Confidence ellipsoids for the primary regression coefficients in m- equation seemingly unrelated regression models

Abstract

For \u00a0a \u00a0m-equation \u00a0seemingly \u00a0unrelated \u00a0regression(SUR) \u00a0model, \u00a0this \u00a0paper \u00a0derives \u00a0two \u00a0basic confidence \u00a0ellipsoids(CEs) \u00a0respectively \u00a0based \u00a0on \u00a0the \u00a0two-stage \u00a0estimation \u00a0and \u00a0maximum \u00a0likelihood estimation(MLE), and corrects the two CEs using the Bartlett correction method, resulting in four new CEs. In \u00a0the \u00a0meantime \u00a0via \u00a0using \u00a0the \u00a0partition \u00a0matrix, \u00a0we \u00a0derive \u00a0a \u00a0new \u00a0matrix-derivative-based \u00a0formulation \u00a0of Fisher's information matrix for calculating the maximum likelihood estimator of the m-equation SUR model. By \u00a0Monte \u00a0Carlo \u00a0simulation, \u00a0the \u00a0coverage \u00a0probabilities \u00a0and \u00a0average \u00a0volumetric \u00a0characteristics \u00a0of \u00a0CEs \u00a0are compared under different sample values and different correlation coefficients. Moreover, it is proved that the CE based on the second bartlett correction method performs better even in the case of small samples. Finally, we apply these CEs to the actual data for analysis. The CEs of the SUR model with multiple equations are found to be more accurate than the case with only two equations.