The data for this question were obtained from Corbae, Lim and Ouliaris (1992) who test for speculative efficiency by considering the equation

\[ s_{t}=\beta_{0}+\beta_{1} f_{t-n}+u_{t} \]

where \(s_{t}\) is the natural logarithm of the spot rate, \(f_{t-n}\) is the natural logarithm of the forward rate lagged \(n\) periods and \(u_{t}\) is a disturbance term. In the case of weekly data and the forward rate is the 1-month rate, \(f_{t-4}\) is an unbiased estimator of \(s_{t}\) if \(\beta_{1}=1\).

(a) Use unit root tests to determine the level of integration of \(s_{t}, f_{t-1}\), \(f_{t-2}\) and \(f_{t-3}\).

(b) Test for cointegration between \(s_{t}\) and \(f_{t-4}\) using Model 2 with \(p=\) 0 lags.

(c) Provided that the two rates are cointegrated, estimate a bivariate VECM for \(s_{t}\) and \(f_{t-4}\) using Model 2 with \(p=0\) lags.

(d) Interpret the coefficients \(\beta_{0}\) and \(\beta_{1}\). In particular, test that \(\beta_{1}=1\).

(e) Repeat these tests for the 3 month and 6 month forward rates.