Supplemental Material for Testing the Unconfoundedness Assumption via Inverse Probability Weighted Estimators of (L)ATT

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Anthony Underwood
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1 Supplemetal Materal for Testg the Ucofoudedess Assumpto va Iverse Probablty Weghted Estmators of (LATT Stephe G. Doald Yu-Ch Hsu Robert P. Lel October 2, 23 Departmet of Ecoomcs, Uversty of Texas, Aust, Isttute of Ecoomcs, Academa Sca, Departmet of Ecoomcs, Cetral Europea Uversty, Budapest ad the Natoal Bak of Hugary,

2 Appedx A. Idetfcato The dervato of equatos ( ad (2 We ca wrte E[W ( W (] = E{[D( D(][Y ( Y (]} = E[Y ( Y ( D( D( = ]P [D( D( = ] = τ E[D( D(], where the secod equalty holds because uder mootocty (Assumpto (v, D( D( s ether zero or oe. Smlarly, E[W ( W ( Z = ] = E{[D( D(][Y ( Y ( Z = ]} = E[Y ( Y ( D( D( =, Z = ]P [D( D( = Z = ] = τ t E[D( D( Z = ], where the thrd equalty holds because uder mootocty, D( D( = mples D = Z. B. The proof of Theorem I order to smplfy otato, we set X = X. Let = { Z Y ˆq(X ( Z } Y { Z D, Γ = ˆq(X ˆq(X ( Z } D. ˆq(X = so that ˆτ = / Γ. The asymptotc propertes of ˆ ad ˆΓ are establshed the followg lemma. = Lemma 3 Uder the codtos of Theorem, ( = = δ(y, D, Z, X +o p ( ad ( Γ Γ = = γ(y, D, Z, X + o p (, where δ(y, D, Z, X = Z Y q(x ( Z ( Y q(x m (X γ(y, D, Z, X = Z D q(x ( Z D q(x Γ q(x + m (X (Z q(x q(x ( µ (X q(x + µ (X (Z q(x q(x The proof of Lemma 3 Recall the defto of W (z. By Assumpto (, t s true that E[W (z Z, X] = E[W (z X], z =,. That s, f we treat Z as the treatmet assgmet ad W (z as the potetal outcomes, W (z ad Z are ucofouded gve X. Also, t s straghtforward to check that Assumptos -5 of Thm. of HIR are satsfed. The result for follows drectly from t. A smlar argumet apples to Γ. Takg a frst order Taylor expaso of ˆ /ˆΓ aroud the pot (, Γ yelds ( (ˆτ τ = Γ = τ ( ( Γ Γ + op (. (4 Γ Γ Γ 2

3 Applyg Lemma 3 to (4 gves (8. Uder Assumpto (, we have E[ψ(Y, D, Z, X] = ad E[ψ 2 (Y, D, Z, X] <. Applyg the Ldeberg-Levy CLT to (8 shows (ˆτ τ d N (, V. Let t = Γ t = { ( Z Y ˆq(X ˆq(X ( Z Y } / ˆq(X, ˆq(X = N { ( Z D ˆq(X ˆq(X ( Z D } / ˆq(X, ˆq(X = = = so that ˆτ t = t / Γ t. The the secod part Theorem (a ca be show after we show the followg lemma: Lemma 4 Uder the codtos of Theorem, ( t t = E(Z ( Γt Γ t = E(Z = = { Z (Y m (X q(x ( Z (Y m (X q(x q(x + (m } (X m (X q Z + o p (, q(x { Z (D µ (X q(x ( Z (D µ (X q(x q(x + (µ (X µ (X Γ q Z q(x } + o p (. The proof of Lemma 4 Theorem of HIR. The proof s detcal to Lemma 3 wth Corollary of HIR place of C. The proof of Theorem 2 ad Lemma The proof of Theorem 2: The proof of Theorem 2 follows from Theorem ad Corollary of HIR. The proof of Lemma : Note that V ar(y ( = mples that P [Y ( = a] = for some a R. Defe Y ( = Y ( a, Y ( = Y ( a ad Y = D(Y ( + ( D(Y (. It s obvous that Y ( =. Hece, ˆτ t ad ˆβ t ca be wrtte as ˆτ t = ˆβ t = = Z Y ( = Z + a D = D Y = ˆp(X 2 + a ( = Z ˆq(X( Z ˆq(X Z D ˆq(X( ZD ˆq(X / = D ˆp(X2( D ˆp(X 2 = ˆp(X 2 = ˆq(X / = ˆq(X, 3

