Fixed and Random Effects a Unobserved individual effects in regression ELyit Xt, CI Notation: yit=X B+C+ 12 T rows K columns a Linear specification Fixed Effects E[C X]=g(i); effects are correlated with included variables. Common CovlXitC]#0 Random Effects E[C X]=F; effects are uncorrelated with included variables. If X, contains a constant term, H=O WLOG Common CovXt C]=O, but ELc X]= His needed for the full modelFixed and Random Effects  Unobserved individual effects in regression: E[yit | xit, ci ] ◼ Notation: ◼  Linear specification: ◼ Fixed Effects: E[ci | Xi ] = g(Xi ); effects are correlated with included variables. Common: Cov[xit,ci ] ≠0 ◼ Random Effects: E[ci | Xi ] = μ; effects are uncorrelated with included variables. If Xi contains a constant term, μ=0 WLOG. Common: Cov[xit,ci ] =0, but E[ci | Xi ] = μ is needed for the full model it it i it y = + c + x  i i1 i2 i i iT T rows, K columns         =          x x X x