and Total (Sw thin Seetueen)-(Swithin B within SBetween B Between) within SEetweeny-Swithin B within +(Swithin Szetweeny-ISEetween B Betwe +F where FWithin =(Sw ithin+SBetween )-ISwithin and WIthin +FBetween=(Swithin+ getween )-(Sw ithin +SBetween )=I. That is the pooling OLS estimator is a matrix weighted average of the within- and between-groups estimator 2 Random effects Consider the model 1,2,…N;t=1,2,,T where there are k regressors including a constant and now the single constant term a is the mean of the unobserved heterogeneity, E(za). The component d is constant through time. We assume further (u)=0; E(2)=a2; E(Eiti)=0 for all i, t, and E(s1s)=0计ft≠sori≠ E(u2u)=0ifi≠j let y; and Xi(including the constant term) be the T observations the ith unit, i be at x 1 column of ones and let m1,n2,…,nr],and S Between xy = S Between xx βˆBetween , we have βˆTotal = (S Within xx + S Between xx ) −1 (S Within xx βˆWithin + S Between xx βˆBetween) = (S Within xx + S Between xx ) −1S Within xx βˆWithin + (S Within xx + S Between xx ) −1S Between xx βˆBetween = F WithinβˆWithin + F BetweenβˆBetween , where F Within = (S Within xx +S Between xx ) −1S Within xx and F Within+F Between = (S Within xx + S Between xx ) −1 (S Within xx +S Between xx ) = I. That is the pooling OLS estimator is a matrix weighted average of the within- and between-groups estimator. 2 Random Effects Consider the model yit = x 0 itβ + α + ui + εit, i = 1, 2, ..., N; t = 1, 2, ..., T. where there are k regressors including a constant and now the single constant term α is the mean of the unobserved heterogeneity, E(z 0 iα). The component ui is the random heterogeneity specific to the ith observation and is constant through time. We assume further E(εit) = E(ui) = 0; E(ε 2 it) = σ 2 ε ; E(u 2 i ) = σ 2 u ; E(εituj) = 0 for all i,t, and j; E(εitεjs) = 0 if t 6= s or i 6= j; E(uiuj) = 0 if i 6= j. Denote ηit = εit + ui , let yi and Xi (including the constant term) be the T observations the ith unit, i be a T × 1 column of ones, and let ηi = [ηi1, ηi2, ...., ηiT ] 0 , 10