Prediction regions for interval-valued time series

We approximate probabilistic forecasts for interval-valued time series by offering alternative approaches. After fitting a possibly non-Gaussian bivariate vector autoregression (VAR) model to the center/log-range system, we transform prediction regions (analytical and bootstrap) for this system into regions for center/range and upper/lower bounds systems. Monte Carlo simulations show that bootstrap methods are preferred according to several new metrics. For daily S&P 500 low/high returns, we build joint conditional prediction regions of the return level and volatility. We illustrate the usefulness of obtaining bootstrap forecasts regions for low/high returns by developing a trading strategy and showing its profitability when compared to using point forecasts.     

An objective function for simulation based inference on exchange rate data

The assessment of models of financial market behaviour requires evaluation tools. When complexity hinders a direct estimation approach, e.g., for agent based microsimulation models, simulation based estimators might provide an alternative. In order to apply such techniques, an objective function is required, which should be based on robust statistics of the time series under consideration. Based on the identification of robust statistics of foreign exchange rate time series in previous research, an objective function is derived. This function takes into account stylized facts about the unconditional distribution of exchange rate returns and properties of the conditional distribution, in particular, autoregressive conditional heteroscedasticity and long memory. A bootstrap procedure is used to obtain an estimate of the variance-covariance matrix of the different moments included in the objective function, which is used as a base for the weighting matrix. Finally, the properties of the objective function are analyzed for two different agent based models of the foreign exchange market, a simple GARCH-model and a stochastic volatility model using the DM/US-$ exchange rate as a benchmark. It is also discussed how the results might be used for inference purposes. Research has been supported by the DFG grant WI 20024/2-1/2. We are indebted to two anonymous referees of this journal, Leigh Tesfatsion, Patrick Burns and other participants of the CEF’06 conference in Limassol for helpful comments on preliminary versions of this paper.     

Robustifying multivariate trend tests to nonstationary volatility

This article studies inference of multivariate trend model when the volatility process is nonstationary. Within a quite general framework we analyze four classes of tests based on least squares estimation, one of which is robust to both weak serial correlation and nonstationary volatility. The existing multivariate trend tests, which either use non-robust standard errors or rely on non-standard distribution theory, are generally non-pivotal involving the unknown time-varying volatility function in the limit. Two-step residual-based i.i.d. bootstrap and wild bootstrap procedures are proposed for the robust tests and are shown to be asymptotically valid. Simulations demonstrate the effects of nonstationary volatility on the trend tests and the good behavior of the robust tests in finite samples.     

Variational Bayes approximation of factor stochastic volatility models

Estimation and prediction in high dimensional multivariate factor stochastic volatility models is an important and active research area, because such models allow a parsimonious representation of multivariate stochastic volatility. Bayesian inference for factor stochastic volatility models is usually done by Markov chain Monte Carlo methods (often by particle Markov chain Monte Carlo methods), which are usually slow for high dimensional or long time series because of the large number of parameters and latent states involved. Our article makes two contributions. The first is to propose a fast and accurate variational Bayes methods to approximate the posterior distribution of the states and parameters in factor stochastic volatility models. The second is to extend this batch methodology to develop fast sequential variational updates for prediction as new observations arrive. The methods are applied to simulated and real datasets, and shown to produce good approximate inference and prediction compared to the latest particle Markov chain Monte Carlo approaches, but are much faster.     

Forecasting the Value-at-Risk of REITs using realized volatility jump models

This paper examines jump risk in the time series of Real Estate Investment Trusts (REITs). Using high-frequency index-level and firm-level data, the econometric model in this paper integrates jumps into the volatility forecast by estimating jump augmented Heterogeneous Autoregressive (HAR) models of realized volatility. To assess the information value of these specifications, their forecasting accuracies for generating one-step ahead daily Value-at-Risk are also compared with other VaR specifications, including those generated from historical returns, bootstrap technique, and severity loss distribution.     

