Consider the bivariate time series records containing monthly U.S. production as measured by the Federal Reserve Board Production Index and monthly unemployment as given in Figure 3.21.

(a) Compute the spectrum and the log spectrum for each series, and identify statistically significant peaks. Explain what might be generating the peaks.

Compute the coherence, and explain what is meant when a high coherence is observed at a particular frequency.

(b) What would be the effect of applying the filter ut = xt − xt−1 followed by vt = ut − ut−12 to the series given above? Plot the predicted frequency responses of the simple difference filter and of the seasonal difference of the first difference.

(c) Apply the filters successively to one of the two series and plot the output. Examine the output after taking a first difference and comment on whether stationarity is a reasonable assumption. Why or why not? Plot after taking the seasonal difference of the first difference. What can be noticed about the output that is consistent with what you have predicted from the frequency response? Verify by computing the spectrum of the output after filtering.