Handbook of Financial Time Series

This paper contains a survey of univariate models of conditional heteroskedasticity. The classical ARCH model is mentioned, and various extensions of the standard Generalized ARCH model are highlighted. This includes the Exponential GARCH model. Stochastic volatility models remain outside this review.

This paper collects some of the well known probabilistic properties of GARCH (p, q) processes. In particular, we address the question of strictly and of weakly stationary solutions. We further investigate moment conditions as well as the strong mixing property of GARCH processes. Some distributional properties such as the tail behaviour and continuity properties of the stationary distribution are also included.

ARCH(∞)-models are a natural nonparametric generalization of the class of GARCH(p, q) models which exhibit a rich covariance structure (in particular, hyperbolic decay of the autocovariance function is possible). We discuss stationarity, long memory properties and the limit behavior of partial sums of ARCH(∞) processes as well as some of their modifications (linear ARCH and bilinear models).

The main estimation methods of the univariate GARCH models are reviewed. A special attention is given to the asymptotic results and the quasi-maximum likelihood method.

This chapter gives a tour through the empirical analysis of univariate GARCH models for financial time series with stops along the way to discuss various practical issues associated with model specification, estimation, diagnostic evaluation and forecasting.

This paper surveys nonparametric approaches to modelling discrete time volatility. We cover functional form, error shape, memory, and relationship between mean and variance.

This paper offers a new method for estimation and forecasting of the volatility of financial time series when the stationarity assumption is violated. We consider varying–coefficient parametric models, such as ARCH and GARCH, whose coefficients may arbitrarily vary with time. This includes global parametric, smooth transition, and change–point models as special cases. The method is based on an adaptive pointwise selection of the largest interval of homogeneity with a given right–end point, which is obtained by a local change–point analysis.We construct locally adaptive volatility estimates that can perform this task and investigate them both from the theoretical point of view and by Monte Carlo simulations. Additionally, the proposed method is applied to stock–index series and shown to outperform the standard parametric GARCH model.

We consider the extreme value theory for a stationary GARCH process with iid innovations. One of the basic ingredients of this theory is the fact that, under general conditions, GARCH processes have power law marginal tails and, more generally, regularly varying finite-dimensional distributions. Distributions with power law tails combined with weak dependence conditions imply that the scaled maxima of a GARCH process converge in distribution to a Fréchet distribution. The dependence structure of a GARCH process is responsible for the clustering of exceedances of a GARCH process above high and low level exceedances. The size of these clusters can be described by the extremal index. We also consider the convergence of the point processes of exceedances of a GARCH process toward a point process whose Laplace functional can be expressed explicitly in terms of the intensity measure of a Poisson process and a cluster distribution.

This article contains a review of multivariate GARCH models. Most common GARCH models are presented and their properties considered. This also includes nonparametric and semiparametric models. Existing specification and misspecification tests are discussed. Finally, there is an empirical example in which several multivariate GARCH models are fitted to the same data set and the results compared.

Stochastic volatility is the main way time-varying volatility is modelled in financial markets. The development of stochastic volatility is reviewed, placing it in a modeling and historical context. Some recent trends in the literature are highlighted.

We collect some of the probabilistic properties of a strictly stationary stochastic volatility process. These include properties about mixing, covariances and correlations, moments, and tail behavior. We also study properties of the autocovariance and autocorrelation functions of stochastic volatility processes and its powers as well as the asymptotic theory of the corresponding sample versions of these functions. In comparison with the GARCH model (see Lindner (2008)) the stochastic volatility model has a much simpler probabilistic structure which contributes to its popularity.

This chapter reviews the possible uses of the Generalized Method of Moments (GMM) to estimate Stochastic Volatility (SV) models. A primary attraction of the method of moments technique is that it is well suited for identifying and estimating volatility models without a complete parametric specification of the probability distributions. Moreover, simulation-based methods of moments are able to exploit a variety of moments, while avoiding limitations due to a lack of closed form expressions. The chapter first highlights the suitability of GMM for popular regression models of volatility forecasting. Then, it reviews the implications of the SV model specification in terms of higher order moments: skewness, kurtosis, variance of the variance, leverage and feedback effects. The chapter examines the ability of a continuous time version of SV models to accommodate data from other sources like option prices or high frequency data on returns and transactions dates. Simulation-based methods are particularly useful for studying continuous time models due to the frequent lack of closed form expressions for their discrete time dynamics. These simulation-based methods of moments are presented within the unifying framework of indirect inference with a special emphasis on misspecification. Likely misspecification of the parametric model used for simulation requires a parsimonious and well-focused choice of the moments to match.

