We propose a sequential monitoring scheme to detect changes in dynamic semiparametric risk models that capture Value-at-Risk (VaR) and Expected Shortfall (ES) jointly. The monitoring scheme is based on a gradient-based detector and a boundary function, and a change is detected when the detector crosses the boundary function. We derive the asymptotic limit of the stopping time of detection under the null hypothesis of no change. Monte Carlo simulations show that the proposed test has reasonable size control under the null hypothesis and high power under alternative hypotheses of various change point scenarios in finite samples. Empirical applications based on the S&P 500 index and the GBP/EUR exchange rate illustrate that our proposed test is able to detect change points in real-time.

We develop a new test procedure for detecting changes in the distribution of the errors in (dynamic) linear models. Our framework accommodates misspecification of the dynamic linear model, thereby allowing for the inclusion of lagged dependent variables as regressors and autocorrelated errors. Under the null hypothesis, the distribution of the errors remains the same throughout the sample period, while there are multiple changes in the distribution of the errors under the alternative. Our procedure is based on the cumulative sum (CUSUM) process that compares the empirical distribution functions of the residuals in the first part of the observations and the whole sample. We derive the asymptotic properties of the proposed test statistics. Monte Carlo simulations show that the proposed test has good size control and high power. We provide empirical applications to Phillips curves and capital asset pricing models.

We propose an online testing framework for detecting distributional changes in serially dependent data with values in a separable metric space. Based on two-sample U-statistics, the framework encompasses sequential analogs of energy distance and maximum mean discrepancy (MMD) procedures while accommodating temporal dependence. We establish asymptotic theory for finite and open-ended monitoring horizons that characterizes the full asymptotic run-length distribution under H_0 and yields asymptotic false-alarm control. We further establish new spectral approximation results for kernel matrices formed from serially dependent observations, and use them to construct a feasible Monte Carlo calibration procedure. Our flexible window construction encompasses classical, Page-type, and full-scan historical-baseline monitoring and can achieve short detection delays for both early and late changepoints, without requiring sub-Gaussianity or high-order moments of the raw observations. Simulations show reliable false-alarm control across linear, nonlinear, high-dimensional, and functional time-series models and further demonstrate that, over a broad range of alternatives and changepoint locations, the proposed method can achieve substantially shorter delays than recent procedures designed specifically for rapid detection. Applications to foreign exchange rates, electricity-market curves, and daily air transportation networks illustrate the methodology across scalar, functional, and network-valued time series.

We consider the problem of estimating and deriving confidence intervals for change points in linear models with heteroscedastic errors. A CUSUM process-based estimator is proposed, and we establish its asymptotic properties when the linear regression model exhibits change points in both the regression parameters and the distribution of the errors. This theory motivates the construction of confidence sets for multiple change points by refining preliminary change point estimators and approximating their distribution in a way that is robust to heteroscedasticity. Monte Carlo experiments indicate that the proposed confidence intervals achieve accurate empirical coverage for change-point locations under both homoscedastic and heteroscedastic error structures. In two data applications, we apply the proposed confidence intervals to examine changes in the flattening of the New Keynesian Phillips curve and in cryptocurrency risk factors.

We propose a novel family of test statistics to detect the presence of changepoints in a sequence of dependent, possibly multivariate, functional-valued observations. Our approach allows to test for a very general class of changepoints, including the "classical" case of changes in the mean, and even changes in the whole distribution. Our statistics are based on a generalisation of the empirical energy distance; we propose weighted functionals of the energy distance process, which are designed in order to enhance the ability to detect breaks occurring at sample endpoints. The limiting distribution of the maximally selected version of our statistics requires only the computation of the eigenvalues of the covariance function, thus being readily implementable in the most commonly employed packages, e.g. R. We show that, under the alternative, our statistics are able to detect changepoints occurring even very close to the beginning/end of the sample. In the presence of multiple changepoints, we propose a binary segmentation algorithm to estimate the number of breaks and the locations thereof. Simulations show that our procedures work very well in finite samples. We complement our theory with applications to financial and temperature data.

