Long-Run Regressions: Theory and Application to US Asset Markets

The question of long-run predictability in the aggregate US stock market is still unsettled. This is due to the lack of a robust method to judge the statistical significance of long-run regressions under the maintained hypothesis. By developing a spectral theory of long-run regressions with both long-run dependent and independent variables, we demonstrate a version of Engle's (1974) conjecture that asymptotically correct standard errors can be computed by multiplying the ordinary least squares standard errors by the square root of 2/3 times the length of the forecast horizon. We generalize Stambaugh's (1999) bias formula to the long-run regression model proposed in this paper. In addition, we find, that for persistent predictive variables, the OLS estimator in our regression model is more efficient than the estimator in the predictive regressions suggested by Campbell and Shiller (1988) and Hodrick (1992). Application of our method shows thatthe long-run earnings yield significantly predicts up to 69% of the variation in the 10-year S&P 500 real return, and up to 49% of long-run bond returns.