Asymptotic Properties of GARCH-X Processes

Abstract

The paper considers the GARCH-X process in which the covariate is generalized as a fractionally integrated process I (d) for 1=2 < d < 1=2 or 1=2 < d < 3=2: We investigate the asymptotic properties of this process, and show how it explains stylized facts of …nancial time series such as the long memory property in volatility, leptokurtosis and IGARCH. If the covariate is a long memory process, regardless that it is stationary or nonstationary, the autocorrelation of the squared process of the model generates the long memory property in volatility by following the trend commonly observed in real data. The asymptotic limit of the sample kurtosis of the GARCH-X process is larger than that of the GARCH(1,1) process unless the covariate is antipersistent. We also analyze the e¤ect of omitting the covariate that is nonstationary as well as persistent. Our analysis shows that, if the relevant covariate is omitted and the usual GARCH(1,1) model is …tted, then the model would be estimated approximately as the IGARCH. This may well explain the ubiquitous evidence of the IGARCH in empirical volatility analysis.

Asymptotic Theory for the QMLE in GARCH-X Models With Stationary and Nonstationary Covariates

“…Hence, the model can accommodate leverage e¤ects catered for by the GJR-GARCH model if f" t g and fv t g are negatively correlated. See Han (2011) for more details on the model and its time series properties. Whether x t is stationary or not, we will require it to be exogeneous in the sense that E [" t jx t 1 ] = 0 and E " 2 t jx t 1 = 1.…”

“…To develop this variance-ratio approximation, we utilize some results derived in Han (2011). We impose the following conditions on the model which are slightly stronger than the ones imposed in the stationary case, but on the other hand allow for non-stationary regressors:…”

“…Our theoretical results for the non-stationary case rely on results developed in Han (2011) who analyzes the time series properties of GARCH-X models with long-memory regressors. He shows how the GARCH-X process explains stylized facts of …nancial time series such as the long memory property in volatility, leptokurtosis and IGARCH.…”

QML Inference for Volatility Models With Covariates

“…Actually, even if practitioners often add exogenous variables to volatility models, the probabilistic properties and the statistical inference of ARCH models with exogenous variables have not been yet extensively studied. Notable exceptions are the papers of Han (2013), Han and Kristensen (2014) and Han andPark (2012, 2014), which studied the inference of the GARCH(1,1) model augmented by an additional covariate which can be persistent. A common assumption to all the references previously given in this section, is that the true value of the parameter belongs to the interior of the parameter space.…”

Misspecification Testing in GARCH-MIDAS Models

“…For this reason, an increasing amount of empirical studies has employed the GARCH-MIDAS framework introduced by Engle et al (2013) (see, e.g., Asgharian et al, 2013, Conrad and Loch, 2014, 2015, Dorion, 2013, Opschoor et al, 2014. In a GARCH-MIDAS specification, the conditional variance consists of two multiplicative components, whereby economic conditions enter through the smooth long-term component around which a short-term unit variance GARCH component fluctuates.…”