Does the absence of cointegration explain the typical findings in long horizon regressions?

One of the stylized facts in financial and international economics is that of increasing predictability of variables such as exchange rates and stock returns at longer horizons. This fact is based upon applications of long horizon regressions, from which the typical findings are that the point estimates of the regression parameter, the associated t-statistic, and the regression R^2 all tend to increase as the horizon increases. Such long horizon regression analyses implicitly assume the existence of cointegration between the variables involved. In this paper, we investigate the consequences of dropping this assumption. In particular, we look upon the long horizon regression as a conditional error-correction model and interpret the test for long horizon predictability as a single equation test for cointegration. We derive the asymptotic distributions of the estimator of the regression parameter and its t-statistic for arbitrary horizons, under the null hypothesis of no cointegration. It is shown that these distributions provide an alternative explanation for at least part of the typical findings. Furthermore, the distributions are used to derive a Phillips-Perron type correction to the ordinary least-squares t-statistic in order to endow it with a stable size for given, arbitrary, horizon. A local asymptotic power analysis reveals that the power of long horizon regression tests does not increase with the horizon. Exchange rate data are used to demonstrate the empirical relevance of our theoretical results.