A note on dynamic spatiotemporal ARCH models: small- and large-sample results

In this paper the student-t autoregressive conditional heteroscedasticity (ARCH) model is considered and a new estimator for the coefficients of the ARCH model and the degree of freedom of the Student-t noise is proposed. ARCH models have been used in numerous applications to model the unpredictability and the strong dependence of the instantaneous variability of a time series on its own past. The usual approach for estimating the ARCH parameters is using QML estimator. This estimator is consistent and asymptotically efficient, but for limited data length it has poor efficiency. Simulation results show that the proposed method has better efficiency even in limited data length case. The price one should pay to obtain this performance is the increased computational load.

In this study, the equitoes’ returns operated in the Istanbul Stock Exchange Market (IMKB) have been considered using the models applied for the series with high frequency. As for the data, the daily IMKB-National 100 Index has been used for the period starting from the beginning of 1988 to the mid of 2000. For this index, the most appropriate asymmetric EGARCH(2,2) model has been attained comparing different models of TGARCH, EGARCH and ARCH-M as different versions of the generalized heteroscedastic models (symmetric or asymmetric GARCH). Later on, the research has focused on whether these variables affect each other or not in the context of the Granger-causality relationship between the stock market return and the stock market return uncertainty. 1.Introduction It is accepted that the series used in the classical models for estimating time series have the same variance whereas some financial series containing high frequency and uncertainty have different variance in each period. The conditional heteroscedasticity is high in highly fluctuating periods while it is low in stagnation periods. The Autoregressive Conditional Heteroscedastic (ARCH) models related to the estimation of series high frequencies and their fluctuations changing over time have been developed first by Engle (1982). Later, Bollerslev (1986) have transferred these models by developing them into the Generalized Autoregressive Conditional Heteroscedastic (GARCH) models. In this study, the most appropriate ARMA(p,q)-GARCH(p,q) return model is produced using the daily National-100 Index of Istanbul Stock Exchange Market (IMKB) for the period starting from the beginning of 1988 to the mid of 2000. Afterwards, the causality between the stock market return uncertainty and the real stock market return has been searched by using the Autoregressive Conditional Heteroscedasticiy of the proposed most appropriate GARCH models. In explaining the Granger-causality, we have followed the method of Grier and Perry (1998) and Nas and Perry (2000) (the later study has applied the model for the Turkish inflation) considering the relationship between inflation and inflation uncertainty.

This paper presents a systematic review of the literature (SRL) on value-at-risk (VaR). More specifically, we review the models that have been applied to estimate VaR, with the following two aims: to find the most used models in the literature and to verify whether their popularity has changed since the 2007–9 financial crisis. The SRL is based on Scopus for the period from 1996 to 2017. Our results show that (generalized) autoregressive conditional heteroscedasticity models and extreme value theory, together with Monte Carlo simulation, historical simulation and variance–covariance, were the most used models. Since the crisis, the autoregressive conditional heteroscedasticity (ARCH) and generalized autoregressive conditional heteroscedasticity (GARCH) models have clearly been the most popular, while no significant difference has been found in the percentage of articles on the other models. This study can be considered the first SRL on VaR models because (to the best of our knowledge) no previous work of a similar nature has been carried out on this topic. This study provides a rich background for researchers and professionals interested in the topic, contributing detailed information about the papers published, classifying them by, for example, the model used, author(s), citation count, journals and year published.

Diagnostics for conditional heteroscedasticity models: some simulation results

We introduce the method of estimating functions to study the class of autoregressive conditional heteroscedasticity (ARCH) models. We derive the optimal estimating functions by combining linear and quadratic estimating functions. The resultant estimators are more efficient than the quasi-maximum likelihood estimator. If the assumption of conditional normality is imposed, the estimator obtained by using the theory of estimating functions is identical to that obtained by using the maximum likelihood method in finite samples. The relative efficiencies of the estimating function (EF) approach in comparison with the quasi-maximum likelihood estimator are developed. We illustrate the EF approach using a univariate GARCH(1,1) model with conditional normal, Student-t, and gamma distributions. The efficiency benefits of the EF approach relative to the quasi-maximum likelihood approach are substantial for the gamma distribution with large skewness. Simulation analysis shows that the finite-sample properties of the estimators from the EF approach are attractive. EF estimators tend to display less bias and root mean squared error than the quasi-maximum likelihood estimator. The efficiency gains are substantial for highly nonnormal distributions. An example demonstrates that implementation of the method is straightforward.

