paolella 3 we emloy four criteria for comaring the goodness of fit of the candidate models. The first is the maximum log-likelihood value L which can be viewed as an overall measure of goodness of fit. The second and the third are the AICC Hurvich and Tsai 989 and the SBC or SIC Schwarz 978 which modify the AIC and are given by AICC L + Tk + T k k logt SBC L + ; 3 T k denotes the number of estimated arameters and T the number of observations. The fourth is the Anderson Darling statistic Anderson and Darling 95 defined as AD T su x R FT x Fx Fx Fx 3 where Fx denotes the estimated arametric cdf of innovation and F T x is the emirical cdf of ex ost innovations i.e. F T x n/t if there are only n ex ostinnovations ẑ t r t m/ σ t less or equal to x for any given x. 6 The return rt in eriod t is defined as r t P t P t /P t where P t is the Level on S&P Comosite Index at time t. 7 The ML estimation is imlemented in Matlab 6. with the command fmincon and initial value φ meanr b where b is given by the variance of returns data multilied by b b.5. Table 3 Goodness-of-fit measures for the AEPD-NGARCH models. L AICC SBC AD AEPD AEPD α SEPD GED The AD statistic is a reasonable measure of the discreancy or distance between the two distributions say the emirical cdf F T x and the hyothetical distribution Fx; this statistic gives aroriate weight to the tails of the distribution so that it can be used to measure goodness of fit in the tails. In our alications since the innovations are assumed to have zero mean and unit variance the estimated cdf of the innovations Fx in 3 can be exressed as Fx FY ω + δx β where F Y β is the cdf of the standard AEPD with β α T β is the ML estimate of β and ω and δ are given by ω ω β and δ δ β which are resectively the estimated mean and standard deviation of F Y β. For simlicity we comute the AD statistic as follows: AD max AD j j AD j FT ẑ jt FẑjT T 3 FẑjT FẑjT { } T where ẑ jt are the sorted in ascending order ex ost j innovations thus F T ẑ jt j/t. Table 3 dislays the four measures of goodness-of-fit for the estimated AEPD-NGARCH models. All measures rank the distribution with full asymmetry as the best followed by asymmetry in tails only skewness only and then symmetry in a descending order Prediction erformance for downside risk To redict the downside risk VaR in the eriod t + j j 3... using the information available in eriod t we must give a j-ste-ahead forecast of the conditional distribution of returns r t+j Ft+j t r t+j. Based on the above models secified in 5 and 6 the conditional distribution is time-varying only due to the time-varying conditional mean and variance. Therefore forecasting the conditional distribution boils down to estimating the arameters of the model using the data available at time t and then forecasting the conditional mean m t+j and variance σ t+j of r t+j. Denote the time-t ML estimates of these arameters by m t bt bt bt ĉ t β t and the estimates of the j-ste-ahead forecasts of m t+j and σ t+j by m t+j t and σ t+j t resectively. Then m t+j t m t for any j and σ t+j t is obtained by substituting the estimated arameters into 7 and 8 σ t+ t bt + bt σ t + bt r t m t ĉ t σ t σ t+j t bt + bt + bt + ĉ t ] σ t+j t j. Now we consider estimation of the j-ste-ahead forecast of the conditional VaR. Note that z t+j r t+j m/σ t+j is assumed to be an AEPD random variable with zero mean and unit variance. Then based on the redicted values m t+j t σ t+j t ω t ω β t and δ t δ β t where ω and δ are defined in 9 we exect at time t that conditional on the information available in eriod t Ŷ AEP ω t + δ t r t+j m t+j t / σ t+j t should aroximately have a standard AEPD density with arameter β t if the model secification is correct. Therefore the j-steahead forecast of the conditional VaR can be estimated as follows: ] VaR t+j t F t+j t m F AEP t+j t + σ β t ω t t+j t. 33 δ t

