The time-averaging problem. (Working, 1960.) Actual data do not give 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 twoperiod 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?

(c) Given your result in part (a), is the change in consumption from one twoperiod interval to the next necessarily uncorrelated with anything known as of the first of these two-period intervals? Is it necessarily uncorrelated with anything known as of the two-period interval immediately preceding the first of the two-period intervals?
(d ) Suppose that measured consumption for a two-period interval is not the average over the interval, but consumption in the second of the two periods. That is, one observes Ct +1, Ct +3, and so on. In this case, is measured consumption a random walk?