[伍德里奇《计量经济学导论》课件]3多元回归分析

Multiple Regression Analysis Estimation 3多元回归分析估计3 y b0 b1x1 b2x2 bkxk u Chapter Outline 本章大纲 Motivation for Multiple Regression 使用多元回归的动因 Mechanics and Interpretation of Ordinary Least Squares 普通最小二乘法的操作和解释 The Expected Values of the OLS Estimators OLS估计量的期望 The Variance of the OLS Estimators OLS估计量的方差 Efficiency of OLS The Gauss-Markov Theorem OLS的有效性高斯马尔科夫定理 Lecture Outline 课堂大纲 The tradeoff of bias and variance in misspecified models 误设模型中偏误和方差间的替代关系 Estimating the error variance 估计误差项方差 The Gauss-Markov Theorem 高斯马尔科夫定理 Goodness of Fit 拟合优度 Sample problems 例题 Variances in Misspecified Models误设模型中的方差 The tradeoff between bias and variance is important for considering whether to include an additional variable in the regression 在考虑一个回归模型中是否该包括一个特定变量的决策中偏误和方差之间的消长关系是重要的 Suppose the true model is y b0 b1x1 b2x2 u then we have 假定真实模型是 y b0 b1x1 b2x2 u 我们有 Variances in Misspecified Models误设模型中的方差 Consider the misspecified model 考虑误设模型是 the estimated variance is 估计的方差是 When x1 and x2 has zero correlation 当x1和x2不相关时 otherwise 否那么 Consequences of Dropping x2 舍弃x2的后果 R120 R120 b20 Both estimates of b1 are unbiased Variances the same 两个对b1的估计都是无偏的方差相同 Both estimates of b1 are unbiased dropping x2 results in smaller variance 两个对b1的估计量都是无偏的舍弃x2使得方差更小 b20 Dropping x2 gives biased estimates of b1but its variance is the same as that from the full model 舍弃x2导致对b1的估计量有偏但方差和从完整模型得到的估计相同 Dropping x2 gives biased estimates of b1but its variance is smaller 舍弃x2导致对b1的估计量有偏但其方差变小 Variances in Misspecified Models误设模型中的方差 If some econometricians prefers comparing the likely size of the bias due to omitting x2 with the reduction in the variance 如果 一些计量经济学家建议将因漏掉x2而导致的偏误的可能大小与方差的降低相比拟以决定漏掉该变量是否重要 Nowadays including x2 is often favored because the induced multicollinearity is less important as the sample size grows but the omitted-variable bias does not necessarily follow any pattern 现在我们更喜欢包含x2 因为随着样本容量的扩大 增加x2导致的多重共线性变得不那么重要但舍弃x2导致的遗漏变量误偏却不一定有任何变化模式 Expectation and Variances of the Estimators under different senarios 不同情形下估计量的期望和方差 估计量 期望 估计量方差 估计量 期望 估计量方差 估计量 期望 估计量方差 模型设定 缺乏时 模型过度 设定时 模型设定正确时 Estimating the Error Variance 估计误差项方差 We wish to form an unbiased estimator of s2 我们希望构造一个s2 的无偏估计量 If we knew u an unbiased estimator of s2 can be formed by calculate the sample average of the u 2 如果我们知道 u通过计算 u 2的样本平均可以构造一个s2的无偏估计量 We dont know what the error variance s2 is because we dont observe the errors ui 我们观察不到误差项 ui 所以我们不知道误差项方差s2 Estimating the Error Variance估计误差项方差 What we observe are the residuals i 我们能观察到的是残差项i We can use the residuals to form an estimate of the error variance 我们可以用残差项构造一个误差项方差的估计 df n k 1 or df n k 1 df ie degrees of freedom is the number of observations number of estimated parameters df自由度是观察点个数被估参数个数 Estimating the Error Variance估计误差项方差 The division of n-k-1 comes from ESum of squared residualsn-k-1 s2 上式中除以n-k-1是因为残差平方和的期望值是n-k-1s2 Why degree of freedom is n-k-1 为什么自由度是n-k-1 Because k1 restrictions are imposed when deriving the OLS estimates That is given n-k-1 residuals the remaining k1 residuals are known hence the degree of freedom is n-k-1 因为推导OLS估计时参加了k1个限制条件也就是说给定n-k-1个残差剩余的k1个残差是知道的因此自由度是n-k-1 Estimating the Error Variance估计误差项方差 Theorem 33 unbiased estimation of s2 Under the Gauss-Markov Assumptions MLR1-5 we have 定理33 s2的无偏估计 在高斯马尔科夫假定 MLR1-5下我们有 Terminology The positive square root of s2 is called standard deviation and the positive square of is called standard error 定义术语 s2 正的平方根称为 标准偏差标准离差SD 正的平方根称为 标准误差标 准 差SE The standard error of is 的标准误差是 Efficiency of OLS The Gauss-Ma