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Econ103fall10lec9

Econ103fall10lec9 - Introduction Polynomial Regression...

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Introduction Polynomial Regression Logarithmic Regression Interactions MR in Practice Econ 103, UCLA, Fall 2010 Introduction to Econometrics Lecture 9: Non-linear Regression Sarolta Laczó October 21, 2010
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Introduction Polynomial Regression Logarithmic Regression Interactions MR in Practice Introduction/1 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 non-linear, e.g. concave, convex, or more complicated. Consider again our Test Scores example. We’ve said that parents’ income might also be important. Let’s look at the data on test scores (testscr) and average per capita incomes in district i (avginc).
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Introduction Polynomial Regression Logarithmic Regression Interactions MR in Practice Introduction/2 600 600 620 620 640 640 660 660 680 680 700 700 Test Scores Test Scores 0 0 20 20 40 40 60 60 Average Income Average Income
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Introduction Polynomial Regression Logarithmic Regression Interactions MR in Practice Introduction/3 In a linear regression model, the effect of a 1-unit change in X ji on the value of Y i is always the same (and is equal to the slope coefficient β j .) In a non-linear regression model, the effect of a 1-unit change in X ji on the value of Y i varies, i.e. the effect is not constant.
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Introduction Polynomial Regression Logarithmic Regression Interactions MR in Practice Introduction/4 Outline: We cover three general types of non-linear regression models: Polynomial regression models Logarithmic regression models Interactions between regressors Regression analysis in practice Which X variables to include in a multiple regression?
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Introduction Polynomial Regression Logarithmic Regression Interactions MR in Practice Polynomial Regression/1 Original model: Y i = β 0 + β 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 = β 0 + β 1 X i + β 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 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 , because when X i changes, X 2 i will necessarily change. So the effect will depend on both β 1 and β 2 .
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Introduction Polynomial Regression Logarithmic Regression Interactions MR in Practice Polynomial Regression/2 How to interpret the estimated coefficients? Let’s consider the example of test scores and incomes. Test Scores i = β 0 + β 1 avginc i + β 2 avginc 2 i + u i This model allows a non-linear relationship between test scores and average per capita income. To estimate the model is Stata: (Open the California schools dataset.) gen avginc2=avginc ˆ 2 regress testscr avginc avginc2, robust
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Introduction Polynomial Regression Logarithmic Regression Interactions MR in Practice Polynomial Regression/3
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Introduction Polynomial Regression Logarithmic Regression Interactions MR in Practice Polynomial Regression/4
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