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Abstract:
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This paper studies the asymptotic properties of least squares estimators for the origination and termination of explosive behavior in a time series. Such estimators are useful for dating the onset and collapse of asset pricing bubbles, an issue of prime importance for policymakers. The dating estimators are based on an autoregressive model in which the origination is modeled as a switch from unit root to explosive behavior while a collapse is indicated by a switch back to unit root behavior. We first show that when the break magnitude is fixed, consistency of the break date estimators only requires the duration of the explosive regime to increase with the sample size. In particular, it is not necessary to assume that the duration is a positive fraction of the sample size, as is typically assumed in the change-point literature. Further, consistency is derived with respect to the break dates and not just the break fractions. To derive the rate of convergence and the ensuing limit distribution, we adopt a mildly explosive representation whereby the break magnitude decreases to zero as the sample size increases. A sharp rate and a tractable limit distribution are obtained in this case.
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