Dynamic Tail Inference with Log-Laplace Volatility
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We propose a family of stochastic volatility models that enable direct estimation of time-varying extreme event probabilities in time series with nonlinear dependence and power law tails. The models are a white noise process with conditionally log-Laplace stochastic volatility. In contrast to other, similar stochastic volatility formalisms, this process has an explicit, closed-form expression for its conditional probability density function, which enables 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. These models thus can accommodate conditional power law-tail behavior ranging from very weakly non-Gaussian to Cauchy-like tails. Closed-form expressions for the models' conditional polynomial moments also allows for volatility modeling. We provide a straightforward, probabilistic method-of-moments estimation procedure that uses an asymptotic result for the process' conditional large deviation probabilities. We demonstrate the estimator's usefulness with a simulation study. We then give empirical applications to financial time series data, which show that this simple modeling method can be effectively used for dynamic tail inference in nonlinear, heavy-tailed time series.
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