In order to provide a methodology to detect bubbles in time series, Phillips, Wu, and Yu [23] proposed the supremum augmented Dickey-Fuller (SADF) test. This applies a series of right-tailed unit root tests to expanding windows of a time series (with a fixed start date), defined by where r2 is the final data point to be considered in each window, starting at r0 which is a fraction representing the smallest allowed window size and expanding to 1(the complete data set).
Although this test successfully detects single isolated bubbles, Phillips, Shi, and Yu [24] acknowledge it may suffer from reduced discriminatory power when applied to time series with multiple occurrences of bubbles. To overcome this weakness, a further enhancement was proposed, as a new method, called a generalized supremum ADF (GSADF) test. This test allows both the start and end points of data subsets to vary, which in turn enables the identification of multiple bubble regimes in one observed time series. The GSADF test is defined by

“Expect very high volatility in crypto over the next few months,” he tweeted on Oct. 21.
Bitcoin is widely regarded as the first modern cryptocurrency — the first publicly used means of exchange to combine decentralized control, user anonymity, record-keeping via a blockchain, and built-in scarcity.