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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 Laczó October 5, 2010 Introduction Properties of OLS Goodness of Fit Linearity Assumption Introduction We consider the linear regression model Y i = β + β 1 X i + u i The OLS estimators are: ˆ β 1 = ∑ i ( X i ¯ X )( Y i ¯ Y ) ∑ i ( X i ¯ X ) 2 = s XY s 2 X ˆ β = ¯ Y ˆ β 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 μ 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 ¯ ˆ u = P i ˆ u i n = 0 (mean value of residuals is zero) 3 ∑ i ˆ u i X i = 0 ( ˆ 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 ˆ β 1 : 1 E ( ˆ β 1 ) = β 1 . In words, ˆ β 1 is an unbiased estimator of β 1 2 As the sample size n increases, ˆ β 1 gets closer and closer to β 1 , i.e. ˆ β 1 is a consistent estimator of β 1 . 3 If n is large, the distribution of ˆ β 1 is well approximated by a normal. In particular, ˆ β 1 ∼ N β 1 ,V ar ( ˆ β 1 ) , where V ar ( ˆ β 1 ) = ˆ σ 2 u ∑ i ( X i ¯ X ) 2 ˆ σ 2 u = ∑ i ˆ u 2 i n 2 = ∑ i ( Y i ˆ Y i ) 2 n 2 For ˆ β : V ar ( ˆ β ) = ∑ i X 2 i n ˆ σ 2 u ∑ i ( X i ¯ X ) 2 Introduction...
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This note was uploaded on 10/23/2010 for the course STAT 10 taught by Professor Davis during the Spring '10 term at UCLA.
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