T-statistic for Autoregressive process

摘要

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 is 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.