PanelDataProblemSet5 - Department of Economics Econometric...

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Econometric Analysis of Panel Data Professor William Greene Phone: 212.998.0876 Office: KMC 7-78 Home Email: [email protected] URL for course web page: Assignment 5 Nonlinear Models Part I. Weibull Regression Model In class, we examined a ‘loglinear,’ exponential regression model, i i i i y 1 f(y | ,1) exp = θ θ i x , θ i = exp( x i ′β ) = E[y i | x i ] The Weibull model is an extension of the exponential model which adds a shape parameter, γ ; 1 i i i i i y y f(y | , ) exp γ γ− γ γ γ = θ θ i x E[y i | x i ]= Γ [( γ +1)/2] θ i = .5*sqr( π ) if γ = 2. The exponential model results when γ = 1. (This distribution looks like, but is not the gamma distribution we discussed in class.) An interesting special case is the Rayleigh distribution, which has γ = 2. The resulting density is 2 i i i 2 i i 2y y f(y | ,2) exp = θ θ i x One of the interesting things about the Rayleigh distribution is that E[y| x i ]= .5 π θ i (compared to θ i for the exponential. .5 π is approximately equal to 0.866.) One difference is the variance. The variance of the exponential variable is θ i 2 . The variance of the Rayleigh variable is [ Γ (2) - Γ 2 (1.5)] θ i 2 . Department of Economics
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