Privacy-Perturbed Parameter Estimation and Inference for Heavy-Tailed GARCH Models

Abstract

Most privacy-aware inference methods are developed for independent data, whereas financial returns combine heavy tails with persistent conditional volatility. This paper examines a Gaussian-perturbed GARCH M-estimation framework and the finite-sample effect of perturbation noise on parameter inference. The idealized large-sample analysis uses stationarity, ergodicity, and a martingale central limit theorem to characterize the sampling and perturbation components. A noise-corrected sandwich covariance estimator and Wald statistic are then constructed. The simulation implementation uses Gaussian QML as the M-estimation center, a projected one-step parameter perturbation, and a working calibration Delta = 2B with B = 2. Across 500 requested replications per design point, the corrected covariance materially improves interval coverage and reduces Wald over-rejection, es-pecially when epsilon is small. The results also reveal finite-sample limitations: projection is non-negligible in the smallest-sample, strongest-perturbation design, and studentized Q-Q plots retain tail departures from normality. Because the initial QML estimate and covariance ingredients use the raw sample and a complete recursive global-sensitivity proof is not provided, the implementation is interpreted as privacy-perturbed and DP-motivated rather than as an end-to-end differentially private release.