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Unformatted text preview: ECON 103, Lecture 11: Nonlinear Regression Maria Casanova May 5 (version 0) Maria Casanova Lecture 11 1. Introduction Our models so far have assumed a linear relationship between X i (or the X i s) and Y i . But often the relationship between variables is nonlinear, e.g. concave, convex, or more complicated. Consider again our Test Scores example. Weve said that parents income might also be important. Lets look at the data on test scores (testscr) and average per capita incomes in district i (avginc). Maria Casanova Lecture 11 1. Introduction 600 600 600 620 620 620 640 640 640 660 660 660 680 680 680 700 700 700 Test Scores Test Scores Test Scores 20 20 20 40 40 40 60 60 60 Average Income Average Income Average Income Maria Casanova Lecture 11 1. Introduction In a linear regression model, the effect of a 1unit change in X ji on the value of Y i is always the same (and is equal to the slope coefficient j .) In a nonlinear regression model, the effect of a 1unit change in X ji on the value of Y i varies, i.e. the effect is not constant. Maria Casanova Lecture 11 1. Introduction Outline: We cover three general types of nonlinear regression models: Polynomial regression models Logarithmic regression models Interactions between regressors Regression analysis in practice Which X variables to include in a multiple regression? Maria Casanova Lecture 11 2. Polynomial Regression Original model: Y i = + 1 X i + u i What if we create a new variable, X 2 i , i.e. the value of X i squared, and consider the multiple regression model: Y i = + 1 X i + 2 X 2 i + u i This model can be estimated by regressing Y i on X i and X 2 i . In this new model, X i and Y i have a nonlinear relationship (unless 2 is zero). Note that the interpretation of the coefficients is different than before. Specifically, 1 does not measure the effect of a one unit change in X i on Y i , because when X i changes, X 2 i will necessarily change. So the effect will depend on both 1 and 2 . Maria Casanova Lecture 11 2. Polynomial Regression How to interpret the estimated coefficients? Lets consider the example of test scores and incomes. Test Scores i = + 1 avginc i + 2 avginc 2 i + u i This model allows a nonlinear relationship between test scores and average per capita income. To estimate the model is Stata: (Open the California schools dataset.) gen avginc2=avginc2 regress testscr avginc avginc2, robust Maria Casanova Lecture 11 2. Polynomial Regression Maria Casanova Lecture 11 2. Polynomial Regression The estimated regression (with robust standard errors) is: \ Test Scores i = 607 . 30 (2 . 902) + 3 . 851 (0 . 268) avginc i . 042 (0 . 005) avginc 2 i It is not even obvious whether avginc positively or negatively affects test scores. One coefficient is positive and the other coefficient is negative......
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This note was uploaded on 09/23/2011 for the course ECON 103 taught by Professor Sandrablack during the Spring '07 term at UCLA.
 Spring '07
 SandraBlack

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