一类相依回归方程限定两步估计的有限样本结果

Some Exact Finite Sample Results of Estimators Based on Restricted Residuals in A Class of Seemingly Unrelated Regressions

  • 摘要: 考虑一类由两个相依回归方程组成的线性回归系统yi=XiBi+ui(i=1,2),其中回归矩阵Xii=1,2)满足P1P2对称,这里Pi=Xi(X'iXi)-1X'i.这种情况包括了以往文献1,2,3,4中考虑的一些情况.记βi∧βi分别表示βi的基于Σ的限定估计S和非限定估计∧S的两步估计.本文推导得到了βi的有限样本方差Vaq(βi),并由此结果得到 \lim _i \rightarrow \infty \operatornameVar\left(\sqrtn \bar\beta_i\right)=\lim _n \rightarrow \infty \operatornameVar\left(\sqrtn \beta_i∧*\right),这里βi*βi的Gauss-Markoff估计.本文讨论了βi相对于LS估计bi和非限定估计∧βi的有效性.本文结果表明,实际中若相关系数ρ的绝对值|ρ|较大,或样本量n较大,估计量βi一般优效于LS估计bi,若|ρ|较小,则限定估计量βi一般优于非限定估计量∧βi(i=1,2)。

     

    Abstract: Consider a system of two seemingly unrelated regression equautionsyi=XiBi+ui(i=1,2), Where Xii=1,2) is the repression matrices for whichP1P2=P2P1Pi=Xi(X'iXi)-1X'i,i=1,2). This system includes some special special models considered in some paperes 1-4.Let βiand ∧βii= l,2) be the β-estimatiors based,respectively, on the restricted estimate S of Σ and on tile unrestricted estimate S of Σ. This paper derives the finite sample variances of the βii = 1, 2), and examines their efficiency with respect to the ∧βiand with respect tile OLS estimator bi obtained directly from the ith equation.It is shown that for large values of the sample size n or the correlation coefficient|ρ|the estimatorβimay be more efficient than bi,and for samll values of|ρ|the restricted estimator βi can work better than the unrestricted estimator ∧βii = 1, 2).

     

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