How the AR(1) Coefficient Controls Mean-Reversion Speed

Summary

The note explains why an AR(1) process with a coefficient closer to zero reverts to its long-run mean faster than one with a coefficient near one, provided the coefficient’s absolute value is below one. In the zero-mean setup, the conditional expected next value is the current value multiplied by the coefficient. A small coefficient therefore pulls that expectation closer to zero in one step, while a coefficient near one leaves it close to the current observation.

This conditional-expectation argument resolves the apparent contradiction in the question’s simulations: larger positive persistence means slower, not faster, mean reversion. The explanation is qualitative and assumes innovations have zero conditional mean. For negative coefficients, the sign also produces oscillation, so speed is governed by the coefficient’s magnitude rather than simply its numerical ordering. The note gives no estimated half-life, sample analysis, or evidence beyond the model relationship.

Key ideas

• In a zero-mean AR(1), the conditional expected next value is the coefficient times the current value.
• A coefficient near zero makes the expected next value close to the mean.
• A positive coefficient near one indicates strong persistence and slower mean reversion.
• For negative coefficients, the magnitude governs persistence while the sign can cause oscillation.

Tags

# Answer by Richard Hardy (score 2, accepted)


# Why does AR(1) model with a small coefficient exhibit faster mean-reversion than one with a greater coefficient (when |$\beta$|<1)?












Suppose we have two mean-reverting AR(1) models, given by

$$X_{t}=\beta X_{t-1}+\epsilon_t,$$

where $|\beta|<1$.

How fast series reverts to its mean is determined by the coefficient $\beta$. As far as I know, greater $\beta$ implies faster mean reversion. However, I simulated two time series using AR(1) model given above with $\beta=0.1$ and $\beta = 0.9$. Presumably, the series with the coefficient of 0.9 should revert to the mean faster than the one with 0.1 coefficient. The plots below, however, suggest the opposite -- AR(1) with 0.1 coefficient is a faster mean-reverting process, than AR(1) with 0.9 coefficient. Why?

## Answer by Richard Hardy (score 2, accepted)

https://quant.stackexchange.com/a/78501

It looks like you got the intuition backwards. The conditional expectation of $Y_t$ given $Y_{t-1}$ is $\beta Y_{t-1}$. That is, $\mathbb{E}(Y_t|Y_{t-1})=\beta Y_{t-1}$. (We used the fact that $\mathbb{E}(\epsilon_t|Y_{t-1})=0$.) If $\beta$ is close to 1, $\mathbb{E}(Y_t|Y_{t-1})=\beta Y_{t-1}$ is close to $1\cdot Y_{t-1}=Y_{t-1}$, thus there is little mean reversion. The series stays close to where it was, on average. If $\beta$ is close to 0, $\mathbb{E}(Y_t|Y_{t-1})=\beta Y_{t-1}$ is close to $0\cdot Y_{t-1}=0$, thus there is quick mean reversion. There series goes to zero (which is its long-term mean), on average.

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