Are we in a bubble? A simple time-series-based diagnostic

Time series with bubble-like patterns display an unbalance between growth and acceleration, in the sense that growth in the upswing is “too fast” and then there is a collapse. In fact, such time series show periods where both the first differences (1-L) and the second differences (1-L)2 of the data are positive-valued, after which period there is a collapse. For a time series without such bubbles, it can be shown that 1-L2 differenced data should be stable. A simple test based on one-step-ahead forecast errors can now be used to timely monitor whether a series experiences a bubble and also whether a collapse is near. Illustration on simulated data and on two housing prices and the Nikkei index illustrates the practical relevance of the new diagnostic. Monte Carlo simulations indicate that the empirical power of the test is high.