Mean Reversion Can Coexist with a Changing Mean

Summary

The document considers whether a mean-reverting time series must be mean stationary. It distinguishes a constant expected value across observations from the possibility of reversion toward a long-run average. In the explanation, the expectation may vary over time, so mean stationarity does not hold, while the average of those time-varying expectations can still converge to a defined long-run level.

This distinction shows why the phrase mean reversion needs care: it depends on what mean is being used, and the question itself is described as vague unless that target is specified. The text offers a conceptual possibility rather than a formal classification of particular processes, empirical evidence, or a test for mean reversion. Researchers should therefore define the relevant long-run mean and separately assess whether the process has time-invariant expectations.

Key ideas

• Mean stationarity requires the expected value to remain constant across observations.
• A process can have time-varying expectations while their average converges to a long-run level.
• Mean reversion may refer to reversion toward that long-run level without mean stationarity.
• The intended meaning of mean reversion should be specified before drawing conclusions.

Tags

# Does mean reverting imply mean stationary?


# Does mean reverting imply mean stationary?












If I have a time series that exhibits mean reverting properties, does it necessarily mean that the time series is mean stationary?

## Answer by Ryogi (score 6)

https://quant.stackexchange.com/a/4545

As pointed out by Brian, the question is vague because generally mean reversion requires a well defined mean. Nevertheless, there are processes which are not mean stationary (mean is not homogenous across observations) for which a concept of mean exists. Let $\mu_t = E(x_t)$. In general you can have $\mu_t \neq \mu_s$ (i.e. violate mean stationarity) but have a well defined long run mean, i.e. the limit

$$\frac1n \sum_{t=1}^n \mu_t \to \bar \mu$$

exists. In such a situation, you can define a concept of mean reversion to the long run mean that applies to non mean stationary processes.

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