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SVNexus · Regimes

Bubble detection (explosive-root, GSADF)

Also called: GSADF, SADF, explosive root test, Phillips–Shi–Yu test

Explosive-root tests such as SADF and GSADF look for periods where prices grow faster than a random walk allows, a statistical signature of bubbles.

Definition

Bubble detection with explosive-root tests asks whether a price series, over some window, behaves like an explosive process, growing at an accelerating rate that a random walk with drift cannot explain. Phillips, Wu and Yu (2011) introduced the SADF test; Phillips, Shi and Yu (2015) extended it to the GSADF test, which handles multiple bubbles in one sample.

The tests also date-stamp episodes, giving estimated start and end dates.

Intuition

Normal prices behave roughly like a random walk: shocks persist but do not compound. During a speculative bubble, prices rise because they have risen, and the series becomes explosive for a while. Standard unit-root tests look for mean reversion (a left-tailed alternative); these tests flip the question and look for the right tail.

Because bubbles start and collapse, a test over the whole sample would miss them. Rolling and expanding windows find the sub-periods where explosiveness is strongest.

How it's calculated

For a window running from fraction r1 to r2 of the sample, estimate the augmented Dickey–Fuller regression on log prices yt:

Δyt = α + β · yt−1 + Σi=1..p ψi · Δyt−i + εt

and record the t-statistic of β, ADFr1r2. The null is a unit root (β = 0); the alternative is explosive (β > 0).

  • SADF: fix r1 = 0 and take the supremum over r2 ≥ r0, a minimum window.
  • GSADF: take the supremum over both r1 and r2.
  • Date-stamping (BSADF): for each end point, the supremum over start points; a bubble is flagged when it exceeds its critical value.

Critical values come from simulation because the statistics do not follow standard distributions.

How to read it

A GSADF statistic above its critical value says explosive behavior occurred somewhere in the sample; the BSADF sequence shows when. Results are statements about statistical behavior, not about fundamentals: an explosive episode can reflect a genuine change in value as well as speculation.

The minimum window and lag order affect results; report them with any finding.

Common mistakes

  • Running on price levels instead of log prices.
  • Using standard Dickey–Fuller critical values instead of simulated right-tail ones.
  • Treating a flagged episode as proof of mispricing.
  • Choosing a minimum window so small that short noisy spikes pass the test.
  • Running the test on returns rather than on log price levels, which removes the explosive behavior the test is designed to detect.

In Saqrid

SVNexus runs change-point and explosive-root tests per asset to detect regime breaks and bubble-like behavior.

Open SVNexus.

  • Change-point detection (BOCPD)Change-point detection finds dates where a series' behavior shifts; Bayesian online change-point detection (BOCPD) tracks the current regime's age.
  • Market regime detectionMarket regime detection classifies periods into states such as calm, trending or stressed, so analysis and risk can adapt to conditions.
  • Hurst exponentThe Hurst exponent measures long-range memory in a time series: about 0.5 for a random walk, above for persistent trends, below for mean reversion.
  • Log returnsA log return is the natural log of the price ratio between two dates; log returns add across time, which makes them the default unit in quant work.

Go deeper

Saqrid Institute lesson: Regime Detection: BOCPD, Hurst & Bubble Tests. Lesson text is open to read without an account.

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