Local Gaussian Correlation in the Tails: A Scarcity Diagnostic, an Optimal Local Bandwidth, and the Limits of Adaptivity

Local Gaussian correlation (LGC) measures dependence locally, making it a natural tool for tail dependence and financial contagion, but its estimates degrade in the joint tails, where they are most needed.

Location-adaptive bandwidths have been tried for LGC and found inferior to a single global bandwidth; we explain why, and map the regime in which adaptivity does help.

First, a diagnostic: across heavy-tailed data-generating processes the parametric marginal pre-transform is inert (it changes the integrated error only in the fourth decimal), while the binding constraint is the local effective sample size, with the replication dispersion following a Fisher variance floor sd ~ (1 - rho^2)/sqrt(eff_n).

Second, theory: specializing the Hjort-Jones local-likelihood asymptotics to the bivariate Gaussian family that LGC fits, we derive the first location-specific AMISE-optimal bandwidth for LGC, b*(x) proportional to [(1 - rho^2)^2 / (f beta^2)]^(1/6) n^(-1/6), and validate its bias expansion directly (bias proportional to b^2 beta, R^2 approximately 0.9, slope-to-beta correlation 0.80).

Third, a regime map: a Monte Carlo across dependence strengths shows the adaptive rule beats the global plug-in only at moderate dependence with curved surfaces.

At weak dependence there is no curvature to exploit; at strong dependence finite-sample bias from the steep surface dominates, and adaptivity performs substantially worse, with an error that grows in the sample size.

This explains the field's experience that global bandwidths are hard to beat, and locates the exception.

Fourth, application: on volatility-filtered equity returns the adaptive estimator yields more stable tail-dependence surfaces under resampling.

The message is cautionary: the binding constraint on tail LGC is data scarcity, not bandwidth placement, and no bandwidth, however optimal, can recover information the data do not contain.