Franses, P.H. (2016). A simple test for a bubble based on growth and acceleration Computational Statistics and Data Analysis, 100:160--169.

Time series with bubble-like patterns display an unbalance between growth and acceleration. When growth in the upswing is “too fast”, then soon there will be a collapse and the bubble bursts. Such time series thus shows periods where both the first differences and the second differences of the data are positive-valued and in unbalance. For a time series without such bubbles, it can be shown that the variable when properly differenced has a stable mean. A simple test based on recursive residuals can now be used to timely discover 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 test. Monte Carlo simulations indicate that the empirical power of the test can be high.