In this paper, the finite-sample properties of symmetric GARCH and asymmetric GJR-GARCH models in the presence of time-varying long term variance are considered. In particular, the deterministic spline-GARCH model is investigated by Monte-Carlo simulation, where the true parameter values are taken from estimated real equity index data. As a proxy for the behaviour of equity indices of developed countries, the S&P500 Index is estimated with the Quasi-Maximum-Likelihood (QML) method for dierent conditional heteroscedastic models (GARCH, GJR-GARCH, spline-GARCH and spline-GJR-GARCH). The estimated S&P500 parameter values are used to simulate a broad range of 6 dierent time-series lengths {100, 500, 1000, 5000, 10000, 25000} and 4 different numbers of spline knots {1,4,9,14}, combining to a total amount of 60 different model setups. To the best of my knowledge, there exist only a few limited simulation studies that focus on the spline-GARCH model. The main contribution of this paper is therefore to highlight the behaviour of the QML estimates when the long-term variance is implemented by the spline-GARCH model. Beside this, the paper provides a least-square approach to get useful starting values for the numerical estimation routine.

The common way to measure the performance of a volatility prediction model is to assess its ability to predict future volatility. However, as volatility is unobservable, there is no natural metric for measuring the accuracy of any particular model. Noh et al. (1994) assessed the performance of a volatility prediction model by devising trading rules to trade options on a daily basis and using forecasts of option prices obtained by the Black & Scholes (BS) option pricing formula. (An option is a security that gives its owner the right, not the obligation, to buy or sell an asset at a fixed price within a specified period of time, subject to certain conditions. The BS formula amounts to buying (selling) an option when its price forecast for tomorrow is higher (lower) than today’s market settlement price.)In this paper, adopting Noh et al.’s (1994) idea, we assess the performance of a number of Autoregressive Conditional Heteroscedasticity (ARCH) models. For, each trading day, the ARCH model, selected on the basis of the prediction error criterion (PEC) introduced by Xekalaki et al. (2003) and suggested by Degiannakis and Xekalaki (1999) in the context of ARCH models, is used to forecast volatility. According to this criterion, the ARCH model with the lowest sum of squared standardized one step ahead prediction errors is selected for forecasting future volatility. A comparative study is made in order to examine which ARCH volatility estimation method yields the highest profits and whether there is any gain in using the PEC model selection algorithm for speculating with financial derivatives. Among a set of model selection algorithms, even marginally, the PEC algorithm appears to achieve the highest rate of return.

ABSTRACTThis paper proposes an adaptive quasi-maximum likelihood estimation (QMLE) when forecasting the volatility of financial data with the generalized autoregressive conditional heteroscedasticity (GARCH) model. When the distribution of volatility data is unspecified or heavy-tailed, we worked out adaptive QMLE based on data by using the scale parameter ηf to identify the discrepancy between wrongly specified innovation density and the true innovation density. With only a few assumptions, this adaptive approach is consistent and asymptotically normal. Moreover, it gains better efficiency under the condition that innovation error is heavy-tailed. Finally, simulation studies and an application show its advantage.

Summary Autoregressive conditional heteroscedastic (ARCH) models and its extensions are widely used in modelling volatility in financial time series. One of the variants, the double-threshold autoregressive conditional heteroscedastic (DTARCH) model, has been proposed to model the conditional mean and the conditional variance that are piecewise linear. The DTARCH model is also useful for modelling conditional heteroscedasticity with nonlinear structures such as asymmetric cycles, jump resonance and amplitude-frequence dependence. Since asset returns often display heavy tails and outliers, it is worth studying robust DTARCH modelling without specific distribution assumption. This paper studies DTARCH structures for conditional scale instead of conditional variance. We examine L1-estimation of the DTARCH model and derive limiting distributions for the proposed estimators. A robust portmanteau statistic based on the L1-norm fit is constructed to test the model adequacy. This approach captures various nonlinear phenomena and stylized facts with desirable robustness. Simulations show that the L1-estimators are robust against innovation distributions and accurate for a moderate sample size, and the proposed test is not only robust against innovation distributions but also powerful in discriminating the delay parameters and ARCH models. It is noted that the quasi-likelihood modelling approach used in ARCH models is inappropriate to DTARCH models in the presence of outliers and heavy tail innovations.

