Time-series analysis if data are randomly missing

Maximum-likelihood (ML) theory presents an elegant asymptotic solution for the estimation of the parameters of time-series models. Unfortunately, the performance of ML algorithms in finite samples is often disappointing, especially in missing-data problems. The likelihood function is symmetric with respect to the unit circle for the estimated zeros of time-series models. 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 nonlinear optimization algorithms that often converge to poor models, with estimated zeros precisely on the unit circle. With ML estimation, it is much easier to estimate a long autoregressive (AR) model with only poles. The parameters of that long AR model can then be used to estimate MA and autoregressive moving-average (ARMA) models for different model orders. 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, the possible convergence to a useful model will depend on the missing fraction and on the specific properties of the data at hand.