We propose nonparametric estimators of the infinitesimal coefficients associated with second-order stochastic differential equations+ We show that under appropriate conditions, the proposed estimators are consistent+ Also, we state conditions ensuring the asymptotic normality of these estimators+ We conclude our paper with a Monte Carlo experiment in which we assess the response of the nonparametric estimators with respect to the step of discretization+ I thank two anonymous referees who made valuable suggestions that led to considerable improvements in the paper+ I am also grateful to Carlos Braumann and Tom Kundert for helpful comments+ This research was supported by the Fundação para a Ciência e a Tecnologia~FCT! and by FEDER0POCI 2010+ Address correspondence to João Nicolau,

Several economic and financial time series are bounded by an upper and lower finite limit (e.g., interest rates). It is not possible to say that these time series are random walks because random walks are limitless with probability one (as time goes to infinity). Yet, some of these time series behave just like random walks. In this paper we propose a new approach that takes into account these ideas. We propose a discrete-time and a continuous-time process (diffusion process) that generate bounded random walks. These paths are almost indistinguishable from random walks, although they are stochastically bounded by an upper and lower finite limit. We derive for both cases the ergodic conditions, and for the diffusion process we present a closed expression for the stationary distribution. This approach suggests that many time series with random walk behavior can in fact be stationarity processes.

This article presents a new simulation-based technique for estimating the likelihood of stochastic differential equations. This technique is based on a result of DacunhaCastelle and Florens-Zmirou. These authors proved that the transition densities of a nonlinear diffusion process with a constant diffusion coefficient can be written in a closed form involving a stochastic integral. We show that this stochastic integral can be easily estimated through simulations and we prove a convergence result. This simulator for the transition density is used to obtain the simulated maximum likelihood (SML) estimator. We show through some Monte Carlo experiments that our technique is highly computationally efficient and the SML estimator converges rapidly to the maximum likelihood estimator.

In this paper, we quantify the asymptotic bias of the Florens-Zmirou~1993, Journal of Applied Probability 30, 790-804! and Jiang and Knight~1997, Econometric Theory 13, 615-645! estimator for the diffusion coefficient when the step of discretization is fixed, and then we propose a bias adjustment that partially compensates for the distortion+ Also, we show that our estimators have all the asymptotic properties of the Florens-Zmirou and Jiang and Knight estimator when the step of discretization goes to zero+ We provide some examples+ I thank the editor Peter C+B+ Phillips and the two referees for comments and suggestions that led to considerable improvement of the paper+ I am also grateful to Carlos Braumann and Tom Kundert for helpful comments+ This research was supported by the Fundação para a Ci

We propose a new model for multivariate Markov chains of order one or higher on the basis of the mixture transition distribution (MTD) model. We call it the MTD‐Probit. The proposed model presents two attractive features: it is completely free of constraints, thereby facilitating the estimation procedure, and it is more precise at estimating the transition probabilities of a multivariate or higher‐order Markov chain than the standard MTD model.