New Methods for Inference in Long-Horizon Regressions

Abstract

I develop new results for long-horizon predictive regressions with overlapping observations. I show that rather than using autocorrelation robust standard errors, the standard t-statistic can simply be divided by the square root of the forecasting horizon to correct for the effects of the overlap in the data. Further, when the regressors are persistent and endogenous, the long-run ordinary least squares (OLS) estimator suffers from the same problems as the short-run OLS estimator, and it is shown how similar corrections and test procedures as those proposed for the short-run case can also be implemented in the long run. An empirical application to stock return predictability shows that, contrary to many popular beliefs, evidence of predictability does not typically become stronger at longer forecasting horizons.

Some curious power properties of long-horizon tests

“…In order to evaluate the power properties of long-horizon tests, I rely on recent test statistics developed by Valkanov (2003) and Hjalmarsson (2011), which have been shown to control size well. Use of overlapping observations induces strong serial correlation in the regression residuals and standard errors that fail to account for this fact lead to biased inference.…”

“…This is true both for Valkanov's (2003) test, but also if one uses, for instance, Newey-West standard errors in a normal t-statistic. The test proposed by Hjalmarsson (2011) specifically for the case of endogenous regressors does not suffer from this problem. These findings add an extra note of caution to the use of long-horizon regressions.…”

Robustness of Multistep Forecasts and Predictive Regressions at Intermediate and Long Horizons

“…In this paper, we propose a local-asymptotic model that builds on the work of Kemp (1999), Valkanov (2003), Torous et al (2004), Chevillon and Hendry (2005) and Hjalmarsson (2011). We prove a new key property of direct multi-step estimators, namely their robustness to misspecification of the serial correlation of the error process.…”

“…The empirical literature has shown that whereas H h=1 0 often does not reject, this is not the case when considering large h, in which case H h 0 may reject and y t−h appears helpful in predicting z t . The question of how large h should be is an empirical one: Hjalmarsson (2011) studies the case where h is fixed, whereas Valkanov (2003), Torous et al (2004) and Hjalmarsson (2011) have considered letting the horizon grow with the sample size T as respectively h = O (T ) and h = o (T ) . In their setting, Torous et al and Hjalmarsson allowed in addition for the error (ε t+1 , t+1 ) to exhibit autocorrelation.…”

“…The prototypical "predictive" or "long horizon" regression will take the form of a regression of z t+h on x t , but h i=1 z t+i or h i=1 x t+i also appear as regressand and regressor in the literature (see e.g. Lanne, 2002, Torous et al, 2004, Valkanov (2003, Boudoukh et al, 2008, Hjalmarsson, 2011, Phillips and Lee, 2013, Phillips, 2015, and the references therein). The long horizon regression literature shares with that on direct multi-step forecasting three key features: (i) the model which is estimated is not a priori that which would most efficiently chose (i.e.…”