GJR Models Research Articles

A note on the Nelson-Cao inequality constraints in the GJR-GARCH model: is there a leverage effect?

The majority of stylized facts of financial time series and several Value-at-Risk measures are modeled via univariate or multivariate GARCH processes. It is not rare that advanced GARCH models fail to converge for computational reasons, and a usual parsimonious approach is the GJR-GARCH model. There is a disagreement in the literature and the specialized econometric software, on which constraints should be used for the parameters, introducing indirectly the distinction between asymmetry and leverage. We show that the approach used by various software packages is not consistent with the Nelson-Cao inequality constraints. Implementing Monte Carlo simulations, despite of the results being empirically correct, the estimated parameters are not theoretically coherent with the Nelson-Cao constraints for ensuring positivity of conditional variances. On the other hand ruling out the leverage hypothesis, the asymmetry term in the GJR model can take negative values when typical constraints like the condition for the existence of the second and fourth moments, are imposed.

The correct regularity condition and interpretation of asymmetry in EGARCH

In the class of univariate conditional volatility models, the three most popular are the generalized autoregressive conditional heteroskedasticity (GARCH) model of Engle (1982) and Bollerslev (1986), the GJR (or threshold GARCH) model of Glosten et al. (1992), and the exponential GARCH (or EGARCH) model of Nelson (1990, 1991). For purposes of deriving the mathematical regularity properties, including invertibility that relates the standardized residuals to the returns shocks, to determine the likelihood function for estimation, and the statistical conditions to establish asymptotic properties, it is essential to understand the stochastic properties underlying the three univariate models. The random coefficient autoregressive process was used to obtain GARCH by Tsay (1987), an extension of which was used by McAleer (2014) to obtain GJR. A random coefficient complex nonlinear moving average process was used by McAleer and Hafner (2014) to obtain EGARCH. These models can be used to capture asymmetry, which denotes the different impacts on conditional volatility of positive and negative shocks of equal magnitude, and possibly also leverage, which is the negative correlation between returns shocks and subsequent shocks to volatility (see Black 1976). McAleer (2014) showed that asymmetry was possible for GJR, but not leverage. McAleer and Hafner (2014) showed that leverage was not possible for EGARCH. Surprisingly, the condition for asymmetry in EGARCH seem to have been ignored in the literature, or has concentrated on the incorrect parametric condition, with no clear explanation, and hence with associated unclear and misleading interpretations. The purpose of the paper is to derive the regularity condition for asymmetry in EGARCH, and to provide the correct interpretation. It is shown that, in practice, EGARCH always displays asymmetry, though not leverage.

Research on the Effect of Chinese Margin Trading on Market Risk: Based on GJR Model and Filtered Historical Simulation

In this paper, we address the issue that the financial institutes need to identify the risk of margin trading, and we analyze the volatility and value at risk of China’s Shanghai-Shenzhen 300 Index before and since the inception of margin trading policy. We first analyze the statistical attributes of the logarithmic return series. Then we build the GJR-GARCH to model the difference of volatility and leverage effect of the two sample time series. After that, we calculate the dynamic value at risk based on the parametric method. Moreover, we apply the filtered historical simulation with the help of Bootstrap technique to obtain the pathway of return and finally calculate the value at risk under the two circumstances. In the end, we propose some reasonable policies to financial risk management department.

The extent of virgin olive-oil prices’ distribution revealing the behavior of market speculators

The major problem facing olive oil producers each winter campaign, contrary to what is expected, is not whether the harvest will be good or not but whether the sale price will allow them to cover production costs and achieve a reasonable margin of profit or not. The aim of this paper is to study the olive oil price formation mechanisms in order to learn about the traders’ behavior in the olive oil market. We econometrically study the price formation by implementing statistical models and we provide an economic explanation for the stylized facts detected in olive oil price series. For prediction purposes, we use the artificial neural network (ANN) approach. Our main findings indicate that the AR(1)-GJR(1,1) model and the Ornstein–Uhlenbeck process with stochastic volatility succeeded to some extent in capturing the series stylized facts. The unstable participants’ behavior creates the volatility clustering, non-linearity dependent and cyclicity phenomena. By imitating each other in some periods of the campaign, different participants contribute to the fat tails observed in the olive oil price distribution. The best prediction model for the olive oil price is based on a back propagation ANN approach with input information based on discrete wavelet decomposition and recent price past history.

A Note on the Nelson Cao Inequality Constraints in the GJR-GARCH Model: Is There a Leverage Effect?

