Estimation of high-dimensional seemingly unrelated regression models

Abstract

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 $T>N\hspace{0.17em}\text{log}\hspace{0.17em}N.$ We confirm these results through finite sample Monte-Carlo simulations.

Keywords:

• Feasible graphical lasso estimator
• Graphical Lasso
• high-dimensional matrix estimation
• precision matrix
• seemingly unrelated regression.

JEL Classification:

• C30;C33

Notes

1 The core firms are linked to each other and to the periphery firms, while the periphery firms are not linked to each other directly but through the core firms.

2 Note $s_{\text{min}}(\Omega )=\frac{1}{s_{\text{max}}(\Sigma )}\ge \frac{1}{\Vert |\Sigma \Vert {|}_{\infty }}.$

3 More precisely, in each replication, we divide the T samples into five folds and use four of them as the training data set and one as the validation set. With each choice of λ, we estimate the ${\widehat{\beta }}_{\text{FGLasso}}$ estimators using the training data, then plug them into the validation set and calculate the mean squared error. We choose λ_n that minimizes the averaged MSE.

4 For four-nearest neighbor lattices design, N can only be square number, so we choose .$N=\{49,100,196,289,400\}.$

5 Here, the $l_{\infty }$ norm is the element-wise maximum norm $\Vert ·{\Vert }_{\infty },$ and the RMSE is defined as $\Vert ·{\Vert }_F/\sqrt{N}.$

6 In practice, the time for deriving estimator also depends on how the penalty parameter is chosen. For example, in five-fold cross validation, it requires 26 times of calculation before deriving the final estimator.

7 For simplicity, we omit the conditional sign in $\mathbb{P}(•|X)$ throughout the proof of Lemma 3.1.