Conditional Heteroskedasticity Process Research Articles

Networked control systems induced delay modeling and stability analysis using Markov-regime-switching GARCH model

This article focuses on the stability analysis of networked control systems (NCSs). It is well-known that network-induced delays pose many control challenges due to their non-uniform and multifractal nature. With the growing integration of communication networks in control systems, the impact of time-varying and random delays on system performance becomes critical. Hence, it is essential to study their impact on the system’s stability. There are several network-induced delay modeling methods, resulting in constant, random, and probabilistic random delays, which provide the most realistic representation according to recent studies. This article introduces a new network-induced delay modeling method based on the Markov-regime-switching generalized autoregressive conditional heteroskedasticity (MRS-GARCH) process, which captures the intricate temporal dependence of random network delays. The resulting model offers a more accurate depiction of delay volatility behavior, accommodating sudden shifts between volatility regimes. Simulation results demonstrate that MRS GARCH-type modeling enables a more realistic representation of delay dynamics. We also conduct a stability analysis of NCSs modeled with the proposed method using linear matrix inequalities LMIs criteria derived from the Lyapunov-Krasovskii theorem. The simulation results show less conservatism than conventional models and provide a larger maximum allowable upper delay bound MAUDB.

Probabilistic properties and estimation methods for periodic threshold autoregressive stochastic volatility

<abstract><p>In an endeavor to encapsulate the dual aspects of volatility progression and periodicity inherent in autocorrelation frameworks demonstrated by various nonlinear time series, a novel conceptualization emerges—the periodic threshold autoregressive stochastic volatility (PTAR-SV) model. This model served as a viable alternative to the conventional periodic threshold generalized autoregressive conditional heteroskedasticity (TGARCH) process. The inherent probabilistic framework of the PTAR-SV model incorporated certain essential features, including strict periodic stationarity, enhancing its analytical robustness. Additionally, this study established the conditions for higher-order moments to exist within the PTAR-SV model. The autocovariance structure pertaining to the powers of the PTAR-SV process has been studied. The process of parameter estimation was scrutinized via the quasi-maximum likelihood technique. This estimation approach involved assessing likelihood using prediction error decomposition and Kalman filtering. Moreover, we extended our analysis to include a Bayesian Markov chain Monte Carlo (MCMC) method based on Griddy-Gibbs sampling, particularly suitable when the distribution of model innovations follows a standard Gaussian. Through a simulation study, we evaluated the performances of both the quasi-maximum likelihood (QML) and Bayesian Griddy Gibbs estimates, providing valuable insights into their respective strengths and weaknesses. Finally, we applied our newly developed methodology to model the spot rates of the euro against the Algerian dinar, demonstrating its applicability and efficacy in real-world financial modeling scenarios.</p></abstract>

Exchange Rate Responses and Volatility Spillover Effects during the COVID-19 Pandemic in Indonesia

This paper aims to assess the impact of the confirmed COVID-19 cases, the timing of the outbreak, and physical measures on the returns and spillover effects of exchange rate in Indonesia. The model will be tested by the exponential generalized autoregressive conditional heteroskedastic (EGARCH) process and the spillover volatility index. The study discovers that the confirmed cases, outbreak news, and the implementation of large-scale social restrictions simultaneously contribute to a leverage effect on the volatility of a direct quote of Indonesian Rupiah to Australian Dollar, Euro, US Dollar, Singapore Dollar, and Great British Pound. To a certain extent, the heatwave as well as the meteor-shower effects as a result of clustering events and intense spillover effects in the currency market of Indonesia are observed.

Hybrid ARMA-GARCH-Neural Networks for intraday strategy exploration in high-frequency trading

The frequency of armed conflicts increased during the last 20 years. The problems of the emergence of military disputes, not only concern social parameters, but also economic and financial dimensions. This study examines the potential impact of global geopolitical events on the stock market prices of the Dow Jones U.S. Aerospace & Defense Index and Foreign Exchange (FOREX) markets movements. We analyse whether defence stocks and exchange rate perform similarly during military incidents or geopolitical crises. We built an Autoregressive Moving Average Model with a Generalized Autoregressive Conditional Heteroskedasticity process (ARMA-GARCH) with the machine learning methods of Neural Networks, Deep Recurrent Convolutional Neural Networks, Deep Neural Decision Trees, Quantum Neural Networks, and Quantum Recurrent Neural Networks, aimed at detecting intraday patterns for forecasting defence stock market and FOREX markets disturbances in a market microstructure framework. The empirical results provide preliminary findings on the foreseeability of market disturbances and small differences are observed before and during geopolitical events. Additionally, we confirm the effectiveness of the hybrid model ARMA-GARCH with the machine learning approaches, being ARMA-GARCH-Quantum Recurrent Neural Network the technique that achieves the best accuracy results. Our work has a large potential impact on investment market agents and portfolio managers, as shocks from geopolitical events could provide a new methodology to support the decision-making process for trading in High-Frequency Trading.

