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In this way, the standard analysis that assumes a differentiable quasi-likelihood function is treated as a special case of our analysis. We also examine and redefine the standard quasi-likelihood ratio, Wald, and Lagrange multiplier test statistics so that their null limit behaviors are regular under our model framework.<\/jats:p>","DOI":"10.1017\/s0266466617000354","type":"journal-article","created":{"date-parts":[[2017,8,22]],"date-time":"2017-08-22T11:39:29Z","timestamp":1503401969000},"page":"1101-1131","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":5,"title":["DIRECTIONALLY DIFFERENTIABLE ECONOMETRIC MODELS"],"prefix":"10.1017","volume":"34","author":[{"given":"Jin Seo","family":"Cho","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Halbert","family":"White","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2017,8,22]]},"reference":[{"key":"S0266466617000354_ref1","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4076(77)90052-5"},{"key":"S0266466617000354_ref13","doi-asserted-by":"publisher","DOI":"10.1111\/j.1468-0262.2007.00809.x"},{"key":"S0266466617000354_ref4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2015.03.041"},{"key":"S0266466617000354_ref11","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199679959.003.0001"},{"key":"S0266466617000354_ref35","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177729952"},{"key":"S0266466617000354_ref15","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2011.02.001"},{"key":"S0266466617000354_ref32","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0737-5"},{"key":"S0266466617000354_ref29","volume-title":"Principles of Mathematical Analysis","author":"Rudin","year":"1976"},{"key":"S0266466617000354_ref34","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1943-0012401-3"},{"key":"S0266466617000354_ref19","first-page":"393","article-title":"Invariance principles for absolutely regular empirical processes","volume":"31","author":"Doukhan","year":"1995","journal-title":"Annales de l\u2019Institut Henri Poincar\u00e9, Probabilites et Statistiques"},{"key":"S0266466617000354_ref12","article-title":"Sequentially Testing Polynomial Model Hypothesis Using the Power Transform of Regressors","author":"Cho","year":"2017","journal-title":"Journal of Applied Econometrics, forthcoming"},{"key":"S0266466617000354_ref7","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177728725"},{"key":"S0266466617000354_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.econlet.2012.05.029"},{"key":"S0266466617000354_ref30","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4076(80)90042-1"},{"key":"S0266466617000354_ref6","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1111\/j.2517-6161.1964.tb00553.x","article-title":"An analysis of transformations","volume":"26","author":"Box","year":"1964","journal-title":"Journal of the Royal Statistical Society, Series B"},{"key":"S0266466617000354_ref36","doi-asserted-by":"publisher","DOI":"10.1162\/NECO_a_00225"},{"key":"S0266466617000354_ref25","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176347498"},{"key":"S0266466617000354_ref18","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":"S0266466617000354_ref33","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2545-2"},{"key":"S0266466617000354_ref14","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2010.03.046"},{"key":"S0266466617000354_ref22","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600006629"},{"key":"S0266466617000354_ref28","first-page":"399","article-title":"The analysis of a cross-section of time series by stochastically convergent parameter regression","volume":"2","author":"Rosenberg","year":"1973","journal-title":"Annals of Economic and Social Measurement"},{"key":"S0266466617000354_ref8","first-page":"89","article-title":"Quasi-maximum likelihood estimation revisited using the distance and direction method","volume":"23","author":"Cho","year":"2011","journal-title":"Journal of Economic Theory and Econometrics"},{"key":"S0266466617000354_ref10","doi-asserted-by":"publisher","DOI":"10.1162\/NECO_a_00117"},{"key":"S0266466617000354_ref16","doi-asserted-by":"crossref","unstructured":"Cho J.S. & White H. 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