Abstract
Consider an Ornstein-Uhlenbeck process driven by a fractional Brownian motion. It is an interesting problem to find criteria for whether the process is stable or has a unit root, given a finite sample of observations. Recently, various asymptotic distributions for estimators of the drift parameter have been developed. We illustrate through computer simulations and through a Stein's bound that these asymptotic distributions are inadequate approximations of the finite-sample distribution for moderate values of the drift and the sample size. We propose a new model to obtain asymptotic distributions near zero and compute the limiting distribution. We show applications to regression analysis and obtain hypothesis tests and their asymptotic power.
Highlights
Stability properties of the ordinary differential equation x t θx t depend on the sign of the parameter θ: the equation is asymptotically stable if θ < 0, neutrally stable if θ 0, and instable if θ > 0
Let θ θ T depend on the observation time T and limT → ∞θ T 0
Strong Consistency We show that the estimator θT,H is a strongly consistent estimator of θ, that is, for all θ ∈ R, Tli→m∞θT,H θ
Summary
Stability properties of the ordinary differential equation x t θx t depend on the sign of the parameter θ: the equation is asymptotically stable if θ < 0, neutrally stable if θ 0, and instable if θ > 0. These stability results carry over to the stochastic process t. The objective of this work is the analysis and implementation of the zero root test for 1.1 when Z W H , the fractional Brownian motion with the Hurst parameter H, and 1/2 ≤ H < 1. Recall that the fractional Brownian motion WHWHt , t ≥ 0, is a Gaussian process with W H 0 0, mean zero, and covariance
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