
T-statistic for Autoregressive process
Benhamou, Eric (2018), T-statistic for Autoregressive process. https://basepub.dauphine.fr/handle/123456789/18922
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Type
Document de travail / Working paperDate
2018Publisher
Preprint Lamsade
Series title
Preprint LamsadePublished in
Paris
Pages
24
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Benhamou, EricLaboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Abstract (EN)
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 independentnormal distribution). We also provide a modified statistic that is follows a non centralt-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.Subjects / Keywords
t-Student; Auto regressive process; Toeplitz matrix; circulant matrix; non centered Student distributionRelated items
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