DISTRIBUTIONS OF ESTIMATES OF COEFFICIENTS OF A SINGLE EQUATION IN A SIMULTANEOUS SYSTEM AND THEIR
DISTRIBUTIONS OF ESTIMATES OF COEFFICIENTS OF A SINGLE EQUATION IN A SIMULTANEOUS SYSTEM AND THEIR
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联立系统中单个方程的系数估计分布及其
DOI：
10.2307/1914090
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发表时间：
1973
期刊：
影响因子：
--
通讯作者：
T. Sawa
中科院分区：
文献类型：
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作者：
Asymptotic Expansions;T. W. Anderson;T. Sawa
有限信息最大似然和两阶段最小二乘估计具有相同的渐近正态分布；普通最小二乘估计具有另一个渐近正态分布。本文考虑了对所谓的“k 类”估计分布的更准确的近似。这种估计分布的渐近展开以 Edgeworth 或 Gram-Charlier 级数（其中首项是正态分布）的形式给出。该发展还允许以多种形式表达精确分布。将两阶段最小二乘和普通最小二乘估计的分布转换为双非中心 F 分布。对第二作者计算的近似分布和精确分布进行了数值比较。已经提出了几种方法来估计联立结构方程的完整系统中单个方程的系数，包括有限信息最大似然法（Anderson 和 Rubin [1]）、两阶段最小二乘法（Basmann [3] 和 Theil [9]）和普通最小二乘法。在适当的一般条件下，前两种方法产生一致的估计；通过样本大小的平方根标准化的两组估计值具有相同的限制联合正态分布（Anderson 和 Rubin [2]）。在特殊情况下，已经获得了估计值的精确分布。特别地，当预定变量是外生的，相关方程中出现两个内生变量，并且指定一个内生变量的系数为1时，在两级最小二乘情况下，Richardson[7]和Sawa[8]以及在有限信息最大似然情况下，由Mariano和Sawa[6]得到了一个内生变量系数估计的精确分布。确切的分布涉及多个无穷级数，很难解释，但 Sawa 根据无穷级数表达式的计算，绘制了两阶段最小二乘估计的一些密度。本文的主要结果是在有两个内生变量的情况下获得了所谓的k类估计（包括两级最小二乘估计和普通最小二乘估计）的分布函数的渐近展开。近似分布的密度是正态密度乘以多项式。正态分布的第一个校正项涉及三次除以样本量的平方根。
The limited information maximum likelihood and two-stage least squares estimates have the same asymptotic normal distribution; the ordinary least squares estimate has another asymptotic normal distribution. This paper considers more accurate approximations to the distributions of the so-called "k-class" estimates. An asymptotic expansion of the distribution of such an estimate is given in terms of an Edgeworth or Gram-Charlier series (of which the leading term is the normal distribution). The development also permits expression of the exact distribution in several forms. The distributions of the two-stage least squares and ordinary least squares estimates are transformed to doubly-noncentral F distributions. Numerical comparisons are made between the approximate distributions and exact distributions calculated by the second author. SEVERAL METHODS HAVE been proposed for estimating the coefficients of a single equation in a complete system of simultaneous structural equations, including limited information maximum likelihood (Anderson and Rubin [1]), two-stage least squares (Basmann [3] and Theil [9]), and ordinary least squares. Under appropriate general conditions the first two methods yield consistent estimates; the two sets of estimates normalized by the square root of the sample size have the same limiting joint normal distributions (Anderson and Rubin [2]). In special cases the exact distributions of the estimates have been obtained. In particular, when the predetermined variables are exogenous, two endogenous variables occur in the relevant equation, and the coefficient of one endogenous variable is specified to be one, the exact distribution of the estimate of the coefficient of one endogenous variable has been obtained by Richardson [7] and Sawa [8] in the case of twostage least squares and by Mariano and Sawa [6] in the case of limited information maximum likelihood. The exact distributions involve multiple infinite series and are hard to interpret, but Sawa has graphed some of the densities of the two-stage least squares estimate on the basis of calculations from an infinite series expression. The main result of this paper is to obtain an asymptotic expansion of the distribution function of the so-called k-class estimate (which includes the twostage least squares estimate and the ordinary least squares estimate) in the case of two endogenous variables. The density of the approximate distribution is a normal density multiplied by a polynomial. The first correction term to the normal distribution involves a cubic divided by the square root of the sample size.