How the instability of ranks under long memory affects large-sample inference

TL;DR

This paper examines how the instability of Hermite ranks under long memory affects large-sample inference, emphasizing the stability of rank one and its implications for statistical analysis.

Contribution

It demonstrates that Hermite rank one is stable under long memory, while higher ranks are unstable, impacting inference accuracy in statistical models.

Findings

Rank one is stable under long memory.

Higher Hermite ranks are unstable and lead to underestimation.

Implications for variance, empirical processes, and estimators.

Abstract

Under long memory, the limit theorems for normalized sums of random variables typically involve a positive integer called "Hermite rank". There is a different limit for each Hermite rank. From a statistical point of view, however, we argue that a rank other than one is unstable, whereas, a rank equal to one is stable. We provide empirical evidence supporting this argument. This has important consequences. Assuming a higher-order rank when it is not really there usually results in underestimating the order of the fluctuations of the statistic of interest. We illustrate this through various examples involving the sample variance, the empirical processes and the Whittle estimator.