The problem of detecting change points in the parameters of a linear regression model with errors and covariates exhibiting heteroscedasticity is considered. Asymptotic results for weighted functionals of the cumulative sum (CUSUM) processes of model residuals are established when the model errors are weakly dependent and non-stationary, allowing for either abrupt or smooth changes in their variance. These theoretical results illuminate how to adapt standard change point test statistics for linear models to this setting. We studied such adapted change-point tests in simulation experiments, along with a finite sample adjustment to the proposed testing procedures. The results suggest that these methods perform well in practice for detecting multiple change points in the linear model parameters and controlling the Type I error rate in the presence of heteroscedasticity. We illustrate the use of these approaches in applications to test for instability in predictive regression models and explanatory asset pricing models.

We propose a family of weighted statistics based on the CUSUM process of the WLS residuals for the online detection of changepoints in a Random Coefficient Autoregressive model, using both the standard CUSUM and the Page-CUSUM process. We derive the asymptotics under the null of no changepoint for all possible weighing schemes, including the case of the standardised CUSUM, for which we derive a Darling-Erdos-type limit theorem; our results guarantee the procedure-wise size control under both an open-ended and a closed-ended monitoring. In addition to considering the standard RCA model with no covariates, we also extend our results to the case of exogenous regressors. Our results can be applied irrespective of (and with no prior knowledge required as to) whether the observations are stationary or not, and irrespective of whether they change into a stationary or nonstationary regime. Hence, our methodology is particularly suited to detect the onset, or the collapse, of a bubble or an epidemic. Our simulations show that our procedures, especially when standardising the CUSUM process, can ensure very good size control and short detection delays. We complement our theory by studying the online detection of breaks in epidemiological and housing prices series.

This article develops a new test to detect changes in generalized autoregressive conditionally heteroscedastic (GARCH(1,1)) processes without imposing a stationary assumption. Specifically, the procedure tests the null hypothesis of a GARCH process with constant parameters, either in (strictly) stationary or explosive regimes, against the alternative hypothesis of parameter changes. We derive the limiting distribution of the test statistics and establish their asymptotic consistency. Monte Carlo simulations show that the proposed test has good size control and high power. We demonstrate a prototype application on a small group of stocks and report a further extensive application to more than ten thousand U.S. stocks.

We develop a new method to detect change points in the distribution of functional data based on integrated CUSUM processes of empirical characteristic functionals. Asymptotic results are presented under conditions allowing for low-order moments and serial dependence in the data establishing the limiting null-distribution of the proposed test statistics, as well as their consistency to detect and localize change points in the distribution of functional data. A key consideration in defining these test statistics is the measure used to integrate the CUSUM process over function space. We show that using a measure generated by Brownian motion leads to generally consistent tests. Further, using this measure allows for computationally simple approximations of the necessary integrals, as well as simulation and permutation-based methods to calibrate detection thresholds for change point analysis. The proposed methods are thoroughly investigated and compared to other existing functional data change point methods in simulation experiments, and are further applied to detect change points in models for continuous electricity demand and high-frequency asset price returns.

We investigate the online detection of changepoints in the distribution of a sequence of observations using a class of degenerate U-statistic-type processes. We consider an ordinary (Chu–Stinchcombe–White-type) detector and a Page-type detector under open- and closed-ended monitoring, and introduce an expanding-baseline Page-type procedure that incorporates sufficiently old monitoring observations into the baseline sample. Under the null, we derive weak limits for all three procedures and justify a Monte Carlo approximation to their critical values. For the ordinary and Page-type detectors, we also establish consistency and limiting distributions for detection delays under both early and late changes. The theory requires only square summability of the eigenvalues associated with the degenerate kernel operator, rather than the stronger absolute-summability condition often imposed in related work. Simulations show competitive performance relative to recent mean-, covariance-, and empirical-CDF-based monitors, and an application to multivariate compressor-sensor data from a metro train illustrates the methodology.