Estimating parameters in a stochastic volatility (SV) model is a challenging task and therefore much research is devoted in this area of estimation. This chapter presents an overview and a practical guide of the quasi-likelihood and the Monte Carlo likelihood methods of estimation. The concepts of the methods are straightforward and the implementation is based on Kalman filter, smoothing, simulation smoothing, mode calculation and Monte Carlo simulation. These methods are general, transparent and computationally fast; therefore, they provide a feasible way for the estimation of parameters in SV models. Various extensions of the SV model are considered and some details are provided for the effective implementation of the Monte Carlo methods. Some empirical illustrations are given to show that the methods can be successful in measuring the unobserved volatility in financial time series.

In this contribution, we consider models in discrete time that contain a latent process for volatility. The most well-known model of this type is the Long-Memory Stochastic Volatility (LMSV) model. We describe its main properties, discuss parametric and semiparametric estimation for these models, and give some generalizations and applications.

We consider extreme value theory for stochastic volatility processes in both cases of light-tailed and heavy-tailed noise. First, the asymptotic behavior of the tails of the marginal distribution is described for the two cases when the noise distribution is Gaussian or heavy-tailed. The sequence of point processes, based on the locations of the suitable normalized observations from a stochastic volatility process, converges in distribution to a Poisson process. From the point process convergence, a variety of limit results for extremes can be derived. Of special note, there is no extremal clustering for stochastic volatility processes in both the light- and heavy-tailed cases. This property is in sharp contrast with GARCH processes which exhibit extremal clustering (i.e., large values of the process come in clusters).

We provide a detailed summary of the large and vibrant emerging literature that deals with the multivariate modeling of conditional volatility of financial time series within the framework of stochastic volatility. The developments and achievements in this area represent one of the great success stories of financial econometrics. Three broad classes of multivariate stochastic volatility models have emerged: one that is a direct extension of the univariate class of stochastic volatility model, another that is related to the factor models of multivariate analysis and a third that is based on the direct modeling of time-varying correlation matrices via matrix exponential transformations, Wishart processes and other means. We discuss each of the various model formulations, provide connections and differences and show how the models are estimated. Given the interest in this area, further significant developments can be expected, perhaps fostered by the overview and details delineated in this paper, especially in the fitting of high-dimensional models.

Discrete-parameter time-series models for financial data have received, and continue to receive, a great deal of attention in the literature. Stochastic volatility models, ARCH and GARCH models and their many generalizations, designed to account for the so-called stylized features of financial time series, have been under development and refinement now for some thirty years. At the same time there has been a rapidly developing interest in continuous-time models, largely as a result of the very successful application of stochastic differential equation models to problems in finance, exemplified by the derivation of the Black-Scholes-Merton (BSM) optionpricing formula and its generalizations. In this overview we start with the BSM option-pricing model in which the asset price is represented by geometric Brownian motion. We then discuss the limitations of the model and survey the various models which have been proposed to provide more realistic representations of empirically observed asset prices. In particular, the observed non-Gaussian distributions of log returns and the appearance of sharp changes in log asset prices which are not consistent with Brownian motion paths have led to an upsurge of interest in Lévy processes and their applications to financial modelling.

