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On the Multivariate Extended Skew-Normal, Normal-Exponential and Normal- Gamma Distributions

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On the Multivariate Extended Skew-Normal, Normal-Exponential and Normal- Gamma Distributions On the Multivariate Extended Skew-Normal, Normal-exponential and Normal- gamma Distributions by CJ Adcock(1) and K Shutes(2) (1) The University of Sheffield (2) Coventry University Business School Abstract This paper presents expressions for the multivariate normal-exponential and normal- gamma distributions. It then presents properties of these distributions. These include conditional distributions and a new extension to Stein’s lemma. It is also shown that the multivariate normal-gamma and normal-exponential distribution are not in general closed under conditioning, although they are closed under linear transformations. The paper also demonstrates that there are relationships between the extended skew-normal distribution and the normal-gamma and normal-exponential distributions. Specifically, it is shown that certain limiting cases of the extended skew-normal distribution are normal-gamma and normal-exponential. One interpretation of these results is that the normal-exponential distribution may considered to be an alternative model to the extended skew-normal in some situations. An alternative point of view, however, is that the normal-exponential distribution is redundant since it may be replicated by a suitable extended skew-normal distribution. The theoretical results are supported by an empirical study of stock returns, which includes use of the multivariate distributions for portfolio selection. Keywords: exponential distribution, gamma distribution, moment generating function, multivariate extended skew-normal distribution, portfolio selection, Stein’s lemma. Correspondence Address: C J Adcock The Management School The University of Sheffield Mappin Street Sheffield, S1 4DT UK Email: [email protected] Tel: +44 (0)114 222 3402 1. Introduction The skew-normal distribution which was introduced by Adelchi Azzalini, Azzalini (1985, 1986), is now almost certainly the best-known example of a general method of generating a skewed distribution. The method has various stochastic representations, there being a summary in Azzalini (2005). One method of generating the skew- normal distribution is to take the bivariate normal distribution of (X,Y) and then to consider the conditional distribution of Xgiven that Y>0 . This representation is, with a change of notation, equivalent to considering the distribution of U +λV , where U has an arbitrary normal distribution and V , which is independent U , has the normal distribution N(0,1 )truncated from below at zero. A generalization in which V has the normal distribution N(t,1 ) truncated from below at zero is generally known as the extended skew-normal. The methods introduced by Azzalini and developed by him and many other authors have led to a wide range of distributions which possess rich theoretical properties. A different method of generating skewed distributions is to consider U +λV , where U has a symmetric distribution, but the non-negative variable V has a specified skewed distribution ab initio rather than being truncated. An example of this is the normal-exponential distribution, which was introduced by Aigner, Lovell and Schmidt (1977). As the name suggests, this specifies that U is normal and that V has an exponential distribution. An extension of this is the normal-gamma distribution which was introduced by Greene (1990) and which includes the normal-exponential as a special case. Although the theoretical foundations of skew-normal and related distributions are now both deep and mature, some authors express the view that applications are less well developed. The book by Genton (2004) rebuts this to some extent. Another exception to this is financial economics in which there has been a number of papers in recent years. This is particularly the case in portfolio theory where the multivariate skew- normal distribution has been used both for empirical studies and for the development of new theoretical results. Examples of the former may be found in Harvey et al (2010) and Adcock (2005). Examples of the latter are in Corns and Satchell (2007) and Adcock (2007), who presents an extension of Stein’s lemma (Stein, 1981) for the skew-normal distribution. A related but separate stream of work may be found in Simaan (1993). He proposes that the n-vector of returns on financial assets should be represented as X=U+λV . The n-vector U has a multivariate elliptically symmetric distribution and is independent of the non-negative scalar variable V , which has an unspecified skewed distribution. The n-vector λ , whose elements may take any real value, induces skewness in the return of individual assets. Multivariate versions of the normal-exponential and normal-gamma distributions are specific cases of the model proposed by Simaan, even though they