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Estimating Ornstein-Uhlenbeck Drift with an AR(1) Regression

Article Quant Q&A · Author: spacemonkey
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Summary

The document asks how to estimate the drift or mean-reversion parameter of an Ornstein-Uhlenbeck process when its long-run mean is known. It proposes expressing the process in discrete time as an AR(1) relation around that mean and fitting the autoregressive coefficient by least squares on historical observations.

The accepted response gives only a brief endorsement of using a rolling AR(1) with the specified mean. It does not provide a full derivation, clarify the mapping from the discrete coefficient to a continuous-time drift parameter, or discuss sampling intervals, residual assumptions, or estimator uncertainty. The material therefore introduces a practical estimation approach but offers little guidance on implementation or validation; those details would be needed before relying on estimates in a trading model.

Key ideas

  • A discretely observed Ornstein-Uhlenbeck process can be represented as an AR(1) around its long-run mean.
  • With the mean specified, least squares can estimate the autoregressive coefficient.
  • A rolling regression can update the coefficient as new observations arrive.
  • The brief answer does not explain how to convert the discrete coefficient into continuous-time drift or assess estimation error.

Tags

StatisticsMean reversion
Full text
# Estimating Ornstein-Uhlenbeck process drift


# Estimating Ornstein-Uhlenbeck process drift












What is the easiest way to obtain a drift parameter of O-U process given I have $\mu$?

Is it ok to linearize the O-U process like so:

$P_{t} = \mu + \phi(P_{t-1}-\mu)+\xi_t$

Form vectors from historic data:

$$ A = \begin{bmatrix} \ P_0-\mu \\ \ P_1-\mu \\ \ \dots \\ \ P_T-1-\mu \\ \end{bmatrix} $$

$$ b = \begin{bmatrix} \ \mu \\ \ \mu \\ \ \dots \\ \ \dots \\ \end{bmatrix} $$

$$ Y = \begin{bmatrix} \ P_1 \\ \ P_2 \\ \ \dots \\ \ P_T \\ \end{bmatrix} $$

And solve $\phi$ with OLS?

Cheers!

## Answer by Anthony Suherli (score 0, accepted)

https://quant.stackexchange.com/a/59389

yes, using a rolling AR-1 and μ

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.