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Unformatted text preview: Nonlinear Regression Functions ECO 4000, Statistical Analysis for Economics and Finance Lecture 13 by Prof: Seyhan Erden Arkonac, PhD 1 2 Interpreting the estimated regression function: (a) Plot the predicted values TestScore = 607.3 + 3.85 Income i 0.0423( Income i ) 2 (2.9) (0.27) (0.0048) 3 Interpreting the estimated regression function, ctd : (b) Compute effects for different values of X TestScore = 607.3 + 3.85 Income i 0.0423( Income i ) 2 (2.9) (0.27) (0.0048) Predicted change in TestScore for a change in income from $5,000 per capita to $6,000 per capita: TestScore = 607.3 + 3.85 6 0.0423 6 2 (607.3 + 3.85 5 0.0423 5 2 ) = 3.4 For a regression Y = + 1 X + 2 X 2 + u we can write Y / X = 1 + 2 2 X Set this equal to zero and solve it for X* X* =  ( 1 / 2 2 ) Similarly; Test Score = 3.85 /[ 2(0.0423)]= 45.508(x$1000 ) Income 4 5 TestScore = 607.3 + 3.85 Income i 0.0423( Income i ) 2 Predicted effects for different values of X : Change in Income ($1000 per capita) TestScore from 5 to 6 3.4 from 25 to 26 1.7 from 45 to 46 0.0 The effect of a change in income is greater at low than high income levels (perhaps, a declining marginal benefit of an increase in school budgets?) Caution! What is the effect of a change from 65 to 66? Dont extrapolate outside the range of the data! 6 The three log regression specifications : Case Population regression function I. linearlog Y i = + 1 ln( X i ) + u i II. loglinear ln( Y i ) = + 1 X i + u i III. loglog ln( Y i ) = + 1 ln( X i ) + u i The interpretation of the slope coefficient differs in each case. The interpretation is found by applying the general before and after rule: figure out the change in Y for a given change in X . 87 88 89 10 Interactions Between Independent Variables (SW Section 8.3) Perhaps a class size reduction is more effective in some circumstances than in others Perhaps smaller classes help more if there are many English learners, who need individual attention That is, TestScore STR might depend on PctEL More generally, 1 Y X might depend on X 2 How to model such interactions between X 1 and X 2 ? We first consider binary X s, then continuous X s 11 (a) Interactions between two binary variables Y i = + 1 D 1 i + 2 D 2 i + u i D 1 i , D 2 i are binary 1 is the effect of changing D 1 =0 to D 1 =1. In this specification, this effect doesnt depend on the value of D 2 . To allow the effect of changing D 1 to depend on D 2 , include the interaction term D 1 i D 2 i as a regressor: Y i = + 1 D 1 i + 2 D 2 i + 3 ( D 1 i D 2 i ) + u i 12 Interpreting the coefficients Y i = + 1 D 1 i + 2 D 2 i + 3 ( D 1 i D 2 i ) + u i General rule: compare the various cases E ( Y i  D 1 i =0, D 2 i = d 2 ) = + 2 d 2 (b) E ( Y i  D 1 i =1, D 2 i = d 2 ) = + 1 + 2 d 2 + 3 d 2 (a) subtract (a) (b): E ( Y...
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This note was uploaded on 05/05/2011 for the course ECON 4000 taught by Professor Arkonac during the Spring '11 term at CUNY Baruch.
 Spring '11
 Arkonac
 Economics, Econometrics

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