Estimation for vector autoregressive model under multivariate skew- <i>t</i> -normal innovations

Abstract

Current procedures for estimating the parameters of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>p</mml:mi> </mml:math> th order vector autoregressive (VAR <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mo>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>)</mml:mo> </mml:math> ) model are usually based on assuming that the ensuing error distribution is multivariate normal. But there exists large body of evidence that several data encountered in real life are skewed; thereby making estimators derived based on normality assumption not suitable in such scenarios. This prompts for the search of appropriate methods for skewed distributions. Therefore, this article proposes estimators for the mean and covariance matrices of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="normal">V</mml:mi> <mml:mi mathvariant="normal">A</mml:mi> <mml:mi mathvariant="normal">R</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>)</mml:mo> </mml:math> model under multivariate skew- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>t</mml:mi> </mml:math> -normal (MSTN) distribution. Also, estimators for the shape and skewness parameters are provided. The expectation conditional maximization (ECM) and its extension the expectation conditional maximization either (ECME) algorithms are the tools used to derive the estimators. The performance of the estimators were examined through extensive simulations, and results show that they compete favourably with other numerical methods especially when the underlying distribution is skewed. The usefulness of our estimators was illustrated using a real data set on some US economic indicators. The VAR <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mo>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>)</mml:mo> </mml:math> model under MSTN distribution provides a good fit, better than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="normal">V</mml:mi> <mml:mi mathvariant="normal">A</mml:mi> <mml:mi mathvariant="normal">R</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo>(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>)</mml:mo> </mml:math> model under the assumption of normality.