do not appear explicitly in the literature. The aim of this paper is as follows. First, it is to present expressions for the multivariate normal-exponential and multivariate normal-gamma distributions. Secondly, it is present some properties of these distributions, specifically those which are of use in portfolio theory. These include conditional distributions, which are of relevance for asset pricing, and a new extension to Stein’s lemma which is relevant for portfolio selection. It is also shown that the multivariate normal-gamma and hence 1 multivariate normal-exponential distribution are not in general closed under conditioning, although they are closed under linear transformations. The paper also demonstrates that there are relationships between the extended skew- normal distribution and the normal-gamma and normal-exponential distributions. Specifically, it is shown that certain limiting cases of the extended skew-normal distribution are normal-gamma and normal-exponential. One interpretation of these results is that the normal-exponential distribution may considered to be an alternative model to the extended skew-normal in some situations. An alternative point of view, however, is that the normal-exponential distribution is redundant since it may be replicated by a suitable extended skew-normal distribution. The paper has five sections. Following a short summary of the multivariate extended skew-normal, henceforth MESN, distribution, the multivariate normal-exponential and normal-gamma distributions, henceforth MNE and MNG respectively, are derived in Section 2. Section 3 contains the main theoretical results of the paper. These describe conditions under which the normal-exponential distribution arises as a limiting case of the extended skew-normal distribution and some further properties of interest. Section 4 present background material for an empirical study of portfolio selection. This section contains the extension to Stein’s lemma for the MNG distribution. Section 5 reports an empirical study of the MESN and MNE distributions. Section 6 concludes. A short appendix contains the proofs of Lemmas reported in Sections 3 and 4. 2. Multivariate extended skew-normal normal-exponential and normal-gamma distributions The multivariate skew-normal distribution was introduced by Azzalini and Dalla Valle (1996). The multivariate extended skew-normal, MESN henceforth, distribution, which was first described in Adcock and Shutes (2001), may be obtained by considering the distribution of an n-vector X=U+λV . The vector U has a full rank multivariate normal distribution with mean vector μ and covariance matrix Σ. The elements of the vector λ may take any real values. The scalar variable V , which is distributed independently of U , has a normal distribution with mean τ and variance 1 truncated from below at zero. The distribution of X is denoted MESN n (μ, Σ, λ, τ) and the probability density function of the distribution is T -1 T ìτ+λΣ (x -μ)ü f()x =φn()x; μ +λτ, Σ +λλ Φí ýΦ()τ, î 1+λTΣ -1 λ þ where Φ(x) is the standard normal distribution function evaluated at x. The notation φn(x;μ,Σ) denotes the probability density function, evaluated at x, of a multivariate normal distribution with mean vector μ and covariance matrix Σ. The moment generating function (MGF) of the distribution, which is required below, is T T T T M X (t) = exp{t (μ + λτ)+ t (Σ + λλ )t 2}Φ(λ t + τ)Φ(τ). The vector of expected values and covariance matrix of X are, respectively 2 T E(X)= μ + λ{τ + ξ1(t )}= δ, var(X)= Σ + λλ {1 + ξ2 ( τ)}= Θ, k k where x k (x) = ¶ lnF(x)¶x , k = 1,2,.. The general notations δ and Θ are used throughout the paper to denote the vector of expected returns and covariance matrix, respectively. Omitting the subscript i, standardised values of skewness and kurtosis of a typical element of X, which are required in Section 5, are 3 2 3 2 4 2 2 sk' = ξ3 (τ)ψ [1 + {}1 + ξ 2 ()τ ψ ], ku' = ξ4 (τ)ψ[1+ {}1+ ξ2 ()τψ],ψ= λσ. A random vector X is said to have a multivariate normal-exponential, henceforth MNE, distribution if it is defined as above except that the variable V has an exponential distribution with scale equal, without loss of generality, to unity. The notation MNEn(μ,Σ,λ) is used. The moment generating function (MGF) of the distribution, which is also required below, is T T T T M X ( t ) = exp(μ t + t Σt/2)(1 - λ t),. λ t < 1. The vector of expected values and covariance matrix of X under the MNE distribution are, respectively E(X)= δ = μ + λ, var(X)= Θ = Σ + λλT. Under the multivariate normal-gamma distribution, henceforth MNG, the variable V has a gamma distribution with scale equal to one and υdegrees of freedom. That is, the probability density function of V is f (v)= v υ-1e -v Γ(υ), υ > 0,v³ 0. The MGF of this distribution is T T T υ T M X ( t ) = exp(μ t + t Σt/2)(1 - λ t), λ t < 1. The vector of expected values and covariance matrix of X under the MNE distribution are, respectively E(X)= δ = μ + λυ, var(X)= Θ = Σ + λλTυ.
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