第5章 多元回归分析：进一步讨论.ppt

1、1,第五章多元回归分析论Multiple Regression Analysis,y = b0 + b1x1 + b2x2 + . . . bkxk + u 4. Further Issues,2,Redefining Variables: An Examplethe determinations of infant birth weight,Variables bwghtkg, child birth weight in kilograms bwghtg, child birth weight in grams bwghtjin, child birth weight in jin cigs

2、, number of cigarettes the mother smoked per day while pregnant packs, packs of cigarettes the mother smoked per day while pregnant and 1 packs=20 cigs faminc, annual family income Model y=b0+b1x+b2faminc+u y stand for bwghtkg, bwghtg, bwghtkjin; x stand for cigs or packs,3,Redefining Variables, con

3、t.,Changing 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 interpretation,Changing the scale of one x variable will lead to a change in the scale of that coefficient and standard error, so no c

4、hange in the significance or interpretation,4,Redefining Variables , cont.,5,Redefining Variables , cont.,Changing the scale of the y variable will lead to a corresponding change in the scale of the coefficients and standard errors, t-stats and R2 is not changed Changing the scale of one x variable

5、will lead to a change in the scale of that coefficient and standard error, t-stats and R2 is not changed,6,Standardized Coefficients (Beta Coefficients),Occasional youll see reference to a “standardized coefficient” or “beta coefficient” which has a specific meaning Idea is to replace y and each x v

6、ariable with a standardized version i.e. subtract mean and divide by standard deviation Coefficient reflects standard deviation of y for a one standard deviation change in x,7,Standardized Coefficients, ExampleThe determinations of wage, wage1.raw,The population model in level-level model wage=b0+b1

7、educ+b2exper+b3tenure+u Estimating the standardized model zwge=0.449zeduc+0.082zexper+0.331ztenure What the meaning of the estimated parameters? The estimated coefficient of zeduc means when the educ changed one standard deviation, the wage will change 0.449 standard deviation. Stata command reg wag

8、e educ exper tenure, beta,8,Functional Form,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 parameters Can take the natural log of x, y or both Can use quadratic forms of x Can use interactions of x variabl

9、es,9,Interpretation of Log Models,If the model is ln(y) = b0 + b1ln(x) + u b1 is the elasticity of y with respect to x If the model is ln(y) = b0 + b1x + u b1 is approximately the percentage change in y given a 1 unit change in x If the model is y = b0 + b1ln(x) + u b1 is approximately the change in

10、 y for a 100 percent change in x Example: the determinations of wages log(wage)=0.084+0.094educ+0.109log(exper)+0.018tenure,10,Why use log models?,Log models are invariant to the scale of the variables since measuring percent changes They give a direct estimate of elasticity For models with y 0, the

11、 conditional distribution is often heteroskedastic or skewed, while ln(y) is much less so The distribution of ln(y) is more narrow, limiting the effect of outliers,11,Some Rules of Thumb,What types of variables are often used in log form? Dollar amounts that must be positive, such as wages, salaries

12、, firm sales, firm market value Very large variables, such as population, number of employees, school enrollment What types of variables are often used in level form? Variables measured in years, such as education, experience, tenure, age Variables that are a proportion or percent, such as unemploym

13、ent rate, interest rate, roe, roa,12,Quadratic Models,For 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, since,13,More on Quadratic Models,Suppose that the coefficient on x is positive an

14、d the coefficient on x2 is negative Then y is increasing in x at first, but will eventually turn around and be decreasing in x,Example: Kuznetz Curve Gini=b0+b1gdppc+b2gdppc2+u,14,More on Quadratic Models,Suppose that the coefficient on x is negative and the coefficient on x2 is positive Then y is d

15、ecreasing in x at first, but will eventually turn around and be increasing in x,y,x,x*,For example: the cost function C(Q)=b0 +b1Q+b2Q2+u,15,More on Quadratic Models, ExampleEffects of Pollution and House Prices,Variables price, median housing price; nox, the amount of nitrogen oxide in the air, in

