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1、第五章多元回归分析论Multiple Regression Analysis y = b0 + b1x1 + b2x2 + . . . bkxk + u 4. Further Issues1多元回归分析进一步讨论课件Redefining Variables: An Examplethe determinations of infant birth weightnVariablesqbwghtkg, child birth weight in kilogramsqbwghtg, child birth weight in gramsqbwghtjin, child birth weight in
2、 jinqcigs, number of cigarettes the mother smoked per day while pregnantqpacks, packs of cigarettes the mother smoked per day while pregnant and 1 packs=20 cigsqfaminc, annual family incomenModelqy=b0+b1x+b2faminc+uqy stand for bwghtkg, bwghtg, bwghtkjin; x stand for cigs or packs2多元回归分析进一步讨论课件Redef
3、ining Variables, cont.Dependent vars(1) bwghtkg(2) bwghtg(3) bwghtjin(4) bwghtjincigs-0.0131492(0.0025985)-5.06-13.1492(2.5985)-5.06-0.0262984(0.005197)-5.06packs-0.5259676(0.1039397)-5.06faminc0.0026322(0.0008252)3.182.6322(0.8282)3.180.0052644(0.0016564)3.180.0052644(0.0016564)3.18Intercept3.31911
4、41(0.029764)111.513319.141(29.7649)111.516.638282(0.595298)111.516.638282(0.595298)111.51Observations1388138813881388R20.02980.02980.02980.0298SSR448.85423444888542331795.416941795.41694S.E0.56928569.281.13861.1386Changing the scale of the y variable will lead to a corresponding change in the scale
5、of the coefficients and standard errors, so no change in the significance or interpretationnChanging the scale 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多元回归分析进一步讨论课件Redefining Variables , cont.4多元回
6、归分析进一步讨论课件Redefining Variables , cont.nChanging the scale of the y variable will lead toqa corresponding change in the scale of the coefficients and standard errors, qt-stats and R2 is not changednChanging the scale of one x variable will lead to qa change in the scale of that coefficient and standa
7、rd error, qt-stats and R2 is not changed5多元回归分析进一步讨论课件Standardized Coefficients (Beta Coefficients)nOccasional 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 mea
8、n and divide by standard deviationnCoefficient reflects standard deviation of y for a one standard deviation change in x 6多元回归分析进一步讨论课件Standardized Coefficients, ExampleThe determinations of wage, wage1.rawnThe population model in level-level modelqwage=b0+b1educ+b2exper+b3tenure+unEstimating the st
9、andardized model qzwge=0.449zeduc+0.082zexper+0.331ztenurenWhat the meaning of the estimated parameters?qThe estimated coefficient of zeduc means when the educ changed one standard deviation, the wage will change 0.449 standard deviation.nStata commandqreg wage educ exper tenure, beta7多元回归分析进一步讨论课件F
10、unctional Formn 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 natural log of x, y or bothn Can use quadratic forms of xn Can use interactions of x variables8多元回归分析进一步讨论课件Interpret
11、ation of Log ModelsnIf the model is ln(y) = b0 + b1ln(x) + uqb1 is the elasticity of y with respect to xnIf the model is ln(y) = b0 + b1x + uqb1 is approximately the percentage change in y given a 1 unit change in x nIf the model is y = b0 + b1ln(x) + uqb1 is approximately the change in y for a 100
12、percent change in xnExample: the determinations of wagesqlog(wage)=0.084+0.094educ+0.109log(exper)+0.018tenure9多元回归分析进一步讨论课件Why use log models?nLog models are invariant to the scale of the variables since measuring percent changesnThey give a direct estimate of elasticitynFor models with y 0, the co
13、nditional distribution is often heteroskedastic or skewed, while ln(y) is much less sonThe distribution of ln(y) is more narrow, limiting the effect of outliers10多元回归分析进一步讨论课件Some Rules of ThumbnWhat types of variables are often used in log form?qDollar amounts that must be positive, such as wages,
14、salaries, firm sales, firm market valueqVery large variables, such as population, number of employees, school enrollmentnWhat types of variables are often used in level form?qVariables measured in years, such as education, experience, tenure, ageqVariables that are a proportion or percent, such as u
15、nemployment rate, interest rate, roe, roa11多元回归分析进一步讨论课件Quadratic ModelsnFor 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, since12多元回归分析进一步讨论课件More on Quadratic ModelsnSuppose that the c
16、oefficient on x is positive and the coefficient on x2 is negativenThen y is increasing in x at first, but will eventually turn around and be decreasing in xExample: Kuznetz CurveGini=b0+b1gdppc+b2gdppc2+u13多元回归分析进一步讨论课件More on Quadratic ModelsnSuppose that the coefficient on x is negative and the co
17、efficient on x2 is positivenThen y is decreasing in x at first, but will eventually turn around and be increasing in xyxx*For example: the cost functionC(Q)=b0 +b1Q+b2Q2+u14多元回归分析进一步讨论课件More on Quadratic Models, ExampleEffects of Pollution and House PricesnVariablesqprice, median housing price;qnox,
