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 p 0, q 0 are the true values and p 0 < P, q 0< 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 p 0, q 0 . However, the asymptotic situation may be very far from that relevant in a practical case. The relevance of overestimation is briefly discussed.