CUSUM-Based Monitoring for Explosive Episodes in Financial Data in the Presence of Time-Varying Volatility

Abstract

We generalize the Homm and Breitung (2012) CUSUM-based procedure for the real-time detection of explosive autoregressive episodes in financial price data to allow for time-varying volatility. Such behavior can heavily inflate the false positive rate (FPR) of the CUSUM-based procedure to spuriously signal the presence of an explosive episode. Our modified procedure involves replacing the standard variance estimate in the CUSUM statistics with a nonparametric kernel-based spot variance estimate. We show that the sequence of modified CUSUM statistics has a joint limiting null distribution which is invariant to any time-varying volatility present in the innovations and that this delivers a real-time monitoring procedure whose theoretical FPR is controlled. Simulations show that the modification is effective in controlling the empirical FPR of the procedure, yet sacrifices only a small amount of power to detect explosive episodes, relative to the standard procedure, when the shocks are homoskedastic. An empirical illustration using Bitcoin price data is provided.

A review of Phillips‐type right‐tailed unit root bubble detection tests

“…Astill et al. (2021) extend the CUSUM‐based procedure of Homm & Breitung (2012) for detecting explosive behavior in financial time series to allow for time‐varying volatility. The CUSUM‐based procedure assumes that shocks are unconditionally homoskedastic, which can be infeasible for many financial time series with time‐varying volatility.…”

“…Astill et al. (2021) modify the standard variance estimate in the CUSUM statistics by a nonparametric kernel‐based variance estimate with an aim to reduce the false positive identification of bubbles. Astill et al.…”

“…Astill et al. (2021) apply the modified version of CUSUM procedure to the daily Bitcoin‐GBP price between January 1, 2017 and November 30, 2017 and identify explosive behavior in the sample period. Monschang and Wilfling (2021) explore the robustness of the PWY and PSY test and heteroscedasticity‐adjusted variants tests of Harvey et al.…”

Testing for explosive bubbles: a review

“…It turns out that the CUSUM test outperforms the ADF-type tests when the bubble emerges early in the monitoring period and the bubble episode is short. Astill et al (2021) extended the CUSUM test of Homm and Breitung (2012) to allow for time-varying volatility. They suggested to replace standard variance estimator by its non-parametric counterpart.…”

“…Table 1. Applications of tests for explosive bubbles stock markets Astill et al (2018Astill et al ( , 2017, Bohl et al (2013), Breitung and Kruse (2013), Chen and Tu (2019), Fulop and Yu (2017), Guo et al (2019), Harvey et al (2017Harvey et al ( , 2019Harvey et al ( , 2015, Kurozumi (2020a), Lin and Tu (2020), Liu and Peng (2019), Monschang and Wilfling (2020), Pavlidis et al (2017), Phillips andShi (2018, 2019), Phillips et al ( , 2015a, Shi (2013), Tao et al (2019), Wang and Yu (2017), Whitehouse (2019) prices of cryptocurrencies Astill et al (2021), Bouri et al (2019), Cheah and Fry (2015), Cheung et al (2015), Corbet et al (2019Corbet et al ( , 2018, Harvey et al (2020b,c) real estate market (housing prices) Anundsen et al (2016), Banerjee et al (2020), Caspi (2016), Chen et al (2017, Das et al (2011), Engsted et al (2016), Escobari and Jafarinejad (2016), Harvey et al (2020a), Horie and Yamamoto (2016), Kivedal (2013), Kurozumi (2020b), Pavlidis et al (2016), Pedersen and Schütte (2020), ,…”