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Unformatted text preview: 1 Regression Continued: Functional Form LIR 832 December 5, 2006 Topics for the Evening 1. Qualitative Variables 2. Nonlinear Estimation Functional Form ¡ Not all relations among variables are linear: ¡ Our basic linear model: y= β + β 1 X 1 + β 2 X 2 +…+ β k X k + e 2 Functional Form ¡ Q: Given that we are using OLS, can we mimic these nonlinear forms? ¡ A: We have a small bag of tricks which we can use with OLS. Functional Form Functional Form 3 Functional Form Functional Form ¡ A first point about functional form: You must have an intercept. ¡ Consider the following case: We estimate a model and test the intercept to determine if it is significantly different than zero. We are not able to reject the null in a hypothesis test and we decide to reestimate the model without an intercept. What is really going on? ¡ Return to our basic model: y= β + β 1 X 1 + β 2 X 2 +…+ β k X k + e ¡ What are we doing when we remove the intercept? y= + β 1 X 1 + β 2 X 2 +…+ β k X k + e Functional Form 4 Functional Form Functional Form /* Regression without an intercept */ Regression Analysis: weekearn versus years ed The regression equation is weekearn = 57.3 years ed 47576 cases used, 7582 cases contain missing values Predictor Coef SE Coef T P Noconstant years ed 57.3005 0.1541 371.96 0.000 S = 534.450 Functional Form /* Regression with an intercept */ Regression Analysis: weekearn versus years ed The regression equation is weekearn =  485 + 87.5 years ed 47576 cases used, 7582 cases contain missing values Predictor Coef SE Coef T P Constant 484.57 18.18 26.65 0.000 years ed 87.492 1.143 76.54 0.000 S = 530.510 RSq = 11.0% RSq(adj) = 11.0% 5 Functional Form ¡ Consequences of forcing through zero: ¡ Unless the intercept is really zero, we are going to bias both the intercept and the slope coefficients. ¡ Remember that we calculate the intercept so that the line passes through the point of means: ¡ Assures that the Σε = 0 ¡ If we impose 0 as the intercept, the line may not pass through the point of means and the sum of the errors may not equal zero. ¡ Biases the coefficients and leads to incorrect estimates of the standard errors of the β s. ¡ Never suppress the intercept, even if your theory suggests that it is not necessary. Functional Form /* What About Those Residuals? */ Descriptive Statistics: RESI1, RESI2 Variable N N* Mean SE Mean StDev Minimum Q1 Median RESI1 47576 7582 8.67 2.45 534.38 1180.31 359.12 122.21 RESI2 47576 7582 0.00 2.43 530.50 1329.77 340.32 107.62 Variable Q3 Maximum RESI1 218.59 2311.61 RESI2 237.69 2494.26 Functional Form ¡ Returning to the issue of nonlinearity… ¡ In our basic model: ¡ β = ∆ Y/ ∆ X = change in Y for a oneunit change in X ¡ Consider the effect of Education on base salary… 6 Functional Form Descriptive Statistics: years ed, Exp Variable N N* Mean SE Mean StDev Minimum Q1 Median Q3 Maximum years ed 55158 0 15.734 0.00941 2.211 1.000 14.000 16.000 18.000 21.00021....
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This note was uploaded on 07/25/2008 for the course LIR 832 taught by Professor Belman during the Spring '07 term at Michigan State University.
 Spring '07
 BELMAN

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