PanelDataNotes-14

PanelDataNotes-14 - Econometric Analysis of Panel Data...

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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 14. Nonlinear Models And Nonlinear Optimization Agenda Nonlinear Models Estimation Theory for Nonlinear Models Estimators Properties M Estimation Nonlinear Least Squares Maximum Likelihood Estimation GMM Estimation Minimum Distance Estimation Minimum Chi-square Estimation Computation – Nonlinear Optimization Nonlinear Least Squares Newton-like Algorithms; Gradient Methods (Background: JW, Chapters 12-14, Greene, Chapters 16-18) What is a ‘Model?’ Unconditional ‘characteristics’ of a population Conditional moments: E[g(y)|x]: median, mean, variance, quantile, correlations, probabilities… Conditional probabilities and densities Conditional means and regressions Fully parametric and semiparametric specifications Parametric specification: Known up to parameter θ Parameter spaces Conditional means: E[ y | x ] = m ( x , θ ) What is a Nonlinear Model? Model: E[g(y)| x ] = m( x , θ ) Objective: Learn about θ from y , X Usually “estimate” θ Linear Model: Closed form; = h( y , X ) Nonlinear Model Not wrt m( x , θ ). E.g., y=exp( θ ’x + ε ) Wrt estimator: Implicitly defined. h( y , X, )=0, E.g., E[y|x]= exp( θ ’x ) ˆ θ θ ˆ What is an Estimator? Point and Interval Classical and Bayesian ˆ f(data| model) ˆ ˆ I( ) sampling variability θ = θ = θ ± ˆ E[ | data,prior f( )] expectation from posterior ˆ I( ) narrowest interval from posterior density containing the specified probability (mass) θ = θ θ = θ = Parameters Model parameters The parameter space Interior of the parameter space Estimators of ‘parameters’ The true parameter(s) i i i i i i i i i exp( y / ) Example : f(y | ) , exp( ) Model parameters : Conditional Mean: E(y | ) exp( )- θ ′ = θ = θ ′ = θ = i i x x β β x x β The Conditional Mean Function 2 y,x m(x, ) E[y | x] for some in . A property of the conditional mean: E (y m(x, )) is minimized by E[y | x] (Proof, pp. 343-344, JW) θ = θ Θ- θ M Estimation Classical estimation method n i i=1 n 2 i i i i=1 1 ˆ arg min q( , ) n Example : Nonlinear Least squares 1 ˆ arg min [y -E(y | , )] n θ = θ θ = θ ∑ ∑ data x An Analogy Principle for M Estimation n i i 1 n P i i 1 1 ˆ The estimator minimizes q= q(data , ) n The true parameter minimizes q* = E[q(data, )] The weak law of large numbers: 1 q= q(data , ) q* = E[q(data, )] n = = θ θ θ θ θ → θ ∑ ∑ Estimation n P i i 1 P P 1 q= q(data , ) q*=E[q(data, )] n ˆ Estimator minimizes q True parameter minimizes q*...
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  • Fall '11
  • WillamGreene
  • Maximum likelihood, log likelihood, conditional mean function, Standard Error |b/St.Er.|P, M Estimation

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PanelDataNotes-14 - Econometric Analysis of Panel Data...

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