6.10 Estimation of Continuous-Time Models
Next, we consider the problem of estimating directly the diffusion equation (i.e., Ito process) from discretely sampled data. Here the drift and volatility functions μ(xt, t) and σ(xt, t) are time varying and may not follow a specific parametric form. This is a topic of considerable interest in recent years. Details of the available methods are beyond the scope of this chapter. Hence, we only outline the approaches proposed in the literature. Interested readers can consult the corresponding references and Lo (1988).
There are several approaches available for estimating a diffusion equation. The first approach is the quasi-maximum-likelihood approach, which makes use of the fact that for a small time interval dwt is normally distributed; see Kessler (1997) and the references therein. The second approach uses methods of moments; see Conley, Hansen, Luttmer, and Scheinkman (1997) and the references therein. The third approach uses nonparametric methods; see Ait-Sahalia (1996, 2002). The fourth approach uses semiparametric and reprojection methods; see Gallant and Long (1997) and Gallant and Tauchen (1997). Recently, many researchers have applied Markov chain Monte Carlo methods to estimate the diffusion equation; see Eraker (2001) and Elerian, Chib, and Shephard (2001).