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.

翻译：我们通过一种新颖的神经网络架构重新激发了最大似然估计在宏观经济密度预测中的应用，该架构包含专用的均值和方差半球。该架构融合了多个关键要素，使最大似然估计在此场景中得以有效运作。首先，两个半球在网络入口处共享一个公共核心，以适应误差方差中各种形式的时间变化。其次，我们引入波动率强调约束，以打破此类过参数化非线性模型中均值/方差的不确定性。第三，我们采用分块袋外现实检验来抑制两个条件矩中的过拟合。第四，该算法利用标准深度学习软件，因此在计算和统计层面均能处理大规模数据集。因此，我们的半球神经网络能够在可行时基于领先指标提供主动波动率预测，并在必要时依据先前预测误差的幅度进行反应性波动率预测。我们通过广泛的样本外实验评估了点预测和密度预测，并与从经典模型到基于机器学习的现代模型等系列基准模型进行了对比。在所有情况下，半球神经网络均表现出色，始终能为所有目标值和预测期提供准确的均值/方差预测。对所得波动率路径的研究揭示了其多功能性，而概率预测评估指标则展示了其令人羡慕的可靠性。最后，我们还通过重新审视Goulet Coulombe（2022）的神经菲利普斯曲线，展示了该机制如何与其他结构化深度学习模型相结合。