Casting vector time series: algorithms for forecasting, imputation, and signal extraction

Abstract

Recursive algorithms, based upon the nested structure of Toeplitz covariance matrices arising from stationary processes, are presented for the efficient computation of multi-step ahead forecast error covariances for nonstationary vector time series. Further, we discuss time reversal to forecast the past, and a filtering algorithm for imputation of missing values. These quantities are required to quantify multi-step ahead forecast error and signal extraction error. An information filter is presented, which provides imputations for arbitrary patterns of missing values (such as ragged edge patterns occurring in mixed frequency data). The methods are applied to multivariate retail data exhibiting trend dynamics and seasonality.