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Exact Likelihood Inference for Autoregressive Gamma Stochastic Volatility Models1

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Exact Likelihood Inference for Autoregressive Gamma Stochastic Volatility Models1 Exact likelihood inference for autoregressive gamma stochastic volatility models1 Drew D. Creal Booth School of Business, University of Chicago First draft: April 10, 2012 Revised: September 12, 2012 Abstract Affine stochastic volatility models are widely applicable and appear regularly in empir- ical finance and macroeconomics. The likelihood function for this class of models is in the form of a high-dimensional integral that does not have a closed-form solution and is difficult to compute accurately. This paper develops a method to compute the likelihood function for discrete-time models that is accurate up to computer tolerance. The key insight is to integrate out the continuous-valued latent variance analytically leaving only a latent discrete variable whose support is over the non-negative integers. The likelihood can be computed using standard formulas for Markov-switching models. The maximum likelihood estimator of the unknown parameters can therefore be calculated as well as the filtered and smoothed estimates of the latent variance. Keywords: stochastic volatility; Markov-switching; Cox-Ingersoll-Ross; affine models; autoregressive-gamma process. JEL classification codes: C32, G32. 1I would like to thank Alan Bester, Chris Hansen, Robert Gramacy, Siem Jan Koopman, Neil Shephard, and Jing Cynthia Wu as well as seminar participants at the Triangle Econometrics Workshop (UNC Chapel Hill), University of Washington, Northwestern University Kellogg School of Management, and Erasmus Universiteit Rotterdam for their helpful comments. Correspondence: Drew D. Creal, University of Chicago Booth School of Business, 5807 South Woodlawn Avenue, Chicago, IL 60637. Email: [email protected] 1 1 Introduction Stochastic volatility models are essential tools in finance and increasingly in macroeconomics for modeling the uncertain variation of asset prices and macroeconomic time series. There exists a literature in econometrics on estimation of both discrete and continuous-time stochastic volatility (SV) models; see, e.g. Shephard (2005) for a survey. The challenge when estimating SV models stems from the fact that the likelihood function of the model is a high-dimensional integral that does not have a closed-form solution. The contribution of this paper is the development of a method that calculates the likelihood function of the model exactly up to computer tolerance for a class of SV models whose stochastic variance follows an autoregressive gamma process, the discrete-time version of the Cox, Ingersoll, and Ross (1985) process. This class of models is popular in finance because it belongs to the affine family which produces closed-form formulas for derivatives prices; see, e.g. Heston (1993). A second contribution of this paper is to illustrate how the ideas developed here can be applied to other affine models, including stochastic intensity, stochastic duration, and stochastic transition models. Linear, Gaussian state space models and finite state Markov-switching models are a corner- stone of research in macroeconomics and finance because the likelihood function for these state space models is known; see, e.g. Kalman (1960) and Hamilton (1989). SV models are non- linear, non-Gaussian state space models which in general do not have closed-form likelihood functions. This has lead to a literature developing methods to approximate the maximum likelihood estimator including importance sampling (Danielsson (1994), Durbin and Koopman (1997), Durham and Gallant (2002)), particle filters (Fern´andez-Villaverde and Rubio-Ram´ırez (2007)), and Markov-chain Monte Carlo (Jacquier, Johannes, and Polson (2007)). Alterna- tively, estimation methods have also been developed that do not directly compute the likeli- hood including Bayesian methods (Jacquier, Polson, and Rossi (1994), Kim, Shephard, and Chib (1998)), the generalized method of moments, simulated method of moments (Duffie and Singleton (1993)), efficient method of moments (Gallant and Tauchen (1996)), and indirect inference (Gouri´eroux,Monfort, and Renault (1993)). The results in this paper follow from the properties of the transition density of the Cox, Ingersoll, and Ross (1985) (CIR) process, which in discrete time is known as the autoregressive gamma (AG) process (Gouri´erouxand Jasiak (2006)). The transition density of the CIR/AG 2 process is a non-central gamma (non-central chi-squared up to scale) distribution, which is a Poisson mixture of gamma random variables. The CIR/AG process has two state variables. One is the continuous valued variance ht which is conditionally a gamma random variable and the other is a discrete mixing variable zt which is conditionally a Poisson random variable. The approach to computing the likelihood function here relies on two key insights about the model. First, it is possible to integrate out the continuous-valued state variable ht analytically. Second, an interesting feature of the CIR/AG process is that once one conditions on the discrete