Estimation of high-dimensional seemingly unrelated regression models

In this article, we investigate seemingly unrelated regression (SUR) models that allow the number of equations (N) to be large and comparable to the number of the observations in each equation (T). It is well known that conventional SUR estimators, for example, the feasible generalized least squares estimator from Zellner (1962) does not perform well in a high-dimensional setting. We propose a new feasible GLS estimator called the feasible graphical lasso (FGLasso) estimator. For a feasible implementation of the GLS estimator, we use the graphical lasso estimation of the precision matrix (the inverse of the covariance matrix of the equation system errors) assuming that the underlying unknown precision matrix is sparse. We show that under certain conditions, FGLasso converges uniformly to GLS even when T < N, and it shares the same asymptotic distribution with the efficient GLS estimator when T > N log N . We confirm these results through finite sample Monte-Carlo simulations.