{"title":"A Simple and Adaptive Confidence Interval when Nuisance Parameters Satisfy an Inequality","authors":"Gregory Fletcher Cox","doi":"arxiv-2409.09962","DOIUrl":null,"url":null,"abstract":"Inequalities may appear in many models. They can be as simple as assuming a\nparameter is nonnegative, possibly a regression coefficient or a treatment\neffect. This paper focuses on the case that there is only one inequality and\nproposes a confidence interval that is particularly attractive, called the\ninequality-imposed confidence interval (IICI). The IICI is simple. It does not\nrequire simulations or tuning parameters. The IICI is adaptive. It reduces to\nthe usual confidence interval (calculated by adding and subtracting the\nstandard error times the $1 - \\alpha/2$ standard normal quantile) when the\ninequality is sufficiently slack. When the inequality is sufficiently violated,\nthe IICI reduces to an equality-imposed confidence interval (the usual\nconfidence interval for the submodel where the inequality holds with equality).\nAlso, the IICI is uniformly valid and has (weakly) shorter length than the\nusual confidence interval; it is never longer. The first empirical application\nconsiders a linear regression when a coefficient is known to be nonpositive. A\nsecond empirical application considers an instrumental variables regression\nwhen the endogeneity of a regressor is known to be nonnegative.","PeriodicalId":501293,"journal":{"name":"arXiv - ECON - Econometrics","volume":"17 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - ECON - Econometrics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09962","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Inequalities may appear in many models. They can be as simple as assuming a
parameter is nonnegative, possibly a regression coefficient or a treatment
effect. This paper focuses on the case that there is only one inequality and
proposes a confidence interval that is particularly attractive, called the
inequality-imposed confidence interval (IICI). The IICI is simple. It does not
require simulations or tuning parameters. The IICI is adaptive. It reduces to
the usual confidence interval (calculated by adding and subtracting the
standard error times the $1 - \alpha/2$ standard normal quantile) when the
inequality is sufficiently slack. When the inequality is sufficiently violated,
the IICI reduces to an equality-imposed confidence interval (the usual
confidence interval for the submodel where the inequality holds with equality).
Also, the IICI is uniformly valid and has (weakly) shorter length than the
usual confidence interval; it is never longer. The first empirical application
considers a linear regression when a coefficient is known to be nonpositive. A
second empirical application considers an instrumental variables regression
when the endogeneity of a regressor is known to be nonnegative.