This paper surveys a class of Generalised Ornstein-Uhlenbeck (GOU) processes associated with Lévy processes, which has been recently much analysed in view of its applications in the financial modelling area, among others. We motivate the Lévy GOU by reviewing the framework already well understood for the “ordinary” (Gaussian) Ornstein-Uhlenbeck process, driven by Brownian motion; thus, defining it in terms of a stochastic differential equation (SDE), as the solution of this SDE, or as a time changed Brownian motion. Each of these approaches has an analogue for the GOU. Only the second approach, where the process is defined in terms of a stochastic integral, has been at all closely studied, and we take this as our definition of the GOU (see Eq. (12) below).The stationarity of the GOU, thus defined, is related to the convergence of a class of “Lévy integrals”, which we also briefly review. The statistical properties of processes related to or derived from the GOU are also currently of great interest, and we mention some of the research in this area. In practise, we can only observe a discrete sample over a finite time interval, and we devote some attention to the associated issues, touching briefly on such topics as an autoregressive representation connected with a discretely sampled GOU, discrete-time perpetuities, self-decomposability, self-similarity, and the Lamperti transform.Some new statistical methodology, derived from a discrete approximation procedure, is applied to a set of financial data, to illustrate the possibilities.

Lévy processes are developed in the more general framework of semimartingale theory with a focus on purely discontinuous processes. The fundamental exponential Lévy model is given, which allows us to describe stock prices or indices in a more realistic way than classical diffusion models. A number of standard examples including generalized hyperbolic and CGMY Lévy processes are considered in detail.

Gaussian ARMA processes with continuous time parameter, otherwise known as stationary continuous-time Gaussian processes with rational spectral density, have been of interest for many years. (See for example the papers of Doob (1944), Bartlett (1946), Phillips (1959), Durbin (1961), Dzhapararidze (1970,1971), Pham-Din-Tuan (1977) and the monograph of Arató (1982).) In the last twenty years there has been a resurgence of interest in continuous-time processes, partly as a result of the very successful application of stochastic differential equation models to problems in finance, exemplified by the derivation of the Black-Scholes option-pricing formula and its generalizations (Hull and White (1987)). Numerous examples of econometric applications of continuous-time models are contained in the book of Bergstrom (1990). Continuous-time models have also been utilized very successfully for the modelling of irregularly-spaced data (Jones (1981, 1985), Jones and Ackerson (1990)). Like their discrete-time counterparts, continuous-time ARMA processes constitute a very convenient parametric family of stationary processes exhibiting a wide range of autocorrelation functions which can be used to model the empirical autocorrelations observed in financial time series analysis. In financial applications it has been observed that jumps play an important role in the realistic modelling of asset prices and derived series such as volatility. This has led to an upsurge of interest in Lévy processes and their applications to financial modelling. In this article we discuss second-order Lévy-driven continuous-time ARMA models, their properties and some of their financial applications. Examples are the modelling of stochastic volatility in the class of models introduced by Barndorff-Nielsen and Shephard (2001) and the construction of a class of continuous-time GARCH models which generalize the COGARCH(1,1) process of Klüppelberg, Lindner and Maller (2004) and which exhibit properties analogous to those of the discretetime GARCH(p,q) process.

We collect some continuous time GARCH models and report on how they approximate discrete time GARCH processes. Similarly, certain continuous time volatility models are viewed as approximations to discrete time volatility models.

This paper overviews maximum likelihood and Gaussian methods of estimating continuous time models used in finance. Since the exact likelihood can be constructed only in special cases, much attention has been devoted to the development of methods designed to approximate the likelihood. These approaches range from crude Euler-type approximations and higher order stochastic Taylor series expansions to more complex polynomial-based expansions and infill approximations to the likelihood based on a continuous time data record. The methods are discussed, their properties are outlined and their relative finite sample performance compared in a simulation experiment with the nonlinear CIR diffusion model, which is popular in empirical finance. Bias correction methods are also considered and particular attention is given to jackknife and indirect inference estimators. The latter retains the good asymptotic properties of ML estimation while removing finite sample bias. This method demonstrates superior performance in finite samples.

A review is given of parametric estimation methods for discretely sampled multivariate diffusion processes. The main focus is on estimating functions and asymptotic results. Maximum likelihood estimation is briefly considered, but the emphasis is on computationally less demanding martingale estimating functions. Particular attention is given to explicit estimating functions. Results on both fixed frequency and high frequency asymptotics are given. When choosing among the many estimators available, guidance is provided by simple criteria for high frequency efficiency and rate optimality that are presented in the framework of approximate martingale estimating functions.

