PanelDataProblemSet1

# PanelDataProblemSet1 - Department of Economics Econometric...

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Econometric Analysis of Panel Data Assignment 1 Part I. Mathematical Statistics The density f( y ) for a nonnegative random variable, y , is exponential with parameter λ , so f( y ) = 1/ λ exp(- y / λ ), y > 0, λ > 0. For this random variable, the mean is E[ y ] = λ . We make this a regression model by formulating the conditional mean (regression) function E[ y | x ] = λ ( x ) = exp( α + β x ). Suppose, further, that x is distributed uniformly with density f( x ) = 1, 0 < x < 1. Note that with this assumption, the joint density of y and x is f( y , x ) = f( y | x ) f( x ) = [1/exp( α + β x )] exp[- y /exp( α + β x )] × 1. 1. Derive the parameters of the linear projection, P ( x ) = δ 0 + δ 1 x , where δ 0 = E[ y ] – δ 1 E[ x ] and δ 1 = Cov[ x , y ]/Var[ x ]. Suppose α = 1/3 and β = 2. What are the values of δ 0 and δ 1 ? Hint: E[ y ] = E x E[ y|x ] = 1 1 0 0 exp( ) 1 exp( ) exp( ) x dx x dx α+β × = α β ∫∫ and Cov[ y , x ] = Cov[ x ,E[ y | x ]] = E x { x × E[ y | x ]} – E[ x ] E[ y ]= 1 0 exp( ) exp( ) x xdx αβ -E[ x ]E[ y ]. Find help at http://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions 2. Consider the linear Taylor series approximation to the conditional mean function. What are the values of θ 0 and θ 1 in the Taylor series: E*[y|x] = θ 0 + θ 1 x when the expansion point is E[ x ] = 1/2 and as before, α = 1/3 and β = 2. Department of Economics

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Part II. Linear Regression Analysis Data for this exercise are on the course website – please use the “Cornwell and Rupert Returns to Schooling Data.” We begin with the linear regression model (using the variable names in the data set) (*) LWAGE it = β 1 + β 2 OCC it + β 3 SMSA it + β 4 MS it + β 5 FEM i + β 6 ED i + β 7 EXP it + ε it The dependent variable is log wage. The RHS variables are defined in the data set. Although this is a panel data set, we are going to ignore that aspect and “pool” the data. 1. Compute the linear least squares regression results and report the coefficients, standard errors, ‘t-ratios,’ R 2 , adjusted R 2 , residual standard deviation, and F statistic for testing the joint significance of all the variables in the equation. 2. Test the hypothesis that neither education (ED) nor experience (EXP) is a significant determinant of the expected log wage. Use an F (Wald), likelihood ratio (assuming normality of ε ), and a Lagrange multiplier (also assuming normality) test. In each case, document in minute detail exactly how you are computing your results and what conclusion you reach.
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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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PanelDataProblemSet1 - Department of Economics Econometric...

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