Consider a fixed effects model: yit=0+1x1it+2x2it+ai+uit. Define the composite error vit=ai+uit, where ai is uncorrelated with uit and the uit have constant variance u2 and are serially uncorrelated. (a) Derive Corr[vit,vis] for t=s. What does it mean for pooled OLS estimators of model coefficients? (b) Does your answer in point (a) above change when ai=0, i.e. when there are no unobservable time-constant characteristics of the cross-sectional units? Explain. (c) Consider now the random effects transformation. Define eit=vitvi, where is as defined during the lecture. (i) Derive E[eit]. (ii) Derive Var[eit]. (iii) Derive Corr[eit,eis] for t=s Consider a fixed effects model: yit=0+1x1it+2x2it+ai+uit. Define the composite error vit=ai+uit, where ai is uncorrelated with uit and the uit have constant variance u2 and are serially uncorrelated. (a) Derive Corr[vit,vis] for t=s. What does it mean for pooled OLS estimators of model coefficients? (b) Does your answer in point (a) above change when ai=0, i.e. when there are no unobservable time-constant characteristics of the cross-sectional units? Explain. (c) Consider now the random effects transformation. Define eit=vitvi, where is as defined during the lecture. (i) Derive E[eit]. (ii) Derive Var[eit]. (iii) Derive Corr[eit,eis] for t=s