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, doald@eco.utexas.edu. Isttute of Ecoomcs, Academa Sca, ychsu@eco.sca.edu.tw Departmet of Ecoomcs, Cetral Europea Uversty, Budapest ad the Natoal Bak of Hugary, rlel@gmal.com.
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
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
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
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. 3.85 5.4 4.84 -. 2.8 2.8 7.89 -. 2.49 2.45 6.2 ˆβ t (ATT.6 2.8 2.9.88.5.7.56 4.68.5.5.48 6.92 Combed.5 2..76.38.5.74.54 4.7.5.5.48 6.38 Pre-tested.3 2.64.88.6.7 3.7.92 2.43. 2.67 2.43 7.6 Power/E(â.75/.8.37/.4.974/. L. quad..5% ˆτ t (LATT.2 7.32.4 53.69. 4.42 4.4 9.53 -. 3.6 2.75 9.37 ˆβ t (ATT.6 2.72 2.4 3.64.5 2.4.67 5.53.5.6.5 56.5 Combed.7 2.56.94 4..5.99.6 5.42.5.62.5 57.2 Pre-tested.7 3.24 2.6 7.98. 3.28.89 6.4. 3.67 2.6 3.57 Power/E(â.62/.8.58/.4.874/. L. 2 quad..5% ˆτ t (LATT. 6.7 5.9 44.97. 4.5 3.88 2.28. 3.8 3.65 4.48 ˆβ t (ATT.3 9.33 4.95 9.47.3 2.93 2.53 6.63.2 2.58 2.5 4.75 Combed.6 4.4 2.7 23.2.3 2.87 2.5 6.44.2 2.59 2.5 4.9 Pre-tested.9 5.3 3. 27.5.7 4.2 2.79 9.57.3 4.76 3.8 24.92 Power/E(â.54/..76/.2.578/. Rat. quad..5% ˆτ t (LATT. 4.66 5.48 2.69 -. 3.24 2.96.58 -. 2.62 2.52 7.32 ˆβ t (ATT.5 2.52 2.8.92.4.86.58 3.3.4.55.48 49.87 Combed.5 2.34.78.33.4.88.55 2.79.4.58.48 49.29 Pre-tested.3 3..93 2.99.6 3.4.95 3.2 -. 2.9 2.48 8.72 Power/E(â.85/.8.279/.4.955 Rat. 2 quad..5% ˆτ t (LATT -. 3.32 3.83. -. 2.69 2.67 7.23 -. 2.5 2.49 6.29 ˆβ t (ATT.5 2.7.89 9.68.4.74.62 3.46.4.6.57 53.72 Combed.4 2..74 9..4.75.62 2.72.4.6.57 52.42 Pre-tested. 2.65.88 9.83.6 3.2.96.53 -. 2.74 2.46 7.5 Power/E(â./.77.328/.4.964/. Rat. 2 cube.5% ˆτ t (LATT..9 54.8 6.3.2 4.33 9.8 8.95. 2.68 2.57 7.8 ˆβ t (ATT.22 8.4 3.86 76.68.7 2.7 4. 2.2.4.65.62 54.22 Combed.32 4.5 5.46 46.6.6 2.56 2.33 9.99.4.7.6 5.63 Pre-tested.33 4.64 5.48 48.2.5 2.92 2.43 9.35. 3.5 2.5 9.39 Power/E(â.4/.23.6/.5.93/. 5
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
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 -.46575.89325-5.23628. HS.22646.498.68273.2427 MINORITY -.64923.5447 -.42293.6743 BELOW3.4593.335.3467.76 HS*MINORITY.48274.8689.8687.445 HS*BELOW3 -.9478.5846 -.2362.92 MINORITY*BELOW3.725.22683.762658.4457 HS*MINORITY*BELOW3 -.23397.2665 -.4632.6434 McFadde R-squared.396 Mea depedet var.46486 S.D. depedet var.49325 S.E. of regresso.492897 Akake fo crtero.35956 Sum squared resd 232.96 Schwarz crtero.369799 Log lkelhood -3447.2 Haa-Qu crter..3637 Devace 6894.42 Restr. devace 694.6 Restr. log lkelhood -3452.3 LR statstc 9.64533 Avg. log lkelhood -.67884 Prob(LR statstc.29876 Obs wth Dep= 2966 Total obs 583 Obs wth Dep= 27,4,2, 8 6 4 2.37.38.39.4.4.42.43.44.45 Seres: PHAT Sample 5 IF SEX= AND GOODOBS= Observatos 583 Mea.46486 Meda.4566 Maxmum.452769 Mmum.3737 Std. Dev..2448 Skewess -.24952 Kurtoss 2.7397 Jarque-Bera 64.92868 Probablty. 7
