Wooldridge PPT ch6

# Is just 1 r 2 n 1 n k 1 but most packages will give

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is just (1 R 2 )( n – 1) / ( n k – 1), but most packages will give you both R 2 and adj- R 2 You can compare the fit of 2 models (with the same y ) by comparing the adj- R 2 You cannot use the adj- R 2 to compare models with different y ’s (e.g. y vs. ln( y ))

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Fall 2008 Under Econometrics Prof. Keunkwan Ryu 17 Goodness of Fit Important not to fixate too much on adj- R 2 and lose sight of theory and common sense If economic theory clearly predicts a variable belongs, generally leave it in Don’t want to include a variable that prohibits a sensible interpretation of the variable of interest – remember ceteris paribus interpretation of multiple regression
Fall 2008 Under Econometrics Prof. Keunkwan Ryu 18 Goodness of Fit (cont) Using Adjusted R-Squared to Choose between Nonnested Models Controlling for Too Many Factors in Regression Analysis Adding Regressors to reduce the Error Variance

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Fall 2008 Under Econometrics Prof. Keunkwan Ryu 19 Standard Errors for Predictions Suppose we want to use our estimates to obtain a specific prediction? First, suppose that we want an estimate of E( y|x 1 =c 1 ,…x k =c k ) = θ 0 = β 0 + β 1 c 1 + …+ β k c k This is easy to obtain by substituting the x ’s in our estimated model with c ’s , but what about a standard error? Really just a test of a linear combination
Fall 2008 Under Econometrics Prof. Keunkwan Ryu 20 Predictions (cont) Can rewrite as β 0 = θ 0 β 1 c 1 – … – β k c k Substitute in to obtain y = θ 0 + β 1 ( x 1 - c 1 ) + … + β k ( x k - c k ) + u So, if you regress y i on ( x ij - c ij ) the intercept will give the predicted value and its standard error Note that the standard error will be smallest when the c ’s equal the means of the x ’s

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Fall 2008 Under Econometrics Prof. Keunkwan Ryu 21 Predictions (cont) This standard error for the expected value is not the same as a standard error for an outcome on y We need to also take into account the variance in the unobserved error. Let the prediction error be ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 [ ] { } 2 1 2 2 0 0 2 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 ˆ ˆ ˆ so , ˆ ˆ ˆ and 0 ˆ ˆ ˆ ˆ σ σ β β β + = + = + = = - + + + + = - = y se e se y Var u Var y Var e Var e E y u x x y y e k k
Fall 2008 Under Econometrics Prof. Keunkwan Ryu 22 Prediction interval ( 29 ( 29 0 025 . 0 0 0 0 0

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