Yue­hao Bai

I am an Assis­tant Pro­fes­sor of Eco­nom­ics at the Uni­ver­sity of Michi­gan. My research inter­ests lie in econo­met­ric the­ory. I received my PhD from the Uni­ver­sity of Chicago in 2020.

Published or forthcoming

1. (2021) “Infer­ence in Exper­i­ments with Matched Pairs” (with J. P. Romano and A. M. Shaikh), forth­com­ing in the Jour­nal of the Amer­i­can Sta­tis­ti­cal Asso­ci­a­tion. abstract sup­ple­ment doi code

   This paper stud­ies infer­ence for the aver­age treat­ment effect in ran­dom­ized con­trolled tri­als where treat­ment sta­tus is deter­mined accord­ing to a “matched pairs” design. By a “matched pairs” design, we mean that units are sam­pled i.i.d. from the pop­u­la­tion of inter­est, paired accord­ing to observed, base­line covari­ates and finally, within each pair, one unit is selected at ran­dom for treat­ment. This type of design is used rou­tinely through­out the sci­ences, but results about its impli­ca­tions for infer­ence about the aver­age treat­ment effect are not avail­able. The main require­ment under­ly­ing our analy­sis is that pairs are formed so that units within pairs are suit­ably “close” in terms of the base­line covari­ates, and we develop novel results to ensure that pairs are formed in a way that sat­is­fies this con­di­tion. Under this assump­tion, we show that, for the prob­lem of test­ing the null hypoth­e­sis that the aver­age treat­ment effect equals a pre-spec­i­fied value in such set­tings, the com­monly used two-sam­ple $t$-test and “matched pairs” $t$-test are con­ser­v­a­tive in the sense that these tests have lim­it­ing rejec­tion prob­a­bil­ity under the null hypoth­e­sis no greater than and typ­i­cally strictly less than the nom­i­nal level. We show, how­ever, that a sim­ple adjust­ment to the stan­dard errors of these tests leads to a test that is asymp­tot­i­cally exact in the sense that its lim­it­ing rejec­tion prob­a­bil­ity under the null hypoth­e­sis equals the nom­i­nal level. We also study the behav­ior of ran­dom­iza­tion tests that arise nat­u­rally in these types of set­tings. When imple­mented appro­pri­ately, we show that this approach also leads to a test that is asymp­tot­i­cally exact in the sense described pre­vi­ously, but addi­tion­ally has finite-sam­ple rejec­tion prob­a­bil­ity no greater than the nom­i­nal level for cer­tain dis­tri­b­u­tions sat­is­fy­ing the null hypoth­e­sis. A sim­u­la­tion study and empir­i­cal illus­tra­tion con­firm the prac­ti­cal rel­e­vance of our the­o­ret­i­cal results.

2. (2021) “A Two-Step Method for Test­ing Many Moment Inequal­i­ties” (with A. San­tos and A. M. Shaikh), forth­com­ing in the Jour­nal of Busi­ness and Eco­nomic Sta­tis­tics. abstract doi code

   This paper con­sid­ers the prob­lem of test­ing a finite num­ber of moment inequal­i­ties. For this prob­lem, Romano et al. (2014) pro­pose a two-step test­ing pro­ce­dure. In the first step, the pro­ce­dure incor­po­rates infor­ma­tion about the loca­tion of moments using a con­fi­dence region. In the sec­ond step, the pro­ce­dure accounts for the use of the con­fi­dence region in the first step by adjust­ing the sig­nif­i­cance level of the test appro­pri­ately. Its jus­ti­fi­ca­tion, how­ever, has so far been lim­ited to set­tings in which the num­ber of moments is fixed with the sam­ple size. In this paper, we pro­vide weak assump­tions under which the same pro­ce­dure remains valid even in set­tings in which there are “many” moments in the sense that the num­ber of moments grows rapidly with the sam­ple size. We con­firm the prac­ti­cal rel­e­vance of our the­o­ret­i­cal guar­an­tees in a sim­u­la­tion study. We addi­tion­ally pro­vide both numer­i­cal and the­o­ret­i­cal evi­dence that the pro­ce­dure com­pares favor­ably with the method pro­posed by Cher­nozhukov et al. (2019), which has also been shown to be valid in such set­tings.

