advanced-algorithmic-trading

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x t = pz t + w x,t (12.1)
y t = qz t + w y,t (12.2)
If we then take a linear combination ax t + by t :
ax t + by t = a(pz t + w x,t ) + b(qz t + w y,t ) (12.3)
= (ap + bq)z t + aw x,t + bw y,t (12.4)
We see that we only achieve a stationary series (that is a combination of white noise terms)
if ap + bq = 0. We can put some numbers to this to make it more concrete. Suppose p = 0.3 and
q = 0.6. After some simple algebra we see that if a = 2 and b = −1 we have that ap + bq = 0,
leading to a stationary series combination. Hence x t and y t are cointegrated when a = 2 and
b = −1.
We can simulate this in R in order to visualise the stationary combination. Firstly, we wish
to create and plot the underlying random walk series, z t :
> set.seed(123)
> z <- rep(0, 1000)
> for (i in 2:1000) z[i] <- z[i-1] + rnorm(1)
> plot(z, type="l")
Figure 12.1: Realisation of a random walk, z t
If we plot both the correlogram of the series and its differences we can see little evidence of