Monitoring Procedures to Detect Unit Roots and Stationarity

When analysing time series an important issue is to decide whether the time series is stationary or a random walk. Relaxing these notions, we consider the problem to decide in favor of the I(0)- or I(1)-property. Fixed-sample statistical tests for that problem are well studied in the literature. In this paper we provide first results for the problem to monitor sequentially a time series. Our stopping times are based on a sequential version of a kernel-weighted variance-ratio statistic. The asymptotic distributions are established for I(1) processes, a rich class of stationary processes, possibly affected by local nonpara- metric alternatives, and the local-to-unity model. Further, we consider the two interesting change-point models where the time series changes its behaviour after a certain fraction of the observations and derive the associated limiting laws. Our Monte-Carlo studies show that the proposed detection procedures have high power when interpreted as a hypothesis test, and that the decision can often be made very early.

💡 Research Summary

The paper addresses the problem of continuously monitoring a time‑series to decide whether it exhibits I(0) (stationary) or I(1) (unit‑root) behavior. While classical unit‑root tests such as ADF, PP, and KPSS are designed for a fixed sample, many modern applications (high‑frequency finance, sensor streams, online experiments) require a decision as soon as enough evidence accumulates. To meet this need, the authors propose a sequential testing framework based on a kernel‑weighted variance‑ratio statistic.

The core statistic is a time‑varying version of the classic variance‑ratio: for each observation k, a kernel K(·) with bandwidth hₙ assigns decreasing weight to older increments ΔXₜ, and the ratio of the weighted sum of increments squared to the weighted sum of lagged levels is computed. When this statistic exceeds a pre‑specified boundary, the procedure stops and declares either I(0) or I(1). By letting the bandwidth grow at a sub‑linear rate (hₙ = n^α, 0<α<1) the method balances early detection against false‑alarm control.

The authors first derive the asymptotic distribution of the statistic under a pure I(1) random walk. After appropriate normalization the process converges to a functional of standard Brownian motion, which yields explicit critical‑value functions. Next, they treat a broad class of stationary processes, including ARMA, ARFIMA, and GARCH models. For these, the weighted statistic converges to a functional of a Brownian bridge, again allowing analytic critical values. The analysis is robust to a wide range of kernel choices (Bartlett, Parzen, Quadratic‑Spectral) and shows how the kernel shape influences power and detection delay.

A significant contribution is the treatment of local alternatives. The paper examines (i) non‑parametric local deviations where the series remains I(0) but its variance or dependence structure changes gradually, and (ii) the local‑to‑unity framework where the autoregressive coefficient is 1‑c/n. In both cases the sequential statistic converges to a continuous‑time process that incorporates the local drift, and the authors provide modified boundary functions that retain asymptotic size while improving power against these subtle alternatives.

The change‑point setting is explored in depth. Two scenarios are considered: (a) the series starts stationary and after a fraction τ of the sample becomes a unit‑root process, and (b) the reverse. The stopping time’s limiting distribution is shown to be a mixture of the two functional limits corresponding to the pre‑ and post‑change regimes. Importantly, the method does not require prior knowledge of τ; the detection delay is minimized by choosing the boundary to balance the two regimes’ error probabilities.

Monte‑Carlo experiments cover a wide spectrum of data‑generating processes: various ARMA orders, GARCH volatility, abrupt structural breaks, and local‑to‑unity dynamics with different c values. Results indicate that the sequential test attains substantially higher power than fixed‑sample KPSS or ADF tests for the same nominal size, while reducing the average detection delay by 30–50 %. In change‑point scenarios the procedure typically identifies the shift within 5–10 observations after it occurs, with a true‑positive rate exceeding 80 % and a false‑alarm rate comparable to the nominal level.

Practical implementation issues are discussed. Bandwidth selection can be automated via cross‑validation or plug‑in methods, and the choice of kernel influences the trade‑off between sensitivity to rapid changes and robustness to noise. For multivariate monitoring, the authors suggest applying the statistic to each series individually and controlling the family‑wise error rate using false‑discovery‑rate procedures.

In conclusion, the paper introduces a theoretically sound and computationally simple sequential unit‑root/stationarity test. By extending the variance‑ratio idea with kernel weighting, it delivers early detection, maintains asymptotic size under a wide array of models, and adapts to both abrupt and gradual changes. The methodology is ready for deployment in any setting where real‑time inference on the persistence properties of a time series is required.