Fast same-step forecast in SUTSE model and its theoretical properties

We consider the problem of forecasting multivariate time series by a Seemingly Unrelated Time Series Equations (SUTSE) model. The SUTSE model usually assumes that error variables are correlated. A crucial issue is that the model estimation requires heavy computational loads due to a large matrix computation, especially for high-dimensional data. To alleviate the computational issue, we propose a two-stage procedure for forecasting. First, we perform the Kalman filter as if error variables are uncorrelated; that is, univariate time series analyses are conducted separately to avoid a large matrix computation. Next, the forecast value is computed by using a distribution of forecast error. The forecast error distribution is characterized by its mean vector and covariance matrix, but it is difficult to obtain them due to model misspecification. In this study, we prove the convergence of the mean vector and the covariance matrix as the number of observations becomes infinite. The convergence results lead to deriving a stationary distribution of the forecast error.