Lecture09-2011

# Lecture09-2011 - Outline Basic Setup Generalize Least Squares Estimator Estimation of Production Functions Random Eﬀects in Panel Data Lecture IX

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Unformatted text preview: Outline Basic Setup Generalize Least Squares Estimator Estimation of Production Functions - Random Eﬀects in Panel Data : Lecture IX Charles B. Moss1 1 University of Florida September 22, 2011 Charles B. Moss Estimation of Production Functions - Random Eﬀects in Panel Outline Basic Setup Generalize Least Squares Estimator 1 Basic Setup Mother of all Heteroscedasticity Variance Matrix 2 Generalize Least Squares Estimator Procedure for Estimation Help Me! System Speciﬁcation Charles B. Moss Estimation of Production Functions - Random Eﬀects in Panel Outline Basic Setup Generalize Least Squares Estimator Mother of all Heteroscedasticity Variance Matrix Basic Setup Regression analysis typically assumes that a large number of factors aﬀect the value of the dependent variable, while some of the variables are measured directly in the model the remaining variables can be summarized by a random distribution yit = α + β xit + γ zit + ￿it (1) = (α + γ E [zit ]) + β xit + (￿it + [zit − E [zit ]]) When “numerous observations” on are observed over time, it is assumed that some of the omitted variables represent factors peculiar to individual and time periods. Going back to the panel speciﬁcation yit = α∗ + β ￿ xit + νit νit = αi + λt + µit Charles B. Moss (2) Estimation of Production Functions - Random Eﬀects in Panel Outline Basic Setup Generalize Least Squares Estimator Mother of all Heteroscedasticity Variance Matrix αi is the “individual variation” in the constant, λt is the “time variation” in the constant, and uit is the “pure error term”. For identiﬁcation purposes, we assume The expected value of each of these components is equal to zero E [αi ] = E [λt ] = E [uit ] = 0 (3) We also assume that each of these error terms are uncorrelated E [αi λt ] = E [αi uit ] = E [λt uit ] = 0 (4) The variance of the “individual eﬀects” is then deﬁned as ￿2 σα if i = j E [αi αj ] = (5) 0 if i ￿= j Charles B. Moss Estimation of Production Functions - Random Eﬀects in Panel Outline Basic Setup Generalize Least Squares Estimator Mother of all Heteroscedasticity Variance Matrix Continued The variance for the “time eﬀects” is then deﬁned as ￿2 σλ if t = s E [ λt λ s ] = 0 if t ￿= s The variance for the “pure residuals” are them ￿2 σu if i = j , t = s E [uit ujs ] = 0 Otherwise (6) (7) Finally, we assume that each of these eﬀects are uncorrelated with the independent variables ￿ ￿ ￿ E [αi xit ] = E [λt xit ] = E [uit xit ] = 0 Charles B. Moss (8) Estimation of Production Functions - Random Eﬀects in Panel Outline Basic Setup Generalize Least Squares Estimator Mother of all Heteroscedasticity Variance Matrix Mother of all Heteroscedasticity The variance of yit on xit based on the assumption above is 2 2 2 2 σy = σα + σλ + σu (9) Thus, this kind of model is typically referred to as a variance-component (or error-components) model. Letting ˜ Xi =￿(e , Xi￿ ) δ = µ, β ￿ νi￿ = (νi 1 , · · · νiT ) νit = αi + µit (10) the panel estimation model can be written in vector form as ˜ yi = Xi δ + νi i = 1, · · · N Charles B. Moss (11) Estimation of Production Functions - Random Eﬀects in Panel Outline Basic Setup Generalize Least Squares Estimator Mother of all Heteroscedasticity Variance Matrix Variance Matrix The expected value of the residual becomes ￿ ￿ 2 2 E νi νi￿ = σu IT + σα ee ￿ = V ￿ ￿ 2 1 σα −1 V = 2 IT − 2 2 σu σu + T σα Using the basic covariance estimator Qyi = Qe µ + QXi β + Qe αi + Qui = QXi β + Qui (12) (13) Whether αi is ﬁxed or random, the covariance estimator is unbaised. However, if αi is random, the covariance estimator is ot the best linear unbiased estimator (BLUE). Instead, a BLUE estimator can be derived using generalized least squares (GLS). Charles B. Moss Estimation of Production Functions - Random Eﬀects in Panel Outline Basic Setup Generalize Least Squares Estimator Procedure for Estimation Help Me! System Speciﬁcation Generalized Least Squares Estimator Because both uit and uis contain αi , they are correlated (by the individual eﬀect) ￿ ￿ δ = µ, β ￿ ￿N ￿ ￿N ￿ ￿ ￿ (14) ˜ ˜ˆ ˜ Xi￿ V −1 Xi δGLS = Xi￿ V −1 yi i =1 i =1 Inverse Variance Matrix V −1 1 =2 σu ￿￿ ￿ ￿ ￿ 1￿ 1 1￿ + ψ ee = 2 Q + ψ ee T T σu 2 σ ψ= 2 u 2 σu + T σα (15) 1 IT − ee ￿ T ￿ Charles B. Moss Estimation of Production Functions - Random Eﬀects in Panel Outline Basic Setup Generalize Least Squares Estimator Procedure for Estimation Help Me! System Speciﬁcation Procedure for Estimation Starting speciﬁcation ￿ WX X + ψ B X X ˜˜ ˜˜ Deﬁning TX X = ˜˜ N ￿ ￿ ￿ µ ˆ ˆ β ˜˜ Xi￿ Xi i =1 BX X ˜˜ ￿ GLS ￿ ￿ = WX y + ψ B X y ˜ ˜ TX y ˜ N ￿ (16) ˜ Xi￿ yi i =1 N ￿ ￿ ￿ ￿ ￿ (17) 1 ￿￿˜￿ ￿ ˜ ￿ ˜ = Xi ee Xi Bx y = 1 N X ee yi ˜ T i =1 i T i =1 WX X = TX X − B X X ˜˜ ˜˜ ˜˜ WX y = TX y − B X y ˜ ˜ ˜ (Hsiao p. 35) Charles B. Moss Estimation of Production Functions - Random Eﬀects in Panel Outline Basic Setup Generalize Least Squares Estimator Procedure for Estimation Help Me! System Speciﬁcation Continued The matrices BX X and BX y contain the sum of squares and ˜˜ ˜ the sum of cross products between groups, WX X and WX y are the corresponding matrices within groups, ˜˜ ˜ and TX X and TX y are the corresponding matrices for total ˜˜ ˜ variation. Charles B. Moss Estimation of Production Functions - Random Eﬀects in Panel Outline Basic Setup Generalize Least Squares Estimator Procedure for Estimation Help Me! System Speciﬁcation Help Me! This looks bad, but think about ￿￿ ￿ ￿ ￿ X QX ￿ = X ￿ Qy β ￿￿ ￿ ￿ ￿ 1 1 X ￿ IT − ee ￿ X β = X ￿ IT − ee ￿ y T T ￿ ￿ 1￿￿ 1￿￿ X ￿ X − X ee X β = X ￿ y − X ee y T T (18) With 1￿￿ X ee X T 1￿￿ ⇒ X ee y T TX X ⇒ X ￿ X ˜˜ BX X ⇒ ˜˜ WX y ⇒ X ￿ y ˜ BX y ˜ Charles B. Moss (19) Estimation of Production Functions - Random Eﬀects in Panel Outline Basic Setup Generalize Least Squares Estimator Procedure for Estimation Help Me! System Speciﬁcation System Speiciﬁcation Solving this system yields ￿ ψT ￿￿ ￿ ￿ µ ˆ ψ T N xi￿ i =1 ¯ ￿N ￿N = ￿ ˆ β GLS ¯ ¯ ¯￿ i =1 xi i =1 Xi QXi + ψ T i =1 xi xi ￿ ￿ ψ NT y ¯ ￿N ￿N ￿ ¯¯ i =1 Xi Qyi + ψ T i =1 xi yi (20) ψ NT ￿N Charles B. Moss Estimation of Production Functions - Random Eﬀects in Panel Outline Basic Setup Generalize Least Squares Estimator Procedure for Estimation Help Me! System Speciﬁcation Using the inverse of a partitioned matrix ￿ N N ￿ 1￿ ￿ ˆ βGLS = Xi QXi + ψ (¯i − x ) (¯i − x )￿ x ¯x ¯ T i =1 i =1 ￿ ￿ N N ￿ 1￿ ￿ × Xi Qyi + ψ (¯i − x ) (¯i − y ) x ¯y ¯ T i =1 i =1 ˆ ˆ = ∆βb + (IK − ∆) βCV µGLS = y − βGLS x ˆ ¯ˆ¯ Charles B. Moss ￿ −1 (21) Estimation of Production Functions - Random Eﬀects in Panel Outline Basic Setup Generalize Least Squares Estimator Procedure for Estimation Help Me! System Speciﬁcation Where ∆ = ψT ˆ βb = ￿ ￿ N ￿ Xi￿ QXi i =1 × N ￿ i =1 ￿ N ￿ i =1 + ψT N ￿ (¯i − x ) (¯i − x ) x ¯x ¯ i =1 ￿ ￿ ￿ −1 (¯i − x ) (¯i − y ) x ¯y ¯ (¯i − x ) (¯i − x )￿ x ¯x ¯ ￿ −1 ￿ N ￿ i =1 (¯i − x ) (¯i − y ) x ¯y ¯ ￿ (22) Where βb is the betweeen estimator. Charles B. Moss Estimation of Production Functions - Random Eﬀects in Panel Outline Basic Setup Generalize Least Squares Estimator Procedure for Estimation Help Me! System Speciﬁcation The variance of the estimator can be written as ￿ ￿ ˆ V βGLS = 2 σu ￿ N ￿ Xi￿ QXi i =1 + Tψ N ￿ i =1 (¯i − x ) (¯i − x ) x ¯x ¯ ￿ ￿−1 (23) Given that we don’t know ψ a priori, we estimate σu = ˆ2 NT ￿￿￿ i =1 t =1 σα = ˆ2 N ￿ i =1 ˆ￿ (yit − yi ) − βCV (xit − xi ) ¯ ¯ N (T − 1) − K ￿ ￿2 yi − µ − β xi ¯ ¯ ¯¯ N − (K + 1) Charles B. Moss − ￿2 (24) 12 σ ˆ Tu Estimation of Production Functions - Random Eﬀects in Panel ...
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## This note was uploaded on 02/01/2012 for the course AEB 6184 taught by Professor Staff during the Fall '09 term at University of Florida.

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