Empirical Likelihood for Regression Discontinuity Design

Transcription

1 Emirical Likeliood for Regression Discontinuity Design Taisuke Otsu Yale University Ke-Li Xu y University of Alberta 8t Marc 2009 Abstract Tis aer rooses emirical likeliood con dence intervals for causal e ects identi ed from regression discontinuity designs. We consider sar and fuzzy discontinuity designs and treat regression functions as nonarametric. Our emirical likeliood con dence intervals ave natural saes and do not require variance estimation unlike Wald-tye con dence intervals. Tese advantages are illustrated by simulation studies and an emirical examle wic evaluates te e ect of class sizes on uils scolastic acievements. Introduction Since Tistletwaite and Cambell (960), regression discontinuity design (RDD) analysis as been used as a fundamental tool to investigate causal e ects of treatment assignments to outcomes of interest. Tere are numerous metodological develoments and emirical alications of RDD analysis articularly in te elds of economics, sycology, and statistics (see, e.g., Trocim (200) and Imbens and Lemieux (2008) for a review). Te main urose of tis aer is to roose a new inference aroac to RDD analysis based on emirical likeliood (Owen (988)). In te literature of RDD analysis, tere are at least two imortant issues tat ave attracted attentions from statisticians. First, altoug RDD analysis were initially discussed in te context of regression analysis, recent researc tries to understand more deely estimated arameters of interest based on teory of causal e ects (Rubin (974), Holland (986), and Angrist, Imbens and Rubin (996)). In articular, deending on te relationsi between te treatment and covariate, RDDs are slit into two categories, te sar and fuzzy RDDs. In te sar case, were te treatment is comletely determined by te covariate tyically based on some cuto value, we can identify and estimate te average causal e ect of te treatment at te cuto value of te covariate. In te fuzzy case, were Cowles Foundation and Deartment of Economics, Yale University, New Haven CT 06520, U.S.A. address: taisuke.otsu@yale.edu. Financial suort from te National Science Foundation (SES ) is gratefully acknowledged. y Deartment of Finance and Management Science, Scool of Business, University of Alberta, Edmonton, Alberta T6G 2R6, Canada. address: keli.xu@ualberta.ca.

2 te treatment is artly determined by te covariate and te treatment assignment robability jums at some cuto value of te covariate, we can identify and estimate te average causal e ect of te treatment for comliers (see, Section 2. and Han, Todd and van der Klaauw (200)). Te resent aer also adots tis framework and focuses on inference roblems for tose average causal e ects in te sar and fuzzy RDDs. Second, since RDD analysis focuses on causal e ects locally at some cuto value of te covariate, te imortance on exible functional forms or nonarametric analysis as been emasized in te statistics literature (Sacks and Ylvisaker (978) and Kna, Sacks and Ylvisaker (985)), wic mostly focuses on robust constructions of con dence intervals for causal e ects. Han, Todd and van der Klaauw (200) and Porter (2003) roosed nonarametric estimators for average causal e ects in te sar and fuzzy RDDs based on local olynomial ttings (Fan and Gijbels (996)). Teir nonarametric estimators sow reasonable convergence rates and are asymtotically normal. However, te asymtotic variances of tose estimators, wic are required to construct Wald-tye con dence intervals, are rater comlicate due to discontinuities in te conditional mean and variance functions. Tyically, in order to estimate te asymtotic variances, we need additional nonarametric regressions to estimate left and rigt limits of te conditional variance and nonarametric density estimation for te covariate. Te resent aer constructs emirical likeliood-based con dence intervals wic allow nonarametric regression functions but do not require comlicate variance estimation. Tis aer also contributes to te raidly growing literature on emirical likeliood (see, Owen (200) for a review). In articular, we extend te likeliood construction by Cen and Qin (2000) to te sar and fuzzy RDD setus, wic incororates local olynomial tting into te emirical likeliood framework. We sow tat te emirical likeliood ratios for causal e ects in te sar and fuzzy RDDs are asymtotically ci-square distributed. Terefore, we can still observe Wilks enomenon (Fan, Zang and Zang (200)) in tis nonarametric RDD setu. Tis aer is organized as follows. Section 2 resents our basic setu and constructs emirical likeliood for causal e ects. Section 3 studies asymtotic roerties of emirical likeliood con dence intervals. Te roosed metods are evaluated by Monte Carlo simulations (Section 4.) and an emirical examle wic evaluates te e ect of class sizes to uils scolastic acievements investigated in Angrist and Lavy (999). Section 5 concludes. Aendix A contains roofs and lemmas. 2 Setu and Metodology 2. Regression Discontinuity Design We rst introduce our basic setu. Let Y i () and Y i (0) be otential outcomes of unit i wit and witout exosure to a treatment, resectively. De ne W i 2 f0; g as an indicator variable for te treatment. We set W i = if unit i is exosed to te treatment and set W i = 0 oterwise. Te observed outcome is Y i = ( W i ) Y i (0) + W i Y i () and we cannot observe Y i (0) and Y i () simultaneously. Our urose is 2

3 to make inference on te causal e ect of te treatment, or more seci cally, robabilistic asects of te di erence of otential outcomes Y i () Y i (0). RDD analysis focuses on te case were te treatment assignment W i is comletely or artly determined by some observable covariate X i. For examle, to study te e ect of class sizes to uils acievements, it is reasonable to consider te following setu: te unit i is scool, Y i is an average exam score, W i is an indicator variable for te class size (W i = 0 for one class and W i = for two classes), and X i is te number of enrollments. Deending on te assignment rule for W i based on X i, we ave two cases, called te sar and fuzzy RDDs. In te sar RDD, te treatment is deterministically assigned based on te value of X i, i.e., W i = I fx i cg ; were I fg is te indicator function and c is a known constant. A arameter of interest in tis case is te average causal e ect at te discontinuity oint c, s = E [Y i () Y i (0)j X i = c] : Since te di erence of otential outcomes Y i () Y i (0) is unobservable, we need a tractable reresentation of s using estimable comonents by data. If te conditional exectations E [Y i ()j X i = x] and E [Y i (0)j X i = x] are continuous at x = c, ten te average causal e ect s can be identi ed as a contrast of te rigt and left limits of te conditional mean E [Y i j X i = x] at x = c, s = lim x#c E [Y i j X i = x] lim E [Y i j X i = x] : () x"c In contrast to te sar RDD, te fuzzy RDD focuses on te case were te covariate X i is not informative enoug to determine te treatment W i but X i can a ect on te treatment robability. In articular, te fuzzy RDD assumes tat te conditional treatment robability of W i jums at X i = c, lim Pr fw i = j X i = xg 6= lim Pr fw i = j X i = xg ; x#c x"c To de ne a reasonable arameter of interest, let W i (x) be a otential treatment for unit i wen te cuto level for te treatment was set at x, and assume tat W i (x) is non-increasing in x at x = c. By using te terminology of Angrist, Imbens and Rubin (996), te unit i is called a comlier if er cuto level is X i, i.e., lim W i (x) = 0; x#x i lim W i (x) = : x"xi A arameter of interest in te fuzzy RDD, suggested by Han, Todd and van der Klaauw (200), is te average causal e ect for comliers at X i = c, f = E [Y i () Y i (0)j i is comlier, X i = c] : If lim x#xi W i (x) = 0 and lim x"xi W i (x) = 0, ten unit i is called a nevertaker. If lim x#xi W i (x) = and lim x"xi W i (x) =, ten unit i is called an alwaystaker. 3

