Heavy Tails and Predictive Ability Testing

TL;DR

This paper investigates how heavy tails in forecast errors affect the asymptotic behavior of predictive accuracy tests, revealing that standard methods can be severely misleading under infinite variance conditions.

Contribution

It introduces a new stable limit theorem for heavy-tailed, strongly mixing time series and proposes a sub-sampling inference method that is robust to tail heaviness.

Findings

Standard tests can over-reject under heavy tails.

A new stable limit theorem for infinite-variance processes.

Sub-sampling provides valid inference regardless of tail heaviness.

Abstract

We study the asymptotic behaviour of widely used tests for evaluating and comparing predictive accuracy when forecast errors exhibit heavy tails. In particular, when loss differentials have infinite variance, the Diebold-Mariano test statistic converges to a nonstandard limit involving non-Gaussian stable random variables. As a consequence, conventional critical values can yield severely distorted inference: a nominal 5$\%$ test may reject a true null as often as 70$\%$ of the time. To establish these results, we develop a new stable limit theorem for strongly mixing, infinite-variance time series processes. Building on this theory, we consider sub-sampling-based inference that remains valid irrespective of tail-heaviness and requires no estimation of long-run variances or tail indices. An application to risk forecasts for emerging-market exchange rates shows that accounting for heavy tails can substantially alter conclusions about predictive performance relative to standard procedures.