Testing for co-integration in vector autoregressions with non-stationary volatility

Many key macroeconomic and financial variables are characterized by permanent changes in unconditional volatility. In this paper we analyse vector autoregressions with non-stationary (unconditional) volatility of a very general form, which includes single and multiple volatility breaks as special cases. We show that the conventional rank statistics computed as in  and  are potentially unreliable. In particular, their large sample distributions depend on the integrated covariation of the underlying multivariate volatility process which impacts on both the size and power of the associated co-integration tests, as we demonstrate numerically. A solution to the identified inference problem is provided by considering wild bootstrap-based implementations of the rank tests. These do not require the practitioner to specify a parametric model for volatility, or to assume that the pattern of volatility is common to, or independent across, the vector of series under analysis. The bootstrap is shown to perform very well in practice.     

Causality effects in return volatility measures with random times

We provide a structural approach to identify instantaneous causality effects between durations and stock price volatility. So far, in the literature, instantaneous causality effects have either been excluded or cannot be identified separately from Granger type causality effects. By giving explicit moment conditions for observed returns over (random) duration intervals, we are able to identify an instantaneous causality effect. The documented causality effect has significant impact on inference for tick-by-tick data. We find that instantaneous volatility forecasts for, e.g., IBM stock returns must be decreased by as much as 40% when not having seen the next quote change before its (conditionally) median time. Also, instantaneous volatilities are found to be much higher than indicated by standard volatility assessment procedures using tick-by-tick data. For IBM, a naive assessment of spot volatility based on observed returns between quote changes would only account for 60% of the actual volatility. For less liquidly traded stocks at NYSE this effect is even stronger.     

Inference with non-Gaussian Ornstein–Uhlenbeck processes for stochastic volatility

Continuous-time stochastic volatility models are becoming an increasingly popular way to describe moderate and high-frequency financial data. Barndorff-Nielsen and Shephard (2001a) proposed a class of models where the volatility behaves according to an Ornstein–Uhlenbeck (OU) process, driven by a positive Lévy process without Gaussian component. These models introduce discontinuities, or jumps, into the volatility process. They also consider superpositions of such processes and we extend that to the inclusion of a jump component in the returns. In addition, we allow for leverage effects and we introduce separate risk pricing for the volatility components. We design and implement practically relevant inference methods for such models, within the Bayesian paradigm. The algorithm is based on Markov chain Monte Carlo (MCMC) methods and we use a series representation of Lévy processes. MCMC methods for such models are complicated by the fact that parameter changes will often induce a change in the distribution of the representation of the process and the associated problem of overconditioning. We avoid this problem by dependent thinning methods. An application to stock price data shows the models perform very well, even in the face of data with rapid changes, especially if a superposition of processes with different risk premiums and a leverage effect is used.     

Stock market and macroeconomic volatility comparison: an US approach

In 2007, as the US subprime mortgage market began to fall down, which reached its peak with the catastrophic collapse of the Lehman Brothers, no one was aware of that this was going to be the worst financial crisis since the Great Depression. Evaluating the advantages and disadvantages connected with financial globalization demands a pure understanding of the influence of financial volatility. Up to the present few researches focused on analyzing macroeconomic volatility of national economies. Therefore, the aim of the paper is to compare the forecast performance of stock market and macroeconomic volatility of US economy between 2007 and 2010. Accordingly, two different types of financial time series were generated, namely weekly stock returns and quarterly return on investment. Firstly, the appropriate model was determined via time series analysis. Secondly, the relevant ARCH-type model was implemented. Finally, conditional variance forecast performance of models was presented with respect to confidence interval. Furthermore, coefficient of correlation between squared residuals and coefficient of conditional variance was given.     

Stick-breaking autoregressive processes

This paper considers the problem of defining a time-dependent nonparametric prior for use in Bayesian nonparametric modelling of time series. A recursive construction allows the definition of priors whose marginals have a general stick-breaking form. The processes with Poisson-Dirichlet and Dirichlet process marginals are investigated in some detail. We develop a general conditional Markov Chain Monte Carlo (MCMC) method for inference in the wide subclass of these models where the parameters of the marginal stick-breaking process are nondecreasing sequences. We derive a generalised Pólya urn scheme type representation of the Dirichlet process construction, which allows us to develop a marginal MCMC method for this case. We apply the proposed methods to financial data to develop a semi-parametric stochastic volatility model with a time-varying nonparametric returns distribution. Finally, we present two examples concerning the analysis of regional GDP and its growth.     

Testing for a unit root in the volatility of asset returns

It is now well established that the volatility of asset returns is time varying and highly persistent. One leading model that is used to represent these features of the data is the stochastic volatility model. The researcher may test for non-stationarity of the volatility process by testing for a unit root in the log-squared time series. This strategy for inference has many advantages, but is not followed in practice because these unit root tests are known to have very poor size properties. In this paper I show that new tests that are robust to negative MA roots allow a reliable test for a unit root in the volatility process to be conducted. In applying these tests to exchange rate and stock returns, strong rejections of non-stationarity in volatility are obtained. Copyright © 1999 John Wiley & Sons, Ltd.     

Measuring financial risk and portfolio reversion with time changed tempered stable Lévy processes

Given that underlying assets in financial markets exhibit stylized facts such as leptokurtosis, asymmetry, clustering properties and heteroskedasticity effect, this paper applies the stochastic volatility models driven by tempered stable Lévy processes to construct time changed tempered stable Lévy processes (TSSV) for financial risk measurement and portfolio reversion. The TSSV model framework permits infinite activity jump behaviors of returns dynamics and time varying volatility consistently observed in financial markets by introducing time changing volatility into tempered stable processes which specially refer to normal tempered stable (NTS) distribution as well as classical tempered stable (CTS) distribution, capturing leptokurtosis, fat tailedness and asymmetry features of returns in addition to volatility clustering effect in stochastic volatility. Through employing the analytical characteristic function and fast Fourier transform (FFT) technique, the closed form formulas for probability density function (PDF) of returns, value at risk (VaR) and conditional value at risk (CVaR) can be derived. Finally, in order to forecast extreme events and volatile market, we perform empirical researches on Hangseng index to measure risks and construct portfolio based on risk adjusted reward risk stock selection criteria employing TSSV models, with the stochastic volatility normal tempered stable (NTSSV) model producing superior performances relative to others.     

Markov switching quantile autoregression

This paper considers the location‐scale quantile autoregression in which the location and scale parameters are subject to regime shifts. The regime changes in lower and upper tails are determined by the outcome of a latent, discrete‐state Markov process. The new method provides direct inference and estimate for different parts of a non‐stationary time series distribution. Bayesian inference for switching regimes within a quantile, via a three‐parameter asymmetric Laplace distribution, is adapted and designed for parameter estimation. Using the Bayesian output, the marginal likelihood is readily available for testing the presence and the number of regimes. The simulation study shows that the predictability of regimes and conditional quantiles by using asymmetric Laplace distribution as the likelihood is fairly comparable with the true model distributions. However, ignoring that autoregressive coefficients might be quantile dependent leads to substantial bias in both regime inference and quantile prediction. The potential of this new approach is illustrated in the empirical applications to the US inflation and real exchange rates for asymmetric dynamics and the S&P 500 index returns of different frequencies for financial market risk assessment.     

Evaluating interval forecasts of high‐frequency financial data

A number of methods of evaluating the validity of interval forecasts of financial data are analysed, and illustrated using intraday FTSE100 index futures returns. Some existing interval forecast evaluation techniques, such as the Markov chain approach of Christoffersen ( 1998 ), are shown to be inappropriate in the presence of periodic heteroscedasticity. Instead, we consider a regression‐based test, and a modified version of Christoffersen's Markov chain test for independence, and analyse their properties when the financial time series exhibit periodic volatility. These approaches lead to different conclusions when interval forecasts of FTSE100 index futures returns generated by various GARCH(1,1) and periodic GARCH(1,1) models are evaluated. Copyright © 2003 John Wiley & Sons, Ltd.     

