Heteroscedastic Time Series Research Articles

A general framework for multi-step ahead adaptive conformal heteroscedastic time series forecasting

This paper introduces a novel model-agnostic algorithm called adaptive ensemble batch multi-input multi-output conformalized quantile regression (AEnbMIMOCQR) that enables forecasters to generate multi-step ahead prediction intervals for a fixed pre-specified miscoverage rate α in a distribution-free manner. Our method is grounded on conformal prediction principles, however, it does not require data splitting and provides close to exact coverage even when the data is not exchangeable. Moreover, the resulting prediction intervals, besides being empirically valid along the forecast horizon, do not neglect heteroscedasticity. AEnbMIMOCQR is designed to be robust to distribution shifts, which means that its prediction intervals remain reliable over an unlimited period of time, without entailing retraining or imposing unrealistic strict assumptions on the data-generating process. Through methodically experimentation, we demonstrate that our approach outperforms other competitive methods on both real-world and synthetic datasets. The code used in the experimental part and a tutorial on how to use AEnbMIMOCQR can be found at the following GitHub repository: https://github.com/Quilograma/AEnbMIMOCQR.

Ensemble Conformalized Quantile Regression for Probabilistic Time Series Forecasting.

This article presents a novel probabilistic forecasting method called ensemble conformalized quantile regression (EnCQR). EnCQR constructs distribution-free and approximately marginally valid prediction intervals (PIs), which are suitable for nonstationary and heteroscedastic time series data. EnCQR can be applied on top of a generic forecasting model, including deep learning architectures. EnCQR exploits a bootstrap ensemble estimator, which enables the use of conformal predictors for time series by removing the requirement of data exchangeability. The ensemble learners are implemented as generic machine learning algorithms performing quantile regression (QR), which allow the length of the PIs to adapt to local variability in the data. In the experiments, we predict time series characterized by a different amount of heteroscedasticity. The results demonstrate that EnCQR outperforms models based only on QR or conformal prediction (CP), and it provides sharper, more informative, and valid PIs.

Nonnegative GARCH-type models with conditional Gamma distributions and their applications

Most of real data are characterized by positive, asymmetric and skewed distributions of various shapes. Modelling and forecasting of such data are addressed by proposing nonnegative conditional heteroscedastic time series models with Gamma distributions. Three types of time-varying parameters of Gamma distributions are adopted to construct the nonnegative GARCH models. A condition for the existence of a stationary Gamma-GARCH model is given. Parameter estimates are discussed via maximum likelihood estimation (MLE) method. A Monte-Carlo study is conducted to illustrate sample paths of the proposed models and to see finite-sample validity of the MLEs, as well as to evaluate model diagnostics using standardized Pearson residuals. Furthermore, out-of-sample forecasting analysis is performed to compute forecasting accuracy measures. Applications to oil price and Bitcoin data are given, respectively.

Sequential monitoring for conditional quantiles of general conditional heteroscedastic time series models

AbstractIn this study, we introduce an online monitoring procedure designed to sequentially detect change points in the conditional quantiles of location‐scale time series models. This statistical process control issue holds great significance in risk management, particularly in measuring the value‐at‐risk or expected shortfall of financial assets. Our approach employs suitable detectors, including cumulative sum statistics. We then define a stopping rule and determine control limits based on asymptotic theorems to signal an anomaly. To further evaluate the proposed methods, we conduct a comprehensive empirical study analyzing various aspects of our monitoring procedures when applied to location‐scale time series models. Additionally, we perform a real data analysis using the daily returns of the Korea Composite Stock Price Index (KOSPI) and EuroStoxx 50 indices to affirm the adequacy of the proposed monitoring procedures in real‐world applications.

Robust control chart for nonlinear conditionally heteroscedastic time series based on Huber support vector regression.