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 -.396333.8798-4.54583. HS.24662.99957 2.47545.37 MINORITY -.77468.43637 -.235526.266 BELOW3.56563.32284.427585.669 HS*MINORITY.7687.66495.6996.2882 HS*BELOW3 -.3384.5348 -.22488.8255 MINORITY*BELOW3 -.27856.228 -.3286.8955 HS*MINORITY*BELOW3.24785.24677.56892.622 McFadde R-squared.3584 Mea depedet var.442888 S.D. depedet var.496768 S.E. of regresso.495842 Akake fo crtero.37934 Sum squared resd 489.66 Schwarz crtero.379783 Log lkelhood -45.728 Haa-Qu crter..3745 Devace 83.456 Restr. devace 833.37 Restr. log lkelhood -465.659 LR statstc 29.8669 Avg. log lkelhood -.68448 Prob(LR statstc. Obs wth Dep= 338 Total obs 667 Obs wth Dep= 2687 3, 2,5 2,,5, 5.36.38.4.42.44.46.48 Seres: PHAT Sample 5 IF SEX= AND GOODOBS= Observatos 667 Mea.442888 Meda.45477 Maxmum.484337 Mmum.3636 Std. Dev..34692 Skewess -.56645 Kurtoss 3.37242 Jarque-Bera 387.66 Probablty. 8
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.636953.944 6.968557. HS.5398.854.499562.674 MINORITY.6598.57344.493.9666 BELOW3.249876.454.778424.753 HS*MINORITY.36439.86896.9497.8454 HS*BELOW3 -.3728.6592 -.885.69 MINORITY*BELOW3 -.25923.234485 -.74.2846 HS*MINORITY*BELOW3.348749.27782.255339.294 McFadde R-squared.96 Mea depedet var.6687 S.D. depedet var.47727 S.E. of regresso.47783 Akake fo crtero.27288 Sum squared resd 24.88 Schwarz crtero.282472 Log lkelhood -3225.267 Haa-Qu crter..275789 Devace 645.533 Restr. devace 6456.383 Restr. log lkelhood -3228.92 LR statstc 5.849964 Avg. log lkelhood -.63452 Prob(LR statstc.557374 Obs wth Dep= 684 Total obs 583 Obs wth Dep= 3399,6,4,2, 8 6 4 2.66.67.68.69.7.7 Seres: QHAT Sample 5 IF SEX= AND GOODOBS= Observatos 583 Mea.6687 Meda.66674 Maxmum.78235 Mmum.65464 Std. Dev..5886 Skewess.69575 Kurtoss 3.672 Jarque-Bera 239.524 Probablty. 9
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.67288.9376 7.427722. HS.996.466.84456.8537 MINORITY -.36365.452 -.939672.3474 BELOW3 -.353.3772 -.257526.7968 HS*MINORITY.245892.7793.43972.5 HS*BELOW3.4975.633.879272.3793 MINORITY*BELOW3.6337.2664.7562.4496 HS*MINORITY*BELOW3 -.255949.254382 -.66.343 McFadde R-squared.67 Mea depedet var.67389 S.D. depedet var.468857 S.E. of regresso.4688 Akake fo crtero.26487 Sum squared resd 33.663 Schwarz crtero.27335 Log lkelhood -3826.9 Haa-Qu crter..267258 Devace 7653.82 Restr. devace 766.992 Restr. log lkelhood -383.996 LR statstc 8.72498 Avg. log lkelhood -.63775 Prob(LR statstc.37629 Obs wth Dep= 979 Total obs 667 Obs wth Dep= 488 2,,6,2 8 4.63.64.65.66.67.68.69 Seres: QHAT Sample 5 IF SEX= AND GOODOBS= Observatos 667 Mea.67389 Meda.66675 Maxmum.69277 Mmum.6363 Std. Dev..7249 Skewess -.62327 Kurtoss 2.754264 Jarque-Bera 47.7626 Probablty.
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 -.46464.862-2.56286.4 A2225 -.88.69248 -.2.99 A2629 -.7449.7725 -.992565.329 A335 -.2333.696 -.6535.545 A3644 -.6735.7562 -.9894.3266 A4554 -.224944.87597 -.9983.235 BLACK.4379.7537.6284.955 HISP.69274.98865.7273.869 MARITAL STATUS.7634.6464 2.865633.42 HIGH SCHOOL.5632.6495.627497.36 WORKED LAST 2WK -.34243.5943 -.576463.5643 CLASS_TR.548423.82963 6.646. OTJ_JSA -.52369.683 -.76865.442 F2SMS -.466.637 -.7379.994 AFDC.4355.4523.256.986 McFadde R-squared.366 Mea depedet var.46486 S.D. depedet var.49325 S.E. of regresso.4896 Akake fo crtero.345687 Sum squared resd 22.39 Schwarz crtero.364968 Log lkelhood -345.63 Haa-Qu crter..352439 Devace 68.25 Restr. devace 694.6 Restr. log lkelhood -3452.3 LR statstc 93.93592 Avg. log lkelhood -.669892 Prob(LR statstc. Obs wth Dep= 2966 Total obs 583 Obs wth Dep= 27 6 5 4 3 2.35.4.45.5.55.6 Seres: PHAT Sample 5 IF SEX= AND GOODOBS= Observatos 583 Mea.46486 Meda.398893 Maxmum.6399 Mmum.35725 Std. Dev..67228 Skewess.4525 Kurtoss 3.7458 Jarque-Bera 92.658 Probablty.
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 -.388253.4643-2.6545.8 A2225 -.843.382 -.58985.566 A2629 -.884.38676 -.57822.563 A335 -.38998.3676 -.2856.7755 A3644 -.93794.3898 -.3957.63 A4554 -.259554.5459 -.678989.932 BLACK -.2487.6472 -.75295.796 HISP.5435.8392.643894.596 MARITAL STATUS.23547.59682 2.798.384 HIGH SCHOOL.26445.6822 4.34728. WORKED LAST 2WK -.5298.55866-2.73725.62 CLASS_TR.38487.6825 5.684. OTJ_JSA -.4947.697-2.68.38 F2SMS.64533.5998.7723.285 AFDC.5.63362.596384.4 McFadde R-squared.655 Mea depedet var.442888 S.D. depedet var.496768 S.E. of regresso.4982 Akake fo crtero.355979 Sum squared resd 463.79 Schwarz crtero.37257 Log lkelhood -498.363 Haa-Qu crter..36738 Devace 896.726 Restr. devace 833.37 Restr. log lkelhood -465.659 LR statstc 34.594 Avg. log lkelhood -.67557 Prob(LR statstc. Obs wth Dep= 338 Total obs 667 Obs wth Dep= 2687 5 4 3 2.25.3.35.4.45.5.55.6 MALES Seres: PHAT Sample 5 IF SEX= AND GOODOBS= Observatos 667 Mea.442888 Meda.435799 Maxmum.62842 Mmum.25687 Std. Dev..73734 Skewess.9635 Kurtoss 2.6597 Jarque-Bera 26.2428 Probablty. 2
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.58533.8968 3.8692.2 A2225 -.45532.7982 -.2543.7994 A2629.24758.8664.36283.896 A335 -.3277.7888 -.832.8546 A3644 -.633.826 -.334687.7379 A4554 -.6826.9654 -.857796.39 BLACK.9934.7338.272548.7852 HISP.4485.4237.429835.6673 MARITAL STATUS.3766.6359.633623.23 HIGH SCHOOL -.438.674 -.2464.83 WORKED LAST 2WK.5657.643.92893.357 CLASS_TR.77956.8762 2.3989.423 OTJ_JSA.7263.69937.3859.299 F2SMS.72452.6597.993.276 McFadde R-squared.86 Mea depedet var.6687 S.D. depedet var.47727 S.E. of regresso.4777 Akake fo crtero.273338 Sum squared resd 23.47 Schwarz crtero.29334 Log lkelhood -3222.89 Haa-Qu crter..27964 Devace 6444.378 Restr. devace 6456.383 Restr. log lkelhood -3228.92 LR statstc 2.565 Avg. log lkelhood -.63395 Prob(LR statstc.5278 Obs wth Dep= 684 Total obs 583 Obs wth Dep= 3399 6 5 4 3 2.6.62.64.66.68.7.72.74 Seres: ZHAT Sample 5 IF SEX= AND GOODOBS= Observatos 583 Mea.6687 Meda.668497 Maxmum.743766 Mmum.599334 Std. Dev..22849 Skewess.6535 Kurtoss 2.829 Jarque-Bera.4648 Probablty.5499 FEMALES Depedet Varable: Z (RANDOM OFFER OF 3
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.65658.53737 4.27332. A2225.22293.45666.5343.8784 A2629.63588.4649.43432.664 A335 -.6629.43969 -.45863.6465 A3644 -.535.4648 -.78924.937 A4554.5763.62.97295.9225 BLACK.9694.6762.42255.552 HISP -.5964.8756 -.6884.496 MARITAL STATUS.88647.636.46742.595 HIGH SCHOOL.33949.62589 2.432.323 WORKED LAST 2WK.29925.5858.5836.695 CLASS_TR -.9383.7229 -.289.974 OTJ_JSA -.587.72556 -.59699.5 F2SMS.22982.634.363988.759 AFDC -.54486.66384 -.82772.48 McFadde R-squared.979 Mea depedet var.67389 S.D. depedet var.468857 S.E. of regresso.46886 Akake fo crtero.265342 Sum squared resd 33.57 Schwarz crtero.28933 Log lkelhood -3823.45 Haa-Qu crter..27 Devace 7646.83 Restr. devace 766.992 Restr. log lkelhood -383.996 LR statstc 5.643 Avg. log lkelhood -.6399 Prob(LR statstc.36729 Obs wth Dep= 979 Total obs 667 Obs wth Dep= 488 7 6 5 4 3 2.6.62.64.66.68.7.72.74 Seres: ZHAT Sample 5 IF SEX= AND GOODOBS= Observatos 667 Mea.67389 Meda.673874 Maxmum.744352 Mmum.59642 Std. Dev..23442 Skewess -.69383 Kurtoss 2.8856 Jarque-Bera 39.363 Probablty. 4