The majority of stylized facts of financial time series and several Value-at-Risk measures are modeled via univariate or multivariate GARCH processes. It is not rare that advanced GARCH models fail to converge for computational reasons, and a usual parsimonious approach is the GJR-GARCH model. There is a disagreement in the literature and the specialized econometric software, on which constraints should be used for the parameters, introducing indirectly the distinction between asymmetry and leverage. We show that the approach used by various software packages is not consistent with the Nelson-Cao inequality constraints. Implementing Monte Carlo simulations, despite of the results being empirically correct, the estimated parameters are not theoretically coherent with the Nelson-Cao constraints for ensuring positivity of conditional variances. On the other hand ruling out the leverage hypothesis, the asymmetry term in the GJR model can take negative values when typical constraints like the condition for the existence of the second and fourth moments, are imposed.

Nucleon structure functions in noncommutative space-time

In the context of noncommutative space-time we investigate the nucleon structure functions which play an important role in identifying the internal structure of nucleons. We use the corrected vertices and employ new vertices that appear in two approaches of noncommutativity and calculate the proton structure functions in terms of the noncommutative tensor theta _{mu nu }. To check our results we plot the nucleon structure function (NSF), F_2(x), and compare it with experimental data and the results from the GRV, GJR and CT10 parametrization models. We show that with the new vertex that arises the noncommutativity correction will lead to a better consistency between theoretical results and experimental data for the NSF. This consistency will be better for small values of the Bjorken variable x. To indicate and confirm the validity of our calculations we also act conversely. We obtain a lower bound for the numerical values of Lambda _{mathrm{NC}} scale which correspond to recent reports.

The Correct Regularity Condition and Interpretation of Asymmetry in EGARCH

In the class of univariate conditional volatility models, the three most popular are the generalized autoregressive conditional heteroskedasticity (GARCH) model of Engle (1982) and Bollerslev (1986), the GJR (or threshold GARCH) model of Glosten, Jagannathan and Runkle (1992), and the exponential GARCH (or EGARCH) model of Nelson (1990, 1991). For purposes of deriving the mathematical regularity properties, including invertibility, to determine the likelihood function for estimation, and the statistical conditions to establish asymptotic properties, it is convenient to understand the stochastic properties underlying the three univariate models. The random coefficient autoregressive process was used to obtain GARCH by Tsay (1987), an extension of which was used by McAleer (2004) to obtain GJR. A random coefficient complex nonlinear moving average process was used by McAleer and Hafner (2014) to obtain EGARCH. These models can be used to capture asymmetry, which denotes the different effects on conditional volatility of positive and negative effects of equal magnitude, and possibly also leverage, which is the negative correlation between returns shocks and subsequent shocks to volatility (see Black 1979). McAleer (2014) showed that asymmetry was possible for GJR, but not leverage. McAleer and Hafner showed that leverage was not possible for EGARCH. Surprisingly, the conditions for asymmetry in EGARCH seem to have been ignored in the literature, or have concentrated on the incorrect conditions, with no clear explanation, and hence with associated misleading interpretations. The purpose of the paper is to derive the regularity condition for asymmetry in EGARCH to provide the correct interpretation. It is shown that, in practice, EGARCH always displays asymmetry, though not leverage.

The role of outliers and oil price shocks on volatility of metal prices

This study investigates the price volatility of metals, using GARCH and GJR models. First, we examine the persistence of volatility and the leverage effect across metal markets taking into account the presence of outliers. Second, we estimate the effects of oil price shocks on the price volatility of metals, allowing for asymmetric responses. We use daily spot prices for aluminum, copper, lead, nickel, tin, zinc, gold, silver, palladium and platinum. We find that returns have a high degree of volatility persistence before and after correcting for outliers, outliers bias the estimation of GARCH and GJR models, and removing outliers improves volatility modelling. However, Student-t-based specifications outperform outliers correction in capturing volatility. Moreover, we document inverse leverage effects for seven metals, leverage effect for copper and no leverage effects for nickel and palladium. Finally, price volatility of metals reacts differently and asymmetrically to oil price shocks.

Robust M-estimate of GJR model with high frequency data

In this paper, we study the GJR scaling model which embeds the intraday return processes into the daily GJR model and propose a class of robust M-estimates for it. The estimation procedures would be more efficient when high-frequency data is taken into the model. However, high-frequency data brings noises and outliers which may lead to big bias of the estimators. Therefore, robust estimates should be taken into consideration. Asymptotic results are derived from the robust M-estimates without the finite fourth moment of the innovations. A simulation study is carried out to assess the performance of the model and its estimates. Robust M-estimate of GJR model is also applied in predicting VaR for real financial time series.