This study proposes a control chart that monitors conditionally heteroscedastic time series by integrating the Huber support vector regression (HSVR) and the one-class classification (OCC) method. For this task, we consider the model that incorporates nonlinearity to the generalized autoregressive conditionally heteroscedastic (GARCH) time series, named HSVR-GARCH, to robustly estimate the conditional volatility when the structure of time series is not specified with parameters. Using the squared residuals, we construct the OCC-based control chart that does not require any posterior modifications of residuals unlike previous studies. Monte Carlo simulations reveal that deploying squared residuals from the HSVR-GARCH model to control charts can be immensely beneficial when the underlying model becomes more complicated and contaminated with noises. Moreover, a real data analysis with the Nasdaq composite index and Korea Composite Stock Price Index (KOSPI) datasets further disclose the validity of using the bootstrap method in constructing control charts.

Test for conditional quantile change in general conditional heteroscedastic time series models

Test for conditional quantile change in general conditional heteroscedastic time series models

Weighted weak convergence of the sequential tail empirical process for heteroscedastic time series with an application to extreme value index estimation

The sequential tail empirical process is analyzed in a stochastic model allowing for serially dependent observations and heteroscedasticity of extremes in the sense of Einmahl et al. (J. R. Stat. Soc. Ser. B. Stat. Methodol. 78(1), 31–51, 2016). Weighted weak convergence of the sequential tail empirical process is established. As an application, a central limit theorem for an estimator of the extreme value index is proven.

Quantile autoregressive conditional heteroscedasticity

Abstract This article proposes a novel conditional heteroscedastic time series model by applying the framework of quantile regression processes to the ARCH(∞) form of the GARCH model. This model can provide varying structures for conditional quantiles of the time series across different quantile levels, while including the commonly used GARCH model as a special case. The strict stationarity of the model is discussed. For robustness against heavy-tailed distributions, a self-weighted quantile regression (QR) estimator is proposed. While QR performs satisfactorily at intermediate quantile levels, its accuracy deteriorates at high quantile levels due to data scarcity. As a remedy, a self-weighted composite quantile regression estimator is further introduced and, based on an approximate GARCH model with a flexible Tukey-lambda distribution for the innovations, we can extrapolate the high quantile levels by borrowing information from intermediate ones. Asymptotic properties for the proposed estimators are established. Simulation experiments are carried out to access the finite sample performance of the proposed methods, and an empirical example is presented to illustrate the usefulness of the new model.

Bootstrapping tests for breaks in mean or variance based on U-statistics

Bootstrapping tests are implemented for the tests U by Schmidt [Detecting changes in the trend function of heteroscedastic time series; 2021. Preprint: arXiv:2108.09206 [math.ST]] for mean break and Schmidt et al. [An asymptotic test for constancy of the variance under short-range dependence. Ann Stat. 2021;49:3460–3481.] for variance break based on U-statistics. The tests U have good powers against epidemic breaks that are common in practice. The test U for variance break is proved to have the nice property of consistency against a general class of alternatives. However, the tests U have non-ignorable finite sample size distortion under serial correlation and/or conditional heteroscedasticity. Our implementation based on autoregressive residual bootstrapping and moving block bootstrapping are shown to remedy the size distortion problems of U for mean break and for variance break, respectively, in a Monte-Carlo experiment. The experiment also demonstrates the power advantages of bootstrapping tests over the original tests and other standard break tests against epidemic breaks, which, however, are accompanied by disadvantages against simple single breaks. The proposed bootstrapping tests are well illustrated by break analyses of means and variances of the log-return and realized the volatility of the US S&P 500.

Forecasting and change point test for nonlinear heteroscedastic time series based on support vector regression.

SVR-ARMA-GARCH models provide flexible model fitting and good predictive powers for nonlinear heteroscedastic time series datasets. In this study, we explore the change point detection problem in the SVR-ARMA-GARCH model using the residual-based CUSUM test. For this task, we propose an alternating recursive estimation (ARE) method to improve the estimation accuracy of residuals. Moreover, we suggest using a new testing method with a time-varying control limit that significantly improves the detection power of the CUSUM test. Our numerical analysis exhibits the merits of the proposed methods in SVR-ARMA-GARCH models. A real data example is also conducted using BDI data for illustration, which also confirms the validity of our methods.

