Problem5

# Problem5 - Department of Economics ECONOMETRICS I Professor...

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ECONOMETRICS I Professor William Greene Phone: 212.998.0876 Office: KMC 7-90 Home page:www.stern.nyu.edu/~wgreene Email: [email protected] URL for course web page: www.stern.nyu.edu/~wgreene/Econometrics/Econometrics.htm Assignment 5 Maximum Likelihood estimation Part I. Econometric Theory 1. The random variable, y, is distributed normally with mean µ and variance σ 2 . a. Obtain the log likelihood function for a sample of n observations, y i ,…,y n for estimation of µ and σ . Derive the likelihood equations (first order conditions) for estimation of µ and σ . b. I am interested in estimating the normalized mean, τ = ( µ / σ ) and the precision, θ = 1/ σ . Derive the log likelihood function for estimation of τ and θ . Derive the first ordere conditions for estimation of τ and θ . How will you compute asymptotic standard errors for your estimator? 2. The continuous, nonnegative random variable y is distributed with a Weibull distribution, f(y) = P λ y P-1 exp(- λ y P ). a. Derive the log likelihood function and the first order conditions for estimation of P and λ . Notice that in computing the log likelihood function, it is necessary to compute log λ . Therefore, we must be sure that λ > 0. In practice, this is done by ‘parameterizing’ λ as λ = exp( α ), where α is the parameter to be estimated. Note that there is no restriction on the value of α . So, we estimate P and α then when done, we compute λ = exp( α ). Derive the log likelihood in terms of P and γ . Obtain the first order conditions for estimation of P and γ . (Note, P must also be positive. However, in practice, it generally works out that the data ensure this, and it is generally not necessary to force the restriction. Finally, note that the case of P = 1 produces the exponential model that we will examine below, f(y) = λ exp(- λ y). How would you test the hypothesis that P = 1 using your estimated Weibull model? Department of Economics

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Part II. Application: (A Hierarchical distribution). The applications below will use the GSOEP healthcare data, healthcare.csv An exponential regression model might be formulated as follows (this is called a “loglinear model”): Let y i be the time until failure of some electronic component. A model that is often used for this phenomenon is the exponential model that we have used in class: f ( y i ) = λ i exp(- λ i y i ), λ i > 0, y i > 0. (Notice that this is a special case of the Weibull model above, in which P = 1.) We believe that the distribution depends on a certain other variable, x i , such that λ i = exp( α + γ x i ). We are interested in estimation of the parameters α and γ and in manipulation of the model after estimation. a.
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Problem5 - Department of Economics ECONOMETRICS I Professor...

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