Large System of Seemingly Unrelated Regressions: A Penalized Quasi-Maximum Likelihood Estimation Perspective

Abstract

In this article, using a shrinkage estimator, we propose a penalized quasi-maximum likelihood estimator (PQMLE) to estimate a large system of equations in seemingly unrelated regression models, where the number of equations is large relative to the sample size. We develop the asymptotic properties of the PQMLE for both the error covariance matrix and model coefficients. In particular, we derive the asymptotic distribution of the coefficient estimator and the convergence rate of the estimated covariance matrix in terms of the Frobenius norm. The model selection consistency of the covariance matrix estimator is also established. Simulation results show that when the number of equations is large relative to the sample size and the error covariance matrix is sparse, the PQMLE outperforms other contemporary estimators.