【正文】 ，擾動(dòng)項(xiàng)的期望或均值為零。4) Assumption 4:The variance of each Ui is constant, or homoscedastic. U的方差為常數(shù)，或同方差。b. The individual Y values are spread around their mean values with the same 。6) Assumption 6:The regression model is correctly 。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(jì)量的方差與標(biāo)準(zhǔn)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(biāo)準(zhǔn)誤l Is simply the standard deviation of the Y values about the estimated regression line. Y值偏離估計(jì)回歸的標(biāo)準(zhǔn)差。 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?yàn)樗?jiǎn)單易行，還因?yàn)樗哂泻軓?qiáng)的理論性質(zhì)。因?yàn)樵谶@些影響中，每種因素對(duì)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(xiàng)服從正態(tài)分布。15. Hypothesis testing 假設(shè)檢驗(yàn)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)。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)。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(gè)95%的置信區(qū)間。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(yàn)的語(yǔ)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(yàn)?zāi)康模汉?jiǎn)單線性回歸中，檢驗(yàn)X對(duì)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(duì)于顯著性水平的臨界值為: （單側(cè)）或（雙側(cè)）計(jì)算的統(tǒng)計(jì)量為:統(tǒng)計(jì)量 t0（大概率事件）（小概率事件）25. ConclusionsSince this interval does not include the nullhypothesized value of 。Put positively, ine does have a relationship to math . scores. 換言之，收入確實(shí)與數(shù)學(xué)SAT有關(guān)系。The decision to accept or reject H0 is made on the basis of the value of the test statistic obtained from the sample 。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 puted P value is sufficiently ，通常求出p值（精確的顯著水平），如果計(jì)算的p值充分小，則拒絕零假設(shè)。31. P valueThe P value of that t statistic of is about . t統(tǒng)計(jì)量（）的p值（概率值）。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. 如果在這個(gè)p值水平之上拒絕零假設(shè)：真實(shí)的斜率系數(shù)為0，則犯錯(cuò)誤的機(jī)會(huì)有萬(wàn)分之六。33. How good is the fitted regression line: the coefficient of determination r2On the basis of t test both the estimated intercept and slope coefficients are statistically significant (. significantly different from zero) suggests that the SRF seems to “fit” the data “reasonably” ，估計(jì)的斜率和結(jié)局都是統(tǒng)計(jì)顯著的。35. Recall 36. Rearrange it37. Deposition 38. In deviation forms 39. Square both sides and sum=the total variation of the actual Y values about their sampling mean Y bar, which may be called the total sum of squares (TSS)總平方和，真實(shí)Y值圍繞其均值的總變異=The total variation of the estimated Y values about their mean value, Y hat bar, which may be called appropriately the sum of squares due to regression (., due to the explanatory variables), or simply called the explained sum of squares (ESS)解釋平方和，估計(jì)的Y值圍繞氣均值的變異，也稱回歸平方和（由解釋變量解釋的部分）40. Put simplyThe total variation in the observed Y values about their mean value can be partitioned into two parts, one attributable to the regression line and the other to random forces, because not all actual Y observations lie on the fitted ：一部分歸于回歸線，另一部分歸于隨機(jī)因素，因?yàn)椴皇撬械恼鎸?shí)觀察值Y都落在你和直線上。b. If the SRF fits the data poorly RSS will be much larger than ，則RSS遠(yuǎn)大于ESS。44. Properties of R2a. it is a nonnegative b. its limits are 0≤ R2 ≤1 since a part (ESS) cannot be greater than the whole (TSS). 0≤ R2 ≤1，因?yàn)椴糠郑‥SS）不可能大于整體（TSS）。An R2 of zero means no relationship between Y and X =0，則表示Y與X之間無(wú)任何關(guān)系。47. As a matter of conventionFrom now on , if we do not specify a specific null hypothesis, then we will assume that it is the zero null ，如果沒(méi)有設(shè)定特殊的零假設(shè)，習(xí)慣地規(guī)定零假設(shè)為：總體參數(shù)為零。The lower the P value, the greater the evidence against the null hypothesis, the lower likelihood the coefficient is ，拒絕假設(shè)的證據(jù)就越充分。If the puted P value is smaller than the critical P value, the null hypothesis can be ，則拒絕原假設(shè)。50. Error term: normality testOur statistical testing procedure is based on the assumption that the error term Ui is normally 。l If the puted Chisquare value exceeds the critical Chisquare value for 2 . at the chosen level of significance, we reject the null hypothesis of normal ，則拒絕正態(tài)分布的零假設(shè)If it does not exceed the critical Chisquare value, we may not reject the null ，則不能拒絕零假設(shè)。 The meaning of partial regression coefficient偏回歸系數(shù)的含義The regression coefficients B2 and B3 are known as partial regression or partial slope coefficients. B2,B3稱為偏回歸系數(shù)或偏斜率系數(shù)1　 The meaning of Partial regression coefficient is as follows: B2 measures the change in the mean value of Y, E(Y), per unit change in X2, holding the value of X3 constant. B2度量了在X3保持不變的情況下，X2單位變動(dòng)引起Y均值E(Y)的變化量。3　 Uniqueness：特殊性質(zhì)In the multiple regression model在多元回歸模型中 we want to find out what part of the change in the average value of Y can be directly attributable to X2 and what part to “直接”來(lái)源于X2，多大比例“直接”來(lái)源于X3。uniquenessThis unique feature