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Figure 10.2: Realisation of AR(1) Model, with α 1 = −0.6 and Associated Correlogram.
> x.ar$ar + c(-1.96, 1.96)*sqrt(x.ar$asy.var)
[1] -0.7538593 -0.4408353
Once again we recover the correct order of the model, with a very good estimate ˆα 1 = −0.597
of α 1 = −0.6. We also see that the true parameter falls within the 95% confidence interval once
again.
AR(2)
Let us add some more complexity to our autoregressive processes by simulating a model of order
2. In particular, we will set α 1 = 0.666, but also set α 2 = −0.333. Here’s the full code to simulate
and plot the realisation, as well as the correlogram for such a series, given in Figure 10.3.
> set.seed(1)
> x <- w <- rnorm(100)
> for (t in 3:100) x[t] <- 0.666*x[t-1] - 0.333*x[t-2] + w[t]
> layout(1:2)
> plot(x, type="l")
> acf(x)
As before we can see that the correlogram differs significantly from that of white noise, as we
would expect. There are statistically significant peaks at k = 1, k = 3 and k = 4.
Once again we are going to use the ar command to fit an AR(p) model to our underlying
AR(2) realisation. The procedure is similar as for the AR(1) fit:
> set.seed(1)