  We reinvigorate maximum likelihood estimation (MLE) for macroeconomic density
forecasting through a novel neural network architecture with dedicated mean and
variance hemispheres. Our architecture features several key ingredients making
MLE work in this context. First, the hemispheres share a common core at the
entrance of the network which accommodates for various forms of time variation
in the error variance. Second, we introduce a volatility emphasis constraint
that breaks mean/variance indeterminacy in this class of overparametrized
nonlinear models. Third, we conduct a blocked out-of-bag reality check to curb
overfitting in both conditional moments. Fourth, the algorithm utilizes
standard deep learning software and thus handles large data sets - both
computationally and statistically. Ergo, our Hemisphere Neural Network (HNN)
provides proactive volatility forecasts based on leading indicators when it
can, and reactive volatility based on the magnitude of previous prediction
errors when it must. We evaluate point and density forecasts with an extensive
out-of-sample experiment and benchmark against a suite of models ranging from
classics to more modern machine learning-based offerings. In all cases, HNN
fares well by consistently providing accurate mean/variance forecasts for all
targets and horizons. Studying the resulting volatility paths reveals its
versatility, while probabilistic forecasting evaluation metrics showcase its
enviable reliability. Finally, we also demonstrate how this machinery can be
merged with other structured deep learning models by revisiting Goulet Coulombe
(2022)'s Neural Phillips Curve.
