Quasi-maximum likelihood estimation for conditional quantiles |
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| Institution: | 1. Department of Economics, University of California, San Diego, 9500 Gilman Dr. La Jolla, CA, 92093, USA;2. School of Economics, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul, 120-749, Republic of Korea;3. Yale University, USA;4. University of Auckland, New Zealand;5. Singapore Management University, Singapore;6. University of Southampton, United Kingdom;1. Department of Applied Economics, Department of Finance, National Chung Hsing University, Taichung, Taiwan;2. Instituto Complutense de Análisis Económico (ICAE), Facultad de Ciencias Económicas y Empresariales, Universidad Complutense de Madrid, Spain;3. Department of Economics, Emory University, USA;4. Department of Finance, Asia University, Taiwan;5. Econometric Institute, Erasmus School of Economics, Erasmus University Rotterdam, the Netherlands;6. Discipline of Business Analytics, University of Sydney Business School, Australia;7. Institute of Advanced Sciences, Yokohama National University, Japan;1. Department of Economics, Indiana University, Wylie Hall, 100 South Woodlawn Avenue, Bloomington, IN 47405, United States;2. Department of Economics, The University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, WI 53201, United States;1. Department of Economics, MIT, 50 Memorial Drive, Cambridge, MA 02142, United States;2. Boston University, Department of Economics, 270 Bay State Road, Boston, MA 02215, United States;3. Department of Economics, Yale University, 37 Hillhouse Avenue, New Haven, CT 06520, United States;4. NBER, 1050 Massachusetts Avenue, Cambridge, MA 02138, United States;1. Department of Economics, University of Iowa, W284 Pappajohn Business Building, 21 E. Market Street, Iowa City, IA 52242, United States;2. Graduate School of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan;1. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, China;2. Department of Mathematics, City University of Hong Kong, Hong Kong;3. South University of Science and Technology, Shenzhen, China;4. Department of Statistics, Chinese University of Hong Kong, Hong Kong |
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| Abstract: | In this paper, we construct a new class of estimators for conditional quantiles in possibly misspecified nonlinear models with time series data. Proposed estimators belong to the family of quasi-maximum likelihood estimators (QMLEs) and are based on a new family of densities which we call ‘tick-exponential’. A well-known member of the tick-exponential family is the asymmetric Laplace density, and the corresponding QMLE reduces to the Koenker and Bassett's (Econometrica 46 (1978) 33) nonlinear quantile regression estimator. We derive primitive conditions under which the tick-exponential QMLEs are consistent and asymptotically normally distributed with an asymptotic covariance matrix that accounts for possible conditional quantile model misspecification and which can be consistently estimated by using the tick-exponential scores and Hessian matrix. Despite its non-differentiability, the tick-exponential quasi-likelihood is easy to maximize by using a ‘minimax’ representation not seen in the earlier work on conditional quantile estimation. |
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