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Unformatted text preview: Introduction Properties of OLS Goodness of Fit Linearity Assumption Econ 103, UCLA, Fall 2010 Introduction to Econometrics Lecture 4: Simple Regression (cont.) Sarolta Lacz October 5, 2010 Introduction Properties of OLS Goodness of Fit Linearity Assumption Introduction We consider the linear regression model Y i = + 1 X i + u i The OLS estimators are: 1 = i ( X i X )( Y i Y ) i ( X i X ) 2 = s XY s 2 X = Y 1 X Introduction Properties of OLS Goodness of Fit Linearity Assumption Introduction/2 Outline: Properties of OLS (just like we did for Y , an estimator of Y ) Goodness of fit of the regression Discussion of the linearity assumption Introduction Properties of OLS Goodness of Fit Linearity Assumption Properties of OLS/1 1 The sample regression function obtained through OLS always passes through the sample mean values of X and Y . 2 u = P i u i n = 0 (mean value of residuals is zero) 3 i u i X i = 0 ( u i and X i are uncorrelated) Note: these results hold by construction, without the OLS assumptions. Introduction Properties of OLS Goodness of Fit Linearity Assumption Properties of OLS/2 Under the OLS assumptions (see Lecture 3), we have the following results for 1 : 1 E ( 1 ) = 1 . In words, 1 is an unbiased estimator of 1 2 As the sample size n increases, 1 gets closer and closer to 1 , i.e. 1 is a consistent estimator of 1 . 3 If n is large, the distribution of 1 is well approximated by a normal. In particular, 1 N 1 ,V ar ( 1 ) , where V ar ( 1 ) = 2 u i ( X i X ) 2 2 u = i u 2 i n 2 = i ( Y i Y i ) 2 n 2 For : V ar ( ) = i X 2 i n 2 u i ( X i X ) 2 Introduction...
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 Spring '10
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