ABSTRACT In this article, we develop two families of sequential monitoring procedure to (timely) detect changes in the parameters of a GARCH(1,1) model. Our statistics can be applied irrespective of whether the historical sample is stationary or not, and indeed without previous knowledge of the regime of the observations before and after the break. In particular, we construct our detectors as the CUSUM process of the quasi‐Fisher scores of the log likelihood function. To ensure timely detection, we then construct our boundary function (exceeding which would indicate a break) by including a weighting sequence which is designed to shorten the detection delay in the presence of a changepoint. We consider two types of weights: a lighter set of weights, which ensures timely detection in the presence of changes occurring “early, but not too early” after the end of the historical sample; and a heavier set of weights, called “Rényi weights” which is designed to ensure timely detection in the presence of changepoints occurring very early in the monitoring horizon. In both cases, we derive the limiting distribution of the detection delays, indicating the expected delay for each set of weights. Our methodologies can be applied for a general analysis of changepoints in GARCH(1,1) sequences; however, they can also be applied to detect changes from stationarity to explosivity or vice versa, thus allowing to check for “volatility bubbles”, upon applying tests for stationarity before and after the identified break. Our theoretical results are validated via a comprehensive set of simulations, and an empirical application to daily returns of individual stocks.

We propose a new functional change point detection procedure, motivated by recent models for commodity futures term structure. We investigate our procedure's properties under the null hypothesis of no change and the alternative. Monte Carlo simulations reveal a reasonable power property in finite sample sizes although size distortion persists. An empirical analysis of oil futures markets identifies two change points near the 2008 financial crisis and 2020 crude oil negative territory. Regression models show that the price behaviour, in general, is exposed to the spot market index and exchange rate from 2007 to 2009. The main drivers of the price term structure are attributed to the trading activities of speculators and financial index innovations between 2017 and 2022.

In this paper, we develop two families of sequential monitoring procedure to (timely) detect changes in a GARCH(1,1) model. Whilst our methodologies can be applied for the general analysis of changepoints in GARCH(1,1) sequences, they are in particular designed to detect changes from stationarity to explosivity or vice versa, thus allowing to check for volatility bubbles. Our statistics can be applied irrespective of whether the historical sample is stationary or not, and indeed without prior knowledge of the regime of the observations before and after the break. In particular, we construct our detectors as the CUSUM process of the quasi-Fisher scores of the log likelihood function. In order to ensure timely detection, we then construct our boundary function (exceeding which would indicate a break) by including a weighting sequence which is designed to shorten the detection delay in the presence of a changepoint. We consider two types of weights: a lighter set of weights, which ensures timely detection in the presence of changes occurring early, but not too early after the end of the historical sample; and a heavier set of weights, called Renyi weights which is designed to ensure timely detection in the presence of changepoints occurring very early in the monitoring horizon. In both cases, we derive the limiting distribution of the detection delays, indicating the expected delay for each set of weights. Our theoretical results are validated via a comprehensive set of simulations, and an empirical application to daily returns of individual stocks.

We consider a regression model with autoregressive terms and propose significance tests for the detection of change points in this model. Our tests are applicable to both low- or moderate dimension and to high-dimension with sparse regressors. The dimension may be high from the practical point of view of economic and business applications, but in our theoretical framework it is fixed. To accommodate practically high dimension, variable selection is incorporated as an integral part of our approach. The regressors and the errors can exhibit general nonlinear dependence and the model incorporates autoregressive dependence. We develop asymptotic justification and evaluate the performance of the tests both on simulated and real economic data. We test for and estimate changes in responses to risk factors of a U.S. energy stocks portfolio and the Industrial Production index. We relate our findings to macroeconomic policy changes and global impact events.

In a recent contribution, we developed a family of cumulative sum-based change-point tests in the context of a random coefficient autoregressive model of order 1. In the current paper, we complement the results in that contribution by studying the (maximally selected) likelihood ratio statistic. We show that this has power versus breaks occurring even as close as periods from the beginning/end of sample; moreover, the use of quasi-maximum likelihood-based estimates yields better power properties, with the added bonus of being nuisance-free. Our test statistic has the same distribution-of the Darling-Erd & odblac;s type-irrespective of whether the data are stationary or not, and can therefore be applied with no prior knowledge of this. Our simulations show that our test has very good power and, when applying a suitable correction to the asymptotic critical values, the correct size. We illustrate the usefulness and generality of our approach through applications to economic and epidemiological time series.