Impact of Trading Volume on Prediction of Stock Market Development

The paper focuses on the influence of trading volume on quality of prediction of stock market development. The main objective of this article is to assess the influence of stock trading volume level on quality of prediction with use of technical analysis. The research was applied on stocks included in the S & P 500 index. Based on average daily trading volume, three aggregate indexes were constructed. The dynamics of index return volatility was modeled by GARCH-class models. GARCH(1,1), GJR and EGARCH models were estimated for each time series. The in-sample evidence indicated that the return volatility of the indexes can be characterized by significant persistence and asymmetric effects. The best estimate of each model was produced for the index of stocks with the highest average trading volume.However the result could differ based on the observed period, the volatility structure of the examined data supports the idea that influential investors respond to various shocks in the market primarily by closing or opening their largest position.The importance of the level of trading volume for the prediction of financial time series development was shown in the paper. This finding could help generate such volatility structure of time series which would allow to explain development of the time series by various models with better results.

The Role of Outliers and Oil Price Shocks on Volatility of Metal Prices

This study investigates the price volatility of metals, using the GARCH and GJR models. First we examine the persistence of volatility and the leverage effect across metal markets taking into account the presence of outliers, and second we estimate the effects of oil price shocks on the price volatility of metals, allowing for the asymmetric responses. We use daily spot prices for the selected metals, including aluminum, copper, lead, nickel, tin, zinc, gold, silver, palladium and platinum. The main findings indicate that, returns show a high degree of volatility persistence before and after correcting outliers, outliers bias the parameters estimation of the GARCH-type models, and removing outliers improves the performance of models in capturing volatility. However in a comparison, Student-t distribution outperforms the approach of correcting outliers in capturing volatility. Moreover, we find the existence of inverse leverage effect for seven metals, the leverage effect for copper and no leverage effect for nickel and palladium. Finally, price volatility of metals differently reacts to oil price shocks and there is an asymmetric reaction of volatility to oil price shocks for seven metals.

High-order moments and extreme value approach for value-at-risk

Abstract We modify a two-step approach by McNeil and Frey (2000) for forecasting Value-at-Risk (VaR). Our approach combines the asymmetric GARCH (GJR) model that allows the high-order moments (i.e., skewness and kurtosis) of the skewed generalized t (SGT) distribution to rely on the past information set to estimate volatility, and the modified Hill estimator (Huisman et al., 2001) for estimating the innovation distribution tail of the GJR model. Using back-testing of the daily return series of 10 stock markets, the empirical results show that our proposed approach could give better one-day VaR forecasts than McNeil and Frey (2000) and the GJR/GARCH models with alternative distributions. In addition, our proposed approach also provides the accuracy of expected shortfall estimates. The evidence demonstrates that our proposed two-step approach that incorporates the modified Hill estimator into the GJR model based on the SGT density with autoregressive conditional skewness and kurtosis provides consistently accurate VaR forecasts in the short and longer sample periods.

Asymmetry and Leverage in Conditional Volatility Models

The three most popular univariate conditional volatility models are the generalized autoregressive conditional heteroskedasticity (GARCH) model of Engle (1982) and Bollerslev (1986), the GJR (or threshold GARCH) model of Glosten, Jagannathan and Runkle (1992), and the exponential GARCH (or EGARCH) model of Nelson (1990, 1991). The underlying stochastic specification to obtain GARCH was demonstrated by Tsay (1987), and that of EGARCH was shown recently in McAleer and Hafner (2014). These models are important in estimating and forecasting volatility, as well as in capturing asymmetry, which is the different effects on conditional volatility of positive and negative effects of equal magnitude, and purportedly in capturing leverage, which is the negative correlation between returns shocks and subsequent shocks to volatility. As there seems to be some confusion in the literature between asymmetry and leverage, as well as which asymmetric models are purported to be able to capture leverage, the purpose of the paper is three-fold, namely, (1) to derive the GJR model from a random coefficient autoregressive process, with appropriate regularity conditions; (2) to show that leverage is not possible in the GJR and EGARCH models; and (3) to present the interpretation of the parameters of the three popular univariate conditional volatility models in a unified manner.

Value at Risk Estimation in Car Insurance by Conditional Heteroskedasticity

Third party insurance is obligatory in Iran. Therefore a large proportion of portfolio in insurance companies dedicates to car policy, especially third party policy. Regarding to this misbalanced portfolio, we decided to represent a suitable model for heteroskedasticity of car insurance data in Dana insurance company. Conditional heteroskedasticity can increase the certitude of value at risk estimation considerably. The results showed that the best models for estimating conditional variance of collision and third party insurance profits are GJR(1,1) and EGARCH(1,1) models alternatively. Then we calculated value at risk with student's t-distribution and observed that this value in third party insurance is about % and much more than collision insurance value at risk. Thus insurance companies are persuaded to sell collision policy and represent an accurate rate to calculate premium. So value at risk can provide an instruction for more profitability in insurance companies.

