Estimation of time series spectra with randomly missing data

Maximum likelihood theory presents an elegant asymptomatic solution for the estimation of the parameters of time series models. Unfortunately, the performance of algorithms is often disappointing in finite samples with missing data. The likelihood function for the estimated zeros of time series models is symmetric with respect to the unit circle. As a consequence, the unit circle is either a local maximum or a local minimum in the likelihood of moving average (MA) models. This is a trap for non-linear optimization algorithms that often converge to poor models. With maximum likelihood estimation, it is easier to estimate a long autoregressive (AR) model with only poles. The parameters of that long AR model can be used to estimate MA and ARMA models for different model orders, with a reduced statistics algorithm known for uninterrupted equidistant data. The accuracy of the estimated AR, MA and ARMA spectra is very good. The robustness is excellent as long as the AR order is less than 10 or 15. For still higher AR orders until about 60, convergence depends on the missing fraction and on the specific properties of the data.