How the Augmented Dickey–Fuller Model Tests Mean Reversion
Summary
The document identifies the displayed linear model of price changes as the Augmented Dickey–Fuller (ADF) regression. It explains the simpler Dickey–Fuller idea: estimate how the current change in a series relates to its previous level. A negative coefficient on that level is consistent with mean reversion, because high values tend to be followed by declines and low values by increases. The estimated coefficients also imply a level at which the expected change is zero.
The augmented form adds a time trend and lagged changes to account for a trend in the series and more complex short-run dynamics. The answer offers an intuitive overview and points toward studying the simpler test first. It does not derive the regression or explain the ADF test’s statistical properties, which it describes as complicated. The example uses interest rates, so applying the same intuition to another price or financial series requires care about the series’ properties and the test’s assumptions.
Key ideas
- The displayed regression is an Augmented Dickey–Fuller model.
- A negative coefficient on the lagged level is consistent with mean reversion.
- The intercept and level coefficient imply a level where the expected change is zero.
- The augmented model includes a time trend and lagged changes to capture additional dynamics.
- The explanation is intuitive and does not cover the test’s statistical properties or derivation.
Tags
Full text
# linear model of price changes
# linear model of price changes
I came across the below equation for linear model of price changes in E.Chan book Algorithmic Trading which is the base for a strategy.
```
Δy(t) = λy(t − 1) + μ + βt + α1Δy(t − 1) + … + αkΔy(t − k) + ∋t (2.1)
```
What is the logic behind the model? Is it a commonly used model ? Does it have its name or author? How it is derived? I am trying to find some source where I could learn more about it, please for some hints.
## Answer by Alex C (score 1, accepted)
https://quant.stackexchange.com/a/46039
The above is just the standard equation for the ADF test.
http://en.wikipedia.org/wiki/Augmented_Dickey%E2%80%93Fuller_test
It looks complicated. You might start by reading about the original, simpler, Dickey Fuller Test, before it became 'Augmented'. Suppose interest rates $y(t)$ are mean reverting: when they are high they tend to come down and when low they increase. You could show this by estimating $\lambda$ and $\mu$ in the equation $\Delta y(t)=\lambda y(t−1)+\mu$. You would expect $\lambda$ to come out negative. And the neutral point where interest rates neither increase nor decrease would be $\tilde{y}$ such that $\lambda \tilde{y}+\mu=0$. Above $\tilde{y}$ changes tend to be negative and below, positive. That's the basic idea of DF test.
The Augmented DF adds to this a trend term βt in case there is a long term trend in the data (certainly the case for interest rates) and the other autoregressive terms in case there are more complicated dynamics in $\Delta y(t)$.
That's just an overview, the statistical properties of the ADF are complicated.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.