LN+5+Quadratic+and+Log+Forms

LN+5+Quadratic+and+Log+Forms - Empirical Methods II...

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Empirical Methods II (API-202A) Kennedy School of Government Harvard University 1 Lecture Notes 5 Non-linear Relationships – Quadratic and Log Forms I – INTRODUCTION So far we have been modeling the relationship between variables as a linear relationship: The associated change in Y given a one-unit increase in X 1 is the same regardless of the value of X 1 . Examples where we have modeled X 1 and Y as a linear relationship: (1) Test scores = 0 ˆ 1 ˆ STR + 2 ˆ AFI + ˆ 2 ˆ _ AFI scores Test (2) Wages = 0 ˆ 1 ˆ Years of Schooling + 2 ˆ Experience + ˆ 2 ˆ Experience Wages Is linearity a reasonable assumption? Not always… Example : The relationship between AFI and Test scores At low levels of income, the positive relationship between Test scores and AFI is very strong At higher levels of income, the positive relationship between Test scores and AFI tends to fade 600 620 640 660 680 700 testscr 0 20 40 60 avginc
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Empirical Methods II (API-202A) Kennedy School of Government Harvard University 2 Decreasing association between Test scores and AFI” How else can we model the relationship between Test scores and AFI to account for this non linear pattern? With a non-linear regression : The predicted change in Y associated with a change in X 1 depends on the value of X 1 . Common alternatives used to model non-linear effects: Quadratic form: Y = 0 ˆ 1 ˆ X 1 + 2 ˆ X 1 2 + ˆ Log form: Y = 0 ˆ 1 ˆ Log X 1 + ˆ Note : We will use “log” to refer to “ln” or natural logarithm. See the Appendix for review of properties of this function. The mechanics to calculate OLS estimates of non-linear effects are essentially the same as before, but the interpretation of the parameters is different.
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Empirical Methods II (API-202A) Kennedy School of Government Harvard University 3 II. QUADRATIC FORM Let’s compare the linear and non linear fit (with quadratic) of the regressions: (1) Test scores = 0 ˆ 1 ˆ AFI + ˆ (2) Test scores = 0 ˆ 1 ˆ AFI + 2 ˆ AFI 2 + ˆ (1) 1 ˆ >0 (2) 1 ˆ >0 and 2 ˆ <0 600 650 700 750 0 20 40 60 avginc Fitted values testscr 620 640 660 680 0 20 40 60 avginc Fitted values testscr Note : Recall that AFI is denominated in thousands of dollars. Adding X 2 to second regression allows us to add curvature to the regression.
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Empirical Methods II (API-202A) Kennedy School of Government Harvard University 4 Interpretation of Coefficients in Quadratic Case (2) Test scores = 0 ˆ 1 ˆ AFI + 2 ˆ AFI 2 + ˆ reg testscr avginc avginc2, robust Linear regression Number of obs = 420 R-squared = 0.5562 ------------------------------------------------------------------------------ | Robust testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- avginc | 3.850995 .2680941 14.36 0.000 3.32401 4.377979 avginc2 | -.0423085 .0047803 -8.85 0.000 -.051705 -.0329119 _cons | 607.3017 2.901754 209.29 0.000 601.5978 613.0056 ------------------------------------------------------------------------------ Interpretation of Coefficients is different than before. o 1 ˆ no longer represents the predicted change in Y given a one-unit change in X. Why not?
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This note was uploaded on 04/12/2009 for the course HKS API202A taught by Professor Levy during the Spring '09 term at Harvard.

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LN+5+Quadratic+and+Log+Forms - Empirical Methods II...

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