# Simple mean reversion measure for binary time series

I am trying to define a simple measure for mean reversion in a stochastic sequence of ones and zeros, which I denote by $x_t$.

Yes, a unit root test on the cumulative sum could be a viable choice, but I started off by correlating the current value of the stochastic process with the deviation of the cumulative sum from its expected trend.

In more detail: let $p_1$ be the probability of $x_t=1$ (and hence $P(x_t=0)=1-p_1$), $n$ the length of the vector $x$.

I then define the above mentioned deviation as  y = cumsum(x) - (1:n)*p1  and my measure of mean reversion as the (negative) correlation between $x_t$ and $y_{t-1}$:  mr = -cor(x[2:n], y[1:(n-1)])  Or as a function:

MeanRevert = function(x,p1){
n=length(x)
#y is the detrended random walk.
y = cumsum(x) - (1:n)*p1
mr = -cor(x[2:n], y[1:(n-1)])
return(mr)
}


Now for a purely random walk, I expected an average zero value for that correlation but there appears to be a bias:

p1 = 0.12;N=100;
MRsample=vector()
for (i in 1:500){
x= sample(c(1,0), N, replace=TRUE, p = c(p1, 1-p1))
MRsample = c(MRsample, MeanRevert(x, p1=p1))
}
hist(MRsample, xlab ="mean reversion", main =paste("p=",p1, "N=",N))


What am I missing, what causes this bias ?

Thanks!

ML

Let me recap your setup: Suppose we have $X,X_{1},\ldots,X_{n}$ i.i.d. with $p=P(X=1)=1-P(X=0)$. So $E X=p$.

Then define $Y_{n}=-np+\sum_{i=1}^{n}X_{i}$. So $E Y_{n}=0$.

So your proposed measure is the correlation between $X_{n}$ and $Y_{n-1}$. Since $X_{n}$ and $Y_{n-1}$ are independent, this correlation is zero.

Yet your sample correlations do not have mean $0$.

I think the issue is that the pairs $(X_i,Y_{i-1})$, for i=2,..,n are not independent, because of the $Y_i$'s. This could be the source of the bias.

** My original response is below, but Markus points out that sample correlation is unbiased when the true correlation is zero:

Sample correlation is a biased estimate of correlation, so maybe this isn't surprising. Take a look at the paper "Bias in Estimation and Hypothesis Testing of Correlation" inPsicológica (2003), 24, 133-158, by Donald W. Zimmerman, Bruno D. Zumbo, and Richard H. Williams.(http://www.uv.es/revispsi/articulos1.03/9.ZUMBO.pdf)

How does it look for sample covariance, which is unbiased (and should be zero)?

• Thanks for the great insight! My only concern is that the paper shows that for zero population correlation, the sample correlation is also zero. The sample covariance is also biased, at least in the simulations. May 22, 2013 at 11:56
• Ok -- I revised my answer with a new proposal. Maybe this one is better ;) May 23, 2013 at 1:56