4 The proof follows easly from the followg four results: ( = D Y = ˆp(X 2 = Z Y = Z = o p (, (5 D ( ( ( D ˆp(X 2( D / ˆp(X 2 Z ˆq(X ( Z / ˆq(X Z D ˆq(X ( Z D ˆq(X /( ˆp(X 2 = o p (, (6 ˆq(X = o p (, (7 p ˆq(X Γt >. (8 We verfy each of these equatos tur. If P [Y ( = ] =, the Y = DY (, ad so ( ZY = ad ( DY =. Hece, = Z Y = Z = D = Z Y = Z D = = D Y = D, where the secod equalty holds sce ZY = ZDY ( = DY ( = DY ad ZD = D. We frst clam that ( = D = ˆp(X 2 = o p (, (9 whch mples ( = D = ˆp(X = o p (. (2 2 To see (9, let L( = ad L( =, the ths case E[L( L( D = ] = ad the ucofoudedess assumpto holds automatcally. The = D / = ˆp(X 2 s a estmator for E[L( L( D = ] ad by Corollary of HIR, ts asymptotc varace equal to because E[L( L( X 2 ] = = AT T, ad V ar(l( X 2 = V ar(l( X 2 = a.s. X 2. Equato (2 further mples that (ˆτt ˆβ t = ( = D Y = = D D Y = ˆp(X 2 = ( = D = ˆp(X ( D Y = o p ( O p ( = o p ( 2 ad ths shows (5. Let L ( =, L ( = ad L = DL (+( DL (. The = (D ˆp(X2( D ˆp(X 2 / ˆp(X 2 s a estmator for E[L ( L ( D = ] =. The result (6 follows from the same argumet we used to establsh (9. Equato (7 ca be show a smlar way ad, fally, (8 follows from Lemma 4. Take together, equatos (5-(8 mply that whether or ot the ucofoudedess assumpto holds, (ˆτt ˆβ t =a ( Z ˆq(X ( Z Z D ˆq(X ( Z D ˆq(X ˆq(X a ( D ˆp(X 2( D + ( = D Y ˆp(X 2 = ˆp(X 2 = Z Y = Z D Ths completes the proof. = o p ( O p ( o p ( + o p ( = o p (. 4

5 D. Addtoal smulato results Here we preset addtoal smulato results o the power propertes of the ucofoudedess test. The results correspod to those preseted Secto 5.2 for b =.5 except here b =.25 ad fewer cases are reported. As the volato of the ucofoudedess assumpto s less severe, power s geerally lower. The table s a partcularly good llustrato of the pheomeo descrbed the last paragraph of Secto 5.2. If ths table was cluded the paper, logcally t would be Table C.5 Appedx C. Table C.5: Propertes of the ucofoudedess test ad the dstrbuto of [ (LAT T (LAT T ] for varous estmators: b =.25 (ucofoudedess does ot hold q Seres (ˆq Trm. Estmator = 25 = 5 = 25 Mea s.e. E(ŝ.e. MSE Mea s.e. E(ŝ.e. MSE Mea s.e. E(ŝ.e. MSE Cost. quad..5% ˆτ t (LATT ˆβ t (ATT Combed Pre-tested Power/E(â.75/.8.37/.4.974/. L. quad..5% ˆτ t (LATT ˆβ t (ATT Combed Pre-tested Power/E(â.62/.8.58/.4.874/. L. 2 quad..5% ˆτ t (LATT ˆβ t (ATT Combed Pre-tested Power/E(â.54/..76/.2.578/. Rat. quad..5% ˆτ t (LATT ˆβ t (ATT Combed Pre-tested Power/E(â.85/.8.279/ Rat. 2 quad..5% ˆτ t (LATT ˆβ t (ATT Combed Pre-tested Power/E(â./ /.4.964/. Rat. 2 cube.5% ˆτ t (LATT ˆβ t (ATT Combed Pre-tested Power/E(â.4/.23.6/.5.93/. 5

6 E. Propesty score estmates We preset the propesty score estmates used to costruct the secod pael of Table 6. (X=three dummes, estmato method=sample splttg. Pages 7 ad 8 show the model estmates for p(x = P (D = X for males ad females ad the dstrbuto of the radom varable p(x. Pages 9 ad show the model estmates for q(x = P (Z = X for males ad females ad the dstrbuto of the radom varable q(x. Of course, as Z s completely radom, ˆq(X = E(Z+estmato error, ad ˆq(X s cetered tghtly aroud E(Z =.67. O pages to 4 we preset the propesty score estmates for the fourth pael of Table 6. the paper. The order s p(x for males, p(x for females, q(x for males, q(x for females. 6

