Cointegration and Tests of Present Value Models

In a model where a variable Y is proportional to the present value, with constant discount rate, of expected future values of a variable y, the "spread" S - Y - qy will be stationary for some q whether or not y must be differenced to induce stationarity. Thus, Y and y are cointegrated. The model implies that S is proportional to the optimal forecast of S*, the present value of future changes in y. We use vector autoregressive methods, and recent literature on cointegrated processes, to test the model. When Y is the long-term interest rate and y the short-term interest rate, we find in postwar United States data that S behaves much like an optimal forecast of S* even though as earlier research has shown it is negatively correlated with next period's change in Y. When Y is a real stock price index and y the corresponding real dividend, using annual United States data for 1871-1986 we obtain less encouraging results for the model, although the results are sensitive to the assumed discount rate.