16、parts per million; dist, a weighted distance of the community from five employment centers, in miles; rooms, the average number of rooms in houses in the community Stratio, the average student-teacher ratio of schools in the community. The estimated model log(prie)=13.39-0.902log(nox)-0.087log(dis)-

17、0.545rooms+0.062rooms 2-0.048stratio (0.57) (0.115) (0.043) (0.0165) (0.013) (0.006) n=506 R2=0.603 The estimated turn point is =0.545/(2*0.062)=4.4,16,Interaction Terms,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

18、x1, we need to take into account b3 as well, since,17,Interaction Terms, cont.Example: wage determinations,Model with interaction terms of educ and tenure wage=b0+b1 educ+b2exper+ b3 tenure+b4eductenure+u Estimated model with interaction terms of educ and tenure wge=-1.097+0.457educ+0.021exper-0.097

19、tenure+0.022eductenure (0.861) (0.063) (0.012) (0.074) (0.006) n=526 R2=0.3247 The effect of educ on wage at the mean of tenure is dwage/deduc=0.457+0.022tenure=0.457+0.0225.105=0.57 Whether the estimate 0.57 is statistically different from zero? That is, whether b1+b4tenure (b1+b45.105) is differen

20、t from zero? wage=b0+(b1 +b45.105)educ+b2exper+ b3 tenure+b4educ(tenure-5.105)+u wge=-1.097+0.570educ+0.021exper-0.097tenure+0.022eductenure (0.861) (0.051) (0.012) (0.074) (0.006) n=526 R2=0.3247 t=11.12, so it different from zero significantly,18,Adjusted R-Squared,R2 is simply an estimate of how

21、much variation in y is explained by x1, x2,xk. That is, Recall that the R2 will always increase as more variables are added to the model The adjusted R2 takes into account the number of variables in a model, and may decrease,19,Adjusted R-Squared (cont),Its easy to see that the adjusted R2 is just (

22、1 R2)(n 1) / (n k 1), but most packages will give you both R2 and adj-R2 You can compare the fit of 2 models (with the same y) by comparing the adj-R2 wge=-3.391+0.644educ+0.070exper adj-R2=0.2222 wge=-2.222+0.569educ+0.190tenure adj-R2=0.2992 You cannot use the adj-R2 to compare models with differe

23、nt ys (e.g. y vs. ln(y) wge=-3.391+0.644educ+0.070exper adj-R2=0.2222 log(wge)=0.404+0.087educ+0.026exper adj-R2=0.3059 Because the variance of the dependent variables is different, the comparation btw them make no sense.,20,Goodness of Fit,Important not to fixate too much on adj-R2 and lose sight o

24、f theory and common sense If economic theory clearly predicts a variable belongs, generally leave it in Dont want to include a variable that prohibits a sensible interpretation of the variable of interest remember ceteris paribus interpretation of multiple regression,21,Standard Errors for Predictio

25、ns,Suppose we want to use our estimates to obtain a specific prediction? First, suppose that we want an estimate of E(y|x1=c1,xk=ck) = q0 = b0+b1c1+ + bkck This is easy to obtain by substituting the xs in our estimated model with cs , but what about a standard error? Really just a test of a linear c

26、ombination,22,Predictions (cont),The original regression model wge=-2.8727+0.5990educ+0.02234exper+0.1693tenure (0.7290) (0.0513) (0.0121) (0.0216) n=526 R2=0.3064 We 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 w

27、e dont know the standard error of the predicted value Now, we reg wage on (educ-12), (exper-12), (tenure-12) wge=6.614+0.5990(educ-12)+0.02234(exper-12)+0.1693(tenure-12) (0.2368) (0.0513) (0.0121) (0.0216) n=526 R2=0.3064,23,Predictions (cont),This standard error for the expected value is not the same as a standard error for an outcome on y We need to also take into account the variance in the unobserved error. Let the prediction error be,24,Prediction interval,25,Residual Analysis,Information can be obtained from lo