We propose an extension of the class of rational expectations bubbles (REBs) to the more general rational beliefs setting of [1, 2]. In a potentially non-stationary but stationarizable environment, among a heterogenous population of agents, it is possible to hold more than one "rational" expectation. When rational but diverse beliefs converge ("correlated beliefs"), they do not cancel each other out in aggregate anymore. This can make them an object of rational speculation. Accounting for the fact that market efficiency has an intrinsic time-dimension, we show that diverse but correlated beliefs can thus account for speculative bubbles, without the need for irrational agents or limits to arbitrage. Many of the shortcomings of REBs that make rational bubbles implausible can be overcome once we relax the ergodicity requirement. In particular, we argue that the hitherto unexplained "bubble component" of REBs corresponds to the extension of the state space in [3].

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It is commonly overlooked that the concept of market efficiency embowers a time-dimension. Illustrating with an example from the class of persistent random walks, we show that a price process can be a martingale on one time-scale but inefficient on another. This means that just as market efficiency can only be defined relative to an information set, it also depends on a timescale. We use this hitherto neglected aspect to propose a new definition of bubbles that does not rely on "fundamental value": A bubble is a violation of the efficient time-scale in that the market starts to "need longer" to reflect the original information set. That is, just as excess volatility is a violation of market efficiency with respect to its filtration, bubbles are a violation of market efficiency with respect to its time-scale.

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