# Regime Change Tests for Mean Reversion

An institutional algo trader recently posed the following question:

Q: So I have data that is mean reverting, but then the mean changes…What are the best tests for me to see when that is happening?

As always, there is no single “correct answer”. Answering this question depends upon broadly generalizing the previous post on Quantile Stability.

Applicable detection methods depend upon the interpretation of “mean changes” in this context. Factors to potentially consider in this context include:

- Signal origin (type and construction)
- Model linearity
- Discontinuity form (linear or non-linear)
- Desired detection frequency

Discontinuity owing to parametric stability benefit from techniques analyzing estimation and decomposition. Low dimensional, linear time-series discontinuities can be detected using classic econometric structural change methods (Chow, Quandt, CUSUM, et al). Quantile regression may be effective for evaluating non-time continuity / stability of parametric linear models. Higher dimension, non-linear time-series discontinuity may be detected using wavelet decomposition. All of these methods can be made pseudo-adaptive, at nearly any frequency, by using the corresponding iterative algorithmic form.

Discontinuity owing to variance or covariance stability benefit from techniques analyzing covariance structure. A very wide variety of techniques are available in this domain, most commonly including: variance-derived tests (e.g. variance ratio to ACF), orthogonal / orthonormal decomposition techniques (e.g. PCA or ICA), and [G]ARCH family. Stochastic volatility models are common in derivative disturbance models.

Depending on methodology, readers may wish to consider modeling the discontinuities or disturbances. Standard disturbance / residual models include such techniques as state space models (generalization of linear filters, such as Kalman).

Subsequent posts will dive into practical use of each of these techniques within the context of quantitative trading.

Nice summary of possible techniques. As u said you will dedicate future posts to the mentioned categories (arch, acf) etc.

I have a question regarding quantile regression. One of the assumptions of quantile regression is that data are iid and stationary. And u r applying it to the prices, not returns. I guess that is because how cointegration is done, Using OLS on prices, not on returns, and then looking at residuals whether they are mean reverting. So instead of OLS u r just applying quantile regression.

Continuing in the same line of argumentation, when u want to estimate beta taking into account (g)arch effects, beta = cov(x,y)/var(y), would u apply garch on pure price series to estimate covariance and variance? Or how would that be done? Because there is apparently no relation between beta estimated on returns and on price series.

I would appreciate your short answer, although u are probably going to dedicate full article to that, Jozef

@jrudy: thanks for your interest; quantile is indeed an ols substitute in this context, focusing on exploring parameter robustness of two ols assumptions: distribution (Guassian vs Laplacian) and moment type (mean vs median / quantile); re iid and stationarity, see the Three Horsemen post for suitable skepticism on those assumptions. Your beta question warrants an entire post, given that is a non-trivial question.