A Note on the Comparison of the Mean Square Error of Inequality Constrained Least Squares and Other Related Estimators
A Note on the Comparison of the Mean Square Error of Inequality Constrained Least Squares and Other Related Estimators
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不等式约束最小二乘均方误差与其他相关估计量比较的注解
DOI：
10.2307/1937963
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发表时间：
1982
期刊：
影响因子：
--
通讯作者：
P. Schmidt
中科院分区：
文献类型：
--
作者：
M. Thomson;P. Schmidt
求出给定XL时Y1对X2的回归系数。考虑一下下面的例子(Zellner，1962)：方程是通用电气和西屋电气公司的投资函数。在每个方程中，因变量v是公司的年度投资；解释变量是每年年初测量的股本(C)和股票价值(F)。与1935-54年期间相对应的观测有20次。泽尔纳指出，他的方法得到的系数不同于普通的最小二乘估计，并将这种变化归因于他的方法的更高精度。他使用渐近方差来测量精度。系数和标准误差如表1所示。该表的第一行和第二行直接取自Zellner。人们还可以假设，这种方法的效果是对估计产生偏差，似乎略微减少了标准误差。这可以通过回归所有四个解释变量的Y2来检验。泽尔纳的论文给出了相关数据，结果列在表格的最后一行。如果假设成立，则在包含四个解释变量的回归中，c1和f1的真实系数应该为零。估计值与此很难相容；在15D.F.的t检验中，F1的系数是显著的(p<；0.05)。将试验方法推广到多个方程的情况是明显的，但自由度会减少，试验可能变得不敏感。那么，对假设的似是而非的先验考虑就变得至关重要了。
ing the regression coefficient of Y1 on X2 given Xl. Consider the following example (Zellner, 1962): The equations were investment functions for the firms, General Electric and Westinghouse. In each equation the dependent variable v was the firm's annual investment; the explanatory variables were capital stock (c) and share value (f), measured at the start of each year. There were twenty observations corresponding to the period 1935-54. Zellner noted that the coefficients obtained by his method differed from the ordinary leastsquares estimates and attributed the change to the greater precision of his method. He measured precision using asymptotic variances. The coefficients and standard errors are shown in table 1. The first and second lines of this table are taken directly from Zellner. One could also hypothesise that the effect of the method was to bias estimates and appear slightly to reduce standard errors. This may be tested by regressing Y2 on all four explanatory variables. Zellner's paper gave the relevant data and the results are given in the last line of the table. If the assumptions are valid, the true coefficients for c1 and f1 in the regression with four explanatory variables ought to be zero. The estimates are hardly compatible with this; the coefficient of f1 is significant (p < .05) on a t-test with 15 d.f. The extension of the test method to the case of several equations is obvious but the degrees of freedom will decrease and the tests may become insensitive. Then a priori considerations about the plausibility of assumptions become most important.