9 94 D. Zhu V. Zinde-Walsh / Journal of Econometrics Table 4 Predictive erformance for the value at risk VaR fj f f5 f f5 f f5 f f5 AEPD AEPD α SEPD GED Note that in the calculation of the conditional VaR we also need to give the exressions for F AEP β i.e. the quantile function see. Therefore the downside risk is determined not only by the secification of the conditional mean and NGARCH equations but also by the distributional choice for the innovations. We can exress the redictive downside risk as follows: after j eriods the return would be less than VaR t+j t with robability. To see redictive erformance out-of-samle we slit the samle in two: N T/ 64. Then we evaluate VaR t+j t N 64 t T j for one and five stes ahead: j 5. We set the shortfall robabilities For each of j if the model is correctly secified we exect % of the observed r t+j -values r N+j... r T to be less than the VaR t+j t imlied by the model. If the observed frequency T j fj {r t+j < VaR t+j t } 34 T N j + tn is lower higher than then the model tends to overestimate underestimate the risk. Table 4 shows the redictive erformance of VaR; the entries in the table are the observed frequency fj given in 34 for one and five stes ahead: j 5 and shortfall robabilities All the models tend to underestimate the value at risk but the models with AEPD errors erform noticeably better than those with a single tail arameter. For small shortfall robabilities..5 the unrestricted AEPD is the best; for the.5 and. the AEPD with α.5 dominates. Thus both by measures of fit and by forecasting erformance the AEPD class rovides a useful model for the error in GARCH tye model of returns on assets. Acknowledgements The suort of the Social Sciences and Humanities Research Council of Canada SSHRC and the Fonds qué becois de la recherche sur la société et la culture FRQSC is gratefully acknowledged. Ɣ x + /Ɣx + xψ x 37 Ɣ x/ɣx Ψ x + Ψ x 38 Ɣ x + /Ɣx Ψ x + xψ x + xψ x 39 where C is Euler s constant k any ositive integer. More roerties and details are in Abramowitz and Stegun Artin and Farrell and Ross 963. III Proerties of EPD based on Box and Tiao 973 and Kotz et al.. For Z with standard EPD density µ σ in the cdf and quantile function are F EP x + signxg x ; ] 4 F EP υ signυ G υ ; ] / 4 Gx; γ is the gamma cdf and G x; γ is the inverse function of Gx; γ. By change of variable 35 for M r see Proosition d the absolute moment is E / Z r r + r/ Ɣ Ɣ/ M r] r r >. 4 The exected shortfall of Z ES EP x E Z Z < x is: ] / Ɣ/ G ES EP x x ; /. 43 Ɣ/ + signxg x ; / For x VaR EP υ F EP υ ES as a function of confidence level υ ES EP υ is ES EP υ { / Ɣ/ G G υ ; ]} ; /. 44 Ɣ/ υ Aendix A The roofs make extensive use of several results. I Integral see Gradshteyn and Ryzhik 994. #3.478 x v ex µx dx v µ v/ Ɣ for µ > v > >. 35 II For the gamma function Ɣx and digamma function Ψ x Ɣ x/ɣx: Ψ x C x + i x + i Ψ k x i k k+ i Ɣx + xɣx 36 x + i k+ Proof of Proosition. The result PX µ α follows directly from. Proofs of other equalities in a b d of Proosition boil down to calculations of d L r and d R r. For d R r of the standard AEPD µ σ by change of variable z x/ α ] and 4 or 35 we have d R r { E X r X > ] } /r { } /r x r f AEP x; α α dx { α z r K EP ex } /r z dz α {E Z r } /r α M r.

