RECURSIVE PREDICTION AND LIKELIHOOD EVALUATION FOR PERIODIC ARMA MODELS By Robert Lund and I. V. Basawa The University of Georgia First version received April 1998 Abstract. This paper explores recursive prediction and likelihood evaluation techniques for periodic autoregressive moving-average (PARMA) time series models. The innovations algorithm is used to develop a simple recursive scheme for computing one-step-ahead predictors and their mean squared errors. The asymptotic form of this recursion is explored. The prediction results are then used to develop an ef®cient (and exact) PARMA likelihood evaluation algorithm for Gaussian series. We then show how a multivariate autoregressive moving average (ARMA) likelihood can be evaluated by writing the multivariate ARMA model in PARMA form. Explicit calculations for PARMA(1, 1) models and periodic autoregressions are included. Keywords. Prediction; likelihood evaluation; innovations algorithm; mean squared error; multivariate ARMA models. 1. INTRODUCTION Time series with periodic statistical structures naturally arise in climatology (Hannan, 1955; Monin, 1963; Jones and Brelsford, 1967; Lund et al., 1995), hydrology (Lawrance and Kottegoda, 1977; Vecchia, 1985a, 1985b; McLeod, 1993), signal processing (Gardner and Franks, 1975) and economics (Parzen and Pagano, 1979). Series with autocovariance periodicities should not be modeled with the seasonal autoregressive moving-average (SARMA) class discussed in Box et al. (1994) and Brockwell and Davis (1991). This is because SARMA models, contrary to their name, are actually stationary models with large (in absolute value) autocovariances at lags that are multiples of the period (see McLeod (1993) for stream¯ow series comparisons and Lund and Basawa (1997) for a mathematical discussion). A ¯exible class of short-memory models that have autocovariance periodicities is the class of periodic autoregressive moving- average (PARMA) models. Despite their applicability, prediction and likelihood evaluation methods for PARMA models remain relatively unexplored, especially when compared with their stationary autoregressive moving-average (ARMA) counterparts. This paper closes this gap by examining recursive prediction and likelihood evaluation techniques for PARMA models. The rest of the paper proceeds as follows. In Section 2, general properties of PARMA models are discussed. Section 3 ®rst considers one-step-ahead linear 0143-9782/00/01 75±93 JOURNAL OF TIME SERIES ANALYSIS Vol. 21, No. 1 # 2000 Blackwell Publishers Ltd., 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA.