Guide

Methodology

An asset price is explosive when it grows faster than a random walk can explain, which means the autoregressive root of the series exceeds one. This is the signature of an inflating bubble and it can be tested directly. Run the same right-tailed unit-root regression over many windows and note where the statistic crosses its critical value.

ADF

01 / 04

Augmented Dickey-Fuller

Full sample, once.

\Delta y_t = \mu + \delta y_{t-1} + \sum_{i=1}^{k} \phi_i \Delta y_{t-i} + \varepsilon_t \qquad H_0\!:\ \delta = 0 \quad\text{vs}\quad H_1\!:\ \delta > 0

The regression is the ordinary ADF regression, but the alternative is right-tailed instead of left-tailed. A standard unit-root test asks whether δ < 0, meaning the series is stationary. Bubble testing asks whether δ > 0, meaning the series is mildly explosive. Run once over the whole sample, ADF cannot say when the explosiveness occurred. Its power also collapses if the bubble has burst and the series has returned to trend by the end of the sample.

SADF

02 / 04

Supremum ADF

Origin fixed at 0. End r₂ grows from r₀ to 1.

\mathrm{SADF} = \sup_{r_2 \in [r_0,\,1]} \mathrm{ADF}_{0}^{\,r_2}

Phillips, Wu & Yu (2011) repeat the ADF regression on an expanding window that always starts at the beginning of the sample, and take the largest statistic over all window endpoints. This locates a single bubble reasonably well. Because the starting point never moves, however, a second and later bubble is diluted by the stable pre-bubble data that remain in every window, and SADF loses power.

GSADF

03 / 04

Generalized SADF

Both r₁ and r₂ float: r₂ ∈ [r₀,1], r₁ ∈ [0, r₂−r₀].

\mathrm{GSADF} = \sup_{\substack{r_2 \in [r_0,\,1] \\ r_1 \in [0,\,r_2 - r_0]}} \mathrm{ADF}_{r_1}^{\,r_2}

Phillips, Shi & Yu (2015) let the start of the window vary as well as its end. The supremum is then taken over every window that could contain a bubble, so an early bubble no longer contaminates the test for a later, separate one. The rest of exuber builds on this statistic.

BSADF

04 / 04

Backward SADF

r₂ fixed at each point in time. r₁ ∈ [0, r₂−r₀].

\mathrm{BSADF}_{r_2} = \sup_{r_1 \in [0,\,r_2 - r_0]} \mathrm{ADF}_{r_1}^{\,r_2}

BSADF takes the same supremum as GSADF but stops one step earlier. It does not take the supremum over r₂ and instead reports one statistic for each time point r₂. The result is a full time series, which is what exuber plots, and GSADF is the maximum of that series. Date-stamping also relies on this sequence.

The recursive window

Where r₁ and r₂ live.

r₀ is the minimum window fraction. If the window is shorter than this, the ADF regression has little power. Every valid GSADF window is a point in the shaded region below, with an end r₂ no earlier than r₀ and a start r₁ no later than r₂ − r₀. The dashed slice contains the windows BSADF uses to produce a single reading at r₂*.

Date-stamping

From a statistic to a calendar date.

Plot BSADF(r₂) against its critical value (which is specific to each r₂) over the whole sample, and the crossings give the bubble dates. Origination is the first r₂ at which the statistic rises above the critical value, and termination is the first later r₂ at which it falls back below. Phillips, Shi & Yu require the excursion to last at least $\delta \log(T)/T$ of the sample before it counts, so a one-period spike above the line is not mistaken for a bubble.

Threshold crossings are only one way to date an episode. exuber also provides break-point estimators that date origination and collapse more precisely (Harvey, Leybourne & Sollis; Harvey, Leybourne & Whitehouse; Pang, Du & Chong; Kejriwal, Nguyen & Perron), confidence intervals for the explosive root itself, and recovery dating. These are described under dating and root inference.

Next, read about the two settings and where the critical values come from.