PanelDataProblemSet5

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 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, i i ii y 1 f(y | ,1) exp ⎛⎞ =− ⎜⎟ θθ ⎝⎠ 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 yy , ) γ γ− γ γ γ= 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 2 2y y ,2) 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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Since Γ (t) = t-1! for integer t, Γ (2) = 1. When t = an integer + .5, we can use the recurrence Γ (t) = (t- 1) Γ (t-1) until we reach Γ (.5) which equals π . Combining terms, then, the variance of the Rayleigh variable is [1-(.5 π ) 2 ] θ i 2 = 0.2146 θ i 2 . a. The parameters β in the Rayleigh model could be estimated either by nonlinear least squares or by maximum likelihood. Which would be more efficient? Explain. b. Form the log likelihood and derive the expressions for the first order conditions for maximizing the log likelihood for the Weibull model. c. How would you test the null hypothesis of the Rayleigh model ( γ =2) against the more general null of the Weibull model ( γ unrestricted)? d. How would you test the null hypothesis of the Rayleigh model ( γ =2) against the alternative of the Exponential model ( γ = 1)? e. Maximum likelihood estimates of the parameters of the three models based on the German health data discussed in class appear below. Carry out the test in part c. Which of the three do you think is the appropriate model given the results below. f. In the Rayleigh model, show how to obtain the three available estimators of the asymptotic covariance matrix of the MLE of β . Remember, you are not estimating γ (it equals 2), and the expected value of y i is still θ i . +---------------------------------------------+ | Weibull (Loglinear) Regression Model | | Dependent variable HHNINC | | Number of observations 27322 | | Log likelihood function 12033.50 | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Parameters in conditional mean function Constant 3.44054643 .02266279 151.815 .0000 EDUC -.10914142 .00147212 -74.139 .0000 11.3201838 MARRIED -.31230818 .00750583 -41.609 .0000 .75869263 AGE .00053144 .00044049 1.206 .2276 43.5271942 Shape parameter for Weibull model P_scale 2.12853619 .00466881 455.905 .0000 +---------------------------------------------+ | Exponential (Loglinear) Regression Model | | Log likelihood function 1539.191 | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Parameters in conditional mean function Constant 1.82555590 .04219675 43.263 .0000 EDUC -.05545277 .00267224 -20.751 .0000 11.3201838 MARRIED -.23664845
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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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PanelDataProblemSet5 - Department of Economics Econometric...

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