计量经济学.pdf

中级计量经济学中级计量经济学 INTERMEDIATE ECONOMETRICS 简单多元回归推断之三简单多元回归推断之三 2 Chapter Outline 本章提纲 Sampling Distributions of the OLS Estimators OLS估计量的样本分布 Testing Hypothesis About a Single Population Parameter The t test 单总体参数的假设检验 t检验 Confidence Intervals 置信区间 Testing Hypotheses About a Single Linear Combination of the Parameters 参数线性组合的假设检验 一维情形 Testing Multiple Linear Restrictions The F Test 多个线性约束的假设检验 F检验 Reporting Regression Results 报告回归结果 3 Multiple Linear Restrictions 多线性约束 In some occasions we need to jointly test multiple hypotheses about parameters 有时需要对参数作多个检验 Typical example testing exclusion restrictions Test if a group of parameters are all equal to zero Whether a set of independent variables have no partial effect on the dependent variable 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 即一组自变量是否对因变量存在局部效应 4 Testing Exclusion Restrictions 检验排除约束 E g The null hypothesis H0 bk q 1 0 bk 0 例如零假设形如H0 bk q 1 0 bk 0 A test of multiple restrictions is called a multiple hypotheses testor a joint hypotheses test 涉及多个限制的检验叫做多重假设检验和联合假设检验 The alternative H1 H0is not true 替代假设H1 H0为假 Is testing each coefficient using t test a good idea 5 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考虑一个解释棒球联赛主力球员工资的模型 log salary b0 b1years b2gamesyr b3 bavg b4hrunsyr b5rbisyr u salary 1993 total salary of major league baseball players 1993年棒球联赛主力球员的总工资年棒球联赛主力球员的总工资 years years in the league在联赛中的年数在联赛中的年数 gamesyr average games played per year每年平均比赛数每年平均比赛数 bavg career batting average职业生涯击球率职业生涯击球率 hrunsyr home runs per year每年本垒打次数每年本垒打次数 rbisyr runs batted in per year每年击球上垒率每年击球上垒率 6 Example H0 b3 0 b4 0 b5 0 H1 H0 is not true The estimated equation 估计方程估计方程 What conclusions will you draw based on t statistic for each parameter 0 29 0 0121 0 0026 0 0011 0 0161 0 0072 2 log 11 10 06890 0126 0 000980 01440 0108 353 183 0 6278 salaryyearsgamesyr bavghrunsyrrbisyr nSSRR 7 Example However if the H0is written as b3 b4 b5 0 then a test on these multiple restrictions could give a different answer 然而 如果H0是b3 b4 b5 0 那么对多个约束的联合检验是否 可以提供不同的答案 Why It is highly possible that because explanatory variables are highly correlated the resulted large standard errors could indicate a parameter to be insignificant while it is indeed significant 为什么 解释变量很可能高度相关 即使变量实际上显著 结果 中的较大的标准误也可能表明参数不显著 8 Testing Exclusion Restrictions 检验排除约束 Must derive a test of multiple restrictions whose distribution is known and tabulated 我们需要一个新的检验 并且 我们知道该检验的分布 也清楚不同分位 的临界值 The sum of squared residuals now turns out to provide a very convenient basis for testing multiple hypotheses We will also show how the R squared can be used in the special case of testing for exclusion restrictions 在这样的背景下 残差平方和成为联合检验的一个良好基础 相应的 我 们也讨论R平方如何用于检验排除性约束 9 Testing Exclusion Restrictions 检验排除约束 Idea because the OLS estimates are chosen to minimize the sum of squared residuals the SSR always increases when variables are dropped from the model 由于OLS是用于最小化残差平方和 当有变量被从模型中舍弃时 SSR 必定上升 If this increase is large