Dynamic Bayesian hierarchical peak over threshold modeling for real-time crash-risk estimation from conflict extremes

Using traffic conflict-based extreme value theory (EVT) models to quantify real-time crash-risk of road facilities is a promising direction for developing proactive traffic safety management strategies. Existing EVT real-time crash-risk analysis studies have only focused on using block maxima models. This study proposes a dynamic Bayesian hierarchical peak over threshold modeling approach to estimate real-time crash-risk based on traffic conflicts. The proposed approach combines quantile regression, dynamic updating approach, Bayesian hierarchical structure, and the peak over threshold method to generate time-varying generalized Pareto distributions to derive real-time crash-risk measures (i.e., crash probability and return level). The derived real-time crash-risk measures are applied to estimate cycle-level crash-risk at three signalized intersections in Surrey, British Columbia. Five approaches are used to dynamically update the model parameters, including time trend model, generalized autoregressive conditional heteroskedasticity process approach, as well as the first-order, second-order, and third-order dynamic linear models. For comparison, static models are also developed. All the developed models are compared in terms of statistical fit and predictive performance. Based on the best fitted dynamic model, cycle-level crash probability and return level are calculated to measure signalized intersection safety at cycle level. The results show that dynamic models considerably outperform static models in terms of statistical fit and predictive performance. Further, the third-order dynamic model has the best performance, which is probably due to that the model incorporates two linear trends to respectively describe the variation of the coefficients as well as its change to better account for the variation in the effect of time-varying covariates. However, it should be noted that the third-order dynamic model development needs more computation time than other dynamic models, which may limit the application of the model.

Testing for parameter change epochs in GARCH time series

Summary We develop a uniform test for detecting and dating the integrated or mildly explosive behaviour of a strictly stationary generalized autoregressive conditional heteroskedasticity (GARCH) process. Namely, we test the null hypothesis of a globally stable GARCH process with constant parameters against the alternative that there is an ‘abnormal’ period with changed parameter values. During this period, the parameter-value change may lead to an integrated or mildly explosive behaviour of the volatility process. It is assumed that both the magnitude and the timing of the breaks are unknown. We develop a double-supreme test for the existence of breaks, and then provide an algorithm to identify the periods of changes. Our theoretical results hold under mild moment assumptions on the innovations of the GARCH process. Technically, the existing properties for the quasi-maximum likelihood estimation in the GARCH model need to be reinvestigated to hold uniformly over all possible periods of change. The key results involve a uniform weak Bahadur representation for the estimated parameters, which leads to weak convergence of the test statistic to the supreme of a Gaussian process. Simulations in the Appendix show that the test has good size and power for reasonably long time series. We apply the test to the conventional early-warning indicators of both the financial market and a representative of the emerging Fintech market, i.e., the Bitcoin returns.

Nonlinear autocorrelation function of functional time series

In functional time series analysis, the functional autocorrelation function (fACF) plays an important role in revealing the temporal dependence structures underlying the dynamics and identifying the lags at which substantial correlation exists. However, akin to its counterpart in the univariate case, the fACF is restricted by linear structure and can be misleading in reflecting nonlinear temporal dependence. This paper proposes a nonlinear alternative to the fACF for analyzing the temporal dependence in functional time series. We consider linear and nonlinear data generating processes: a functional autoregressive process and a functional generalized autoregressive conditional heteroskedasticity process. We demonstrate that when the process exhibits linear temporal structures, the inference obtained from our proposed nonlinear fACF is consistent with that from the fACF. When the underlying process exhibits nonlinear temporal dependence, our nonlinear fACF has a superior capability in uncovering the nonlinear structure that the fACF misleads. An empirical data analysis highlights its applications in unveiling nonlinear temporal structures in the daily curves of the intraday volatility dynamics of the foreign exchange rate.

