Algorithm AS 154: An Algorithm for Exact Maximum Likelihood Estimation of Autoregressive-Moving Average Models by Means of Kalman Filtering

Abstract

LANGUAGEFortran 66 DESCRIPTION AND PURPOSE The algorithm presented here enables the exact likelihood function of a stationary autoregressive-moving average (ARM A) process to be calculated by means ofthe Kalman filter; see Harvey andPhillips (1976, 1979). Two subroutines are basic to the algorithm. The first, subroutine STARMA, casts the ARMA model into the "state space" form necessary for Kalman filtering, and computes the covariance matrix associated with the initial value ofthe state vector. The second subroutine, KARMA, carries out the recursions and produces a set of standardized prediction errors, together with the determinant of the covariance matrix of the observations. These two quantities together yield the exact likelihood, and this may be maximized by an iterative procedure based on a numerical optimization algorithm which does not require analytic derivatives.Subroutine KARMA contains a device whereby the likelihood may be approximated to a level of accuracy which is under the control of the user. This enables a considerable amount of computing time to be saved, with very little attendant loss in precision.Finally, another subroutine, KALFO R, may be used to compute predictions offuture values of the series, together with the associated conditional mean square errors. THEORY An autoregressive-moving average process is defined by (1) where the sr's are normally and independently distributed with mean zero and variance (J2, and WI is observable. Such a process will be referred to as an ARMA(p, q) process and the set of parameters (¢1>""¢P, O!, ...,8 q ) will be denoted by (¢,O).An ARMA(p, q) process may be put in "state space" form by defining an r x 1 vector, lXI' which obeys the "transition equation"where r = max(p,q+ 1) and 312 T= APPLIED STATISTICS , and R = (3) Note that, unless p = q + 1, some of the <jJ/s or O/s will be identically equal to zero. The associated "measurement equation" isEquations (2) and (4)constitute a linear dynamic model. Given at-I' an estimate ofthe state vector at time t-l, together with a matrix P t -1 defined bya prediction of IXt, atIt _ I' may be made. This may then be updated once the tth observations, w" becomes available. The prediction and updating are carried out by means of a set of recursive equations known as the "Kalman filter". The parmeter (J2 does not appear in these recursions.In order to start the recursions, an initial estimator of the state 1X 0 is needed, together with the associated matrix Po. The best estimator of 1X 0 for the ARMA model is a o = 0, and the matrix Po is therefore given by (J-2 £[1X0 1X0]. The evaluation of Po constitutes a key feature ofthe present algorithm, and the method employed is discussed at some length in the next section.Application of the recursive formulae yields a set of n standardized residuals, denoted by V t , t = 1,..., n, together with a set of n quantitiea j, t = 1,..., n, proportional to the one-step prediction mean square errors. The log-likelihood function may then be maximized with respect to (<jJ, 0) by mini...

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