Indirect estimation of randomized generalized autoregressive conditional heteroskedastic models

The class of generalized autoregressive conditional heteroskedastic (GARCH) models can be used to describe the volatility with less parameters than autoregressive conditional heteroskedastic (ARCH)-type models, their distributions are heavy-tailed, with time-dependent conditional variance, and are able to model clustering of volatility. Despite all these facts, the way that GARCH models are built imposes limits on the heaviness of the tails of their unconditional distribution. The class of randomized generalized autoregressive conditional heteroskedastic (R-GARCH) models includes the ARCH and GARCH models allowing the use of stable innovations. Estimation methods and empirical analysis of R-GARCH models are the focus of this work. We present the indirect inference method to estimate the R-GARCH models, some simulations and an empirical application.     

The Stochastic Volatility in Mean Model With Time-Varying Parameters: An Application to Inflation Modeling

This article generalizes the popular stochastic volatility in mean model to allow for time-varying parameters in the conditional mean. The estimation of this extension is nontrival since the volatility appears in both the conditional mean and the conditional variance, and its coefficient in the former is time-varying. We develop an efficient Markov chain Monte Carlo algorithm based on band and sparse matrix algorithms instead of the Kalman filter to estimate this more general variant. The methodology is illustrated with an application that involves U.S., U.K., and Germany inflation. The estimation results show substantial time-variation in the coefficient associated with the volatility, highlighting the empirical relevance of the proposed extension. Moreover, in a pseudo out-of-sample forecasting exercise, the proposed variant also forecasts better than various standard benchmarks.     

A Markov Model of Switching-Regime ARCH

In this article I present a new approach to model more realistically the variability of financial time series. I develop a Markov-ARCH model that incorporates the features of both Hamilton's switching-regime model and Engle's autoregressive conditional heteroscedasticity (ARCH) model to examine the issue of volatility persistence in the monthly excess returns of the three-month treasury bill. The issue can be resolved by taking into account occasional shifts in the asymptotic variance of the Markov-ARCH process that cause the spurious persistence of the volatility process. I identify two periods during which there is a regime shift, the 1974:2–1974:8 period associated with the oil shock and the 1979:9–1982:8 period associated with the Federal Reserve's policy change. The variance approached asymptotically in these two episodes is more than 10 times as high as the asymptotic variance for the remainder of the sample. I conclude that regime shifts have a greater impact on the properties of the data, and I cannot reject the null hypothesis of no ARCH effects within the regimes. As a consequence of the striking findings in this article, previous empirical results that adopt an ARCH approach in modeling monthly or lower frequency interest-rate dynamics are rendered questionable.     

Detecting parameter shift in garch models

This paper applies recent theories of testing for parameter constancy to the conditional variance in a GARCH model. The supremum Lagrange multiplier test for conditional Gaussian GARCH models and its robustified variants are discussed. The asymptotic null distribution of the test statistics are derived from the weak convergence of the scores, and the critical values from the hitting probability of squared Bessel process.

Monte Carlo studies on the finite sample size and power performance of the supremum LM tests are conducted. Applications of these tests to S&P 500 indicate that the hypothesis of stable conditional variance parameters can be rejected.     

Detecting parameter shift in garch models

This paper applies recent theories of testing for parameter constancy to the conditional variance in a GARCH model. The supremum Lagrange multiplier test for conditional Gaussian GARCH models and its robustified variants are discussed. The asymptotic null distribution of the test statistics are derived from the weak convergence of the scores, and the critical values from the hitting probability of squared Bessel process.

Monte Carlo studies on the finite sample size and power performance of the supremum LM tests are conducted. Applications of these tests to S&P 500 indicate that the hypothesis of stable conditional variance parameters can be rejected.     

Fractionally integrated GARCH model with tempered stable distribution: a simulation study

With the growing availability of high-frequency data, long memory has become a popular topic in finance research. Fractionally Integrated GARCH (FIGARCH) model is a standard approach to study the long memory of financial volatility. The original specification of FIGARCH model is developed using Normal distribution, which cannot accommodate fat-tailed properties commonly existing in financial time series. Traditionally, the Student-t distribution and General Error Distribution (GED) are used instead to solve that problem. However, a recent study points out that the Student-t lacks stability. Instead, the Stable distribution is introduced. The issue of this distribution is that its second moment does not exist. To overcome this new problem, the tempered stable distribution, which retains most attractive characteristics of the Stable distribution and has defined moments, is a natural candidate. In this paper, we describe the estimation procedure of the FIGARCH model with tempered stable distribution and conduct a series of simulation studies to demonstrate that it consistently outperforms FIGARCH models with the Normal, Student-t and GED distributions. An empirical evidence of the S&P 500 hourly return is also provided with robust results. Therefore, we argue that the tempered stable distribution could be a widely useful tool for modelling the high-frequency financial volatility in general contexts with a FIGARCH-type specification.     

