Steady state distributions for models of locally explosive regimes: Existence and econometric implications

The purpose of this paper is to examine the properties of locally explosive regimes in the light of steady state results for threshold auto-regressive (TAR) models recently derived by Knight and Satchell (2011) [Journal of Time Series Econometrics, 3]. We study the conditions under which a steady state distribution of deviations of asset prices from fair value can be obtained using our simple model based on our particular definition of a bubble, noting that it is applicable to locally explosive regimes. After deriving general results, the analysis is further extended by considering the steady state distribution in three cases of a normally distributed error process, a non normally (exponentially) distributed steady-state process and a switching random walk with a fairly general iid error process. Then, the issues related to unit root testing for the presence of bubbles using standard econometric procedures are examined. Our results shed light on the ubiquitous finding of no bubbles in the econometric literature.