An attempt has been made in this paper to explain the stock market volatility at the individual script level and at the aggregate indices level. The empirical analysis has been done by using Autoregressive conditional heteroscedasticity model (ARCH), Generalised autoregressive conditional heteroscedasticity (GARCH) model and ARCH in Mean model and it is based on daily data for the time period from January 1990 to November 2004. The analysis reveals the same trend of volatility in the case of aggregate indices and five different sectors such as electrical, machinery, mining, non-metalic and power plant sector. The GARCH (1,1) model is persistent for all the five aggregate indices and individual company.

Modeling streamflow time series using nonlinear SETAR-GARCH models

Stocks are time series data in the financial sector, which usually have a tendency to fluctuate rapidly from time to time so that the variance of the error will always change over time or is not constant, or is often called a case of heteroscedasticity. The time series model to model this condition is the Autoregressive Conditional Heteroscedasticity (ARCH) model. ARCH models require large orders in modeling variations because financial data has a large level of volatility. To overcome orders that are too large in the ARCH model, a generalization of the ARCH model is carried out, this model is known as Generalized Autoregressive Conditional Heteroscedasticity (GARCH). The results of the research show that the general picture of the closing price of BRI shares in the period January 2015 to December 2018 experienced unstable fluctuations and the highest closing price was in 2017. The best model used to print BRI share returns is the ARIMA model (29.0, 1) GARCH (1,1) so that the resulting model is as follows: Z_t=〖-0.084777e〗_(t-26) and σ_t^2=0.000241+〖0.149753α〗_(t-1)^2+ 〖0.334886σ〗_(t-1)^2. From the results of forecasting the return value, there was one period that experienced a loss and four periods that experienced a profit, with a loss in the 11/01/2018 period of 0.000116 and the highest profit in the 10/31/2018 period of 0.000039.

Estimating multivariate ARCH parameters by two-stage least-squares method

This paper makes a significant contribution by focusing on estimating the coefficients of a sample of non-linear time series, a subject well-established in the statistical literature, using bilinear time series. Specifically, this study delves into a subset of bilinear models where Generalized Autoregressive Conditional Heteroscedastic (GARCH) models serve as the white noise component. The methodology involves applying the Klimko–Nilsen theorem, which plays a crucial role in extracting the asymptotic behavior of the estimators. In this context, the Generalized Autoregressive Conditional Heteroscedastic model of order (1,1) noted that the GARCH (1,1) model is defined as the white noise for the coefficients of the example models. Notably, this GARCH model satisfies the condition of having time-varying coefficients. This study meticulously outlines the essential stationarity conditions required for these models. The estimation of coefficients is accomplished by applying the least squares method. One of the key contributions lies in utilizing the fundamental theorem of Klimko and Nilsen, to prove the asymptotic behavior of the estimators, particularly how they vary with changes in the sample size. This paper illuminates the impact of estimators and their approximations based on varying sample sizes. Extending our study to include the estimation of bilinear models alongside GARCH and GARCH symmetric coefficients adds depth to our analysis and provides valuable insights into modeling financial time series data. Furthermore, this study sheds light on the influence of the GARCH white noise trace on the estimation of model coefficients. The results establish a clear connection between the model characteristics and the nature of the white noise, contributing to a more profound understanding of the relationship between these elements.

Time series data can be modeled using the Autoregressive Intergrated Moving Average (ARIMA) model. ARIMA model requires the assumption of homoscedasticity. But on financial data, like stock data, has a high fluctuation so that the variant is heteroscedastic. To solve this problem, Autoregressive Conditional Heteroscedasticity (ARCH) or Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model is used. The ARCH process introduced in Engle allows the conditional variance to change over time as a function of past errors leaving the unconditional variance constant. The ARCH model was generalized to the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model and introduced in Bollerslev allowing for a much more flexible lag structure.Another model that can be used is the Elman Recurrent Neural Network model (ERNN) which is a nonlinear model. The purpose of this research is to determine the best model between ARCH / GARCH model and ERNN model based on MSE value. The data used is the return of daily closing price of BRI stocks period of January 1st, 2015 to 17th February 2017. In this study, the best model obtained is GARCH (1,1) which has the smallest MSE value.