# Mean Reversion Tests on DARWIN $DWC In a previous post – Quantitative Modeling for Algorithmic Traders – we discussed the importance of Expectation, Variance, Standard Deviation, Covariance and Correlation. In this post we’ll discuss how those concepts can be applied to DARWIN assets. As a practical example, we will employ a series of statistical tests to assess if DARWIN$DWC is a Mean Reverting time series or otherwise.

These will include:

1) Hurst Exponent
3) Half-life of Mean Reversion

In case you missed it, the mean reverting nature of DARWIN $DWC was discussed in our most recent post here. Tests will be conducted on 1-Minute returns from$DWC, results and interpretation being published along the way. As always, please share your comments, feedback and suggestions in the comments at the end.
Note: Different statistical tests don’t always lead to similar outcomes, therefore it’s considered good practice to use at least two when evaluating mean reversion or any other statistical properties.
Before proceeding further, it’s important that we understand what Autocorrelation and Stationarity are.

## Autocorrelation:

Also referred to as Serial Correlation.
It is a measure of the similarity or relationship between a time series and a delayed or “lagged” version of the same time series, over successive periods in time.

## Stationarity:

A time series is considered stationary if its core statistical attributes remain constant over time.
These include mean, variance, standard deviation, autocorrelation, etc.
Stationary series demonstrate high predictability.

If a time series (e.g. DARWIN) can be mathematically transformed to approximately stationary, future Quotes of the time series (or trade entry direction / entries) can be reverse engineered from future points in its forecasted stationary series.
More on this in future blog posts.

## Prior Assumptions:

Prior to conducting these tests on $DWC data, we are expecting to see a reasonable degree of mean reversion for the following reasons: 1. There is visual confirmation (see below) that mean reverting tendency may exist. 2. As$DWC behaves in relation to real time trader sentiment, it is reasonable to assume that it could exhibit cyclical behaviour.

## Mean Reversion Test #1: Hurst Exponent

Mean Reversion in a time series can be assessed in terms of its rate of diffusion from inception.

For a time series X to be considered mean reverting:

##### Rate of Diffusion (X) < Rate of Diffusion of a Geometric Random Walk (GBM)

This rate of diffusion can be measured as the variance of the logarithm of the time series, at a random time interval T:

##### $Var(T) = \left \langle \left | log(t + T) – log(t) \right |^{2} \right \rangle$

If a time series is a GBM, then Var(T) ~ T, as T gets larger:

##### $\left \langle \left | log(t + T) – log(t) \right |^{2} \right \rangle$ ~ T

If a time series is either trending or mean reverting, then:

##### $\left \langle \left | log(t + T) – log(t) \right |^{2} \right \rangle$ ~ $T^{2H}$

.. where H is the Hurst Exponent, a measure of the extent to which the time series trends or mean reverts.

Hurst Exponent Interpretation:
If H > 0.5, the time series is TRENDING
If H < 0.5, the time series is Mean Reverting
If H = 0.5, the time series is a Geometric Random Walk

The DWC’s Hurst Exponent can be easily calculated in R, using the “pracma” library.
Note: For all code examples in this blog post, we have pre-loaded M1 data as “DWC.M1” to save time.

library(pracma)# Print M1 data Hurst Exponent > hurstexp(log(DWC.M1$quote))Simple R/S Hurst estimation: 0.8962816 Corrected R over S Hurst exponent: 0.9945418 Empirical Hurst exponent: 1.001317 Corrected empirical Hurst exponent: 0.9938308 Theoretical Hurst exponent: 0.520278 This first test shows that though this sample of DWC data is not demonstrating mean reverting behaviour (Theoretical Hurst Exponent > 0.5), it is not trending significantly either -> i.e. it is almost behaving like a GBM as per this test’s results (H = 0.520278), reducing the probability of DWC being a non-stationary random walk process. ## Mean Reversion Test #2: Augmented Dickey-Fuller Test If the$DWC time series is not a random walk (non-stationary series), then any Quote in the series will have a proportional relationship with the Quote immediately before it.

### Interpretation of ADF Test Results

Referring back to the autoregressive model earlier:

###### $\Delta x_{t} = \alpha + \beta t + \gamma x_{t-1} + \delta _{1}\Delta x_{t-1} + … + \delta _{p-1}\Delta x_{t-p+1} + \epsilon _{t}$

z.lag.1 = The value of the test-statistic γ (gamma) in the above equation.
tau2 = Critical values corresponding to the null hypothesis (γ = 0)
In order to reject the null hypothesis (γ = 0 – i.e. to reject that DWC is a non-stationary random walk), the value of the test statistic must be smaller than the critical values in tau2 (1%, 5% and 10% confidence intervals).
As z.lag.1 is -3.0566 (smaller than the critical values for the 5% and 10% confidence intervals), the null hypothesis can be rejected at the 90% and 95% confidence intervals, i.e. the probability of DWC being stationary (or not a random walk) is very high.

