Error Bounds for Asymptotic Expansion of the Conditional Variance of the Scale Mixtures of the Multivariate Normal Distribution

Fotopoulos, S.; He, Lijian

doi:10.1023/a:1004039431669

Cited by 7 publications

(1 citation statement)

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“…Since the multi‐variate t distribution is a scale mixture of normal distributions as in (11) for a particular choice of A , it is desirable to know if the exact likelihood equations and analogue of (8) and (18) can be derived for other scale mixtures of normals. In this direction, all we know at this time (Cambanis et al ., 2000) is that However, the function g n ( X )= E ( A | X ) is much harder to analyse (Fotopoulos and He, 1999) than the function g n , ν ( X ) appearing as the multiplier in (8). More research is needed to understand the dependence of g n ( X ) on n and the possibility of finding GARCH‐type models for the prediction error variance of other scale mixtures of normal random vectors.…”

Section: Discussionmentioning

confidence: 99%

Multi‐variatetAutoregressions: Innovations, Prediction Variances and Exact Likelihood Equations

Tarami

Pourahmadi

2003

Journal Time Series Analysis

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The multi-variate t distribution provides a viable framework for modelling volatile time-series data; it includes the multi-variate Cauchy and normal distributions as special cases. For multi-variate t autoregressive models, we study the nature of the innovation distribution and the prediction error variance; the latter is nonconstant and satisfies a kind of generalized autoregressive conditionally heteroscedastic model. We derive the exact likelihood equations for the model parameters, they are related to the Yule-Walker equations and involve simple functions of the data, the model parameters and the autocovariances up to the order of the model. The maximum likelihood estimators are obtained by alternately solving two linear systems and illustrated using the lynx data. The simplicity of these equations contributes greatly to our theoretical understanding of the likelihood function and the ensuing estimators. Their range of applications are not limited to the parameters of autoregressive models; in fact, they are applicable to the parameters of ARMA models and covariance matrices of stochastic processes whose finite-dimensional distributions are multi-variate t.

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