It is often claimed that "di?erencing" the regression model will reduce the degree of autocorrelation. Considerthis claim by using as the model:

yt = xt + ut and ut = ut1 + t

where (t) is white noise with mean zero and variance 2, and ||  1. We want to compare the autocorrelation of (ut ) in thismodel with the

autocorrelation of (vt) in the following di?erenced model:

yt yt1 = (xt xt1 ) + vt

where vt = ut ut1. Answer the following:

(a) Show that the autocorrelation of ut, call it u(k), in the original model is k .

(b) Find the autocorrelation of vt at the first lag, i.e., v (1), in the differenced model.

(c) Evaluatethe claim by comparing u(1) and v (1) over the range of f.