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

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Abstract
Abstract. The multi‐variatetdistribution provides a viable framework for modelling volatile time‐series data; it includes the multi‐variate Cauchy and normal distributions as special cases. For multi‐variatetautoregressive 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‐variatet.