10 D. Zhu V. Zinde-Walsh / Journal of Econometrics To rove that ξ r K EP M r is strictly increasing in r we evaluate the derivative of ln M r with resect to r and show >. Note that ln Mr r ln M r r r Ψ r + log Ɣr + / log Ɣ/]. r By the mean value theorem we have ln M r ] r + εr + Ψ Ψ r r where < ε <. Since Ψ x is ositive for any x > see Abramowitz and Stegun ln Mr it follows that > for any r > and r >. To show that ξ r is strictly decreasing in for any given r > write ln ξ r Ψ + / + r Ψ r + Ψ r + ] and note that the second art of the above exression denoted by h r h r C + i r g i ] r + g i 45 where g i x x/i x/i + x C is Euler s constant and 45 for h r is based on 36. From the mean value theorem and g i x > g i x > for any x > and i it follows that ] h i r g r i g i r+ is strictly decreasing in r for every i ; so h r is a decreasing function of r. Therefore for r > ln ξ r < Ψ + / + lim r ] Ψ r + Ψ 3 i r + Ψ i + / < ] r + since Ψ + x Ψ x + /x by 37; the last equality uses 36 for Ψ x. To rove Proosition c define an increasing function r r c ξ c for a given. Note that ξ r lb as r and ξ r + as r + by using Equality 6.. in Abramowitz and Stegun 97. When c > lb r ξ c > and thus r c > and r c > for any c > max{lb lb }. Using definition of r c and equalities in Proosition d we get Proosition c. Proof of Proosition. The exressions for kur L r and kur R r in Proosition are easily obtained using equalities in Proosition d. Here we rove only that kr Ɣ r+ Ɣ /Ɣ r+ is strictly decreasing in and increasing in r. The second oint follows from ln kr ] r + r + Ψ Ψ > r > r and Ψ x > see 36 imlying >. The first oint follows from ln kr r + ρ ρ ln kr r > for any r > + ρ r + ] where ρx xψ x is strictly convex in + from 36 ρ x i i/i + x3 ln kr > for any x >. Then < for any > and r >. Proof of Proosition 3. The entroy of a distribution with density f x; θ is by definition + Hf f x; θ ln f x; θdx. A straightforward calculation shows that for AEPD Hf ln σ + α + α. By Theorem 3.. of Kagan et al. 973 also see Proosition.4.6 in Kotz et al. for all densities x suorted on + that satisfy: + + Lx; θ] xdx α ; Rx; θ] xdx α 46 the maximum entroy is attained by distributions with the densities of the form ME x ex{ λ λ Lx; θ] λ Rx; θ] } and only by them if constants λ λ and λ such that ME x > for all x + and satisfies 46 exist. We find a unique set {λ λ λ } such that ME x is the AEPD density 6. From + the conditions in 46 and MExdx straightforward calculations show that ασ λ / imlying ασ + λ / σ λ +/ σ λ +/ exλ αλ + αλ λ +/ λ +/. 47 This uniquely determines λ λ because λ as a function of λ is strictly decreasing by the first equation in 47 and increasing by the second and thus λ ln σ. Aendix B Aendix B is devoted to the derivation of the information matrix and to verifying the information matrix equality. Exectations are always taken with resect to the true underlying distribution f y; θ where θ α µ σ. Suose that y t t... T are i.i.d. observations from the AEPD whose density f y t ; θ with θ Θ is defined in 6. Let L Ly t ; θ Ɣ + / µ y t y t < µ ασ R Ry t ; θ Ɣ + / y t µ y t > µ. ασ Then the log-density function ln f y t ; θ ln σ Ly t ; θ] Ry t ; θ] and the score vector for observation t θ ln f y t; θ

