Average Treatment Effect

Transcription

1 Average Treatment ffect L Gan Aprl, 28 We are nterested n average canges n outcome after a polc cange. As n te case of te medcal feld, e are nterested n te outcomes of a ne medcne, ncludng ts effectveness and ts sde effect. To do tat, t s tpcal tat e randoml dvde patents nto to groups, te treatment group, and te control group. Te treatment group s gven te medcne le te control group s gven te placebo. Consder an economc eample. An mportant polc queston s o to elp need famles. Income transferrng programs, suc as Ad to Famles t Cldren (AFDC) creates dsncentves of orkng. One alternatve metod s te arned Income Ta Credt (ITC). lgblt for ITC: Gross ncome belo a specfed amount (n 27, te amount s $39,783 f ou cldren and $4,59 f ou do not ave cldren). Benefts (n 27): mamum benefts: $428 f no cldren; $2,853 f one cld; and $4,76 f to cldren. Te Ta Reform Act of 986 ncludes an epanson of earned ncome ta credt. We are nterested f epanson of ITC elps ncreasng labor suppl. Denote f t treatment (eperences epansons of ITC), and tout treatment (not affected b epansons of ITC). Average Treatment ffect (AT) s defned as: AT ( ) () Te dffcult n estmatng () s tat e observe eter or, not bot, for eac person.. Regresson metod: te Dfference-n-dfference metod In ts case, e potentall can observe outcomes before te treatment and after te treatment for te same person or for dfferent persons. We ave to tme perods, sa ear and ear. One ould sa tat e can smpl appl (). In te case of ITC epanson, ear s before 986, and ear s after 986. Hoever, ts s not entrel approprate snce tere ma be oter factors tat affect treated people as ell. Te dfference n labor suppl before 986 and after 986 ma be due to overall economc envronment. Terefore, e need a control group. Tere are to groups, te control group (denoted as group A), and te treatment group (denoted as B). At perod, no treatment

2 for bot groups. At perod, te treatment group eperences polc cange (treatment) le te control group does not. Let D denote a dumm varable for tme perod, and D B B denote te treatment group. Te smplest regresson for analzng te mpact of te polc cange s: β δ D β D B B δd *D BB u (2) It s eas to so tat: ˆ δ (3) B, B, A, A, Ts s te regresson of (2) s often called te dfference-n-dfference. In te eample of te epansons of ITC, ssa and Lebman use sngle omen tout cldren as te control group, and sngle omen t cldren as te treatment group. Ter tme perods are: as tme, and as tme. Te regresson, terefore, s: ( lfp ) Φ( α βz γ CldrenDumm γ post86 γ ( CldrenDumm post86 )) Pr t t t 2 ssa and Lebman (996) fnd tat sngle omen t cldren ncreased ter relatve labor force partcpaton b up to 2.8% percentage ponts. t 2. Regresson metod More generall, suppose e do not observe outcome varables before and after treatment. Ten: let f treatment. Te observed outcome can be rtten as: (-). (4) If s ndependent of, ten: ( - ) ( - ) ( ) ( ) In fact, e onl need te eaker assumpton (rater tan ndependence): mean ndependence: ( ) ( ), ( ) ( ). In te ssa and Lebman (996) eample, te treatment group s te sngle omen t cldren le te control group s te sngle omen tout cldren. It s entrel possble te treatment varable s NOT ndependent t te outcome varable (labor force partcpaton). No let: μ v, (v ) μ v, (v ) 2

3 Terefore, (4) can be rtten as: (-) μ (μ - μ ) v (v -v ) (5) Frst, assume condtonal mean ndependence: Assumpton (AT ): (a) (,) ( ), and (b) (,) ( ) Intuton: even toug and ma be correlated t, te are uncorrelated t f e partal out. Takng epectaton of (5) (and t AT ): (,) μ α g () (g ()- g ()), (6) ere αμ - μ s te AT, and g ()(v ). Lnearzaton of g (): g () β. (,) μ α β (β -β ). Rerte t: (,) μ α β (-ψ) δ, ere ψ(), and δ β -β. Te last term s to ensure tat g ()- g (). So te regresson to estmate AT α s: on,,, ( ) Here te control functons nvolve not just, but also te nteractons of te covarates t te treatment varable. We can estmate treatment effect condtonal on : AT ˆ 3. Propenst Score: Let p() Pr(). ˆ α ( )δˆ ( p()) ( p())( (-) ) p() (-) p() Take condtonal epectaton t respect to : 3

4 [( p()),] m () p() (-) m () p()m (), ere ( j,) ( j )m j (). Takng epectaton t respect to : Terefore, { [( p()),]} [m () p() (-) m () p()m ()] p()m () p() (- p()) m () p() p()m () m ()p()(-p())- m ()p()(- p()) (m ()-m ())p()(-p()) AT m m (( p( ) ) ) p( )( p( )) A smple and popular estmator n program evaluaton s obtaned from OLS regresson: on,, pˆ ( ) ere coeffcent for s te estmate of te treatment effect. In oter ords, te estmated propenst score plas te role of te control functon. 4. Dumm ndogenous Varables Procedure : Consder te model agan: (, ) μ α β u, (7) s endogenous. Agan, f treated, and oterse. Assume tat Pr(,z) G(, z; γ) () stmate te bnar response model Pr(,z ) G(,z ;γ), and obtan te ftted values. Ĝ (2) stmate (4) usng nstruments, Ĝ and. Procedure as mportant robustness propert: (a) Because e use Ĝ as an IV, te model Pr(,z ) G(,z ;γ) does not ave to be correctl specfed. 4

