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A new process—the factorial hidden Markov volatility (FHMV) model—is proposed to model financial returns or realized variances. Its dynamics are driven by a latent volatility process specified as a product of three components: a Markov chain controlling volatility persistence, an independent discrete process capable of generating jumps in the volatility, and a predictable (data-driven) process capturing the leverage effect. An economic interpretation is attached to each one of these components. Moreover, the Markov chain and jump components allow volatility to switch abruptly between thousands of states, and the transition matrix of the model is structured to generate a high degree of volatility persistence. An empirical study on six financial time series shows that the FHMV process compares favorably to state-of-the-art volatility models in terms of in-sample fit and out-of-sample forecasting performance over time horizons ranging from 1 to 100 days. Supplementary materials for this article are available online.

We present an estimation and forecasting method, based on a differential evolution MCMC method, for inference in GARCH models subjected to an unknown number of structural breaks at unknown dates. We treat break dates as parameters and determine the number of breaks by computing the marginal likelihoods of competing models. We allow for both recurrent and non-recurrent (change-point) regime specifications.We illustrate the estimation method through simulations and apply it to seven financial time series of daily returns. We find structural breaks in the volatility dynamics of all series and recurrent regimes in nearly all series. Finally, we carry out a forecasting exercise to evaluate the usefulness of structural break models.

This paper compares the forecasting performance of models that have been proposed for forecasting in the presence of structural breaks. They differ in their treatment of the break process, the model applied in each regime and the out-of-sample probability of a break. In an extensive empirical evaluation, we demonstrate the presence of breaks and their importance for forecasting. We find no single model that consistently works best in the presence of breaks. In many cases, the formal modeling of the break process is important in achieving a good forecast performance. However, there are also many cases where rolling window forecasts perform well.

The Multiplicative MIDAS Realized DCC (MMReDCC) model simultaneously accounts for short and long term dynamics in the conditional (co)volatilities of asset returns, in line with the empirical evidence suggesting that their level is changing over time as a function of economic conditions. Herein the applicability of the model is improved along two directions. First, by proposing an algorithm that relies on the maximization of an iteratively re-computed moment-based profile likelihood function and keeps estimation feasible in large dimensions by mitigating the incidental parameter problem. Second, by illustrating a conditional bootstrap procedure to generate multi-step ahead predictions from the model. In an empirical application on a dataset of forty-six equities, the MMReDCC model is found to statistically outperform the selected benchmarks in terms of in-sample fit as well as in terms of out-of-sample covariance predictions. The latter are mostly significant in periods of high market volatility.

The deregulation of European electricity markets has led to an increasing need in understanding the volatility and correlation structure of electricity prices. We model a multivariate futures series of the European Energy Exchange (EEX) index, using an asymmetric GARCH model for volatilities and augmented dynamic conditional correlation (DCC) models for correlations. In particular, we allow for smooth changes in the unconditional volatilities and correlations through a multiplicative component that we estimate non-parametrically. We also introduce exogenous variables in our new multiplicative DCC model to account for congestion in short-term conditional volatilities. We find different correlation dynamics for long and short-term contracts and the new model achieves higher forecasting performance compared to a standard DCC model.

A portfolio selection model which allocates a portfolio of currencies by maximizing the expected return subject to Value-at-Risk (VaR) constraint is designed and implemented. Based on an econometric implementation using intradaily data, the optimal portfolio allocation is forecasted at regular time intervals. For the estimation of the conditional variance from which the VaR is computed, univariate and multivariate GARCH models are used. Model evaluation is done using two economic criteria and two statistical tests. The result for each model is given by the best forecasted intradaily investment recommendations in terms of the optimal weights of the currencies in the risky portfolio. The results show that estimating the VaR from multivariate GARCH models improves the results of the forecasted optimal portfolio allocation, compared to using a univariate model.

Modelling multivariate time series of possibly high dimension calls for appropriate dimension-reduction, e.g. by some factor modelling, additive modelling, or some simplified parametric structure for the dynam- ics (i.e. the serial dependence) of the time series. This workshop aimed to bring together experts in this field in order to discuss recent methodology for multivariate time series dynamics which are changing over time: by an abrupt switch between two (or more) different regimes or rather smoothly evolving over time. The emphasis has been on mathematical methods for semiparametric modelling and estimation, where "semiparametric" is to be understood in a rather broad sense: parametric models where the parameters are themselves nonparametric functions (of time), regime-switching nonpara- metric models with a parametric specification of the transition mechanism, and alike. An ultimate goal of these models to be applied to economic and financial time series is prediction. Another emphasis has been on comparing Bayesian with frequentist approaches, and to cover both theoretical aspects of estimation, such as consistency and efficiency, and computational aspects.

A new model - the factorial hidden Markov volatility (FHMV) model - is proposed for financial returns and their latent variances. It is also applicable to model directly realized variances. Volatility is modeled as a product of three components: a Markov chain driving volatility persistence, an independent discrete process capable of generating jumps in the volatility, and a predictable (data-driven) process capturing the leverage effect. An economic interpretation is attached to each one of these components. Moreover, the Markov chain and jump components allow volatility to switch abruptly between thousands of states, and the transition matrix of the model is structured in such a way as to generate a high degree of volatility persistence. In-sample results on six financial time series highlight that the FHMV process compares favorably to state-of-the-art volatility models. A forecasting experiment shows that it also outperforms its competitors when predicting volatility over time horizons ranging from one to one hundred days.

Several models have been developed to capture the dynamics of the conditional correlations between time series of financial returns, but few studies have investigated the determinants of the correlation dynamics. A common opinion is that the market volatility is a major determinant of the correlations. We extend some models to capture explicitly the dependence of the correlations on the volatility of the market of interest. The models differ in the way by which the volatility influences the correlations, which can be transmitted through linear or nonlinear, and direct or indirect effects. They are applied to different data sets to verify the presence and possible regularity of the volatility impact on correlations.

We estimate by Bayesian inference the mixed conditional heteroskedasticity model of Haas et al. (2004a Journal of Financial Econometrics 2, 211–50). We construct a Gibbs sampler algorithm to compute posterior and predictive densities. The number of mixture components is selected by the marginal likelihood criterion. We apply the model to the SP500 daily returns.

This paper compares the forecasting performance of different models which have been proposed for forecasting in the presence of structural breaks. These models differ in their treatment of the break process, the model which applies in each regime and the out-of-sample probability of a break occurring. In an extensive empirical evaluation involving 60 macroeconomic quarterly and monthly time series, we demonstrate the presence of structural breaks and their importance for forecasting in the vast majority of cases. We find no single forecasting model consistently works best in the presence of structural breaks. In many cases, the formal modeling of the break process is important in achieving good forecast performance.\ud However, there are also many cases where simple, rolling window based forecasts perform well.

In this article, we introduce the so-called stochastic conditional intensity (SCI) model by extending Russell’s (1999) autoregressive conditional intensity (ACI) model by a latent common dynamic factor that jointly drives the individual intensity components. We show by simulations that the proposed model allows for a wide range of (cross-)autocorrelation structures in multivariate point processes. The model is estimated by simulated maximum likelihood (SML) using the efficient importance sampling (EIS) technique. By modeling price intensities based on NYSE trading, we provide significant evidence for a joint latent factor and show that its inclusion allows for an improved and more parsimonious specification of the multivariate intensity process.

We propose a practical and flexible method to introduce skewness in multivariate symmetric distributions. Applying this procedure to the multivariate Student density leads to a “multivariate skew-Student” density in which each marginal has a specific asymmetry coefficient. Combined with a multivariate generalized autoregressive conditional heteroscedasticity model, this new family of distributions is found to be more useful than its symmetric counterpart for modeling stock returns and especially for forecasting the value-at-risk of portfolios.

Using density forecasts, we compare the predictive performance of duration models that have been developed for modelling intra-day data on stock markets. Our model portfolio encompasses the autoregressive conditional duration (ACD) model, its logarithmic version (Log-ACD), the threshold ACD (TACD) model - in each case with alternative error distributions -, the stochastic conditional duration model (SCD), and the stochastic volatility duration model (SVD). The evaluation is done on transaction, price, and volume durations of four stocks listed at the NYSE. The results lead us to conclude that the ACD/log-ACD/TACD/SCD models capture the dynamic dependence in the data in a satisfactory way. They fit correctly the conditional distribution of volume durations, but fail to do so for trade durations. The evidence is mixed for price durations and ACDbased models, poor for the SCDmo del. The SVDmo del in its original version performs worse than the (Log-)ACDmo dels on the dynamicsof trade durations, and offers no improvement with respect to the distributional aspect. The SVDis not suitable to model volume durations. Regarding price durations the performance of the SVDis comparable to those of (Log-)ACD specifications that provide the best results.

