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.     

Assessment of diagnostic procedures in symmetrical nonlinear regression models

The aim of this paper is to derive diagnostic procedures based on case-deletion model for symmetrical nonlinear regression models, which complements Galea et al. (2005) that developed local influence diagnostics under some perturbation schemes. This class of models includes all symmetric continuous distributions for errors covering both light- and heavy-tailed distributions such as Student-t, logistic-I and -II, power exponential, generalized Student-t, generalized logistic and contaminated normal, among others. Thus, these models can be checked for robustness to outliers in the response variable and diagnostic methods may be a useful tool for an appropriate choice. First, an iterative process for the parameter estimation as well as some inferential results are presented. Besides, we present the results of a simulation study in which the characteristics of heavy-tailed models are evaluated in the presence of outliers. Then, we derive some diagnostic measures such as Cook distance, W-K statistic, one-step approach and likelihood displacement, generalizing results obtained for normal nonlinear regression models. Also, we present simulation studies that illustrate the behavior of diagnostic measures proposed. Finally, we consider two real data sets previously analyzed under normal nonlinear regression models. The diagnostic analysis indicates that a Student-t nonlinear regression model seems to fit the data better than the normal nonlinear regression model as well as other symmetrical nonlinear models in the sense of robustness against extreme observations.     

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.     

Bayesian Robust PCA of Incomplete Data

We present a probabilistic model for robust factor analysis and principal component analysis in which the observation noise is modeled by Student-t distributions in order to reduce the negative effect of outliers. The Student-t distributions are modeled independently for each data dimensions, which is different from previous works using multivariate Student-t distributions. We compare methods using the proposed noise distribution, the multivariate Student-t and the Laplace distribution. Intractability of evaluating the posterior probability density is solved by using variational Bayesian approximation methods. We demonstrate that the assumed noise model can yield accurate reconstructions because corrupted elements of a bad quality sample can be reconstructed using the other elements of the same data vector. Experiments on an artificial dataset and a weather dataset show that the dimensional independency and the flexibility of the proposed Student-t noise model can make it superior in some applications.     

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.     

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.     

Empirical distributions of daily equity index returns: A comparison

The normality assumption concerning the distribution of equity returns has long been challenged both empirically and theoretically. Alternative distributions have been proposed to better capture the characteristics of equity return data. This paper investigates the ability of five alternative distributions to represent the behavior of daily equity index returns over the period 1979–2014: the skewed Student-t distribution, the generalized lambda distribution, the Johnson system of distributions, the normal inverse Gaussian distribution, and the g-and-h distribution. We find that the generalized lambda distribution is a prominent alternative for modeling the behavior of daily equity index returns.     

Indirect estimation of elliptical stable distributions

An indirect estimation approach for elliptical stable distributions is presented. The auxiliary model is another elliptical distribution, the multivariate Student-t distribution. It has parameters that have a one-to-one relationship with those of the elliptical stable, making the proposed indirect approach particularly suitable. The finite sample behaviour of the estimators is analyzed using a comprehensive Monte Carlo study. An application to 27 emerging markets stock indexes concludes the paper.     

Robust Bayesian analysis of heavy-tailed stochastic volatility models using scale mixtures of normal distributions

A Bayesian analysis of stochastic volatility (SV) models using the class of symmetric scale mixtures of normal (SMN) distributions is considered. In the face of non-normality, this provides an appealing robust alternative to the routine use of the normal distribution. Specific distributions examined include the normal, student-t, slash and the variance gamma distributions. Using a Bayesian paradigm, an efficient Markov chain Monte Carlo (MCMC) algorithm is introduced for parameter estimation. Moreover, the mixing parameters obtained as a by-product of the scale mixture representation can be used to identify outliers. The methods developed are applied to analyze daily stock returns data on S&P500 index. Bayesian model selection criteria as well as out-of-sample forecasting results reveal that the SV models based on heavy-tailed SMN distributions provide significant improvement in model fit as well as prediction to the S&P500 index data over the usual normal model.     

