[Paper Review] Parametric inference and forecasting in continuously invertible volatility models

Olivier Wintenberger, Sixiang Cai|arXiv (Cornell University)|Jun 24, 2011

Financial Risk and Volatility Modeling30 references3 citations

TL;DR

This paper establishes the strong consistency and asymptotic normality of M-estimators in continuously invertible volatility models via a novel framework of continuous invertibility for Stochastic Recurrence Equations (SREs). It provides the first rigorous justification for the quasi-likelihood estimation procedure used in practice for EGARCH(1,1) models, deriving necessary and sufficient conditions for asymptotic normality under minimal moment assumptions.

ABSTRACT

We introduce the notion of continuously invertible volatility models that relies on some Lyapunov condition and some regularity condition. We show that it is almost equivalent to the ability of the volatilities forecasting using the parametric inference approach based on the SRE given in [16]. Under very weak assumptions, we prove the strong consistency and the asymptotic normality of the parametric inference. Based on this parametric estimation, a natural strongly consistent forecast of the volatility is given. We apply successfully this approach to recover known results on univariate and multivariate GARCH type models and to the EGARCH(1,1) model. We prove the strong consistency of the forecasting as soon as the model is invertible and the asymptotic normality of the parametric inference as soon as the limiting variance exists. Finally, we give some encouraging empirical results of our approach on simulations and real data.

Motivation & Objective

• To address the lack of theoretical justification for quasi-likelihood estimation in non-linear volatility models, particularly EGARCH(1,1), used empirically since Nelson (1991).
• To introduce and formalize the concept of continuous invertibility on a compact set for volatility models driven by Stochastic Recurrence Equations (SREs).
• To prove the strong consistency and asymptotic normality of the M-estimator under the new continuous invertibility condition.
• To recover known results for GARCH-type models and extend them to asymmetric and non-linear models like EGARCH.
• To provide a necessary and sufficient condition for the existence of the limiting covariance matrix in the EGARCH(1,1) model.

Proposed method

• Introduces the notion of continuous invertibility on a compact parameter set for SRE-driven volatility models.
• Applies the M-estimator framework using the Quasi-Likelihood (QLIK) criterion to ensure consistency and asymptotic normality.
• Establishes the convergence of the SRE driven by observed data $X_t$ via a new SRE formulation $ (h(Σ_k))_{k\leq t} = \phi_t((h(Σ_k))_{k\leq t-1}, \theta_0) $, replacing the innovation-driven SRE.
• Uses Lyapunov-type conditions and logarithmic moment assumptions to ensure stability and invertibility of the SRE under the new framework.
• Applies functional central limit theory to derive asymptotic normality, relying on differentiability and moment conditions on the score function.
• Derives explicit conditions for the limiting covariance matrix in EGARCH(1,1), showing that $\mathbb{E}[Z_0^4] < \infty$ and $\mathbb{E}[(\beta_0 - \frac{1}{2}(\gamma_0 Z_0 + \delta_0 |Z_0|))^2] < 1$ are necessary and sufficient.

Experimental results

Research questions

• RQ1 Under what conditions is the M-estimator for the Quasi-Likelihood in continuously invertible volatility models strongly consistent and asymptotically normal?
• RQ2 What is the necessary and sufficient condition for the existence of the limiting covariance matrix in the EGARCH(1,1) model?
• RQ3 How can the quasi-likelihood estimation procedure used in practice for EGARCH(1,1) be theoretically justified?
• RQ4 Can the framework of continuous invertibility be applied to recover known results for GARCH-type models?
• RQ5 What are the minimal moment and differentiability conditions required to ensure asymptotic normality of the estimator in non-linear volatility models?

Key findings

• The M-estimator based on the Quasi-Likelihood criterion is strongly consistent and asymptotically normal under the continuous invertibility condition.
• For EGARCH(1,1), the limiting covariance matrix exists if and only if $\mathbb{E}[Z_0^4] < \infty$ and $\mathbb{E}[(\beta_0 - \frac{1}{2}(\gamma_0 Z_0 + \delta_0 |Z_0|))^2] < 1$, which is both necessary and sufficient.
• The paper provides the first rigorous justification for the empirical estimation procedure used in Nelson (1991) for EGARCH(1,1), which lacked theoretical support prior to this work.
• The framework recovers known results for univariate and multivariate GARCH models, where the M-estimator coincides with the classical QMLE.
• The strong consistency of the estimator holds under the condition $\mathbb{E}[\log \Lambda(\phi_t(\cdot, \theta))] < 0$ and integrability of logarithmic moments.
• The asymptotic normality result holds under differentiability of the SRE and moment conditions, with the limiting variance-covariance matrix being invertible when $\theta_0$ is in the interior of the parameter space $\Theta$.