Exercise

Dealing with trend and heteroscedasticity

Here, we will coerce nonstationary data to stationarity by calculating the return or growth rate as follows.

Often time series are generated as $$X_t = (1 + p_t) X_{t-1}$$ meaning that the value of the time series observed at time \(t\) equals the value observed at time \(t-1\) and a small percent change \(p_t\) at time \(t\).

A simple deterministic example is putting money into a bank with a fixed interest \(p\). In this case, \(X_t\) is the value of the account at time period \(t\) with an initial deposit of \(X_0\).

Typically, \(p_t\) is referred to as the return or growth rate of a time series, and this process is often stable.

For reasons that are outside the scope of this course, it can be shown that the growth rate \(p_t\) can be approximated by $$Y_t = \log X_t - \log X_{t-1} \approx p_t.$$

In R, \(p_t\) is often calculated as diff(log(x)) and plotting it can be done in one line plot(diff(log(x))).

Instructions

100 XP
  • As before, the packages astsa and xts are preloaded.
  • Generate a multifigure plot to (1) plot the quarterly US GNP (gnp) data and notice it is not stationary, and (2) plot the approximate growth rate of the US GNP using diff() and log().
  • Use a multifigure plot to (1) plot the daily DJIA closings (djia$Close) and notice that it is not stationary. The data are an xts object. Then (2) plot the approximate DJIA returns using diff() and log(). How does this compare to the growth rate of the GNP?