We introduce a class of models for the analysis of durations, which we call stochastic conditional duration (SCD) models. These models are based on the assumption that the durations are generated by a dynamic stochastic latent variable. The model yields a wide range of shapes of hazard functions. The estimation of the parameters is performed by quasi-maximum likelihood and using the Kalman filter. The model is applied to trade, price and volume durations of stocks traded at NYSE. We also investigate the relation between price durations, spread, trade intensity and volume. (C) 2003 Elsevier B.V. All rights reserved.

This paper proposes a class of asymmetric Autoregressive Conditional Duration models, which extends the ACD model of Engle and Russell (1997). The asymmetry consists of letting the duration process depend on the state of the price process in the beginning and at the end of each duration. If the price has increased, the parameters of the ACD can differ from what they are if the price has decreased. Thus, the model is also a transition model for the price process, with durations following an ACD process. The logarithmic version of the model is applied to the bid/ask price revision process by the specialist for the IBM stock on the New York Stock Exchange. The empirical evidence in favour of asymmetry is compelling.

Markov-switching models are usually specified under the assumption that all the parameters change when a regime switch occurs. Relaxing this hypothesis and being able to detect which parameters evolve over time is relevant for interpreting the changes in the dynamics of the series, for specifying models parsimoniously, and may be helpful in forecasting. We propose the class of sticky infinite hidden Markov-switching autoregressive moving average models, in which we disentangle the break dynamics of the mean and the variance parameters. In this class, the number of regimes is possibly infinite and is determined when estimating the model, thus avoiding the need to set this number by a model choice criterion. We develop a new Markov chain Monte Carlo estimation method that solves the path dependence issue due to the moving average component. Empirical results on macroeconomic series illustrate that the proposed class of models dominates the model with fixed parameters in terms of point and density forecasts. Appendix available at: https://ssrn.com/abstract=2965668

This Appendix contains details on several technical points and additional empirical results. Sections in this Appendix are indexed by letters and formulas/tables/figures by a letter followed by a number (e.g. A.1). Sections and formulas/tables/figures of the paper are referenced by numbers. In Section A, we expose the estimation procedure of the models employed in the paper. In Section B, we explain the approximation used to avoid the path dependence problem in Markov Switching models. In Section C, we provide the formulas of the marginal impact of volatility on correlations for each VDCC model. In Section D, we detail the estimation method and the results behind our findings about the long-run and short-run effects of the volatility on the correlations. In Section E, we report the application to the three stock data set (and other subsets of stocks) of the method explained in Section 4.1 to evaluate the performance of the constrained RSDC models. In Section F, we include and comment the estimates of the VDCC models for the thirty stock data set used in Section 4.2. Full paper is available at: https://ssrn.com/abstract=2965487

This Appendix contains additional empirical results with respect to the published article. In Section 1, the posterior results for the HDP parameters of the IHMS- ARMA models are presented for the U.S. GDP growth rate and inflation series. In Section 2, we report additional in-sample and forecasting results for the same series. In Section 3, some results for a different truncation choice of the number of regimes in the approximate model are reported. Full paper available at: https://ssrn.com/abstract=2965441

This paper shows how one can compute option prices from a Bayesian inference viewpoint, using a GARCH model for the dynamics of the volatility of the underlying asset. The proposed evaluation of an option is the predictive expectation of its payoff function. The predictive distribution of this function provides a natural metric, provided it is neutralised with respect to risk, for gauging the predictive option price or other option evaluations. The proposed method is compared to the Black and Scholes evaluation, in which a marginal mean volatility is plugged, but which does not provide a natural metric. The methods are illustrated using symmetric, asymmetric and smooth transition GARCH models with data on a stock index in Brussels. D 2002 Elsevier Science B.V. All rights reserved.