Methodology Using Nonlinear Regression Models with Kelly Criterion to Determine Optimal ETF Investment Strategies: An Application to the S&P 500 Index

How to allocate the proportion of money to invest into the stock market to earn optimal profit at lower risk? To answer the question, this study uses the Kelly Criterion to help develop a methodology to determine the optimal proportional investment strategies for ETFs, based on the forecasting results of the Generalized Auto Regressive Conditional Heteroskedasticity (GARCH) and the Glosten-Jagannathan-Runkie GARCH (GJR-GARCH) models, which together consider the volatility of fund indices. We use six different GARCH and GJR models to forecast volatilities, and five different estimated long-term returns in Kelly Criterion to calculate final profit; then we determine the profitable investment strategies. The necessary inputs for the GARCH, GJR, and Kelly Criterion are the closing prices of the S&amp;P 500 Index. While we use the closing prices of the S&amp;P 500 Index from 1950 to 2011 as the inputs of the methodology, we analyze mainly the forecasting results from the representative years from 1981 to 2011. The methodology seems to be robust, as optimal investment strategies can be deduced from it. <b>TOPICS:</b>Exchange-traded funds and applications, statistical methods, equity portfolio management

Geometric ergodicity of asymmetric volatility models with stochastic parameters

In this paper, we consider a general family of asymmetric volatility models with stationary and ergodic coefficients. This family can nest several non-linear asymmetric GARCH models with stochastic parameters into its ambit. It also generalizes Markov-switching GARCH and GJR models. The geometric ergodicity of the proposed process is established. Sufficient conditions for stationarity and existence of moments have also been investigated. Geometric ergodicity of various volatility models with stochastic parameters has been discussed as special cases.

Modelling the Effects of Oil Prices on Global Fertilizer Prices and Volatility

The main purpose of this paper is to evaluate the effect of crude oil price on global fertilizer prices in both the mean and volatility. The endogenous structural breakpoint unit root test, ARDL model, and alternative volatility models, including GARCH, EGARCH, and GJR models, are used to investigate the relationship between crude oil price and six global fertilizer prices. The empirical results from ARDL show that most fertilizer prices are significantly affected by the crude oil price while the volatility of global fertilizer prices and crude oil price from March to December 2008 are higher than in other periods.

Variance Risk Premium and VIX Pricing: A Simple GARCH Approach

This paper assesses variance risk premium and forecasts out-of-sample VIX under GARCH(1,1), GJR, and Heston-Nandi models. With the date-t GARCH parameters estimated in a moving window fashion from 3,500 daily returns of the SP these risk-neutral parameters forecast the date-t VIX accurately with errors of not more than 0.2% on average.

Revisiting Several Popular GARCH Models with Leverage Effect: Differences and Similarities

In this paper, we analyze five of the most popular models proposed to represent conditional heteroscedasticity with leverage effect, namely, GQARCH, TGARCH, GJR, EGARCH, and APARCH. We show that when the parameters satisfy the positivity, stationarity, and finite kurtosis conditions, the dynamics that the GJR and GQARCH models can represent are heavily limited while those of the TGARCH and EGARCH models are less restricted. Alternatively, when the parameters of the GQARCH and GJR models are estimated without restrictions, one can often conclude that the unconditional variance or kurtosis of returns do not exist even when they truly do. Copyright The Author, 2012. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org, Oxford University Press.

International Evidence on GFC‐Robust Forecasts for Risk Management under the Basel Accord

ABSTRACTA risk management strategy designed to be robust to the global financial crisis (GFC), in the sense of selecting a value‐at‐risk (VaR) forecast that combines the forecasts of different VaR models, was proposed by McAleer and coworkers in 2010. The robust forecast is based on the median of the point VaR forecasts of a set of conditional volatility models. Such a risk management strategy is robust to the GFC in the sense that, while maintaining the same risk management strategy before, during and after a financial crisis, it will lead to comparatively low daily capital charges and violation penalties for the entire period. This paper presents evidence to support the claim that the median point forecast of VaR is generally GFC robust. We investigate the performance of a variety of single and combined VaR forecasts in terms of daily capital requirements and violation penalties under the Basel II Accord, as well as other criteria. In the empirical analysis we choose several major indexes, namely French CAC, German DAX, US Dow Jones, UK FTSE100, Hong Kong Hang Seng, Spanish Ibex 35, Japanese Nikkei, Swiss SMI and US S&amp;P 500. The GARCH, EGARCH, GJR and RiskMetrics models as well as several other strategies, are used in the comparison. Backtesting is performed on each of these indexes using the Basel II Accord regulations for 2008–10 to examine the performance of the median strategy in terms of the number of violations and daily capital charges, among other criteria. The median is shown to be a profitable and safe strategy for risk management, both in calm and turbulent periods, as it provides a reasonable number of violations and daily capital charges. The median also performs well when both total losses and the asymmetric linear tick loss function are considered Copyright © 2012 John Wiley &amp; Sons, Ltd.