Multiple values-inflated time series of counts: modeling and inference based on INGARCH scheme

ABSTRACT In this study, we consider integer-valued autoregressive conditional heteroscedastic (INGARCH) time series models of counts whose conditional density follows an multiple values-inflated one parameter exponential family. We check the stationarity and ergodicity of the models and verify the consistency and asymptotic normality of the conditional maximum likelihood estimator (CMLE) under regularity conditions. Then, as an application, we consider a parameter change test using the cumulative sum (CUSUM) test based on (standardized) residuals and squares of those residuals, then subsequently derive their limiting null distributions. A simulation study is conducted focusing on the 0-1-inflated time series of counts, which affirms the validity of the proposed methods. Moreover, a real data analysis using the number of monthly drug cases in the 32nd police car beat in Pittsburgh is also provided to further strengthen the validity of our methods.

BRC-GARCH-X model: the empirical evidence in stock returns

A covariate-driven random coefficient generalized conditional heteroscedasticity (GARCH) time series model with the form of the buffered autoregression (BRC-GARCH-X) for modeling financial time series data is considered. As an extension of the classical two-regime threshold process, the buffered autoregression enjoys a more flexible regime-switching mechanism. Furthermore, the main feature of this model is that the threshold variable for regime-switching is formulated as a weighted average of important auxiliary variables. The estimator for regression parameters is obtained by the quasi-maximum exponential likelihood (QMEL) estimator and the corresponding asymptotic properties are established. Moreover, a mixed portmanteau test is developed for diagnostic checking. And a reasonable method for selecting search ranges for thresholds is also proposed and simulation studies are considered. As an application, we bring attention to some features of of stock returns of SP500 which shows that our model is feasible.

Monitoring parameter change for time series models with application to location-Scale heteroscedastic models

In this study, we consider an on-line monitoring procedure to detect a parameter change in general time series models, featuring location-scale heteroscedastic time series models and their conditional quantiles. To resolve this statistical process control (SPC) problem, we employ a residual-based cumulative sum (CUSUM) process specially designed to effectively detect both upward and downward changes in the conditional mean, variance, and quantiles of time series. To attain control limits analytically, limit theorems are provided for the proposed CUSUM monitoring process. A simulation study and real data analysis are conducted to illustrate its validity empirically.

Risk measurement for conditionally heteroscedastic location-scale time series models with ASTD and AEPD innovations

This study examines the asymptotic properties of a class of conditionally heteroscedastic location-scale time series models with innovations following a generalized asymmetric Student-t distribution (ASTD) or an asymmetric exponential power distribution (AEPD). We first show the consistency and asymptotic normality of the conditional maximum likelihood estimator of the model parameters under certain regularity conditions. Then, based on the maximum likelihood estimator, we estimate conditional value-at-risk (VaR) and expected shortfall (ES) by using their closed forms induced from the model. Their performance is finally compared with that of conditional autoregressive VaR and expectile methods. A simulation study and real data analysis are provided as an illustration.

ECG Signal Modeling Using Volatility Properties: Its Application in Sleep Apnea Syndrome.

This study presents and evaluates the mathematical model to estimate the mean and variance of single-lead ECG signals in sleep apnea syndrome. Our objective is to use the volatility property of the ECG signal for modeling. ECG signal is a stochastic signal whose mean and variance are time-varying. So, we propose to decompose this nonstationarity into two additive components; a homoscedastic Autoregressive Integrated Moving Average (ARIMA) and a heteroscedastic time series in terms of Exponential Generalized Autoregressive Conditional Heteroskedasticity (EGARCH), where the former captures the linearity property and the latter the nonlinear characteristics of the ECG signal. First, ECG signals are segmented into one-minute segments. The heteroskedasticity property is then examined through various tests such as the ARCH/GARCH test, kurtosis, skewness, and histograms. Next, the ARIMA model is applied to signals as a linear model and EGARCH as a nonlinear model. The appropriate orders of models are estimated by using the Bayesian Information Criterion (BIC). We assess the effectiveness of our model in terms of mean square error (MSE), root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE). The data in this article is obtained from the Physionet Apnea-ECG database. Results show that the ARIMA-EGARCH model performs better than other models for modeling both apneic and normal ECG signals in sleep apnea syndrome.

