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.
Campbell, John Y. and Shiller, Robert J., "Cointegration and Tests of Present Value Models" (1986). Cowles Foundation Discussion Papers. 1028.