Random Coefficient Continuous Systems: Testing for Extreme Sample Path Behaviour

Abstract

This paper studies a continuous time dynamic system with a random persistence parameter. The exact discrete time representation is obtained and related to several discrete time random coefficient models currently in the literature. The model distinguishes various forms of unstable and explosive behaviour according to specific regions of the parameter space that open up the potential for testing these forms of extreme behaviour. A two-stage approach that employs realized volatility is proposed for the continuous system estimation, the asymptotic theory is developed, and test statistics to identify the different forms of extreme sample path behaviour are proposed. Simulations show that the proposed estimators work well in empirically realistic settings and that the tests have good size and power properties in discriminating characteristics in the data that differ from typical unit root behaviour. The theory is extended to cover models where the random persistence parameter is endogenously determined. An empirical application based on daily real S&P500 index data over 1928-2018 reveals strong evidence against parameter constancy over the whole sample period leading to a long duration of what the model characterizes as extreme behaviour in real stock prices.

Keywords: Continuous time models, Explosive path, Extreme behaviour, Random coefficient autoregression, Infill asymptotics, Bubble testing

JEL Classification: C13, C22, G13