Non-linearities in financial bubbles: Theory and Bayesian evidence from S&P500

•We capture the presence of non-linearity in modeling financial bubbles.•The length of the bubble may vary.•The paper uses Bayesian methods on S&P500.•We approximate very satisfactorily all smooth non-linearities in bubbles.•The model provides evidence of an early warning mechanism (EWM).

The modeling process of bubbles, using advanced mathematical and econometric techniques, is a young field of research. In this context, significant model misspecification could result from ignoring potential non-linearities. More precisely, the present paper attempts to detect and date non-linear bubble episodes. To do so, we use Neural Networks to capture the neglected non-linearities. Also, we provide a recursive dating procedure for bubble episodes. When using data on stock price-dividend ratio S&P500 (1871.1–2014.6), employing Bayesian techniques, the proposed approach identifies more episodes than other bubble tests in the literature, while the common episodes are, in general, found to have a longer duration, which is evidence of an early warning mechanism (EWM) that could have important policy implications.