midterm2review

# midterm2review - Econ 139 Midterm 2 Review Andrew Sweeting...

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Econ 139: Midterm 2 Review Andrew Sweeting 1 Department of Economics Duke University Spring 2011 Econ 139 Review 2 (Duke) Review 2 Spring 2011 1 / 33

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1 Multivariate Regression (esp. Testing) 2 Modelling nonlinearities 3 Maximum Likelihood and Limited Dependent Variables Econ 139 Review 2 (Duke) Review 2 Spring 2011 2 / 33
Multivariate Regression motivation for including more variables examples: Wage = β 0 + β 1 Educ + β 2 Ability + u TestScr = β 0 + β 1 STR + β 2 El _ Pct + u Econ 139 Review 2 (Duke) Review 2 Spring 2011 3 / 33

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Interpretation of the Multiple Regression Model E ( Y i j X 1 i , ..., X ki ) = β 0 + β 1 X 1 i + ... + β k X ki As in the univariate case, β 0 is the intercept and β k is the "slope coe¢ cient" of X k . β 0 (the intercept) is the expected value of Y i when all the regressors ( X ki °s) are zero. β 1 (the slope coe¢ cient of X 1 ) is the e/ect on Y (the expected change in Y ) of a one unit change in X 1 , holding all other variables constant (or ±controlling for all other variables²). β 1 may also be described as the partial e/ect of X on Y Econ 139 Review 2 (Duke) Review 2 Spring 2011 4 / 33
Interpreting Changes in Coe¢ cients: Omitted Variables Bias when we add regressors the coe¢ cients on existing variables sometimes change e.g., when include controls for ability the estimated returns to eduction fall e.g., when include state ³xed e/ects the e/ect of beer tax changes sign OVB in univariate case if truth: Y i = β 0 + β 1 X 1 i + β 2 X 2 i + u i estimate: Y i = α 0 + α 1 X 1 i + e i b α 1 p ! β 1 + β 2 σ X 1 X 2 σ 2 X 1 Econ 139 Review 2 (Duke) Review 2 Spring 2011 5 / 33

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OLS Assumptions assumptions for OLS in multiple regression are as before OLS Assumption 1 Correct Speci³cation 1 E ( u i j X 1 i , ..., X ki ) = 0 OLS Assumption 2 Simple random sample ( Y i , X 1 i , ..., X ki ) ° iid OLS Assumption 3 No extreme outliers X 1 i , ..., X ki , u i have non-zero & ³nite fourth moments plus OLS Assumption 4 No perfect collinearity Regressors are not linear combinations of each other if these are true, OLS is unbiased, consistent and asymptotically normal 1 Note that this condition also implies that the conditional expectation is zero given any subset of regressors. For example, E ° u i j X 1 i ± = E h u i j X ji i = E ° u i j X 1 i , X 2 i ± = 0 . Why? This is due to a ±more complicated² version of LIE, according to which E [ Y j X ] = E [ E [ Y j X , Z ] j X ] Econ 139 Review 2 (Duke) Review 2 Spring 2011 6 / 33
Collinearity when we are using a set of dummy variables (e.g., for states) and a constant, we drop one of the dummies to avoid perfect collinearity if have w sets of dummies (e.g., states & years) and a constant, then we drop one dummy of each type if we have highly (but not perfectly) collinear variables, OLS will work but we may end up with large standard errors (imprecise estimates) Econ 139 Review 2 (Duke) Review 2 Spring 2011 7 / 33

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Goodness of Fit use adjusted R 2 R 2 = 1 ± n ± 1 n ± k ± 1 SSR TSS = 1 ± s 2 b u s 2 Y Notice that: 1 R 2 is always less than R 2 : n ± 1 n ± k ± 1 > 1 = ) R 2 < R 2 2 Adding a regressor has two e/ects on R 2 : 1.) SSR falls, but, 2.) n ± 1 n ± k ± 1 increases. So the total e/ect on R 2 depends on which e/ect is bigger.
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