Invariant quantile regression for heterogeneous environments

In this paper, we propose an invariant quantile regression (IQR) framework specifically designed for multi-environment datasets, which captures the invariance across different environments. This model is closely related to transfer learning, causal inference, and fair machine learning, and is motivated by scenarios in which the conditional probability of the response given covariates varies, while certain key features remain invariant. This perspective differs notably from previous works that restrict attention to the conditional mean, which is often insufficient in heterogeneous environments and the resulting estimators can become sensitive to ``bad" environments or changes in noise distributional shape. In contrast, quantile-based invariance naturally accommodates heterogeneity, and aligns more closely with structural causal models, in which variables invariant across environments at one or multiple quantile levels naturally indicate potential and stable causal predictors. Moreover, the set of endogenous variables under the IQR framework can be larger than that under the conditional mean framework typically, which in turn promotes more effective exclusion of spurious (no-causal) predictors provided that endogenous variables are not incorporated. To achieve this, we introduce a Kernel-Smoothed Focused Invariance Quantile Regression (KSFIQR) estimator, which leverages the underlying invariance structure and heterogeneity among environments, ensuring stable estimation across multiple environments. We establish the causal discovery properties of our method, demonstrate its ability to overcome the ``curse of endogeneity", and derive an $\ell_2$ error bound for our estimator in the low-dimensional regime, all in a non-asymptotic framework. From an algorithmic perspective, we implement the L-BFGS-B method and the Gumbel trick, with our numerical studies validating the proposed approach.