发布时间 : 星期四 文章计量经济学（英文）重点知识点考试必备更新完毕开始阅读7d58c83ab52acfc789ebc9df
? Do it by Stata. =invnorm(uniform())
12.Central Limit Theorem’s 中心极限定理
If there is a large number of independent and identically distributed (iid) random variables, then, with a few exceptions , the distribution of their sum tends to be a normal distribution as the number of such variables increases indefinitely. 随着变量个数的无限增加，独立同分布随机变量近似服从正态分布 13.Recall
U, the error term represents the influence of all those forces that affect Y but are not specifically included in the regression model because there are so many of them and the individual effect of any one such force on Y may be too minor.
误差项代表了未纳入回归模型的其他所有因素的影响。因为在这些影响中，每种因素对Y的影响都很微弱
If all these forces are random, if we let U represent the sum of all these forces, then by invoking the CLT, we can assume that the error term U follows the normal
distribution.如果所有这些影响因素都是随机的，用U代表所有这些影响因素之和，那么根据中心极限定理，可以假定误差项服从正态分布。 14.Another property of normal distribution另一个正态分布的性质
Any linear function of a normally distributed variable is itself normally distributed. 正态变量的性质函数仍服从正态分布。 15.Hypothesis testing 假设检验
Having known the distribution of OLS estimators b1 and b2, we can proceed the topic of hypothesis testing. 16.Null hypothesis 零假设
“zero” null hypothesis is deliberately chosen to find out whether Y is related to X al all, which is also called straw man hypothesis.之所以选择这样一个假设是为了确定Y是否与X有关，也称为稻草人假设。
17.We need some formal testing procedure to reject or receive the null hypothesis and make the skeptical guys shut up.需要正规的检验过程拒绝或接受零假设
18. If our null hypothesis is B2=0 and the computed b2=0.0013, we can find out the probability of obtaining such a value from the Z, the standard normal distribution.如果零假设为B2=0，计算得到b2=0.0013，那么根据标准正态分布Z，能够求得获此b2值的概率If the probability is very small, we can reject the null hypothesis.如果这个概率非常小，则拒绝零假设。If the probability is larger, say , greater than 10 percent, we may not reject the null hypothesis.如果这概率比较大，比如大于10%，就不拒绝零假设。
19.We don’t know the σ2
2We must know the true σ2, but we can estimate it by using ?
20.What will happen if we replace σby its estimator σ-hat
b2?B2?tn?2?x2ior,more?generally
b2?B2se(b2)tn?2
21.Let us assume that α, the level of significance or the probability of committing a type I error, is fixed at 5 percent.假定α，显著水平成犯第一类错误的概率为5%。 22.red area = rejection region for 2-sided test
f(t)
a/2
(1-a)
a/2
0 t t-t
23.Loop and ball c c a. This is a 95% confidence interval for B2 给出了B2的一个95%的置信区间。
b. in repeated applications 95 out of 100 such intervals will include the true B2重复上述过程，
100个这样的区间中将有95个包括真实的B2。
c. Such a confidence interval is known as the region of acceptance (of H0) and the area outside
the confidence interval is known as the rejection region (of H0)用假设检验的语言把这样的置信区间称为（H0的）接受区域，把置信区间以外的区间成为（H0的）拒绝区域 24.回归系数的假设检验
目的：简单线性回归中，检验X对Y是否真有显著影响 基本概念回顾: 临界值与概率、大概率事件与小概率事件 相对于显著性水平?的临界值为: t?（单侧）或t?2（双侧）
*t计算的统计量为:
（大概率事件）1? ? 统计
量 t （小概率事件）
??t?20
t*t?225.Conclusions
Since this interval does not include the null-hypothesized value of 0.因为这个区间没有包括零假设值0。We can reject the null hypothesis that annual family income is not related to math S.A.T. Scores.所以拒绝假设：家庭年收入对数学SAT没有影响。Put positively, income does have a relationship to math S.A.T. scores. 换言之，收入确实与数学SAT有关系。
26.A cautionary note
Although the statement given is true, we cannot say that the probability is 95 percent that the particular interval includes B2, for this interval is not a random interval, it is fixed, therefore, the probability is either 1 ore 0 that the interval includes B2.虽然式子3.26为真，但不能说某个特定区间式3.27包括真实B2的概率为95%，因为与式子3.26不同，式3.27是固定的，而不是一根随机区间，所以区间3.27包括B2的概率为1或0.We can only say that if we construct 100 intervals like this interval, 95 out of 100 such intervals will include the true B2.我们只能说，如果建立100个像式3.27这样的区间，则有95个区间包括真实的B2.We can not guarantee that this particular interval will necessarily includes B2.并不能保证某个区间一定有B2.