11 96 D. Zhu V. Zinde-Walsh / Journal of Econometrics is given by ln f α α Ly t; θ] α Ry t; θ] 48 ln f Ψ + / ln Ly t ; θ Ly t ; θ] ln f Ψ + / ln Ry t ; θ Ry t ; θ] ln f µ Ɣ/ Ly t ; θ] + Ɣ/ ασ ασ Ry t; θ] ln f σ σ Ly t; θ] + σ Ry t; θ] σ where for x and > set x ln x. To derive the information matrix Iθ E θ ln f y t θ θ ln f y t ; θ ] and the Hessian Hθ E θ θ ln f y t ; θ ] and to verify the information matrix equality Iθ Hθ we first give the following Lemmas. Lemma 8. For any real number r > and integer m we have ELy t ; θ ] r ln Ly t ; θ ] m y t < µ α Ɣ m + r/ 49 m+ Ɣ + / ERy t ; θ ] r ln Ry t ; θ ] m y t > µ α m+ Ɣ m + r/ 5 Ɣ + / where Ɣ m is the mth order derivative of Ɣ and Ɣ means Ɣ. Proof. 8 Both equalities 49 and 5 are similarly. Here we only show 49. Denote by EL the exectation of the left hand side of 49 then µ EL Ly; θ] r ln Ly; θ] m f y; θdy µ Ly; θ] r ln Ly; θ] m σ ex{ Ly; θ] }dy. Then a change of variable x Ly; θ] results in + α EL x +r/ ln x m ex xdx m+ Ɣ + / α m+ Ɣ + / Ɣm + r/ for derivatives of gamma function see Farrell and Ross 963. Lemma 9. The score vector for observation t θ ln f y t; θ satisfies E θ ln f y t; θ. 5 8 For simlicity we omit the subscrit on the true arameters in all the following roofs. Proof. To verify this we use 49 5 and in Aendix A- II. Here we show E ln f ] ; other calculations are similar. In fact ln f E Ψ + / ELy t ; θ] ELy t ; θ] ln Ly t ; θ Ψ + / αɣ + / Ɣ + / αɣ + / Ɣ + / α Ψ + / α Ψ + /. Proof of Proosition 5. We derive exressions for E ln f y t ; θ / θ i θ j ] and E ln f y t ; θ/ θ i ] ln f y t ; θ/ θ j ] searately and then verify ln f yt ; θ E ln f y ] t; θ ln f y t ; θ E θ i θ j θ i θ j i j In the roof we use y t < µy t > µ and make use of 49 5 and in Aendix A-II. Here we show only the equality associated with φ 44 ; the others are roved similarly. In fact ln f E µ Ɣ/ ELy t ; θ] ασ Ɣ/ + ERy t ; θ] ασ Ɣ/ αɣ / ασ Ɣ + / Ɣ/ αɣ / + ασ Ɣ + / Ɣ/ Ɣ / + Ɣ/ Ɣ / ; ασ α σ also ln f E µ Ɣ + / Ɣ EL ] ασ Ɣ + / + Ɣ ER ] ασ ] Ɣ/ αɣ / ασ Ɣ + / Ɣ/ αɣ / + ασ Ɣ + / Ɣ/ / Ɣ / σ α + Ɣ/ ] / Ɣ / α Ɣ/ Ɣ / + Ɣ/ ] Ɣ /. σ α α

12 D. Zhu V. Zinde-Walsh / Journal of Econometrics Remark. Note that calculation of E ln f / µ ] requires i > to ensure / i > in the domain + of definition of Ɣx. If considering the Gamma function Ɣz defined on the comlex lane excet z... however the information matrix equality may hold for all > and > excet for oints /n n of and. We restrict and to satisfy > / and > / since Iθ and Hθ are undefined and thus discontinuous at oints /n n ; and the information matrix equality has no significance for and in intervals n n+ n 3... as E ln f µ ] is negative when both and are in these intervals see the exression for E ln f µ ] and roerties of the gamma function. The existence of E ln f / µ ] at and or is due to the fact that xɣx or sinx/x as x. Aendix C Aendix C establishes consistency and asymtotic normality of MLE of all arameters in AEPD. The following lemma is used in roof of Proosition 7. Lemma. a For any ε > there exists a ositive constant M that may deend on ε such that ln x M + x ε + x ε for any x >. 