Bootstrapping realized multivariate volatility measures

We propose a bootstrap method for statistics that are a function of multivariate high frequency returns such as realized regression, covariance and correlation coefficients. We show that the finite sample performance of the bootstrap is superior to the existing first-order asymptotic theory. Nevertheless, and contrary to the existing results in the bootstrap literature for regression models subject to error heteroskedasticity, the Edgeworth expansion for the pairs bootstrap that we develop here shows that this method is not second-order accurate. We argue that this is due to the fact that the conditional mean parameters of realized regression models are heterogeneous under stochastic volatility.     

Functional data analysis for volatility

We introduce a functional volatility process for modeling volatility trajectories for high frequency observations in financial markets and describe functional representations and data-based recovery of the process from repeated observations. A study of its asymptotic properties, as the frequency of observed trades increases, is complemented by simulations and an application to the analysis of intra-day volatility patterns of the S&P 500 index. The proposed volatility model is found to be useful to identify recurring patterns of volatility and for successful prediction of future volatility, through the application of functional regression and prediction techniques.     

Stock market volatility and the business cycle

This paper investigates the joint time series behavior of monthly stock returns and growth in industrial production. We find that stock returns are well characterized by year-long episodes of high volatility, separated by longer quiet periods. Real output growth, on the other hand, is subject to abrupt changes in the mean associated with economic recessions. We study a bivariate model in which these two changes are driven by related unobserved variables, and conclude that economic recessions are the primary factor that drives fluctuations in the volatility of stock returns. This framework proves useful both for forecasting stock volatility and for identifying and forecasting economic turning points.     

Realising the future: forecasting with high‐frequency‐based volatility (HEAVY) models

This paper studies in some detail a class of high‐frequency‐based volatility (HEAVY) models. These models are direct models of daily asset return volatility based on realised measures constructed from high‐frequency data. Our analysis identifies that the models have momentum and mean reversion effects, and that they adjust quickly to structural breaks in the level of the volatility process. We study how to estimate the models and how they perform through the credit crunch, comparing their fit to more traditional GARCH models. We analyse a model‐based bootstrap which allows us to estimate the entire predictive distribution of returns. We also provide an analysis of missing data in the context of these models. Copyright © 2010 John Wiley & Sons, Ltd.     

Forecasting cryptocurrency volatility

This paper studies the behavior of cryptocurrencies’ financial time series, of which Bitcoin is the most prominent example. The dynamics of these series are quite complex, displaying extreme observations, asymmetries, and several nonlinear characteristics that are difficult to model and forecast. We develop a new dynamic model that is able to account for long memory and asymmetries in the volatility process, as well as for the presence of time-varying skewness and kurtosis. The empirical application, carried out on 606 cryptocurrencies, indicates that a robust filter for the volatility of cryptocurrencies is strongly required. Forecasting results show that the inclusion of time-varying skewness systematically improves volatility, density, and quantile predictions at different horizons.     

Bayesian forecasting and portfolio decisions using dynamic dependent sparse factor models

We extend the recently introduced latent threshold dynamic models to include dependencies among the dynamic latent factors which underlie multivariate volatility. With an ability to induce time-varying sparsity in factor loadings, these models now also allow time-varying correlations among factors, which may be exploited in order to improve volatility forecasts. We couple multi-period, out-of-sample forecasting with portfolio analysis using standard and novel benchmark neutral portfolios. Detailed studies of stock index and FX time series include: multi-period, out-of-sample forecasting, statistical model comparisons, and portfolio performance testing using raw returns, risk-adjusted returns and portfolio volatility. We find uniform improvements on all measures relative to standard dynamic factor models. This is due to the parsimony of latent threshold models and their ability to exploit between-factor correlations so as to improve the characterization and prediction of volatility. These advances will be of interest to financial analysts, investors and practitioners, as well as to modeling researchers.