An Improved Procedure for Retrospectively Dating the Emergence and Collapse of Bubbles

This article proposes a new ordinary least squares (OLS)-based procedure for retrospectively dating the emergence and collapse of bubbles. We first consider a data generating process that entails a switch from a unit root regime to an explosive regime followed by a collapse and subsequent return to unit root behavior. We demonstrate analytically that the standard OLS estimates are inconsistent and date both the origination and implosion points with a delay in large samples. A simple modification that involves omitting the residual corresponding to the implosion date is shown to yield consistent estimates. We also develop an efficient dating algorithm that can accommodate a framework with multiple bubbles. The algorithm exploits the explicit form of the unit root restrictions to directly embed them into the recursive optimization problem which obviates the need to rely on an iterative scheme that requires initial values. Extensive simulation experiments indicate that our proposed procedure typically delivers estimates with lower bias and root mean squared error relative to competing alternatives. An empirical illustration is included.