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lect12_103_2010_Compatibility_Mode_

# lect12_103_2010_Compatibility_Mode_ - Polynomial Regression...

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5/17/2010 1 Polynomial Regression Models - I Original model: What if we create a new variable, X i 2 , i.e. the value of X i squared, and consider the multiple regression model: This model can be estimated by regressing Y i on X i and X i 2 . 0 1 = i i i Y X u β β + 2 0 1 2 = i i i i Y X X u β β β + + Polynomial Regression Models - II In this new model, X i and Y i have a non-linear relationship (unless β 2 is zero!) Note that the interpretation of the coefficients is different than before. Specifically, β 1 does not measure the effect of a one unit change in X i on Y i . Why? When X i changes, X i 2 will necessarily change. So the effect of a unit change in X i on Y i will depend on both β 1 and β 2 . Assessing the meaning of the coefficients in this model is a little tougher than before. Let’s do it with an example. 2 0 1 2 ˆ ˆ ˆ ˆ = i i i Y X X β β β + +

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5/17/2010 2 Polynomial Regression Models - III This model allows a non-linear relationship between test scores and average income. To estimate model in STATA, – 1) Open the dataset (relevant variables are testscr and avginc) – 2) “gen avginc2 = avginc^2” – 2) “regress testscr avginc avginc2” 2 0 1 2 Testscore = avginc avginc i i i i u β β β + + Polynomial Regression Models - IV . regress testscr avginc avginc2 Source | SS df MS Number of obs = 420 -------------+------------------------------ F( 2, 417) = 261.28 Model | 84599.2786 2 42299.6393 Prob > F = 0.0000 Residual | 67510.3151 417 161.89524 R-squared = 0.5562 -------------+------------------------------ Adj R-squared = 0.5540 Total | 152109.594 419 363.030056 Root MSE = 12.724 ------------------------------------------------------------------------------ testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- avginc | 3.850995 .3042617 12.66 0.000 3.252917 4.449073 avginc2 | -.0423085 .0062601 -6.76 0.000 -.0546137 -.0300033 _cons | 607.3017 3.046219 199.36 0.000 601.3139 613.2896 ---------------------------------------------------------------------------
5/17/2010 3 Polynomial Regression Models - V Note that in this estimated model, it is not even obvious whether avginc positively or negatively affects testscores. Why? One coefficient is positive and the other coefficient is negative.

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lect12_103_2010_Compatibility_Mode_ - Polynomial Regression...

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