Consider two time series xt = wt − wt−1, yt = 1 2 (wt + wt−1), formed from the white noise series wt with variance σ2 w = 1.

(a) Are xt and yt jointly stationary? Recall the cross-covariance function must also be a function only of the lag h and cannot depend on time.

(b) Compute the spectra fy(ω) and fx(ω), and comment on the difference between the two results.

(c) Suppose sample spectral estimators ¯fy(.10) are computed for the series using L = 3. Find a and b such that P



a ≤ ¯fy(.10) ≤ b



= .90.

This expression gives two points that will contain 90% of the sample spectral values. Put 5% of the area in each tail.