Technicl Appendix: Childhood Fmily Structure nd Schooling Outcomes: Evidence for Germny Mrco Frncesconi* Stephen P. Jenkins Thoms Siedler Universy of Essex nd Universy of Essex Universy of Essex Instute for Fiscl Studies nd DIW Berlin nd DIW Berlin Decemer 008
Identifiction Issues nd Estimtion Methods A centrl concern of this study is tht n estimted effect of childhood fmily structure on eduction my e spurious due to the mutul ssocion etween fmily structure nd children s schooling chievements nd some unmesured true cusl fctor. For exmple the ssocion etween hving experience of life in non-intct fmily nd lower eductionl ttinment my not e the result of fmily structure during childhood; differences in ttinment my simply reflect the chrcteristics of fmilies in which children of lone mothers re rought up. Our econometric strtegy is to pply numer of different techniques which plce different ssumptions on the dt in order to identify the effects of experiencing life in nonintct fmily during childhood. Ech of these techniques hs dvntges nd disdvntges. Although the methods differ they shre numer of common elements. For this reson in wht follows we strip down the corresponding sttisticl models in order to etter highlight the difference in their identifying restrictions. Siling Difference Model Let index fmilies nd i index young dults (or children). For convenience ssume tht the reltionship we estimte is (A.) S = β F + u where S is eduction F is vrile tht indictes childhood fmily structure (e.g. ever lived in non-intct fmily in the first 0 yers of life) nd u is rndom shock wh zero men. In eqution (A.) which for the moment excludes other determinnts of schooling β is the prmeter of interest. Consistent estimtion of β requires tht F e uncorrelted wh the disturnce term. u We investigte this issue using frmework suggested y Behrmn et l. (994) nd Rosenzweig nd Wolpin (995). Consider two-child fmily. For the i-th child in fmily wh siling k u cn e decomposed s follows: (A.) u = δ ε + δ ε k + ε + η In our empiricl nlysis the reltionship (A.) is expnded to include set of child- nd fmily-specific vriles tht my e fixed or time-vrying. In this formultion β is ssumed to e the sme for ll individuls. Arguly the effect of fmily structure is heterogeneous (i.e. some children might e etter off in non-intct fmily while others might e worse off). The siling difference pproch would pply even if one specifies rndom-coefficients model in which β = β + ζ nd E( ζ u ) = E( ζ X ) = 0. However my not e fesile to estimte such model ecuse repeted oservtions wh ech fmily re needed wheres most of the fmilies in our smple consist of only two or three silings.
whereε nd ε k re the endowments or ily of ech siling ε denotes the genetic endowments tht re common to oth children of fmily nd η is rndom shock tht is specific to i in inclusive of mesurement error in schooling. Endowments of oth silings re likely to e trnsmted cross genertions in Glton-type lw of herily (Becker nd Tomes 986): (A.) ε = ρε + ψ where ψ is child-specific idiosyncrtic disturnce wh zero men nd uncorrelted wh other unoservles (including ψ ik the corresponding rndom term for siling k). Assuming tht 0 ρ < implies tht endowments regress towrds the men cross genertions. Finlly the fmily structure vrile F is self function of unoserved vriles tht pertin to the fmily ( φ ) nd to the two silings ( μ nd μ k) : (A.4) F = γ μ + γ μ k + πφ + θ where θ is disturnce tht ffects F ut does not ffect S except indirectly through F. It is well known tht the prmeter β is not identified wh equtions (A.) (A.4) if π is not zero ρ is not zero nd if eher δ nd γ or δ nd γ re not zero even if orthogonly restrictions on the moments involving η ψ nd θ re imposed. Tht is β is estimted wh is if eqution (A.) is estimted cross individuls wh different vlues of fmily nd children s endowments. 4 The siling difference model estimtes β y compring eductionl outcomes mong silings ccording to whether they experienced life in non-intct fmily during childhood. In our two-child fmily cse the siling difference estimtor is computed from (A.5) ΔS = β ΔF + Δu where Δr = r r k for ny term r in eqution (A.5). The whin-fmily covrince etween fmily structure differences nd the disturnce term in (A.5) is thus given y (A.6) cov( ΔF Δu) = ( γ γ )( δ δ ) E( ΔμΔθ ) + ( γ γ ) E( ΔμΔη). Therefore sufficient condion for β to e identified is tht γ = γ ; tht is prents respond to their children s idiosyncrtic endowments eqully. A stronger condion would e to These would e the selection-on-oservles ssumptions which re relevnt for ll cross-sectionl estimtors including those sed on propensy score mtching methods. 4 This conclusion pplies to propensy score mtching estimtes since these too rely on the selection on oservles ssumption of the cross-sectionl model (A.) (A.4).
