Quasi-maximum likelihood estimation in GARCH processes when some coefficients are equal to zero

GARCH models can contain outliers, such as Additive Level Outliers (ALO) and/or Additive Volatility Outliers (AVO), so a robust parameter estimation is used to overcome the presence of outliers. This study uses Quasi-Maximum Likelihood Estimation (QMLE) and M-estimation to estimate the parameters of the GARCH model containing outliers. Hierarchical and K-means clustering algorithms are then used to cluster time-series data based on distances calculated from these parameter estimates. The proposed approach applies hierarchical and K-means clustering using distances derived from QMLE and M-estimation results for GARCH (1,1) models containing outliers for clustering with simulation data and a case study using stock data listed in the Indonesia Stock Exchange. The results show that hierarchical and K-means clustering with distance between two GARCH (1,1) models based on M-estimation produces good clusters with smaller C index than QMLE estimations for simulation data and the case study of stock data in Indonesia.

To be utilized in the smart charging of plug-in electric vehicles, this paper proposes semiparametric conditional mean and variance models for the daily density forecasting of the electricity load. The mean is modeled by means of an autoregressive moving average model with exogenous inputs (ARMAX), whereas several options for the variance evolution are investigated, starting with modeling the variance as a power of the conditional mean, then as a piecewise constant function, and finally as a generalized autoregressive conditional heteroskedasticity (GARCH) model. Due to the possible non-Gaussianity of the distribution of the stochastic components, a quasi-maximum likelihood estimation (QMLE) with the Pearson type IV (P-IV) distribution is also considered, apart from the estimation with the Gaussianity assumption. Moreover, the daily density forecasts are generated in a non-parametric manner by propagating samples from the stochastic components iteratively through the available model. These strategies, involving different options for variance modeling and estimation, are compared in terms of their forecast performances on two representative phase currents from the low voltage cables of medium-to-low voltage transformers in the Netherlands. The results indicate that the QMLE with the Gaussianity assumption performs better due to the additional complexity of the P-IV distribution on estimation. Concerning the variance models, the piecewise constant and GARCH models are more preferable when processing phase currents exhibiting only daily seasonality, and the model as a power of the conditional mean outperforms the others if both daily and weekly seasonality and hence more complexity is present.

Matrix-variate time series data are largely available in applications. However, no attempt has been made to study their conditional heteroscedasticity that is often observed in economic and financial data. To address this gap, we propose a novel matrix generalized autoregressive conditional heteroscedasticity (GARCH) model to capture the dynamics of conditional row and column covariance matrices of matrix time series. The key innovation of the matrix GARCH model is the use of a univariate GARCH specification for the trace of conditional row or column covariance matrix, which allows for the model identification. Moreover, we introduce a quasi-maximum likelihood estimator (QMLE) for model estimation and develop a portmanteau test for model diagnostic checking. Simulation studies are conducted to assess the finite-sample performance of the QMLE and portmanteau test. To handle large dimensional matrix time series, we also propose a matrix factor GARCH model, and establish its theoretical properties. Finally, we demonstrate the superiority of the matrix GARCH and matrix factor GARCH models over existing multivariate GARCH-type models in volatility forecasting and portfolio allocations using three applications on credit default swap prices, global stock sector indices, and future prices. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

We study the properties of the quasi-maximum likelihood estimator (QMLE) and related test statistics in dynamic models that jointly parameterize conditional means and conditional covariances, when a normal log-likelihood os maximized but the assumption of normality is violated. Because the score of the normal log-likelihood has the martingale difference property when the forst two conditional moments are correctly specified, the QMLE is generally Consistent and has a limiting normal destribution. We provide easily computable formulas for asymptotic standard errors that are valid under nonnormality. Further, we show how robust LM tests for the adequacy of the jointly parameterized mean and variance can be computed from simple auxiliary regressions. An appealing feature of these robyst inference procedures is that only first derivatives of the conditional mean and variance functions are needed. A monte Carlo study indicates that the asymptotic results carry over to finite samples. Estimation of several AR and AR-GARCH time series models reveals that in most sotuations the robust test statistics compare favorably to the two standard (nonrobust) formulations of the Wald and IM tests. Also, for the GARCH models and the sample sizes analyzed here, the bias in the QMLE appears to be relatively small. An empirical application to stock return volatility illustrates the potential imprtance of computing robust statistics in practice.

This paper investigates the asymptotic theory for a vector autoregressive moving average–generalized autoregressive conditional heteroskedasticity (ARMA-GARCH) model. The conditions for the strict stationarity, the ergodicity, and the higher order moments of the model are established. Consistency of the quasi-maximum-likelihood estimator (QMLE) is proved under only the second-order moment condition. This consistency result is new, even for the univariate autoregressive conditional heteroskedasticity (ARCH) and GARCH models. Moreover, the asymptotic normality of the QMLE for the vector ARCH model is obtained under only the second-order moment of the unconditional errors and the finite fourth-order moment of the conditional errors. Under additional moment conditions, the asymptotic normality of the QMLE is also obtained for the vector ARMA-ARCH and ARMA-GARCH models and also a consistent estimator of the asymptotic covariance.The authors thank the co-Editor, Bruce Hansen, and two referees for very helpful comments and suggestions and acknowledge the financial support of the Australian Research Council.