4 Han, Todd and van der Klaauw (200) sowed tat under mild conditions te arameter f can be identi ed by te ratio of te jum in te conditional mean of Y i to te jum in te conditional treatment robability at X i = c, i.e., f = lim x#c E [Y i j X i = x] lim x"c E [Y i j X i = x] lim x#c Pr fw i = j X i = xg lim x"c Pr fw i = j X i = xg : (2) If additional covariates Z i are available, te above identi cation results go troug by sligtly modifying te assumtions and adding conditioning variables Z i = z to te conditional means and robabilities. Te main focus of tis aer is ow to make inference on tese average causal e ects s and f, articularly ow to construct valid con dence intervals for s and f. To estimate te arameters s and s, it is common to aly nonarametric regression tecniques (see, e.g., Han, Todd and van der Klaauw (200) and Porter (2003)). For examle, te arameter s can be estimated by using te local linear regression estimators (Fan and Gijbels (996)): ^s = ^ r ^ l ; were ^ l and ^ r are estimators for te left and rigt limits of te conditional mean l = lim x"c E [Y i j X i = x] and r = lim x#c E [Y i j X i = x], resectively, and are obtained as solutions to te weigted least square roblems wit resect to a l and a r min a l ;b l min a r;b r X i:x i <c X i:x i c Xi K Xi K c c (Y i a l b l (X i c)) 2 ; (3) (Y i a r b r (X i c)) 2 ; resectively, wit a kernel function K and bandwidt satisfying! 0 as n!. nonarametric estimator for f can be obtained as Similarly, a ^f = ^ r ^ l ^ wr ^ wl ; were ^ wl and ^ wr are estimators for te rigt and left limits of te conditional treatment robabilities wl = lim x"c Pr fw i = j X i = xg and wr = lim x#c Pr fw i = j X i = xg, resectively, and are obtained as solutions to te weigted least square roblems wit resect to a wl and a wr, X Xi c min K (W i a wl b wl (X i c)) 2 ; (4) a wl ;b wl i:x i <c X Xi c K (W i a wr b wr (X i c)) 2 ; min a wr;b wr i:x i c resectively. Altoug te kernel functions and bandwidts in (3) and (4) can be di erent, we assume tey are identical to simlify te resentation. Porter (2003) sowed tat te nonarametric estimators ^ s and ^ f are asymtotically normal wit some nonarametric convergence rates. Based on te derived asymtotic distributions, it is ossible to 4

5 construct asymtotically valid Wald-tye con dence intervals for s and f. However, to tis end, we need to estimate te asymtotic variances of te estimators wic take rater comlicate forms: tey deend on te left and rigt limits of te conditional variances of Y i and W i at X i = c and te density function of X i evaluated at c. Also te saes of te Wald-tye con dence intervals are restricted to be symmetric around te estimators ^ s and ^ f. Tis aer develos alternative con dence intervals for s and f based on emirical likeliood wic avoid comlicate variance estimation and determine te saes of te con dence intervals by data emasis. 2.2 Emirical Likeliood We now construct emirical likeliood functions for te average causal e ect arameters s and f. Our construction can be considered as an extension of Cen and Qin (2000) to te sar and fuzzy RDDs, wic introduced an emirical likeliood aroac to local linear tting for te conditional mean function. Let I i = I fx i cg be an indicator for weter te covariate X i exceeds te cuto level c. Note tat altoug W i = I i in te sar RDD, W i 6= I i in te fuzzy RDD. We rst consider te sar RDD case. Observe tat te local linear estimators ^ l and ^ r de ned in (3) satisfy te rst-order conditions (see, Fan and Gijbels (996,. 20)) were ( I i ) K li (Y i ^ l ) = 0; Xi K li = K Xi K ri = K 8 c < : 8 c < : I i K ri (Y i ^ r ) = 0; (5) P n ( I i) K Xi c 2 Xi c Xi c Xi c Xi c 2 9 Xi c Xi c = Xi c Xi c ; : P n ( I i) K P n I ik Xi c P n I ik 9 = ; ; If we regard (5) as estimating equations for E [^ l ] and E [^ r ], te emirical likeliood function at te candidate values (t; a) for (E [^ r ] s.t. 0 i, i =, E [^ l ] ; E [^ l ]) is de ned as L s (t; a) = su f i g n i ( I i ) K li (Y i a) = 0, ny i ; (6) i I i K ri (Y i t a) = 0: By alying te Lagrange multilier metod under certain regularity conditions (see, Newey and Smit (2004, Teorem 2.2)), we can obtain te dual roblem of (6). Te dual form of te log emirical likeliood ratio is written as `s (t; a) = 2 flog L s (t; a) + n log ng = 2 su 2 n(t;a) 5 log + 0 g i (t; a) ; (7)

6 were n (t; a) = 2 R 2 : 0 g i (t; a) 2 V for i = ; : : : ; n, V is an oen interval containing 0, and g i (t; a) = (( I i ) K li (Y i a) ; I i K ri (Y i t a)) 0 : (8) Also, te concentrated emirical likeliood function for t is de ned as `s (t) = min a2a `s (t; a) ; (9) were A is a arameter sace of l. In ractice, we use te dual reresentations in (7) and (9) to imlement emirical likeliood inference. Note tat te otimization roblem for te Lagrange multilier in (7) is two-dimensional and te objective function is tyically concave in. Terefore, te comutational cost to evaluate te emirical likeliood ratio `s (t; a) based on (7) is not exensive. Also, note tat tis construction gives us te emirical likeliood ratio for E [^ r ] tan for s = r E [^ l ] and E [^ l ], rater l and l. However, by introducing undersmooting (i.e., coose a relatively fast convergence rate for to zero), we can asymtotically neglect te bias comonents s (E [^ r ] E [^ l ]) and l E [^ l ]. Terefore, te functions (7) and (9) can be emloyed as valid emirical likeliood ratios for te arameters s and l. We next consider te fuzzy RDD case. Similar to (6), we consider te following likeliood maximization roblem: s.t. 0 i ; i = ; L f (t; a; a wl ; a wr ) = max f i g n i ( I i ) K li (Y i a) = 0, i ( I i ) K li (W i a wl ) = 0, ny i ; (0) i I i K ri (Y i t (a wr a wl ) a) = 0; i I i K ri (W i a wr ) = 0: Note tat te last two conditions come from te rst-order conditions for te local linear estimators of wl and wr. Te dual form of te emirical likeliood ratio is written as `f (t; a; a wl ; a wr ) = 2 flog L f (t; a; a wl ; a wr ) + n log ng = 2 su 2 n(t;a;a wl ;a wr) log + 0 i (t; a; a wl ; a wr ) ; () were n (t; a) = 2 R 4 : 0 i (t; a; a wl ; a wr ) 2 V for i = ; : : : ; n, V is an oen interval containing 0, and i (t; a; a wl ; a wr ) = (( I i ) K li (Y i a) ; I i K ri (Y i t (a wr a wl ) a) ; (2) ( I i ) K li (W i a wl ) ; I i K ri (W i a wr )) 0 : Also, te concentrated emirical likeliood function for t is de ned as `f (t) = min (a;a wl ;a `f (t; a; a wl ; a wr ) ; (3) wr)2a[0;][0;] By undersmooting, te functions () and (3) can be emloyed as valid emirical likeliood ratios for te arameters f, l, wl, and wr. 6

