Indirect inference methods for stochastic volatility models based on non-Gaussian Ornstein–Uhlenbeck processes
Introduction
There has been much research activity in the field of statistical modeling of high-frequency financial data based on non-Gaussian Ornstein–Uhlenbeck (OU) processes during the present decade. Among the most important contributions are three articles by Barndorff-Nielsen and Shephard, 2001, Barndorff-Nielsen and Shephard, 2002, Barndorff-Nielsen and Shephard, 2003 (hereafter BS) and Barndorff-Nielsen et al. (2001). Overviews of recent developments in the field of financial econometrics are given by Harvey et al. (2004), Shephard (2005) and Andersen et al. (2009). Traditional likelihood-based methods are generally not applicable to non-Gaussian stochastic volatility models, and we propose an estimation method based on indirect inference (see Gourieroux et al., 1993 and Gallant and Tauchen, 1996). We apply this method to daily Euro/Norwegian krone (NOK) and US Dollar/NOK exchange rate data.
While the statistical properties of OU processes and their implications for derivative pricing have been examined by BS (2001) and others (e.g., Nicolato and Venardos, 2003), many issues regarding the practical implementation and estimation remain unsolved. Moreover, non-Gaussian OU processes have hardly been tested in applications. This paper examines the use of indirect inference methods in this context. In general, indirect inference combines the estimation of an approximative model with simulations from an underlying “true” data generating model: First, the auxiliary model is estimated from the real data. In our case, this is done by maximizing a Gaussian quasi-likelihood function corresponding to a linear state space representation for returns and squared returns. Then, simulations are made from the underlying OU model for given parameter values. We apply a method of moments version of indirect inference. That is, the indirect inference estimator is the value of the parameter vector in the OU model that minimizes, in a weighted mean squared error sense, the score vector of the quasi-likelihood function for the simulated data, when this score vector is evaluated at the quasi-likelihood estimator obtained from the real data. Our estimation method should be seen as an alternative to the Bayesian Markov Chain Monte Carlo (MCMC) approach proposed by Griffin and Steel (2006) and as being complementary to pure quasi-likelihood estimation. The quasi-likelihood function is constructed by means of the Kalman filter by assuming that the actual volatility process is a Gaussian latent (state) variable. Our Gaussian quasi-likelihood function treats the optimal linear predictors of returns and squared returns as if they are conditional expectations, which they are not. We investigate the consequences of this simplification for statistical inference. We also provide software in the form of a user-friendly R-package that interfaces efficient C++ code (see http://folk.uio.no/skare/SV/ for software and user documentation).
In the applied part of the paper, we analyze exchange rate volatility by using daily data from 1.1.1989–4.2.2010 on the US Dollar/NOK exchange rate and data from 1.1.1999–4.2.2010 on the Euro/NOK exchange rate (1.1.1999 is the date of the introduction of the Euro).
There exists a large literature on exchange rate dynamics, especially regarding the role of purchasing power parity and uncovered interest parity. While there is some evidence that economic fundamentals may govern the behavior of exchange rates in the long run (see MacDonald, 1999), it is now generally accepted that exchange rates at daily (or intradaily) frequencies cannot be explained by monetary economic theory. In fact, the well-known study of Meese and Rogoff (1983) demonstrates that a wide range of exchange rate models based on economic fundamentals were unable to outperform a simple random walk model. Subsequent work in this area shows that even if a random walk is a good approximation of the conditional mean process, there is strong evidence of heteroscedasticity in the errors, in the sense that large changes tend to be followed by large changes, and small ones by small changes, which causes periods of prolonged high volatility to be followed by periods of relative stability (see, e.g., Diebold and Nerlove, 1989). Thus, the error terms may be uncorrelated but not independent. Generally, modeling the volatility of a stochastic process, which is a second-order property, is much more difficult than modeling the conditional mean (a first-order property).
The rest of this paper is organized as follows. In Section 2, we present the formal modeling framework and introduce the notation. In Section 3, we describe the estimation method. In Section 4, we present the empirical application and provide simulation studies of the statistical and computational properties of the proposed estimator. Section 5 concludes the paper.