Realized volatility is a nonparametric ex-post estimate of the return variation. The most obvious realized volatility measure is the sum of finely-sampled squared return realizations over a fixed time interval. In a frictionless market the estimate achieves consistency for the underlying quadratic return variation when returns are sampled at increasingly higher frequency. We begin with an account of how and why the procedure works in a simplified setting and then extend the discussion to a more general framework. Along the way we clarify how the realized volatility and quadratic return variation relate to the more commonly applied concept of conditional return variance. We then review a set of related and useful notions of return variation along with practical measurement issues (e.g., discretization error and microstructure noise) before briefly touching on the existing empirical applications.

This chapter reviews our recent work on disentangling high frequency volatility estimators from market microstructure noise, based on maximum-likelihood in the parametric case and two (or more) scales realized volatility (TSRV) in the nonparametric case. We discuss the basic theory, its extensions and the practical implementation of the estimators.

This chapter reviews basic concepts of derivative pricing in financial mathematics.We distinguish market prices and individual values of a potential seller. We focus mainly on arbitrage theory. In addition, two hedgingbased valuation approaches are discussed. The first relies on quadratic hedging whereas the second involves a first-order approximation to utility indifference prices.

In this paper we give a short overview of some basic topics in interest rate theory, from the point of view of arbitrage free pricing. We cover short rate models, affine term structure models, inversion of the yield curve, the Musiela parameterization, and the potential approach to positive interest rates. The text is essentially self contained.

In this paper we present a review on the extremal behavior of stationary continuous-time processes with emphasis on generalized Ornstein-Uhlenbeck processes. We restrict our attention to heavy-tailed models like heavy-tailed Ornstein-Uhlenbeck processes or continuous-time GARCH processes. The survey includes the tail behavior of the stationary distribution, the tail behavior of the sample maximum and the asymptotic behavior of sample maxima of our models.

This article presents a survey of the analysis of cointegration using the vector autoregressive model. After a few illustrative economic examples, the three model based approaches to the analysis of cointegration are discussed.The vector autoregressive model is defined and the moving average representation of the solution, the Granger representation, is given. Next the interpretation of the model and its parameters and likelihood based inference follows using reduced rank regression. The asymptotic analysis includes the distribution of the Gaussian maximum likelihood estimators, the rank test, and test for hypotheses on the cointegrating vectors. Finally, some applications and extensions of the basic model are mentioned and the survey concludes with some open problems.

This paper reviews some of the developments of the unit root and near unit root time series. It gives an overview of this important topic and describes the impact of some of the recent progress on subsequent research.

We describe a variety of seimparametric models and estimators for fractional cointegration. All of the estimators we consider are based on the discrete Fourier transform of the data. This includes the ordinary least squares estimator as a special case.We make a distinction between Type I and Type II models, which differ from each other in terms of assumptions about initialization, and which lead to different functional limit laws for the partial sum processes. We compare the estimators in terms of rate of convergence. We briefly discuss the problems of testing for cointegration and determining the cointegrating rank. We also discuss relevant modeling issues, such as the local parametrization of the phase function.

Over the last twenty years, the financial industry has developed numerous tools for the quantitative measurement of risk. The need for this was mainly due to changing market conditions and regulatory guidelines. In this article we review these processes and summarize the most important risk categories considered.

In this chapter, we build first a univariate and then a multivariate filtered historical simulation (FHS) model for financial risk management. Both the univariate and multivariate methods simulate future returns from a model using historical return innovations. While the former relies on portfolio returns filtered by a dynamic variance model, the latter uses individual or base asset return innovations from dynamic variance and correlation models. The univariate model is suitable for passive risk management or risk measurement whereas the multivariate model is useful for active risk management such as optimal portfolio allocation. Both models are constructed in such a way as to capture the stylized facts in daily asset returns and to be simple to estimate. The FHS approach enables the risk manager to easily compute Value-at-Risk and other risk measures including Expected Shortfall for various investment horizons that are conditional on current market conditions. The chapter also lists various alternatives to the suggested FHS approach.

