{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T16:43:59Z","timestamp":1759941839210},"reference-count":72,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2020,6,2]],"date-time":"2020-06-02T00:00:00Z","timestamp":1591056000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2021,6]]},"abstract":"<jats:p>We present a new robust bootstrap method for a test when there is a nuisance parameter under the alternative, and some parameters are possibly weakly or nonidentified. We focus on a Bierens (1990,<jats:italic>Econometrica<\/jats:italic>58, 1443\u20131458)-type conditional moment test of omitted nonlinearity for convenience. Existing methods include the supremum<jats:italic>p<\/jats:italic>-value which promotes a conservative test that is generally not consistent, and test statistic transforms like the supremum and average for which bootstrap methods are not valid under weak identification. We propose a new wild bootstrap method for<jats:italic>p<\/jats:italic>-value computation by targeting specific identification cases. We then combine bootstrapped<jats:italic>p<\/jats:italic>-values across polar identification cases to form an asymptotically valid<jats:italic>p<\/jats:italic>-value approximation that is robust to any identification case. Our wild bootstrap procedure does not require knowledge of the covariance structure of the bootstrapped processes, whereas Andrews and Cheng\u2019s (2012a,<jats:italic>Econometrica<\/jats:italic>80, 2153\u20132211; 2013,<jats:italic>Journal of Econometrics<\/jats:italic>173, 36\u201356; 2014,<jats:italic>Econometric Theory<\/jats:italic>30, 287\u2013333) simulation approach generally does. Our method allows for robust bootstrap critical value computation as well. Our bootstrap method (like conventional ones) does not lead to a consistent<jats:italic>p<\/jats:italic>-value approximation for test statistic functions like the supremum and average. Therefore, we smooth over the robust bootstrapped<jats:italic>p<\/jats:italic>-value as the basis for several tests which achieve the correct asymptotic level, and are consistent, for any degree of identification. They also achieve uniform size control. A simulation study reveals possibly large empirical size distortions in nonrobust tests when weak or nonidentification arises. One of our smoothed<jats:italic>p<\/jats:italic>-value tests, however, dominates all other tests by delivering accurate empirical size and comparatively high power.<\/jats:p>","DOI":"10.1017\/s0266466620000201","type":"journal-article","created":{"date-parts":[[2020,6,2]],"date-time":"2020-06-02T10:00:58Z","timestamp":1591092058000},"page":"409-463","source":"Crossref","is-referenced-by-count":2,"title":["WEAK-IDENTIFICATION ROBUST WILD BOOTSTRAP APPLIED TO A CONSISTENT MODEL SPECIFICATION TEST"],"prefix":"10.1017","volume":"37","author":[{"given":"Jonathan B.","family":"Hill","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2020,6,2]]},"reference":[{"key":"S0266466620000201_r7","doi-asserted-by":"publisher","DOI":"10.2307\/2951753"},{"key":"S0266466620000201_r66","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1111\/j.2517-6161.1980.tb01126.x","article-title":"Threshold autoregression, limit cycles and cyclical data (with discussion)","volume":"42","author":"Tong","year":"1980","journal-title":"Journal of the Royal Statistical Society, Series B"},{"key":"S0266466620000201_r22","first-page":"33","article-title":"Hypothesis testing when a nuisance parameter is present only under the alternative","volume":"74","author":"Davies","year":"1987","journal-title":"Biometrika"},{"key":"S0266466620000201_r32","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0099111"},{"key":"S0266466620000201_r58","doi-asserted-by":"publisher","DOI":"10.2307\/3314764"},{"key":"S0266466620000201_r26","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1018031266"},{"key":"S0266466620000201_r42","unstructured":"Hoffman-J\u00f8rgensen, J. 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