{"title":"矩阵秩的改进推理","authors":"Qihui Chen, Z. Fang","doi":"10.2139/ssrn.3177681","DOIUrl":null,"url":null,"abstract":"This paper develops a general framework for conducting inference on the rank of an unknown matrixΠ0. A defining feature of our setup is the null hypothesis of the formH0:rank(Π0)≤r. The problem is of first‐order importance because the previous literature focuses onH0′:rank(Π0)=rby implicitly assuming awayrank(Π0)<r, which may lead to invalid rank tests due to overrejections. In particular, we show that limiting distributions of test statistics underH0′may not stochastically dominate those underrank(Π0)<r. A multiple test on the nullsrank(Π0)=0,…,r, though valid, may be substantially conservative. We employ a testing statistic whose limiting distributions underH0are highly nonstandard due to the inherent irregular natures of the problem, and then construct bootstrap critical values that deliver size control and improved power. Since our procedure relies on a tuning parameter, a two‐step procedure is designed to mitigate concerns on this nuisance. We additionally argue that our setup is also important for estimation. We illustrate the empirical relevance of our results through testing identification in linear IV models that allows for clustered data and inference on sorting dimensions in a two‐sided matching model with transferrable utility.","PeriodicalId":420730,"journal":{"name":"ERN: Bargaining Theory (Topic)","volume":"19 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2018-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"22","resultStr":"{\"title\":\"Improved Inference on the Rank of a Matrix\",\"authors\":\"Qihui Chen, Z. Fang\",\"doi\":\"10.2139/ssrn.3177681\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper develops a general framework for conducting inference on the rank of an unknown matrixΠ0. A defining feature of our setup is the null hypothesis of the formH0:rank(Π0)≤r. The problem is of first‐order importance because the previous literature focuses onH0′:rank(Π0)=rby implicitly assuming awayrank(Π0)<r, which may lead to invalid rank tests due to overrejections. In particular, we show that limiting distributions of test statistics underH0′may not stochastically dominate those underrank(Π0)<r. A multiple test on the nullsrank(Π0)=0,…,r, though valid, may be substantially conservative. We employ a testing statistic whose limiting distributions underH0are highly nonstandard due to the inherent irregular natures of the problem, and then construct bootstrap critical values that deliver size control and improved power. Since our procedure relies on a tuning parameter, a two‐step procedure is designed to mitigate concerns on this nuisance. We additionally argue that our setup is also important for estimation. We illustrate the empirical relevance of our results through testing identification in linear IV models that allows for clustered data and inference on sorting dimensions in a two‐sided matching model with transferrable utility.\",\"PeriodicalId\":420730,\"journal\":{\"name\":\"ERN: Bargaining Theory (Topic)\",\"volume\":\"19 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2018-12-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"22\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ERN: Bargaining Theory (Topic)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2139/ssrn.3177681\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ERN: Bargaining Theory (Topic)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2139/ssrn.3177681","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
This paper develops a general framework for conducting inference on the rank of an unknown matrixΠ0. A defining feature of our setup is the null hypothesis of the formH0:rank(Π0)≤r. The problem is of first‐order importance because the previous literature focuses onH0′:rank(Π0)=rby implicitly assuming awayrank(Π0)
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