5 b For any µ q such that q q d and µ µ d the following inequalities hold: µ y q + µ + d y q+d + µ d y q d if y < µ d; 53 µ y q + µ + d y q+d if q > y < µ d; 54 y µ q + y µ + d q+d + y µ d q d if y > µ + d; 55 y µ q + y µ + d q+d if q > y > µ + d. 56 c Suose that Y is an AEPD r.v. with density f y θ defined in 6 where θ α µ σ. Then for any µ R and r > the following inequality holds: + r E Y µ r M µ r; θ Ɣ + r + M µ r; θ Ɣ 57 where M ; θ and M ; θ are two ositive continuous functions. Proof. Part a is immediate since for any ε > x ε ln x as x + and ln x /x ε as x +. Part b is easy for the cases of q > and q < ; the cases of µ ± d y > and µ ± d y < are considered subsequently. Part c is roved by using the c r -inequality see Loève y µ y µ µ µ for and then change of variable. For r > E Y µ r D µ θ + D µ θ µ y µ r ex C θ y µ ] dy + µ y µ r ex C θ y µ ] dy M i µ r; θ i i + x +r i e x dx + r M i µ r; θ Ɣ i where C i θ D i µ θ and M i µ r; θ i are certain functions deending on arameters in brackets. Proof of Proosition 4 Consistency. The consistency of the MLE θ T can be shown by verifying the conditions of Theorem.5 in Newey and McFadden 994 which holds under conditions that are rimitive and also quite weak. Condition ii of Theorem.5 comactness of the arameter set is ensured by considering comact Θ. Condition iii requires that the log-likelihood ln f y θ be continuous at each θ Θ with robability one. This holds by insection. We only need to check the identification condition and dominance condition conditions i and iv of Theorem.5. The identification condition imlies that if θ θ then Pr{f y θ f y θ } >. It is sufficient to show that for any given θ Θ; θ θ there exists a set of ositive robability Sθ such that ln f y θ ln f y θ a.e. y Sθ. 58 The roof uses the fact that any AEPD random variable Y has ositive robability on any interval. If µ µ say µ > µ then for y µ µ] the function ln f y θ is strictly increasing but ln f y θ strictly decreases so 58 always holds on µ µ]. Now suose µ µ. We show that 58 is true almost everywhere in µ ] or µ + if or. Suose. Then for y µ + ln f y θ ln σ C θy µ since µ µ and ln f y θ ln σ C θ y µ where C θ Ɣ + / / ασ. Since both functions on µ + are ower functions they intersect at no more than two oints imlying that 58 holds for Sθ µ +. Similarly for µ µ and it is easy to show that 58 holds if α α or σ σ see Newey and McFadden The dominance condition of Theorem.5 Esu θ Θ ln f Y θ ] < can be verified here by the comactness of arameter set Θ and the boundedness of the th absolute moment of a standard AEPD r.v. where is the suremum of and in Θ. Since the arameter set Θ is comact any continuous function of θ is bounded on Θ. By using the c r -inequality see Loève i.e. a + b r c r a r + c r b r with c r or r as < r or r we have ln f Y θ K + K X for all θ Θ for K and K ositive constants and X σ Y µ a standard AEPD r.v. with arameters α. Since E X ] < by 5 the dominance condition is satisfied. Proof of Proosition 7 Asymtotic Normality. The roof of the asymtotic normality result roceeds by verifying the conditions of Theorem 3 as well as its corollary in Huber 967. Following the ln f yθ notation of Huber 967 let ψy θ the score vector θ and set λθ Eψy θ uy θ d su θ D ψy θ ψy θ 59 where D {θ θ θ d} and all exectations are with resect to the true underlying distribution f y; θ with θ α µ σ. The condition N- i.e. for each fixed θ ψy θ is measurable and searable in Assumtion A- of Huber 967 is immediate; conditions N- and N-4: λθ and E ψy θ ] < hold by 5 and the fact that φ ii in are finite. For the MLE θ T we have t ψy t θ ; then Equation 7 of Huber 967 holds. Since consistency has been roved the