7 3 Asymtotic Proerties Tis section investigates asymtotic roerties of te emirical likeliood ratios roosed in te last section and constructs asymtotically valid emirical likeliood con dence intervals for te average causal e ects s and f identi ed from te sar and fuzzy RDDs. First, we consider te emirical likeliood ratios `s (t; a) in (7) and `s (t) in (9) for te sar RDD. We imose te following assumtions. Assumtion 3.. (i) fy i ; W i ; X i g n is i.i.d. (ii) Tere exists a neigborood N around c suc tat (a) te density function f of X i is continuously di erentiable and bounded away from zero in N, (b) E [Y i j X i = x] s I fx cg is continuously di erentiable in N n fcg and is continuous at c wit nite left and rigt and derivatives, (c) E Yi 2 X i = x is continuous in N n fcg and as nite left and rigt and limits at c, and (d) E jy i j i Xi = x is uniformly bounded on N for some 4. Also, V l and V r de ned in (7) are ositive. (iii) K is a symmetric and bounded density function wit suort [ k; k] for some k 2 (0; ). (iv) As n!,! 0,!, 3! 0, and n = =2 =2! 0. (v) A is comact and l 2 int (A). Assumtion 3. (i) is on te data structure. Since RDD analysis is tyically alied to cross section data, tis assumtion is reasonable. Assumtion 3. (ii) restricts te local sae of te data distribution form around x = c. Tis assumtion allows discontinuity of te conditional moments E [Y i j X i = x], E Y 2 Xi = x, and E jy i j i Xi = x at x = c. Assumtion 3. (iii) is on te kernel function K and i imlies te second-order kernel. Assumtion 3. (iv) is on te bandwidt arameter. If / n, tis assumtion is satis ed for 2 3 ; 2. Te assumtion 3! 0 corresonds to an undersmooting condition to remove te bias comonents in te construction of emirical likeliood. Assumtion 3. (v) is required for te concentrated emirical likeliood ratio `s ( s ). Under tese assumtions, te asymtotic distributions of te emirical likeliood ratios `s ( s ; l ) and `s ( s ) are obtained as follows. Teorem 3.. (i) Under Assumtion 3. (i)-(iv), `s ( s ; l ) d! 2 (2). (ii) Under Assumtion 3., `s ( s ) d! 2 (). 7

8 See Aendix A. for te roof of tis teorem. Tis teorem says tat te emirical likeliood ratios `s ( s ; l ) and `s ( s ) are asymtotically ivotal and converge to te ci-square distributions, i.e., Wilks enomenon emerges in tis nonarametric RDD context. Based on Teorem 3. (ii), te 00 ( ) % emirical likeliood con dence interval for s can be obtained as ELCI s; = t : `s (t) 2 () ; were 2 () is te 00 ( ) % critical " value for te 2 () distribution. Comared to te r \ # conventional Wald-tye con dence interval ^s z =2 Asy:V ar ^s wit te 00 ( =2) % standard normal critical value z =2, te emirical likeliood con dence interval does not require te estimator for te asymtotic variance of ^ s and is not necessarily symmetric around ^ s. Furtermore, te emirical likeliood con dence interval is generally range reserving and transformation resecting. Next, we consider te emirical likeliood ratios `f (t; a; a wl ; a wr ) in () and `f (t) in (3) for te fuzzy RDD. For tis case, we add te following assumtion. Assumtion 3.2. Tere exists a neigborood N 0 around c suc tat E [W i j X i = x] ( wr wl ) I fx cg is continuously di erentiable in N 0 n fcg and is continuous at c wit nite left and rigt and derivatives. Also, wl ; wr 2 (0; ). Tis assumtion corresonds to Assumtion 3. (ii) in te sar RDD case. Te asymtotic roerties of te emirical likeliood ratios `f ( s ; l ; wl ; wr ) and `f ( f ) are obtained as follows. Teorem 3.2. (i) Under Assumtions 3. (i)-(iv) and 3.2, `f ( s ; l ; wl ; wr ) d! 2 (4). (ii) Under Assumtion 3. and 3.2, `f ( f ) d! 2 (). Since te roof is similar to tat of Teorem 3., it is omitted. Based on Teorem 3.2 (ii), te 00 ( ) % emirical likeliood con dence interval for f can be constructed as ELCI f; = t : `f (t) 2 () : Note tat in te fuzzy RDD case, imlementation of te Wald-tye con dence interval becomes more comlicate tan te sar RDD case because of a linear aroximation to te ratio ^ f = ^r ^ l ^ wr ^ wl, estimation of te asymtotic variance of (^ wr ; ^ wl ), and estimation of te asymtotic covariance between (^ r ; ^ l ) and (^ wr ; ^ wl ). Finally, we discuss two extensions of te resent results: additional covariates and arametric side information. If we ave additional covariates Z i wic a ect on Y i or W i in linear ways, we simly modify te estimating functions g i (t; a) in (8) or i (t; a; a wl ; a wr ) in (2) as g i (t; a; d) = ( I i ) K li Y i a d 0 Z i ; Ii K ri Y i t a d 0 Z i 0 ; 8

