  We propose two robust methods for testing hypotheses on unknown parameters of
predictive regression models under heterogeneous and persistent volatility as
well as endogenous, persistent and/or fat-tailed regressors and errors. The
proposed robust testing approaches are applicable both in the case of discrete
and continuous time models. Both of the methods use the Cauchy estimator to
effectively handle the problems of endogeneity, persistence and/or
fat-tailedness in regressors and errors. The difference between our two methods
is how the heterogeneous volatility is controlled. The first method relies on
robust t-statistic inference using group estimators of a regression parameter
of interest proposed in Ibragimov and Muller, 2010. It is simple to implement,
but requires the exogenous volatility assumption. To relax the exogenous
volatility assumption, we propose another method which relies on the
nonparametric correction of volatility. The proposed methods perform well
compared with widely used alternative inference procedures in terms of their
finite sample properties.
