计量经济第四章4-2.ppt

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 information 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 Intervals 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 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-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,Introductory 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 constructing 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 = aj , 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 whether 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,Introductory 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 = q1 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 Restrictions 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 “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, . , 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: home runs per year 每年本垒打次数 Rbisyr: runs batted in per year 每年击球上垒率,Introductory Econometrics, Lijun Jia,15,Example,H0: b3 =0, b4 = 0, b5 = 0; （4.29） H1 : H0 is not true（4.30） The estimated equation估计方程（4.28） （4.31）,Introductory Econometrics, Lijun Jia,16,Consider the regression with 5 regressors to be the unrestricted model and then estimate the restricted model, we get 考虑一个回归，无约束情况下有5个自变量，然后估计带约束的模型，得到（4.33）,Introductory Econometrics, Lijun Jia,17,General Case 一般情况,Unrestricted model y = b0 + b1x1 + b2x2 + . . . bkxk + u （4.34） Null Hypothesis H0: bk-q+1 =0, . . . , bk = 0; (4.35) Restricted model y = b0 + b1x1 + b2x2 + . . . bk-qxk-q + u （4.36),Introductory Econometrics, Lijun Jia,18,General Case 一般情况,A test of multiple restrictions is a joint (or multiple) hypotheses test. 对多个约束的检验称为联合假设检验 To do the test we need to estimate the “restricted model” without xk-q+1, , xk included, as well as the “unrestricted model” with all xs included 为进行检验，我们需要排除xk-q+1, , xk进行“约束回归”，也要包括所有的x进行“无约束”回归。 Intuitively, we want to know if the change in SSR is big enough to warrant inclusion of xk-q+1, , xk 直觉上，我们想知道加入xk-q+1, , xk来降低SSR是否值得,定义F为公式4.37,r 表示约束，ur表示无约束，q是约束个数or dfr dfur ,n k 1 = dfur,Introductory Econometrics, Lijun Jia,19,0,c,a,(1 - a),f(F),F,The F statistic (Figure 4.7),reject,fail to reject,Reject H0 at a significance level if F c 如果F c则在a显著水平上拒绝H0,Introductory Econometrics, Lijun Jia,20,Example,In this example, we have n-k-1=347, q=3, so the F statistic 本例中， n-k-1=347, q=3，故F统计量 F=(198.311-183.186)/3/(183.186/347) =9.553.78, the 1% level critical value 其中3.78为1%水平的临界值 Reject the joint hypothesis. 拒绝联合假设,Introductory Econometrics, Lijun Jia,21,Steps of calculating F statistics 计算F统计量的步骤,Estimate the unrestricted model, obtain the SSR and degrees of freedom in this model. 估计无约束模型，得到此模型的SSR和自由度 Count how many variables are excluded in the restricted model, this is q. 计算约束模型中排除的变量数目q Estimate the restricted model, obtain the related SSR. 估计约束模型，得到对应的SSR Use the F statistic formula to do the calculations. 利用F统计量公式进行计算,Introductory Econometrics, Lijun Jia,22,The R2 form of the F statistic 用R2构造 F 统计量,Because the SSRs may be large and unwieldy, an alternative form of the formula is useful 由于SSR可能很大而不易处理，我们有另一个有用的公式 We use the fact that SSR = SST(1 R2) for any regression, so can substitute in for SSRu and SSRur 因为SSR = SST(1 R2) 对任何回归成立，替换SSRu和SSRur（回忆R2=SSE/SST=1-(SSR/SST)）可得(4.41),Introductory Econometrics, Lijun Jia,23,The Example Continued,If we calculate the F statistic for the example using the