Efficacy of GARCH-EVT Model in Intraday Risk Management: Evidence from Severely Pandemic-affected Countries

Countries around the world have experienced severe-to-moderate economic restrictions during the first two waves of COVID-19 pandemic. The present article captures the time frame of this unprecedented turmoil to test the efficacy of the conditional extreme value theory (EVT) model to forecast value at risk (VaR) and expected shortfall (ES). The article considers the eight most-affected countries and applies the generalized autoregressive conditional heteroskedasticity (GARCH) process in appropriate GARCH-EVT models assuming Student’s t distribution to accommodate the fat-tailed behaviour of financial return series. The study confirms that our model is substantially superior in forecasting intraday VaR and ES in comparison to other conditional EVT models with a classical assumption of normal distribution as well as an unconditional EVT model. Contrary to more commonly used 5-minute-frequency data in intraday risk management, we use hourly data, which is much less researched in the context of stock market volatility forecasting. Moreover, our study adds to the reliability of conditional EVT models because of its accuracy in predicting intraday risk during extreme economic uncertainty in severely pandemic-hit countries.

Asymptotic independence ex machina: Extreme value theory for the diagonal SRE model

We consider multivariate stationary processes satisfying a stochastic recurrence equation of the form , where are i.i.d. random vectors and are i.i.d. diagonal matrices and (Mt) are i.i.d. random variables. We obtain a full characterization of the vector scaling regular variation properties of , proving that some coordinates Xt, i and Xt, j are asymptotically independent even though all coordinates rely on the same random input (Mt). We prove the asynchrony of extreme clusters among marginals with different tail indices. Our results are applied to some multivariate autoregressive conditional heteroskedastic (BEKK‐ARCH and CCC‐GARCH) processes and to log‐returns. Angular measure inference shows evidences of asymptotic independence among marginals of diagonal SRE with different tail indices.

Implications of the state assistance program in the province of Quebec: the case of lamb production

  This study investigated the impact of the Farm Income Stabilization Insurance (ASRA) on the adoption of price risk management strategies by lamb producers in the province of Quebec. This study employed a Generalized Autoregressive Conditional Heteroskedasticity (GARCH) process to model price risks. The results indicated that the application of the Farm Income Stabilization Insurance in Quebec generates crowding-out effects on price risk management strategies, which decreases the efficiency of this program. On the other hand, the product-specific nature of ASRA leads to some challenges such as a modification in the revenue distribution across the farm, increased production, increased indebtedness of farmers and increased financial burden on governments’ shoulders. Finally, the results revealed an asymmetric impact of positive and negative shocks on the production decision of lamb producers. This asymmetric impact of negative and positive shocks generated by ASRA results in an increasing risk-aversion of producers over the periods of decreased prices and a decreasing risk aversion over the periods of increased prices.

Temporal aggregation and systematic sampling for INGARCH processes

The temporal aggregation (TA) and systematic sampling (SS) in continuous-valued time series were widely studied, while the TA and SS in integer-valued time series obtain very little attention. In this paper, we thoroughly discuss them based on the integer-valued generalized autoregressive conditional heteroskedasticity (INGARCH) process. On one hand, the TA and SS for INGARCH processes are studied, and related predictors are discussed. We also illustrate that the TA or SS provides approaches to estimate INGARCH processes with different frequencies. On the other hand, definitions of the strong and weak INGARCH processes are given. The aggregated INGARCH and sampled INGARCH processes are shown to be weak, that is, they satisfy a weak characterization of INGARCH processes in terms of linear projection. Quasi maximum likelihood and nonlinear least squares estimation methods are considered for the weak INGARCH processes. Related asymptotic properties and simulation results are discussed. Finally, an empirical example illustrates the applicability of our results.

Does Uncertainty Matter for Trade - Economic Growth Nexus in Indonesia?

This paper aims to test the impact of uncertainty on the causal relationship among exports, imports, and economic growth in Indonesia. The relationship is constructed by examining the presence of FDI-adjusted exports and imports (trade) and the output link using conditional variances-covariances derived from the generalized autoregressive conditional heteroskedastic (GARCH) process in a vector error correction model (VEC-GARCH model). Using evidence in Indonesia, the model exposes the uni-directional nexus from trade performance to trade-adjusted output growth in the absence of uncertainty. The volatility effects are evident in the causal relationship between trade and output. The finding shows that the uncertainty effects hamper the trade-economic growth nexus. Incorporated with the long-run causality, trade still causes output even after containing the contributions of volatility. The significant role of imports highlights the higher demand for intermediate capital products and the inclusion of technology in strengthening economic growth.