mixing variable zt and the data yt, the variance ht is independent of its own leads and lags. The combination of these properties means that the original SV model can be reformulated as a state space model whose only state-variable is discrete-valued with support over the set of non-negative integers. I show that the Markov transition distribution of the discrete variable zt is a Sichel distribution (Sichel (1974, 1975)). Like the Poisson distribution from which it is built, the Sichel distribution assigns increasingly small probability mass to large integers and the probabilities eventually converge to zero. Although the support of the state variable zt is technically infinite, it is finite for all practical purposes. In this paper, I propose to compute the likelihood for the affine SV model by representing it as a finite state Markov-switching model. The likelihood of a finite-state Markov switching model can be computed from known recursions; see, e.g. Baum and Petrie (1966), Baum, Petrie, Soules, and Weiss (1970), and Hamilton (1989). In addition to computing the maximum likelihood estimator, I also show how to compute moments and quantiles of the filtered and smoothed distributions of the latent variance ht. The filtering distribution characterizes the information about the state variable at time t given information up to and including time t whereas the smoothing distribution conditions on all the available data. Solving the filtering and smoothing problem for the variance ht reduces to a two step problem. First, the marginal filtering and smoothing distributions for the discrete variable zt are calculated using Markov-switching algorithms. Then, the marginal filtering and smoothing distributions of the variance ht conditional on zt and the data yt are shown to be generalized inverse Gaussian (GIG) distributions whose moments can be computed analyti- cally. Moments and quantiles of the filtering distribution for the continuous-variable ht can be computed by averaging the moments and quantiles of the conditional GIG distribution by the marginal probabilities of zt calculated from the Markov-switching algorithm. 3 Another method to approximate the likelihood and filtering distribution for this class of models are sequential Monte Carlo methods also known as particle filters. Particle filters were introduced into the economics literature by Kim, Shephard, and Chib (1998). Creal (2012a) provides a recent review of the literature on particle filters. In this paper, I describe how the filtering recursions based on the finite state Markov-switching model are related to the particle filter. Essentially, the finite state Markov-switching model can be interpreted as a particle filter but where there are no Monte Carlo methods involved. This paper continues as follows. In Section 2, I describe the autoregressive gamma stochastic volatility models and demonstrate how to reformulate it with the latent variance integrated out. The model reduces to a Markov-switching model, whose transition distribution is characterized. In Section 3, I provide the details of the filtering and smoothing algorithms and a discussion of how to compute them efficiently. In Section 4, I compare standard particle filterings to the new algorithms and illustrate their relative accuracy. The new methods are then applied to several financial time series. Section 5 concludes. 2 Autoregressive gamma stochastic volatility models The discrete time autoregressive gamma stochastic volatility (AG-SV) model for an observable time series yt is given by p yt = µ + βht + ht"t;"t ∼ N(0; 1); (1) ht ∼ Gamma (ν + zt; c) ; (2) φh z ∼ Poisson t−1 ; (3) t c where µ controls the location, β determines the skewness, and φ is the first-order autocorrelation of ht. The restriction φ < 1 is required for stationarity. The AG-SV model has two state variables ht and zt, where the latter can be regarded as an \auxiliary" variable in the sense that it is not of direct interest. From the relationship to the CIR process discussed below, it is possible to deduce the restriction ν > 1 which is the well-known Feller condition that guarantees that the process ht never reaches zero. The model is completed by specifying an c initial condition that h0 is a draw from the stationary distribution h0 ∼ Gamma ν; 1−φ with 4 νc mean E [h0] = 1−φ . The stochastic process for ht given by (2) and (3) is known as the autoregressive gamma (AG) process of Gouri´erouxand Jasiak (2006), who developed many of its properties. In particular, they illustrated how it is related to the continuous-time model of Cox, Ingersoll, and Ross (1985) for the short term interest rate. Consider an interval of time of length τ. The autoregressive gamma process (2) and (3) converges in the continuous-time limit as τ ! 0 to p dht = κ (θh − ht) dt + σ htdWt: (4) 2 where the discrete and continuous time parameters are related as φ = exp (−κτ) ; ν = 2κθh/σ , and c = σ2 [1 − exp (−κτ)] =2κ. In the continuous-time parameterization, the unconditional mean of the variance is θh and κ controls the spead of mean reversion. The continuous-time process (4) was originally analyzed by Feller (1951) and is sometimes called a Feller or square- root process.
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