计量经济第二章2-3.ppt

The Simple Regression Model 简单二元回归 (3) y = b0 + b1x + u Chapter Outline本章大纲 Definition of the Simple Regression Model 二元回归模型的定义 Deriving the Ordinary Least Squares Estimates 推导普通最小二乘法的估计量 Mechanics of OLS OLS的操作技巧 Unites of Measurement and Functional Form 测量单位和函数形式 Expected Values and Variances of the OLS estimators OLS估计量的期望值和方差 Regression through the Origin 过原点回归 Expected Values and Variances of the OLS EstimatorsOLS估计量的期望值和方差 We will study the properties of the distributions of OLS estimators over different random samples from the population. 从总体中抽取的不同的随机样本可得到不同的OLS估计量，我们将研究这些OLS估计量的分布。 We begin by establishing the unbiasedness of OLS under a set of assumptions. 首先，我们在一些假定下证明OLS的无偏性。 Assumption SLR.1 (Linear in Parameters):假定SLR.1 （关于参数是线性的） In the population model, the dependent variable y is related to the independent variable x and the error u as 在总体模型中，因变量 y 和自变量 x 和残差 u 的关系可写作 y = b0 + b1x + u ,(2.47) where b0 and b1 are the population intercept and slope parameters respectively. 其中 b0 和 b1 分别是总体的截距参数和斜率参数 Assumption SLR.2 (Random Sampling):假定SLR.2 (随机抽样): Assume we can use a random sample of size n, {(xi, yi): i=1, 2, …, n}, from the population model. Thus we can write the sample model yi = b0 + b1xi + ui 假定我们从总体模型随机抽取容量为n的样本, {(xi, yi): i=1, 2, …, n}, 那么可以写出样本模型为 yi = b0 + b1xi + ui (2.48) Assumptions SLR.3 and SLR.4假定 SLR.3 和 SLR.4 SLR.3 (Sample Variation in the Independent Variable): In the sample, the independent variables x‘s are not all equal to the same constant. SLR.3 (自变量中的样本变动): 在样本中，自变量 x 并不等于一个不变常数。 SLR.4 , Zero Conditional Mean: SLR.4, 零条件期望： Assume E(u|x) = 0 and thus in a random sample E(ui|xi) = 0 假定 E(u|x) = 0 . 那么在随机样本中我们有 E(ui|xi) = 0 Theorem 2.1 (Unbiasedness of OLS)定理2.1 ( OLS的无偏性) Using assumptions SLR.1 through SLR.4, we have the expected value of the OLS estimates of b0, and b1 equals the true values of b0 and b1 , respecti