Conditional Correlation Models of Autoregressive Conditional Heteroscedasticity With Nonstationary GARCH Equations

In this article, we investigate the effects of careful modeling the long-run dynamics of the volatilities of stock market returns on the conditional correlation structure. To this end, we allow the individual unconditional variances in conditional correlation generalized autoregressive conditional heteroscedasticity (CC-GARCH) models to change smoothly over time by incorporating a nonstationary component in the variance equations such as the spline-GARCH model and the time-varying (TV)-GARCH model. The variance equations combine the long-run and the short-run dynamic behavior of the volatilities. The structure of the conditional correlation matrix is assumed to be either time independent or to vary over time. We apply our model to pairs of seven daily stock returns belonging to the S&P 500 composite index and traded at the New York Stock Exchange. The results suggest that accounting for deterministic changes in the unconditional variances improves the fit of the multivariate CC-GARCH models to the data. The effect of careful specification of the variance equations on the estimated correlations is variable: in some cases rather small, in others more discernible. We also show empirically that the CC-GARCH models with time-varying unconditional variances using the TV-GARCH model outperform the other models under study in terms of out-of-sample forecasting performance. In addition, we find that portfolio volatility-timing strategies based on time-varying unconditional variances often outperform the unmodeled long-run variances strategy out-of-sample. As a by-product, we generalize news impact surfaces to the situation in which both the GARCH equations and the conditional correlations contain a deterministic component that is a function of time.     

A Multivariate GARCH Model Incorporating the Direct and Indirect Transmission of Shocks

Theoretical models of contagion and spillovers allow for asset-specific shocks that can be directly transmitted from one asset to another, as well as indirectly transmitted across uncorrelated assets through some intermediary mechanism. Standard multivariate Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models, however, provide estimates of volatilities and correlations based only on the direct transmission of shocks across assets. As such, spillover effects via an intermediary asset or market are not considered. In this article, a multivariate GARCH model is constructed that provides estimates of volatilities and correlations based on both directly and indirectly transmitted shocks. The model is applied to exchange rate and equity returns data. The results suggest that if a spillover component is observed in the data, the spillover augmented models provide significantly different volatility estimates compared to standard multivariate GARCH models.     

A Lagrange multiplier test for testing the adequacy of constant conditional correlation GARCH model

ABSTRACT

A Lagrange multiplier test for testing the parametric structure of a constant conditional correlation-generalized autoregressive conditional heteroskedasticity (CCC-GARCH) model is proposed. The test is based on decomposing the CCC-GARCH model multiplicatively into two components, one of which represents the null model, whereas the other one describes the misspecification. A simulation study shows that the test has good finite sample properties. We compare the test with other tests for misspecification of multivariate GARCH models. The test has high power against alternatives where the misspecification is in the GARCH parameters and is superior to other tests. The test is not greatly affected by misspecification in the conditional correlations and is therefore well suited for considering misspecification of GARCH equations.     

Estimation of the parameters of symmetric stable ARMA and ARMA–GARCH models

In this article, we first propose the modified Hannan–Rissanen Method for estimating the parameters of autoregressive moving average (ARMA) process with symmetric stable noise and symmetric stable generalized autoregressive conditional heteroskedastic (GARCH) noise. Next, we propose the modified empirical characteristic function method for the estimation of GARCH parameters with symmetric stable noise. Further, we show the efficiency, accuracy and simplicity of our methods with Monte-Carlo simulation. Finally, we apply our proposed methods to model the financial data.     

Evaluating Multivariate GARCH Models in the Nordic Electricity Markets

ABSTRACT

This article considers a variety of specification tests for multivariate GARCH models that are used for dynamic hedging in electricity markets. The test statistics include the robust conditional moments tests for sign-size bias along with the recently introduced copula tests for an appropriate dependence structure. We consider this effort worthwhile, since quite often the tests of multivariate GARCH models are omitted and the models become selected ad hoc depending on the results they generate. Hedging performance comparisons, in terms of unconditional and conditional ex-post variance portfolio reduction, are conducted.     

Evidence for hedge fund predictability from a multivariate Student's t full-factor GARCH model

Extending previous work on hedge fund return predictability, this paper introduces the idea of modelling the conditional distribution of hedge fund returns using Student's t full-factor multivariate GARCH models. This class of models takes into account the stylized facts of hedge fund return series, that is, heteroskedasticity, fat tails and deviations from normality. For the proposed class of multivariate predictive regression models, we derive analytic expressions for the score and the Hessian matrix, which can be used within classical and Bayesian inferential procedures to estimate the model parameters, as well as to compare different predictive regression models. We propose a Bayesian approach to model comparison which provides posterior probabilities for various predictive models that can be used for model averaging. Our empirical application indicates that accounting for fat tails and time-varying covariances/correlations provides a more appropriate modelling approach of the underlying dynamics of financial series and improves our ability to predict hedge fund returns.     

Robust estimates for GARCH models

In this paper we present two robust estimates for GARCH models. The first is defined by the minimization of a conveniently modified likelihood and the second is similarly defined, but includes an additional mechanism for restricting the propagation of the effect of one outlier on the next estimated conditional variances. We study the asymptotic properties of our estimates proving consistency and asymptotic normality. A Monte Carlo study shows that the proposed estimates compare favorably with respect to other robust estimates. Moreover, we consider some real examples with financial data that illustrate the behavior of these estimates.     

Modelling asymmetric volatility dynamics by multivariate BL-GARCH models

The class of Multivariate BiLinear GARCH (MBL-GARCH) models is proposed and its statistical properties are investigated. The model can be regarded as a generalization to a multivariate setting of the univariate BL-GARCH model proposed by Storti and Vitale (Stat Methods Appl 12:19–40, 2003a; Comput Stat 18:387–400, 2003b). It is shown how MBL-GARCH models allow to account for asymmetric effects in both conditional variances and correlations. An EM algorithm for the maximum likelihood estimation of the model parameters is derived. Furthermore, in order to test for the appropriateness of the conditional variance and covariance specifications, a set of robust conditional moments test statistics are defined. Finally, the effectiveness of MBL-GARCH models in a risk management setting is assessed by means of an application to the estimation of the optimal hedge ratio in futures hedging.     

Generalized EGARCH Random Effect Models Application to Financial Time Series

In this article, we propose a simple alternative model to analyze the volatility of the financial time series. In the applications, the performance of this model is compared with the performance of the GARCH type models. Using GARCH, EGARCH, and the proposed models, we analyze the time series of the Bovespa and Dow Jones Industrial Average indexes. In the applications we can see that the proposed models have good performance compared with the usual GARCH type model.     

Stepwise local influence in generalized autoregressive conditional heteroskedasticity models

Detection of outliers or influential observations is an important work in statistical modeling, especially for the correlated time series data. In this paper we propose a new procedure to detect patch of influential observations in the generalized autoregressive conditional heteroskedasticity (GARCH) model. Firstly we compare the performance of innovative perturbation scheme, additive perturbation scheme and data perturbation scheme in local influence analysis. We find that the innovative perturbation scheme give better result than other two schemes although this perturbation scheme may suffer from masking effects. Then we use the stepwise local influence method under innovative perturbation scheme to detect patch of influential observations and uncover the masking effects. The simulated studies show that the new technique can successfully detect a patch of influential observations or outliers under innovative perturbation scheme. The analysis based on simulation studies and two real data sets show that the stepwise local influence method under innovative perturbation scheme is efficient for detecting multiple influential observations and dealing with masking effects in the GARCH model.     

Markov switching component GARCH model: Stability and forecasting

ABSTRACT

This paper introduces an extension of the Markov switching GARCH model where the volatility in each state is a convex combination of two different GARCH components with time varying weights. This model has the dynamic behavior to capture the variants of shocks. The asymptotic behavior of the second moment is investigated and an appropriate upper bound for it is evaluated. Using the Bayesian method via Gibbs sampling algorithm, a dynamic method for the estimation of the parameters is proposed. Finally, we illustrate the efficiency of the model by simulation and also by considering two different set of empirical financial data. We show that this model provides much better forecasts of the volatility than the Markov switching GARCH model.     

Exploring the Impact of Multivariate GARCH Innovations on Hypothesis Testing for Cointegrating Vectors

This article investigates the impact of multivariate generalized autoregressive conditional heteroskedastic (GARCH) errors on hypothesis testing for cointegrating vectors. The study reviews a cointegrated vector autoregressive model incorporating multivariate GARCH innovations and a regularity condition required for valid asymptotic inferences. Monte Carlo experiments are then conducted on a test statistic for a hypothesis on the cointegrating vectors. The experiments demonstrate that the regularity condition plays a critical role in rendering the hypothesis testing operational. It is also shown that Bartlett-type correction and wild bootstrap are useful in improving the small-sample size and power performance of the test statistic of interest.     

A dynamic double asymmetric copula generalized autoregressive conditional heteroskedasticity model: application to China's and US stock market

Modeling the relationship between multiple financial markets has had a great deal of attention in both literature and real-life applications. One state-of-the-art technique is that the individual financial market is modeled by generalized autoregressive conditional heteroskedasticity (GARCH) process, while market dependence is modeled by copula, e.g. dynamic asymmetric copula-GARCH. As an extension, we propose a dynamic double asymmetric copula (DDAC)-GARCH model to allow for the joint asymmetry caused by the negative shocks as well as by the copula model. Furthermore, our model adopts a more intuitive way of constructing the sample correlation matrix. Our new model yet satisfies the positive-definite condition as found in dynamic conditional correlation-GARCH and constant conditional correlation-GARCH models. The simulation study shows the performance of the maximum likelihood estimate for DDAC-GARCH model. As a case study, we apply this model to examine the dependence between China and US stock markets since 1990s. We conduct a series of likelihood ratio test tests that demonstrate our extension (dynamic double joint asymmetry) is adequate in dynamic dependence modeling. Also, we propose a simulation method involving the DDAC-GARCH model to estimate value at risk (VaR) of a portfolio. Our study shows that the proposed method depicts VaR much better than well-established variance–covariance method.     

Normal Mixture Quasi-maximum Likelihood Estimator for GARCH Models

Abstract.  The generalized autoregressive conditional heteroscedastic (GARCH) model has been popular in the analysis of financial time series data with high volatility. Conventionally, the parameter estimation in GARCH models has been performed based on the Gaussian quasi-maximum likelihood. However, when the innovation terms have either heavy-tailed or skewed distributions, the quasi-maximum likelihood estimator (QMLE) does not function well. In order to remedy this defect, we propose the normal mixture QMLE (NM-QMLE), which is obtained from the normal mixture quasi-likelihood, and demonstrate that the NM-QMLE is consistent and asymptotically normal. Finally, we present simulation results and a real data analysis in order to illustrate our findings.     

Quasi-maximum exponential likelihood estimation for a non stationary GARCH(1,1) model

ABSTRACT

This article investigates a quasi-maximum exponential likelihood estimator(QMELE) for a non stationary generalized autoregressive conditional heteroscedastic (GARCH(1,1)) model. Asymptotic normality of this estimator is derived under a non stationary condition. A simulation study and a real example are given to evaluate the performance of QMELE for this model.     

A comparison of Bayesian model selection based on MCMC with an application to GARCH-type models

This paper presents a comprehensive review and comparison of five computational methods for Bayesian model selection, based on MCMC simulations from posterior model parameter distributions. We apply these methods to a well-known and important class of models in financial time series analysis, namely GARCH and GARCH-t models for conditional return distributions (assuming normal and t-distributions). We compare their performance with the more common maximum likelihood-based model selection for simulated and real market data. All five MCMC methods proved reliable in the simulation study, although differing in their computational demands. Results on simulated data also show that for large degrees of freedom (where the t-distribution becomes more similar to a normal one), Bayesian model selection results in better decisions in favor of the true model than maximum likelihood. Results on market data show the instability of the harmonic mean estimator and reliability of the advanced model selection methods.     

Surveillance of the covariance matrix of multivariate nonlinear time series

In this paper, sequential procedures for the surveillance of the covariance matrices of multivariate nonlinear time series are introduced. Two different types of control charts are proposed. The first type is based on the exponential smoothing of each component of a local measure for the covariances. The control statistic is equal to the Mahalanobis distance of this quantity with its in-control mean. In our second approach, the Mahalanobis distance is first determined and after that it is exponentially smoothed. We discuss three examples of local measures.

Several properties of the proposed schemes are discussed assuming the target process to be generated by a multivariate GARCH(1, 1) model. The generalization to the family of spherical distributions allows the modelling of frequently observed fat tails in financial data. Some results of an extensive Monte Carlo simulation study are provided in order to judge the performance of the presented control schemes. As a performance measure we use the average run length. An empirical example illustrates the importance of the fast detection of the changes in the covariance structure of the returns of financial assets.     

ASSESSING AND TESTING FOR THRESHOLD NONLINEARITY IN STOCK RETURNS

This paper proposes a test for threshold nonlinearity in a time series with generalized autore‐gressive conditional heteroscedasticity (GARCH) volatility dynamics. This test is used to examine whether financial returns on market indices exhibit asymmetric mean and volatility around a threshold value, using a double‐threshold GARCH model. The test adopts the reversible‐jump Markov chain Monte Carlo idea of Green, proposed in 1995, to calculate the posterior probabilities for a conventional GARCH model and a double‐threshold GARCH model. Posterior evidence favouring the threshold GARCH model indicates threshold nonlinearity with asymmetric behaviour of the mean and volatility. Simulation experiments demonstrate that the test works very well in distinguishing between the conventional GARCH and the double‐threshold GARCH models. In an application to eight international financial market indices, including the G‐7 countries, clear evidence supporting the hypothesis of threshold nonlinearity is discovered, simultaneously indicating an uneven mean‐reverting pattern and volatility asymmetry around a threshold return value.     

Impact study of volatility modelling of Bangladesh stock index using non-normal density

This article examines a wide variety of popular volatility models for stock index return, including the random walk (RW), autoregressive, generalized autoregressive conditional heteroscedasticity (GARCH), and asymmetric GARCH models with normal and non-normal (Student's t and generalized error) distributional assumption. Fitting these models to the Chittagong stock index return data from the period 2 January 1999 to 29 December 2005, we found that the asymmetric GARCH/GARCH model fits better under the assumption of non-normal distribution than under normal distribution. Non-parametric specification tests show that the RW-GARCH, RW-TGARCH, RW-EGARCH, and RW-APARCH models under the Student's t-distributional assumption are significant at the 5% level. Finally, the study suggests that these four models are suitable for the Chittagong Stock Exchange of Bangladesh. We believe that this study would be of great benefit to investors and policy makers at home and abroad.     

Estimation and prediction of time-varying GARCH models through a state-space representation: a computational approach

We propose a state-space approach for GARCH models with time-varying parameters able to deal with non-stationarity that is usually observed in a wide variety of time series. The parameters of the non-stationary model are allowed to vary smoothly over time through non-negative deterministic functions. We implement the estimation of the time-varying parameters in the time domain through Kalman filter recursive equations, finding a state-space representation of a class of time-varying GARCH models. We provide prediction intervals for time-varying GARCH models and, additionally, we propose a simple methodology for handling missing values. Finally, the proposed methodology is applied to the Chilean Stock Market (IPSA) and to the American Standard&Poor's 500 index (S&P500).