Non-linear properties of conditional returns under scale mixtures

This article provides a framework for analyzing multifactor financial returns that violate the Gaussian distributional assumption. Analytical expressions are provided for the non-linear regression equation and its prediction error (heteroscedasticity) by modeling the returns of financial assets as scale mixtures of the multivariate normal distribution. The expressions involve conditional moments of the mixing variable. These conditional moments are explicitly derived when the mixing variable belongs to the generalized inverse Gaussian family, of which gamma, inverse gamma and the inverse Gaussian distributions are distinguished members. The derived expressions are non-linear in the parameters and involve the modified Bessel function of the third kind. The effects of the non-linear model, in terms of both the regression equation and heteroscedasticity against the corresponding values for the standard linear regression model, are captured through simulations for the gamma, inverse gamma and inverse Gaussian distributions. The proposed scale mixture models extend the well-known arbitrage pricing theory (APT) in financial modeling to non-Gaussian cases. The methodology is applied to analyze the intra-day log returns quarterly data for DELL and COKE regressed against S&P 500 for the years 1998-2000.     

GARCH dependence in extreme value models with Bayesian inference

Extreme value methods are widely used in financial applications such as risk analysis, forecasting and pricing models. One of the challenges with their application in finance is accounting for the temporal dependence between the observations, for example the stylised fact that financial time series exhibit volatility clustering. Various approaches have been proposed to capture the dependence. Commonly a two-stage approach is taken, where the volatility dependence is removed using a volatility model like a GARCH (or one of its many incarnations) followed by application of standard extreme value models to the assumed independent residual innovations.This study examines an alternative one stage approach, which makes parameter estimation and accounting for the associated uncertainties more straightforward than the two-stage approach. The location and scale parameters of the extreme value distribution are defined to follow a conditional autoregressive heteroscedasticity process. Essentially, the model implements GARCH volatility via the extreme value model parameters. Bayesian inference is used and implemented via Markov chain Monte Carlo, to permit all sources of uncertainty to be accounted for. The model is applied to both simulated and empirical data to demonstrate performance in extrapolating the extreme quantiles and quantifying the associated uncertainty.     

Modeling Hong Kong’s stock index with the Student t-mixture autoregressive model

It is well known that financial returns are usually not normally distributed, but rather exhibit excess kurtosis. This implies that there is greater probability mass at the tails of the marginal or conditional distribution. Mixture-type time series models are potentially useful for modeling financial returns. However, most of these models make the assumption that the return series in each component is conditionally Gaussian, which may result in underestimates of the occurrence of extreme financial events, such as market crashes. In this paper, we apply the class of Student t-mixture autoregressive (TMAR) models to the return series of the Hong Kong Hang Seng Index. A TMAR model consists of a mixture of g autoregressive components with Student t-error distributions. Several interesting properties make the TMAR process a promising candidate for financial time series modeling. These models are able to capture serial correlations, time-varying means and volatilities, and the shape of the conditional distributions can be time-varied from short- to long-tailed or from unimodal to multi-modal. The use of Student t-distributed errors in each component of the model allows for conditional leptokurtic distribution, which can account for the commonly observed unconditional kurtosis in financial data.     

A generalized Dynamic Conditional Correlation model for portfolio risk evaluation

We propose a generalization of the Dynamic Conditional Correlation multivariate GARCH model of Engle [R.F. Engle, Dynamic conditional correlation: a simple class of multivariate generalized autoregressive conditional heteroskedasticity models, Journal of Business and Economic Statistics 20 (2002) 339–350] and of the Asymmetric Dynamic Conditional Correlation model of Cappiello et al.[L. Cappiello, R.F. Engle, K. Sheppard, Asymmetric dynamics in the correlations of global equity and bond returns, Journal of Financial Econometrics 25 (2006) 537–572]. The model we propose introduces a block structure in parameter matrices that allows for interdependence with a reduced number of parameters. Our model nests the Flexible Dynamic Conditional Correlation model of Billio et al. [M. Billio, M. Caporin, M. Gobbo, Flexible dynamic conditional correlation multivariate GARCH for asset allocation, Applied Financial Economics Letters 2 (2006) 123–130] and is named Quadratic Flexible Dynamic Conditional Correlation Multivariate GARCH. In the paper, we provide conditions for positive definiteness of the conditional correlations. We also present an empirical application to the Italian stock market comparing alternative correlation models for portfolio risk evaluation.     

Heavy-tailed mixture GARCH volatility modeling and Value-at-Risk estimation

This paper presents a heavy-tailed mixture model for describing time-varying conditional distributions in time series of returns on prices. Student-t component distributions are taken to capture the heavy tails typically encountered in such financial data. We design a mixture MT(m)-GARCH(p, q) volatility model for returns, and develop an EM algorithm for maximum likelihood estimation of its parameters. This includes formulation of proper temporal derivatives for the volatility parameters. The experiments with a low order MT(2)-GARCH(1, 1) show that it yields results with improved statistical characteristics and economic performance compared to linear and nonlinear heavy-tail GARCH, as well as normal mixture GARCH. We demonstrate that our model leads to reliable Value-at-Risk performance in short and long trading positions across different confidence levels.     

Comparison of alternative ACD models via density and interval forecasts: Evidence from the Australian stock market

In this paper a number of alternative autoregressive conditional duration (ACD) models are compared using a sample of data for three major companies traded on the Australian Stock Exchange. The comparison is performed by employing the methodology for evaluating density and interval forecasts, developed by Diebold et al. [F. Diebold, A. Gunther, S. Tay, Evaluating density forecasts with applications to financial risk management, International Economic Review 39 (1998) 863–883] and Christoffersen [P. Christoffersen, Evaluating interval forecasts, International Economic Review 39 (1998) 841–862], respectively. Our main finding is that the generalized gamma and log-normal distributions for the error terms have similar performance and perform better that the exponential and Weibull distributions. Additionally, there seems to be no substantial difference between the standard ACD specification of Engle and Russel [R. Engle, J. Russell, Autoregressive conditional duration: a new model for irregularly-spaced transaction data, Econometrica 66 (1998) 1127–1162] and the log-ACD specification of Bauwens and Giot [L. Bauwens, P. Giot, The logarithmic ACD model: an application to the bid-ask quote process of three NYSE stocks, Annales d’Economie et de Statistique 60 (2000) 117–150].     

Bayesian analysis of stochastic volatility models with mixture-of-normal distributions

Stochastic volatility (SV) models usually assume that the distribution of asset returns conditional on the latent volatility is normal. This article analyzes SV models with a mixture-of-normal distributions in order to compare with other heavy-tailed distributions such as the Student-t distribution and generalized error distribution (GED). A Bayesian method via Markov-chain Monte Carlo (MCMC) techniques is used to estimate parameters and Bayes factors are calculated to compare the fit of distributions. The method is illustrated by analyzing daily data from the Yen/Dollar exchange rate and the Tokyo stock price index (TOPIX). According to Bayes factors, we find that while the t distribution fits the TOPIX better than the normal, the GED and the normal mixture, the mixture-of-normal distributions give a better fit to the Yen/Dollar exchange rate than other models. The effects of the specification of error distributions on the Bayesian confidence intervals of future returns are also examined. Comparison of SV with GARCH models shows that there are cases that the SV model with the normal distribution is less effective to capture leptokurtosis than the GARCH with heavy-tailed distributions.     

On pseudo maximum likelihood estimation for multivariate time series models with conditional heteroskedasticity

We consider a general multivariate conditional heteroskedastic model under a conditional distribution that is not necessarily normal. This model contains autoregressive conditional heteroskedastic (ARCH) models as a special class. We use the pseudo maximum likelihood estimation method and derive a new estimator of the asymptotic variance matrix for the pseudo maximum likelihood estimator. We also study four special cases in this class, which are conditional heteroskedastic autoregressive moving-average models, regression models with ARCH errors, models with constant conditional correlations, and ARCH in mean models.     

ARCH模型的研究与探讨

自回归条件异方差(ARCH)模型是近年来新发展起来的时间序列模型,它反映了随机 过程的一种特殊特性：即方差随时间变化而变化,且具有丛集性、波动性．ARCH模型已广泛 地应用于经济领域的建模及研究过程中．本文介绍了ARCH模型的特点,它的参数估计和检验 ,以及ARCH模型的发展情况．     

Fat tails and asymmetry in financial volatility models

Although the generalised autoregressive conditional heteroskedasticity (GARCH) model has been quite successful in capturing important empirical aspects of financial data, particularly for the symmetric effects of volatility, it has had far less success in capturing the effects of extreme observations, outliers and skewness in returns. This paper examines the GARCH model under various non-normal error distributions in order to evaluate skewness and leptokurtosis. The empirical results show that GARCH models estimated using asymmetric leptokurtic distributions are superior to their counterparts estimated under normality, in terms of: (i) capturing skewness and leptokurtosis; (ii) the maximized log-likelihood values; and (iii) isolating the ARCH and GARCH parameter estimates from the adverse effects of outliers. Overall, the flexible asymmetric Student’s t-distribution performs best in capturing the non-normal aspects of the data.     

Testing for multivariate autoregressive conditional heteroskedasticity using wavelets

Test statistics for autoregressive conditional heteroskedasticity (ARCH) in the residuals from a possibly nonlinear and dynamic multivariate regression model are considered. The new approach is based on estimation of the multivariate spectral density of squared and cross-residuals. A simple wavelet-based spectral density estimator is advocated, which is a particularly suitable analytic tool when the spectral density exhibits peaks or kinks that may arise from strong cross-dependence, seasonal patterns and other forms of periodic behaviors. In several circumstances, the spectral density may have peaks at various frequencies, such as seasonal frequencies, and the wavelet method may capture them effectively. Compared to kernel-based test statistics for multivariate ARCH effects, the weighting scheme offered by the new wavelet-based test statistics differs in several important aspects. An asymptotic analysis under the null hypothesis of no ARCH effects shows that the wavelet-based test statistic converges in distribution to a convenient standard normal distribution. Under fixed alternatives, the consistency of the wavelet-based test statistics is established in a class of static regression models with uncorrelated but dependent errors. In a Monte Carlo study comparisons are made under various alternatives between the proposed wavelet-based test statistics, the kernel-based test statistics for ARCH effects, and several popular portmanteau test statistics for ARCH effects available in the literature.     

On asymmetric generalised t stochastic volatility models

In stochastic volatility (SV) models, asset returns conditional on the latent volatility are usually assumed to have a normal, Student-t or exponential power (EP) distribution. An earlier study uses a generalised t (GT) distribution for the conditional returns and the results indicate that the GT distribution provides a better model fit to the Australian Dollar/Japanese Yen daily exchange rate than the Student-t distribution. In fact, the GT family nests a number of well-known distributions including the commonly used normal, Student-t and EP distributions. This paper extends the SV model with a GT distribution by incorporating general volatility asymmetry. We compare the empirical performance of nested distributions of the GT distribution as well as different volatility asymmetry specifications. The new asymmetric GT SV models are estimated using the Bayesian Markov chain Monte Carlo (MCMC) method to obtain parameter and log-volatility estimates. By using daily returns from the Standard and Poors (S&P) 500 index, we investigate the effects of the specification of error distributions as well as volatility asymmetry on parameter and volatility estimates. Results show that the choice of error distributions has a major influence on volatility estimation only when volatility asymmetry is not accounted for.     

Mixture periodic autoregressive conditional heteroskedastic models

Mixture Periodically Correlated Autoregressive Conditionally Heteroskedastic (MPARCH) model, which extends the ARCH model, is proposed. The primary motivation behind this extension is to make the model consistent with high kurtosis, outliers and extreme events, and at the same time, able to capture the periodicity feature exhibited by the autocovariance structure. The second and the fourth moment periodically stationary conditions and their closed-forms are derived. Maximum likelihood estimation is obtained via the iterative Expectation Maximization algorithm and the performance of this algorithm is shown via a simulation studies and the MPARCH models are fitted to a real data set.     

Mixture periodic autoregressive conditional heteroskedastic models

Mixture Periodically Correlated Autoregressive Conditionally Heteroskedastic (MPARCH) model, which extends the ARCH model, is proposed. The primary motivation behind this extension is to make the model consistent with high kurtosis, outliers and extreme events, and at the same time, able to capture the periodicity feature exhibited by the autocovariance structure. The second and the fourth moment periodically stationary conditions and their closed-forms are derived. Maximum likelihood estimation is obtained via the iterative Expectation Maximization algorithm and the performance of this algorithm is shown via a simulation studies and the MPARCH models are fitted to a real data set.     

Adaptive signal processing of asset price dynamics with predictability analysis

In this paper we illustrate the optimal filtering of log returns of commodity prices in which both the mean and volatility are modulated by a hidden Markov chain with finite state space. The optimal estimate of the Markov chain and the parameters of the price model are given in terms of discrete-time recursive filters. We provide an application on a set of high frequency gold price data for the period 1973-2006 and analyse the h-step ahead price predictions against the Diebold-Kilian metric. Within the modelling framework where the mean and volatility are switching regimes, our findings suggest that a two-state hidden Markov model is sufficient to describe the dynamics of the data and the gold price is predictable up to a certain extent in the short term but almost impossible to predict in the long term. The proposed model is also benchmarked with ARCH and GARCH models with respect to price predictability and forecasting errors.     