7 MALES Depedet Varable: D (PARTICIPATION DUMMY Method: ML - Bary Logt (Quadratc hll clmbg Date: 2/5/2 Tme: 5:47 Sample: 5 IF SEX= AND GOODOBS= Icluded observatos: 583 Covergece acheved after 4 teratos Covarace matrx computed usg secod dervatves Varable Coeffcet Std. Error z-statstc Prob. C HS MINORITY BELOW HS*MINORITY HS*BELOW MINORITY*BELOW HS*MINORITY*BELOW McFadde R-squared.396 Mea depedet var S.D. depedet var S.E. of regresso Akake fo crtero Sum squared resd Schwarz crtero Log lkelhood Haa-Qu crter Devace Restr. devace Restr. log lkelhood LR statstc Avg. log lkelhood Prob(LR statstc Obs wth Dep= 2966 Total obs 583 Obs wth Dep= 27,4,2, Seres: PHAT Sample 5 IF SEX= AND GOODOBS= Observatos 583 Mea Meda.4566 Maxmum Mmum.3737 Std. Dev Skewess Kurtoss Jarque-Bera Probablty. 7

8 FEMALES Depedet Varable: D (PARTICIPATION DUMMY Method: ML - Bary Logt (Quadratc hll clmbg Date: 2/5/2 Tme: 5:5 Sample: 5 IF SEX= AND GOODOBS= Icluded observatos: 667 Covergece acheved after 4 teratos Covarace matrx computed usg secod dervatves Varable Coeffcet Std. Error z-statstc Prob. C HS MINORITY BELOW HS*MINORITY HS*BELOW MINORITY*BELOW HS*MINORITY*BELOW McFadde R-squared.3584 Mea depedet var S.D. depedet var S.E. of regresso Akake fo crtero Sum squared resd Schwarz crtero Log lkelhood Haa-Qu crter Devace Restr. devace Restr. log lkelhood LR statstc Avg. log lkelhood Prob(LR statstc. Obs wth Dep= 338 Total obs 667 Obs wth Dep= , 2,5 2,,5, Seres: PHAT Sample 5 IF SEX= AND GOODOBS= Observatos 667 Mea Meda Maxmum Mmum.3636 Std. Dev Skewess Kurtoss Jarque-Bera Probablty. 8

9 MALES Depedet Varable: Z (RANDOM OFFER OF SERVICES Method: ML - Bary Logt (Quadratc hll clmbg Date: 2/5/2 Tme: 6: Sample: 5 IF SEX= AND GOODOBS= Icluded observatos: 583 Covergece acheved after 4 teratos Covarace matrx computed usg secod dervatves Varable Coeffcet Std. Error z-statstc Prob. C HS MINORITY BELOW HS*MINORITY HS*BELOW MINORITY*BELOW HS*MINORITY*BELOW McFadde R-squared.96 Mea depedet var.6687 S.D. depedet var S.E. of regresso Akake fo crtero Sum squared resd Schwarz crtero Log lkelhood Haa-Qu crter Devace Restr. devace Restr. log lkelhood LR statstc Avg. log lkelhood Prob(LR statstc Obs wth Dep= 684 Total obs 583 Obs wth Dep= 3399,6,4,2, Seres: QHAT Sample 5 IF SEX= AND GOODOBS= Observatos 583 Mea.6687 Meda Maxmum Mmum Std. Dev Skewess Kurtoss Jarque-Bera Probablty. 9

10 FEMALES Depedet Varable: Z (RANDOM OFFER OF SERVICES Method: ML - Bary Logt (Quadratc hll clmbg Date: 2/5/2 Tme: 6:2 Sample: 5 IF SEX= AND GOODOBS= Icluded observatos: 667 Covergece acheved after 4 teratos Covarace matrx computed usg secod dervatves Varable Coeffcet Std. Error z-statstc Prob. C HS MINORITY BELOW HS*MINORITY HS*BELOW MINORITY*BELOW HS*MINORITY*BELOW McFadde R-squared.67 Mea depedet var S.D. depedet var S.E. of regresso.4688 Akake fo crtero Sum squared resd Schwarz crtero Log lkelhood Haa-Qu crter Devace Restr. devace Restr. log lkelhood LR statstc Avg. log lkelhood Prob(LR statstc Obs wth Dep= 979 Total obs 667 Obs wth Dep= 488 2,,6, Seres: QHAT Sample 5 IF SEX= AND GOODOBS= Observatos 667 Mea Meda Maxmum Mmum.6363 Std. Dev Skewess Kurtoss Jarque-Bera Probablty.

