Multiple Regression Analysis:Inference亠多元回归分析：推断 申1)y=Bo + Pixi + 0为+ 0/k + ULecture Outline 本课提纲 CLM assumptions and Sampling Distributions of the OLS Estimators经典假设与OLS估计量的样本分布 Background review of hypothesis testing 假设检验的背景知识 One-sided and two-sided t tests单边与双边t检验 Calculating the p values计算p值Assumption MLR.6 (Normality) 假设MLR.6 (正态) So far, we know that given the Gauss-Markov assumptions, OLS is BLUE,我们已经知道当Gauss-Markov假设成立时，OLS是最优线性 无偏估计。 In order to do classical hypothesis testing, we need to add another assumption (beyond the Gauss-Markov assumptions)为了进行经典的假设检验，我们要在Gauss-Markov假设之外 增加另一假设。 Assumption MLR.6 (Normality): Assume that uisindependent of Xy., jrana uis normally distributed with zero mean and variance o2:Normal(Oo2)蹩強舷霹w.”丽服从均值CLM Assumptions屋典线性模型假设 Assumptions MLR.1 一 MLR.6 are called the classical linear modelCLM assumptions.假设MLR.1-MLR.6被称为经典线性模型假设 We refer to the model under these six assumptions as the classical linear mode/.我们将满足这六个假设的模型称为经典线性模型Under CLM, OLS is not only BLUE, but also the minimum variance unbiased estimator that isz amonq linear and nonlinear estimators, OLS estimator gives tne smallest variance.r=的方差。CLM Assumptions屋典线性模型假设 We can summarize the population assumptions of CLM as follows我们对总体的经典线性模型假设做个总结 yxz Normal + 角七 +. + fa2) While for now we just assume normality sometimes this is not the case尽管现在我们假设了正态，但有时候并不是这种情况CLM Assumptions屋典线性模型假设 What should we do when the normality assumption fails?如果正态假设不成立怎么办？Using a transformation, especially taking the log, often yields a distribution that is closer to normal.通过变换，特别是通过取自然对数，往往可以 得到接近于正态的分布。A服从正态分布，因为谟误差的线性组合Theorem 4-1 Normal Sampling Distributions占理4.1正态样木分布Under thdCLM assumption, conditionion thesamplevaluesofso thattheindependert variable 0,NormNorma（0,1）A服从正态分布，因为谟误差的线性组合0）is distributed normallybecause 让 is a linearcombinatian of theerrors 在CLM假设下，条件于解释的样本值有龙故Norma（0,l）A服从正态分布，因为谟误差的线性组合4.2 Testing Hypotheses about a Single Population Parameter: the f-test 知单个总体参数的假设检验：t检验 Consider a population model (4.2)J = A)+ Px 十十 Pkxk + uwhich satisfies the CLM assumptions.We now study how to test hypotheses about a particular p.考虑总本中濟足CLM白勿莫型y =优+0內+卩kXk+u我们现在研究如何对一个特定的0；进行假设检验A服从正态分布，因为谟误差的线性组合Background Review景知识回顾 The hypothesis to be tested is called the /7t/Z hypothesis 被检验的假设称为零假设 Hypothesis testing entails using data to compare the null hypothesis with a second hypothesis, i.e.z the alternative hypothesis 假设检验利用数据将零假设和另一个假设（替 代假设）进行比较Background Review The alternative hypothesis specifies what is true if the null hypothesis is not.替代假设给出在零假设不成立时的真实情况。 Our goal: use the evidence in a randomly selected sample of data to decide whether to accept the null hypothesis我们的目的：利用一个随机选取的样本提供给 我们的证据来决定是否应当接受零假设。景知识回顾 Two kinds of mistakes are possible in hypothesis testing.在假设检验中存在两种可能的错误。 Type I error: reject the null hypothesis when it is in fact true.第一类错误：当零假设为真时拒绝零假设（弃真） Type IIerrors fail to reject the null when it is actually false.第二类错误：当零假设为假时未拒绝零假设（取伪）Background Review背景知识回顾 Hypothesis testing rules are constructed to make the probability of committing type I error fairly small.黠敝些假设检验的规则使发生第一类错误的概 The sianificance level (level) of a test is the probability of a Type I error.一个检验的显著性水平是发生第一类错误的概率。 Commonly specified significance levels: 0.1z0.05z 0.01. If it equals 0.05z it means the researcher is willing to falsely reject the null at 5% of the time.假设。景知识回顾 The critical value of the test statistic is thevalue of the statistic for which the test just reject the null hypothesis at the given significance level.鱷狒 器零假设刚好在给定显著性 The set of values of the test statistic for which the test rejects the null is the rejection region, and the values of the test statistic for which it does not reject the null is the acceptance region.量药敢借范嵐成为彥交銃Background Review背景知识回顾 A test statistic （7） is some function of the random sample. When we compute the statistic for a particular sample, we obtain an outcome of the test statistic （幼一个检验统计量（T）是关于随机样本的一个 函数。当我们用某一特定样本计算此统计量时, 我们得到这个检验统计量的一个实现（t）。Theorem 4-2 fDistribution for theStandardized Estimators514.2:标准化估计量的t分布Under thdCLMassumptionsU-）（4.3）Notethisis a t distributfon （vsnormal）because we havetoestimatecr2byNotethedegreesof freedom: n-Z:-l检验The /-Test (cont) Knowing the sampling distribution for the standardized estimator allows us to carry out hypothesis tests 知道标准化估计量的样本分布后，便可以进行假设检验 Start with a null hypothesis由零假设出发 For example, Ho: /J=0 (4.4)例如，Hq：月=0 If accept null, then accept that 巧 has no partial effect on yt controlling for other ”s如果接受零假设，则认为控制x其它分量后，吶y没有边际影响。Useful information about p-values些关于p值的信息ecause it is a probabilit