  The approach for testing equal predictive accuracy for pairs of forecasting
models proposed by Giacomini and White (2006) assumes that the parameters of
the underlying forecasting models are estimated using a rolling window of fixed
width and incorporates the effect of parameter estimation in the null
hypothesis. We show that a necessary and sufficient condition for the
conditionally expected loss differential of two forecasting models to be a
martingale difference sequence is that the outcome is a simple average of the
two forecasts. When the forecasts contain parameter estimation errors, this
means that the conditional mean of the outcome has to be a function of past
estimation errors--a condition that fails in many situations. We also show that
the null can fail even in the absence of parameter estimation for many types of
stochastic processes in common use.
