第4章n 多元回归估计与假设检验.ppt
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1、1,第4章 多元回归分析:估计与假设检验 Multiple Regression Analysis,y = b0 + b1x1 + b2x2 + . . . bkxk + u Estimation and Inference,2,Parallels with Simple Regression,Yi = b0 + b1Xi1 + b2Xi2 + . . . bkXik + ui b0 is still the intercept b1 to bk all called slope parameters, also called partial regression coefficients a
2、nd any coefficient bj denote the change of Y with the changes of Xj as all the other independent variables fixed. u is still the error term (or disturbance) Still minimizing the sum of squared residuals, so have k+1 first order conditions,3,Obtaining OLS Estimates,4,Obtaining OLS Estimates, cont.,Th
3、e above estimated equation is called the OLS regression line or the sample regression function (SRF) the above equation is the estimated equation, is not the really equation. The really equation is population regression line which we dont know. We only estimate it. So, using a different sample, we c
4、an get another different estimated equation line. The population regression line is,5,Interpreting Multiple Regression,6,An Example (Wooldridge, p76),The determination of wage (dollars per hour), wage: Years of education, educ Years of labor market experience, exper Years with the current employer,
5、tenure The relationship btw. wage and educ, exper, tenure: wage=b0+b1educ+b2exper+b3tenure+u The estimated equation as below: wage=-2.873+0.599educ+0.022exper+0.169tenure,7,A “Partialling Out” Interpretation,8,A “Partialling Out” Interpretation,9,“Partialling Out” continued,Previous equation implies
6、 that regressing Y on X1 and X2 gives same effect of X1 as regressing Y on residuals from a regression of X1 on X2 This means only the part of Xi1 that is uncorrelated with Xi2 are being related to Yi , so were estimating the effect of X1 on Y after X2 has been “partialled out”,10,The wage determina
7、tions,The estimated equation as below: wage=-2.873+0.599educ+0.022exper+0.169tenure Now, we first regress educ on exper and tenure to patial out the exper and tenures effects. Then we regress wage on the residuals of educ on exper and tenure. Whether we get the same result.? educ=13.575-0.0738exper+
8、0.048tenure denote residuals resid wage=5.896+0.599resid We can see that the coefficient of resid is the same of the coefficien of the variable educ in the first estimated equation. So is in the second equation.,11,Goodness-of-Fit: R2,12,Goodness-of-Fit,13,Goodness-of-Fit (continued),How do we think
9、 about how well our sample regression line fits our sample data? Can compute the fraction of the total sum of squares (SST) that is explained by the model, call this the R-squared of regression R2 = ESS/TSS = 1 RSS/TSS,14,Goodness-of-Fit (continued),15,More about R-squared,R2 can never decrease when
10、 another independent variable is added to a regression, and usually will increase Because R2 will usually increase with the number of independent variables, it is not a good way to compare models,16,An Example,Using wage determination model to show that when we add another new independent variable w
11、ill increase the value of R2.,17,Adjusted R-Squared,R2 is simply an estimate of how 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
12、decrease,18,Adjusted R-Squared (cont),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
13、with different 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.,19,Assumptions for Unbiasedness,20,Assumptions for Unbiasedness,Po
14、pulation model is linear in parameters: Y = b0 + b1X1 + b2X2 + bkXk + u We can use a sample of size n, (Xi1, Xi2, Xik, Yi): i=1, 2, , n, from the population model, so that the sample model is Yi = b0 + b1Xi1 + b2Xi2 + bkXik + ui Cov(uXi)=0, E(uXi)=0 , i=1, 2, , n. E(u|X1, X2, Xk) = 0, implying that
15、all of the explanatory variables are exogenous. E(u|X)=0, where X= (X1, X2, Xk), which will reduce to E(u)=0 if independent variables X are not random variables. None of the Xs is constant, and there are no exact linear relationships among them.,The new additional assumption.,21,About multicollinear
16、ity,It does allow the independent variables to be correlated; they just cannot be perfectly linear correlated. Student performance: colGPA=b0+b1 hsGPA+b2ACT+ b3 skipped + u Consumption function: consum=b0+b1inc+b2inc2+u But, the following is invalid: log(consum)=b0+b1log(inc)+b2log(inc2)+u In this c
17、ase, we can not estimate the regression coefficients b1, b2 .,22,Unbiasedness of OLS estimation,Under the three assumptions above, we can get,23,Too Many or Too Few Variables,24,Too Many or Too Few Variables,What happens if we include variables in our specification that dont belong? There is no effe
18、ct on our parameter estimate, and OLS remains unbiased What if we exclude a variable from our specification that does belong? OLS will usually be biased,25,Omitted Variable Bias,26,Omitted Variable Bias (cont),27,Omitted Variable Bias (cont),28,Omitted Variable Bias (cont),There are two cases where
19、the estimated parameter is unbiased: If b2=0, so that X2 does not appear in the true model If tilde of d1=0, the tilde b1 is unbiased for b1,29,Summary of Direction of Bias,30,An example,The estimated equation as below: wage=-3.391+0.644educ+0.070exper the correlation between educ and exper corr(edu
20、c, exper)=-0.2295 Estimate them again without exper wage=-0.905+0.541educ,31,Omitted Variable Bias Summary,Two cases where bias is equal to zero b2 = 0, that is X2 doesnt really belong in model X1 and X2 are uncorrelated in the sample If correlation between X2 , X1 and X2 , Y is the same direction,
21、bias will be positive If correlation between X2 , X1 and X2 , Y is the opposite direction, bias will be negative,32,The More General Case,When there are multiple regressors in the estimated model, to derive the omitted variable bias is more difficult. Because correlation btw a single explanatory var
22、iable and the error generally results in all OLS estimators being biased.,33,Variance of the OLS Estimators,34,Variance of the OLS Estimators,Now we know that the sampling distribution of our estimate is centered around the true parameter Want to think about how spread out this distribution is Much
23、easier to think about this variance under an additional assumption, so Assume Var(u|X1, X2, Xk) = s2 , or Var(u)= s2 (Homoskedasticity),35,Variance of OLS (cont),Yi = b0 + b1Xi1 + b2Xi2 + . . . bkXik + ui Let X stand for (X1, X2,Xk) Assuming that Var(u|X) = s2 also implies that Var(Y| X) = s2, we ca
24、n rewrite them as Var(u)= s2 and Var(Y)= s2 if X are not random variables.,36,Variance of OLS (cont),VIF,37,Variance of OLS (cont),38,Components of OLS Variances,The error variance: a larger s2 implies a larger variance for the OLS estimators The total sample variation: a larger TSSj implies a small
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