The Causal-Noncausal Tail Processes

TL;DR

This paper analyzes the extremal behavior of mixed causal/noncausal autoregressive processes with heavy tails, focusing on their tail dynamics, prediction of turning points, and diagnostic tools for bubble episodes.

Contribution

It introduces a detailed description of the tail process dynamics in mixed causal/noncausal AR models and proposes diagnostic methods for identifying bubble episodes.

Findings

Characterization of tail process dynamics during extremal events

Method for updating tail process during speculative bubbles

Diagnostic plots for bubble episode detection

Abstract

This paper considers one-dimensional mixed causal/noncausal autoregressive (MAR) processes with heavy tail, usually introduced to model trajectories with patterns including asymmetric peaks and throughs, speculative bubbles, flash crashes, or jumps. We especially focus on the extremal behaviour of these processes when at a given date the process is above a large threshold and emphasize the roles of pure causal and noncausal components of the tail process. We provide the dynamic of the tail process and explain how it can be updated during the life of a speculative bubble. In particular we discuss the prediction of the turning point(s) and introduce pure residual plots as a diagnostic for the bubble episodes.