Mixture Autoregressive Models Based On Skew T Distributions

Mixture time series models have received considerable attention because of their ability to deal with multi-modality.On one hand, mixture autoregressive models proposed by Wong and Li (2000) and Wong et al. (2009) are based on symmetric distributions, which can handle with multi-modality and extreme observations, but they may not be ideal in the presence of highly asymmetric observations. In general, the skewness parameters could be unduly affected by extreme observations in the frame of t mixture autoregressive models. On the other hand, skew normal distributions and skew t distributions are successful in dealing with asymmetry in regression models, see Lin et al. (2007a, b).Based on the above facts, we propose a skew t mixture autoregressive model. Statis-tical mixture modeling based on mormal (Wong and Li,2000), t (Wong et al.,2009) and skew normal distributions can be viewed as special cases of the skew t model. The new model can deal with multi-modality, asymmetry, extreme observations and other char-acteristics endowed by time series data. The first-and second-order stationary, strictly stationary and geometrically ergodic conditions are given, and the autocovariance func-tions are also given. Statistical estimating procedures based on the EM algorithm are considered, and three modifications of the EM algorithm are discussed, and a new method to compute the standard errors of the estimators is also given.In the following we introduce the main results of this thesis.Let the distribution function of X-ST(ζ,σ2,λ,v) be FST(x|ζ,σ2,λ,v). The pro-posed model is defined by whereα1+…+αK=1,αK＞0, k= 1,…, K, F(Xt|Ft-1) is the conditional cumulative distribution function of Xt given the past information, Ft-1 is theσ-field generated by {Xt-i,Xt-2,…}.Without loss of generality, assume p1=…=pK=p. Theorem 1 A necessary and sufficient condition for model (1) to be first-order stationary is that all roots of the equation lie inside the unit circle.Theorem 2 Suppose that model (1) satisfies the first-order stationary condition. A necessary and sufficient condition for model (1) to be second-order stationary is that all roots of the equation 1-C1z-1-…-Cpz-p=0 lie inside the unit circle, where for u,l =1,…,p-1, where A and A-1 are (p - 1)×(p - 1) matrices such that A=(θij)i,j=1 p-1,A-1=(bij)i,j=1 p-1Let{ηt}be a sequence of independently and identically distributed random variables, such that P(ηt = k) =αk, andεt,k be jointly independent random variables, whose distribution functions are FST(x|0, 1,λk, vk). Assume that the choice of the component at time t (i.e.ηt) does not depend on Ft-1 and {εt,k, t≥1, 1≤k≤K}. LetThen model (1) has the following stochastic difference equation representation where Bt = Ct + Bt*. Define the top Lyapunov exponent as where‖·‖denotes any arbitrary matrix norm in Rp.Theorem 3 (1) (sufficiency) A sufficient condition for model (2) to have a strictly stationary solution is that 7< 0. Moreover, the solution is unique and is given by where the above series converges almost surely.(2) (necessity) Assume that model (1) is irreducible in the sense of Bougerol and Picard (1992, definition 2.3), then the conditionγ＜0 is also necessary.Theorem 4 If 7< 0, then the unique and stationary solution given by (3) of model (2) is geometrically ergodic and hence isβ-mixing with geometric rate.Define Uk= E(At(?)At\ηt= k), The spectral radius of U is denoted by p(U).Theorem 5 A sufficient condition for model (2) to have a unique square-integrable stationary solution given by (3) is that p(U)< 1, where the series in (3) converges in the squared mean. Moreover, the solution is strictly stationary and ergodic.The autocorrelation coefficients of model (1) satisfies where pj is the lag j autocorrelation.We estimate the parameters by using the EM algorithm. Letα= (α1,…,αK-1)T,σ2= (σ12,…,σK2)T,λ=(λ1,…,λK)T,v=(v1,…,vK)T,βk=(βk0,βk1,…,βkpk)T,θ=(αT,β1T,…,βKT,σ2T,λT,vT)T. Let Z= (Z1,…,Zn), where Zt= (Z1t,…,ZKt)T,Zkt equal to 1 if Xt comes the kth component and equal to 0 otherwise.Model (1) can be hierarchically represented by The complete data log-likelihood function, ignoring additive constant terms, is given by whereηkt= (Xt-μkt*)σk*We have considered three modifications of the EM algorithm:ECM, ECME and PX-EM algorithms.Let whereThenΩ-1 is the asymptotic covariance matrix for the estimators, from which we can ob-tain the standard errors. In practice, we can choose L=10.