Balanced and unbalanced panels 口 Distinction o a notation to help with mechanics a The role of the assumption Mathematical and notational convenience 口 Balanced,NT Unbalanced: 2L T Is the fixed Ti assumption ever necessary? sUr models
Balanced and Unbalanced Panels Distinction A notation to help with mechanics zi,t, i = 1,…,N; t = 1,…,Ti The role of the assumption ◼ Mathematical and notational convenience: Balanced, NT Unbalanced: ◼ Is the fixed Ti assumption ever necessary? SUR models. N i i=1 T
Benefits of panel data a Time and individual variation in behavior unobservable in cross sections or aggregate t ime series a observable and unobservable individual heterogeneity a Rich hierarchical structures a Dynamics in economic behavior
Benefits of Panel Data Time and individual variation in behavior unobservable in cross sections or aggregate time series Observable and unobservable individual heterogeneity Rich hierarchical structures Dynamics in economic behavior
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 model
Fixed 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
Convenient notation 口 Fixed Effects Yt=01+x1β+ct Individual specific constant terms 口 Random effects Yt=X1β+ct+u Compound c"disturbance;error components
Convenient Notation Fixed Effects Random Effects it i it it y = + + x Individual specific constant terms. it it it i y = + + u x Compound (“composed”) disturbance; “error components
Assumptions for Asymptotics o Convergence of moments involving cross section X o Increasing T or Ti assumed fixed Fixed T asymptotics"(see text, p. 196) Time series characteristics are not relevant( may be nonstationary) If T is also growing need to treat as multivariate time series o Ranks of matrices. X must have full column rank. x may not if Ti< k) a Strict exogeneity and dynamics. If Xit contains yi t-1 then xit cannot be strictly exogenous. Xit will be correlated with the unobservables in period t-1.(To be revisited later.) o Empirical characteristics of microeconomic data
Assumptions for Asymptotics Convergence of moments involving cross section Xi . N increasing, T or Ti assumed fixed. ◼ “Fixed T asymptotics” (see text, p. 196) ◼ Time series characteristics are not relevant (may be nonstationary) ◼ If T is also growing, need to treat as multivariate time series. Ranks of matrices. X must have full column rank. (Xi may not, if Ti < K.) Strict exogeneity and dynamics. If xit contains yi,t-1 then xit cannot be strictly exogenous. Xit will be correlated with the unobservables in period t-1. (To be revisited later.) Empirical characteristics of microeconomic data