Novel model specifications that include a time-varying long run component in the dynamics of realized covariance matrices are proposed. The adopted modeling framework allows the secular component to enter the model structure either in an additive fashion or as a multiplicative factor, and to be specified parametrically, using a MIDAS filter, or non-parametrically. Estimation is performed by maximizing a Wishart quasi-likelihood function. The one-step ahead forecasting performance of the models is assessed by means of three approaches: the Model Confidence Set, (global) minimum variance portfolios and Value-at-Risk. The results provide evidence in favour of the hypothesis that the proposed models outperform benchmarks incorporating a constant long run component, both in and out-of sample.

Summary This paper explains how the Gibbs sampler can be used to perform Bayesian inference on GARCH models. Although the Gibbs sampler is usually based on the analytical knowledge of the full conditional posterior densities, such knowledge is not available in regression models with GARCH errors. We show that the Gibbs sampler can be combined with a unidimensional deterministic integration rule applied to each coordinate of the posterior density. The full conditional densities are evaluated and inverted numerically to obtain random draws of the joint posterior. The method is shown to be feasible and competitive compared with importance sampling and the Metropolis‐Hastings algorithm. It is applied to estimate an asymmetric Student‐GARCH model for the return on a stock exchange index, and to compute predictive option prices on the index. We prove, moreover, that a flat prior on the degrees of freedom parameter leads to an improper posterior density.

We develop an easy-to-implement method for forecasting a stationary autoregressive fractionally integrated moving average (ARFIMA) process subject to structural breaks with unknown break dates. We show that an ARFIMA process subject to a mean shift and a change in the long memory parameter can be well approximated by an autoregressive (AR) model and suggest using an information criterion (AIC or Mallows' C_p) to choose the order of the approximate AR model. Our methods avoids the issue of estimation inaccuracy of the long memory parameter and the issue of spurious breaks in finite sample. Insights from our theoretical analysis are confirmed by Monte Carlo experiments, through which we also find that our method provides a substantial improvement over existing prediction methods. An empirical application to the realized volatility of three exchange rates illustrates the usefulness of our forecasting procedure. The empirical success of the HAR-RV model can be explained, from an econometric perspective, by our theoretical and simulation results.

Eliminating negative end-use or appliance-consumption estimates and incorporating direct-metering information into the process of generating these estimates—these are two important aspects of conditional demand analysis that will be the focus of this article. In both cases a Bayesian approach seems a natural way of proceeding. What needs to be investigated is whether it is also a viable and effective approach. In addition, such a framework naturally lends itself to prediction. Our application involves the estimation of electrical-appliance consumptions for a sample of Australian households. This application is designed to illustrate the viability of a full Bayesian analysis of the problem.

This paper compares the forecasting performance of different models which have been proposed for forecasting in the presence of structural breaks. These models differ in their treatment of the break process, the parameters defining the model which applies in each regime and the out-of-sample probability of a break occurring. In an extensive empirical evaluation involving many important macroeconomic time series, we demonstrate the presence of structural breaks and their importance for forecasting in the vast majority of cases. However, we find no single forecasting model consistently works best in the presence of structural breaks. In many cases, the formal modeling of the break process is important in achieving good forecast performance. However, there are also many cases where simple, rolling OLS forecasts perform well.

The first three appendices contain details about the implementation of the estimation and forecasting of the structural break models named PPT and KP in the paper. These models are explained in Section 2 of the paper and information about the forecasting implementation of these models is presented in Section 4 of the paper. The fourth appendix contains tables that show detailed results that are summarized and discussed in Section 5 of the paper.

Online Appendix to "MULTIVARIATE VOLATILITY MODELING OF ELECTRICITY FUTURES". Full paper available at: https://ssrn.com/abstract=2965508

GARCH volatility models with fixed parameters are too restrictive for long time series due to breaks in the volatility process. Flexible alternatives are Markov-switching GARCH and change-point GARCH models. They require estimation by MCMC methods due to the path dependence problem. An unsolved issue is the computation of their marginal likelihood, which is essential for determining the number of regimes or change-points. We solve the problem by using particle MCMC, a technique proposed by Andrieu, Doucet, and Holenstein (2010). We examine the performance of this new method on simulated data, and we illustrate its use on several return series.

