[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-201987-105":59,"doc-detail-201987-en":130},{"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":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","whats-new-in-econometrics-lecture-14-quantile-methods","What’s New in Econometrics - Lecture 14 - Quantile Methods","","Econometrics lecture 14 introduces quantile-focused estimation methods and contrasts them with mean-based approaches such as OLS and least absolute deviations. The material reviews conditions under which OLS and LAD identify similar parameters, emphasizes how asymmetric and heteroskedastic error structures affect consistency, and clarifies resilience to outliers versus identification differences. It then extends quantile regression using endogenous explanatory variables, panel-data settings, and censored-data methods, highlighting useful asymptotic results and quantile regression foundations.",{"@graph":69,"@context":122},[70,84,105],{"@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/whats-new-in-econometrics-lecture-14-quantile-methods/201987/",{"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/whats-new-in-econometrics-lecture-14-quantile-methods/201987.png","ImageObject",300,407,{"name":92,"@type":93},"Violet","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-09-18","2026-09-04",true,{"@type":102,"interactionType":103,"userInteractionCount":19},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"When do OLS and LAD provide similar parameter estimates?","Question",{"text":112,"@type":113},"If the conditional distribution of the error is symmetric about zero, or if the error is independent of covariates with mean zero (with the corresponding normalization), both OLS and LAD consistently estimate the location and slope parameters.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Why can OLS and LAD estimates differ even for thin-tailed distributions?",{"text":117,"@type":113},"If the error distribution is asymmetric and varies with covariates, the identified quantities of least squares (conditional mean) and least absolute deviations (conditional median) differ, so similar estimates should not be expected.",{"name":119,"@type":110,"acceptedAnswer":120},"How is quantile regression formulated and estimated in the lecture?",{"text":121,"@type":113},"For a target quantile, covariates affect quantiles under a linearity assumption. Estimation uses minimizing the quantile “check” function, with weights determined by the chosen quantile level, yielding consistency under standard arguments.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},201987,1788540696,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":19,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144},4398048950312,"https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908","What’s New in Econometrics?  \nLecture 14 Quantile Methods  \nJeff Wooldridge NBER Summer Institute, 2007  \n1 . Reminders About Means, Medians, and Quantiles  \n2 . Some Useful Asymptotic Results  \n3 . Quantile Regression with Endogenous Explanatory Variables  \n4 . Quantile Regression for Panel Data  \n5 . Quantile Methods for “Censored” Data  \n1 . Reminders About Means, Medians, and Quantiles  \n∙ Consider the standard linear model in a population, with intercept 􀀩 and K 􀂕 1 slopes 􀀪:  \ny 􀀝 􀀩 􀀎 x􀀪 􀀎 u . (1)  \nAssume E􀂟 u 2 􀂠 􀀜 􀀮, so that the distribution of u is not too spread out. Given a large random sample, when should we expect ordinary least squares, which solves  \nma, 􀂟yi − a − xi b 􀂠 2 , (2)  \nand least absolute deviations (LAD), which solves  \nma, |yi − a − xi b | , (3)  \nto provide similar parameter estimates? There are two important cases . If  \nD 􀂟 u|x􀂠 is symmetric about zero (4)  \nthen OLS and LAD both consistently estimate 􀀩 and 􀀪 . If  \nu is independent of x with E 􀂟 u 􀂠 􀀝 0, (5)  \nwhere E 􀂟 u 􀂠 􀀝 0 is the normalization that identifies 􀀩, then OLS and LAD both consistently estimate the slopes, 􀀪 . If u has an asymmetric distribution, then Med 􀂟 u 􀂠 ≡ 􀀱 ≠ 0, and LAD converges to 􀀩 􀀎 􀀱 because Med 􀂟y |x􀂠 􀀝 􀀩 􀀎 x􀀪 􀀎 Med􀂟 u|x􀂠 􀀝 􀀩 􀀎 x􀀪 􀀎 􀀱 .  \n∙ In many applications, neither (4) nor (5) is likely to be true . For example, y may be a measure of wealth, in which case the error distribution is probably asymmetric and Var 􀂟 u|x􀂠 not constant.  \n∙ Therefore, it is important to remember that if D􀂟 u|x􀂠 is asymmetric and changes with x, then we should not expect OLS and LAD to deliver similar estimates of 􀀪, even for “thin-tailed” distributions. It is important to separate discussions of resiliency to outliers from the  \ndifferent quantities identified by least squares (E 􀂟y |x􀂠) and least absolution deviations 􀂟Med 􀂟y |x􀂠􀂠 .  \n∙ Of course, LAD is much more resilient to changes in extreme values because, as a measure of central tendency, the median is much less sensitive than the mean to changes in extreme values. But it does not follow that a large difference in OLS and LAD estimates means something is “wrong” with OLS.  \n∙ Big advantage for median over mean: the median passes through monotonic functions. For example, if log􀂟y 􀂠 􀀝 􀀩 􀀎 x􀀪 􀀎 u and Med 􀂟 u|x􀂠 􀀝 0, then Med􀂟y |x􀂠 􀀝 exp 􀂟 Med 􀂡log􀂟y 􀂠|x􀂢􀂠 􀀝 exp 􀂟 􀀩 􀀎 x􀀪􀂠 . By contrast, we cannot generally find E􀂟y |x􀂠 􀀝 exp 􀂟 􀀩 􀀎 x􀀪􀂠E 􀂡 exp 􀂟 u 􀂠|x􀂢 .  \n∙ But the expectation operator has useful properties that the median  \ndoes not: linearity and the law of iterated expectations. Suppose we begin with a random coefficient model  \nyi 􀀝 ai 􀀎 xi bi , (6)  \nIf 􀂟ai , bi 􀂠 is independent of xi , then  \nE 􀂟yi |xi 􀂠 􀀝 E 􀂟 ai |xi 􀂠 􀀎 xiE 􀂟 bi |xi 􀂠 ≡ 􀀩 􀀎 xi 􀀪 , (7)  \nwhere 􀀩 􀀝 E 􀂟 ai 􀂠 and 􀀪 􀀝 E 􀂟 bi 􀂠 . So OLS consistently estimates 􀀩 and 􀀪 . By contrast, no way to derive Med􀂟yi |xi 􀂠 without imposing more restrictions.  \n∙ What can we add so that LAD estimates something of interest in (7)? If u i is a vector, then its distribution conditional on xi is centrally  \nsymmetric if D􀂟ui |xi 􀂠 􀀝 D 􀂟−ui |xi 􀂠, which implies that, if gi is any vector function of xi , D􀂟g ′iui |xi 􀂠 has a univariate distribution that is symmetric about zero. This implies E􀂟 ui |xi 􀂠 􀀝 0 .  \n∙ Apply central symmetry to random coefficient model by writing ci 􀀝 􀂟 ai , bi 􀂠 with 􀀫 􀀝 E 􀂟 ci 􀂠, and let di 􀀝 ci − 􀀫 . Then  \nyi 􀀝 􀀩 􀀎 xi 􀀪 􀀎 􀂟 ai − 􀀩􀂠 􀀎 xi 􀂟 bi − 􀀪􀂠 (8)  \nwith gi 􀀝 􀂟1, xi 􀂠 . If ci given xi is centrally symmetric about 􀀫 , then  \nMed 􀂟 g ′i􀂟 ci − 􀀫􀂠|xi 􀂠 􀀝 0, and LAD applied to the usual model yi 􀀝 􀀩 􀀎 xi 􀀪 􀀎 ui consistently estimates 􀀩 and 􀀪 .  \n∙ For 0 􀀜 􀁁 􀀜 1, q 􀂟 􀁁 􀂠 is the 􀁁 th quantile of yi if P 􀂟yi ≤ q 􀂟 􀁁 􀂠􀂠 ≥ 􀁁 and  \nP 􀂟yi ≥ q 􀂟 􀁁 􀂠􀂠 ≥ 1 − 􀁁 .  \n∙ Usually, we are interested in how covariates affect quantiles (of  \nwhich the median is the special case with 􀁁 􀀝 1/2􀂠 . Under linearity,  \nQuant 􀁁 􀂟yi |xi 􀂠 􀀝 􀀩 􀂟 􀁁 􀂠 􀀎 xi 􀀪􀂟􀁁 􀂠 . (9)  \nUnder (9), consistent estimators of 􀀩􀂟 􀁁 􀂠 and 􀀪􀂟􀁁 􀂠 are obtained ","cbCaisCf6XxL9wV5","https://ap.wps.com/l/cbCaisCf6XxL9wV5","pdf",117546,34,"English","# 1. Reminders About Means, Medians, and Quantiles\n## OLS vs LAD identification under symmetry and independence\n## Median invariance under monotonic transformations\n## Quantiles, check function, and consistency\n# 2. Some Useful Asymptotic Results\n## Misspecification and probability limits\n## Linear quantile approximation and estimator characterization","[{\"question\":\"When do OLS and LAD provide similar parameter estimates?\",\"answer\":\"If the conditional distribution of the error is symmetric about zero, or if the error is independent of covariates with mean zero (with the corresponding normalization), both OLS and LAD consistently estimate the location and slope parameters.\"},{\"question\":\"Why can OLS and LAD estimates differ even for thin-tailed distributions?\",\"answer\":\"If the error distribution is asymmetric and varies with covariates, the identified quantities of least squares (conditional mean) and least absolute deviations (conditional median) differ, so similar estimates should not be expected.\"},{\"question\":\"How is quantile regression formulated and estimated in the lecture?\",\"answer\":\"For a target quantile, covariates affect quantiles under a linearity assumption. Estimation uses minimizing the quantile “check” function, with weights determined by the chosen quantile level, yielding consistency under standard arguments.\"}]","What’s New in Econometrics - Lecture 14 - Quantile Methods | PDF",86]