Restricted Estimator in Two Seemingly Unrelated Regression Model

This article is concerned with the estimation problem of multicollinearity in two seemingly unrelated regression (SUR) equations with linear restrictions. We propose a restricted feasible SUR estimates of the regression coefficients of this model and compare with feasible generalized least squares (FGLS) estimator and the estimator proposed by Revankar (1974) in the matrix mean square error sense. The ideas in the article are evaluated using Monte Carlo simulation.


Introduction
The seemingly unrelated regression model, introduced by Zellner (1962) improves the estimation efficiency by combining several equations into a single equation. The SUR model has simulated a countless theoretical and empirical results in many fields, such as econometrics, industry, biological sciences and etc. Zellner (1963), Revankar (1974), Kariya (1981), Srivastava and Giles (1987), Liu (2002), Wang and Veraverbeke (2008), Ma and Ye (2010) and Wang et all (2011) discussed an efficient estimation procedure for a system of two SUR equations.  by Revankar (1974). This estimator is also the special case of the two-stage covariance improved estimator proposed by Wang (1989) when there are only two linear equations in the system of seemingly unrelated regression (Ma and Ye, 2010).

Consider a system of two SUR equations
In practical regression analysis, researchers often encounter the problem of multicollinearity. In case of multicollinearity we know that when the correlation matrix has one or more small eigenvalues, the estimates of the regression coefficients can be large in absolute value. The least squares estimator performs poorly in the presence of multicollinearity. One of the methods to overcome the multicollinearity problem is to consider parameter estimation in addition to the sample information such as under some exact or stochastic restrictions on the unknown parameters (Rao et al., 2008). Alkhamisi (2010) proposed two SUR type estimators based on combining the SUR ridge regression and the restricted least squares methods.
The object of the present paper is to consider the problem of multicollinearity and its statistical consequences for two seemingly unrelated regression (SUR) model when additional linear restrictions are assumed to hold. In Section 2, the restricted feasible GLS estimator is introduced under condition (1.4), and covariance matrix of an estimator is obtained in this section. In Section 3, we give a Monte Carlo experiment to compare the estimators. The conclusions of the paper are presented in Section 4.

Proposing Estimator
The availability of prior information in the form of exact linear restrictions is utilized in the estimation of the parameters of a linear regression model. If we have prior information for each equation, we predict to be useful in SUR model estimation problem. Let us assume that the prior information is such that it can be written in the form of linear equalities where  is an s-dimensional vector of Lagrange multipliers. Differentiating function   , L  with respect to β and λ, respectively, gives the normal equations (

P C KC C KC C LC C NC C MC C KC C LC C MC C KC
. Clearly under the 0 We investigate the efficiency of the restricted feasible SUR estimator of 1  , as compared to the unrestricted estimator in Revankar (1974). The expected value of (2.8) is equal to This means that   1FR  is biased estimator of 1  . Thus, the covariance matrix for (2.8) is given by Following Zellner (1963), Revankar (1974)  I D I P X X X X X X P X X X I D Tn x . To study the effect of small and large sample on the properties of estimators of β we considered samples of sizes 15 and 100 for small and large samples respectively. The restriction matrices are given in Table 1. equations. The increase in the correlation among the explanatory variables leads to an increase in the scalar MSEs for all the estimators. We observed that as the correlation among equations increases, the scalar MSEs of the estimators increases.

Conclusion remarks
In this article, we proposed a restricted feasible SUR estimator for the vector of parameters in two seemingly unrelated regression models when additional linear restrictions on the parameter vector are assumed to hold. The restricted feasible estimator of the  parameter vector is then compared with the FGLS estimator and the estimator proposed by Revankar (1974) in terms of MSE criterion. The investigation has been done by means of Monte Carlo simulations. The results have shown that our proposed estimator produce smaller MSEs.