27.The test of significance approach to hypothesis testing 假设检验的显著性检验方法
Hypothesis testing is that of a test statistic and the sampling distribution of the test statistic under the null hypothesis, H0.假设检验方法涉及两个重要的概念检验统计量和零假设下检验统计量的抽样分布。The decision to accept or reject H0 is made on the basis of the value of the test statistic obtained from the sample data.根据从样本数据求得的检验统计量的值决定接受或拒绝零假设。 28.T test
We can use the t value computed here ad the test statistic, which follows the t distribution with (n-2) d.f.可以计算出t值作为检验统计量，它服从自由度为（n-2）的t 分布。
29.Instead of arbitrarily choosing the α value , we can find the p value (the exact level of significance) and reject the null hypothesis if the computed P value is sufficiently low.为了避免选择显著水平的随意性，通常求出p值（精确的显著水平），如果计算的p值充分小，则拒绝零假设。
30.Conclusions
In the case of two-sided t test 双边检验情况中If the computed |t|, the absolute value of t, exceeds the critical t value at the chosen level of significance, we can reject the null hypothesis.如果计算得到的|t|值超过临界t值，则拒绝零假设。 31.P value
The P value of that t statistic of 5.4354 is about 0.0006. t统计量（5.4354）的p值（概率值）约为0.0006。The smaller the p value, the more confident we are when reject the null hypothesis.p值越小，在拒绝零假设的时候就越有自信。Thus if we were to reject the null hypothesis that the true slope coefficient is zero at this P value, we would be wrong in six out of ten thousand occasions. 如果在这个p值水平之上拒绝零假设：真实的斜率系数为0，则犯错误的机会有万分之六。
32.How can we computed t
We first compute the t value as if the null hypothesis were that B2=0, we still get the t
t?0.0013?0?5.43540.000245首先计算在零假设B2=0下的t值Since this value exceeds any of the critical
values shown in the preceding table, following the rules laid down. t值大与上表给出的任何临界值，附录D表D-2列出的规则，We can reject the hypothesis that annual family income has no relationship to math S.A.T. Scores.拒绝零假设：家庭年收入对数学SAT没有影响。 33.How good is the fitted regression line: the coefficient of determination r2
On the basis of t test both the estimated intercept and slope coefficients are statistically significant (i.e. significantly different from zero) suggests that the SRF seems to “fit” the data “reasonably” well.根据t检验，估计的斜率和结局都是统计显著的，这说明样本回归函数式3.16很好地拟
合了样本数据。
34.Coefficient of determination
Can we develop an overall measure of “goodness of fit ” that will tell us how well the estimated regression line fits the actual Y values?能否建立一个“拟合优度”的判定规则，从而辨别估计的回归线拟合真实Y值的优劣程度呢？Such a measure has been developed and is known as the coefficient of determination.称之为判定系数。 35.Recall
Yi?Yi?ei
36.Rearrange it Yi?Yi?ei?Yi?Yi?eiYi?Y?Yi?Y?ei(Yi?Y)?(Yi?Y)?(Yi?Yi)
37.Decomposition
(Yi?Y):variation?in?Yi1、3、
(Yi?Y):vaiationr?in?Yiexplained?byX.?(Yiaround)2、 ?its?mean?value(note:Y?Y)
from?its?mean?value(Yi?Yi):unexplained?or?residual?variation
38.In deviation forms
(Yi?Y)?(Yi?Y)?(Yi?Yi)Y?Y?(Yi?Y)?(Yi?Y)?(Yi?Yi)yi?yi?eiyi?yi?ei?(Yi?Y)?ei?(b1?b2Xi)?(b1?b2X)?ei?b2(Xi?X)?ei 2、
1、yi?b2xi?ei39.Square both sides and sum
2?yi?b2xi?ei
?yi2??yi??ei2?y?y2i22?b2xi??ei2
2i=the total variation of the actual Y values about their sampling mean Y bar, which may be
called the total sum of squares (TSS)总平方和，真实Y值围绕其均值的总变异
?y=The total variation of the estimated Y values about their mean value, Y hat bar, which may
i2be called appropriately the sum of squares due to regression (i.e., due to the explanatory variables), or simply called the explained sum of squares (ESS)解释平方和，估计的Y值围绕气均值的变异，也称回归平方和（由解释变量解释的部分） 40.Put simply
TSS?ESS?RSSThe total variation in the observed Y values about their mean value can be