9 or i (t; a; a wl ; a wr ; d; d w ) = ( I i ) K li Y i a d 0 Z i ; Ii K ri Y i t (a wr a wl ) a d 0 Z i ; ( I i ) K li W i a wl d 0 wz i ; Ii K ri W i a wr d 0 wz i 0 ; resectively. Te emirical likeliood ratios are analogously de ned, and under similar conditions tose ratios are asymtotically ci-square distributed at te true arameter values. So far, we do not imose any arametric functional form on te conditional mean E [Y i j X i = x] and conditional treatment robability Pr fw i = j X i = xg. Our emirical likeliood aroac can naturally accommodate arametric side information. For examle, consider te fuzzy RDD wit te cuto value c = 0, and secify te regression model for Y i as Y i = 0 + I i + 0 l P l (X i ) ( I i ) + 0 rp r (X i ) I i + u i ; E [u i j X i ; I i ] = 0; (4) were P l (X i ) and P r (X i ) are nite dimensional vectors of olynomials of X i witout constant terms. Tis seci cation allows te regression functions to ave di erent left and rigt limits at te tresold x = 0. In tis case, te numerator of f in (2) is identi ed as lim x#0 E [Y i jx i = x] lim x"0 E [Y i jx i = x] =. Te regression model (4) can be estimated by te two stage least squares wit instrumental variables V i, for examle. Tyical candidates for V i are te indicator variable I i and olynomials of X i. To incororate arametric information in (4), we can modify te estimating function i (t; a; a wl ; a wr ) in (2) as i (t; b 0 ; b l ; b r ; a wl ; a wr ) = V 0 i Y i b 0 t (a wr a wl ) I i b 0 l P l (X i ) ( I i ) b 0 rp r (X i ) I i ; ( I i ) K li (W i a wl ) ; I i K ri (W i a wr )) 0 : Te emirical likeliood ratios and teir asymtotic ci-square distributions can be obtained under analogous conditions. By alying te same argument, it is also ossible to incororate arametric information on te conditional treatment robability Pr fw i = j X i = xg, suc as te logit or robit functions. 4 Numerical Examles In tis section we study te nite samle erformance of te roosed emirical likeliood metods troug simulations and an emirical alication, and comare wit te conventional t-tests based on te asymtotic normality of te average causal e ect estimators ^ s and ^ f. 4. Simulations Consider te following data generating rocess of te sar RDD: Y i = (X i ) + s W i + (X i ) " i ; (5) 9

10 were (x) = x 2, W i = I fx i cg, X i iid Uniform [ 2; 2], " i iid N (0; ), and (x) = 2:5 ex ( jxj) I fx cg + :4 ( I fx cg) : (6) A jum in te conditional mean of Y i occurs at c = 0:5 wit te jum size (te average causal e ect) s = 3 sifting from l = 0:25 to r = 3:25. A reresentative samle wit 00 observations is dislayed in Figure (a). As mentioned above, te t-test and resulting Wald-tye con dence interval deend on consistent estimators of 2 r (c) = lim x#c 2 (x) and 2 l (c) = lim x"c 2 (x) at te discontinuity oint c. We coose (x) as in (6), viz. wit increasing variances toward te discontinuity oint on te rigt side and omoskedastic on te left side, so tat we are able to ceck ow well te variance function is estimated at te jum oint, te key comonent in te construction of te t-test. We consider two t-tests using te Porter s (2003) residual-based kernel estimator of te variance function on boundaries and its imroved version, te local linear estimation of te variances as in Ruert et al. (997) and Fan and Yao (998), denoted as AN and AN2 resectively, togeter wit te emirical likeliood test (EL) introduced in tis aer. Local linear tting is generally referred in estimating te nonarametric function, esecially at te boundary oints as ere because of its nice roerty of automatic boundary bias correction. But in nite samles te local linear tting may give negative estimates of variances occasionally 2. Coice of te kernel function a ects te construction of te t-tests and te associated con dence intervals troug a scale constant C(K) aeared in te estimated asymtotic variances (c.f. Porter, 2003). Table gives te values of C(K) for various commonly used kernel functions wit bounded suorts. It sows tat wen te Eanecnikov kernel function is used, te jum size estimator b s as te smallest variance and te con dence interval for s is te tigtest. We use tis kernel function in our simulations and te emirical alication. Table and Figure Here Comared to te coice of te kernel function, it is well known tat nonarametric estimation and inference is muc more sensitive to te selection of te smooting bandwidt. In our exeriments, we use te six xed bandwidts ranging from = 0:8 to = :3 wen te samle size is 00 and from = 0:7 to = :2 wen te samle size is 200. We also consider a data-deendent bandwidt selected via cross validation, in wic we discard 50% of te observations on eac side of te tresold wen constructing te sum of squared deviations in te cross validation criterion function, as recommended by Imbens and Lemieux (2008, Section 5.). 2 Tis is because local linear tting may assign negative weigts to some squared residuals. e.g. wen a small bandwidt is used or te design oint is close to te boundary. In suc cases te resultant variance estimates may be volatile enoug to reac te negativity region. In our simulations, te ercentages of negative local linear estimates of ^ 2 r (c) or ^ 2 l (c) found range from 5.8% to 0.8% for six bandwidts considered wen n = 00; and from.% to 0.% wen n = 200: But we didn t nd negative estimates for ^ 2 r (c) + ^ 2 l (c). 0

11 Tables 2 and 3 reort te rejection rates of te two t-tests (AN and AN2) and emirical likeliood test (EL) over 000 relications wit te nominal sizes 5% and 0%, wen te samle sizes are 00 and 200 resectively. In addition, we reort te averages and standard errors (over relications) of estimates ^ r and ^ l of te rigt and left means, and tose of te estimates ^ 2 r (c) and ^ 2 l (c) of te rigt and left variances. We also record te averages and variances (over relications) of te estimate ^s of s in te columns labeled as ^ s and var ^s. Te column labeled as var \ ^s gives te averages and standard errors (over relications) of te estimated asymtotic variances were ^ 2 r (c) and ^ 2 l (c) are estimated by te kernel (AN) or te local linear metod (AN2). It sould be comared wit var ^s to see ow te nite-samle variance of ^ s is aroximated asymtotically. Tables 2-3 and Figure 2 Here Several observations are in order. Noticeable biases are observed for ^ r and ^ l ; esecially wen large bandwidts are used due to te small samle e ects. But tey aen to be biased towards te same direction so tat teir di erence, ^ s ; estimates te true value of s very accurately wit negligible bias. Te variance of ^ s is quite close to te sum of te variances of ^ r and ^ l : Te tree tests (AN, AN2, and EL) about s are generally oversized wit te bandwidts considered. Marked size distortions of te t-test AN are largely exlained by te fact tat te variance of ^ s is oorly estimated by var \ ^s wen ^ 2 r (c) and ^ 2 l (c) are estimated using te kernel metod. In articular, ^2 r (c) is very seriously biased, wit te average (over relications) just about te alf of te true value of 2 r (c), altoug 2 l (c) aears to be estimated satisfactorily. Take te case wen n = 00 and = :0 for examle. Te average of 000 estimates of 2 r (c) is.7 wit standard error 0.48, wic is far below te true value 2.3, wile te average of 000 estimates of 2 l (c) is.38 wit standard error 0.47, wic is fairly close to te true value.4. Consequently, te average of estimated asymtotic variances of ^ s is 0.46 wit standard error 0.3, wic underestimates te actual nite-samle variance Tis exlains te serious over-rejection of te t-test AN. Te similar observations occur for oter bandwidts and wen te samle size n = 200: Tis is not surrising in view of our design of te variance function (wit signi cant non-zero derivatives on te rigt side but zero derivatives of any order on te left \ side). In contrast, var ^s is considerably better estimated by var ^s wen we use te local linear estimators of 2 r (c) and 2 l (c) (still wit areciable downward bias for ^2 r (c)). Tis leads to te better size roerty of AN2 comared to tat of AN. Over all bandwidts considered, te EL test aears to ave te least size distortions among tree tests. Wen te larger samle size is used, te emirical sizes of te tree tests are closer to te nominal ones, wit te largest imrovement observed for te EL test. Figure 2 comares every nite-samle ercentile (% - 99% ercentiles) of eac of te two squared t test statistics and te EL test statistic (not just te 5% and 0% quantiles as reorted in Tables 2 and 3) wit te teoretical ercentiles of te 2 () distribution, wen (n; ) = (00; :0) and (200; 0:9). It is clear tat all ercentiles of te EL test statistic are closer to te 45 degree line, wic means te 2 () distribution serves as a better

