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Lecture 7 February 22, 2007 Simple Linear Regression What do the ols estimator s tell us? Sample covariance between x and y Sample variance of x The slope coefficient is of primary interest, as: y = 0 + 1x dy = 1 dx Mathpnl.dta When will the estimator be undefined? When denominator is zero we have a problem: Suppose Denominator becomes: Algebraic Properties of OLS u = 0 i x =0 iui The point ( x , y ) is always on the OLS regression line. Sum of Squares We know that yi = yi + ui It can be shown (see p 42) that Goodness of Fit Start with the property that SST=SSE+SSR: Define R-squared: Assumptions of OLS SLR1: Linear in Parameters y = 0 + 1x + u SLR2: Random Sampling We are sampling {yi, xi} from the population model SLR3: Sample Variation in the Explanatory Variable We must have xi c Thinking about the zero conditional mean assumption Assumptions of OLS (cont) SLR4: Zero Conditional Mean Error has an expected value of zero, given x: E(u|x) +ui -ui Why we can assume E(u)=0 In the simple linear regression model y = 0 + 1x + u Suppose that E (u ) 0 If we let 0 = E (u ) we can show that the model can be written with a new intercept and error term, where the new error term has an expected value of zero. Unbiasedness of OLS Using assumptions SLR1-SLR4, we can prove that the ols estimator s are unbiased: E ( 0) = 0, E ( 1) = 1 See proof in pdf file: Unbiasedness of ols estimator s How likely is it that one of our assumptions may fail and we wind up with biased estimators? SLR1: Transformation of Functions Cobb-Douglas production function: CES production function: Assumption concerning variance of ols estimator s SLR5: Homoskedasticity- constant variance versus Heteroskedasticity Homoskedasticity Under assumptions SLR1-SLR5 Using these five assumptions, we can say that the ols estimator s are the best linear unbiased estimators Best Linear Unbiased This is known as the Gauss-Markov Theorem Sampling Variance of ols estimator s Under assumptions SLR1-SLR5: Error variance Total variation in x s Problem? We don t often know 2.

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