We survey the modelling of financial markets transaction data characterized by irregular spacing in time, in particular so-called financial durations.We begin by reviewing the important concepts of point process theory, such as intensity functions, compensators and hazard rates, and then the intensity, duration, and counting representations of point processes. Next, in two separate sections, we review dynamic duration models, especially autoregressive conditional duration models, and dynamic intensity models (Hawkes and autoregressive intensity processes). In each section, we discuss model specification, statistical inference and applications.

Change-point models are useful for modeling time series subject to structural breaks. For interpretation and forecasting, it is essential to estimate correctly the number of change points in this class of models. In Bayesian inference, the number of change points is typically chosen by the marginal likelihood criterion, computed by Chib's method. This method requires one to select a value in the parameter space at which the computation is performed. Bayesian inference for a change-point dynamic regression model and the computation of its marginal likelihood are explained. Motivated by results from three empirical illustrations, a simulation study shows that Chib's method is robust with respect to the choice of the parameter value used in the computations, among posterior mean, mode and quartiles. However, taking into account the precision of the marginal likelihood estimator, the overall recommendation is to use the posterior mode or median. Furthermore, the performance of the Bayesian information criterion, which is based on maximum likelihood estimates, in selecting the correct model is comparable to that of the marginal likelihood.

We develop a Markov-switching GARCH model (MS-GARCH) wherein the conditional mean and variance switch in time from one GARCH process to another. The switching is governed by a hidden Markov chain. We provide sufficient conditions for geometric ergodicity and existence of moments of the process. Because of path dependence, maximum likelihood estimation is not feasible. By enlarging the parameter space to include the state variables, Bayesian estimation using a Gibbs sampling algorithm is feasible. We illustrate the model on S&P500 daily returns.

The general-to-specific (GETS) methodology is widely employed in the modelling of economic series, but less so in financial volatility modelling due to computational complexity when many explanatory variables are involved. This study proposes a simple way of avoiding this problem when the conditional mean can appropriately be restricted to zero, and undertakes an out-of-sample forecast evaluation of the methodology applied to the modelling of weekly exchange rate volatility. Our findings suggest that GETS specifications perform comparatively well in both ex post and ex ante forecasting as long as sufficient care is taken with respect to functional form and with respect to how the conditioning information is used. Also, our forecast comparison provides an example of a discrete time explanatory model being more accurate than realised volatility ex post in 1 step forecasting.

We design and implement optimal foreign exchange portfolio allocations. An optimal allocation maximizes the expected return sub ject to a Value-at-Risk (VaR) constraint. Based on intradaily data, the optimization procedure is carried out at regular time intervals. For the estimation of the conditional variance from which the VaR is computed, we use univariate and multivariate GARCH models. The result for each model is given by the best intradaily investment recom- mendations in terms of the optimal weights of the currencies in the risky portfolio

We develop univariate regime-switching GARCH (RS-GARCH) models wherein the conditional variance switches in time from one GARCH process to another. The switching is governed by a time-varying probability, specified as a function of past information. We provide sufficient conditions for stationarity and existence of moments. Because of path dependence, maximum likelihood estimation is infeasible. By enlarging the parameter space to include the state variables, Bayesian estimation using a Gibbs sampling algorithm is feasible. We apply this model using the NASDAQ daily return series.

This paper sheds new light on the mixture of distribution hypothesis by means of a study of the weekly exchange rate volatility of the Norwegian krone. In line with other studies we find that the impact of information arrival on exchange rate volatility is positive and statistically significant, and that the hypothesis that an increase in the number of traders reduces exchange rate volatility is not supported. The novelties of our study consist in documenting that the positive impact of information arrival on volatility is relatively stable across three different exchange rate regimes, and in that the impact is relatively similar for both weekly volatility and weekly realised volatility. It is not given that the former should be the case since exchange rate stabilisation was actively pursued by the central bank in parts of the study period. We also report a case in which undesirable residual properties attained within traditional frameworks are easily removed by applying the log-transformation on volatilities.

We analyze statistically inter-trade durations of four stocks listed on the Tokyo Stock Exchange in 2003. We find that these data display the usual stylized facts (intra-daily seasonality clustering and overdispersion) found for similar data of the New York Stock Exchange, but with some differences. We also estimate autoregressive conditional duration models for fitting the durations. We find that, like for comparable data of the NYSE, some models fit in a satisfactory way the dynamic properties of the durations, but do not always fit well the conditional distribution of the data.