A Student-<Emphasis Type="Italic">t</Emphasis> Full Factor Multivariate GARCH Model

We extend the full-factor multivariate GARCH model of Vrontos et al. (Econom J 6:312–334, 2003a) to account for fat tails in the conditional distribution of financial returns, using a multivariate Student-t error distribution. For the new class of Student-t full factor multivariate GARCH models, we derive analytical expressions for the score, the Hessian matrix and the Information matrix. These expressions can be used within classical inferential procedures in order to obtain maximum likelihood estimates for the model parameters. This fact, combined with the parsimonious parameterization of the covariance matrix under the full factor multivariate GARCH models, enables us to apply the models in high dimensional problems. We provide implementation details and illustrations using financial time series on eight stocks of the US market.     

Modeling of Asymmetry between Gasoline and Crude Oil Prices: A Monte Carlo Comparison

An Engle–Granger two-step procedure is commonly used to estimate cointegrating vectors and consequently asymmetric error-correction models. This study uses Monte Carlo methods and demonstrates that the Engle–Granger two-step method leads to biased estimates of asymmetric parameters and in some cases suggests symmetry in the asymmetric data generating process (DGP). The single equation error correction models (SEECM) based on ordinary least squares (OLS) and nonlinear least squares (NLS) are employed for simultaneous estimation of the cointegrating vector and the ECM. The SEECMs perform better than Engle–Granger two-step procedures in estimating the asymmetry and making inferences on its existence in various DGPs. We show that SEECM estimations are less biased and inferences are less likely to be misleading compared to the Engle–Granger two-step procedure. Unlike the asymmetric specifications based on Engle–Granger two-step approach, the asymmetric SEECMs do not refute the possibility of long-run asymmetry by allowing different cointegrating vectors for positive and negative regimes. Examining the model with real data also supports the Monte Carlo results. While the conventional approaches imply symmetry, the proposed asymmetric SEECM, which has been embedded in a Threshold Autoregressive model, uncovers asymmetry at the presence of different cointegrating vectors for positive and negative regimes.     

Simplified specifications of a multivariate generalized autoregressive conditional heteroscedasticity model

Recent developments in multivariate volatility modeling suggest that the conditional correlation matrix can be described by a time series recursion, where the total number of parameters grows by the power-of-two of the dimension of financial returns. The power of two computational requirement makes high-dimensional multivariate volatility modeling very time consuming. In this paper, we propose two simplified specifications in a multivariate autoregressive conditional heteroscedasticity model. The first specification computes an unconditional correlation matrix from standardized residuals of the model. The second specification restricts the sum of the weights in a time-varying conditional correlation equation to be one. Applying a Bayesian sampling scheme allows the number of parameters to be reduced from the power of two of the dimension to the linear order of the dimension only and simultaneously provides us a framework for model comparison. We test our simplified specifications using simulated and real data from three sectoral indices in Hong Kong, three market indices and four exchange rates. The results suggest that our simplified specifications are more effective than the original formulation.     

Modeling threshold conditional heteroscedasticity with regime-dependent skewness and kurtosis

Construction of nonlinear time series models with a flexible probabilistic structure is an important challenge for statisticians. Applications of such a time series model include ecology, economics and finance. In this paper we consider a threshold model for all the first four conditional moments of a time series. The nonlinear structure in the conditional mean is specified by a threshold autoregression and that of the conditional variance by a threshold generalized autoregressive conditional heteroscedastic (GARCH) model. There are many options for the conditional innovation density in the modeling of the skewness and kurtosis such as the Gram-Charlier (GC) density and the skewed-t density. The Gram-Charlier (GC) density allows explicit modeling of the skewness and kurtosis parameters and therefore is the main focus of this paper. However, its performance is compared with that of Hansen’s skewed-t distribution in the data analysis section of the paper. The regime-dependent feature for the first four conditional moments allows more flexibility in modeling and provides better insights into the structure of a time series. A Lagrange multiplier (LM) test is developed for testing for the presence of threshold structure. The test statistic is similar to the classical tests for the presence of a threshold structure but allowing for a more general regime-dependent structure. The new model and the LM test are illustrated using the Dow Jones Industrial Average, the Hong Kong Hang Seng Index and the Yen/US exchange rate.     

利用PSO对上证指数ARCH模型的实证研究

针对ARCH模型传统估计方法的不足,提出了利用微粒群算法及其改进的算法快速精确的估计ARCH模型的参数,最后利用微粒群算法实证建立了上证指数收益的ARCH模型,并且对以后的情况进行了预测。