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.

Do Stock Prices Move Too Much to Be Justified by Subsequent Changes in Dividends?

Shiller tests the “efficient markets model” of stock prices – that the real price equals the present value of rationally expected future real dividends discounted at a constant real rate – by comparing the actual detrended real price series p to the “ex post rational price” p*, the present value of the dividends that were actually paid afterward. Because p is the market’s optimal forecast of p* under the model, and an optimal forecast’s error must be uncorrelated with the forecast itself, the model implies the variance of p cannot exceed the variance of p*; using U.S. data on the S&P Composite (1871-1979) and a modified Dow Jones Industrial Average (1928-1979), Shiller finds this and two related inequalities are violated dramatically, with the observed volatility of prices five to thirteen times too large relative to the bound implied by the actual subsequent variation in dividends. He shows the gap cannot plausibly be closed by allowing the real discount rate to vary over time instead of holding it constant: the required standard deviation of the ex ante real interest rate would have to be at least 4.4-7.4 percentage points, implying a real rate ranging as wide as -8 to +17 percent over the sample, far more volatile than observed nominal interest rates. Shiller concludes that this volatility evidence is a different and, in his view, more robust way of rejecting the simple efficient-markets model than standard tests for forecastable returns, since it does not require confining the data to short, carefully measured samples, though he acknowledges the remaining “academic” escape route – that the market feared much larger dividend movements than ever materialized – cannot be evaluated statistically because it rests on an unobservable.

Robert J. Shiller | Macro Paper Warehouse

Cointegration and Tests of Present Value Models

Do Stock Prices Move Too Much to Be Justified by Subsequent Changes in Dividends?