Wavelets are wavelike functions used to transform signals into a representation which has time and frequency domain components. Visually wavelets appear as wave-like oscillations with an amplitude that starts at zero, increases, then returns to zero. Another way to consider a wavelet is as a bandpass filter, which can be applied to a time series under investigation, letting through only components of the time series within a finite range of frequencies to different extents depending on the energy spectrum of the wavelet. Wavelets take the form:
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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
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