Distribution of the mean reversion estimator in the Ornstein–Uhlenbeck process

ABSTRACT

We derive the exact distribution of the maximum likelihood estimator of the mean reversion parameter (κ) in the Ornstein–Uhlenbeck process using numerical integration through analytical evaluation of a joint characteristic function. Different scenarios are considered: known or unknown drift term, fixed or random start-up value, and zero or positive κ. Monte Carlo results demonstrate the remarkably reliable performance of our exact approach across all the scenarios. In comparison, misleading results may arise under the asymptotic distributions, including the advocated infill asymptotic distribution, which performs poorly in the tails when there is no intercept in the regression and the starting value of the process is nonzero.

KEYWORDS:

• Distribution
• mean reversion estimator
• Ornstein–Uhlenbeck process

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• C22
• C46
• C58

Acknowledgments

We are grateful to Peter Phillips and three anonymous referees for their constructive feedback. We also thank Raymond Kan for his discussion of the numerical algorithm presented in this paper.

Notes

¹Since we are interested in studying the finite-sample properties of $\widehat{\kappa },$ the initial condition x₀ matters and we include it in the estimation procedure. This stands in contrast to the convention of Hurwicz (1950). For the case of known μ, if μ≠0, one can simply define $y_i=x_i-\mu$ and work with y_i.

²For 0<ϕ≤1, $Pr(\widehat{\varphi }\le 0)\to 0$ asymptotically, since $\widehat{\varphi }$ is consistent. In other words, $\widehat{\kappa }$ as in (2.5) is always well defined asymptotically, and so is its asymptotic distribution. We may define an inequality constrained estimator of κ following the spirit of Judge and Takayama (1966), but this approach is not pursued in this paper.

³Note that (2.6) holds regardless of the distributional assumption.

⁴If we discard x₀ in formulating $\widehat{\varphi }$, then the Imhof (1961) technique is still applicable, as we can define $\widehat{\varphi }$ in terms of quadratic forms in the random vector ${(x_1,x_2,\dots ,x_n)}^{\prime }.$

⁵As emphasized in Zhou and Yu (2015), under the infill asymptotic regime, the same notation is used for any value of κ, demonstrating that the infill distribution is continuous in κ. When κ = 0, the terms $A_1\left({\gamma }_0,c\right)$, $B_1\left({\gamma }_0,c\right)$, $A_2\left({\gamma }_0,c\right),$ and $B_2\left({\gamma }_0,c\right)$ should be interpreted as their limiting values as κ→0.

⁶http://www.krannert.purdue.edu/faculty/ybao/OU.zip. The file names are self-suggestive.

⁷In simulating the asymptotic (nonnormal) results, we used a sample size of 5,000 and 100,000 replications to approximate the integrals involving the Brownian motion by the discrete Riemann sums.

⁸More specifically, let $g\left(t\right)=\sqrt{a\left(t\right)+ib\left(t\right)}$ denote the integrand function in question with t∈[l,u]. quadgk requires the integrand function to accept a vector $\left(t_1,t_2,\dots ,t_n\right)$ and returns a vector of output. Let ${𝜃}_i=arg\left(a\left(t_i\right)+ib\left(t_i\right)\right)\in [-\pi ,\pi ]$ and denote $a_i=a\left(t_i\right)$, $b_i=b\left(t_i\right),$ and $g_i=g\left(t_i\right).$ The algorithm is as follows: (i) Start with t₁ and set $g_1=sqrt\left(a_1+i{b}_1\right).$ Set k = 0. (ii) Beginning with t₂, if a_i<0, a_i−1<0, and $b_i{b}_{i-1}<=0,$ set k = k+1; otherwise, k is unchanged. Set $g_i=\sqrt{a_i^2+b_i^2}\left(cos\left({𝜃}_i^{\ast }∕2\right)+isin\left({𝜃}_i^{\ast }∕2\right)\right),$ where ${𝜃}_i^{\ast }={𝜃}_i+2k\pi .$

⁹Phillips (1980) provided analytical characteristic functions involving the more general cases of linear and quadratic forms.

¹⁰$\widehat{\varphi }$ is independent of μ because $A_{n+1}^M{\mathbf{1}}_{n+1}={\mathbf{0}}_{n+1}$, $A_{n+1}^{M\prime }{\mathbf{1}}_{n+1}={\mathbf{0}}_{n+1}$, and $B_{n+1}^M{\mathbf{1}}_{n+1}={\mathbf{0}}_{n+1}$.

¹¹This is different from the case when no intercept is included in the AR(1) model.