Cointegration and Tests of Present Value Models

This paper develops a way to test “present value” models – in which a variable Y (a long rate, a stock price) equals a constant times the discounted present value of expected future values of another variable y (a short rate, a dividend) – when y and Y are each nonstationary but their appropriately weighted difference, which the authors call the “spread,” is stationary; in that case y and Y are cointegrated, and the model’s restrictions can be tested as restrictions on a vector autoregression (VAR) in the spread and the change in y, without the bias from detrending nonstationary data that earlier tests risked. Applying this method to postwar U.S. interest rates (1959-78 and 1959-83), the authors find that the model is rejected at conventional statistical significance levels, yet the actual spread between long and short rates tracks the VAR’s unrestricted optimal forecast of future short-rate changes closely (correlations of 0.978 and 0.956 in the two samples) and deviations from the model do not appear persistent – so the statistical rejection overstates how badly the expectations theory of the term structure fits. Applying the same method to U.S. stock prices and dividends (1871-1986), they find “less encouraging” results: the model is again rejected, the spread between stock prices and dividends appears to move more than the present-value model implies (an innovations variance ratio of 11.27, with a large standard error, at the sample-mean 8.2 percent discount rate), and this excess-volatility finding is “sensitive to the assumed discount rate” – it is much weaker when a lower, 3.2 percent discount rate estimated from the data’s own cointegrating relationship is used instead.

John Y. Campbell | Macro Paper Warehouse

Cointegration and Tests of Present Value Models