Evaluation of the effect of stochastic explanatory variables in regression models when the underlying distribution is in the class of elliptical distributions

1.得到了回归参数的LSE的渐近协方差矩阵的性质，通过模拟研究了它的小样本性质，并以资产定价模型为例，研究了回归模型中解释变量为随机变量的情形。特别地，I(1)推导了解释变量和因变量的联合分布为椭圆时LSE协方差矩阵的渐近和小样本性质，(2)证明了金融市场模型中α和β的LSE的渐近方差如何依赖于联合分布的偏度和峰度，(3)发现当因素和收益为独立同分布时，非正态对K因素资产定价模型中期望资产收益的无条件LSE推断没有影响。研究了解释变量和误差项不相关但不独立时最小二乘估计协方差矩阵的小样本性质，并与存在异方差时的小样本性质进行了比较。在椭圆分布类中，当基础分布为非正态分布时，我提出了弱异质性的新定义。在新定义的基础上，通过模拟比较了斯潘诺斯(1994)研究的t-线性异方差回归模型参数的联合最大似然估计和条件最大似然估计。研究发现，条件最大似然估计对条件模型的回归参数效果较好，但对其他参数则不适用。

1. I obtained the properties of the asymptotic covariance matrix of the LSE of regression parameters, investigated its small sample properties by simulation, and studied asset pricing models as examples, when explanatory variables are random in regression models. In particular, I (1) derived the asymptotic and small sample properties of the covariance matrix of the LSE when the joint distribution of explanatory and dependent variables is elliptical, (2) showed how the asymptotic variances of the LSE of alpha and beta depend on skewness and kurtosis of the joint distribution in the market model in finance without assuming any specific distribution, (3) found there is no effect of nonnormality on the unconditional LSE inference of expected asset returns in K-factor asset pricing models when factors and returns are i.i.d., and (4) studied the small sample properties of the covariance matrix of the LSE when explanatory variables and error term are uncorrelated but not independent and compared them to those when there is heteroskedasticity.2. I proposed a new definition of weak exogeneity when the underlying distribution is nonnormal in the class of elliptical distributions. Based on the new definition, I compared by simulation the joint and conditional MLEs of parameters for the student's t linear heteroskedastic regression model studied by Spanos (1994). The conditional MLE was found to work fine for regression parameters of the conditional model but not for other parameters.