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.

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Lobato and Robinson (1998) develop semiparametric tests for the null hypothesis that a series is weakly autocorrelated, or I(0), about a constant level, against fractionally integrated alternatives. These tests have the advantage that the user is not required to specify a parametric model for any weak autocorrelation present in the series. We extend this approach in two distinct ways. First we show that it can be generalised to allow for testing of the null hypothesis that a series is I(δ) for any δ lying in the usual stationary and invertible region of the parameter space. The second extension is the more substantive and addresses the well known issue in the literature that long memory and level breaks can be mistaken for one another, with unmodelled level breaks rendering fractional integration tests highly unreliable. To deal with this inference problem we extend the Lobato and Robinson (1998) approach to allow for the possibility of changes in level at unknown points in the series. We show that the resulting statistics have standard limiting null distributions, and that the tests based on these statistics attain the same asymptotic local power functions as infeasible tests based on the unobserved errors, and hence there is no loss in asymptotic local power from allowing for level breaks, even where none is present. We report results from a Monte Carlo study into the finite-sample behaviour of our proposed tests, as well as several empirical examples.