ssume tht children s endowments do not ffect prents ehviour (fmily structure) or lterntively there re no intrfmily responses (i.e. γ = γ 0). = 5 Before-After Comprisons nd Qusi-Experiments To see the ssumptions needed for identifiction in this cse we modify our empiricl frmework slightly nd explicly llow prents ehviour nd child unoservles to differ over time so tht: (A.7) F F + F nd u u + u wh the suscripts nd indicting some time period efore nd fter specific event occurs (e.g. deth of prent nd the introduction of divorce lw reform). Equtions (A.) nd (A.4) respectively ecome: (A.') u = δ ε + ε + ε + η τ τ τ τ nd (A.4') F = γ μ + μ + πφ + θ τ τ τ τ for τ =. A strightforwrd efore-fter comprison of outcomes for the sme individul requires repeted informtion on S which is prolemtic when schooling is mesured in term of highest eductionl ttinment. Bering this in mind nd imposing orthogonly restrictions on ll moments involving η τ ψ nd θ τ (A.) (A.') (A.) nd (A.4') will imply (A.8) fixed-effects estimtor sed on cov( Δ F Δu) = γ δ σ γ δ σ + γ δ σ γ δ σ where σ cov( p q) for p q = ε ε μ nd μ. Notice tht β cnnot e identified even pq = if we ssume tht the correltions etween ε s nd μ s re the sme efore nd fter the chnge of interest. Identifiction insted cn e gurnteed if there re no intrfmily responses (i.e. γ = γ = 0) or if child endowments re not regime specific (i.e. δ = δ = 0). The ssumption tht δ δ = 0 is perhps more credile when some exogenous = events re tken s instruments (e.g. the pssge of divorce lw regultions) rther thn others such s remrrige or prentl deth since specific relistions of ε (nd 5 If the fmily lloctes schooling so s to reinforce endowment differences etween silings then γ > 0 nd γ < 0; if insted the fmily compenstes for child-specific endowment differentils then γ < 0 nd γ > 0 (see Berhmn et l. 994). The ssumption of no intrfmily responses ws imposed y Rosenzweig nd Wolpin (99) nd Currie nd Cole (99) to estimte the determinnts of irth outcomes. 4
expecttions out ε ) my ultimtely led to such events. Even when those more credile exogenous circumstnces pply one concern is tht the regultions my e endogenous in the sense tht there my e trends in eductionl ttinments of children of divorced prents tht re correlted wh the introduction of specific divorce lw (Gruer 004). To illustrte this point let denote dummy vrile tht is equl to if individul i s prents divorced during his/her childhood nd 0 otherwise nd let q e the time period in which the divorce lw reform occurred. Suppose our outcome of interest the following specifiction (A.9) S 6 D i = α + α D α α D t α t q βd t q) + ξ 0 i + ( + i) + 4I( ) + ii( S is determined y wh E( ξ D t) = 0 where E ( ) is the expecttion opertor nd the term I( w) is function indicting tht the event w occurs (i.e. the post-reform period) so tht in eqution (A.9) β is our prmeter of interest. The prmeters α nd α reflect two different time trends in eductionl chievement for children in intct fmilies nd children in divorced fmilies respectively while α 4 reflects common ump in S from the time the reform is introduced onwrds. From (A.9) difference-in-difference estimte of β is given y β + α ( k + k' ) where k + k' represents the verge numer of clendr periods (sy yers) etween the post-reform nd pre-reform period oservtions in the smple. 7 Unless α = 0 this is clerly ised estimte of β. The is rises precisely ecuse the time evolution of S differs etween children in intct fmilies nd children of divorced prents. Wh this pproch therefore modelling group-specific trends will e crucil. 6 The fmily suscript hs een dropped for convenience. 7 Wh the fmily structure vrile nd the residuls in (A.9) following the sme structures s those specified in (A.') nd (A.4') necessry condion for identifiction of β is s efore tht child endowments re independent of the reform (or tht there re no intrfmily responses). 5