Self-Normalized Inference for Relevance-Based Functional Granger Causality
Published:
Status: Revise and resubmit at the Journal of Financial Econometrics.
This working paper connects two strands of econometric methodology that often need to be combined in practice: functional time series causality and tuning-free self-normalized inference. In functional settings, predictors and responses are curves (or surfaces), so causal effects are naturally operator-valued and are best summarized through interpretable effect sizes rather than single coefficients.
A second theme is relevance-based testing. Classical point-null Granger-causality tests ask whether the effect is exactly zero; with rich functional predictors, this can lead to rejections that are statistically significant but economically trivial. We instead test whether the causal effect exceeds a relevance threshold (∆), so the conclusion targets the operational question: is the predictive effect large enough to matter?
For inference under dependence and heteroskedasticity, we use self-normalization to avoid long-run variance estimation and its fragile tuning choices. Alongside the standard quadratic self-normalizer, we incorporate an adjusted-range-based self-normalizer built from the partial-sum range. This alternative normalizer can deliver sharper local power and robust finite-sample behavior, while remaining fully tuning-free.
Summary and Contribution:
We develop self-normalized tests for relevance-based functional Granger causality for stationary functional time series in a separable Hilbert space. The procedure yields pivotal limits (and hence usable critical values) without any bandwidth, kernel, or block-length selection, and it accommodates economically interpretable relevance thresholds.
We provide asymptotic theory for relevant hypotheses (∆ > 0), including Pitman local power, and we also address the degenerate point-null case (∆ = 0) by building inference on the underlying linear Hilbert-space parameter rather than the quadratic effect size.
Evidence:
Monte Carlo experiments based on functional autoregressive designs (including dependent and heavy-tailed innovations) show that the proposed self-normalized procedures deliver reliable size control at the relevance boundary and strong power against relevant alternatives, with the adjusted-range self-normalizer often improving sensitivity.
Empirically, we study economically relevant volatility spillovers between BTC spot realized volatility (Bitfinex) and option-market functionals from Deribit (daily implied-volatility smiles and amount/flow smiles, 2025). Using RMS-correlation benchmarks to parameterize relevance, the full-sample results support economically meaningful predictive content from options to next-day spot volatility at modest relevance levels, while stricter relevance thresholds and reverse-direction channels are not supported; rolling-window analysis highlights that lead–lag patterns can be time-varying.
Availability:
The manuscript is not hosted on this website due to copyright and journal submission policy.