Maximum likelihood estimation of dynamic factor models using general cross sectional covariance

The existing literature on large dynamic factor models invariably assumes that the cross sectional covariance matrix is diagonal. This is due to the curse of dimensionality which means that many parameters need to be estimated for large data sets. This paper introduces a novel maximum likelihood approach which relaxes this diagonal assumption. All of the parameters are concentrated out so the parameters are jointly estimated with the factors. Importantly, the cross sectional covariance matrix is concentrated out so does not need to be explicitly estimated. The approach uses a neat simplification of the log-likelihood which makes estimation for large dimensional data feasible. Implementation of the general covariance approach is by numerical optimisation of the concentrated log-likelihood with respect to the factors. A diagonal version of the general covariance approach is also introduced, mainly for comparative reasons. Out of sample tests using Monte Carlo simulations shows the new general approach performs well, with smaller prediction errors overall compared to a range of existing diagonal approaches. Understandably, the general approach does particularly well for high cross sectional covariance. This is most apparent for low numbers of factors. This paper opens up the literature to new ways of estimating dynamic factor models and improvements in inference and forecasting for big data.