PanelDataNotes-19

PanelDataNotes-19 - Econometric Analysis of Panel Data...

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Econometric Analysis of Panel Data William Greene Department of Economics Stern School of Business
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Econometric Analysis of Panel Data 19. Limited Dependent Variables And Models for Count Data
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Censoring and Corner Solution Models Censoring model: T(y*) = 0 if y* <  0 and y*  otherwise.  Corner solution: y = 0 if some exogenous  condition is met; y = g(x)+e if the condition is  not met.  We then model P(y=0) and E[y|x,y>0].   (See text, pp. 518-519)     Application: Fair, R., “A Theory of Extramarital     Affairs,” JPE, 1978. Same structural form
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The Tobit Model Y* = x’ β + ε , y = Max(0,y*),  ε  ~ N[0, σ 2 ] Other censoring limits:         y=Max(L,y*) => y-L=Max(0,y*-L).  Change constant term and LHS  variable Upper censoring:        y=Min(U,y*) => U-y = Max(0,U-Y*). Change constant and LHS variable  and swap signs of estimated coefficients. Censoring limit is person specific: Same as above: Constant term  becomes the variable with a coefficient fixed at 1. Censoring at both extremes: y=Max(L,Min(y*,U)). A minor extension of  the model.  Easy to accommodate. (Already done in major software.) Interval censoring over the entire range.  See yesterday’s notes. (Tobin:  “Estimation of Relationships for Limited Dependent  Variables,” Econometrica, 1958. Tobin’s probit?)
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Conditional Mean Functions 2 y* ,   ~N[0, ] y Max(0, y* ) E[y* | ] x E[y| ]=Prob[y=0| ]×0+Prob[y>0| ]E[y|y>0, ]          =Prob[y* >0| ]  E[y* |y* >0, ] ( )          =           ( )          =   ( ) = + ε ε σ = = β φ σ Φ × σ ÷ σ Φ σ Φ σ x β x x x x x x x x x / β β x + β x / β x / x   β β ( ) ( ) " Inverse Mills ratio" ( ) ( ) E[y| ,y>0]= ( ) σφ σ φ σ = Φ σ φ σ σ Φ σ x / β x / β x / β x / β x x + β x / β
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Conditional Means XB -1.20 -.40 .40 1.20 2.00 -2.00 -1.20 -.40 .40 1.20 2.00 -2.00 EYSTAR EY Variable
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Predictions and Residuals What variable do we want to predict? y*?  Probably not – not relevant y?  Randomly drawn observation from the population y | y>0?  Maybe. Depends on the desired function What is the residual? y – prediction?  Probably not.  What do you do with  the zeros? Anything - x β ? Probably not. x β  is not the mean. Generalized residuals – coming below.
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OLS is Inconsistent - Attenuation E[y | ] =   ( ) ( ) Nonlinear function of x.  What is estimated by OLS regression of y on x? Slopes of the linear projection are approximately
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This note was uploaded on 01/05/2012 for the course B 55.9912 taught by Professor Willamgreene during the Fall '11 term at NYU.

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PanelDataNotes-19 - Econometric Analysis of Panel Data...

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