enough relative to the SSR in the model with all of the variables then we can reject the null hypothesis 如果相对于包括所有解释变量的模型 新的SSR的上升足够大的话 那 么我们就能拒绝原假设 10 Testing Exclusion Restrictions 检验排除约束 To do the test we need to estimate the restricted model without xk q 1 xkincluded as well as the unrestricted model with all x s included and then the F statistic can be constructed and calculated 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 然后 我们就可以构造并计算F统计 量 where 1 r is restricted and ur is unrestricted q is of restrictions rur ur SSRSSRq F SSRnk 11 The F statistic F 统计量 The F statistic is always positive since the SSR from the restricted model can not be less than the SSR from the unrestricted F统计量总是正的 因为约束模型的SSR不会小于无约束模型的 SSR Essentially the F statistic is measuring the relative increase in SSR when moving from the unrestricted to restricted model 本质上 F统计量度量的是从无约束模型变为约束模型导致的SSR 的相对增量 q numerator degrees of freedom dfr dfur n k 1 denominator degrees of freedom dfur q是分子自由度 n k 1是分母自由度 12 The F statistic cont F统计量 To decide if the increase in SSR when we move to a restricted model is big enough to reject the exclusions we need to know about the sampling distribution of our F statistic 使用约束模型导致SSR增加是否足够大使我们可以拒绝排除假设 为了决定这一点 我们需要知道F统计量的样本分布 Under the null and the CLM assumptions F Fq n k 1 where q is referred to as the numerator degrees of freedom and n k 1 as the denominator degrees of freedom 自然 F Fq n k 1其中q 代表分子的自由度 n k 1代表分母的 自由度 13 The F statistic cont F统计量 Generally the numerator degrees of freedom q will be notably smaller than the denominator degrees of freedom n k 1 一般而言 q 要比n k 1小得多 14 The F statistic cont F统计量 When the denominator df reaches about 120 the F distribution is no longer sensitive to it A similar statement holds for a very large numerator df which however rarely occurs in applications 当分母自由度达120后 F分布就对分母自由度的增大就不 再敏感了 F分布对大的分子自由度也不太敏感 但是在实 证中 很大的分子自由度是很少见的 15 The F statistic cont F统计量 Rejection Rule 1 Set significance level 2 obtain the corresponding critical value c for the F distribution 3 reject H0if F c 在决定了显著性水平后 我们可以从F分布表中找到相应的 临界值 如果F c 则拒绝原假设 16 The F statistic cont F统计量 If H0is not rejected the variables are jointly insignificant justifies dropping them from the model If H0is rejected say that xk q 1 xkare jointly statistically significant or jointly significant at the given significance level 如果原假设未被拒绝 则称这些解释变量联合不显著 我们 通常就可以将这些变量从模型中舍去 如果原假设被拒绝 则称这些解释变量在给定显著性水平上联合显著 17 The F Distribution 18 Example cont Consider the regression with 5 regressors to be the unrestricted model and then estimate the restricted model we get 继续上例继续上例 现在我们从五个自变量减至两个进行回归现在我们从五个自变量减至两个进行回归 即估计带约束的模型 得到即估计带约束的模型 得到 0 11 0 0125 0 0013 2 log 11 220 07130 0202 353 198 3 0 5971 salaryyearsgamesyr nSSRR 19 Example cont 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 55 Because 9 55 3 78 the 1 level critical value we reject the joint hypothesis 因为9 55 3 78 其中3 78为1 水平的临界值 我们拒绝联合假设 20 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 which 