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 time- scale. 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.