Mathieu Kessler

Estimation of an Ergodic Diffusion from Discrete Observations

We consider a one-dimensional diffusion process X, with ergodic property, with drift b(x, ) and diffusion coefficient a(x, U) depending on unknown parameters and U. We are interested in the joint estimation of ( , U). For that purpose, we dispose of a discretized trajectory, observed at n equidistant times t n a ihn ,1 < i < n. We assume that hn! 0 and nhn!1. Under the condition nh p ! 0 for an arbitrary integer p ,w e exhibit a contrast dependent on p which provides us with an asymptotically normal and efficient estimator of ( , U).

Estimating equations based on eigenfunctions for a discretely observed diffusion process

A new type of martingale estimating function is proposed for inference about classes of diffusion processes based on discrete-time observations. These estimating functions can be tailored to a particular class of diffusion processes by utilizing a martingale property of the eigenfunctions of the generators of the diffusions. Optimal estimating functions in the sense of Godambe and Heyde are found. Inference based on these is invariant under transformations of data. A result on consistency and asymptotic normality of the estimators is given for ergodic diffusions. The theory is illustrated by several examples and by a simulation study

Asymptotic Likelihood Based Inference for Co‐integrated Homogenous Gaussian Diffusions

In this paper we consider inference for a multivariate Gaussian homogenous diffusion which is co-integrated, i.e. admits a representation in terms of stable relations (ergodic diffusions) plus Brownian motions. We show that inference on co-integration rank (the number of stable relations) in continuous time can be based on existing asymptotic distributions from discrete time co-integration analysis. Likewise the asymptotic distributions of the co-integration parameters are shown to be mixed Gaussian. Special attention is given to the parametrization of the drift terms. It is shown that the asymptotic distribution of the co-integration rank test statistic does not depend on the level of the process as a result of the chosen parametrization.

Identification and Inference for Multivariate Cointegrated and Ergodic Gaussian Diffusions

Inference is considered in the multivariate continuous time Gaussian Ornstein-Uhlenbeck (OU) model on the basis of observations in discrete time. Under the hypothesis of ergodicity as well as cointegration, the classical identification or ‘aliasing’ problem is re-addressed and new results given. Exact conditions are given for (i) identification of individual parameters, as well as results for, (ii) identification of rank and cointegration parameters, and, furthermore (iii) for the existence of a continuous time OU process which embeds a discrete time vector autoregression. Estimation and cointegration rank inference are discussed. An empirical illustration is given in which the ‘cost-of-carry’ hypothesis is investigated.

Computational Aspects Related to Martingale Estimating Functions for a Discretely Observed Diffusion

Martingale estimating functions for a discretely observed diffusion have turned out to provide estimators with nice asymptotic properties. However, their expression usually involves some conditional expectation that has to be evaluated through Monte Carlo simulations giving rise to an approximated estimator. In this work we study, for ergodic models, the asymptotic properties of the approximated estimator and describe the influence of the number of independent simulated trajectories involved in the Monte Carlo method as well as of the approximation scheme used. Our results are of practical relevance to assess the implementation of martingale estimating functions for discretely observed diffusions.

The information in the marginal law of a Markov chain

If we have a parametric model for the invariant distribution of a Markov chain but cannot or do not want to use any information about the transition distribution (except, perhaps, that the chain is reversible) — what, then, is the best use we can make of the observations? We determine a lower bound for the asymptotic variance of regular estimators and show constructively that the bound is attainable. The results apply to discretely observed diffusions. AMS 1991 subject classifications. Primary 62G20, 62M05; secondary 62F12.

On Intrinsic Priors for Nonnested Models

Model selection problems involving nonnested models are considered. Bayes factor based solution to these problems needs prior distributions for the parameters in the alternative models. When the prior information on these parameters is vague default priors are available but, unfortunately, these priors are usually imporper which yields a calibration problem that makes the Bayes factor to be defined up to a multiplicative constant. Intrinsic priors have been introduced for solving this difficulty. While these priors are well established for nested models, their construction for nonnested models is still an open problem.In this latter setting this paper studies the system of functional equations that defines the intrinsic priors. It is shown that the solutions to these equations are obtained from the solutions to a single homogeneous linear functional equation. The Bayes factors associated with these solutions are analyzed. Some illustrative examples are provided and, in particular, location, scale, and location-scale models are considered.