advanced-algorithmic-trading

113
⎧
1 if k = 0
⎪⎨ q−k
∑
q∑
ACF: ρ k = β i β i+k / if k = 1, . . . , q
i=0
i=0
β 2 i
⎪⎩
0 if k > q
Where β 0 = 1.
We are now going to generate some simulated data and use it to create correlograms. This
will make the above formula for ρ k somewhat more concrete.
10.5.4 Simulations and Correlograms
MA(1)
Let us start with a MA(1) process. If we set β 1 = 0.6 we obtain the following model:
x t = w t + 0.6w t−1 (10.15)
As with the AR(p) model we can use R to simulate such a series and then plot the correlogram.
Since we have had a lot of practice in the previous sections of carrying out plots, I will write the
R code in full, rather than splitting it up:
> set.seed(1)
> x <- w <- rnorm(100)
> for (t in 2:100) x[t] <- w[t] + 0.6*w[t-1]
> layout(1:2)
> plot(x, type="l")
> acf(x)
The output is given in Figure 10.10.
As we saw above in the formula for ρ k , for k > q, all autocorrelations should be zero. Since
q = 1, we should see a significant peak at k = 1 and then insignificant peaks subsequent to that.
However, due to sampling bias we should expect to see 5% (marginally) significant peaks on a
sample autocorrelation plot.
This is precisely what the correlogram shows us in this case. We have a significant peak
at k = 1 and then insignificant peaks for k > 1, except at k = 4 where we have a marginally
significant peak.
This is a useful way of determining whether an MA(q) model is appropriate. By taking a
look at the correlogram of a particular series we can see how many sequential non-zero lags exist.
If q such lags exist then we can legitimately attempt to fit a MA(q) model to a particular series.
Since we have evidence from our simulated data of a MA(1) process we are now going to try
and fit a MA(1) model to our simulated data. Unfortunately there is no equivalent ma command
to the autoregressive model ar command in R.
Instead we must use the more general arima command and set the autoregressive and integrated
components to zero. We do this by creating a 3-vector and setting the first two components
(the autogressive and integrated parameters, respectively) to zero: