Abstract
We generalize the Homm and Breitung (2012) CUSUM-based procedure for the realtime 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. The presence of historical asset price bubbles, in which asset prices rise well above their fundamental value at a particular point in time, is widely documented. Well-known examples include the South Sea bubble of 1720, the Dot-Com bubble that originated in the mid-1990s, and the U.S. housing market bubble of the late 1990s and early 2000s, while the Bitcoin price has been argued to constitute a more recent example. In all instances, asset prices, having risen to unsustainable levels, were subject to large crashes, causing significant economic damage. Given the damage caused by their collapse, it is of vital importance for policy makers to be able to identify asset price bubbles as they occur to attempt to limit their economic damage. Accordingly, a large literature has developed in the last decade or so around testing for the presence of historical explosive rational asset bubbles in price series. An early contribution is Diba and Grossman (1988) who applied orthodox left-tailed unit root tests (i.e., tests against stationary autoregressive alternatives) to the price and dividend series in levels and first-differenced forms to investigate the presence of asset bubbles in stock price data. They adopt this approach based on the observation that if the bubble component of the stock price evolves as an explosive autoregressive process then, as an explosive autoregressive process cannot be differenced to stationarity, a finding of non-stationarity for the price and dividend series when the series are in levels, but stationarity when the series are in first differences, is indicative that an explosive rational bubble does not exist. However, Evans (1991) argues that the tests adopted in Diba and Grossman (1988) will have little, if any, power to detect periodically collapsing bubbles. Consequently, the recent focus in the literature has been on the use of right-tailed unit root tests, that is, tests against explosive autoregressive alternatives, applied to the levels of a series. The first such contribution was made by Phillips, Wu, and Yu (2011) , who developed a test of the null of no explosive behavior against the alternative of explosivity based on a sequence of forward recursive right-tailed augmented Dickey-Fuller (DF) statistics. Further contributions using sub-sample testing methods have been developed in Homm and Breitung (2012) , Harvey, Leybourne, and Sollis (2015) , Harvey et al. (2016) , Harvey, Leybourne, and Zu (2019, 2020) , Phillips, Shi, and Yu (2015) , Astill et al. (2017) , Phillips and Shi (2018) , among others. Applications of these methods have uncovered evidence of historical asset price bubbles in stock prices, commodities futures prices, real estate prices, exchange rates, and many other price series; see Homm and Breitung (2012) for a detailed review. A feature of the procedures outlined above, however, is that they are designed to detect speculative bubbles within a fixed historical dataset. In practice, it would seem to be of much greater practical relevance to sequentially monitor for the emergence of an asset price bubble as new data points are obtained using a real-time monitoring procedure. While sequential application of the tests of Phillips, Shi, and Yu (2015) or Astill et al. (2017) , both of which are designed to detect an end-of-sample explosive autoregressive episode, could be used to do this, one could not use the critical values appropriate for their use as one-shot tests in such a monitoring exercise as these would not be size controlled. In particular, the overall false-positive rate (FPR) of such a procedure would be unknown and, as discussed in the context of a generic monitoring exercise in Chu, Stinchcombe, and White (1996) and also for the specific case of monitoring for the emergence of an explosive episode in Homm and Breitung (2012) and Astill et al. (2018) , would increase monotonically as the monitoring horizon grows. Here, we define the FPR as the probability of at least one test in the monitoring sequence rejecting when the null was true and, hence, no explosive episode was present. Homm and Breitung (2012) and Astill et al. (2018) develop real-time monitoring procedures for explosive episodes which are such that the theoretical FPR can be controlled by the practitioner. Astill et al. (2018) develop a real-time monitoring procedure based on sequential application of the end of sample test of Astill et al. (2017) . Their preferred procedure signals the presence of an explosive episode if any statistic in the monitoring period exceeds the largest value of the statistic calculated over a training period of data. Homm and Breitung (2012) propose two real-time monitoring procedures, one based on standard cumulative sum (CUSUM) statistics and the other on (unaugmented) DF unit root statistics. Adopting the methodology of Chu, Stinchcombe, and White (1996) , the CUSUM and DF statistics are calculated sequentially across a given monitoring period with a decision rule designed to control the theoretical FPR of the procedure. A key assumption underlying the large sample validity of the real-time monitoring procedures of Homm and Breitung (2012) is that the shocks driving the series being monitored are unconditionally homoskedastic. This assumption is not innocuous and indeed is likely to be infeasible for many financial price series which display clear patterns of time-varying volatility. In particular, many applied studies have found strong evidence of structural breaks in the unconditional variance of asset returns, often linked to major financial and macroeconomic crises such as the 1970s oil price shocks, the East Asian currency crisis in the late-1990s, the dot-com crash in 2001, and the recent global financial crisis in [2007] [2008] [2009] . In a number of these studies very large structural breaks have been detected; for example, Rapach, Strauss, and Wohar (2008) and McMillan and Wohar (2011) detect breaks in the unconditional variance of the returns of some major stock market indices and sectoral stock price indices, finding that the unconditional variance in some sub-samples can be larger than that in other sub-samples by a factor of about 10. For commodity returns, both Calvo-Gonzalez, Shankar, and Trezzi (2010) and Vivian and Wohar (2012) find statistically significant evidence of structural breaks in unconditional volatility. Volatility changes in innovations to price series processes could be induced by the presence of a speculative bubble, but equally it could be the case that changes in volatility occur without an explosive bubble period being present. It is therefore important to develop reliable methods for detecting an emerging explosive period in a series that is robust to the presence of timevarying volatility. Using Monte Carlo simulation, Astill et al. (2018) show that the empirical FPR of the CUSUM-based procedure of Homm and Breitung (2012) cannot be adequately controlled in the presence of time-varying volatility and can differ quite drastically from the theoretical FPR which obtains under homoskedasticity. In contrast they show that the empirical FPR of their maximum-based procedure is robust to a wide range of time-varying patterns of volatility. However, as we show in the simulation results in this paper, in the case where the innovations are homoskedastic, such that its FPR is controlled, the CUSUM-based procedure displays a very clear advantage over the procedure of Astill et al. (2018) in terms of its empirical true positive rate (TPR) to detect an emergent explosive episode, where the TPR is defined as the probability of at least one test in the monitoring sequence rejecting when an explosive period is present. Given that our aim is to develop real-time monitoring procedures which have both a controlled FPR and strong power to detect an emerging explosive episode, it therefore seems worthwhile developing a heteroskedasticity-robust version of the CUSUM-based procedure. To that end, we propose a modification to the CUSUM-based procedure which replaces the standard full sample first-difference-based variance estimate used by Homm and Breitung (2012) in calculating the CUSUM statistics with a nonparametric kernel-based spot variance estimate, designed to model the unknown variance path of the underlying innovations. Under quite general conditions we show that the resulting 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, as a result, this delivers a real-time monitoring procedure whose theoretical FPR is controlled. Indeed, these quantities are shown to coincide with those which obtain for the standard CUSUM procedure in the case of homoskedastic innovations. Monte Carlo methods are used to examine the empirical FPR and TPR of our proposed monitoring procedure. These results show that the empirical FPR of the modified procedure is well controlled in practice. Moreover, the efficacy of the modified procedure to detect an explosive episode, as measured by the empirical TPR, is shown to be little altered in the homoskedastic case, so that the cost (in terms of ability to detect an emerging explosive episode) of this additional robustness to time-varying volatility appears relatively small. We also show here that the presence of an explosive episode prior to the start of the monitoring period has little impact on the properties of our modified CUSUM procedure but can very substantially lower the empirical TPR of both the CUSUM-based procedure and the procedure of Astill et al. (2018) . The remainder of the paper is organized as follows. Section 1 outlines the autoregressive data generating process (DGP) we work with and outlines the assumptions under which our analysis will be conducted. In Section 2, we briefly review the CUSUM-based procedure of Homm and Breitung (2012) and demonstrate that it does not, in general, have a controlled FPR when time-varying volatility is present in the innovations. We then outline our modified CUSUM procedure and establish the large sample validity of this procedure. Issues concerning its practical implementation, including the selection of the bandwidth and kernel used in the context of the nonparametric spot variance estimator, are also discussed in this section. Our Monte Carlo study is reported in Section 3. An empirical illustration of our modified CUSUM monitoring procedure, using Bitcoin price data, is provided in Section 4. Section 5 concludes.
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