Bias Reduction in Nonparametric Diffusion Coefficient Estimation

Abstract

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

Nonparametric estimation of scalar diffusion models of interest rates using asymmetric kernels

“…(2002), Bandi and Phillips (2003), Fan and Zhang (2003), Nicolau (2003), and Arapis and Gao (2006). An interesting situation that often arises in economics and finance is when the conditioning variables are nonnegative (i.e., they have a natural boundary at the origin) such as nominal interest rates, volatility, etc.…”

An explicit form of all‐pole filter function with decreasing envelope of the summed sensitivity function

“…Nicolau (2003) proposes an estimator for the diffusion coefficient b 2 ( z ) in which he subtracts the influence on bias of the drift coefficient (or its nonparametric estimator). He shows that this estimator, compared with the Florens‐Zmirou (1993) and the Jiang and Knight (1997) estimator for the diffusion coefficient, when the step of discretization is fixed, results in a bias adjustment that partially compensates for the distortion and still maintains all the nice asymptotic properties of the Florens‐Zmirou and Jiang and Knight estimators.…”

“…Bandi and Philips (2003) have developed an asymptotic theory for nonparametric estimates of the drift and diffusion coefficients. Nicolau (2003) quantified the asymptotic bias of the Florens‐Zmirou (1993) and the Jiang and Knight (1997) estimator for the diffusion coefficient when the discretization step is fixed, and proposed a diffusion estimator in which bias is corrected by subtracting the influence of the drift coefficient or its nonparametric estimator, showing that it enables a reduction of the asymptotic bias.…”

“…However, as Jiang and Knight (1997) says, the results can be derived for nonequidistant data, the only complication being notational. We present in Section 3 the estimators we have used, which are adaptations to nonequidistant data of the drift coefficient estimator proposed by Bandi and Philips (2003) and of the diffusion coefficient estimator suggested by Nicolau (2003). However, these estimators were developed for data from a single trajectory with abundant data points, typical of financial data.…”

Local Linear Estimation of Recurrent Jump—Diffusion Models

“…Second, functions to be estimated are directly estimated by the method of the kernel regression which locally approximates a target function by constant or linear functions (c.f., Fan and Gijbels, 1996). The local constant kernel regression, the so-called Nadaraya-Watson kernel estimation method, was originally applied to estimation of diffusion coefficients by Florens-Zmirou (1993), and since then its related theoretical and empirical studies have been carried out by many authors (i.e., Arapis and Gao, 2006;Bandi and Phillps, 2003;Jiang and Knight, 1997;Nicolau, 2003;Stanton, 1997). dX t = X t dt + X t dW t (1.1)…”