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Testing for autocorrelation and Akaike's criterion

Published online by Cambridge University Press:  14 July 2016

E. J. Hannan
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Abstract

The problem initially considered is that of testing whether ρ = 0 in a model y(n) = x(n)+ η (n), x(n) = ρx(n – 1) + ξ (n) where only y(n) is observed and η (n), ξ (n) are white noise. This is equivalent to distinguishing between an ARMA (1, 1) model and white noise. The asymptotic distribution of the likelihood ratio criterion is derived. This is shown to be of an unusual form. This result is then used to discuss the asymptotic properties of Akaike's procedure for estimating (p, q) in an ARMA (p, q) model. If p0, q0 are the true values and p0 < P, q0< Q, when P, Q are the maximum values considered, then it is shown that, in a certain asymptotic sense, the procedure is sure to overestimate p0, q0. However, the asymptotic situation may be very far from that relevant in a practical case. The relevance of overestimation is briefly discussed.

Keywords

AUTOCORRELATION TESTAKAIKE'S CRITERIONARMA MODELSWHITE NOISEASYMPTOTIC PROPERTIESOVERESTIMATION
Type
Part 7 — Time Series
Information
Journal of Applied Probability , Volume 19 , Issue A , 1982 , pp. 403 - 412
Copyright
Copyright © 1982 Applied Probability Trust 

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