This paper introduces a new framework for the dynamic modelling of univariate and multivariate point processes. The so-called latent factor intensity (LFI) model is based on the assumption that the intensity function consists of univariate or multivariate observation driven dynamic components and a univariate dynamic latent factor. In this sense, the model corresponds to a dynamic extension of a doubly stochastic Poisson process. We illustrate alternative parameterizations of the observation driven component based on autoregressive conditional intensity (ACI) specifications, as well as Hawkes types models. Based on simulation studies, it is shown that the proposed model provides a flexible tool to capture the joint dynamics of multivariate point processes. Since the latent component has to be integrated out, the model is estimated by simulated maximum likelihood based upon efficient importance sampling techniques. Applications of univariate and bivariate LFI models to transaction data extracted from the German XETRA trading system provide evidence for an improvement of the econometric specification when observable as well as unobservable dynamic components are taken into account.

This paper surveys the most important developments in multivariate ARCH-type modelling. It reviews the model specifications, the inference methods, and the main areas of application of these models in financial econometrics.

We provide existence conditions and analytical expressions of the moments of logarithmic autoregressive conditional duration (Log-ACD) models. We focus on the dispersion index and the autocorrelation function and compare them with those of ACD (Engle and Russell 1998) and SCD models. Using duration data for several stocks traded on the New York Stock Exchange, we compare the models in terms of their ability at fitting some stylized facts.

In this chapter we consider two extensions of the models presented in Chapter 3. Firstly, we use the intraday duration models to test some market microstructure effects. The models presented in Chapter 3 introduce a new way of modelling the times between market events and are directly related to quantitative tools developed in labour econometrics. While the models can be used to characterize the point processes defined by trade, quote, price or volume durations, they can also be used to test market microstructure effects such as presented in Chapter 1. This has been one of the main motivations for the use of such intraday duration models, and most recent papers focusing on the econometric modelling of the durations usually include a market microstructure section (Engle and Russell, 1997, 1998; Engle and Lunde, 1998; Bauwens and Giot, 1998, 2000; Grammig and Wellner, 1999). In Section 2 of this chapter, we review some possible applications of the Log-ACD model to the testing of market microstructure issues which deal with the (bid-ask quotes) updating behavior of a market maker with respect to the information flow.

When intraday data are modelled, at least two different approaches can be used. As detailed in Chapter 3 and 4, a first possibility is to deal directly with the irregularly time-spaced data and thus use duration models, or joint models for durations and associated marks (such as the return over the duration). This approach fits well with the literature on market microstructure, which stresses the importance of the times between market events, since they supposedly convey important information.

In Chapter 2 we have defined different types of durations that are of interest for the analysis of stock market intraday activity from the viewpoint of characterizing the trading frequency, the volatility, and the liquidity of a market. We also highlighted the main stylized facts of durations: a regular intraday seasonality, and after correcting the data for this pattern, clustering and overdispersion (underdispersion for volume durations). The object of this chapter is to set out econometric models that are compatible with the stylized facts. In Section 2, we introduce some basic statistical concepts relevant to the analysis of durations and make briefly the link with point processes, i.e. collections of points randomly distributed along a time axis. In Section 3, we introduce the class of autoregressive conditional duration models and their logarithmic versions. The models are carefully defined, and their properties are investigated, using a mix of analytical and numerical methods. The estimation of these models by the maximum likelihood method is briefly reviewed, and tools for testing the specification of a model are developed. In Section 4, we provide empirical results that illustrate the estimation and testing procedures. The illustration is done on trade, price, and volume durations of several stocks listed on the NYSE. Applications of the same models to test microstructure effects are pursued in Chapter 4.

SummaryPublic sales art catalogues include low and high pre-sale price estimates by experts. This makes it possible to analyze whether pre-sale estimates are unbiased predictors of realized prices. Unbiasedness is tested using a sample of some 1,600 lots of English silver auctioned by Christie’s and Sotheby’s. Results show that estimates are slightly (but significantly) biased and that experts do not use all the information that is available to them when they make their estimates.