Predicting in multivariate incomplete time series. Application of the expectation-maximisation algorithm supplemented by the Newton-Raphson method

Statistical practice requires various imperfections resulting from the nature of data to be addressed. Data containing different types of measurement errors and irregularities, such as missing observations, have to be modelled. The study presented in the paper concerns the application of the expectation-maximisation (EM) algorithm to calculate maximum likelihood estimates, using an autoregressive model as an example. The model allows describing a process observed only through measurements with certain level of precision and through more than one data series. The studied series are affected by a measurement error and interrupted in some time periods, which causes the information for parameters estimation and later for prediction to be less precise. The presented technique aims to compensate for missing data in time series. The missing data appear in the form of breaks in the source of the signal. The adjustment has been performed by the EM algorithm to a hybrid version, supplemented by the Newton-Raphson method. This technique allows the estimation of more complex models. The formulation of the substantive model of an autoregressive process affected by noise is outlined, as well as the adjustment introduced to overcome the issue of missing data. The extended version of the algorithm has been verified using sampled data from a model serving as an example for the examined process. The verification demonstrated that the joint EM and Newton-Raphson algorithms converged with a relatively small number of iterations and resulted in the restoration of the information lost due to missing data, providing more accurate predictions than the original algorithm. The study also features an example of the application of the supplemented algorithm to some empirical data (in the calculation of a forecasted demand for newspapers).