# Ornstein versus AR(1) for modeling stationary data

I've come across several posts regarding parameter estimation for O-U models given some stationary data (say, some sort of mean reverting spread), but I can't seem to find an answer as to why modeling the data as a continuous O-U bears a benefit over modeling it as an AR(1) process. Are the parameters more robust/precise when treating the process as O-U versus AR(1)? I suppose O-U may give better estimates at higher frequencies. Any insight would be great.

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## 1 Answer

O-U is continuous time mean reverting process, hence used to model stationary series. It has closed form analytic solution. This allows insight into stationary processes and act like asymptotic limiting case for calculating coefficients that matter.

EDIT: You can see AR(1) below $$x_{k+1} = c + a x_k + b\varepsilon_k$$ and by substituting c=θμΔt, a=−θΔt and $b = \sigma\sqrt{\Delta t} \space$ you will get OU $$x_{k+1} = \theta(\mu - x_k)\Delta t + \sigma \varepsilon_k\sqrt{\Delta t}$$

This is simple discretization to show they are same and how the parameters can be translated. O-U can be used to detect the steady state parameters. As you see paramaters are interchangeable, frequency used in AR and O-U should be same, then it will be frequency agnostic. I am doing some work on pair trading using O-U I will re-edit at some later time.

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You're not really addressing the @user7889's point with respect to OU versus AR(1). –  John Apr 24 '14 at 16:13
My comment assumed by AR(1) the OP is modeling a stationary process in the context of pair trading suitable for O-U. AR can be non-stationary, in which case that wont be suitable for O-U model. The focus of the question is why is O-U process used and what benefits it provides over discrete process like AR. –  user12348 Apr 24 '14 at 16:50
Thanks for the helpful comments. To clarify, you are correct- I'm assuming here the AR(1) process has a coefficient <1; therefore both the AR(1) model and O-U model would both be suitable to model the stationary data. So I understand that the continuous time model has a nice analytic form, but from a practical point of view, what advantage (or disadvantage) does the O-U model have over AR(1) for modeling a stationary (discrete) process? –  user7889 Apr 24 '14 at 22:08
@John does it address your point? –  user12348 Apr 29 '14 at 23:11
Yeah, I think that's an improvement. –  John Apr 29 '14 at 23:26