Dual Interpretation of Machine Learning Forecasts
Philippe Goulet Coulombe, Maximilian Goebel, Karin Klieber
TL;DR
This paper addresses interpretability in macro forecasting by introducing a dual interpretation that decomposes out-of-sample predictions into proximity-based contributions from training observations, i.e., an exact representation $\hat{y}_j=\sum_i w_{ji} y_i$. It shows how data-portfolio weights $\mathbf{w}_j$ can be recovered for linear models via the dual solution and extended to kernel ridge regression, neural networks with linear output layers, random forests, and boosting, leveraging the representer theorem and the AXIL algorithm. The authors apply the method to post-pandemic inflation, Great Recession GDP growth, unemployment, and post-Covid recession probabilities, revealing how models leverage historical analogies and enabling diagnostics such as forecast concentration, short positions, leverage, turnover, and OHI. The approach offers a practical bridge between narrative insights and quantitative forecasts, providing actionable diagnostics for policymakers and researchers while offering an efficient alternative to Shapley-value explanations.
Abstract
Machine learning predictions are typically interpreted as the sum of contributions of predictors. Yet, each out-of-sample prediction can also be expressed as a linear combination of in-sample values of the predicted variable, with weights corresponding to pairwise proximity scores between current and past economic events. While this dual route leads nowhere in some contexts (e.g., large cross-sectional datasets), it provides sparser interpretations in settings with many regressors and little training data-like macroeconomic forecasting. In this case, the sequence of contributions can be visualized as a time series, allowing analysts to explain predictions as quantifiable combinations of historical analogies. Moreover, the weights can be viewed as those of a data portfolio, inspiring new diagnostic measures such as forecast concentration, short position, and turnover. We show how weights can be retrieved seamlessly for (kernel) ridge regression, random forest, boosted trees, and neural networks. Then, we apply these tools to analyze post-pandemic forecasts of inflation, GDP growth, and recession probabilities. In all cases, the approach opens the black box from a new angle and demonstrates how machine learning models leverage history partly repeating itself.