Mixture GARCH Models Research Articles

On Mixture GARCH Models: Long, Short Memory and Application in Finance

In this work, we study the famous model of volatility; called model of conditional heteroscedastic autoregressive with mixed memory MMGARCH for modeling nonlinear time series. The MMGARCH model has two mixing components, one is a GARCH short memory and the other is GARCH long memory. the main objective of this search for finds the best model between mixtures of the models we made (long memory with long memory, short memory with short memory and short memory with long memory) Also, the existence of its stationary solution is discussed. The Monte Carlo experiments demonstrate we discovered theoretical. In addition, the empirical application of the MMGARCH model (1, 1) to the daily index DOW and NASDAQ illustrates its capabilities; we find that for the mixture between APARCH and EGARCH is superior to any other model tested because it produces the smallest errors.

Statistical inference for mixture GARCH models with financial application

In this paper we consider mixture generalized autoregressive conditional heteroskedastic models, and propose a new iteration algorithm of type EM for the estimation of model parameters. The maximum likelihood estimates are shown to be consistent, and their asymptotic properties are investigated. More precisely, we derive simple expressions in closed form for the asymptotic covariance matrix and the expected Fisher information matrix of the ML estimator. Finally, we study the model selection and propose testing procedures. A simulation study and an application to financial real-series illustrate the results.

Time-varying mixture GARCH models and asymmetric volatility

The class of mixed normal conditional heteroskedastic (MixN-GARCH) models, which couples a mixed normal distributional structure with GARCH-type dynamics, has been shown to offer a plausible decomposition of the contributions to volatility, as well as excellent out-of-sample forecasting performance, for financial asset returns. In this paper, we generalize the MixN-GARCH model by relaxing the assumption of constant mixing weights. Two different specifications with time-varying mixing weights are considered. In particular, by relating current weights to past returns and realized (component-wise) likelihood values, an empirically reasonable representation of Engle and Ng's (1993) news impact curve with an asymmetric impact of unexpected return shocks on future volatility is obtained. An empirical out-of-sample study confirms the usefulness of the new approach and gives evidence that the leverage effect in financial returns data is closely connected, in a non-linear fashion, to the time-varying interplay of mixture components representing, for example, various groups of market participants.

Time-Varying Mixture GARCH Models and Asymmetric Volatility

The class of mixed normal conditional heteroskedastic (MixN-GARCH) models, which couples a mixed normal distributional structure with GARCH-type dynamics, has been shown to offer a plausible decomposition of the contributions to volatility, as well as excellent out-of-sample forecasting performance, for financial asset returns. In this paper, we generalize the MixN-GARCH model by relaxing the assumption of constant mixing weights. Two different specifications with time-varying mixing weights are considered. In particular, by relating current weights to past returns and realized (component-wise) likelihood values, an empirically reasonable representation of Engle and Ng's (1993) news impact curve with an asymmetric impact of unexpected return shocks on future volatility is obtained. An empirical out-of-sample study confirms the usefulness of the new approach and gives evidence that the leverage effect in financial returns data is closely connected, in a non-linear fashion, to the time-varying interplay of mixture components representing, for example, various groups of market participants.

Stable mixture GARCH models

A new model class for univariate asset returns is proposed which involves the use of mixtures of stable Paretian distributions, and readily lends itself to use in a multivariate context for portfolio selection. The model nests numerous ones currently in use, and is shown to outperform all its special cases. In particular, an extensive out-of-sample risk forecasting exercise for seven major FX and equity indices confirms the superiority of the general model compared to its special cases and other competitors. Estimation issues related to problems associated with mixture models are discussed, and a new, general, method is proposed to successfully circumvent these. The model is straightforwardly extended to the multivariate setting by using an independent component analysis framework. The tractability of the relevant characteristic function then facilitates portfolio optimization using expected shortfall as the downside risk measure.

A class of adaptive importance sampling weighted EM algorithms for efficient and robust posterior and predictive simulation

A class of adaptive sampling methods is introduced for efficient posterior and predictive simulation. The proposed methods are robust in the sense that they can handle target distributions that exhibit non-elliptical shapes such as multimodality and skewness. The basic method makes use of sequences of importance weighted Expectation Maximization steps in order to efficiently construct a mixture of Student-t densities that approximates accurately the target distribution–typically a posterior distribution, of which we only require a kernel–in the sense that the Kullback–Leibler divergence between target and mixture is minimized. We label this approach Mixture oftby Importance Sampling weighted Expectation Maximization (MitISEM). The constructed mixture is used as a candidate density for quick and reliable application of either Importance Sampling (IS) or the Metropolis–Hastings (MH) method. We also introduce three extensions of the basic MitISEM approach. First, we propose a method for applying MitISEM in a sequential manner, so that the candidate distribution for posterior simulation is cleverly updated when new data become available. Our results show that the computational effort reduces enormously, while the quality of the approximation remains almost unchanged. This sequential approach can be combined with a tempering approach, which facilitates the simulation from densities with multiple modes that are far apart. Second, we introduce a permutation-augmented MitISEM approach. This is useful for importance or Metropolis–Hastings sampling from posterior distributions in mixture models without the requirement of imposing identification restrictions on the model’s mixture regimes’ parameters. Third, we propose a partial MitISEM approach, which aims at approximating the joint distribution by estimating a product of marginal and conditional distributions. This division can substantially reduce the dimension of the approximation problem, which facilitates the application of adaptive importance sampling for posterior simulation in more complex models with larger numbers of parameters. Our results indicate that the proposed methods can substantially reduce the computational burden in econometric models like DCC or mixture GARCH models and a mixture instrumental variables model.

A Class of Adaptive Importance Sampling Weighted EM Algorithms for Efficient and Robust Posterior and Predictive Simulation

A class of adaptive sampling methods is introduced for efficient posterior and predictive simulation. The proposed methods are robust in the sense that they can handle target distributions that exhibit non-elliptical shapes such as multimodality and skewness. The basic method makes use of sequences of importance weighted Expectation Maximization steps in order to efficiently construct a mixture of Student-t densities that approximates accurately the target distribution - typically a posterior distribution, of which we only require a kernel - in the sense that the Kullback-Leibler divergence between target and mixture is minimized. We label this approach Mixture of t by Importance Sampling and Expectation Maximization (MitISEM). The constructed mixture is used as a candidate density for quick and reliable application of either Importance Sampling (IS) or the Metropolis-Hastings (MH) method. We also introduce three extensions of the basic MitISEM approach. First, we propose a method for applying MitISEM in a sequential manner. Second, we introduce a permutation-augmented MitISEM approach. Third, we propose a partial MitISEM approach, which aims at approximating the joint distribution by estimating a product of marginal and conditional distributions. This division can substantially reduce the dimension of the approximation problem, which facilitates the application of adaptive importance sampling for posterior simulation in more complex models with larger numbers of parameters. Our results indicate that the proposed methods can substantially reduce the computational burden in econometric models like DCC or mixture GARCH models and a mixture instrumental variables model.

A comparative study of Monte Carlo methods for efficient evaluation of marginal likelihood

Strategic choices for efficient and accurate evaluation of marginal likelihoods by means of Monte Carlo simulation methods are studied for the case of highly non-elliptical posterior distributions. A comparative analysis is presented of possible advantages and limitations of different simulation techniques; of possible choices of candidate distributions and choices of target or warped target distributions; and finally of numerical standard errors. The importance of a robust and flexible estimation strategy is demonstrated where the complete posterior distribution is explored. Given an appropriately yet quickly tuned adaptive candidate, straightforward importance sampling provides a computationally efficient estimator of the marginal likelihood (and a reliable and easily computed corresponding numerical standard error) in the cases investigated, which include a non-linear regression model and a mixture GARCH model. Warping the posterior density can lead to a further gain in efficiency, but it is more important that the posterior kernel be appropriately wrapped by the candidate distribution than that it is warped.

On Some Models for Value-At-Risk

The idea of statistical learning can be applied in financial risk management. In recent years, value-at-risk (VaR) has become the standard tool for market risk measurement and management. For better VaR estimation, Engle and Manganelli (2004) introduced the conditional autoregressive value-at-risk (CAViaR) model to estimate the VaR directly by quantile regression. To entertain the nonlinearity and structural change in the VaR, we extend the CAViaR idea using two approaches: the threshold GARCH (TGARCH) and the mixture-GARCH models. The estimation method of these models are proposed. Our models should possess all the advantages of the CAViaR model and enhance the nonlinear structure. The methods are applied to the S&P500, Hang Seng, Nikkei and Nasdaq indices to illustrate our models.

