Consider a moving average of white noise, \(x_{t}=\frac{1}{k} \sum_{j=1}^{k} w_{t-j}\), where the \(w_{i}\) have independent \(\mathrm{t}\) distributions with \(4 \mathrm{df}\).

Generate random samples of length 1,000 of this process with \(k=1\) (which is just white noise) and with \(k=2, k=4\), and \(k=8\).

In each case compute and plot the ACF. How does the ACF relate to the window width, \(k\) ?