12 aroximation of te nite samle distribution of te te EL test statistic tan tat of te two squared t-test statistics. Similar observations are obtained for oter bandwidts. Te usage of te cross validated bandwidt does not el muc in reducing te size distortions. Figure (b) lots te distribution (over relications) of te bandwidts selected for two samle sizes. Again, wit suc data-deendent bandwidts te erformance of te two t-tests is dominated by tat of te emirical likeliood-based test. Figures 3 and 4 sow te size-adjusted owers of te tree tests under te alternative H A : s = A ; based on critical values from reliminary simulations suc tat tey resectively ave te correct sizes (c.f. Table 4). We observe tat all tests are more owerful wen a larger bandwidt is used. Two t-tests ave generally similar owers excet tat AN2 is less owerful for small bandwidts due to te relatively iger variability of te local linear variance estimates. It is clear from te gures tat te EL test as te dominant owers for all bandwidts considered over its cometitors excet wen te value of A is at te far rigt side of te null yotesis. Tis excetion disaears wen te samle size is 200 and te EL test as te uniformly igest owers among all te tree tests considered. Simulation evidences reorted suggest tat te emirical likeliood metod is very romising in constructing tests wic ave better sizes and iger owers over te conventional t-tests and tus is recommended for use in ractice. Table 4 and Figures 3-4 Here 4.2 An Emirical Alication We use te data of Angrist and Lavy (999) to study te e ect of te number of classes on uils scolastic acievement. In Israeli ublic scools, Maimonides s rule, wic stiulates tat te class be slit wen it as more tan 40 students, as been used to determine te division of enrollment coorts into classes. Here we only consider scools wic ave one or two classes and focus on te 4t graders, altoug Angrist and Lavy s original analysis involved scools wit u to six classes and studied te 3rd, 4t and 5t graders. We end u wit a samle wit 77 observations (after removing 2 observations wit missing values), wit 307 scools aving only one class (te controlled grou) and 870 scools aving two classes (te treated grou). Te lots of te average mat scores and verbal scores against te enrollment sizes are dislayed in Figures 6 and 7 resectively, wit te round circles denoting te controlled grou and te entagrams for te treated grou. Te actual class sizes may not be te same as wat would be redicted by a strict alication of te Maimonides s rule, tat is, tere could be te cases tat more tan 40 students are crowded into one class and some classes ave less tan 20 students due to various administrative constraints. Tis is clear from te gures tat te scools wit enrollments near te cuto oint 40 aearing bot in te treated and controlled grou, making it an fuzzy RDD. Te local linear ts are also lotted for te two grous. We use te bandwidt = 0 for illustration, wic is selected via cross validation. Te jum size in te average verbal scores seems to be larger tan tat in te average mat scores. Te local linear estimate of te roensity 2

13 score function is lotted in Figure 5, wit treatment assignments, wic are jiggled wit random noises in te lot so tat overlaed observations are distinguisable. A discontinuity at te enrollment count 40 is clearly identi ed. We construct te con dence intervals for te average causal treatment e ect f in (2) for te fuzzy RDD by inverting te t-test (te AN CIs) and te emirical likeliood test (te EL CIs) wit con dence level 90%. Tey are sown in Figures 8 (b) and 9 for te mat score and te verbal score, resectively, togeter wit te local linear oint estimates using a grou of bandwidts. Te treatment e ect is estimated ranging from.8 to 7.4 for te mat score and from 5.0 to 2.0 for te verbal score. Te corresonding estimates and con dence intervals for te discontinuity size in te roensity score function, wic can be tougt as te jum in a sar RDD design, are lotted in Figure 8 (a). Te estimates of te discontinuity sizes are between 0.54 and Te AN CIs are arti cially symmetric around te oint estimates, and we nd tat in all cases, te AN CIs are overly otimistic wic are otentially subject to under-coverage. In contrast, Te EL CIs are generally wider tan te AN CIs and are automatically saed by te data. Tey are skewed downwards for te discontinuity size in te roensity score function but strongly uwards for te bot average causal treatment e ects. Bot con dence intervals sow tat te e ect of slitting a large class into two small classes is signi cant on imroving te uils verbal scores, but not on teir mat scores. Figures 5-9 Here 5 Conclusion Tis aer rooses emirical likeliood inference for average causal e ects in regression discontinuity designs. Our metods allow sar and fuzzy regression discontinuity designs and do not need to secify arametric functional forms on te regression functions. Comared to te conventional Wald-tye con dence interval, our emirical likeliood ratios do not require te asymtotic variance estimation and can be asymmetric around te estimators. Monte Carlo simulations and an emirical examle on te evaluation of class sizes to uils erformances illustrate te bene ts of te roosed metods. 3

