A two-stage realized volatility approach to estimation of diffusion processes with discrete data

Abstract

This paper motivates and introduces a two-stage method of estimating di¤u-sion processes based on discretely sampled observations. In the …rst stage we make use of the feasible central limit theory for realized volatility, as developed in Jacod (1994) and Barndor¤-Nielsen and Shephard (2002), to provide a regression model for estimating the parameters in the di¤usion function. In the second stage the in-…ll likelihood function is derived by means of the Girsanov theorem and then used to estimate the parameters in the drift function. Consistency and asymptotic distribution theory for these estimates are established in various contexts. The …nite sample performance of the proposed method is compared with that of the approximate maximum likelihood method of Aït-Sahalia (2002).JEL Classi…cation: C13, C22, E43, G13

Quasi‐maximum likelihood estimation of discretely observed diffusions

“…These simulation‐based methods can also achieve high numerical precision but the computation cost is high. More recently, Durham and Gallant (2002) use the Brownian bridge sampler to improve SMLE; Phillips and Yu (2009) propose a two‐stage realized volatility approach; Beskos et al (2009) suggest a simultaneous acceptance method (SAM) by estimating each conditional likelihood independently. However, SAM is applicable only to a restricted class of diffusions.…”

On Gaussian HJM framework for Eurodollar Futures

“…Now, it becomes convenient to write down the likelihood function in terms of one unknown parameter, namely δ and obtain estimates. This method is particularly useful when we deal with two‐factor models, for example if we want to estimate σ 1 and σ 2 , when the coefficient of volatility is taken to be Florens‐Zmirou 25, Yoshida 26 and Philip and Yu 13 use similar method for estimating parameters of diffusion equation from discrete data.…”

“…In this approach, we make use of both realized volatility and method of ML to compute estimates. The method is similar to the one proposed by Phillips and Yu 13, in which they estimate the drift and diffusion term of a diffusion process.…”

Quasi-maximum likelihood estimation of multivariate diffusions

“…Various methods have been developed to estimate the parameters in univariate parametric SDEs (see Aït-Sahalia, 2007 andHurn et al, 2007 for recent reviews). More recently, Beskos et al (2009) propose a Monte Carlo MLE for discretely observed diffusions and Phillips and Yu (2009) introduce a two-stage estimator.…”