Real-time monitoring with RCA models

Technical Summary: Real-Time Monitoring with RCA Models

Problem Statement
The paper addresses the challenge of online, real-time detection of changepoints in persistence and regime shifts (e.g., from stationarity to explosiveness or vice versa) within time series data. Traditional financial and epidemiological applications often involve periods of "tame" fluctuations followed by explosive growth (bubbles, pandemics) or sudden collapses. While existing literature offers robust ex-post detection methods and some real-time procedures based on Autoregressive (AR) models, these often rely on unweighted CUSUM statistics that do not guarantee optimal detection timing. Furthermore, many existing sequential monitoring methods for Random Coefficient Autoregressive (RCA) models assume stationarity, rendering them incapable of detecting transitions into or out of explosive regimes. The authors aim to fill this gap by developing a monitoring framework that functions irrespective of whether the underlying process is stationary, non-stationary, or explosive, and without requiring prior knowledge of the regime.

Methodology
The authors propose a family of weighted statistics based on the Cumulative Sum (CUSUM) process of Weighted Least Squares (WLS) residuals derived from an RCA model. The core model is defined as:
$$y_i = (\beta_i + \epsilon_{i,1})y_{i-1} + \epsilon_{i,2}$$
where $\beta_i$ is the autoregressive parameter. The methodology extends to include exogenous regressors ($x_i$) in a dynamic model:
$$y_i = (\beta_i + \epsilon_{i,1})y_{i-1} + \lambda_0^\top x_i + \epsilon_{i,2}$$

The detection procedure operates as follows:

1. Training Phase: A WLS estimator ($\hat{\beta}_m$) is computed using a training sample $\{y_1, \dots, y_m\}$.
2. Monitoring Phase: As new data arrives, a detector is constructed based on the weighted sum of residuals. The authors define two types of detectors:
   • Standard CUSUM: $Z_m(k) = \left| \sum_{i=m+1}^{m+k} (y_i - \hat{\beta}_m y_{i-1}) \frac{y_{i-1}}{1+y_{i-1}^2} \right|$
   • Page-CUSUM: $Z_m^\dagger(k) = \max_{1 \le \ell \le k} \left| \sum_{i=m+\ell}^{m+k} (y_i - \hat{\beta}_m y_{i-1}) \frac{y_{i-1}}{1+y_{i-1}^2} \right|$
     The Page-CUSUM is designed to search for the "worst-case scenario" at each time point to reduce detection delay.
3. Weighting Schemes: The detectors are compared against a boundary function $g_{m,\psi}(k)$ that includes a weighting parameter $\psi \in [0, 1/2]$.
   • $\psi < 1/2$: Weighted CUSUM.
   • $\psi = 1/2$: Standardised CUSUM (weights proportional to standard deviation).
4. Monitoring Horizons: The framework covers both "open-ended" monitoring (indefinite) and "closed-ended" monitoring (stopping at a pre-specified time $m^*$), including cases where $m^*$ is very short relative to the training sample ($m^* = o(m)$).

Key Contributions
The paper makes five primary contributions to the literature on online changepoint detection:

1. Closed-Ended Limiting Distributions: The authors derive the limiting distribution of the test statistics for the closed-ended case. They note that standard practice often uses open-ended critical values for closed-ended tests (a conservative choice), which leads to a loss of power. Their derived distributions allow for more precise critical values.
2. Short Monitoring Horizons: They adapt the boundary function and derive the limiting distribution for "very short" monitoring horizons ($m^* = o(m)$), a scenario previously unaddressed in the literature but practically relevant for researchers who wish to restart procedures after short intervals.
3. Improved Critical Values for Standardised CUSUM: For the case $\psi = 1/2$, the standard asymptotic critical values (based on the Extreme Value Distribution) converge slowly and are inaccurate in finite samples. The authors propose a Darling-Erdős-type approximation (Equation 3.8) that offers superior size control and accuracy even in small samples.
4. Weighted Page-CUSUM Theory: They derive the limiting distribution for the weighted Page-CUSUM statistics, demonstrating that the limit is a well-defined random variable.
5. Extension to Covariates: The theory is extended to RCA models with exogenous regressors. This is the first contribution to deal with online detection in RCA models with covariates, allowing for the inclusion of control variables (e.g., interest rates, economic indicators) without requiring modifications to the procedure based on the stationarity of the data.

Results

• Asymptotics: Under the null hypothesis of no changepoint, the authors establish weak convergence to functionals of Wiener processes. The limiting distributions are nuisance-free, allowing critical values to be obtained via simulation. Under the alternative hypothesis (a change in $\beta$), the procedures are shown to be consistent (probability of detection approaches 1) provided the magnitude of the change does not decay too rapidly.
• Simulations: Monte Carlo experiments across stationary, mildly explosive, and Stochastic Unit Root (STUR) regimes indicate:
  • The standardised CUSUM ($\psi = 1/2$) generally provides the best balance of size control and detection delay, particularly when using the proposed approximation for critical values ($\hat{c}_{\alpha, 0.5}$).
  • In explosive regimes, weighted CUSUMs with $\psi < 1/2$ tend to be less conservative than the standardised version.
  • In the presence of covariates, the inclusion of relevant variables (e.g., economic activity indices) can improve detection delays, though care must be taken with size control in STUR cases.
• Empirical Applications:
  • Covid-19 Hospitalisations: Applied to daily UK hospitalisation data, the procedure successfully detected the onset of the second wave (August 2020) and subsequent lockdowns with short delays, often preceding official announcements or ex-post detection dates.
  • Housing Prices: Applied to Los Angeles housing prices, the method detected the stabilization of prices following the 2008 crash. The inclusion of covariates (interest rates, volatility, economic activity) yielded meaningful improvements in detection timing compared to the pure RCA model.

Significance
The paper claims that its methodology is particularly suited for detecting the onset or collapse of bubbles and epidemics because it requires no prior knowledge of whether the observations are stationary or non-stationary. The ability to switch between regimes (stationary to explosive, or vice versa) without modifying the estimators or the monitoring scheme is a distinct advantage over standard AR-based approaches. By providing a unified framework that handles both open and closed monitoring, short horizons, and exogenous regressors, the authors offer a robust tool for public health authorities and policymakers to respond timely to structural breaks in persistence. The proposed approximation for critical values in the standardised case is highlighted as a practical improvement over existing asymptotic theory, ensuring better size control in finite samples.