Estimating Mean-Reversion Half-Life from a Discrete-Time Regression

# Why is my mean-reversion half-life completely wrong?


# Why is my mean-reversion half-life completely wrong?












I am using a couple of resources (here and here) to calculate the mean reversion half-life of a time series. This method of calculating it is also presented in Ernest Chan's Algorithmic Trading on page 47.

The method, as I understand it, is to use the formula: halflife = -log(2)/G

Where G is the coefficient of a linear regression where the dependent variable is the differenced series (Yt - Y(t-1)) and the dependent variable is the lagged series (Y(t-1)).

Here is my code:

```
a <- sin(0.2*seq(1, 100, 0.25))
y <- diff(a)
x <- a[1:(length(a)-1)]
lm <- lm(y~x)
halflife <- -log(2)/lm$coefficients[2]
```

I wanted to try it out with a clearly cyclical series, so I used a sine function. Here is a graph of the function:

The result I get, however, is: -1064 .

How is this possible? First of all, the value should be positive. This means that my coefficient is not negative, which suggests the series is not mean reverting. But it clearly is, right?

I'd greatly appreciate any assistance with understanding what's going on here.

## Answer by Jamie Ballingall (score 3)

https://quant.stackexchange.com/a/70417

I think the main problem is that, as @KurtG says, your example sine function exhibits strong momentum (albeit with reversals) and fitting an Ornstein-Uhlenbeck process to it results in strange parameters.

However, it might be nice to check that your code is correct. To do so, let's simulate a process that we know to be OU with an obvious half-life:

```
e <- rnorm(n, 0, 1)
a <- array(data = NA, dim = n)
a[1] = 0
for (i in 2:n) { a[i] <- 0.5 * a[i - 1] + 0.1 * e[i] }
```

Now `a` contains a large sample from an OU process with mean 0 and half-life of 1.

We run your fitting code:

```
y <- diff(a)
x <- a[1:(length(a)-1)]
lm <- lm(y~x)
halflife <- -log(2)/lm$coefficients[2]
```

and get a halflife of (with my random numbers) about 1.38. Which is close, but wrong and too far from the true value to be explained by just randomness because of the large sample size.

Let's revisit the formulas for halflife. First, and easiest to derive, if we have the OU process (with mean zero) $$ dX_t = -\theta X_t dt + \sigma dW_t $$ with $\theta > 0$ and $\sigma > 0$ then the half-life is $$ h = \frac{\log \left( 2 \right)}{\theta} $$ If we write this in discrete time we have: $$ X_t = \phi X_{t-1} + \epsilon $$ with $0 < \phi < 1$. Then the half-life is $$ h = -\frac{\log \left( 2 \right)}{\log \left( \phi \right)} $$ But your regression uses the differenced form of: $$ X_t - X_{t-1} = \left(\phi - 1 \right) X_{t-1} + \epsilon $$ We can define $\beta = \phi - 1$ (and hence $\phi = 1 + \beta$) to get: $$ X_t - X_{t-1} = \beta X_{t-1} + \epsilon $$ which is the regression you are running. Specifically, `lm$coefficients[2]` is giving you $\beta$. Rewriting a previous formula we get: $$ h = -\frac{\log \left( 2 \right)}{\log \left( 1 + \beta \right)} $$ If we try that:

```
halflife <- -log(2)/log(1 + lm$coefficients[2])
```

we get a number that is almost exactly 1, as expected. So that formula is probably the best one to use.

Where did your original formula come from? Well, the function $\beta \mapsto \log \left( 1 + \beta \right)$ is approximately linear near zero, so replacing $\log \left( 1 + \beta \right)$ with $\beta$ is a reasonable approximation. Nevertheless, using the more exact formula is probably worth it most of the time since it isn't expensive to evaluate.