Tests for an end-of-sample bubble in financial time series

ABSTRACT

In this paper, we examine the issue of detecting explosive behavior in economic and financial time series when an explosive episode is both ongoing at the end of the sample and of finite length. We propose a testing strategy based on a subsampling method in which a suitable test statistic is calculated on a finite number of end-of-sample observations, with a critical value obtained using subsample test statistics calculated on the remaining observations. This approach also has the practical advantage that, by virtue of how the critical values are obtained, it can deliver tests which are robust to, among other things, conditional heteroskedasticity and serial correlation in the driving shocks. We also explore modifications of the raw statistics to account for unconditional heteroskedasticity using studentization and a White-type correction. We evaluate the finite sample size and power properties of our proposed procedures and find that they offer promising levels of power, suggesting the possibility for earlier detection of end-of-sample bubble episodes compared to existing procedures.

KEYWORDS:

• Explosive autoregression
• rational bubble
• right-tailed unit root testing: sub-sampling

JEL CLASSIFICATION:

• C22
• C12
• G14

Acknowledgment

We are grateful to the Guest Editors, Peter Phillips and Aman Ullah, and two anonymous referees for their helpful and constructive comments on earlier versions of this paper.

Notes

¹Note that the BSADF results depend on ϕ but do not of course change across the settings for m^′; the results are simply repeated for ease of comparison. Figure 2 also reports results for tests that will be introduced and discussed later in the paper.

²In a companion discussion paper version of this paper (Astill et al., 2016), we also considered DGPs where a bubble abruptly collapses after a number of periods and also examined the impact of a previously collapsed bubble on the power of the tests to detect an end-of-sample bubble. Overall, we find similar power patterns to those in Fig. 2, apart from when a previously collapsed bubble is relatively close to the end-of-sample bubble. In this latter case, the $S_{m^{\prime }}$ tests recover their properties from the collapse of the first bubble much more rapidly than the BSADF test, which has relatively poor power to detect the second bubble. Of course, a prior bubble of long duration can adversely affect the powers of the $S_{m^{\prime }}$ tests, since a large proportion of subsample statistics used for computing the critical values are affected by the earlier bubble, hence caution should be exercized if a long bubble is present in the sample period used to obtain critical values.

³The HLST dating methodology only identifies valid bubble dates where the end of bubble date (denoted $y_{T^{\ast }}$ here) exceeds the start of bubble date (denoted y_T here). In cases where this is not satisfied for any T, we revert to using Δy_t for the model-based residuals.