##### The tests above were also conducted on 30-minute, 1-hour, 2-hour, 4-hour and Daily precision $DWC data. 1. Daily precision lead to the null hypothesis for the presence of a unit root being rejected at the 90% confidence interval. This test will be repeated periodically as more data is accrued over time. 2. 30-minute, 1-hour, 2-hour and 4-hour tests all lead to the null hypothesis for the presence of a unit root being rejected at the 95% confidence interval. ## Mean Reversion Test #3: Half-life of Mean Reversion An alternative to the autoregressive linear model described above, is to consider how long any particular time series takes “to mean revert”. By definition, a change in the next periodic value of a mean-reverting time series is proportional to the difference between the historical mean of the series and the current value. Such time series are referred to as Ornstein-Uhlenbeck processes. The differential of the earlier model leads us to the expected value of x(t): ##### $E(x_{t}) = x_{0}e^{\gamma t} – \frac{\mu }{1 – e^{\gamma t}}$ If DWC is a mean reverting series, and has a negative $\gamma$, then the equation above tells us that DWC prices decay exponentially, with a half-life of $\frac {-log(2)}{\gamma}$. ### This means we now have two tasks ahead of us: 1. Find $\gamma$ and check if it is negative. 2. Calculate the half-life and assess whether it is a practical length of time for traders to consider a mean reverting strategy on DWC. Once again, we can easily conduct both steps in R. ### Step 1: Calculate $\gamma$ and check sign. > M1.data <- as.ts(DWC.M1$quote) > M1.data.lag <- lag(M1.data, -1) > M1.data.delta <- diff(M1.data)> M1.data.frame <- cbind(M1.data, M1.data.lag, M1.data.delta) > M1.data.frame <- M1.data.frame[-1,]
> M1.regression <- lm(M1.data.delta ~ M1.data.lag, data=as.data.frame(M1.data.frame))
> gamma <- summary(M1.regression)$coefficients[2] > print(gamma) [1] -0.0003588994 $\gamma$ is negative (-0.0003588994), so this$DWC 1-minute data sample can be considered mean reverting.

### Step 2: Calculate half-life and assess practicality of mean reversion strategy.

> M1.data.half.life <- -log(2) / gamma> print(paste("Half-life: ", M1.data.half.life, " minutes, or ", M1.data.half.life/60, " Hours", sep="")) [1] "Half-life: 1931.31306610404 minutes, or 32.1885511017341 Hours"
The half-life calculated for this $DWC 1-minute data sample is 32 hours. ##### Another important feature of the calculated half-life, is that it can be used as the period of a moving average employed in a mean reverting trading strategy[1]. If we plot a Simple Moving Average of period 1931 (in minutes, not hours), we get: ## Summary: 1. We conducted three statistical tests to ascertain the degree of mean reversion in$DWC 1-minute data, namely Hurst Exponent, Augmented Dickey-Fuller (ADF) and Half-Life of Mean Reversion.
2. Hurst Exponent did not indicate mean reverting behaviour in the $DWC, but a rather close estimate for possible GBM behaviour. 3. The Augmented Dickey-Fuller test results indicated stationary behaviour at the 95% confidence interval. 4. The Half-life of Mean Reversion test indicated$DWC possesses mean reverting properties.
5. We used the half-life calculated above as the period for a moving average, which when plotted on the chart revealed mean reverting Quote behaviour.

##### References:

[1] Chan, Ernest, 2013. Algorithmic Trading: Winning Strategies and Their Rationale, John Wiley and Sons.

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• ###### Martyn Tinsley
Posted July 22, 2017 at 8:42 am

A very interesting and well written article. Clearly a very complex subject. Luckily due to my own research around cointegration between pairs of instruments and the mean reversion that exists there, I managed to follow most of the message.
However, a very important (and more practical) message that I (and other traders) need to hear if they are to use DWC in real life are things like:
1.How do we know the investment ratio of DWC we should use compared to the other traditional DARWINs in our portfolios. Is is 50% DWC, 50% other? Or is it more complex? If so how do we calculate?
2. Is the strategy around DWC buy and hold? I.e. It always forms a component of an investors portfolio. Or is it a case of buying DWC following the dips in anticipation of traditional DARWINs struggling and then selling when DWC reaches a high?
This current article in invaluable in terms of the background rationale and proof. However I expect many would have struggled with the maths, and therefore would not have benefited from the message. What I think the community would now really benefit from is a more grounded and practical article on its everyday use 🙂 Something that as a minimum should clearly cover my 2 points above, but I’m sure you can also provide many more practical insights and use cases.
Keep up the good work
Martyn

• Post Author
###### The Market Owl
Posted September 13, 2017 at 2:06 pm

Many thanks for the kind comments Martyn – more applied content on its way!

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