Publication Date: December 2017
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, 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&P 500 index data over 1964-2015 reveals strong evidence against parameter constancy after early 1980, which strengthens after July 1997, leading to a long duration of what the model characterizes as extreme behaviour in real stock prices.
Continuous time models, Explosive path, Extreme behaviour, Random coeﬀicient autoregression, Inﬁll asymptotics, Bubble testing
JEL Classification Codes: C13, C22, G13