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 shortterm interest rates into the future plus a term premium that is \(I(0)\). Specifically, let \(R k_{t}\) denote a \(k\)-period interest rate, let \(R 1_{t}\) denote a one-period interest rate, and let \(e_{t}\) denote an \(I(0)\) term premium. Then \(R k_{t}=\frac{1}{k} \sum_{i=0}^{k-1} R 1_{t+i \mid t}+e_{t}\), where \(R 1_{t+i \mid t}\) 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 \(R 1_{t}=R 1_{t-1}+u_{t}\).

a. Show that \(R k_{t}=R 1_{t}+e_{t}\).

b. Show that \(R k_{t}\) and \(R 1_{t}\) are cointegrated. What is the cointegrating coefficient?

c. Now suppose that \(\Delta R 1_{t}=0.5 \Delta R 1_{t-1}+u_{t}\). 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?