This paper uses fractional integration to examine the long-run dynamics and cyclical structure of US inflation, real risk-free rate, real stock returns, equity premium and price/dividend ratio, annually from 1871 to 2000. It implements a procedure which allows consideration of unit roots with possibly fractional orders of integration both at zero (long-run) and cyclical frequencies. When focusing exclusively on the former, the estimated order of integration varies considerably, and non-stationarity is found only for the price/dividend ratio. When the cyclical component is also taken into account, the series appear to be stationary but to exhibit long memory with respect to both components in almost all cases. The exception is the price/dividend ratio, whose order of integration is higher than 0.5 but smaller than 1 for the long-run frequency, and is between 0 and 0.5 for the cyclical component. Also, mean reversion occurs in all cases. Finally, six different criteria are applied to compare the forecasting performance of the fractional (at both zero and cyclical frequencies) models with others based on fractional and integer differentiation only at the zero frequency. The results, based on a 15-year horizon, show that the former outperforms the others in a number of cases.
Long run and cyclical dynamics in the US stock market
Luis Alberiko Gil-Alaña
Guglielmo Maria Caporale
Journal of Forecasting Vol. 33, No. 2