18、 the amount of nitrogen oxide in the air, in parts per million;qdist, a weighted distance of the community from five employment centers, in miles;qrooms, the average number of rooms in houses in the communityqStratio, the average student-teacher ratio of schools in the community.nThe estimated model
19、qlog(prie)=13.39-0.902log(nox)-0.087log(dis)-0.545rooms+0.062rooms 2-0.048stratioq (0.57) (0.115) (0.043) (0.0165) (0.013) (0.006)q n=506 R2=0.603qThe estimated turn point is =0.545/(2*0.062)=4.44.4roomslog(price)15多元回归分析进一步讨论课件Interaction TermsnFor a model of the form qy = b0 + b1x1 + b2x2 + b3x1x2
20、 + u nwe cant interpret b1 alone as measuring the change in y with respect to x1, we need to take into account b3 as well, since Interaction term16多元回归分析进一步讨论课件Interaction Terms, cont.Example: wage determinationsnModel with interaction terms of educ and tenureqwage=b0+b1 educ+b2exper+ b3 tenure+b4ed
21、uctenure+unEstimated model with interaction terms of educ and tenureqwge=-1.097+0.457educ+0.021exper-0.097tenure+0.022eductenureq (0.861) (0.063) (0.012) (0.074) (0.006)q n=526 R2=0.3247nThe effect of educ on wage at the mean of tenure isqdwage/deduc=0.457+0.022tenure=0.457+0.0225.105=0.57qWhether t
22、he estimate 0.57 is statistically different from zero? That is, whether b1+b4tenure (b1+b45.105) is different from zero?qwage=b0+(b1 +b45.105)educ+b2exper+ b3 tenure+b4educ(tenure-5.105)+uqwge=-1.097+0.570educ+0.021exper-0.097tenure+0.022eductenureq (0.861) (0.051) (0.012) (0.074) (0.006)q n=526 R2=
23、0.3247qt=11.12, so it different from zero significantly17多元回归分析进一步讨论课件Adjusted R-SquarednR2 is simply an estimate of how much variation in y is explained by x1, x2,xk. That is,nRecall that the R2 will always increase as more variables are added to the modelnThe adjusted R2 takes into account the num
24、ber of variables in a model, and may decrease18多元回归分析进一步讨论课件Adjusted R-Squared (cont)nIts easy to see that the adjusted 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-R2qwge=-3.391+0.644
25、educ+0.070exper adj-R2=0.2222qwge=-2.222+0.569educ+0.190tenure adj-R2=0.2992n You cannot use the adj-R2 to compare models with different ys (e.g. y vs. ln(y)qwge=-3.391+0.644educ+0.070exper adj-R2=0.2222qlog(wge)=0.404+0.087educ+0.026exper adj-R2=0.3059qBecause the variance of the dependent variable
26、s is different, the comparation btw them make no sense.19多元回归分析进一步讨论课件Goodness of FitnImportant not to fixate too much on adj-R2 and lose sight of theory and common sensenIf economic theory clearly predicts a variable belongs, generally leave it innDont want to include a variable that prohibits a se
27、nsible interpretation of the variable of interest remember ceteris paribus interpretation of multiple regression20多元回归分析进一步讨论课件Standard Errors for PredictionsnSuppose we want to use our estimates to obtain a specific prediction?nFirst, suppose that we want an estimate of E(y|x1=c1,xk=ck) = q0 = b0+b
28、1c1+ + bkcknThis is easy to obtain by substituting the xs in our estimated model with cs , but what about a standard error?nReally just a test of a linear combination21多元回归分析进一步讨论课件Predictions (cont)nThe original regression modelqwge=-2.8727+0.5990educ+0.02234exper+0.1693tenureq (0.7290) (0.0513) (0
29、.0121) (0.0216) q n=526 R2=0.3064nWe want predict the wages of educ=exper=tenure=12, we can easily put the value in the above estimated equation, and get wge=6.614, but we dont know the standard error of the predicted valuenNow, we reg wage on (educ-12), (exper-12), (tenure-12)qwge=6.614+0.5990(educ
30、-12)+0.02234(exper-12)+0.1693(tenure-12)q (0.2368) (0.0513) (0.0121) (0.0216) q n=526 R2=0.3064 22多元回归分析进一步讨论课件Predictions (cont)nThis standard error for the expected value is not the same as a standard error for an outcome on ynWe need to also take into account the variance in the unobserved error.
31、 Let the prediction error be23多元回归分析进一步讨论课件Prediction interval24多元回归分析进一步讨论课件Residual AnalysisnInformation can be obtained from looking at the residuals (i.e. predicted vs. observed)nExample: Regress price of cars on characteristics big negative residuals indicate a good dealnExample: Regress averag
32、e earnings for students from a school on student characteristics big positive residuals indicate greatest value-added25多元回归分析进一步讨论课件Predicting y in a log model26多元回归分析进一步讨论课件Predicting y in a log modelnSTATA commandqreg y x1 x2 xkqpredict logybar, xbqgen mbar=exp(logybar)qreg y mbarqgen ybar=_bmbar*
33、mbarnExample(wage1.raw)qlog(wage)=b0+b1educ+b2exper+b3tenure+u27多元回归分析进一步讨论课件Comparing log and level modelsnA by-product of the previous procedure is a method to compare a model in logs with one in levels. nTake the fitted values from the auxiliary regression, and find the sample correlation between this and y nCompare the R2 from the levels regression with this correlation squarednExample(wage1.raw)nwage=b0+b1educ+b2exper+b3tenure+unlog(wage)=b0+b1educ+b2exper+b3tenure+u28多元回归分析进一步讨论课件Homework29多元回归分析进一步讨论课件
