103lect12hand

103lect12hand - Polynomial Regression Models - I Original...

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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 . 01 = ii i YX u β ++ 2 2 = i i X u ββ + 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 2 ˆˆ ˆ ˆ = i X 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” “regress testscr avginc avginc2” 2 2 Testscore = avginc avginc i i u +
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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 --------------------------------------------------------------------------- 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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103lect12hand - Polynomial Regression Models - I Original...

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