Section snippets
Technical aspects of OU processes
Stochastic volatility models based on OU processes. In the classical contributions to modern financial theory, the log price or log exchange rate, , is modeled as a Brownian motion with drift: where is the volatility parameter, is the drift term and is a standard Brownian motion. Assume that the process is observed at discrete time points , for some , and . Then, integrated returns i.e., the changes in the log price over
Estimation
BS (2001) provide an approximate state space representation of the OU model for volatility discussed above, by using the first- and second-order properties of and . In particular, they show that and where for . In Section 3.2 we extend the state space representation proposed by BS (2001) to the case with superposition (10). Then, cf. (11). The state space form allows us to formulate a Gaussian quasi-likelihood function, to make
Application to exchange rate data
We estimate models by using Euro/NOK and US Dollar/NOK daily exchange rate data for the periods 1.1.1999–4.2.2010 and 1.1.1989–4.2.2010, respectively. That is, , for , are the daily changes in the log prices of either the Euro or US Dollar, measured in Norwegian kroner. This gives for the US Dollar/NOK data and for the Euro/NOK data. These returns series are depicted in Fig. 2; before 1.1.1999, the “Euro/NOK” data are those of the European Currency Unit (ECU); i.e., a
Conclusions
In this paper, we examined the statistical and computational properties of an indirect inference estimator for a class of stochastic volatility models for financial data based on non-Gaussian Ornstein–Uhlenbeck (OU) processes, originally proposed in this context by Barndorff-Nielsen and Shephard (2001). In this class of models, the volatility is driven by Levy jump processes. In the literature, there are many analytical results relating to the distribution and dependence structure of integrated
Acknowledgements
We are especially indebted to two anonymous referees for making many suggestions that have substantially improved the paper. Comments from Terje Skjerpen, Anders Rygh Swensen and participants at the Third International Conference on Computational and Financial Econometrics (CFE) held on 29–31 October 2009, in Limassol, Cyprus, are also appreciated. Financial support from the Norwegian Research Council (“Finansmarkedsfondet”) is gratefully acknowledged.
References (36)
- et al.
ARCH models
- et al.
Inference with non-Gaussian Ornstein–Uhlenbeck processes for stochastic volatility
Journal of Econometrics
(2006) - et al.
Bayesian inference with stochastic volatility models using continuous superpositions of non-Gaussian Ornstein–Uhlenbeck processes
Computational Statistics and Data Analysis
(2010) - et al.
Simulations of Levy-driven Ornstein–Uhlenbeck processes with given marginal distributions
Computational Statistics and Data Analysis
(2009) - et al.
The distribution of realized exchange rate volatility
Journal of the American Statistical Association
(2001) - et al.
Estimating continuous time stochastic volatility models of the short term interest rate
Journal of Econometrics
(1997) - et al.
Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics
Journal of the Royal Statistical Society, Series B. Statistical Methodology
(2001) - et al.
Econometric analysis of realized volatility and its use in estimating stochastic volatility models
Journal of the Royal Statistical Society, Series B. Statistical Methodology
(2002)
Integrated OU processes and non-Gaussian OU-based stochastic volatility models
Scandinavian Journal of Statistics
On the Levy measure of the lognormal and the log Cauchy distributions
Methodology and Computing in Applied Probability
Algorithms for Minimization Without Derivatives
Quasi-indirect inference for diffusion processes
Econometric Theory
Control variants for variance reduction in indirect inference: interest rate models in continuous time
Econometrics Journal
Empirical Modeling of Exchange Rate Dynamics
The dynamics of exchange rate volatility: a multivariate latent factor ARCH model
Journal of Applied Econometrics
Cited by (14)
Exit dynamics of start-up firms: Structural estimation using indirect inference
2018, Journal of EconometricsCitation Excerpt :Indirect inference seems appropriate for our study because it is not possible to compute the exact likelihood, whereas simulation of the model is feasible. Indirect inference is commonly used in financial econometrics; some examples include stochastic volatility-, exchange rate-, asset price- and interest rate modeling; for example, see Andersen and Lund (1997), Andersen et al. (1999), Bansal et al. (2007), and Raknerud and Skare (2012). Other examples of application of indirect inference include Magnac et al. (1995) and An and Liu (2000) on labor market transitions, Nagypál (2007) on learning by employees, Collard-Wexler (2013) on the role of demand shocks in the US ready-mix concrete industry, and Li and Zhang (2015) on bidding by heterogeneous actors.
The Split-SV model
2016, Computational Statistics and Data AnalysisPricing Asian options in a stochastic volatility model with jumps
2014, Applied Mathematics and ComputationFitting general stochastic volatility models using Laplace accelerated sequential importance sampling
2012, Computational Statistics and Data AnalysisCitation Excerpt :Parameter estimation in such models is made difficult by the presence of a latent volatility process. The recent approaches follow essentially five lines of attack for integrating out the volatility: simulated maximum likelihood (SML) (e.g. Danielsson and Richard, 1993, Danielsson, 1994, Shephard and Pitt, 1997, Sandmann and Koopman, 1998, Durbin and Koopman, 2000, Liesenfeld and Richard, 2003, 2006, Durham, 2006, 2007, Jungbacker and Koopman, 2007 and Richard and Zhang, 2007), Markov chain Monte Carlo (e.g. Jacquier et al., 1994, Eraker et al., 2003 and Omori et al., 2007 and references therein), sequential importance sampling (SIS) (e.g. Pitt and Shephard, 1999 and Durham and Gallant, 2002; Durham, 2006 and references therein), the efficient method of moments (e.g. Gallant and Tauchen, 1997, Gallant et al., 1997) and indirect inference (Raknerud and Skare, 2012). A number of other alternatives to Laplace based SML have been described in literature.
Indirect inference for time series using the empirical characteristic function and control variates
2021, Journal of Time Series AnalysisApplication of iterated filtering to stochastic volatility models based on non-Gaussian Ornstein-Uhlenbeck process
2020, Statistics in Transition New Series