  Though out-of-sample forecast evaluation is systematically employed with
modern machine learning methods and there exists a well-established classic
inference theory for predictive ability, see, e.g., West (1996, Asymptotic
Inference About Predictive Ability, Econometrica, 64, 1067-1084), such theory
is not directly applicable to modern machine learners such as the Lasso in the
high dimensional setting. We investigate under which conditions such extensions
are possible. Two key properties for standard out-of-sample asymptotic
inference to be valid with machine learning are (i) a zero-mean condition for
the score of the prediction loss function; and (ii) a fast rate of convergence
for the machine learner. Monte Carlo simulations confirm our theoretical
findings. We recommend a small out-of-sample vs in-sample size ratio for
accurate finite sample inferences with machine learning. We illustrate the wide
applicability of our results with a new out-of-sample test for the Martingale
Difference Hypothesis (MDH). We obtain the asymptotic null distribution of our
test and use it to evaluate the MDH of some major exchange rates at daily and
higher frequencies.
