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1、Multiple Regression Analysis y b0 b1x1 b2x2 bkxk u 1 Estimation 1Economics 20 Prof Anderson Parallels with Simple Regression w b0 is still the intercept w b1 to bk all called slope parameters w u is still the error term or disturbance w Still need to make a zero conditional mean assumption so now as
2、sume that w E u x1 x2 xk 0 w Still minimizing the sum of squared residuals so have k 1 first order conditions 2Economics 20 Prof Anderson Interpreting Multiple Regression 3Economics 20 Prof Anderson A Partialling Out Interpretation 4Economics 20 Prof Anderson Partialling Out continued w Previous equ
3、ation implies that regressing y on x1 and x2 gives same effect of x1 as regressing y on residuals from a regression of x1 on x2 w This means only the part of xi1 that is uncorrelated with xi2 are being related to yi so we re estimating the effect of x1 on y after x2 has been partialled out 5Economic
4、s 20 Prof Anderson Simple vs Multiple Reg Estimate 6Economics 20 Prof Anderson Goodness of Fit 7Economics 20 Prof Anderson Goodness of Fit continued How do we think about how well our sample regression line fits our sample data w Can compute the fraction of the total sum of squares SST that is expla
5、ined by the model call this the R squared of regression w R2 SSE SST 1 SSR SST 8Economics 20 Prof Anderson Goodness of Fit continued 9Economics 20 Prof Anderson More about R squared w R2 can never decrease when another independent variable is added to a regression and usually will increase w Because
6、 R2 will usually increase with the number of independent variables it is not a good way to compare models 10Economics 20 Prof Anderson Assumptions for Unbiasedness Population model is linear in parameters y b0 b1x1 b2x2 bkxk u We can use a random sample of size n xi1 xi2 xik yi i 1 2 n from the popu
7、lation model so that the sample model is yi b0 b1xi1 b2xi2 bkxik ui E u x1 x2 xk 0 implying that all of the explanatory variables are exogenous None of the x s is constant and there are no exact linear relationships among them 11Economics 20 Prof Anderson Too Many or Too Few Variables w What happens
8、 if we include variables in our specification that don t belong w There is no effect on our parameter estimate and OLS remains unbiased wWhat if we exclude a variable from our specification that does belong w OLS will usually be biased 12Economics 20 Prof Anderson Omitted Variable Bias 13Economics 2
9、0 Prof Anderson Omitted Variable Bias cont 14Economics 20 Prof Anderson Omitted Variable Bias cont 15Economics 20 Prof Anderson Omitted Variable Bias cont 16Economics 20 Prof Anderson Summary of Direction of Bias Corr x1 x2 0 Corr x1 x2 0Positive biasNegative bias b2 0Negative biasPositive bias 17Ec
10、onomics 20 Prof Anderson Omitted Variable Bias Summary w Two cases where bias is equal to zero nb2 0 that is x2 doesn t really belong in model nx1 and x2 are uncorrelated in the sample w If correlation between x2 x1 and x2 y is the same direction bias will be positive w If correlation between x2 x1
11、and x2 y is the opposite direction bias will be negative 18Economics 20 Prof Anderson The More General Case w Technically can only sign the bias for the more general case if all of the included x s are uncorrelated w Typically then we work through the bias assuming the x s are uncorrelated as a usef
12、ul guide even if this assumption is not strictly true 19Economics 20 Prof Anderson 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 easier to think about this v
13、ariance under an additional assumption so Assume Var u x1 x2 xk s2 Homoskedasticity 20Economics 20 Prof Anderson Variance of OLS cont w Let x stand for x1 x2 xk w Assuming that Var u x s2 also implies that Var y x s2 w The 4 assumptions for unbiasedness plus this homoskedasticity assumption are know
14、n as the Gauss Markov assumptions 21Economics 20 Prof Anderson Variance of OLS cont 22Economics 20 Prof Anderson Components of OLS Variances w The error variance a larger s2 implies a larger variance for the OLS estimators w The total sample variation a larger SSTj implies a smaller variance for the
15、 estimators w Linear relationships among the independent variables a larger Rj2 implies a larger variance for the estimators 23Economics 20 Prof Anderson Misspecified Models 24Economics 20 Prof Anderson Misspecified Models cont w While the variance of the estimator is smaller for the misspecified mo
16、del unless b2 0 the misspecified model is biased w As the sample size grows the variance of each estimator shrinks to zero making the variance difference less important 25Economics 20 Prof Anderson Estimating the Error Variance We don t know what the error variance s2 is because we don t observe the errors ui What we observe are the residuals i We can use the residuals to form an estimate of the error variance 26Economics 20 Prof Anderson Error Variance Estimate cont w df n k 1 or df n k 1 w df
