Some finite sample results for a system of seemingly unrelated regression equations

Abstract

In this paper, we consider a system of seemingly unrelated regression equations: $y_i=X_i{\beta }_i+{\epsilon }_i,i=1,\dots ,m,m\ge 2,$ with $\mu \left\{X_1\right\}\supseteq \cdots \supseteq \mu \left\{X_m\right\},$ where the symbol $\mu \left\{X_i\right\}$ denotes the linear space spanned by all the column vectors of matrix $X_i,1\le i\le m.$ Some exact finite sample results of Zellner’s two-stage β-estimators are extended under multivariate normal and multivariate t distributions of the disturbances, respectively. It is found that when $(n-p_1-1){\rho }_{(i)j}^2/(m-j+1)>1,$ the relevant information among $\{y_j,\dots ,y_m\}$ can increase the efficiency of the two-stage estimator of β_i, and this efficiency increases with the sample size n, where ${\rho }_{(i)j}$ is the multiple correlation coefficient between y_i and $\{y_j,\dots ,y_m\},j>i.$

Keywords:

• Seemingly unrelated regressions
• Zellner’s two-stage estimator
• multivariate t distribution
• finite sample property

AMS Subject classification:

• 62H12
• 62F11
• 62J02

Additional information

Funding

This research was supported by the National Natural Science Foundation of China (No. 91746102, 11171117) and Ministry of Education in China Project of Humanities and Social Sciences (No.17YJA910002).