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Likelihood Based Estimation in a Panel Setting: Robustness, Redundancy and Validity of Copulas

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Department of Economics Working Paper Series Likelihood Based Estimation in a Panel Setting: Robustness, Redundancy and Validity of Copulas 09-002 Artem Prokhorov Concordia University Peter Schmidt Michigan State University Department of Economics, 1455 De Maisonneuve Blvd. West, Montreal, Quebec, Canada H3G 1M8 Tel 514–848–2424 # 3900 · Fax 514–848–4536 · [email protected] · alcor.concordia.ca/~econ/repec Likelihood Based Estimation in a Panel Setting: Robustness, Redundancy and Validity of Copulas Artem Prokhorov∗ Peter Schmidt† Abstract This paper considers estimation of likelihood-based models in a panel setting. That is, we have panel data, and for each time period separately we have a correctly specified model that could be estimated by MLE. We want to allow non-independence over time. This paper shows how to improve on the QMLE. It then considers MLE based on joint distributions constructed using copulas. It discusses the efficiency gain from using the true copula, and shows that knowledge of the true copula is redundant only if the vari- ance matrix of the relevant set of moment conditions is singular. It also discusses the question of robustness against misspecification of the copula, and proposes a test of the validity of the copula. GMM methods are argued to be useful analytically, and also for reasons of efficiency if the copula is robust but not correct. JEL Classification: C13 Keywords: Copula, MLE, QMLE, GMM, panel data ∗Department of Economics, Concordia University, Montreal, PQ H3G1M8 Canada; email: [email protected] †Department of Economics, Michigan State University, East Lansing, MI 48824 USA; email: [email protected] 1 Introduction In this paper we suppose that we have panel data, with individuals i = 1, . . . , N and time periods t = 1, . . . , T . We presume that N is large and T is small, and we will consider asymptotic arguments as N → ∞ with T fixed. In this setting, suppose t

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