[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-201824-105":59,"doc-detail-201824-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","quantile-based-test-for-heterogeneous-treatment-effects","Quantile-based Test for Heterogeneous Treatment Effects","","Permutation test methodology for heterogeneous treatment effects is developed using the quantile process. Quantile-based procedures are shown to be threatened by estimated nuisance parameters that distort asymptotic behavior, motivating a solution via Khmaladze’s martingale transformation. The paper proves asymptotic size control for a permutation test built on the transformed statistic, independent of underlying distribution features. Extensive numerical results demonstrate strong size and power versus related quantile-based tests, and the work provides a fast implementation algorithm with an application to welfare-reform experimental data.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/quantile-based-test-for-heterogeneous-treatment-effects/201824/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/quantile-based-test-for-heterogeneous-treatment-effects/201824.png","ImageObject",300,407,{"name":92,"@type":93},"Jake","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-29","2026-09-04",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",13,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What problem does the paper address in quantile-based tests for heterogeneous treatment effects?","Question",{"text":113,"@type":114},"Quantile-based tests can lose validity because the estimation of nuisance parameters affects the limiting behavior of the test statistic, leading to an asymptotic distribution that is not tractable.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How does the paper restore asymptotic validity for the permutation test?",{"text":118,"@type":114},"It applies Khmaladze’s martingale transformation to the quantile process, producing an asymptotically pivotal transformed statistic whose permutation distribution matches the sampling distribution in large samples.",{"name":120,"@type":111,"acceptedAnswer":121},"What do the numerical and empirical parts show?",{"text":122,"@type":114},"Numerical evidence indicates good size and power relative to other quantile-based tests, and an application to a welfare reform experiment finds strong evidence against constant treatment effects across multiple subgroups.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},201824,1788539084,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":130,"read_time":145,"preview_page_count":4},962084928904,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Quantile-based Test for Heterogeneous Treatment Effects  \nEunYi Chung†  \nDepartment of Economics University of Illinois, Urbana–Champaign  \n[eunyi@illinois. edu](eunyi@illinois. edu)  \nMauricio Olivares Department of Statistics LMU Munich  \n[m. olivares@lmu. de](m. olivares@lmu. de)  \nMarch 1, 2023  \nAbstract  \nWe introduce a permutation test for heterogeneous treatment effects based on the quantile process. However, tests based on the quantile process often suffer from estimated nuisance parameters that jeopardize their validity, even in large samples. To overcome this problem, we use Khmaladze’s martingale transformation. We show that the permutation test based on the transformed statistic controls size asymptotically. Numerical evidence asserts the good size and power performance of our test procedure compared to other popular quantile-based tests. We discuss a fast implementation algorithm and illustrate our method using experimental data from a welfare reform.  \nKeywords: Permutation Test, Quantile Treatment Effects, Heterogeneous Treatment Effects.  \nJEL Classification: C12, C14, C46 .  \n†This paper supersedes the paper by the authors which was circulated with the title Comment on “Can  \nVariation in Subgroups’ Average Treatment Effects Explain Treatment Effect Heterogeneity? Evidence from a Social Experiment.” Mauricio Olivares acknowledges support from the European Research Council (Starting Grant No. 852332) . We are grateful to MDRC for granting access to the experimental data we use in this paper. All errors are our own.  \n1 Introduction  \nThe study of heterogeneous treatment effects (HTE) plays an important role in program evaluation. A popular approach to studying HTE involves a form of subgroup analysis: divide the sample into subgroups defined by covariates, and then estimate the average treatment effects (ATEs) across subgroups. Then, this approach detects HTE by checking whether the ATEs vary significantly across subgroups.1  \nWhile the ATE subgroup analysis might detect some HTE, it has important limitations because it focuses on mean impacts (Bitler, Gelbach, and Hoynes, 2006) . Therefore, we take an alternative route to HTE analysis based on the comparison of quantile treatment effect (QTE) at different quantiles. Under this framework, we test for HTE by checking if the QTEs are constant across quantiles, i. e. , whether the QTEs are equal to an unknown constant γ for all quantiles. Thus, by looking at the entire distribution and not only the mean, the QTE approach complements the ATE subgroup analysis by giving the researcher more tools to investigate HTE.  \nHowever, tests of HTE based on QTEs come with their own challenges. In particular, when we estimate the nuisance parameter γ to compute the test statistic, the estimation error influences the limit behavior of the test statistic. In fact, its asymptotic distribution becomes intractable because of the dependence on the (unknown) probability distribution generating the data. This phenomenon is called the Durbin problem in the literature.  \nIn this paper, we introduce a new permutation test for HTE based on the quantile process. Importantly, we establish the asymptotic validity of our method despite the presence of the nuisance parameter. To this end, we apply Khmaladze’s (1981) martingale transformation of the quantile process. Simply put, the Khmaladze transformation removes the estimation effects by residualizing the quantile process. This transformation yields an asymptotically pivotal statistic, i. e . , a statistic whose limiting distribution does not depend on the fundamentals. Our main theoretical result shows the permutation distribution of the transformed statistic and  \n1 According to a survey of papers in top economics journals, 40% of the studies report at least one treatment effect this way (Chernozhukov et al. , 2018) . This practice sparked the development of tests for HTE. For example, Crump et al. (2008) test whether the ATEs conditional","cbCaigMqM70ulqNa","https://ap.wps.com/l/cbCaigMqM70ulqNa","pdf",523159,35,"English","# Introduction\n## Heterogeneous treatment effects and subgroup analysis\n## Quantile treatment effects and QTE-constancy testing\n## Durbin problem and nuisance-parameter challenges\n## New permutation test via Khmaladze martingale transformation\n## Multiple-testing implementation and welfare-reform application","[{\"question\":\"What problem does the paper address in quantile-based tests for heterogeneous treatment effects?\",\"answer\":\"Quantile-based tests can lose validity because the estimation of nuisance parameters affects the limiting behavior of the test statistic, leading to an asymptotic distribution that is not tractable.\"},{\"question\":\"How does the paper restore asymptotic validity for the permutation test?\",\"answer\":\"It applies Khmaladze’s martingale transformation to the quantile process, producing an asymptotically pivotal transformed statistic whose permutation distribution matches the sampling distribution in large samples.\"},{\"question\":\"What do the numerical and empirical parts show?\",\"answer\":\"Numerical evidence indicates good size and power relative to other quantile-based tests, and an application to a welfare reform experiment finds strong evidence against constant treatment effects across multiple subgroups.\"}]","Quantile-based Test for Heterogeneous Treatment Effects | PDF",88]