Empirical Likelihood Estimation of Continuous-Time Models With Conditional Moment Restrictions

The EL estimators have some favorable higher order asymptotic properties. We extend the EL method proposed by Donald et al. (2003) to estimate non-i:i:d: continuous-time models with the known functional form of the conditional characteristic function. In many cases even the MLE method can not be performed, the EL method can do. More over, not only does the EL method resolve the problem of covariance matrix singularity in the regular GMM but also utilize the information in the conditional moment conditions fully. The EL method can be applied to many popular nancial models such as some of diffusion models, jump diffusion models and stochastic volatility models. By way of a Monte Carlo comparison, we show that the EL method has better nite sample properties than C-GMM introduced by Carrasco et al. (2004). Work in the area of EL has been initiated by Owen (1988). Qin and Lawless (1994) proposed the EL method for general estimation equations. Donald et al. (2003) and Kitamura et al. (2004) propound some estimation methods with conditional moment restrictions. Our method is an extension of the EL method by Donald et al. (2003). According to Newey and Smith (2004), The EL estimators have some favorable higher order asymptotic properties. In particular, the higher order asymptotic bias of the EL will be less than that of the GMM, when there are manymoment restrictions. Such theoretical advantage can lead to better results to the empirical analysis with many moment restrictions. Although the method of Kitamura et al. (2004) can also work, we do not select the method of Kitamura et al. (2004), because of its computational burdens. When we know a closed-form and the tractable expression of the likelihood function of the model, the maximum likelihood estimation method is the best option to estimate parameters of a model. Unfortunately, in many cases we often fail to derive a tractable form of likelihood function, especially when it comes to models with jumps, derivation of the tractable form becomes more dif cult. Since in many cases the characteristic function often has a tractable form even when likelihood function does not, the characteristic function may be employed as an available substitute of likelihood function. Once we obtained a tractable form of the characteristic function, we can exploit conditional moment conditions using the conditional characteristic function (CCF) and the empirical conditional characteristic function (ECCF) for non-i:i:d: processes. Based on the conditional moment conditions derived by the way mentioned above, one can perform the EL estimation. A great deal of effort has been made on the estimation using the GMM approach in this area (see, for example, Singleton 2001, Chacko and Viceira 2003 and Yu 2004). What seems to be a grave drawback, however, is the singularity of the covariance matrix, which occurs when we possess many moment conditions. Carrasco et al. (2003) have introduced the C-GMM (GMM with a continuum of moment conditions) to overcome such a drawback of GMM approach. For the same purpose, we propose a different estimation method adopting the maximum empirical likelihood (EL) approach. We carry out a Monte-Carlo experiment with the CIR model and a jump diffusion model to compare our method with C-GMM. As a result, the EL method shows us some surpassing nite sample properties, which are shown by the Monte-Carlo simulation.