Forecasting financial volatility of the Athens stock exchange daily returns: an application of the asymmetric normal mixture GARCH model

AbstractIn this paper we model the return volatility of stocks traded in the Athens Stock Exchange using alternative GARCH models. We employ daily data for the period January 1998 to November 2008 allowing us to capture possible positive and negative effects that may be due to either contagion or idiosyncratic sources. The econometric analysis is based on the estimation of a class of five GARCH models under alternative assumptions with respect to the error distribution. The main findings of our analysis are: first, based on a battery of diagnostic tests it is shown that the normal mixture asymmetric GARCH (NM‐AGARCH) models perform better in modeling the volatility of stock returns. Second, it is shown that with the use of the Kupiec's tests for in‐sample and out‐of‐sample forecasting performance the evidence is mixed as the choice of the appropriate volatility model depends on the trading position under consideration. Third, at the 99% confidence interval the NM‐AGARCH model with skewed Student‐distribution outperforms all other competing models both for in‐sample and out‐of‐sample forecasting performance. This increase in predictive performance for higher confidence intervals of the NM‐AGARCH model with skewed Student‐distribution makes this specification consistent with the requirements of the Basel II agreement. Copyright © 2010 John Wiley & Sons, Ltd.

Skew-Normal Mixture and Markov-Switching GARCH Processes

This paper introduces skew-normal (SN) mixture and Markov-switching (MS) GARCH processes for capturing the skewness in the distribution of stock returns. The model class is motivated by the fact that the common way of incorporating asymmetries into Gaussian MS GARCH models, i.e., regime-dependent means, leads to autocorrelated raw returns, which may not be desirable. The appearance of the SN distribution can be explained by a pre-asymptotic behavior of daily stock returns, and can still be viewed as "generic." The dynamic properties of the process are derived, and its in- and out-of-sample performance is compared with that of several competing models in an application to three major European stock markets over a period covering the recent financial turmoil. It turns out that parsimoniously parameterized SN mixture GARCH processes perform best overall. In particular, they outperform both a skewed t GARCH specification as well as normal mixture GARCH models with skewness generated via nonzero component means.

On a dynamic mixture GARCH model

AbstractThis paper proposes a new mixture GARCH model with a dynamic mixture proportion. The mixture Gaussian distribution of the error can vary from time to time. The Bayesian Information Criterion and the EM algorithm are used to estimate the number of parameters as well as the model parameters and their standard errors. The new model is applied to the S&P500 Index and Hang Seng Index and compared with GARCH models with Gaussian error and Student's t error. The result shows that the IGARCH effect in these index returns could be the result of the mixture of one stationary volatility component with another non‐stationary volatility component. The VaR based on the new model performs better than traditional GARCH‐based VaRs, especially in unstable stock markets. Copyright © 2008 John Wiley & Sons, Ltd.

Option Valuation with Normal Mixture GARCH Models

The class of mixture GARCH models introduced by Haas, Mittnik and Paollela (2004) and Alexander and Lazar (2006) provides a better alternative for fitting financial data than various other GARCH models driven by the normal or skewed t-distribution. In this paper we propose different option pricing methodologies when the underlying stock dynamic is modeled by an asymmetric normal mixture GARCH model with K volatility components. Since under GARCH models the market is incomplete there are an infinite number of martingale measures one can use for pricing. For our mixture setting we analyze the impact of three risk-neutral candidates: a generalized local risk neutral valuation relationship, an Esscher transform and an extended Girsanov principle. We investigate the out-of-sample performance of an asymmetric GARCH model with a mixture density of two normals for Call options written on the S&P 500 Index. The performance under all three transformations is quite impressive when compared to the benchmark GARCH model with normal driving noise. The overall improvement is explained not only by the skewness and leptokurtosis exhibited by the innovation mixture distribution, but also by the richer parametrization used in modeling the dynamics of the multi-component conditional volatility.

Asymmetric multivariate normal mixture GARCH

An asymmetric multivariate generalization of the recently proposed class of normal mixture GARCH models is developed. Issues of parametrization and estimation are discussed. Conditions for covariance stationarity and the existence of the fourth moment are derived, and expressions for the dynamic correlation structure of the process are provided. In an application to stock market returns, it is shown that the disaggregation of the conditional (co)variance process generated by the model provides substantial intuition. Moreover, the model exhibits a strong performance in calculating out-of-sample Value-at-Risk measures.

