Abstract
In this paper, we propose an efficient weighted average estimator in Seemingly Unrelated Regressions. This average estimator shrinks a generalized least squares (GLS) estimator towards a restricted GLS estimator, where the restrictions represent possible parameter homogeneity specifications. The shrinkage weight is inversely proportional to a weighted quadratic loss function. The approximate bias and second moment matrix of the average estimator using the large-sample approximations are provided. We give the conditions under which the average estimator dominates the GLS estimator on the basis of their mean squared errors. We illustrate our estimator by applying it to a cost system for United States (U.S.) Commercial banks, over the period from 2000 to 2018. Our results indicate that on average most of the banks have been operating under increasing returns to scale. We find that over the recent years, scale economies are a plausible reason for the growth in average size of banks and the tendency toward increasing scale is likely to continue
Highlights
Unrelated regressions (SUR) was introduced by (Zellner 1962) and is one of the econometric developments that has been widely used in applied work
The weight is inversely related to a quadratic loss function which measures the weighted distance between the unrestricted and the restricted generalized least squares (GLS) estimators
The superiority conditions of the average estimator in terms of the weighted mean squared error is given for any user-specific symmetric positive definite weight matrix, and is not limited to the case where the weight is the inverse of the variance-covariance matrix of the unrestricted
Summary
Unrelated regressions (SUR) was introduced by (Zellner 1962) and is one of the econometric developments that has been widely used in applied work. The parameter homogeneity assumption causes higher efficiency but could be at the cost of estimation bias and inconsistency of estimators, which is supported by an increasing number of studies due to a better forecast performance of the estimators under this assumption (see, for example, Maddala 1991; Maddala and Hu 1996; Baltagi and Griffin 1984; Baltagi et al 2000; Hoogstrate et al 2000) This question and the results of the mentioned research show the typical bias-variance trade-off that needs to be considered in choosing the restrictions. We show the dominance properties in terms of mean squared error (MSE) of the estimators, which ensures that the proposed estimator is robust against arbitrary deviations from the restrictions This is an advantage of our method relative to the “local asymptotic” argument that some previous studies rely on (see, for example, Hansen 2016).
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