Simultaneous Detection of Structural Breaks and Outliers in Time Series

Journal of Time Series Analysis · 2025-07-24 · 1 citations

ABSTRACT This article considers the problem of modeling a class of nonstationary time series using piecewise autoregressive (AR) processes in the presence of outliers. The number and locations of the piecewise AR segments, as well as the orders of the respective AR processes, are assumed to be unknown. In addition, each piece may contain an unknown number of innovational and/or additive outliers. The minimum description length (MDL) principle is applied to compare various segmented AR fits to the data. The goal is to find the “best” combination of the number of segments, the lengths of the segments, the orders of the piecewise AR processes, and the number and type of outliers. Such a “best” combination is implicitly defined as the optimizer of an MDL criterion. Since the optimization is carried over a large number of configurations of segments and positions of outliers, a genetic algorithm is used to find optimal or near‐optimal solutions. Numerical results from simulation experiments and real data analyses show that the procedure enjoys excellent empirical properties.

Mathematics, Statistics, and Geometry of Extreme Events in High Dimensions

Oberwolfach Reports · 2025-02-14

The workshop brought together researchers contributing to various recent topics in Extreme Value Theory. Discussions and talks included recent probabilistic development in the theory of regular variation, advances in multivariate representations compatible with sparsity structures, statistical inference in both high dimensional and time series frameworks, and novel applications and emerging directions that leverage recent advances in deep learning.

Sample Splitting and Assessing Goodness-of-fit of Time Series

arXiv (Cornell University) · 2024-03-11

A fundamental and often final step in time series modeling is to assess the quality of fit of a proposed model to the data. Since the underlying distribution of the innovations that generate a model is often not prescribed, goodness-of-fit tests typically take the form of testing the fitted residuals for serial independence. However, these fitted residuals are inherently dependent since they are based on the same parameter estimates and thus standard tests of serial independence, such as those based on the autocorrelation function (ACF) or distance correlation function (ADCF) of the fitted residuals need to be adjusted. The sample splitting procedure in Pfister et al.~(2018) is one such fix for the case of models for independent data, but fails to work in the dependent setting. In this paper sample splitting is leveraged in the time series setting to perform tests of serial dependence of fitted residuals using the ACF and ADCF. Here the first $f_n$ of the data points are used to estimate the parameters of the model and then using these parameter estimates, the last $l_n$ of the data points are used to compute the estimated residuals. Tests for serial independence are then based on these $l_n$ residuals. As long as the overlap between the $f_n$ and $l_n$ data splits is asymptotically 1/2, the ACF and ADCF tests of serial independence tests often have the same limit distributions as though the underlying residuals are indeed iid. In particular if the first half of the data is used to estimate the parameters and the estimated residuals are computed for the entire data set based on these parameter estimates, then the ACF and ADCF can have the same limit distributions as though the residuals were iid. This procedure ameliorates the need for adjustment in the construction of confidence bounds for both the ACF and ADCF in goodness-of-fit testing.

Discrete Extremes

Journal of Data Science · 2024-01-01 · 6 citations

Our contribution is to widen the scope of extreme value analysis applied to discrete-valued data. Extreme values of a random variable are commonly modeled using the generalized Pareto distribution, a peak-over-threshold method that often gives good results in practice. When data is discrete, we propose two other methods using a discrete generalized Pareto and a generalized Zipf distribution respectively. Both are theoretically motivated and we show that they perform well in estimating rare events in several simulated and real data cases such as word frequency, tornado outbreaks and multiple births.

Time series estimation of the dynamic effects of disaster-type shocks

Journal of Econometrics · 2022 · 24 citations

• Econometrics
• Mathematics
• Economics

COVID-19 cases and deaths in the United States follow Taylor’s law for heavy-tailed distributions with infinite variance

Proceedings of the National Academy of Sciences · 2022-09-12 · 10 citations

The spatial and temporal patterns of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) cases and COVID-19 deaths in the United States are poorly understood. We show that variations in the cumulative reported cases and deaths by county, state, and date exemplify Taylor's law of fluctuation scaling. Specifically, on day 1 of each month from April 2020 through June 2021, each state's variance (across its counties) of cases is nearly proportional to its squared mean of cases. COVID-19 deaths behave similarly. The lower 99% of counts of cases and deaths across all counties are approximately lognormally distributed. Unexpectedly, the largest 1% of counts are approximately Pareto distributed, with a tail index that implies a finite mean and an infinite variance. We explain why the counts across the entire distribution conform to Taylor's law with exponent two using models and mathematics. The finding of infinite variance has practical consequences. Local jurisdictions (counties, states, and countries) that are planning for prevention and care of largely unvaccinated populations should anticipate the rare but extremely high counts of cases and deaths that occur in distributions with infinite variance. Jurisdictions should prepare collaborative responses across boundaries, because extremely high local counts of cases and deaths may vary beyond the resources of any local jurisdiction.

Indirect inference for time series using the empirical characteristic function and control variates

Journal of Time Series Analysis · 2021-01-04 · 1 citations

We estimate the parameter of a stationary time series process by minimizing the integrated weighted mean squared error between the empirical and simulated characteristic function, when the true characteristic functions cannot be explicitly computed. Motivated by Indirect Inference, we use a Monte Carlo approximation of the characteristic function based on i.i.d. simulated blocks. As a classical variance reduction technique, we propose the use of control variates for reducing the variance of this Monte Carlo approximation. These two approximations yield two new estimators that are applicable to a large class of time series processes. We show consistency and asymptotic normality of the parameter estimators under strong mixing, moment conditions, and smoothness of the simulated blocks with respect to its parameter. In a simulation study we show the good performance of these new simulation based estimators, and the superiority of the control variates based estimator for Poisson driven time series of counts.

Handling missing extremes in tail estimation

Extremes · 2021-12-23 · 2 citations

Time Series Estimation of the Dynamic Effects of Disaster-Type Shock

arXiv (Cornell University) · 2021-07-14 · 1 citations

This paper provides three results for SVARs under the assumption that the primitive shocks are mutually independent. First, a framework is proposed to accommodate a disaster-type variable with infinite variance into a SVAR. We show that the least squares estimates of the SVAR are consistent but have non-standard asymptotics. Second, the disaster shock is identified as the component with the largest kurtosis and whose impact effect is negative. An estimator that is robust to infinite variance is used to recover the mutually independent components. Third, an independence test on the residuals pre-whitened by the Choleski decomposition is proposed to test the restrictions imposed on a SVAR. The test can be applied whether the data have fat or thin tails, and to over as well as exactly identified models. Three applications are considered. In the first, the independence test is used to shed light on the conflicting evidence regarding the role of uncertainty in economic fluctuations. In the second, disaster shocks are shown to have short term economic impact arising mostly from feedback dynamics. The third uses the framework to study the dynamic effects of economic shocks post-covid.

Goodness-of-fit testing for time series models via distance covariance

Journal of Econometrics · 2020-08-10 · 9 citations