ABSTRACT
The log-GARCH model provides a exible framework for the modelling of economic uncertainty, financial volatility and other positively valued variables. Its exponential specification ensures fitted volatilities are positive, allows for exible dynamics, simplifies inference when parameters are equal to zero under the null, and the logtransform makes the model robust to jumps and outliers. An additional advantage is that the model admits ARMA-like representations. This means log-GARCH models can readily be estimated by means of widely available software, and enables a vast range of well-known time-series results and methods. This chapter provides an overview of the log-GARCH model and its ARMA representation(s), and of how estimation can be implemented in practice. After the Introduction, we provide an overview of univariate models in Section 2. We start by outlining an asymmetric specification, before we turn to its ARMA representation. Next we add stochastic conditioning covariates (“X”), before turning to how estimates of the coeficientcovariances can be obtained in numerical software. We complete the Section by empirical illustrations of the log-GARCH(1,1) model. Section 3 provides an overview of multivariate models. Again, we start by outlining the asymmetric specification and its corresponding VARMA and VARMA-X representations. Next, we turn to specifications that are amenable to equation-by-equation estimation, both stationary and non-stationary versions, even in the presence of Dynamic Conditional Correlations (DCCs) of unknown form. The focus on multivariate specifications that can be estimated equation-by-equation is motivated by the fact estimation becomes infeasible in practice as the dimension grows too large. We end the section with a short note on how models of Dynamic Conditional Correlations (DCCs) can be estimated subsequently. Section 4 provides some suggestions on how to handle zeros in practice, whereas Section 5 outlines how log-GARCH models can be used to model positively valued variables. Finally, we conclude the chapter and provide suggestions for further research.
JEL Classification: C22, C32, C51, C58
