ABSTRACT
CONTENTS 10.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 Part 1: Deterministic Conditional Volatility Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 10.2 Notions of Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
10.2.1 Realized Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 10.2.2 Conditional Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 10.2.3 Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
10.3 Autoregressive Conditional Heteroscedasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 10.3.1 Moments of a General ARCH Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
10.4 Estimation of ARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 10.4.1 ML Estimation of an ARCH Regression Model . . . . . . . . . . . . . . . . . . . . . . 646 10.4.2 Estimation of ARCH Models: Practical Aspects . . . . . . . . . . . . . . . . . . . . . 647 10.4.3 Testing for ARCH Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 10.4.4 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648
10.5 Generalized Autoregressive Conditional Heteroscedasticity . . . . . . . . . . . . . . . . . . . 649 10.5.1 Estimation of GARCH(p, q) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 10.5.2 Testing for GARCH Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653
10.6 Stationary ARMA-GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 10.6.1 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654
10.7 Nonstationary ARMA-GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 10.8 Forecasting ARCH and GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654
10.8.1 Forecasting an ARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654 10.8.2 Forecasting a GARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655
10.9 Integrated GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 10.9.1 Advantage of IGARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657
10.10 Asymmetric GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658 10.10.1 EGARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658 10.10.2 GJR-GARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 10.10.3 An Alternative Form of the EGARCH Model . . . . . . . . . . . . . . . . . . . . . . . 661 10.10.4 An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661
10.11 News Impact Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 10.12 Some Alternative Predictable Volatility Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 10.13 Diagnostic Tests Based on News Impact Curves: Test of Asymmetry . . . . . . . . . . . . . 666
10.13.1 How to Conduct These Tests? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666 10.14 Nesting the GARCH Family of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 10.15 Forecasting an EGARCH Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 Part 2: Stochastic Volatility Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 10.16 Stochastic Volatility Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673
10.16.1 Alternative Forms of the SV Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674 10.16.2 Properties of Returns in SV Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674
10.17 Standard SV Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 10.17.1 An Alternative Parameterization of the Log-Normal SV Model . . . . . . . . . . . 676 10.17.2 Basic Properties of the Alternative Log-Normal SV Model . . . . . . . . . . . . . . 676 10.17.3 Autocorrelations of lt , σt , at , st . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678
10.18 Estimation of SV Models: State Space Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 10.19 MCMC-Based Estimation of the Standard SV Model . . . . . . . . . . . . . . . . . . . . . . . . 680 10.20 Bayesian Inference and MCMC Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681
10.20.1 Gibbs Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 10.20.2 Metropolis-Hastings Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682
10.21 MCMC-Based Estimation of the Standard SV Model (Continued) . . . . . . . . . . . . . . . 683 10.21.1 Sampling of μ, φ, and σ2η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 10.21.2 Sampling of the Latent Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 10.21.3 Simple Rejection Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686 10.21.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687
10.22 GARCH versus SV Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 10.23 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690
ABSTRACT Accounting for and predicting volatility, especially in asset prices, occupy a central role in finance. The importance of correctly specifying conditional volatility has deep implications for quantifying risk. Naturally, this led to a plethora of volatility models, both deterministic and stochastic. Such volatility models have been estimated both in univariate and multivariate contexts. This chapter focuses on univariate volatility models. In the first part, we begin listing the various notions of volatility and concentrate on some of the pioneering statistical models that have been discussed in the literature to quantify conditional volatility. This part is a tour to understand both the theoretical underpinnings and the practical aspects in the estimation of deterministic volatility models such as the linear ARCH and GARCH family of models and their nonlinear counterparts. Our discussion will revolve around models that explain stylized facts such as high kurtosis and fat tails in financial returns, the presence of positive autocorrelations in both the absolute and squared returns, indicating substantially more linear dependence than autocorrelations of returns. Along the way, we outline the practical issues involved in model building and forecasting, from a practitioner’s viewpoint. In the second part, we describe stochastic volatility models, where conditional volatility is described as a latent variable that depends not on past observations but on some unobserved past latent structure. Steps involved in estimating such models using state space and Markov chain Monte Carlo methods are outlined. A number of examples will highlight the important practical issues involved.