Markov switch smooth transition HYGARCH model: Stability and estimation

Abstract

HYGARCH model is basically used to model long-range dependence in volatility. We propose Markov switch smooth-transition HYGARCH model, where the volatility in each state is a time-dependent convex combination of GARCH and FIGARCH. This model provides a flexible structure to capture different levels of volatilities and also short and long memory effects. The necessary and sufficient condition for the asymptotic stability is derived. Forecast of conditional variance is studied by using all past information through a parsimonious way. Bayesian estimations based on Gibbs sampling are provided. A simulation study has been given to evaluate the estimations and model stability. The competitive performance of the proposed model is shown by comparing it with the HYGARCH and smooth-transition HYGARCH models for some period of the S&P500 and Dow Jones industrial average indices based on volatility and value-at-risk forecasts.     

A wavelet-based approach for modelling exchange rates

This paper proposes a new approach, based on the recent developments of the wavelet theory, to model the dynamic of the exchange rate. First, we consider the maximum overlap discrete wavelet transform (MODWT) to decompose the level exchange rates into several scales. Second, we focus on modelling the conditional mean of the detrended series as well as their volatilities. In particular, we consider the generalized fractional, one-factor, Gegenbauer process (GARMA) to model the conditional mean and the fractionally integrated generalized autoregressive conditional heteroskedasticity process (FIGARCH) to model the conditional variance. Moreover, we estimate the GARMA-FIGARCH model using the wavelet-based maximum likelihood estimator (Whitcher in Technometrics 46:225–238, 2004). To illustrate the usefulness of our methodology, we carry out an empirical application using the daily Tunisian exchange rates relative to the American Dollar, the Euro and the Japanese Yen. The empirical results show the relevance of the selected modelling approach which contributes to a better forecasting performance of the exchange rate series.     

Factor Stochastic Volatility in Mean Models: A GMM Approach

This paper provides a semiparametric framework for modeling multivariate conditional heteroskedasticity. We put forward latent stochastic volatility (SV) factors as capturing the commonality in the joint conditional variance matrix of asset returns. This approach is in line with common features as studied by Engle and Kozicki (1993), and it allows us to focus on identication of factors and factor loadings through first- and second-order conditional moments only. We assume that the time-varying part of risk premiums is based on constant prices of factor risks, and we consider a factor SV in mean model. Additional specification of both expectations and volatility of future volatility of factors provides conditional moment restrictions, through which the parameters of the model are all identied. These conditional moment restrictions pave the way for instrumental variables estimation and GMM inference.     

A Stochastic Volatility Model With Realized Measures for Option Pricing

Abstract

Based on the fact that realized measures of volatility are affected by measurement errors, we introduce a new family of discrete-time stochastic volatility models having two measurement equations relating both observed returns and realized measures to the latent conditional variance. A semi-analytical option pricing framework is developed for this class of models. In addition, we provide analytical filtering and smoothing recursions for the basic specification of the model, and an effective MCMC algorithm for its richer variants. The empirical analysis shows the effectiveness of filtering and smoothing realized measures in inflating the latent volatility persistence—the crucial parameter in pricing Standard and Poor’s 500 Index options.     

INDEX TO VOLUME 4 (1986)

This article provides an empirical investigation of the risk-neutral variance process and the market price of variance risk implied in the foreign-currency options market. There are three principal contributions. First, the parameters of Heston's mean-reverting square-root stochastic volatility model are estimated using dollar/mark option prices from 1987 to 1992. Second, it is shown that these implied parameters can be combined with historical moments of the dollar/mark exchange rate to deduce an estimate of the market price of variance risk. These estimates are found to be nonzero, time varying, and of sufficient magnitude to imply that the compensation for variance risk is a significant component of the risk premia in the currency market. Finally, the out-of-sample test suggests that the historical variance and the Hull and White implied variance contain no more information than that imbedded in the Heston implied variance.     

Stochastic volatility demand systems

We address the estimation of stochastic volatility demand systems. In particular, we relax the homoscedasticity assumption and instead assume that the covariance matrix of the errors of demand systems is time-varying. Since most economic and financial time series are nonlinear, we achieve superior modeling using parametric nonlinear demand systems in which the unconditional variance is constant but the conditional variance, like the conditional mean, is also a random variable depending on current and past information. We also prove an important practical result of invariance of the maximum likelihood estimator with respect to the choice of equation eliminated from a singular demand system. An empirical application is provided, using the BEKK specification to model the conditional covariance matrix of the errors of the basic translog demand system.     

