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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

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.

Ask a homework question - tutors are online