8.3. The time-averaging problem. (Working, 1960.) Actual data give not consumption at a point in time, but average consumption over an extended period, such as a quarter. This problem asks you to examine the effects of this fact.

Suppose that consumption follows a random walk: Ct = Ct −1 + et, where e is white noise. Suppose, however, that the data provide average consumption over two-period intervals; that is, one observes (Ct + Ct +1)/2, (Ct +2 + Ct +3)/2, and so on.

(a) Find an expression for the change in measured consumption from one two-period interval to the next in terms of the e’s.

(b) Is the change in measured consumption uncorrelated with the previous value of the change in measured consumption? In light of this, is measured consumption a random walk?