Pairwise Fisher transformation of a conditional correlation matrix

C. Francq${}^a$, S. Laurent${}^b$, M. Plazzogna${}^c$ and J-M. Zakoian${}^a$

${}^a$CREST-ENSAE and Lille University, ${}^b$Aix-Marseille University, ${}^c$University of Geneva

This paper introduces a novel Multivariate GARCH (MGARCH) framework for modeling dynamic conditional correlation matrices using a generalized pairwise Fisher transformation. We propose a flexible multivariate volatility model where the conditional correlation between any two assets is governed by a stochastic recurrence equation mapped through a known bijection to the interval $(-1,1)$.

We establish the theoretical conditions necessary for the existence of a unique, strictly stationary, and ergodic solution for both bivariate and general $N$-variate specifications. To tackle the curse of dimensionality inherent in MGARCH models, we develop a multi-step quasi-maximum likelihood estimation (QMLE) procedure that estimates GARCH-type volatilities equation-by-equation and correlations pair-by-pair. We formally derive the strong consistency and asymptotic normality of these estimators.

Furthermore, because the independent assembly of pairwise correlations does not inherently guarantee a positive definite global correlation matrix, we implement an ex-post metric projection step. Crucially, we prove that the entrywise perturbation introduced by this nearest-positive-definite projection is bounded by the asymptotic estimation error, thereby preserving the $\sqrt{T}$-consistency of the initial multi-step estimators.

Keywords: MGARCH, Conditional Correlation, Multi-step QMLE.