Modelling nonlinearities and heavy tails via threshold normal mixture GARCH models

A new class of flexible threshold normal mixture GARCH models is proposed for the analysis and modelling of the stylized facts appeared in many financial time series. A Bayesian stochastic method is developed and presented for the analysis of the proposed model allowing for automatic model determination and estimation of the thresholds and their unknown number. A computationally feasible algorithm that explores the posterior distribution of the threshold models is designed using Markov chain Monte Carlo stochastic search methods. A simulation study is conducted to assess the performance of the proposed method, and an empirical application of the proposed model is illustrated using real data.

Normal mixture GARCH(1,1): applications to exchange rate modelling

AbstractSome recent specifications for GARCH error processes explicitly assume a conditional variance that is generated by a mixture of normal components, albeit with some parameter restrictions. This paper analyses the general normal mixture GARCH(1,1) model which can capture time variation in both conditional skewness and kurtosis. A main focus of the paper is to provide evidence that, for modelling exchange rates, generalized two‐component normal mixture GARCH(1,1) models perform better than those with three or more components, and better than symmetric and skewed Student'st‐GARCH models. In addition to the extensive empirical results based on simulation and on historical data on three US dollar foreign exchange rates (British pound, euro and Japanese yen), we derive: expressions for the conditional and unconditional moments of all models; parameter conditions to ensure that the second and fourth conditional and unconditional moments are positive and finite; and analytic derivatives for the maximum likelihood estimation of the model parameters and standard errors of the estimates. Copyright © 2006 John Wiley & Sons, Ltd.

Random coefficient mixture (RCM) GARCH models

In financial modelling, it has been constantly pointed out that volatility clustering and conditional nonnormality induced leptokurtosis are observed in high frequency data. Financial time series data are not adequately modelled by normal distribution and empirical evidence on the nonnormality assumption is well documented in the financial literature (see [1,2] for details). An ARMA representation has been used in [3] to derive the kurtosis of the various class of GARCH models such as power GARCH, non-Gaussian GARCH, and nonstationary and random coefficient GARCH. Several empirical studies have shown that mixture distributions are more likely to capture heteroscedasticity observed in high frequency data than normal distribution. This paper derives the moments for a class of hidden Markov models including Markov switching models under mixture distribution. ARCH-type bilinear models considered by Giraitis and Surgailis [4] with mixture errors are also discussed in some details.

Symmetric Normal Mixture GARCH

Normal mixture GARCH models offer a more intuitive and tractable framework for risk analysis and option pricing than student's t-GARCH models. We present a general, symmetric parameterisation for normal mixture GARCH(1,1) models, with analytic derivatives for the maximum likelihood estimation of the model parameters, deriving also the standard errors of the estimates and expressions for the unconditional moments. Also, we formulate specific conditions on the model parameters to ensure positive, finite conditional and unconditional second and fourth moments. Simulations quantify the potential bias and inefficiency of parameter estimates as a function of the weight of the variance component in the mixture. We show that there is a substantial bias of a definite sign on parameter estimates for variance components having very low weight in the mixing law. An empirical application uses moment specification tests and information criteria to determine the optimal number of normal densities in the mixture. For daily returns on three US dollar foreign exchange rates (British pound, Euro and Japanese yen) we find that, whilst normal GARCH(1,1) models fail the specification tests, a simple mixture of two normal densities is found sufficient to capture the conditional and unconditional excess kurtosis in the data. According to our chosen criteria, and given our simulation results, we conclude that a symmetric NM(2)-GARCH(1,1) model, which has the additional benefit of quantifying volatility corresponding to 'normal' and 'exceptional' market circumstances, is optimal for these exchange rate data.

Normal Mixture GARCH(1,1): Applications to Exchange Rate Modelling

Some recent specifications for GARCH error processes explicitly assume a conditional variance that is generated by a mixture of normal components, albeit with some parameter restrictions. This paper analyses the general normal mixture GARCH(1,1) model which can capture time-variation in both conditional skewness and kurtosis. A main focus of the paper is to provide conclusive evidence that, for modelling exchange rates, generalized two component normal mixture GARCH(1,1) models perform better than those with three or more components, and better than symmetric and skewed Student's t-GARCH models. In addition to the extensive empirical results based on simulation and on historical data on three US dollar foreign exchange rates (British pound, Euro and Japanese yen) we derive: expressions for the conditional and unconditional moments of all models; parameter conditions to ensure that the second and fourth conditional and unconditional moments are positive and finite; and analytic derivatives for the maximum likelihood estimation of the model parameters and standard errors of the estimates.