ITERATIONS OF DEPENDENT RANDOM MAPS AND EXOGENEITY IN NONLINEAR DYNAMICS

We discuss the existence and uniqueness of stationary and ergodic nonlinear autoregressive processes when exogenous regressors are incorporated into the dynamic. To this end, we consider the convergence of the backward iterations of dependent random maps. In particular, we give a new result when the classical condition of contraction on average is replaced with a contraction in conditional expectation. Under some conditions, we also discuss the dependence properties of these processes using the functional dependence measure of Wu (2005, Proceedings of the National Academy of Sciences 102, 14150–14154) that delivers a central limit theorem giving a wide range of applications. Our results are illustrated with conditional heteroscedastic autoregressive nonlinear models, Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) processes, count time series, binary choice models, and categorical time series for which we provide many extensions of existing results.

Testing for parameter change epochs in GARCH time series

We develop a uniform test for detecting and dating the integrated or mildly explosive behaviour of a strictly stationary generalized autoregressive conditional heteroskedasticity (GARCH) process. Namely, we test the null hypothesis of a globally stable GARCH process with constant parameters against {the alternative that} there is an ``abnormal period with changed parameter values. During this period, the parameter-value change may lead to an integrated or mildly explosive behaviour of the volatility process. It is assumed that both the magnitude and the timing of the breaks are unknown. We develop a double-supreme test for the existence of breaks, and then provide an algorithm to identify the periods of changes. Our theoretical results hold under mild moment assumptions on the innovations of the GARCH process. Technically, the existing properties for the quasi-maximum likelihood estimation (QMLE) in the GARCH model need to be reinvestigated to hold uniformly over all possible periods of change. The key results involve a uniform weak Bahadur representation for the estimated parameters, which leads to weak convergence of the test statistic to the supreme of a Gaussian process. In simulations we show that the test has good size and power for reasonably long time series. We apply the test to the conventional early-warning indicators of both the financial market and a representative of the emerging Fintech market, i.e. the Bitcoin returns.

Influence of teleconnections on night-time minimum temperature variability in the Southwestern U.S.

Night-time minimum temperatures are increasing, faster than day-time maximum temperatures. They are also strongly connected to key outcomes such as health, vegetation, and crop productivity, and therefore changes in the distributions of night-time minimum temperatures have strong impacts on living systems. Night-time minimum temperatures are also related to teleconnections, which are indices that capture climate phenomenon occurring on a large spatial and temporal scale. As autocorrelated sequences, teleconnections are predictable and therefore inform distributions of weather in the near future. In this paper, we link four well-known teleconnections to night-time minimum temperatures in the American Southwest. These teleconnections are covariates for the mean process, but also for the variance, which is modelled as an exponential generalized autoregressive conditional heteroskedastic process (AR-EGARCH). The resulting models are used to estimate the probability of a low temperature event across the phases of each teleconnection, demonstrating the impact each teleconnection has on variability of night-time minimum temperatures.

The Tail Behavior due to the Presence of the Risk Premium in AR-GARCH-in-Mean, GARCH-AR, and Double-Autoregressive-in-Mean Models

Abstract We extend the results in Borkovec (2000), Basrak, David, and Mikosch (2002a), Lange (2011), and Francq and Zakoïan (2015) by describing the tail behavior when a risk premium component is added in the mean equation of different conditional heteroskedastic processes. We study three types of parametric models: the traditional generalized autoregressive conditional heteroskedastic (GARCH)-M model, the double autoregressive (AR) model with risk premium, and the GARCH-AR model. We find that if an AR process is introduced in the mean equation of a traditional GARCH-M process, the tail behavior is the same as if it is not introduced. However, if we add a risk premium component to the double AR model, then the tail behavior changes with respect to the GARCH-M. The GARCH-AR model also has a different tail index than the traditional AR-GARCH model. In a simulation study, we show that larger tail indexes are associated with the traditional GARCH-M model. When the size of the risk premium component increases, the tail index tends to fall. The only exception to this rule occurs in the double AR model when the risk premium depends on log-volatility. Parameter configurations where the strong stationarity condition of the risk premium models fails are also illustrated and discussed.