Markov Switching in GARCH Processes and Mean-Reverting Stock-Market Volatility

This article introduces four models of conditional heteroscedasticity that contain Markov-switching parameters to examine their multiperiod stock-market volatility forecasts as predictions of options-implied volatilities. The volatility model that best predicts the behavior of the options-implied volatilities allows the Student-t degrees-of-freedom parameter to switch such that the conditional variance and kurtosis are subject to discrete shifts. The half-life of the most leptokurtic state is estimated to be a week, so expected market volatility reverts to near-normal levels fairly quickly following a spike.     

Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances

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.     

Conditional correlation in asset return and GARCH intensity model

In an asset return series, there is a conditional asymmetric dependence between current return and past volatility depending on the current return’s sign. To take into account the conditional asymmetry, we introduce new models for asset return dynamics in which frequencies of the up and down movements of asset price have conditionally independent Poisson distributions with stochastic intensities. The intensities are assumed to be stochastic recurrence equations of the GARCH type to capture the volatility clustering and the leverage effect. We provide an important linkage between our model and existing GARCH, explain how to apply maximum likelihood estimation to determine the parameters in the intensity model and show empirical results with the S&P 500 index return series.     

Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances

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.     

PRICING AUSTRALIAN S&P200 OPTIONS: A BAYESIAN APPROACH BASED ON GENERALIZED DISTRIBUTIONAL FORMS

This paper develops a new class of option price models and applies it to options on the Australian S&P200 Index. The class of models generalizes the traditional Black‐Scholes framework by accommodating time‐varying conditional volatility, skewness and excess kurtosis in the underlying returns process. An important property of these more general pricing models is that the computational requirements are essentially the same as those associated with the Black‐Scholes model, with both methods being based on one‐dimensional integrals. Bayesian inferential methods are used to evaluate a range of models nested in the general framework, using observed market option prices. The evaluation is based on posterior parameter distributions, as well as posterior model probabilities. Various fit and predictive measures, plus implied volatility graphs, are also used to rank the alternative models. The empirical results provide evidence that time‐varying volatility, leptokurtosis and a small degree of negative skewness are priced in Australian stock market options.     

A fractionally integrated Wishart stochastic volatility model

There has recently been growing interest in modeling and estimating alternative continuous time multivariate stochastic volatility models. We propose a continuous time fractionally integrated Wishart stochastic volatility (FIWSV) process, and derive the conditional Laplace transform of the FIWSV model in order to obtain a closed form expression of moments. A two-step procedure is used, namely estimating the parameter of fractional integration via the local Whittle estimator in the first step, and estimating the remaining parameters via the generalized method of moments in the second step. Monte Carlo results for the procedure show a reasonable performance in finite samples. The empirical results for the S&P 500 and FTSE 100 indexes show that the data favor the new FIWSV process rather than the one-factor and two-factor models of the Wishart autoregressive process for the covariance structure.     

BL-GARCH models and asymmetries in volatility

In this paper the class of Bilinear GARCH (BL-GARCH) models is proposed. BL-GARCH models allow to capture asymmetries in the conditional variance of financial and economic time series by means of interactions between past shocks and volatilities. The availability of likelihood based inference is an attractive feature of BL-GARCH models. Under the assumption of conditional normality, the log-likelihood function can be maximized by means of an EM type algorithm. The main reason for using the EM algorithm is that it allows to obtain parameter estimates which naturally guarantee the positive definiteness of the conditional variance with no need for additional parameter constraints. We also derive a robust LM test statistic which can be used for model identification. Finally, the effectiveness of BL-GARCH models in capturing asymmetric volatility patterns in financial time series is assessed by means of an application to a time series of daily returns on the NASDAQ Composite stock market index.     

Specification testing in nonparametric AR‐ARCH models

In this paper, an autoregressive time series model with conditional heteroscedasticity is considered, where both conditional mean and conditional variance function are modeled nonparametrically. Tests for the model assumption of independence of innovations from past time series values are suggested. Tests based on weighted L²‐distances of empirical characteristic functions are considered as well as a Cramér–von Mises‐type test. The asymptotic distributions under the null hypothesis of independence are derived, and the consistency against fixed alternatives is shown. A smooth autoregressive residual bootstrap procedure is suggested, and its performance is shown in a simulation study.