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DIRECTIONALLY DIFFERENTIABLE ECONOMETRIC MODELS

  • Jin Seo Cho (a1) and Halbert White (a2)
Abstract

The current article examines the limit distribution of the quasi-maximum likelihood estimator obtained from a directionally differentiable quasi-likelihood function and represents its limit distribution as a functional of a Gaussian stochastic process indexed by direction. 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.

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Corresponding author
*Address correspondence to Jin Seo Cho, School of Economics, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 03722, Korea; e-mail: jinseocho@yonsei.ac.kr.
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The co-editor, Yoon-Jae Whang, and two anonymous referees provided very helpful comments for which we are most grateful. The second author (Halbert White) passed away while the submission version was written. He formed the outline of the article and also provided an exemplary guide for writing a quality research paper. The authors benefited from discussions with Yoichi Arai, Seung Chan Ahn, In Choi, Horag Choi, Robert Davies, Graham Elliott, Chirok Han, John Hillas, Jung Hur, Hide Ichimura, Isao Ishida, Yongho Jeon, Estate Khmaladze, Chang-Jin Kim, Chang Sik Kim, Tae-Hwan Kim, Naoto Kunitomo, Hon Ho Kwok, Taesuk Lee, Bruce Lehmann, Mark Machina, Jaesun Noh, Kosuke Oya, Taeyoung Park, Peter Phillips, Erwann Sbai, Juwon Seo, Donggyu Sul, Denis Tkachenko, Albert K.C. Tsui, Hung-Jen Wang, Yoshihiro Yajima, Byung Sam Yoo, Ping Yu, and other seminar participants at Sogang University, the University of Auckland, the University of Hong Kong, the University of Tokyo, Osaka University, the Econometrics Study Group of the Korean Econometric Society, VUW, Yonsei University, and other conference participants at NZESG (Auckland, 2013). Cho acknowledges support from the Yonsei University Future-leading Research Initiative of 2017 (2017-22-0090).

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References
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Aigner D., Lovell C., & Schmidt P. (1977) Formulation and estimation of stochastic frontier production function models. Journal of Econometrics 6, 2137.
Andrews D. (1999) Estimation when a parameter is on a boundary. Econometrica 67, 543563.
Andrews D. (2001) Testing when a parameter is on the boundary of the maintained hypothesis. Econometrica 69, 683734.
Baek Y., Cho J.S., & Phillips P.C.B. (2015) Testing linearity using power transforms of regressors. Journal of Econometrics 187, 376384.
Billingsley P. (1999) Convergence of Probability Measures. Wiley.
Box G. & Cox D. (1964) An analysis of transformations. Journal of the Royal Statistical Society, Series B 26, 211252.
Chernoff H. (1954) On the distribution of the likelihood ratio. The Annals of Mathematical Statistics 54, 573578.
Cho J.S. (2011) Quasi-maximum likelihood estimation revisited using the distance and direction method. Journal of Economic Theory and Econometrics 23, 89112.
Cho J.S. & Ishida I. (2012) Testing for the effects of omitted power transformations. Economics Letters 117, 287290.
Cho J.S., Ishida I., & White H. (2011) Revisiting tests for neglected nonlinearity using artificial neural networks. Neural Computation 23, 11331186.
Cho J.S., Ishida I., & White H. (2014) Testing for neglected nonlinearity using twofold unidentified models under the null and hexic expansions. In Haldrup N., Meitz M., & Saikkonen P., eds., Essays in Nonlinear Time Series Econometrics, pp. 327. Oxford University Press.
Cho J.S. & Phillips P.C.B. (2017) Sequentially Testing Polynomial Model Hypothesis Using the Power Transform of Regressors. Journal of Applied Econometrics, forthcoming.
Cho J.S. & White H. (2007) Testing for regime switching. Econometrica 75, 16711720.
Cho J.S. & White H. (2010) Testing for unobserved heterogeneity in exponential and weibull duration models. Journal of Econometrics 157, 458480.
Cho J.S. & White H. (2011) Generalized runs tests for the IID hypothesis. Journal of Econometrics 162, 326344.
Cho J.S. & White H. (2017) Supplements to “Directionally Differentiable Econometric Models”. School of Economics, Yonsei University. Available at: http://web.yonsei.ac.kr/jinseocho/pardiff.htm.
Davies R. (1977) Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 64, 247254.
Davies R. (1987) Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 74, 3343.
Doukhan P., Massart P., & Rio E. (1995) Invariance principles for absolutely regular empirical processes. Annales de l’Institut Henri Poincaré, Probabilites et Statistiques 31, 393427.
Duofur J.-M. (2006) Monte carlo tests with nuisance parameters: A general approach to fininte-sample inference and nonstandard asymptotics in ecoonometrics. Journal of Econometrics 133, 443477.
Fang Z. & Santos A. (2014) Inference on Directionally Differentiable Functions, ArXiv preprint arXiv: 1404.3763.
Hansen B. (1996a) Stochastic equicontinuity for unbounded dependent heterigeneous arrays. Econometric Theory 12, 347359.
Hansen B. (1996b) Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64, 413440.
King M. & Shively T. (1993) Locally optimal testing when a nuisance parameter is present only under the alternative. The Review of Economics and Statistics 75, 17.
Kim J. & Pollard D. (1990) Cube root asymptotics. Annals of Statistics 18, 191219.
Liu X. & Shao Y. (2003) Asymptotics for likelihood ratio tests under loss of identifiability. Annals of Statistics 31, 807832.
Pollard D. (1985) New ways to prove central limit theorem. Econometric Theory 1, 295–131.
Rosenberg B. (1973) The analysis of a cross-section of time series by stochastically convergent parameter regression. Annals of Economic and Social Measurement 2, 399428.
Rudin W. (1976) Principles of Mathematical Analysis. McGraw-Hill.
Stevenson R. (1980) Likelihood functions for generalized stochastic frontier estimation. Journal of Econometrics 13, 5766.
Stout W. (1974) Almost Sure Convergence. Academic Press.
Troutman J. (1996) Variational Calculus and Optimal Control. Springer-Verlag.
van der Vaart A. & Weller J. (1996) Weak Convergence and Empirical Processes with Applications to Statistics. Springer-Verlag.
Wald A. (1943) Tests of statistical hypotheses concerning several parameters when the number of observations is large. Transactions of the American Mathematical Society 54, 426482.
Wald A. (1949) Note on the consistency of the maximum likelihood estimate. The Annals of Mathematical Statistics 20, 596601.
White H. & Cho J.S. (2012) Higher-order approximations for testing neglected nonlinearity. Neural Computation 24, 273287.
Wooldridge J. & White H. (1988) Some invariance principles and central limit theorems for dependent heterogeneous processes. Econometric Theory 4, 210230.
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Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
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