T-statistic for Autoregressive process

TL;DR

This paper derives the distribution of the t-statistic for normal autoregressive processes, especially AR(1), providing explicit formulas and asymptotic behavior, with applications in finance and other fields.

Contribution

It introduces a new distribution for the t-statistic under AR(1) assumptions, generalizing classical results and providing explicit formulas and a modified non-central t-distribution.

Findings

Derived the exact distribution of the t-statistic for AR(1) processes.

Provided an explicit formula as a ratio of dependent distributions.

Confirmed asymptotic convergence to a normal distribution.

Abstract

In this paper, we discuss the distribution of the t-statistic under the assumption of normal autoregressive distribution for the underlying discrete time process. This result generalizes the classical result of the traditional t-distribution where the underlying discrete time process follows an uncorrelated normal distribution. However, for AR(1), the underlying process is correlated. All traditional results break down and the resulting t-statistic is a new distribution that converges asymptotically to a normal. We give an explicit formula for this new distribution obtained as the ratio of two dependent distribution (a normal and the distribution of the norm of another independent normal distribution). We also provide a modified statistic that follows a non central t-distribution. Its derivation comes from finding an orthogonal basis for the the initial circulant Toeplitz covariance matrix. Our findings are consistent with the asymptotic distribution for the t-statistic derived for the asympotic case of large number of observations or zero correlation. This exact finding of this distribution has applications in multiple fields and in particular provides a way to derive the exact distribution of the Sharpe ratio under normal AR(1) assumptions.