An empirical evaluation of fat-tailed distributions in modeling financial time series

There is substantial evidence that many financial time series exhibit leptokurtosis and volatility clustering. We compare the two most commonly used statistical distributions in empirical analysis to capture these features: the t distribution and the generalized error distribution (GED). A Bayesian approach using a reversible-jump Markov chain Monte Carlo method and a forecasting evaluation method are adopted for the comparison. In the Bayesian evaluation of eight daily market returns, we find that the fitted t error distribution outperforms the GED. In terms of volatility forecasting, models with t innovations also demonstrate superior out-of-sample performance.     

Deep recurrent Gaussian processes for outlier-robust system identification

Gaussian Processes (GP) comprise a powerful kernel-based machine learning paradigm which has recently attracted the attention of the nonlinear system identification community, specially due to its inherent Bayesian-style treatment of the uncertainty. However, since standard GP models assume a Gaussian distribution for the observation noise, i.e., a Gaussian likelihood, the learning and predictive capabilities of such models can be severely degraded when outliers are present in the data. In this paper, motivated by our previous work on GP learning with data containing outliers and recent advances in hierarchical (deep GPs) and recurrent GP (RGP) approaches, we introduce an outlier-robust recurrent GP model, the RGP-t. Our approach explicitly models the observation layer, which includes a heavy-tailed Student-t likelihood, and allows for a hierarchy of multiple transition layers to learn the system dynamics directly from estimation data contaminated by outliers. In addition, we modify the original variational framework of standard RGP in order to perform inference with the new RGP-t model. The proposed approach is comprehensively evaluated using six artificial benchmarks, within several outlier contamination levels, and two datasets related to process industry systems (pH neutralization and heat exchanger), whose estimation data undergo large contamination rates. The simulation results obtained by the RGP-t model indicates an impressive resilience to outliers and a superior capability to learn nonlinear dynamics directly from highly outlier-contaminated data in comparison to existing GP models.     

Modelling financial time series based on heavy-tailed market microstructure models with scale mixtures of normal distributions

This paper presents a type of heavy-tailed market microstructure models with the scale mixtures of normal distributions (MM-SMN), which include two specific sub-classes, viz. the slash and the Student-t distributions. Under a Bayesian perspective, the Markov Chain Monte Carlo (MCMC) method is constructed to estimate all the parameters and latent variables in the proposed MM-SMN models. Two evaluating indices, namely the deviance information criterion (DIC) and the test of white noise hypothesis on the standardised residual, are used to compare the MM-SMN models with the classic normal market microstructure (MM-N) model and the stochastic volatility models with the scale mixtures of normal distributions (SV-SMN). Empirical studies on daily stock return data show that the MM-SMN models can accommodate possible outliers in the observed returns by use of the mixing latent variable. These results also indicate that the heavy-tailed MM-SMN models have better model fitting than the MM-N model, and the market microstructure model with slash distribution (MM-s) has the best model fitting. Finally, the two evaluating indices indicate that the market microstructure models with three different distributions are superior to the corresponding stochastic volatility models.     

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.     

Bayesian testing for non-linearity in volatility modeling

Neural networks provide a tool for describing non-linearity in volatility processes of financial data and help to answer the question “how much” non-linearity is present in the data. Non-linearity is studied under three different specifications of the conditional distribution: Gaussian, Student-t and mixture of Gaussians. To rank the volatility models, a Bayesian framework is adopted to perform a Bayesian model selection within the different classes of models. In the empirical analysis, the return series of the Dow Jones Industrial Average index, FTSE 100 and NIKKEI 225 indices over a period of 16 years are studied. The results show different behavior across the three markets. In general, if a statistical model accounts for non-normality and explains most of the fat tails in the conditional distribution, then there is less need for complex non-linear specifications.     

Robust M-estimation of multivariate GARCH models

The Gaussian quasi-maximum likelihood estimator of Multivariate GARCH models is shown to be very sensitive to outliers in the data. A class of robust M-estimators for MGARCH models is developed. To increase the robustness of the estimators, the use of volatility models with the property of bounded innovation propagation is recommended. The Monte Carlo study and an empirical application to stock returns document the good robustness properties of the M-estimator with a fat-tailed Student t loss function.     

Testing for ellipsoidal symmetry: A comparison study

The focus of this paper is the methodology for testing ellipsoidal symmetry, which was recently proposed by Koltchinskii and Sakhanenko [Koltchinskii, V., Sakhanenko, L. 2000. Testing for ellipsoidal symmetry of a multivariate distribution. In: Giné, E., Mason, D., Wellner, J. (Eds.), High Dimensional Probability II. In: Progress in Probability, Birkhäuser, Boston, pp. 493–510]. It is a class of omnibus bootstrap tests that are affine invariant and consistent against any fixed alternative. First, we study their behavior under a sequence of local alternatives. Secondly, a finite sample comparison study of this new class of tests with other popular methods given by Beran, Manzotti et al., and Huffer et al. is carried out. We find that the new tests outperform other methods in preserving the level and have superior power for the most of the chosen alternatives. We also suggest a tool for identifying periods of financial instability and crises when these tests are applied to the distribution of the return rates of stock market indices. These tests can be used in place of tests for normality of asset return distributions since ellipsoidally symmetric distributions are the natural extensions of multivariate normal distributions, so that the capital asset pricing model holds.     

Testing for ellipsoidal symmetry: A comparison study

The focus of this paper is the methodology for testing ellipsoidal symmetry, which was recently proposed by Koltchinskii and Sakhanenko [Koltchinskii, V., Sakhanenko, L. 2000. Testing for ellipsoidal symmetry of a multivariate distribution. In: Giné, E., Mason, D., Wellner, J. (Eds.), High Dimensional Probability II. In: Progress in Probability, Birkhäuser, Boston, pp. 493-510]. It is a class of omnibus bootstrap tests that are affine invariant and consistent against any fixed alternative. First, we study their behavior under a sequence of local alternatives. Secondly, a finite sample comparison study of this new class of tests with other popular methods given by Beran, Manzotti et al., and Huffer et al. is carried out. We find that the new tests outperform other methods in preserving the level and have superior power for the most of the chosen alternatives. We also suggest a tool for identifying periods of financial instability and crises when these tests are applied to the distribution of the return rates of stock market indices. These tests can be used in place of tests for normality of asset return distributions since ellipsoidally symmetric distributions are the natural extensions of multivariate normal distributions, so that the capital asset pricing model holds.     

Distributed robust Gaussian Process regression

We study distributed and robust Gaussian Processes where robustness is introduced by a Gaussian Process prior on the function values combined with a Student-t likelihood. The posterior distribution is approximated by a Laplace Approximation, and together with concepts from Bayesian Committee Machines, we efficiently distribute the computations and render robust GPs on huge data sets feasible. We provide a detailed derivation and report on empirical results. Our findings on real and artificial data show that our approach outperforms existing baselines in the presence of outliers by using all available data.     

Wavelet-based detection of outliers in financial time series

Outliers in financial data can lead to model parameter estimation biases, invalid inferences and poor volatility forecasts. Therefore, their detection and correction should be taken seriously when modeling financial data. The present paper focuses on these issues and proposes a general detection and correction method based on wavelets that can be applied to a large class of volatility models. The effectiveness of the new proposal is tested by an intensive Monte Carlo study for six well-known volatility models and compared to alternative proposals in the literature, before it is applied to three daily stock market indices. The Monte Carlo experiments show that the new method is both very effective in detecting isolated outliers and outlier patches and much more reliable than other alternatives, since it detects a significantly smaller number of false outliers. Correcting the data of outliers reduces the skewness and the excess kurtosis of the return series distributions and allows for more accurate return prediction intervals compared to those obtained when the existence of outliers is ignored.     

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.