Indirect inference methods for stochastic volatility models based on non-Gaussian Ornstein–Uhlenbeck processes

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Abstract

An indirect inference method is implemented for a class of stochastic volatility models for financial data based on non-Gaussian Ornstein–Uhlenbeck (OU) processes. First, a quasi-likelihood estimator is derived from an approximative Gaussian state space representation of the OU model. Next, data are simulated from the OU model for given parameter values. The indirect inference estimator is then obtained by minimizing, 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. The method is applied to Euro/Norwegian krone (NOK) and US Dollar/NOK daily exchange rate data. A simulation study reveals that the quasi-likelihood estimator may have a large bias even in large samples, but that the indirect inference estimator substantially reduces this bias. The accompanying R-package, which interfaces C++ code, is documented and can be downloaded.

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, y(t), is modeled as a Brownian motion with drift: dy(t)=μdt+σdw(t), where σ is the volatility parameter, μ is the drift term and w(t) is a standard Brownian motion. Assume that the process is observed at discrete time points tn=nΔ, for some Δ>0, and n=1,2,,N. Then, integrated returns yny(nΔ)y((n1)Δ), 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 yn and yn2. In particular, they show that yn=μΔ+u1n and yn2=μ2Δ2+σn2+u2n, where E(ui1|σn)=0 for i=1,2. In Section 3.2 we extend the state space representation proposed by BS (2001) to the case with superposition (10). Then, σn2=j=1mσjn2, 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, yn, for n=1,,N, are the daily changes in the log prices of either the Euro or US Dollar, measured in Norwegian kroner. This gives N=5497 for the US Dollar/NOK data and N=2890 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.

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