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1、中级计量经济学 INTERMEDIATE ECONOMETRICS 简单多元回归推断之一 Chapter Outline Sampling Distributions of the OLS Estimators Testing Hypothesis About a Single Population Parameter: The ttest Confidence Intervals Testing Hypotheses About a Single Linear Combination of the Parameters Testing Multiple Linear Restrictions
2、: The FTest Reporting Regression Results 北京大学中国经济研究中心 沈艳 2 Lecture Outline Sampling Distributions: Review CLM assumptions and Sampling Distributions of the OLS Estimators Background review of hypothesis testing One-sided and two-sided t tests Calculating the p values 北京大学中国经济研究中心 沈艳 3 Sampling Distr
3、ibution: Review 抽样分布:复习 Sampling distributions play a central role in the development of statistical and econometric procedures. It is the probability distribution of an estimator over all possible outcomes. There are two approaches charactering sampling distributions: an “exact” approach and an “ap
4、proximate” approach. 抽样分布在统计学和计量经济学发展中具有核心地位.指一个估 计量在其所有可能取值上的概率分布.刻画抽样分布有两种方式： “准确”方式和“近似”方式 北京大学中国经济研究中心 沈艳 4 Sampling Distribution: finite-sample distribution The “exact” approach: derives a formula for the sampling distribution that holds for any value of n. Such distribution is called the exact
5、 distributionor finite-sample distribution. For example, if yis normally distributed, and y1, y2, , ynare i.i.d, then their average has an exact distribution of normal distribution. “准确”方式需要对任何n的取值都得到样本分布的精确表达式.这样的分 布被称为准确分布戒者有限样本分布。例如，如果y服从正态分布，且 y1, y2, , yn独立同分布，则其均值恰好服从正态分布 北京大学中国经济研究中心 沈艳 5 Sam
6、pling Distribution: the asymptotic distribution The “approximate” approach uses approximations to the sampling distributions that rely on the sample size being large. The asymptotic distribution: The large sample approximation to the sampling distribution. The asymptotic distributions can be counted
7、 on to provide good approximations to the exact sampling distribution, as long as the sample size is large. “近似”方式对样本分布进行大样本下的近似。对样本分布的大样本 近似常称为渐近分布。只要样本量足够大，渐近分布就是对准确分布 的很好的近似。 北京大学中国经济研究中心 沈艳 6 Sampling Distribution: the asymptotic distribution Two key tools: the law of large numbers (LLN), and th
8、e central limit theorem (CLT). 两个重要工具：大数定律，中心极限定理。 Why these two theorems are important? Most estimators encountered in statistics and econometrics can be written as functions of sample averages, hence can apply these two theorem to get the asymptotic distribution. Come back to them in Chapter 5. 北京
9、大学中国经济研究中心 沈艳 7 Sampling Distribution of OLS Estimators OLS估计量的样本分布 We have discussed the expected value and variances of the OLS estimators, but to perform statistical inference, we wish to know the sampling distribution. The sampling distributions of the OLS estimators depend on the underlying dis
10、tribution of the errors. 我们已经讨论了OLS估计量的期望和方差，但是为了进行 统计推断，我们需要完全了解估计量的抽样分布。OLS估 计量的抽样分布依赖于误差项的分布。 北京大学中国经济研究中心 沈艳 8 抽样在实证使用中的沿革 早期：普查 1936起：社会调查可以通过选取部分有代表性的样本完成。从 美国发源，政治、商业；之后多国。 谁能当选总统？ 1824-1936：“大” 1895，挪威国家统计局，Anders Niscolai Kiaer 1936: George Gallop. Literary Digest之 240万vs. Gallup之 5000. Gon
11、e with the Wind 好莱坞历史上的第一次 北京大学中国经济研究中心 沈艳 9 Assumption MLR.6 (Normality) 假定MLR.6 （正态） So far, we know that given the Gauss-Markov assumptions, OLS is BLUE. An additional assumption needed for classical hypothesis testing. Assumption MLR.6 (Normality): Assume that uis independent of x1, x2, xkand ui
12、s normally distributed with zero mean and variance s2: u Normal(0,s2) 我们已经知道当GaussMarkov假定成立时，OLS是最优线性无偏估 计。为了进行经典假设检验，需要增加一个假定。假定MLR.6 （正 态）：假设u不x1, x2, xk独立，且u服从均值为0，方差为s2的正态分 布。 北京大学中国经济研究中心 沈艳 10 CLM Assumptions What do we assume when normality of the error term assumption is invoked? One can co
13、nsider u as the sum of many different unobserved factors affecting y, hence can invoke the CLT to conclude that uhas an approximate normal distribution. It assumes that all unobserved factors affect yin a separate, additive fashion. Strong. To be relaxed when large sample is available. 北京大学中国经济研究中心
14、沈艳 11 CLM Assumptions Classical linear model(CLM) assumptions : Assumptions MLR.1 MLR.6. Summarize the population assumptions of CLM as follows y|x Normal(b0+ b1x1+ bkxk, s2) Minimum variance unbiased estimator: Under CLM, OLS is not only BLUE, but also MVUE the OLS estimator gives the smallest vari
15、ance among all unbiased estimators. 假定MLR.1-MLR.6被称为经典线性模型假定。在经典线性模型 假设下，OLS丌仅是BLUE，而且是最小方差无偏估计量，即在 所有无偏估计量中，OLS估计量具有最小的方差。 北京大学中国经济研究中心 沈艳 12 Theorem 4.1 Normal Sampling Distributions 2 jjjj 2 jj jj j j Under the CLM assumptions, conditional on the sample values of the independent variables, Normal
16、 ,Var , where Var = SST (1-R ) - Therefore, Normal 0,1 sd is distributed norm 在假设下，条件于解释变量的样本值有故 服从正态分布，因为它是误差的线性组合 jjj jj j j ally because it is a linear combination of the errors CLM Normal ,Var , - Normal 0,1 sd 北京大学中国经济研究中心 沈艳 13 Brief Proof of Theorem 4.1 证明提要 jjjj i j From previous lectures we know E( )= and Var( ). If we can pr
