ec20-lecture3-v7

ec20-lecture3-v7 - Econometrics Lecture #3: More on...

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Economics 20 - Prof. Lewis 1 Econometrics Lecture #3: More on Bivariate Regression Multiple Regression
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Economics 20 - Prof. Lewis 2 Continue with Last Time Simple Linear Regression (SLR) or “bivariate” model = one y, one x. Like all econometric models, untestable assumptions needed for causal inference Least Squares (OLS) estimation procedure
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What is true in the simple linear regression model? 1. To allow causal inference, we need the errors to be zero 2. To allow causal inference, we need the residuals, û, to be uncorrelated with the independent variable 3. The residuals are always uncorrelated with the independent variable 4. More than one of these 7% 55% 0% 38% Economics 20 - Prof. Lewis 3
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Economics 20 - Prof. Lewis 4 Continue with Last Time Simple Linear Regression (SLR) or “bivariate” model = one y, one x. Like all econometric models, untestable assumptions needed for causal inference E[u|X] = 0, or (1) Cov(u,x) = 0 and (2) E[u]=0 Least Squares (OLS) estimation procedure Slope and intercept estimate come from imposing (1) and (2) on the residuals , û, the observable sample analog of the error, u
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Economics 20 - Prof. Lewis 5 This Time Continue with derivation of OLS estimate Demonstrate assumptions, especially key assn E[u|X] = 0, sufficient for causal inference Talk about estimation situations when getting causal inference is less important Multivariate regression model Probably next time: Interpretation of slope estimates The effect of adding/removing variables
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Economics 20 - Prof. Lewis 6 Recap of derivation of OLS from last time… Intercept solves Equivalent to Slope solves Equivalent to Are these expressions familiar? ( 29 ( 29 + - = = n i i i x y n 1 2 1 0 1 ˆ ˆ 1 ˆ 0 β ( 29 ( 29 + - = = n i i i x y n 1 2 1 0 0 ˆ ˆ 1 ˆ 0 0 ˆ = u ( 29 0 ˆ , cov = i i u x
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Economics 20 - Prof. Lewis 7 Method of Moments These expressions are the sample analogs of what we assumed was true of the error in the population analog of E[u i ] = 0 analog of Cov(u i ,x i ) = 0 Another motivation for least squares: “method of moments” estimator Imposes population assumptions on sample This is the approach described in the textbook 0 ˆ = u ( 29 0 ˆ , cov = i i u x
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Economics 20 - Prof. Lewis 8 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Some lines consistent with 0 ˆ = u Solving for least-squares line…
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Economics 20 - Prof. Lewis 9 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Lines consistent with and 0 ˆ = u Solving for least-squares line… ( 29 0 ˆ , cov = i i u x
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Economics 20 - Prof. Lewis 10 Mathematically: Solving for intercept Least squares intercept chosen so that mean X, mean Y on the line 0 ˆ ˆ ˆ 1 0 = - - = X Y u β 0 ˆ = u X Y 1 0 ˆ ˆ - =
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Economics 20 - Prof. Lewis 11 Solving for slope The least-squares estimator of the slope X Y u X Y u i i i 1 0 1 0 ˆ ˆ ˆ ˆ ˆ ˆ β - - = - - - = ( 29 ( 29 ( 29 X X Y Y u u i i i - - - = - 1 ˆ ˆ ˆ ( 29 ( 29 0 ˆ ˆ = - - u u X X i i ( 29 ( 29 ( 29 ( 29 0 ˆ 1 = - - - - - X X X X Y Y X X i i i i ( 29 ( 29 ( 29 - = - - 2 1 ˆ X X Y Y X X i i i ( 29 ( 29 ( 29 ) ( ) , ( ˆ 2 2 1 X Var Y X Cov s s X X Y Y X X x xy i i i = = - - - =
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Economics 20 - Prof. Lewis 12 Textbook’s version of SLRM SLR.1 Pop. relationship
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ec20-lecture3-v7 - Econometrics Lecture #3: More on...

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