第六讲 多元回归分析深入专题

1、1Multiple Regression Analysis: Further Issues y = b0 + b1x1 + b2x2 + . . . bkxk + u2.6.1 数据的测度单位对OLS统计量的影响nChanging the scale of the y variable will lead to a corresponding change in the scale of the coefficients and standard errors, so no change in the significance or interpretationnChanging the sc

2、ale of one x variable will lead to a change in the scale of that coefficient and standard error, so no change in the significance or interpretation3. 1111122112221122122,var,1/,/11nniiiiiiniiiniinxiijjjjjrurjjurxxyyr yrxxyySSERsSSTyytseSSTRseSSRSSRqFc seSSRnkSERSSRnkbbbbbbbb.n因变量或自变量以对数形式出现，改

3、变度量单位不会影响斜率系数，只会改变截距项。 11logloglogiic ycy 11logloglogjjc xcx8.Beta CoefficientsnOccasional youll see reference to a “standardized coefficient” or “beta coefficient” which has a specific meaningnIdea is to replace y and each x variable with a standardized version i.e. subtract mean and divide by stan

4、dard deviationnCoefficient reflects standard deviation of y for a one standard deviation change in x 9.10.Beta Coefficients (cont)1 12211j.where denote the z-score of , is the z score of ,and so on. Andb(/) for 1,., are called beta coefficients.ykkyjyjzb zb zb zerrorzy zxjkb11.nExample 6.1 :Effects

5、of Pollution on Housing Prices（数据名：HPRICE2）Stata 命令语句： reg price nox crime rooms dist stradio,beta (标准化后的回归分析)12.6.2 对函数形式的进一步讨论n OLS can be used for relationships that are not strictly linear in x and y by using nonlinear functions of x and y will still be linear in the parametersn Can take the nat

6、ural log of x, y or bothn Can use quadratic forms of xn Can use interactions of x variables13.Interpretation of Log Modelsnln(y) = b0 + b1ln(x) + u b1 is the elasticity of y with respect to xnln(y) = b0 + b1x + u b1 is approximately the percentage change in y given a 1 unit change in x ny = b0 + b1l

7、n(x) + u b1 is approximately the change in y for a 100 percent change in x14.Why use log models?n1.使用自然对数使得对系数的解释颇具有吸引力，可以直接以弹性的形式体现出来。n2.斜率系数不随测量单位的变化而变化。n3.取对数后，即使不能消除异方差的影响，但可以使之有所缓解。n4.取对数通常会缩小变量的取值范围，在某些情况下还相当可观。n5.缺点：变量不能取零和负值；更难预测原变量的值，原模型使我们预测log(y),而不是y。15.Quadratic Models（二次式模型）（二次式模型）n Fo

8、r a model of the form y = b0 + b1x + b2x2 + u we cant interpret b1 alone as measuring the change in y with respect to x, we need to take into account b2 as well, since1212122, so 20,maxmin2yxxyxxywhenthat isxy reach itsi orvaluexbbbbbb 16.More on Quadratic Modelsn Suppose that the coefficient on x i

9、s positive and the coefficient on x2 is negativen Then y is increasing in x at first, but will eventually turn around and be decreasing in x21*212at be willpoint turning the0 and 0For bbbbx17.More on Quadratic Modelsn Suppose that the coefficient on x is negative and the coefficient on x2 is positiv

10、en Then y is decreasing in x at first, but will eventually turn around and be increasing in x0 and 0 when as same theis which ,2at be willpoint turning the0 and 0For 2121*21bbbbbbx18.How to describe decreasing effect19.How to describe increasing effect20.二次项模型案例：污染对住房价格的影响（数据名：HPRICE2）2log(price) 13

11、.39 0.902log(nox) 0.087log(dist) 0.545rooms 0.0620.048roomsstratio （0.57）（0.115） （0.043） （0.165） （0.013） （0.006）N=506 ,R2=0.603 log(price)(0.54520.062) room sroom s% (price)100( 0.5452 0.062)rooms rooms 100( 0.5452 12.4)rooms rooms 在room*=0.545/(2*0.062)4.4的右边，增加一个卧室对价格的百分比变化具有递增的影响。21.n比如，rooms从5增加

12、到6会导致价格提高约为 -54.5+12.4*5=7.5%; rooms从6增加到7会导致价格提高约为-54.5+12.4*6=19.9%。这是一个很强的递增影响。STATA命令语句：gen rooms2=rooms*rooms gen ldist=log(dist)reg lprice lnox ldist rooms rooms2 stratiodisplay -1*_brooms/(2*_brooms2)（求转折点）display 100*(_brooms+2*_brooms2*6（求rooms从6增加到7会导致价格提高的百分比）。22.Interaction Terms（有交互作用项的

13、模型）（有交互作用项的模型）n For a model of the form y = b0 + b1x1 + b2x2 + b3x1x2 + u we cant interpret b1 alone as measuring the change in y with respect to x1, we need to take into account b3 as well, since 132112, so to summarizethe effect of on we typicallyevaluate the above at yxxxyxbb23.n交互效应通常需将模型重新参数化：n

14、原模型：01 1223 12yxxx xbbbb2b 是x2=0时，X2对y的偏效应，这通常没有什么意义，我们转而将模型重新参数化为：01 1221122()(x)yxxx其中， 和 分别是x1和x2的总体均值。很容易计算出：我们立即得到在均值的偏效应。 122231bb 24.nExample: Effects of Attendance on Final Exam Performance（数据名：ATTEND.DTA）natndrte系数为负，是否意味着听课对期末考试分数具有负面影响？ b1仅考虑了priGPA=0时的影响。natndrte和priGPAatndrte系数估计值t值不显著，

