Econometric Analysis of Panel DataProfessor William Greene Phone: 212.998.0876 Office: KMC 7-78 Home page:www.stern.nyu.edu/~wgreene Email: [email protected]URL for course web page: www.stern.nyu.edu/~wgreene/Econometrics/PanelDataEconometrics.htm Assignment 5 Nonlinear Models Part I. Weibull Regression Model In class, we examined a ‘loglinear,’ exponential regression model, iiiiy1f(y |,1)exp⎛⎞=−⎜⎟θθ⎝⎠ix, θi= exp(xi′β) = E[yi|xi] The Weibull model is an extension of the exponential model which adds a shape parameter, γ; 1iiiiiyyf(y |, )expγγ−γ⎛⎞⎡⎤γ⎜⎟γ=−⎢⎥⎜⎟θθ⎣⎦⎝⎠ixE[yi|xi]=Γ[(γ+1)/2] θi= .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 2iii2ii2yyf(y |,2)exp⎛⎞⎡⎤⎜⎟=−⎢⎥⎜⎟θθ⎣⎦⎝⎠ixOne of the interesting things about the Rayleigh distribution is that E[y|xi]= .5πθi(compared to θifor the exponential. .5πis approximately equal to 0.866.) One difference is the variance. The variance of the exponential variable is θi2. The variance of the Rayleigh variable is [Γ(2) - Γ2(1.5)]θi2. Department of Economics
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