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 21 The R2form of the F statistic 用R2构造 F 统计量 Because the SSR s may be large and unwieldy an alternative form of the formula is useful 由于由于SSR可能很大而不易处理 我们有另一个有用的公式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 SSRuand SSRur 因为因为SSR SST 1 R2 对任何回归成立 替换对任何回归成立 替换SSRu和和SSRur 可得可得 22 2 where again 11 is restricted and is unrestricted urr ur RRq F Rnk rur rur 其中 代表约束 代表无约束 22 Overall Significance 总体显著性 A special case of exclusion restrictions is to test H0 b1 b2 bk 0 排除约束的一个特殊情况是检验H0 b1 b2 bk 0 Since the R2from a model with only an intercept will be zero the F statistic is simply 由于只带常数项的回归得到的R2为0 此时的F 统计量应为 2 2 11 R k F Rnk 23 Example cont If we calculate the F statistic for the example using the R squared form of the F statistic we get a very close value 如果我们用拟合优度来计算如果我们用拟合优度来计算F 统计量 我们可统计量 我们可 以得到非常接近的值以得到非常接近的值 0 62780 5971 347 9 54 10 6278 3 F 24 Computing p values for F tests p values are especially useful for reporting the outcomes of F tests Why Let denote an F random variable with q n k 1 degrees of freedom and F is the actual value of the test statistic In the F testing context the p value is defined as p value Pr F p值对于报告F检验的结果尤其有用 此处p值定义为p value Pr F 25 Computing p values for F tests Interpretation it is the probability of observing a value of the F at least as large as we did given that the null hypothesis is true p值的含义 如果原假设为真 那么我们能够看得统计值至 少有估计的那么大的概率是多少 26 Computing p values for F tests A small p value is strong evidence against H0 For example p value 0 005 means that the chance of observing a value of F as large as we did when the null hypothesis was true is only 0 5 we usually reject H0in such cases If the p value 897 then the chance of observing a value of the F statistic as large as we did under the null hypothesis is 89 7 小p值是拒绝原假设的强有力的证据 比如 如果我们算的p值为0 005 那么如果原假设为真 我们得到大于或等于现有F统计量的概率为0 5 27 Relationship between F and t Statistics The F statistic for testing exclusion of a single variable is equal to the square of the corresponding t statistic if the test is two sided F统计量可用于检验单个约束 我们可以证明 如果检验是双边检验 就有F t2 However the F statistic is intended to detect whether any linear combination of coefficients is different from zero never the best test for a single coefficient F统计量为检验联合约束而生 并非检验单个约束的最佳统计量 Relationship between F and t Statistics The t statistic is best suited for testing a single hypothesis It t is also more flexible since it can be used to test against one sided alternatives t统计量检验单个约束最好 而且 由于它可用于单边检验 使用上也 比F更灵活 28 29北京大学中国经济研究中心 沈艳29 例子 环境 vs 经济发展 coverage b0 b1lgdp b2g3 b3bus u Stata会自动报告检验overall significance的F统计量 及其p value 其他参数线性约束的检验可用在reg命令结束后用test命令实施 例 test g3 0 bus 0 or test g3 bus 0 检验 H0 b2 b3 0 test g3 1 bus 0 检验 H0 b2 1 and b3 0 M Mo od de el l 1 14 49 97 77 7 3 34 47 79 9 3 3 4 49 99 92 2 4 44 49 93 31 1 P Pr ro ob b F F 0 0 0 00 00 00 0 F F 3 3 2 26 64 4 1 11 1 8 81 1 S So ou ur rc ce e S SS S d df f M MS S N Nu um mb be er r o of f o ob bs s 2 26 68 8 r