Testing error distribution by kernelized Stein discrepancy in multivariate time series models

TL;DR

This paper introduces a new kernelized Stein discrepancy-based test for error distribution in multivariate time series, capable of detecting deviations from assumed distributions including heavy-tailed and skewed types, with practical bootstrap implementation.

Contribution

It develops a versatile, consistent testing method for general error distributions in multivariate time series, extending beyond the normal distribution assumption.

Findings

The test accurately detects distribution mis-specification in simulations.

Bootstrap method effectively computes critical values.

Application to real data demonstrates practical utility.

Abstract

Knowing the error distribution is important in many multivariate time series applications. To alleviate the risk of error distribution mis-specification, testing methodologies are needed to detect whether the chosen error distribution is correct. However, the majority of the existing tests only deal with the multivariate normal distribution for some special multivariate time series models, and they thus can not be used to testing for the often observed heavy-tailed and skewed error distributions in applications. In this paper, we construct a new consistent test for general multivariate time series models, based on the kernelized Stein discrepancy. To account for the estimation uncertainty and unobserved initial values, a bootstrap method is provided to calculate the critical values. Our new test is easy-to-implement for a large scope of multivariate error distributions, and its importance is illustrated by simulated and real data.