3. (2021) “Infer­ence for Sup­port Vec­tor Regres­sion under $\ell_1$ Reg­u­lar­iza­tion” (with H. Ho, G. A. Pouliot, and J. K. C. Shea), AEA Papers and Pro­ceed­ings, vol. 111, pp. 611-615. abstract doi

   We show that sup­port vec­tor regres­sion (SVR) con­sis­tently esti­mates lin­ear median regres­sion func­tions and we develop a large sam­ple infer­ence method based on the inver­sion of a novel test sta­tis­tic in order to pro­duce error bars for SVR with $\ell_1$-norm reg­u­lar­iza­tion. Under a homoskedas­tic­ity assump­tion com­monly imposed in the quan­tile regres­sion lit­er­a­ture, the pro­ce­dure does not involve esti­ma­tion of den­si­ties. It is thus unique amongst large sam­ple infer­ence meth­ods for SVR in that it cir­cum­vents the need to select a band­width para­me­ter. Sim­u­la­tion stud­ies sug­gest that our pro­ce­dure pro­duces nar­rower error bars than does the stan­dard infer­ence method in quan­tile regres­sion.

Working papers

1. (2020) “Opti­mal­ity of Matched-Pair Designs in Ran­dom­ized Con­trolled Tri­als,” revi­sion requested by the Amer­i­can Eco­nomic Review. abstract sup­ple­ment slides

   This paper stud­ies the opti­mal­ity of matched-pair designs in ran­dom­ized con­trolled tri­als (RCTs). Matched-pair designs are exam­ples of strat­i­fied ran­dom­iza­tion, in which the researcher par­ti­tions a set of units into strata based on their observed covari­ates and assign a frac­tion of units in each stra­tum to treat­ment. A matched-pair design is such a pro­ce­dure with two units per stra­tum. Despite the preva­lence of strat­i­fied ran­dom­iza­tion in RCTs, imple­men­ta­tions dif­fer vastly. We pro­vide an econo­met­ric frame­work in which, among all strat­i­fied ran­dom­iza­tion pro­ce­dures, the opti­mal one in terms of the mean-squared error of the dif­fer­ence-in-means esti­ma­tor is a matched-pair design that orders units accord­ing to a scalar func­tion of their covari­ates and matches adja­cent units. Our frame­work cap­tures a lead­ing moti­va­tion for strat­i­fy­ing in the sense that it shows that the pro­posed matched-pair design addi­tion­ally min­i­mizes the mag­ni­tude of the ex-post bias, i.e., the bias of the esti­ma­tor con­di­tional on real­ized treat­ment sta­tus. We then con­sider empir­i­cal coun­ter­parts to the opti­mal strat­i­fi­ca­tion using data from pilot exper­i­ments and pro­vide two dif­fer­ent pro­ce­dures depend­ing on whether the sam­ple size of the pilot is large or small. For each pro­ce­dure, we develop meth­ods for test­ing the null hypoth­e­sis that the aver­age treat­ment effect equals a pre­spec­i­fied value. Each test we pro­vide is asymp­tot­i­cally exact in the sense that the lim­it­ing rejec­tion prob­a­bil­ity under the null equals the nom­i­nal level. We run an exper­i­ment on the Ama­zon Mechan­i­cal Turk using one of the pro­posed pro­ce­dures, repli­cat­ing one of the treat­ment arms in Della­Vigna and Pope (2018), and find the stan­dard error decreases by 29%, so that only half of the sam­ple size is required to attain the same stan­dard error.