re eg g c co ov ve er ra ag ge e l lg gd dp p g g3 3 b bu us s 例子 环境 vs 经济发展 test命令实施单参数的显著性检验时 此时 stata报告的F检验和t检验等价 且F统计量 为t统计量的平方 例 test lgdp 检验 H0 b1 0 30 P Pr ro ob b F F 0 0 0 00 00 00 0 F F 1 1 2 26 64 4 3 30 0 0 07 7 1 1 l lg gd dp p 0 0 t te es st t l lg gd dp p l lg gd dp p 1 12 2 1 10 01 18 89 9 2 2 2 20 06 69 92 22 2 5 5 4 48 8 0 0 0 00 00 0 7 7 7 75 56 64 48 83 3 1 16 6 4 44 47 73 3 c co ov ve er ra ag ge e C Co oe ef f S St td d E Er rr r t t P P t t 9 95 5 C Co on nf f I In nt te er rv va al l 31 General Linear Restrictions 一般线性约束 The basic form of the F statistic will work for any set of linear restrictions F 统计量的基本形式对任何线性约束组适用 First estimate the unrestricted model and then estimate the restricted model 先估计无约束模型 再估计约束模型 In each case record the SSR 每一步记录下SSR Imposing the restrictions can be tricky will likely have to redefine variables 设置约束可能需要技巧 有时要重新定义变量 32 General Linear Restrictions 一般线性模型 Notice that if the dependent variable is transformed due to restrictions on the explanatory variables for example if in 注意 如果由于自变量系数受到约束而使因变量发生改变 如 y b0 b1x1 b2x2 bkxk u b1is restricted to 1 and the new dependent variable is y x then the R squared form of the F statistic cannot be used Only the SSR form can be used b1被限制为1 从而新的因变量为y x 则R方构造的F统计量不再 适用 只有SSR形式的公式适用 Example Use the same voting model as before 再次考虑投票模型 Model is voteA b0 b1log expendA b2log expendB b3prtystrA u Now null hypothesis is 零假设为 H0 b1 1 b3 0 Substituting in the restrictions voteA b0 log expendA b2log expendB u so Use voteA log expendA b0 b2log expendB u as restricted model 将voteA log expendA b0 b2log expendB u作为约束模 型 34北京大学中国经济研究中心 沈艳34 例子 环境 vs 经济发展 coverage b0 b1lgdp b2g3 b3bus u 考虑检验 H0 b2 1 and b3 0 回归无约束方程 coverage b0 b1lgdp b2g3 b3bus u 得到无约束回归的残差平方和SSRur 受约束的回归方程为 coverage b0 lgdp b2g3 u 可构造新被解释变量y coverage lgdp 此时用new对g3回归 得到受约束回归残差平方和的SSRr 由F SSRr SSRur q SSRur n k 1 可构造检验统计量 35北京大学中国经济研究中心 沈艳35 例子 环境 vs 经济发展 请根据如下两个stata报告计算上述F统计量 请验证用R square 与用SSR计算的F统计量不同 R Re es si id du ua al l 1 11 11 15 55 57 7 4 47 74 4 2 26 64 4 4 42 22 2 5 56 66 61 18 89 9 R R s sq qu ua ar re ed d 0 0 1 11 18 84 4 M Mo od de el l 1 14 49 97 77 7 3 34 47 79 9 3 3 4 49 99 92 2 4 44 49 93 31 1 P Pr ro ob b F F 0 0 0 00 00 00 0 F F 3 3 2 26 64 4 1 11 1 8 81 1 S So ou ur rc ce e S SS S d df f M MS S N Nu um mb be er r o of f o ob bs s 2 26 68 8 r re eg g c co ov ve er ra ag ge e l lg gd dp p g g3 3 b bu us s R Re es si id du ua al l 1 12 23 33 39 94 4 9 98 88 8 2 26 66 6 4 46 63 3 8 89 90 09 93 32 2 R R s sq qu ua ar re ed d 0 0 0 00 01 15 5 M Mo od de el l 1 18 82 2 0 00 00 05 58 84 4 1 1 1 18 82 2 0 00 00 05 58 84 4 P Pr ro ob b F F 0 0 5 53 31 16 6 F F 1 1 2 26 66 6 0 0 3 39 9 S So ou ur rc ce e S SS S d df f M MS S N Nu um mb be er r o of f o ob bs s 2 26 68 8 r re eg g y y g g3 3 36北京大学中国经济研究中心 沈艳36 例子 环境 vs 经济发展 由test命令得出的结果 由SSR计算得出的结果 由R square计算得出的结果 P Pr ro ob b F F 0 0 0 00 00 00 0 F F 2 2 2 26 64 4 1 14 4 0 01 1 2 2 b bu us s 0 0 1 1 l lg gd dp p 1 1 t te es st t l lg gd dp p