5 (b) Tecncall, α and β are dentfed even f e do not ave etra varables ecluded from. In oter ords, e do not ave z. In ts case, te dentfcaton s completel comng from nonlnear functon of. Hoever, e can rarel justf te estmator n ts case. Suppose tat gven follos a probt model (no z). Because G(, γ) Φ(γ γ ), s a nonlnear functon of, t s not perfectl correlated t, so t can be used as IV for tecncall. (c) In prncple, t mportant to recognze tat Procedure s not te same as usng G as a regressor n place of. on, Ĝ and. Consstenc of te OLS estmators from te regresson: u δ αgˆ β (8) reles on G(.) to be correctl specfed. Note tat (8) also as problems t standard errors tat need to be corrected. Furter, f e allo te nteracton term: Procedure 2: ( ) e δ δ α β (9) (a) stmate Pr(,z ) G(,z ;γ) (b) Use, and Gˆ as IVs. Ĝ, and Dscussons are te same as before. 5. Regresson dscontnut Consder an eample: e are nterested n effect of class sze on te performance of students. A tpcal assumpton s tat a smaller class sze elps learnng. Te problem of a tpcal stud, obvousl, s tat te class sze s not eogenous. It s often determned partl b te student performance. In Angrst and Lav (Quarterl Journal of conomcs, 999. Usng Mamondes Rule to stmate te ffect of Class Sze on Scolastc Acevement), suppose tere s a la requrng class sze to be less tan 4; f a scool as 2 students, ten tere ll be 5

6 four classes. Hoever, f a scool as 2 students, ten tere ould be fve classes. Te class sze s smaller for scools t 2 students tan scools for 2 students. Tese to scools sould be smlar n ever aspect oter tan class sze. Oter eamples: Ant-povert program targeted to ouseolds belo a gven povert nde. Penson program targeted to populaton above a certan age. Ta rebate ncome loer tan a certan tresold. It s useful to dstngus beteen to general settngs, te Sarp and te Fuzz Regresson Dscontnut desgns. Sarp Desgn: ( > ) Te assgnment s a determnstc functon of one of te covarates, te forcng (or treatment-determnng) varable. All unts t > are assgned to te treatment group (and partcpaton s mandator for tese ndvduals), and all unts t are assgned to te control group. In ts sarp desgn, e look at te dscontnut n te condtonal epectaton of te outcome gven te covarates to uncover te AT: AT Fuzz desgn: lm [ ] lm [ ] ( ) ( ) Pr( ) s dscontnuous at knon value. In ts case, t s observed tat f a person s treated or not. ( s observed). Hoever, t s no longer te case tat all > belongs to te treatment group. Some ll ave. Assumpton: () lm ( ) and lm ( ) () - est. Te sarp and fuzz desgns dffer n tat n te sarp desgn te treatment assgnment s determnstc gven, le te fuzz desgn te treatment assgnment ma depend on addtonal factors unobserved b econometrcan. In bot desgns, te dscontnut pont s knon. In Angrst and Lav (999), t s a sarp desgn, t knon beng at te multples of 4,.e., 4, 8, 2,. Assumpton: ( ) s contnuous n at. 6

7 Ts s assumpton s vald ere e ave reason to beleve tat person close to tresold c are smlar and tus ould eperence smlar outcome absent treatment. Teorem: AT, denoted as α: α Proof: Let Δ to be a small postve number. ( Δ Δ Δ Δ Δ Δ Δ Δ Δ Δ ) Δ Δ α As Δ, e ave: α Here e use te fact (assumpton) tat ( ) s contnuous at tout treatment. Te concluson follos. Gven ts teorem, e can obtan an estmate of α b estmatng, -,, and -. Tere are several as to estmate ts. Te most popular a s to do t nonparametrcall. In practce, ˆ ˆ Note for a sarp desgn RD, - -. For a fuzz desgn RD, ˆ ˆ 7

8 ere s te banddt. An nterestng note s tat ts s numercall equvalent to an IV estmator for te regresson: on for people n te subsample ( ) usng ( ) as te IV. Te regresson metod can be useful because one can add control varables n te. regresson. Note for te fuzz desgn, t s not necessar tat Practcall, for a sarp desgn,. Grap te data b computng te average value of te outcome varable over a set of bns. Te banddt as to be large enoug to ave suffcent amount of precson so tat te plots look smoot on eter sde of te cutoff value, but at te same tme small enoug to make te jump around te cutoff value clear. 2. stmate te treatment effect b runnng lnear regresson on bot sdes of te cutoff pont. Snce e propose to use a rectangular kernel, tese are just standard regresson estmated tn a bn of dt on bot sdes of cutoff pont. Note tat:. Standard errors can be computed usng standard least square metods (robust standard errors). Te optmal banddt can be cosen usng cross-valdaton metods. Fuzz desgn:. Grap te average outcome over a set of bns as n te case of SRD, but also grap te probablt of treatment. 2. stmatng te treatment effect usng TSLS. 8

9 Fgures and 2 sos tat tout treatment, ( ) ( ) 5.. Here s Fgure 2 9

10 Fgure 3 sos tat te effect of te treatment. ( ) ( ) Fgure 4 sos te Average Treatment ffect.