11 MALES Depedet Varable: D (PARTICIPATION DUMMY Method: ML - Bary Logt (Quadratc hll clmbg Date: 2/5/2 Tme: 5:37 Sample: 5 IF SEX= AND GOODOBS= Icluded observatos: 583 Covergece acheved after 4 teratos Covarace matrx computed usg secod dervatves Varable Coeffcet Std. Error z-statstc Prob. C A A A A A BLACK HISP MARITAL STATUS HIGH SCHOOL WORKED LAST 2WK CLASS_TR OTJ_JSA F2SMS AFDC McFadde R-squared.366 Mea depedet var S.D. depedet var S.E. of regresso.4896 Akake fo crtero Sum squared resd Schwarz crtero Log lkelhood Haa-Qu crter Devace Restr. devace Restr. log lkelhood LR statstc Avg. log lkelhood Prob(LR statstc. Obs wth Dep= 2966 Total obs 583 Obs wth Dep= Seres: PHAT Sample 5 IF SEX= AND GOODOBS= Observatos 583 Mea Meda Maxmum.6399 Mmum Std. Dev Skewess.4525 Kurtoss Jarque-Bera Probablty.

12 FEMALES Depedet Varable: D (PARTICIPATION DUMMY Method: ML - Bary Logt (Quadratc hll clmbg Date: 2/5/2 Tme: 5:28 Sample: 5 IF SEX= AND GOODOBS= Icluded observatos: 667 Covergece acheved after 4 teratos Covarace matrx computed usg secod dervatves Varable Coeffcet Std. Error z-statstc Prob. C A A A A A BLACK HISP MARITAL STATUS HIGH SCHOOL WORKED LAST 2WK CLASS_TR OTJ_JSA F2SMS AFDC McFadde R-squared.655 Mea depedet var S.D. depedet var S.E. of regresso.4982 Akake fo crtero Sum squared resd Schwarz crtero Log lkelhood Haa-Qu crter Devace Restr. devace Restr. log lkelhood LR statstc Avg. log lkelhood Prob(LR statstc. Obs wth Dep= 338 Total obs 667 Obs wth Dep= MALES Seres: PHAT Sample 5 IF SEX= AND GOODOBS= Observatos 667 Mea Meda Maxmum Mmum Std. Dev Skewess.9635 Kurtoss Jarque-Bera Probablty. 2

13 Depedet Varable: Z (RANDOM OFFER OF SERVICES Method: ML - Bary Logt (Quadratc hll clmbg Date: 2/5/2 Tme: 6:2 Sample: 5 IF SEX= AND GOODOBS= Icluded observatos: 583 Covergece acheved after 4 teratos Covarace matrx computed usg secod dervatves Varable Coeffcet Std. Error z-statstc Prob. C A A A A A BLACK HISP MARITAL STATUS HIGH SCHOOL WORKED LAST 2WK CLASS_TR OTJ_JSA F2SMS McFadde R-squared.86 Mea depedet var.6687 S.D. depedet var S.E. of regresso.4777 Akake fo crtero Sum squared resd Schwarz crtero Log lkelhood Haa-Qu crter Devace Restr. devace Restr. log lkelhood LR statstc Avg. log lkelhood Prob(LR statstc.5278 Obs wth Dep= 684 Total obs 583 Obs wth Dep= Seres: ZHAT Sample 5 IF SEX= AND GOODOBS= Observatos 583 Mea.6687 Meda Maxmum Mmum Std. Dev Skewess.6535 Kurtoss Jarque-Bera.4648 Probablty.5499 FEMALES Depedet Varable: Z (RANDOM OFFER OF 3

14 SERVICES Method: ML - Bary Logt (Quadratc hll clmbg Date: 2/5/2 Tme: 6: Sample: 5 IF SEX= AND GOODOBS= Icluded observatos: 667 Covergece acheved after 4 teratos Covarace matrx computed usg secod dervatves Varable Coeffcet Std. Error z-statstc Prob. C A A A A A BLACK HISP MARITAL STATUS HIGH SCHOOL WORKED LAST 2WK CLASS_TR OTJ_JSA F2SMS AFDC McFadde R-squared.979 Mea depedet var S.D. depedet var S.E. of regresso Akake fo crtero Sum squared resd Schwarz crtero Log lkelhood Haa-Qu crter..27 Devace Restr. devace Restr. log lkelhood LR statstc Avg. log lkelhood Prob(LR statstc Obs wth Dep= 979 Total obs 667 Obs wth Dep= Seres: ZHAT Sample 5 IF SEX= AND GOODOBS= Observatos 667 Mea Meda Maxmum Mmum Std. Dev Skewess Kurtoss Jarque-Bera Probablty. 4

Complete Convergence and Some Maximal Inequalities for Weighted Sums of Random Variables

Complete Convergence and Some Maximal Inequalities for Weighted Sums of Random Variables Joural of Sceces, Islamc Republc of Ira 8(4): -6 (007) Uversty of Tehra, ISSN 06-04 http://sceces.ut.ac.r Complete Covergece ad Some Maxmal Iequaltes for Weghted Sums of Radom Varables M. Am,,* H.R. Nl