Confidence sets for the emergence, collapse, and recovery dates of a bubble

We propose constructing confidence sets for the emergence, collapse, and recovery dates of a bubble separately by inverting tests for the location of the break date. We examine both likelihood ratio-type tests and the Elliott-Müller-type (2007) tests for detecting break locations. The limiting distributions of these tests are derived under the null hypothesis, and their asymptotic consistency under the alternative is established. Finite-sample properties are evaluated through Monte Carlo simulations. The results indicate that combining different types of tests effectively controls the empirical coverage rate while maintaining a reasonably small length of the confidence set. The confidence sets for the bubble dates are empirically useful for quantifying the uncertainty surrounding the point estimates, as demonstrated through an application to Japanese stock price data.