Detecting Structural Breaks and Explosive Dynamics

A long financial series may not follow one stable law. The notebooks use recursive right-tailed unit-root statistics to mark explosive episodes and PELT to divide a series into segments with different behavior.

The central relation

The right-tailed augmented Dickey-Fuller regression is

\Delta y_t=\alpha+\beta y_{t-1} +\sum_{j=1}^{k}\psi_j\Delta y_{t-j}+\varepsilon_t, \qquad H_0:\beta=0,\qquad H_1:\beta>0.

$y_t$ is the logged level, $\Delta y_t$ is its change, $k$ is the lag order, and $\psi_j$ are lag coefficients. The test asks whether $\beta$ lies in the explosive right tail rather than below zero.

Locating changes in financial time series

A controlled two-bubble path checks episode detection against known intervals. The public segmentation shows the dates selected by the PELT objective.

A controlled series with two explosive episodes and the intervals detected by recursive statistics. — The controlled path provides known explosive intervals, allowing the recursive test to be checked for missed and extra detections.

Public financial observations divided into segments by the PELT change-point method. — The PELT result divides the public series according to the chosen cost and penalty. The dates describe this fitted segmentation rather than a universal market chronology.

Break dates depend on the statistic, lag choice, minimum segment length, and penalty. The notebooks make those choices visible and test them under controlled cases.

Notebooks


1. Recursive Right-Tailed Unit-Root Statistics
2. Detecting explosive episodes with right-tailed tests
3. Detecting change points with PELT