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Unformatted text preview: Econometric Analysis of Panel Data William Greene Department of Economics Stern School of Business Econometric Analysis of Panel Data 2. Econometric Methods A Statistical Relationship A relationship of interest: Number of hospital visits: H = 0,1,2,… Covariates: x 1 =Age, x 2 =Sex, x 3 =Income, x 4 =Health … Causality and covariation Theoretical implications of ‘causation’ Comovement and association Intervention of omitted or ‘latent’ variables Models Conditional mean function: E[y  x ] Other conditional characteristics – what is ‘the model?’ Conditional variance function: Var[y  x ] Conditional quantiles, e.g., median [y  x ] Other conditional moments Conditional probabilities: P(y x ) What is the sense in which “y varies with x?” Using the Model Understanding the relationship: Estimation of quantities of interest such as elasticities Prediction of the outcome of interest Control of the path of the outcome of interest Representing Covariation Conditional mean function: E[y  x ] = g( x ) Linear approximation to the conditional mean function: Linear Taylor series The linear projection (linear regression?) δ δ k K k=1 k k k K k=1 k k ˆ g( ) = g( ) + [g  = ](x x ) Σ = + (x x ) Σ x x x x = γ + Σ γ γ = K k k 1 k k1 g* (x)= (x E[x ]) E[y] Var[ ]} { Cov[ ,y]} x x γ={ Projection and Regression The linear projection is not the regression, and is not the Taylor series. Example: (Derivation and demonstration in problem set 1) α β ≤ ≤ f(yx)=[1/ (x)]exp[y/ (x)] λ λ (x)=exp( + x)=E[yx] λ x~ U[0,1]; f(x)=1, 0 x 1 For the Example: α=1, β=2 Linear Projection X 4 .1 8 8 . 3 5 1 2 . 5 3 1 6 . 7 0 2 0 .8 8 .0 0 .2 0 .4 0 . 6 0 .8 0 1 .0 0 . 0 0 E Y _ X P R O J E C T N T A Y L O R Variable Linear Projection Conditional Mean Taylor Series What About the Linear Projection? What we do when we linearly regress a variable on a set of variables Assuming there exists a conditional mean There usually exists a linear projection. Requires finite variance of y. Approximation to the conditional mean If the conditional mean is linear Taylor series equals the conditional mean Linear projection equals the conditional mean Application: Doctor Visits German Individual Health Care data: N=27,236 Model for number of visits to the doctor: Poisson regression (fit by maximum likelihood) E[VIncome]=exp(1.412  .0745*income) Linear regression: g*(Income)=3.917  .208*income H i s t o g r a m f o r N u m b e r o f D o c t o r V i s i t s Frequency D O C V I S 2 7 8 8 5 5 7 6 8 3 6 4 1 1 1 5 2 4 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6 4 0 4 4 4 8 5 2 5 6 6 0 6 4 6 8 7 2 7 6 8 0 8 4 8 8 9 2 9 6 1 0 0 1 0 4 1 0 8 1 1 2 1 1 6 1 2 0 Conditional Mean and Projection...
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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.
 Fall '11
 WillamGreene
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