【正文】 cation error in the 。l This assumption can be explained informally as follows. An econometric investigation begins with the specification of the econometric model underlying the phenomenon of interest. and Standard errors of OLS estimators普通最小二乘估計量的方差與標準誤:One immediate result of the assumptions introduced is that they enable us to estimate the variances and standard errors of the OLS estimators given in Eq.() and (). should know:l Variances of the estimatorsl Standard errors of the estimators is the value of σl The homoscedastic σ is estimated from formula Error of the Regression (SER) 回歸標準誤l Is simply the standard deviation of the Y values about the estimated regression line. Y值偏離估計回歸的標準差。 of math function1) Interpretationl The standard deviation, or standard error, is , is a measure of variability of b2 from sample to sample.l If we can say that our puted b2 lies within a certain number of standard deviation units from the true B2, we can state with some confidence how good the puted SRF is as an estimator of the true PRF.2）Sampling Distribution 抽樣分布Once we determine the sampling distribution of our two estimators, the task of hypothesis testing bees ，那么假設(shè)檢驗就是舉手之勞的事情。 do we use OLS ?l The properties of OLS estimatorsl The method of OLS is used popularly not only because it is easy to use but also because it has some strong theoretical properties. OLS法得到廣泛使用，不僅是因為它簡單易行，還因為它具有很強的理論性質(zhì)。 theorem 高斯馬爾科夫定理Given the assumptions of the classical linear regression model (CLRM), the OLS estimators have minimum variance in the class of linear OLS estimators are BLUE (best linear unbiased estimators)滿足古典線性模型的基本假定，則在所有線性據(jù)計量中，OLS估計兩具有最小方差性，即OLS是最優(yōu)線性無偏估計量（BLUE）10. BLUE property 最優(yōu)線性無偏估計量的性質(zhì)1) B1 and B2 are linear estimators. B1和B2是線性估計量2) They are unbiased , that is E(b1)=B1, E(b2)=B2. B1和B2是無偏估計兩3) The OLS estimator of the error variance is 4) b1 and b2 are efficient Var(b1) is less than the variance of any other linear unbiased estimator of B1Var(b2) is less than the variance of any other linear unbiased estimator of B211. Monte Carlo simulation 蒙特卡洛模擬l Do the experiment at labl Do it by Excell. =NORMINV(RAND(),0,2)l Do it by matlab.= NORMINV(uniform(),MU,SIGMA)l Do it by Stata. =invnorm(uniform())12. Central Limit Theorem’s 中心極限定理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. 隨著變量個數(shù)的無限增加，獨立同分布隨機變量近似服從正態(tài)分布13. RecallU, 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. 誤差項代表了未納入回歸模型的其他所有因素的影響。因為在這些影響中，每種因素對Y的影響都很微弱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 ，用U代表所有這些影響因素之和，那么根據(jù)中心極限定理，可以假定誤差項服從正態(tài)分布。14. Another property of normal distribution另一個正態(tài)分布的性質(zhì)Any linear function of a normally distributed variable is itself normally distributed.正態(tài)變量的性質(zhì)函數(shù)仍服從正態(tài)分布。15. Hypothesis testing 假設(shè)檢驗Having known the distribution of OLS estimators b1 and b2, we can proceed the topic of hypothesis testing.16. Null hypothesis 零假設(shè)“zero” null hypothesis is deliberately chosen to find out whether Y is related to X al all, which is also called straw man ，也稱為稻草人假設(shè)。17. We need some formal testing procedure to reject or receive the null hypothesis and make the skeptical guys shut 18. If our null hypothesis is B2=0 and the puted b2=, we can find out the probability of obtaining such a value from the Z, the standard normal =0，計算得到b2=，那么根據(jù)標準正態(tài)分布Z，能夠求得獲此b2值的概率If the probability is very small, we can reject the null ，則拒絕零假設(shè)。If the probability is larger, say , greater than 10 percent, we may not reject the null ，比如大于10%，就不拒絕零假設(shè)。19. We don’t know the σ2We must know the true σ2, but we can estimate it by using 20. What will happen if we replace σby its estimator σhat 21. Let us assume that α, the level of significance or the probability of mitting a type I error, is fixed at 5 ，顯著水平成犯第一類錯誤的概率為5%。22. red area = rejection region for 2sided test
(1a)t0f(t)tctca/2a/223. Loop and balla. This is a 95% confidence interval for B2 給出了B2的一個95%的置信區(qū)間。b. in repeated applications 95 out of 100 such intervals will include the true B2重復(fù)上述過程，100個這樣的區(qū)間中將有95個包括真實的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)用假設(shè)檢驗的語言把這樣的置信區(qū)間稱為（H0的）接受區(qū)域，把置信區(qū)間以外的區(qū)間成為（H0的）拒絕區(qū)域24. 回歸系數(shù)的假設(shè)檢驗?zāi)康模汉唵尉€性回歸中，檢驗X對Y是否真有顯著影響基本概念回顧: 臨界值與概率、大概率事件與小概率事件相對于顯著性水平的臨界值為: （單側(cè)）或（雙側(cè)）計算的統(tǒng)計量為:統(tǒng)計量 t0（大概率事件）（小概率事件）25. ConclusionsSince this interval does not include the nullhypothesized value of 。We can reject the null hypothesis that annual family ine is not related to math . ：家庭年收入對數(shù)學(xué)SAT沒有影響。Put positively, ine does have a relationship to math . scores. 換言之，收入確實與數(shù)學(xué)SAT有關(guān)系。26. A cautionary noteAlthough 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 ，%，，而不是一根隨機區(qū)間， can only say that if we construct 100 intervals like this interval, 95 out of 100 such intervals will include the true ， can not guarantee that this particular interval will necessarily includes .27. The test of significance approach to hypothesis testing 假設(shè)檢驗的顯著性檢驗方法Hypothesis testing is that of a test statistic and the sampling distribution of the test statistic under the null hypothesis, 。The decision to accept or reject