14 A Matematical Aendix De ne ^ = arg min a2a `s ( s ; a), 2 l = lim E (Y i l ) 2 i Xi = x ; 2 r = lim E (Y i s l ) 2 i Xi = x ; x"c x#c V l = 2 l s 2 l;2 s l;20 2s l;2 s l; s l;2 + s 2 l; s l;22 ; Vr = 2 r s 2 r;2s r;20 2s r;2 s r; s r;2 + s 2 r;s r;22 ;! V l 0 V = : (7) 0 V r A. Proof of Teorem 3. Proof of (i). From Lemma A. (iii), te rst-order condition for ^ ( s ; l ), wic solves te otimization roblem in (7), satis es 0 = g i ( s ; l ) + ^ ( s ; l ) 0 g i ( s ; l ) = w..a. (wit robability aroacing one), were ^V = P n g i ( s ; l ) ^V^ (s ; l ) ; (8) g i ( s; l )g i ( s; l ) 0 + _ 0 g i ( s; l ) 2, te second equality follows from an exansion around ^ ( s ; l ) = 0, and _ is a oint on te line joining ^ ( s ; l ) and 0. Since ^V V max in 2 P n g i ( s ; l ) g i ( s ; l ) 0 V! 0 (by Lemma + _ 0 g i ( s; l ) A. (ii) and (iii)) and V is ositive de nite (Assumtion 3. (ii)), ^V is invertible w..a.. Tus, we ave ^ ( s ; l ) = ^V P n g i ( s ; l ) w..a., and a second-order exansion of `s ( s ; l ) = 2 P n log + ^ ( s ; l ) 0 g i ( s ; l ) w..a. (by Lemma A. (iii)) around ^ ( s ; l ) = 0 yields `s ( s ; l ) = 2^ ( s ; l ) 0 g i ( s ; l ) ^ (s ; l ) 0 ^V2^ (s ; l ) = w..a., were ^V 2 = P n ^V 2 V! 0 g i ( s ; l ) 2 ^V g i ( s; l )g i ( s; l ) g i ( s; l )! 0 by te same argument to ^V, we ave 2 ^V (ii) imlies te conclusion. ^V i ^V 2 ^V! g i ( s ; l ) ; (9) and is a oint on te line joining ^ ( s ; l ) and 0. Since ^V ^V 2 ^V! V. Terefore, Lemma A. Proof of (ii). Based on Lemma A.2, we can aly te same argument to derive (9), wic yields! 0! `s ( s ) = g i ( s ; ^) 2 V ~ i ~V ~V 2 V ~ g i ( s ; ^) ; (20) w..a.., were ~ V = P n g i ( s;^)g i ( s;^) 0 + _ 0 g i ( s;^) 2, ~ V 2 = P n g i ( s;^)g i ( s;^) g i ( s;^) te line joining ^ ( s ; ^) and 0. Also, Lemma A.2 imlies 2 ~ V 4 2, and _ and are oints on ~V ~V 2 V ~! V.

15 We now derive te asymtotic distribution of P n g i ( s ; ^). From Lemma A.2 (ii), ^ ( s ; ^) satis es te rst-order condition w..a.. 0 = g i ( s ; ^) + ^ ( s ; ^) 0 g i ( s ; ^) ; (2) Since te derivative of tis condition wit resect to ^ ( s ; ^) converges in robability to te ositive de nite matrix V (by Lemma A.2), we can aly te imlicit function teorem, i.e., ^ ( s ; a) is continuously di erentiable wit resect to a in a neigborood of ^ w..a.. i( = (( I i ) K li ; I i K ri ) 0 = G i. Te enveloe teorem imlies 0 = G 0 i^ ( s ; ^) + ^ ( s ; ^) 0 g i ( s ; ^) = ^G 0 ^ ( s ; ^) ; (22) w..a., were ^G is imlicitly de ned. On te oter and, an exansion of (2) around ( l ; 0) yields were 0 = ~; ~ = g i ( s ; l ) + G i (^ l ) + ~ 0 g i ( s ; ~) g i ( s ; ~) g i ( s ; ~) 0 + ~ 0 2 ^ (s ; ^) g i ( s ; ~) ^; ^ ( s ; ^) = g i ( s ; l ) ^G2 (^ l ) ^V3^ (s ; ^) ; (23) is a oint on te line joining de ned. Combining (22) and (23), 0 = 0 P n g i ( s ; l )! + ^M ^; ^ ( s ; ^) and ( l ; 0), and ^G 2 and ^V 3 are imlicitly! ^ l ; were ^ ^M = ( s ; ^) 0 ^G0 ^G 2 ^V3! : (24) Lemma A.2 imlies ^V 3! V, ^G! G, and ^G2! G, were G = f(c) 2 (; ) 0. Tus, ^M is invertible w..a.. By solving (24) for (^ l ), we ave (^ l ) = G 0 V G G 0 V P n g i ( s ; l )+ o (). From tis and an exansion of P n g i ( s ; ^) around ^ = l, g i ( s ; ^) = I G G 0 V G G 0 V i From (20), (25), and P n g i ( s ; l ) d! N (0; V ) (by Lemma A. (ii)), g i ( s ; l ) + o () : (25) `s ( s ) d! 0 V =2 I G G 0 V G G 0 V i 0 V I G G 0 V G G 0 V i V =2 = 0 I A A 0 A A 0 i = 2 () ; were N (0; I) and A = V =2 G. Terefore, te conclusion is obtained. 5

16 A.2 Lemmas Denote S ln;j = s l;j j 2 = f (c) Z 0 Xi ( I i ) K k K (z) j z j 2 dz; (x) = E [Y i j X i = x] s I fx cg ; 0 l = lim "0 Note tat K li = K Xi c n S ln;2 Xi c Lemma A.. Suose tat Assumtion 3. (i)-(iv) olds. Ten (i) S ln;! sl;, S ln;2! sl;2, S rn;! sr;, and S rn;2! sr;2, c Xi c j ; S rn;j = Xi c Xi c j I i K ; Z k s r;j j 2 = f (c) K (z) j z j 2 dz; 0 (c + ) (c) ; 0 (c + ) (c) r = lim : #0 o n o S ln; and K ri = K Xi c S Xi c rn;2 S rn;. (ii) P n g i ( s ; l ) g i ( s ; l ) 0! V, and P n g i ( s ; l )! d N (0; V ), (iii) tere exists ^ ( s ; l ) 2 int ( n ( s ; l )) satisfying P n log + ^ ( s ; l ) 0 P g i ( s ; l ) = su n 2n(s; l ) log + 0 g i ( s ; l ) w..a., ^ ( s ; l ) = O () =2, and max in ^ (s ; l ) 0 g i ( s ; l )! 0. Proof of (i). We only rove te rst statement. Te oter statements can be sown in te same manner. By te cange of variables and Assumtion 3. (i)-(iv), E [S ln; ] = Var (S ln; ) Z 0 K (z) zf (c + z) dz! s l; ; k " 2 E Xi c 2 Xi ( I i ) K # c 2 = Z 0 K (z) 2 z 2 f (c + z) dz! 0:(26) k Terefore, Cebysev s inequality yields te conclusion. Proof of (ii). Proof of te rst statement. It is su cient to sow tat ( I i ) Kli 2 (Y i l ) 2! Vl ; I i Kri 2 (Y i s l ) 2! Vr : Since te roofs are similar, we only sow te rst statement. By te de nition of K 2 li, ( I i ) Kli 2 (Y i l ) 2 = Sln;2 2 2S ln;2 S ln; Xi ( I i ) K Xi ( I i ) K c 2 (Y i l ) 2 + S 2 c 2 Xi 6 ln; c Xi c 2 Xi c 2 ( I i ) K (Y i l ) 2 (Y i l ) 2 : (27)