In this study, we introduce a novel approach, called the quasi maximum likelihood estimation (Q-MLE), for estimating large-dimensional matrix factor models. In contrast to the principal component analysis based approach, Q-MLE considers the heteroscedasticity of the idiosyncratic error term, the heteroscedasticity of which is simultaneously estimated with other parameters. Interestingly, under the homoscedasticity assumption of the idiosyncratic error, the Q-MLE estimator encompassed the projected estimator (PE) as a special case. We provide the convergence rates and asymptotic distributions of the Q-MLE estimators under mild conditions. Extensive numerical experiments demonstrate that the Q-MLE method performs better, especially when heteroscedasticity exists. Furthermore, two real examples in finance and macroeconomics reveal factor patterns across rows and columns, which coincide with financial, economic, or geographical interpretations.

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.

This paper tests for long memory in volatility of fixed‐income returns; specifically, South Africa's local currency 10‐year government bond, given that the characterisation of stochastic long‐memory volatility is of interest and importance in portfolio and risk management. The long‐memory parameter is estimated using methods based on wavelets, which have gained prominence in recent years. Evidence of long memory in fixed‐income return volatility is conclusively demonstrated across a variety of volatility measures and wavelet forms. This finding suggests a pattern of time dependence, which may potentially be exploited to generate improved volatility forecasting performance especially over long horizons. This paper further extends the extant literature by comparing the predictive power of long‐memory forecasts with those obtained from a standard (short‐memory) generalised autoregressive conditional heteroskedasticity (GARCH) process. The results of this exercise suggest that the information content of long‐memory models does not lead to improved forecast accuracy. The GARCH(1,1) model is shown to provide the best forecasts across most horizons (i.e.daily, weekly and monthly). Forecast performance is further revealed to be sensitive to the choice of volatility proxy used. Finally, the derived volatility forecasts are generally very close, and in some cases, almost indistinguishable.

Quasi maximum likelihood estimation and inference in multivariate volatility models remains a challenging computational task if, for example, the dimension of the parameter space is high. One of the reasons is that typically numerical procedures are used to compute the score and the Hessian, and often they are numerically unstable. We provide analytical formulae for the score and the Hessian for a variety of multivariate GARCH models including the Vec and BEKK specifications as well as the recent dynamic conditional correlation model. By means of a Monte Carlo investigation of the BEKK–GARCH model we illustrate that employing analytical derivatives for inference is clearly preferable to numerical methods.

This paper collects a total of 541 groups of the RMB exchange rate to build the model of the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) for forecasting of RMB exchange rate against US dollar. The empirical results show that the GARCH models can fit the series well and the prediction effects of both the GARCH (1, 1) model and the GARCH-M model are good.

Stock price data at State Gas Company is defined as the time-series data comprising varying volatility and heteroscedasticity. One of the best models used to solve the problem of heteroscedasticity is the GARCH (generalized autoregressive conditional heteroscedasticity) model. Therefore, this study aims to build the most suitable model for predicting the 186 days before and 176 days after the Covid-19 pandemic, as well as to provide recommendations to reduce the impact of daily stock price movements. Data were obtained by examining the daily stock price data in Indonesian National Gas Companies from 2019 to 2020. The study also discusses the Event Window, with the best model identified as AR (1) -GARCH (1,1). The result showed that an error of less than 0.0015 is AR (1) - GARCH (1,1), provided the best model for price forecasting of Indonesian National Gas Companies.Keywords: Stock Price, Heteroscedasticity, GARCH Model, Event WindowJEL Classifications: C5, O42, Q4, Q47DOI: https://doi.org/10.32479/ijeep.10999

Crude oil is an important energy commodity to mankind. The fluctuation of crude oil prices has affected many related sectors and stock market indices. Hence, forecasting the crude oil prices is essential to avoid the future prices of the non-renewable natural resources to raise sky-rocket. In this study, daily West Texas Intermediate (WTI) crude oil prices data is obtained from Energy Information Administration (EIA) from 2nd January 1986 to 30th September 2009. This study uses the Box-Jenkins methodology and Generalized Autoregressive Conditional Heteroscedasticity (GARCH) approach in analyzing the crude oil prices. ARIMA(1,2,1) and GARCH(1,1) are found to be the appropriate models under model identification, parameter estimation, diagnostic checking and forecasting future prices. In this study, the analyses are done with the aid of EViews software where the potential of this software in forecasting daily crude oil prices time series data is explored. Finally, using several measures, comparison performances between ARIMA(1, 2, 1) and GARCH(1,1) models are made. GARCH(1,1) is found to be a better model than ARIMA(1, 2, 1) model. Based on the study, it is concluded that ARIMA(1,2,1) model is able to produce good forecast based on a description of history patterns in crude oil prices. However, the GARCH(1,1) is the better model for daily crude oil prices due to its ability to capture the volatility by the non-constant of conditional variance.

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.

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.

Stock is an investment in the capital market that is very promising for investors. Investors can also get high returns from the shares invested. However, this stock price is not always stable, it can go up and down drastically. The purpose of this study is to predict stock prices because they often experience instability. The method used in this research is using the Exponential generalized autoregressive conditional heteroscedasticity (EGARCH) model with the Quasi Maximum Likelihood (QML) method. The result of this research is the implementation of this model. The EGARCH model used is the stock price index model that is formed, namely the autoregressive integrated moving average (ARIMA) (0, 1, 2) EGARCH (1.4). The conclusion from the results of the research that predictions using the ARIMA model (0, 1, 2) EGARCH (1, 4) is the best model in accommodating the asymmetric nature of the volatility of the stock price index. The results of this egarch model show more optimal prediction results seen from an error of 3% compared to other modes such as the arch model and the GARCH model.