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1、Introductory Econometrics, Lijun Jia,1,Multiple Regression Analysis: Inference 多元回归分析：推断 (2),y = b0 + b1x1 + b2x2 + . . . bkxk + u,Introductory Econometrics, Lijun Jia,2,Confidence Intervals 置信区间,Because of random sampling error, it is impossible to learn the exact value of b using only the informat
2、ion in the sample. 由于随机取样误差的存在，我们不可能通过样本知道b 的准确值。 But it is possible to use data from a random sample to construct a set of values that contains the true value with a certain specified probability. 但是利用来自随机样本的数据构造一个取值的集合，使得真值在给定概率下属于这个集合是可能的。,Introductory Econometrics, Lijun Jia,3,Confidence Interva
3、ls 置信区间,Such a set is called a confidence set, and the pre-specified probability that the true value is contained in this set is called the confidence level. 这样的集合称为置信集，预先设定的真值属于此集合的概率称为置信水平（置信度）。 The confidence set turns out to be all the possible values between a lower and an upper limit, so that
4、the confidence set is an interval, i.e., confidence interval. 置信集是下限和上限之间所有可能的取值，故置信集为一个区间，称为置信区间,Introductory Econometrics, Lijun Jia,4,Confidence Intervals for b b 的置信区间,The above analysis can be extended to construct a confidence interval for b using the same critical value as was used for a two-
5、sided test. 通过对上述分析进行扩展，我们可以利用双边检验的临界值来构造 b 的置信区间。 If has a t distribution with n-k-1 degrees of freedom, simple manipulation leads to a confidence intervals for the unknown bj 如果 服从n-k-1自由度的 t 分布，简单的运算可以得到关于未知的 bj 的置信区间,Introductory Econometrics, Lijun Jia,5,Confidence Intervals for b b 的置信区间,Intro
6、ductory Econometrics, Lijun Jia,6,Confidence Intervals for b b 的置信区间,For df=n-k-1=25, a 95% confidence interval for any bj is given by 如果自由度为25，那么对任意bj ，95的置信区间为 When n-k-1120, the t(n-k-1) distribution is close enough to normal to use the 97.5th percentile in a standard normal distribution for cons
7、tructing a 95% CI: 当n-k-1120， t(n-k-1) 分布与正态分布充分接近，可以用标准正态分布的97.5分位数来构造95%置信区间,Introductory Econometrics, Lijun Jia,7,Confidence Intervals for b b 的置信区间,Once a confidence interval is constructed, one can carry out two-tailed hypotheses tests. 构造了置信区间之后，可以进行双尾假设检验 If the null hypothesis is H0: bj = a
8、j , then it is rejected against H1: bj = aj at the 5% significance level if and only if aj is not in the 95% confidence interval. 零假设为H0: bj = aj，当且仅当aj不在95的置信区间内时，零假设相对于H1: bj = aj在5的显著水平上被拒绝。,Introductory Econometrics, Lijun Jia,8,Testing a Linear Combination 检验线性组合,Suppose instead of testing whet
9、her b1 is equal to a constant, you want to test if it is equal to another parameter, that is H0 : b1 = b2 假设我们要检验是否一个参数等于另一个参数H0 : b1 = b2，而不是检验b1是否等于一个常数。 Use same basic procedure for forming a t statistic 应用与构造t统计量相同的程序,Introductory Econometrics, Lijun Jia,9,Testing Linear Combo (cont) 检验线性组合,Intr
10、oductory Econometrics, Lijun Jia,10,Example: 例子：,Suppose you are interested in the effect of campaign expenditures on outcomes: 假设你感兴趣的是竞选支出对选举结果的影响 voteA = b0 + b1log(expendA) + b2log(expendB) + b3prtystrA + u H0: b1 = - b2, or H0: q1 = b1 + b2 = 0 b1 = q1 b2, so substitute in and rearrange 令b1 = q
11、1 b2, 带入并移项可得 voteA = b0 + q1log(expendA) + b2log(expendB - expendA) + b3prtystrA + u,Introductory Econometrics, Lijun Jia,11,Example (cont):,This is the same model as originally, but now you get a standard error for b1 b2 = q1 directly from the basic regression 这个模型与原模型相同，但是此时可以直接从回归中得到b1 b2 = q1的标
12、准差 Any linear combination of parameters could be tested in a similar manner 参数的任何线性组合都可以用类似的手段进行检验。 Other examples of hypotheses about a single linear combination of parameters: 关于检验参数的单个线性组合的其它例子 b1 = 1 + b2 ; b1 = 5b2 ; b1 = -1/2b2 ; etc,Introductory Econometrics, Lijun Jia,12,Multiple Linear Rest
13、rictions 多线性约束The F test,Everything weve done so far has involved testing a single linear restriction, (e.g. b1 = 0 or b1 = b2 ) 目前为止，我们讨论了对单个线性约束的假设检验(例如， b1 = 0 或 b1 = b2 ) However, we may want to jointly test multiple hypotheses about our parameters 然而，我们也想对我们的参数作多个检验 A typical example is testing
14、 “exclusion restrictions” we want to know if a group of parameters are all equal to zero 一个典型的例子是检验“排除约束”我们想知道是不是一组参数都等于0,Introductory Econometrics, Lijun Jia,13,Testing Exclusion Restrictions 检验排除约束,Now the null hypothesis might be something like H0: bk-q+1 = 0, . , bk = 0 此时，零假设形如H0: bk-q+1 = 0, .
15、 , bk = 0(4.35) The alternative is just H1: H0 is not true 替代假设H1: H0 为假 Cant just check each t statistic separately, because it is possible for none to be individually significant at the specified significance level, but the joint test gives a different result. 不能分别进行 t 检验，因为存在这样的可能性：在给定显著水平下，所有的参数
16、都不显著，但是联合检验显著。,Introductory Econometrics, Lijun Jia,14,Example,Consider a model explaining the major league baseball players salaries: 考虑一个解释棒球联赛主力球员工资的模型 log(salary)= b0+ b1years+ b2 gamesyr+ b3 bavg+b4 hrunsyr+ b5 rbisyr+u, （4.28） salary: 1993 total salary of major league baseball players 1993年棒球联赛主力球员的总工资 Years: years in the league 在联赛中的年数 Gamesyr: average games played per year 每年平均比赛数 Bavg: career batting average 职业生涯击球率 Hrunsyr: hom