17 By te same argument to derive (26), " Xi E ( I i ) K Var Xi ( I i ) K c 2 (Y i # l ) 2 c 2 (Y i! l ) 2! 2 l s l;20;! 0; (28) Tus, from Cebysev s inequality and Lemma A. (i), te robability limit of te rst term in (27) is 2 l s2 l;2 s l;20. By alying te same argument to te second and tird terms of (27), we obtain te conclusion. Proof of te second statement. From te de nition of g i ( s ; l ), it is su cient to sow tat ( I i ) K li (Y i l )! d N (0; V l ) ; I i K ri (Y i s l )! d N (0; V r ) : Since te roofs are similar, we only sow te rst statement. From te de nition of K li, ( I i ) K li (Y i l ) = (S ln;2 s l;2 ) Xi c ( I i ) K (Y i l ) Xi c Xi c (S ln; s l; ) ( I i ) K (Y i l ) 8 n o 9 + < ( I i ) K Xi c s Xi c l;2 s l; (Y i l ) = n o i : E ( I i ) K Xi c s Xi c l;2 s l; (Y i l ) ; r n + E Xi c Xi c ( I i ) K s l;2 s l; (Y i l ) = T T 2 + T 3 + T 4 : For T, Lyaunov s central limit teorem imlies Xi ( I i ) K c Xi (Y i l ) E ( I i ) K c d (Y i l )! N 0; 2 l s l;20 ; and te cange of variables and Assumtion 3. (ii)-(iv) imly Xi E ( I i ) K c Z 0 (Y i l ) = K (z) (E [Y i j X i = c + z] l ) f (c + z) dz = 2 0 l s l;0+o 3 : k Tus, from Lemma A. (i) and n =2 3=2! 0 (Assumtion 3. (iv)), we ave T = o (). Similarly, we can sow tat T 2 = o (). For T 4, te cange of variables and Assumtion 3. (ii)-(iv) yield T 4 = Z 0 K (z) (s l;2 s l; z) (E [Y i j X i = c + z] l ) f (c + z) dz k = 0 l s l;2 s l;0 s 2 l; + O 2! 0: 7

18 For T 3, note tat E T3 2 = Z 0 k K (z) 2 (s l;2 Z 0 k s l; z) 2 E (Y i l ) 2 i Xi = c + z f (c + z) dz 2 K (z) (s l;2 s l; z) (E [Y i j X i = c + z] l ) f (c + z) dz! 2 l s 2 l;2 s l;20 2s l;2 s l; s l;2 + s 2 l; s l;22 = Vl ; were te convergence follows from a similar argument to (28). Terefore, Lyaunov s central limit d teorem imlies T 3! N (0; Vl ). Combining tese results, we obtain te conclusion. Proof of (iii). Since te roof is similar to Newey and Smit (2004, Lemmas A and A2), it is omitted. Lemma A.2. Suose tat Assumtion 3. olds. Ten (i) S ln;0! sl;0, and S rn;0! sr;0, (ii) P n g i ( s ; ^) g i ( s ; ^) 0! V, and P n g i ( s ; ^) = O () =2, (iii) tere exists ^ ( s ; ^) 2 int ( n ( s ; ^)) satisfying P n log + ^ ( s ; ^) 0 P g i ( s ; ^) = su n 2n(s;^) log + 0 g i ( s ; ^) w..a., ^ ( s ; ^) = O () =2, and max in ^ (s ; ^) 0 g i ( s ; ^)! 0. Detailed roofs are available from te autors uon request. Te roof of Lemma A.2 (i) is similar to tat of Lemma A. (i). Te second statement of Lemma A.2 (ii) follows from a similar argument to te roof of Newey and Smit (2004, Lemma A3) combined wit Lemma A.. Since tis statement imlies te weak consistency of ^ to l, Lemma A. (ii) imlies te rst statement of Lemma A.2 (ii). Also, given te consistency of ^ and Lemma A.2 (ii), a similar argument to te roof of Newey and Smit (2004, Lemma A2) imlies Lemma A.2 (iii). 8

19 References [] Angrist, J. D. and Lavy, V. (999) Using Maimonides rule to estimate te e ect of class size on scolastic acievement. Quarterly Journal of Economics, 4, [2] Angrist, J. D., Imbens, G. W. and Rubin, D. B. (996) Identi cation of causal e ects using instrumental variables. Journal of te American Statistical Association, 9, [3] Cen, S. X. and Qin, Y. S. (2000) Emirical likeliood con dence intervals for local linear smooters. Biometrika, 87, [4] Fan, J. and Gijbels, I. (996) Local Polynomial Modelling and Its Alications. New York: Caman & Hall. [5] Fan, J., Zang, C. and Zang, J. (200) Generalized likeliood ratio statistics and Wilks enomenon. Annals of Statistics, 29, [6] Fan, J. and Yao, Q. (998) E cient estimation of conditional variance functions in stocastic regression. Biometrika, 85, [7] Han, J., Todd, P. and van der Klaauw, W. (200) Identi cation and estimation of treatment e ects wit a regression discontinuity design. Econometrica, 69, [8] Holland, P. (986) Statistics and causal inference. Journal of te American Statistical Association, 8, [9] Imbens, G. W. and Lemieux, T. (2008) Regression discontinuity designs: a guide to ractice. Journal of Econometrics, 42, [0] Kna, G., Sacks, J. and Ylvisaker, D. (985) Con dence bands for regression functions. Journal of te American Statistical Association, 80, [] Newey, W. K. and Smit, R. J. (2004) Higer order roerties of GMM and generalized emirical likeliood estimators. Econometrica, 72, [2] Owen, A. B.(988) Emirical likeliood ratio con dence intervals for a single functional. Biometrika, 75, [3] Owen, A. B. (200) Emirical Likeliood. New York: Caman & Hall. [4] Porter, J. (2003) Estimation in te regression discontinuity model. Working aer, Deartment of Economics, University of Wisconsin. [5] Rubin, D. (974) Estimating causal e ects of treatments in randomized and non-randomized studies. Journal of Educational Psycology, 5,

20 [6] Ruert, D., Wand, M. P., Holst, U. and Hössjer, O. (997) Local olynomial variance function estimation. Tecnometrics, 39, [7] Sacks, J. and Ylvisaker, D. (978) Linear estimates for aroximately linear models. Annals of Statistics, 6, [8] Tistletwaite, D. and Cambell, D. (960) Regression-discontinuity analysis: an alternative to te ex-ost factor exeriment. Journal of Educational Psycology, 5, [9] Trocim, W. (200) Regression-discontinuity design. In N. J. Smelser and P. B. Baltes (eds.), International Encycloedia of te Social and Beavioral Sciences, vol. 9, , Oxford, UK: Elsevier. 20

21 Table : Te constant C(K) in te asymtotic variances of b s and b f (c.f. Porter, 2003) Kernel K(u) C(K) Uniform I [ 0:5;0:5] Bartlett ( juj)i [ ;] Eanecnikov 3 4 ( u2 )I [ ;] Biweigt 5 6 ( u2 ) 2 I [ ;] 5. Triweigt ( u2 ) 3 I [ ;]

