Online Monitoring of Distributional Granger Causality
Published:
This working paper develops a unified framework for online monitoring of distributional Granger causality after a forecasting model has already been trained and deployed. The goal is to detect, in real time, whether an omitted predictor block begins to add incremental predictive content for a conditional quantile, expectile, or other elicitable functional of the response distribution.
The general backbone is a restricted-model score architecture: estimate the baseline X-only model on a stable Phase-I training sample, freeze the estimator, construct excluded-block score contributions, and monitor them sequentially in Phase II. For broad online inference under weak dependence and conditional heteroskedasticity, the paper develops self-normalized KS and CvM detectors that avoid long-run variance estimation and deliver pivotal stopping rules.
For quantile and Value-at-Risk monitoring, the paper adds a more specialized branch based on conditional calibration of exceedance hits. By betting on the hit process using predictable features extracted from the omitted block, it builds anytime-valid quantile e-processes that remain valid under optional stopping and admit mixture, adaptive, and restart extensions. This gives the monitoring problem a direct tail-risk interpretation and complements the broader self-normalized approach.
Summary and Contribution:
We formulate online distributional predictive causality in a predictive-regression setting that nests familiar VAR-style specifications but is not restricted to them, and we distinguish clearly between the structural null of no incremental predictive content and the operational null implied by a frozen Phase-I forecaster. The paper provides a unified theory: pivotal null limits and local-power results for the self-normalized monitoring statistics, together with finite-sample anytime-valid guarantees for the quantile e-process branch.
More broadly, the paper shows how a common omitted-block score process can support two complementary monitoring strategies. The self-normalized branch is the general econometric solution for quantiles, expectiles, and other elicitable functionals, while the e-process branch is a quantile/VaR-specific refinement that exploits additional calibration structure when exact sequential validity is especially important.
Evidence:
Monte Carlo experiments study a range of monitoring environments, including abrupt location breaks, symmetric scale changes, downside-tail alternatives, gradual switch-on, and contaminated-training designs. The results show a clear division of labor across procedures: HAC/RSMS KS rules are strong broad-purpose detectors, late-weight CvM rules are especially useful when predictive content is tail-localized or accumulates gradually, and quantile e-processes are especially attractive for exact size control, open-end monitoring, and contaminated-training scenarios.
Empirically, the paper studies hourly BTC downside-risk monitoring using BTC spot-market controls together with Deribit option-market signals. The illustration shows that different detectors can imply economically distinct alarm profiles: on this sample the e-process branch remains conservative and silent, while more reactive CvM alarms generate different alarm-to-trade performance profiles relative to buy-and-hold.
Availability:
The manuscript is not hosted on this website due to copyright considerations.