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We test for price bubbles in 14 national real estate investment trust (REIT) markets, and measure the degree of their convergence towards a common trend. Our methodology consists of the recently developed test of Phillips, Shi and Yu (2014) for mildly explosive processes, and the Phillips and Sul (2007) method for modelling convergence among random variables. We find evidence of explosive behaviour in index levels of seven of the 14 markets. In contrast, explosive dynamics are found in only one price/dividend ratio. More than half of the episodes of explosive behaviour are date-stamped to periods prior to the 2007–2009 financial crisis. We also discover a number of periods over which the markets converge towards a common trend. Interestingly, all of the convergence intervals coincide either with periods of crisis, or with periods of market exuberance. For instance, evidence of convergence is found during the 2000 dot-com crash, the 2007–2009 subprime crisis and the 2010–2013 European sovereign debt crisis, as well as over the bubble period of 2004–2005.