Dynamic tail inference with log-Laplace volatility

Abstract

We propose a family of models that enable predictive estimation of time-varying extreme event probabilities in heavy-tailed and nonlinearly dependent time series. The models are a white noise process with conditionally log-Laplace stochastic volatility. In contrast to other, similar stochastic volatility formalisms, this process has analytic expressions for its conditional probabilistic structure that enable straightforward estimation of dynamically changing extreme event probabilities. The process and volatility are conditionally Pareto-tailed, with tail exponent given by the reciprocal of the log-volatility's mean absolute innovation. This formalism can accommodate a wide variety of nonlinear dependence, as well as conditional power law-tail behavior ranging from weakly non-Gaussian to Cauchy-like tails. We provide a computationally straightforward estimation procedure that uses an asymptotic approximation of the process' dynamic large deviation probabilities. We demonstrate the estimator's utility with a simulation study. We then show the method's predictive capabilities on a simulated nonlinear time series where the volatility is driven by the chaotic Lorenz system. Lastly we provide an empirical application, which shows that this simple modeling method can be effectively used for dynamic and predictive tail inference in financial time series.