Introduction to Econometrics 3rd Edition by James Stock, Mark Watson

Exercise 3

One version of the expectations theory of the term structure of interest rates holds that a long-term rate equals the average of the expected values of short-term interest rates into the future, plus a term premium that is I (0). Specifically, let Rk t denote a k -period interest rate, let R l t denote a one-period interest rate, and let e t denote an I (0) term premium. Then
, where
, is the forecast made at date t of the value of R 1 at date t + i. Suppose that R 1 t follows a random walk so that

a. Show that

b. Show that Rk t and R 1 t are cointegrated. What is the cointegrating coefficient
c. Now suppose that R 1 t = 0.5 R 1 t 1 + u t. How does your answer to (b) change
d. Now suppose that R 1 t = 0.5 R 1 t 1 + u t. How does your answer to (b) change