The present value model predicts the following relationship between the two series

\[ p_{t}=\beta_{0}+\beta_{1} d_{t}+u_{t} \]

where \(p_{t}\) is the natural logarithm of real price of equities, \(d_{t}\) is the natural logarithm of real dividend payments, \(u_{t}\) is a disturbance term and \(\beta_{1}\) is the discount rate and \(\beta_{1}=1\).

(a) Test for cointegration between \(p_{t}\) and \(d_{t}\) using Model 3 and \(p=1\) lags.

(b) Given the results in part (a) estimate a bivariate ECM for \(p_{t}\) and \(d_{t}\) using Model 3 with \(p=1\) lag. Interpret the results paying particular attention to the long-run parameter estimates, \(\beta_{0}\) and \(\beta_{1}\) and the error correction parameter estimates, \(\widehat{\alpha}_{i}\).

(c) Derive an estimate of the long-run real discount rate from \(R=\) \(\exp \left(-\beta_{0}\right)\) and interpret the result.

(d) Test the restriction \(H_{0}: \beta_{1}=1\).

(e) Discuss whether the empirical results support the present value model.