This paper presents an overview of the literature on applications of copulas in the modelling of financial time series. Copulas have been used both in multivariate time series analysis, where they are used to characterize the (conditional) cross-sectional dependence between individual time series, and in univariate time series analysis, where they are used to characterize the dependence between a sequence of observations of a scalar time series process. The paper includes a broad, brief, review of the many applications of copulas in finance and economics.

The chapter gives a broad outline of the central themes of credit risk modeling starting with the modeling of default probabilities, ratings and recovery.We present the two main frameworks for pricing credit risky instruments and credit derivatives. The key credit derivative - the Credit Default Swap - is introduced. The premium on this contract provides a meausure of the credit spread of the reference issuer. We then provide some key empirical works looking at credit spreads thorugh CDS contracts and bonds and finish with a description of the role of correlation in credit risk modeling.

This chapter considers the problems of evaluation and comparison of volatility forecasts, both univariate (variance) and multivariate (covariance matrix and/or correlation). We pay explicit attention to the fact that the object of interest in these applications is unobservable, even ex post, and so the evaluation and comparison of volatility forecasts often rely on the use of a “volatility proxy”, i.e. an observable variable that is related to the latent variable of interest. We focus on methods that are robust to the presence of measurement error in the volatility proxy, and to the conditional distribution of returns.

This paper reviews the literature on structural breaks in financial time series. The second section discusses the implications of structural breaks in financial time series for statistical inference purposes. In the third section we discuss change-point tests in financial time series, including historical and sequential tests as well as single and multiple break tests. The fourth section focuses on structural break tests of financial asset returns and volatility using the parametric versus nonparametric classification as well as tests in the long memory and the distribution of financial time series. In concluding we provide some areas of future research in the subject.

A survey is given on regime switching in econometric time series modelling. Numerous references to applied as well as methodological literature are presented. A distinction between observation switching (OS) and Markov switching (MS) models is suggested, where in OS models, the switching probabilities depend on functions of lagged observations. In contrast, in MS models the switching is a latent unobserved exogenous process. With an emphasis on OS and MS ARCH and cointegrated models, stationarity and ergodicity properties are discussed as well as likelihood-based estimation, asymptotic theory and hypothesis testing.

We provide an overview of the vast and rapidly growing area of model selection in statistics and econometrics.

In this chapter, we deal with nonparametric methods for discretely observed financial data. The main ideas of nonparametric kernel smoothing are explained in the rather simple situation of density estimation and regression. For financial data, a rather relevant topic is nonparametric estimation of a volatility function in a continuous-time model such as a homogeneous diffusion model. We review results on nonparametric estimation for discretely observed processes, sampled at high or at low frequency. We also discuss application of nonparametric methods to testing, especially model validation and goodness-of-fit testing. In risk measurement for financial time series, conditional quantiles play an important role and nonparametric methods have been successfully applied in this field too. At the end of the chapter we discuss Grenander’s sieve methods and other more recent advanced nonparametric approaches.

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.

We review different methods of bootstrapping or subsampling financial time series.We first discuss methods that can be applied to generate pseudo-series of log-returns which mimic closely the essential dependence characteristics of the observed series. We then review methods that apply the bootstrap in order to infer properties of statistics based on financial times series. Such methods do not work by generating new pseudo-series of the observed log-returns but by generating pseudo-replicates of the statistic of interest. Finally, we discuss subsampling and self-normalization methods applied to financial data.

This chapter provides an overview of Markov Chain Monte Carlo (MCMC) methods. MCMC methods provide samples from high-dimensional distributions that commonly arise in Bayesian inference problems. We review the theoretical underpinnings used to construct the algorithms, the Metropolis-Hastings algorithm, the Gibbs sampler, Markov Chain convergence, and provide a number of examples in financial econometrics.

This chapter provides an overview of particle filters. Particle filters generate approximations to filtering distributions and are commonly used in non-linear and/or non-Gaussian state space models. We discuss general concepts associated with particle filtering, provide an overview of the main particle filtering algorithms, and provide an empirical example of filtering volatility from noisy asset price data.