Predictive Regression: A two-staged generalized Cochrane-Orcutt transformation with AR approximation approach

Predictive regressions in which a time series of one variable is regressed on the lagged time series of another variable are frequently applied to examine the relationship between financial returns and financial variable in lag, and are often used as a base to test the predictability of the lagged financial variable. These include predicting equity premium using lagged dividend yield. However, it is shown that ordinary least square (OLS) produces biased coefcient estimates in small samples when the regressor in lag is a Gaussian first-order autoregressive (AR) process with errors that are correlated with the error series of the dependent variable. That may lead researchers to erroneously conclude that the next-period value of the dependent variable can be forecasted by the current value of the predictive regressor. The bias will be even more severe when it is highly persistent or even nonstationary. What's more, it is now common empirical practice to allow for unknown persistence in the predictive regressor. An estimator that can deal with different level of persistence and has a standard distribution will be of value to the practitioners. Existing literature shows that stationary, nonstationary or long memory process with or without breaks can be well approximated by the autoregressive process with finite order, where the order can be selected by some information criteria for instance, Mallow's criterion or Akaike Information Criterion. Three new estimators for the coefficient of the predictive regressor are proposed in this project. First, we propose to use at wo-stage generalized Cochrane-Orcutt transformation estimator based on an auto regressive approximation framework (COAR)to estimate the lagged regressor. The filtered innovation is then included in the augmented regression model as in Amihud and Hurvich(2004) to obtain the coefficient in the predictive regression. Second, when both the dependent, independent and the innovations are all nonstationary, we consider applying the COAR estimator directly to the predictive regression. Third, we can apply the COAR approach to the model twice: first to the predictor process, and then to the augmented predictive regression itself. We also find that its t-statistic used for predictability test follows traditional t-distribution. The implementation of this methodology is relatively easy. A range of Monte Carlo studies that mimic the properties of financial time series indicate that the finite sample performance of our methodology is promising We shall also compare the robustness of our estimator to the existing ones through an application for predicting equity premium.