Assume that Y = X???? + ????, with ???? ∼ N(????, ????2I), and X full rank, of size T × k. Let R̆

1,…, R̆

m be the elements of the SACF computed from the regression residuals based on X, but using the recursive residuals, as discussed in Section 1.5. That is,

where As is given in (8.9), but is of size (T − k)×(T − k), and ̂???? = CY = C???? are the T − k recursive residuals. Show analytically that the density of R̆
s is symmetric about zero. Verify this computationally by comparing FR̆
s (z) with 1 − FR̆
s (−z) over a grid of z values. Similarly, show computationally that the Rs, i.e., the SACF elements based on the usual regression residuals, are not symmetric about their mean (which is nonzero).

= E 'A d'