Maximum Likelihood Estimatioll of Misspecified Models Halbert矶ThiteEconometrica, Volume 50, Issue 1 (Jan., 1982), 1-26 Stable URL http://links .j st01"α'g/sici ?sici=OO 12-9682%28198201 %2950%3Al %3C 1% 3AMLEOMM%%3B2-C Your use of the JSTOR archive indicates your acceptance of JSTOR’s Terms and Conditions of Use, available at JSTOR’s Terms and Conditions ofUse provides, in part, that unl巳切yoohave obtained prior permission, you may not download an entire issue of a joumal or multiple copies of articI町，皿dyou may use content in the JSTOR archive only for your pαrsonal， r丁u-c饥nmercialuse Each copy of any part of a JSTOR transmission must contain the same copyright notice that app国rson the screen or printed page of such tr皿smlSSlonECOfωmetrica is pubIished by The Society. Please contact由epublisher for further permissions regarding the use of this work. Publisher contact information may be obtained at http://www .j stor .org/joumals/econosoc .html Econometrica 。1982The Society JSTOR and由eJSTOR logo are trademarks of JSTOR, and are Registered in the . Patent and Trademark Office For more information on JSTOR contactjstor-info@ 。2003JSTOR 趾h'"ψp)f八w、τ'huAp严r321:15:162ο03
ECONOMETRICA VOLUME 50 NUMBE混lJANUARY, 1982 MAXIMUM LIKELIHOOD ESTIMATION OF MISSPECIFIED MODELS 1 By HALBERT WHlTETlus paper the ∞ and detectton of model m四P'α U剑ngmaximum likelihood teεhnique'5 for estlmalωn and mference. The quas山町aXlmumllkelihood esl˛malor (QML罔cú[0 a well defined limil, a叭dmay or may nO[ b~ cOQSistent for particular阳rametersof interest. Standard tes凶(Wald，Lagrange Multiplier or Llkellhood Ratlo) afe invalid m lhe presence of ml创pecifica!\on，blJt more general statistics are given which aUow inferences to be drawn robustly. The prope阳Sof the QMLE a时也emformatωn matr :: are exp[oiled 10 yield severa[ lIseful tes[s f.ωmode[ misspedfication l. INTRODUcrION SINCE R. A FrsHER advocated the method of maximum likelihood in his inftuential papers [13, 14], it has become one of the most important tools for estimat on and inference available to statistiαans. A fllndamental assllmpt on Ilnderiying classical reslllts on the properti出ofthe maximllm likelihood由ttmator (., Wald [3坷;LeCam [23]) is that the stochastic law which determines the behavior of the phenomena investigated (出e"trlle" strllctllre) is known to lLe within a specified parametric family of probability distributions (the model). In other words, the probability model is assumed to be "correctly specified." In many (if not most) circumstances, one may not have complete confidence that this is so If one does not assume that the probabil ty model is correctly specified, t is natural to ask what happens to the properti时ofthe maximum likelihood estimator. 00因此stiUconverge to some 1im t asymptotically, and does this lim t have any meaning? If the estimator is somehow consistent, is it also asymptoti cally normal?口。因theestimator have propert es which can be Ilsed to decide whether or not the the specified family of probabihty distributions do出contamthe true structllre? This paper provides a Ilnified framework within which specific answers to each of these equations can be given The consistency question was apparently first considered independently by Berk [7, 8] and Huber [201. Berk tak口aBayesian approach and ment ons in passing the information theoretic interpretation emphasized here. Hllber’s ap proach is classical; he provid臼verygeneral conditio时，bui1ding on those of Wald [321, under which the maximllm likelihood est˛mator converges to a well-defined limit, even when the probability model is not correctly specified 11 am lndebted 10 Jon飞N"eUner，Tom Rothenberg. lhe referees. and the partlcipaQ臼oflhe HarvardfMIT fCOQOmelrics worbhop for helpful CommeQlS aQd
