Two-dimensional Weighted Maximum Likelihood Estimation applied in cross-sectionally restricted VAR model for improving forecasts under model misspecifications

In this paper, we investigate the Weighted Maximum Likelihood Estimation (WMLE), which weights each observation differently depending on the population and the time it is observed. The WMLE can effectively handle situations where the model parameters are not static over time. Additionally, utilizing cross-sectional weights, the WMLE can find the optimal balance for forecasting between variance reduction and biasedness triggered by cross-sectional misspecification. This paper gives a cross-validation algorithm that determines these weights to produce optimal forecasts along with their asymptotic properties. Additionally, the paper provides insights into finite sample properties of the weighting parameters through simulation studies. In application to the quarterly GDP growth rate data, it is shown that the WMLE can outperform conventional models in out-of-sample predictions for certain forecasting horizons.