Real-Time Algorithms for the Detection of Changes in the Variance of Video Content Popularity

As video content is responsible for more than 70% of the global IP traffic, related resource allocation approaches, e.g., using content caching, become increasingly important. In this context, to avoid under-provisioning, it is important to rapidly detect and respond to changes in content popularity dynamics, including volatility, i.e., changes in the second order moment of the underlying process. In this paper, we focus on the early identification of changes in the variance of video content popularity, which we address as a statistical change point (CP) detection problem. Unlike changes in the mean that can be well captured by non-parametric statistical approaches, to address this more demanding problem, we construct a hypothesis test that uses in the test statistic both parametric and non-parametric approaches. In the context of parametric models, we consider linear, in the form of autoregressive moving average (ARMA), and, nonlinear, in the form of generalized autoregressive conditional heteroskedasticity (GARCH) processes. We propose an integrated algorithm that combines off-line and on-line CP schemes, with the off-line scheme used as a training (learning) phase. The algorithm is first assessed over synthetic data; our analysis demonstrates that non parametric and GARCH model based approaches can better generalize and are better suited for content views time series with unknown statistics. Finally, the non-parametric and the GARCH based variations of our proposed integrated algorithm are applied on real YouTube video content views time series, to illustrate the performance of the proposed approach of volatility change detection.

South African Historical Interest Rate Volatility - Evidence of Regime- Switching

Accurate estimates of volatility are important for the valuation and risk management of financial assets. The benchmark interest rate for many assets in South Africa, the 3-month Jibar, is found to exhibit a high volatility persistence when modelled with a standard generalised autoregressive conditional heteroskedasticity (GARCH) process. The literature suggests that unobserved regime-switching in interest rate data may lead to an overestimation of volatility persistence. In this study 120 GARCH-type volatility models are tested to determine which model, conditional distribution and number of regimes best fit the data in order to extract the most accurate in-sample estimation of historical volatility for asset pricing. The data analysed consists of 877 weekly observations of 3-month Jibar in total, spanning from September 2001 to July 2018. The switching between regimes is governed by a Markov chain process which produces state-dependent transition probabilities for the unobserved regimes. The study finds that a standard single regime GARCH model may not capture the underlying volatility dynamics of the interest rate. The best performing model in this study is the 4 State Threshold-GARCH indicating that in addition to evidence of regime-switching in the data, there is an asymmetric reaction to negative information.

Nested Conditional Value-at-Risk portfolio selection: A model with temporal dependence driven by market-index volatility

In a multistage stochastic programming framework, we develop a new method for finding an approximated portfolio allocation solution to the nested Conditional Value-at-Risk model when asset log returns are stagewise dependent. We describe asset log returns through a single-factor model where the driving factor is the market-index log return modeled by a Generalized Autoregressive Conditional Heteroskedasticity process to take into account the serial dependence usually observed. To solve the nested Conditional Value-at-Risk model, we implement a backward induction scheme coupled with cubic spline interpolation that reduces the computational complexity of the optimal portfolio allocation and allows to treat problems otherwise unmanageable.

Modeling and forecasting the oil volatility index

AbstractThe increase in oil price volatility in recent years has raised the importance of forecasting it accurately for valuing and hedging investments. The paper models and forecasts the crude oil exchange‐traded funds (ETF) volatility index, which has been used in the last years as an important alternative measure to track and analyze the volatility of future oil prices. Analysis of the oil volatility index suggests that it presents features similar to those of the daily market volatility index, such as long memory, which is modeled using well‐known heterogeneous autoregressive (HAR) specifications and new extensions that are based on net and scaled measures of oil price changes. The aim is to improve the forecasting performance of the traditional HAR models by including predictors that capture the impact of oil price changes on the economy. The performance of the new proposals and benchmarks is evaluated with the model confidence set (MCS) and the Generalized‐AutoContouR (G‐ACR) tests in terms of point forecasts and density forecasting, respectively. We find that including the leverage in the conditional mean or variance of the basic HAR model increases its predictive ability. Furthermore, when considering density forecasting, the best models are a conditional heteroskedastic HAR model that includes a scaled measure of oil price changes, and a HAR model with errors following an exponential generalized autoregressive conditional heteroskedasticity specification. In both cases, we consider a flexible distribution for the errors of the conditional heteroskedastic process.