22 8 6 χ 2 () quantiles χ 2 () quantiles (a).6 (b).2 n=200 n=00 4 Y i µ(x)=x X i Figure : (a) A reresentative samle wit 00 observations; (b) Te distributions of te bandwidts selected by cross validation over 000 relications wen te samle sizes are 00 and (a) n=00, =.0 7 (b) n=200, = Emirical quantiles 3 2 AN AN2 45 o line EL Emirical quantiles Figure 2: Quantile-Quantile lot of te 2 () distribution against te nite-samle distribution of eac of te two squared t-test statistics (AN and AN2) and emirical likeliood-basd test statistic (EL). 22

23 Table 2: Te emirical sizes of two t-tests and te emirical likeliood-based test wit various xed bandwidts and te one selected via cross validation, wen te nominal sizes are 5% and 0% and te samle size is 00. (Standard errors are in te arenteses.) Bandwidt T ests 5% Sizes 0% Sizes b r b l b s var( b s ) \ var( b s ) b 2 r b 2 l T rue values of arameters 3: = 0:8 AN :5 (0:69) 0:20 (0:59) :60 (0:9) AN :70 (0:38) EL = 0:9 AN :7 (0:62) 0:6 (0:57) :53 (0:5) AN :62 (0:3) EL = :0 AN :3 (0:59) 0:3 (0:52) :46 (0:3) AN :57 (0:25) EL = : AN : (0:55) 0:2 (0:54) :42 (0:) AN :52 (0:22) EL = :2 AN :07 (0:52) 0:06 (0:48) :39 (0:09) AN :49 (0:9) EL = :3 AN :05 (0:50) 0:06 (0:45) :36 (0:08) AN :45 (0:8) EL cv AN : (0:63) 0: (0:58) :47 (0:25) AN :55 (0:34) EL :29 (0:57) :83 (:4) :22 (0:53) :83 (:09) :7 (0:48) :82 (0:97) :6 (0:47) :79 (0:90) :4 (0:44) :87 (0:88) :2 (0:42) :82 (0:89) :22 (0:5) :84 (:00) :32 (0:52) :24 (0:99) :35 (0:48) :22 (0:88) :38 (0:47) :30 (0:79) :39 (0:46) :35 (0:77) :42 (0:43) :39 (0:80) :45 (0:4) :37 (0:75) :39 (0:47) :33 (0:83) 23

24 Table 3: Te emirical sizes of two t-tests and te emirical likeliood-based tests wit various xed bandwidts and te one selected via cross validation, wen te nominal sizes are 5% and 0% and te samle size is 200. (Standard errors are in te arenteses.) Bandwidt T ests 5% Sizes 0% Sizes b r b l b s var( b s ) \ var( b s ) b 2 r b 2 l T rue values of arameters 3: = 0:7 AN :8 (0:5) 0:9 (0:43) :35 (0:08) AN :42 (0:6) EL = 0:8 AN :6 (0:47) 0:8 (0:4) :3 (0:07) AN :37 (0:3) EL = 0:9 AN :4 (0:42) 0:8 (0:37) :27 (0:05) AN :33 (0:) EL = :0 AN :4 (0:39) 0:5 (0:36) :24 (0:05) AN :30 (0:09) EL = : AN :2 (0:40) 0: (0:35) :2 (0:04) AN :28 (0:08) EL = :2 AN :05 (0:37) 0:09 (0:32) :20 (0:03) AN :25 (0:07) EL cv AN :7 (0:5) 0:4 (0:45) :29 (0:6) AN :36 (0:9) EL :39 (0:45) :94 (0:88) :32 (0:42) :96 (0:84) :26 (0:38) :98 (0:82) :23 (0:35) :92 (0:72) :6 (0:33) :97 (0:70) :5 (0:3) :95 (0:70) :28 (0:47) :96 (0:87) :34 (0:39) :30 (0:69) :37 (0:38) :34 (0:67) :38 (0:36) :34 (0:6) :40 (0:34) :37 (0:58) :44 (0:33) :39 (0:58) :46 (0:30) :43 (0:54) :40 (0:37) :38 (0:63) 24

25 Size adjusted ower Size adjusted ower Size adjusted ower Size adjusted ower (a) =0.8 (b) = AN EL AN θ in te alternative s (c) = θ in te alternative s θ in te alternative s (d) = cv θ in te alternative s Figure 3: Te size-adjusted owers of te two t-tests (AN and AN2) and te emirical likeliood-based test (EL) for various xed bandwidts and te one selected via cross validation, wen te nominal size is 0% and te samle size is

26 Size adjusted ower Size adjusted ower Size adjusted ower Size adjusted ower.0 (a) =0.7.0 (b) = θ in te alternative s (c) = θ in te alternative s θ in te alternative s (d) = cv AN EL AN θ in te alternative s Figure 4: Te size-adjusted owers of te two t-tests (AN and AN2) and te emirical likeliood-based test (EL) for various xed bandwidts and te one selected via cross validation, wen te nominal size is 0% and te samle size is

27 .2 Assignment Probabilities Enrollment Count Figure 5: Te lot of te assignment (wit imosed random noises) by te enrollment count, and te local linear estimates of te conditional robabilities of getting treated (slitting into two classes) given te enrollment counts for te controlled samle (enrollment 40) and te treatment samle (enrollment> 40). 27

28 Mat Score Enrollment Count Figure 6: Te lot of te average mat scores by te enrollment counts, and te local linear ts for te controlled samle (enrollment 40) and te treatment samle (enrollment> 40). 28

29 Verbal Score Enrollment Count Figure 7: Te lot of te average verbal scores by te enrollment counts, and te local linear ts for te controlled samle (enrollment 40) and te treatment samle (enrollment> 40). 29

30 Jum in assignment robabilities Table 4: Te adjusted 0% critical values (used in obtaining te size-adjusted owers) of te two squared t tests (AN and AN2) and te EL test. n = 00 = 0:8 = :0 = :2 cv unadjusted AN 5: AN EL 3: n = 200 = 0:7 = 0:9 = : cv unadjusted AN 4: AN EL 3: (a) θ f (Mat score) (b) Estimates 90% AN CIs 90% AN CIs 90% EL CIs 90% EL CIs Bandwidt Bandwidt Figure 8: Te local linear estimates and te 90% asymtotic normality con dence intervals (AN CIs) and emirical likeliood con dence intervals (AN CIs) of (a) te jum in te roensity score and (b) te average causal treatment e ect of slitting into two classes on uils mat score. 30

31 θ f (Verbal score) Estimates 90% AN CIs 90% AN CIs 90% EL CIs 90% EL CIs Bandwidt Figure 9: Te local linear estimates and te 90% asymtotic normality con dence intervals (AN CIs) and emirical likeliood con dence intervals (AN CIs) of te average causal treatment e ect of slitting into two classes on uils verbal score. 3