Regression Procedures for ARMA Estimation

Abstract

Abstract Regression procedures for parameter estimation in autoregression moving average (ARMA) models are discussed, mainly for providing initial estimates for iterative maximization of a Gaussian likelihood. An iterative procedure of Spliid (1983) is compared to a procedure of Hannan and Rissanen (1982), and a global convergence result is established for an iterative modification of Spliid's procedure. Spliid's iteration does not always converge; when it does, it has the same asymptotic distribution as the second stage of the Hannan-Rissanen procedure. This second-stage iteration gives the same problems as Spliid's procedure, so it is preferable to go immediately to the third stage, which is asymptotically efficient in the Gaussian case. An example is provided by a first-order ARMA model, y(t) + αy(t − 1) = ε(t) + βε(t −1), under the usual regularity conditions of stationarity and invertibility. Spliid's procedure fits an autoregression of predetermined order (i.e., 2) to obtain estimates of the ε(t), and then regresses y(t) on y(t − 1) and to obtain estimates and . These estimates are used to filter the data to obtain a further estimate ; the procedure is iterated until the sequence of estimates of α and β converges. Convergence will not take place unless αβ < ½. If the order of the autoregression in Spliid's procedure is chosen by the information criterion and only the first regression on y(t − 1) and is carried out, the first two stages of the Hannan-Rissanen procedure result. Then and converge strongly as T → ∞ to α and β, and they also obey a central limit theorem. The modified Spliid procedure replaces by the residuals from the first regression, and so on. In a restricted sense, this iteration always converges in the ARMA case, and will generally do so if the positive real condition for the moving average transfer function is satisfied [see (1.5) in Sec. 1]. Simulations bear out the theory. In practice, there will be no true ARMA system; in any case, the ARMA orders will need to be determined from the data. Thus the results are of suggestive value only. Nevertheless, these suggestions favor the Hannan-Rissanen procedure.