## Abstract

Speculative bubbles have played an important role ever since in financial economics. During an ongoing bubble it is relevant for investors and policy-makers to know whether the bubble continues to grow or whether it is already collapsing. Prices are typically well approximated by a random walk in absence of bubbles, while periods of bubbles are characterised by explosive price paths. In this paper we first propose a conventional Chow-type testing procedure for a structural break from an explosive to a random walk regime. It is shown that under the null hypothesis of a mildly explosive process a suitably modified Chow-type statistic possesses a standard normal limiting distribution. Second, a monitoring procedure based on the CUSUM statistic is suggested. It timely indicates such a structural change. Asymptotic results are derived and small-sample properties are studied via Monte Carlo simulations. Finally, two empirical applications illustrate the merits and limitations of our suggested procedures.

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## Notes

Phillips et al. (2011c) suggest critical values that slowly tend to infinity in order to ensure that the break dates are estimated consistently. In finite samples, however, the distribution of estimated break dates is severely skewed leading to a considerable delay of 5–15 periods [cf. Homm and Breitung (2010)].

Note that on exchange markets the stock price immediately adjusts for dividends so that the price represents the

*ex dividend*price of the share. It follows that in periods with dividend payment the ex dividend price also have the martingale difference property.To simplify the exposition we neglect a possible constant or time trend in the autoregressive representation. In our empirical examples a constant is included.

As pointed out by Ploberger and Krämer (1992), the forecast residuals can be replaced by the ordinary (in-sample) residuals. In the explosive model, however, the distribution of the cumulated residuals depends on the unknown parameter \(c\). We therefore do not consider the OLS CUSUM approach in what follows.

We also tried out various other rules such as the ones suggested by Schwert (1989). Overall the square-root rule performs best in our simulations.

Please note that this estimate differs from the one reported in Phillips et al. (2011c) as we consider a sub-sample running from June 1993 to June 2003.

Note that there is an obvious typo in Theorem 4.3(a) of Phillips and Magdalinos (2007), cf. their Eq. (9).

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## Acknowledgments

We would like to thank an anonymous referee and a Guest Editor for their helpful comments. Robinson Kruse gratefully acknowledges financial support from CREATES funded by the Danish National Research Foundation.

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## Appendix: Proof of Theorem 1

### Appendix: Proof of Theorem 1

(i) Under the null hypothesis we have

where

We now show that \(\kappa _T=1 - o_p(1)\). From Phillips and Magdalinos (2007) we have as \(T\rightarrow \infty \)

and, thus,

Using

for any \(\gamma <1\), it follows that \(\kappa _T \stackrel{p}{\rightarrow } 1\).

In a similar manner it can be shown that

Therefore, \(\sum \nolimits _{t=[\tau T]+1}^T \varepsilon _t y_{t-1}\) is \(O_p\left(\varrho ^{T}T^{\gamma } \right)\) (cf. Phillips and Magdalinos (2007), Theorem 4.3a). It follows that

Using \((\tau T)^{-1} \sum _{t=2}^{[\tau T]} (\tilde{e}_t^0)^2 \stackrel{p}{\rightarrow } \sigma _u^2\) we obtain

and, therefore, \(\psi _\tau \) converges to zero in probability as \(T\rightarrow \infty \) and \(c>0\).

(ii) From Theorem 4.3 of Phillips and Magdalinos (2007) it follows that^{Footnote 7}

for all \(0<\tau <1\), where \(X\) and \(Y\) are independent \(\mathcal{N}(0,\sigma _\varepsilon ^2/2c)\) random variables. Thus,

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Breitung, J., Kruse, R. When bubbles burst: econometric tests based on structural breaks.
*Stat Papers* **54**, 911–930 (2013). https://doi.org/10.1007/s00362-012-0497-3

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DOI: https://doi.org/10.1007/s00362-012-0497-3