2. (2020) “Why Ran­dom­ize? Min­i­max Opti­mal­ity under Per­mu­ta­tion Invari­ance,” revi­sion requested by the Jour­nal of Econo­met­rics. abstract

   This paper stud­ies finite sam­ple min­i­max opti­mal ran­dom­iza­tion schemes and esti­ma­tion schemes in esti­mat­ing para­me­ters includ­ing the aver­age treat­ment effect, when treat­ment effects are het­ero­ge­neous. A ran­dom­iza­tion scheme is a dis­tri­bu­tion over a group of per­mu­ta­tions of a given treat­ment assign­ment vec­tor. An esti­ma­tion scheme is a joint dis­tri­bu­tion over assign­ment vec­tors, lin­ear esti­ma­tors, and per­mu­ta­tions of assign­ment vec­tors. The key ele­ment in the min­i­max prob­lem is that the worst case is over a class of dis­tri­b­u­tions of the data which is invari­ant to a group of per­mu­ta­tions. First, I show that given any assign­ment vec­tor and any esti­ma­tor, the uni­form dis­tri­bu­tion over the same group of per­mu­ta­tions, namely the com­plete ran­dom­iza­tion scheme, is min­i­max opti­mal. Sec­ond, under fur­ther assump­tions on the class of dis­tri­b­u­tions and the objec­tive func­tion, I show the min­i­max opti­mal esti­ma­tion scheme involves com­pletely ran­dom­iz­ing an assign­ment vec­tor, while the opti­mal esti­ma­tor is the dif­fer­ence-in-means under com­plete invari­ance and a weighted aver­age of within-block dif­fer­ences under a block struc­ture, and the num­bers of treated and untreated units are deter­mined by Ney­man allo­ca­tions.

3. (2021) “Par­tial Iden­ti­fi­ca­tion of Treat­ment Effect Rank­ings with Instru­men­tal Vari­ables” (with A. M. Shaikh and E. J. Vyt­lacil), work­ing paper. abstract slides

   This paper devel­ops par­tial-iden­ti­fi­ca­tion and infer­ence for treat­ment effect para­me­ters and the rank­ings of treat­ments in an instru­men­tal vari­able frame­work while impos­ing alter­na­tive monot­o­nic­ity restric­tions. In par­tic­u­lar, we con­sider a dis­crete, multi-val­ued treat­ment, a binary out­come, and a dis­crete, pos­si­bly multi-val­ued instru­ment. We use a lin­ear pro­gram­ming for­mu­la­tion to present a flex­i­ble frame­work and to develop gen­eral results for char­ac­ter­iz­ing the testable restric­tions and the sharp iden­ti­fi­ca­tion of treat­ment effect para­me­ters and the rank­ings of treat­ments in terms of these para­me­ters that fol­low from impos­ing instru­ment exo­gene­ity while addi­tion­ally impos­ing alter­na­tive monot­o­nic­ity restric­tions on how the treat­ments depend on the instru­ments and how the out­comes depend on the treat­ments. Our results nest both ordered and unordered treat­ments. We fur­ther char­ac­ter­ize lead­ing spe­cial cases of our gen­eral analy­sis. We develop meth­ods for simul­ta­ne­ous infer­ence about the con­sis­tency of the observed data with our restric­tions and the treat­ment effect rank­ing when the dis­tri­bu­tion of the observed data is con­sis­tent with our restric­tions. We illus­trate our method­ol­ogy with empir­i­cal appli­ca­tions to the encour­age­ment design of Behaghel, Cre­pon and Gur­gand (2014) inves­ti­gat­ing the effects of pub­lic vs pri­vate job search assis­tance; the RCTs with one-sided non-com­pli­ance of Angrist, Lang and Ore­opou­los (2009) inves­ti­gat­ing the effects of alter­na­tive strate­gies on aca­d­e­mic per­for­mance of col­lege stu­dents and of Blattman, Jami­son, and Sheri­dan (2017) inves­ti­gat­ing the effects cash incen­tives and ther­apy on reduc­ing crime in Liberia; and the RCT with close sub­sti­tutes of Kline and Wal­ters (2016) inves­ti­gat­ing the effects of alter­na­tive early child­hood pro­grams.