Econometrics 17. Linear Models for panel data
Econometrics 17. Linear Models for Panel Data
Panel Data sets 口 Longitudinal data National longitudinal survey of youth (nls) British household panel survey ( Bhps Panel Study of Income Dynamics( PsiD a Cross section time series Grunfeld's investment data Penn world tables a Financial data by firm year t-rt=β(rmt-rt)+ti=1,,many;t=1,…many EXchange rate data, essentially infinite T, large n Effects:β=β+
Panel Data Sets Longitudinal data ◼ National longitudinal survey of youth (NLS) ◼ British household panel survey (BHPS) ◼ Panel Study of Income Dynamics (PSID) Cross section time series ◼ Grunfeld’s investment data ◼ Penn world tables Financial data by firm, year ◼ rit – rft = i (rmt - rft) + εit, i = 1,…,many; t=1,…many ◼ Exchange rate data, essentially infinite T, large N ◼ Effects: i= + vi
Terms of art a Cross sectional vs, time series variation (history consumption function studies Heterogeneity a group effects(individual effects) Fixed effects and or random effects Substantive differences? Is it possible to tell them apart in observed data?
Terms of Art Cross sectional vs. time series variation - (history: consumption function studies) Heterogeneity Group effects (individual effects) Fixed effects and/or random effects ◼ Substantive differences? ◼ Is it possible to tell them apart in observed data?
Panel data u Rotating panels: Spanish household survey Spanish income study http://www.cemfi.es/valbarran/0008r.pdf) Efficiency analysis: Efficiency measurement in rotating panel data,Heshmati, A, Applied conomics30,1998,pp.919930 o Hierarchical(nested )data sets: Student outcome, by year, district, school, teacher
Panel Data Rotating panels: Spanish household survey ◼ Spanish income study (http://www.cemfi.es/~albarran/0008r.pdf) ◼ Efficiency analysis: “Efficiency measurement in rotating panel data,” Heshmati, A, Applied Economics, 30, 1998, pp. 919-930 Hierarchical (nested) data sets: Student outcome, by year, district, school, teacher
Nested panel data a Antweiler. w nested random effects Journal of econometrics 101, 2001, 295-313 Sulfide concentration year, country, station B+B,(logGDP/kmst+B3log(k/Lct+B4Communist+ +阝sog(O∥ Pr ice)2+βt+st+vs+Ws
Nested Panel Data Antweiler, W., Nested Random Effects…” Journal of Econometrics, 101, 2001, 295-313 2 1 2 c,s,t 3 c,t 4 8 9 , , , Sulfide concentration(year,country,station=t,c,s) = β +β (logGDP/km ) +β log(K/L) ... + log( Pr ) c t c s t c s s Communist Oil ice t v w + + + + + +
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
