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Statistical inference for multi-regime threshold Ornstein-Uhlenbeck processes

Yuecai Han, Dingwen Zhang

Abstract

In this paper, we investigate the parameter estimation for threshold Ornstein$\mathit{-}$Uhlenbeck processes. Least squares method is used to obtain continuous-type and discrete-type estimators for the drift parameters based on continuous and discrete observations, respectively. The strong consistency and asymptotic normality of the proposed least squares estimators are studied. We also propose a modified quadratic variation estimator based on the long-time observations for the diffusion parameters and prove its consistency. Our simulation results suggest that the performance of our proposed estimators for the drift parameters may show improvements compared to generalized moment estimators. Additionally, the proposed modified quadratic variation estimator exhibits potential advantages over the usual quadratic variation estimator with relatively small sample sizes. In particular, our method can be applied to the multi-regime cases ($m>2$), while the generalized moment method only deals with the two regime cases ($m=2$). The U.S. treasury rate data is used to illustrate the theoretical results.

Statistical inference for multi-regime threshold Ornstein-Uhlenbeck processes

Abstract

In this paper, we investigate the parameter estimation for threshold OrnsteinUhlenbeck processes. Least squares method is used to obtain continuous-type and discrete-type estimators for the drift parameters based on continuous and discrete observations, respectively. The strong consistency and asymptotic normality of the proposed least squares estimators are studied. We also propose a modified quadratic variation estimator based on the long-time observations for the diffusion parameters and prove its consistency. Our simulation results suggest that the performance of our proposed estimators for the drift parameters may show improvements compared to generalized moment estimators. Additionally, the proposed modified quadratic variation estimator exhibits potential advantages over the usual quadratic variation estimator with relatively small sample sizes. In particular, our method can be applied to the multi-regime cases (), while the generalized moment method only deals with the two regime cases (). The U.S. treasury rate data is used to illustrate the theoretical results.
Paper Structure (13 sections, 11 theorems, 138 equations, 2 figures, 5 tables)

This paper contains 13 sections, 11 theorems, 138 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The LSE based on continuous sampling
  4. Estimate $\alpha_{j}$ for known $\beta_{j}=0$ and $\Theta$ $(j=1, \cdots, m)$
  5. Estimate $\alpha_{j}$ and $\beta_{j}$ for known $\Theta$ $(j=1,\cdots,m)$
  6. The LSE based on discrete sampling
  7. Estimate $\alpha_{j}$ for known $\beta_{j}=0$ and $\Theta$ $(j=1, \cdots, m)$
  8. Estimate $\alpha_{j}$ and $\beta_{j}$ for known $\Theta$ $(j=1, \cdots, m)$
  9. Parameter estimation for $\sigma$
  10. Numerical Results and Applications
  11. Numerical results
  12. An application in U.S. treasury rate
  13. Disscussion

Key Result

Proposition 1

If the drift parameters $\alpha_{1}$, $\alpha_{m}$, $\beta_{1}$, and $\beta_{m}$ satisfy the unique invariant density of the process $\{X_{t}\}_{t\geq0}$ is given by where $k_{j}$ are uniquely determined by the following equations

Figures (2)

  • Figure 1: Normal QQ plot for $1000$ samples of the drift parameters with $n = 5000$ and $h=0.1$.
  • Figure 2: Daily U.S. treasury rate (solid black line) with a horizontal dashed blue line being the thresholds.

Theorems & Definitions (18)

  • Proposition 1
  • Proposition 2
  • Remark 1
  • Proposition 3
  • Remark 2
  • Theorem 1
  • Theorem 2
  • Remark 3
  • Remark 4
  • Theorem 3
  • ...and 8 more