Maximum Likelihood Fitting of ARMA Models to Time Series With Missing Observations

The method of calculating the exact likelihood function of a stationary autoregressive moving average (ARMA) time series based on Akaike's Markovian representation and using Kalman recursive estimation is reviewed. This state space approach involves matrices and vectors with dimensions equal to Max (p, q + 1) where p is the order of the autoregression and q is the order of the moving average, rather than matrices with dimensions equal to the number of observations. A key to the calculation of the exact likelihood function is the proper calculation of the initial state covariance matrix. The inclusion of observational error into the model is discussed as is the extension to missing observations. The use of a nonlinear optimization program gives the maximum likelihood estimates of the parameters and allows for model identification based on Akaike's Information Criterion (AIC). An example is presented fitting models to western United States drought data.