15、是否意味着两者对期末考试分数无影响？ F检验的p值为0.014.25.nAtndrte对stndfnl的偏效应： 其含义是：在priGPA的平均水平（2.59）上，atndrte提高10个百分点，使stndfnl比期末考试平均分数高出0.078倍。 0.00670.0056 2.590.00781161162.59,2.59bbbb01660612.592.59stndfnlatndrtepriGPA atndrteuatndrtepriGPAatndrteubbbbb10.0078 0.00263t26.STATA命令语句：nsum priGPAngen priGPA2=priGPA*pri

16、GPAngen ACT2=ACT*ACTngen priatn=priGPA*atndrtenreg stndfnl atndrte priGPA priGPA2 ACT2 priatn27.6.3 拟合优度和回归元选择的进一步探讨n Recall that the R2 will always increase as more variables are added to the modelnThe adjusted R2 takes into account the number of variables in a model, and may decrease 2222221111111

17、1111uySSR nRSST nSSRnkRSSTnSSTnRnnk 28.n调整R方的作用：为在一个模型中另外增加自变量施加了惩罚。 随着一个新的自变量加入回归方程，SSR下降，但回归中的自由度df=n-k-1也下降。因此，SSR/(n-k-1)可能上升，也可能下降。n作为一个结论有： 在回归中增加一个新变量，当且仅当新变量的t统计量在绝对值上大于1，调整R方才会有所提高； 在回归中增加一组变量时，当且仅当这组新变量联合显著性的F统计量大于1，调整R方才会有所提高。29.Adjusted R-Squared (cont)n Its easy to see that the adjusted

18、 R2 is just (1 R2)(n 1) / (n k 1), but most packages will give you both R2 and adj-R2n You can compare the fit of 2 models (with the same y) by comparing the adj-R2n You cannot use the adj-R2 to compare models with different ys (e.g. y vs. ln(y)30.Using adjusted R-squared to choose between nonnested

19、 models.20.6211R 20.6226R 220.061,0.03RR220.148,0.09RR31.391732982salarySST66.72lsalarySST32.Controlling too many factors in regression analysis回归分析中控制了过多的因素nImportant not to fixate too much on adj-R2 and lose sight of theory and common sensenIf economic theory clearly predicts a variable belongs, g

20、enerally leave it innDont want to include a variable that prohibits a sensible interpretation of the variable of interest remember ceteris paribus interpretation of multiple regression33.u在研究啤酒税对交通死亡率影响的回归模型中， 是否应该将人均啤酒消费量变量包括在模型之中？u在保持beercons不变的情况下，死亡率因tax提高 1个百分点而导致的差异。这一说法是否有意义？34.Adding regress

21、ors to reduce the error of variance-增加回归元以减少误差方差 n在回归中增加一个新的自变量会加剧多重共线性问题；另一方面，从误差项中取出一些因素作为解释变量可以减少误差方差。n应该将那些影响y而又与所有我们关心的自变量都无关的自变量包括进来。35.6.4 预测和残差分析n Suppose we want to use our estimates to obtain a specific prediction.n First, suppose that we want an estimate of E(y|x1=c1,xk=ck) = 0 = b0+b1c1+

22、 + bkckn This is easy to obtain by substituting the xs in our estimated model with cs , but what about a standard error?n Really just a test of a linear combination36.Predictions (cont)n Can rewrite as b0 = 0 b1c1 bkckn Substitute in to obtain y = 0 + b1 (x1 - c1) + + bk (xk - ck) + u n So, if you r

23、egress yi on (xij - cij) the intercept will give the predicted value and its standard errorn Note that the standard error will be smallest when the cs equal the means of the xs37.Example: CI for predicted college GPA2.7 1.96 0.020: 2.66 2.74CI01200,030,05satsathsperchsperchsizehsize38.Predictions (c

24、ont)n This standard error for the expected value is not the same as a standard error for an new outcome on yn We need to also take into account the variance in the unobserved error. 39. 00000001 1000001 10000110001 10000012200202000 , so kkkkkkkkeyyxxuyyxxE yEExExxxE eE uVar eVar yVar usVar yye ese

25、yebbbbbbbbbbbbbj的方差有两个来源：第一个是 的抽样误差，来自我们对20var1jjsecnynb的估计。因为，所以与成比例。第二个是总体误差的方差，它不随样本容量的变化而变化。40. 00000CLM10.95,95%j000.0250.025ueese enktP -tese etyb在经典线性模型（）假定下，和都是正态分布，所以也是正态分布的。服从一个自由度为的 分布。则 的一个的预测区间： 000.025ytse e41.42.Prediction interval 0025.00000100for interval prediction 95% a have we gi

26、ven that so ,esetyyyyeteseeknnUsually the estimate of s2 is much larger than the variance of the prediction, thusnThis prediction interval will be a lot wider than the simple confidence interval for the prediction43.Residual AnalysisnInformation can be obtained from looking at the residuals (i.e. pr

27、edicted vs. observed)nExample: Regress price of cars on characteristics big negative residuals indicate a good dealnExample: Regress average earnings for students from a school on student characteristics big positive residuals indicate greatest value-addediiiuyy44.n例如，HPRICE1.RAW的住房价格模型中。01238181811

28、20.206pricelotsizesqrftbdrmsuupricepricebbbb 45.46.Predicting y in a log modelnSimple exponentiation of the predicted ln(y) will underestimate the expected value of ynInstead need to scale this up by an estimate of the expected value of exp(u)47. 01 101 101 101 1200121logexpexpexp|expexpIn this case can predict y as followsexp2 exp