Figure 2.14 plots the monthly changes in the US S&P stock market index over 1965 to 2016. Consider an AR(1) model as a very simple exploratory model for understanding local dependencies but not for forecasting more than a month or two ahead. We know there is a great deal of variation across the years in the market economy and that we might expect change that an AR(1) model does not capture. To ex plore this, we can simply t the AR(1) model to shorter sections of the data and examine the resulting inferences on parameters to see if they seem to vary across time. Do this as follows. The full series has T = 621 months of data; look at many separate time series by selecting a month mandtaking some number k months either side; for example, you might take k = 84 and for any month m analyze the data over the windowed period from m k to m+k inclusive. Repeat this for each month m running from m = k+1 to m = T k. These repeated analyses will de ne a trajectory of AR(1) analyses over time, one for each sub-series.
For each sub-series, subtract the sub-series mean (to roughly center the sub-series series about zero) and then compute the summaries of the ref erence posterior for an AR(1) model to just those 2k+1 time points just treating each selected sub-series separately. Using the theoretical poste rior T distribution for the parameter, compute and compare (graphi cally) the exact posterior 90% credible intervals.

(a) Comment on what you see in the plot and comparison, and what you might conclude in terms of changes over time.

(b) Do you believe that short-term changes in S&P have shown real changes in month-month dependencies since 1965?

(c) How would you suggest also addressing the question of whether or not the underlying mean of the series is stable over time?

(d) What about the innovations variance?

(e) What does this suggest for more general models that might do a better job of imitating this data?