AdaVol: An Adaptive Recursive Volatility Prediction Method

Quasi-Maximum Likelihood (QML) procedures are theoretically appealing and widely used for statistical inference. While there are extensive references on QML estimation in batch settings, it has attracted little attention in streaming settings until recently. An investigation of the convergence properties of the QML procedure in a general conditionally heteroscedastic time series model is conducted, and the classical batch optimization routines extended to the framework of streaming and large-scale problems. An adaptive recursive estimation routine for GARCH models named AdaVol is presented. The AdaVol procedure relies on stochastic approximations combined with the technique of Variance Targeting Estimation (VTE). This recursive method has computationally efficient properties, while VTE alleviates some convergence difficulties encountered by the usual QML estimation due to a lack of convexity. Empirical results demonstrate a favorable trade-off between AdaVol’s stability and the ability to adapt to time-varying estimates for real-life data.

An indirect proof for the asymptotic properties of VARMA model estimators

Strong consistency and asymptotic normality of a Gaussian quasi-maximum likelihood estimator for the parameters of a causal, invertible, and identifiable vector autoregressive-moving average (VARMA) model are established in an indirect way. The proof is based on similar results for a much wider class of VARMA models with time-dependent coefficients, hence in the context of non-stationary and heteroscedastic time series. For that reason, the proof avoids spectral analysis arguments and does not make use of ergodicity. The results presented are also applicable to ARMA models.

Generalized autoregressive conditional heteroscedasticity (GARCH) for predicting volatility in Stock Market

Volatility plays an important role in financial markets and has held the attention of academics and practitioners. Using Generalized autoregressive conditional heteroscedasticity (GARCH) is one of the effective perspective for forecasting volatility. This study focused on comparing GARCH (P, Q) model with GJR-GARCH (P, Q) model and EGARCH (P, Q) model to make prediction more reliable and accurate. The results suggested that both GARCH (P, Q) model and GJR-GARCH (P, Q) model are good choices for forecasting volatility in financial market, especially for describing heteroscedastic time series. GARCH models are consistent with various forms of efficient market theory. These theories state that asset returns observed in the past cannot improve the forecasts of asset returns in the future.

Nonparametric volatility change detection

AbstractWe consider a nonparametric heteroscedastic time series regression model and suggest testing procedures to detect changes in the conditional variance function. The tests are based on a sequential marked empirical process and thus combine classical CUSUM tests from change point analysis with marked empirical process approaches known from goodness‐of‐fit testing. The tests are consistent against general alternatives of a change in the conditional variance function, a feature that classical CUSUM tests are lacking. We derive a simple limiting distribution and in the case of univariate covariates even obtain asymptotically distribution‐free tests. We demonstrate the good performance of the tests in a simulation study and consider exchange rates as a real data application.

A developed short-term electricity price and load forecasting method based on data processing, support vector machine, and virus colony search

Surely, electricity market participants need an accurate estimate for the price and load signals to properly manage their programs to increase their profits. Regarding the two-way interaction between suppliers and consumers, they are able to manage their required profile. Therefore, the individual price or load forecasting is not acceptable and cannot capture their related pattern. Therefore, this paper presents a new forecasting method for price and load signals in simultaneous configuration. This method consists of three main parts: the first part proposes two data preprocessing methods based on wavelet transform to remove noisy term and mutual information to select best features with maximum relevancy and minimum redundancy. The second part suggests a learning engine based on a least square support vector machine (LSSVM) with self-adaptive kernel functions and generalized autoregressive conditional heteroscedasticity (GARCH) time series to extract the linear and nonlinear patterns. Finally, the last part employs a new modified virus colony search algorithm (VCS) to efficiently set the LSSVM control parameters to use its all capacity. Simulations are carried out for three well-known Australia’s, New York’s, and Ontario’s electricity markets. The obtained results show the acceptable results.