advanced-algorithmic-trading

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9.4.2 Correlogram
The autocorrelation of a random walk (which is also time-dependent) can be derived as follows:
ρ k (t) =
Cov(x t , x t+k )
√
Var(xt )Var(x t+k ) = tσ 2
√
tσ2 (t + k)σ = 1
√ (9.7)
2 1 + k/t
Notice that if we are considering a long time series, with short term lags, then we get an
autocorrelation that is almost unity. That is, we have extremely high autocorrelation that does
not decrease very rapidly as the lag increases. We can simulate such a series using R.
Firstly, we set the seed so that you can replicate my results exactly. Then we create two
sequences of random draws (x and w), each of which has the same value (as defined by the seed).
We then loop through every element of x and assign it the value of the previous value of
x plus the current value of w. This gives us the random walk. We then plot the results using
type="l" to give us a line plot, rather than a plot of circular points, see Figure 9.2.
> set.seed(4)
> x <- w <- rnorm(1000)
> for (t in 2:1000) x[t] <- x[t-1] + w[t]
> plot(x, type="l")
Figure 9.2: Realisation of a Random Walk with 1000 timesteps.
It is simple enough to draw the correlogram too, see Figure 9.3.
> acf(x)