GARCH Process with Persistent Covariates

The paper considers a volatility model that includes a persistent, integrated or nearly integrated, covariate in the standard GARCH(1,1) model. First, we establish the asymptotic distribution theory of the quasi-maximum likelihood estimator (QMLE) for such a model. In general, the QMLE is consistent, and its limit distribution is non-Gaussian and represented as a functional of Brownian motions. However, it becomes Gaussian if the covariate is strictly exogenous or the volatility function is linear in parameter. Second, we provide asymptotic theories, which show that the model successfully explains stylized facts of nancial time series, such as long memory property in volatility, IGARCH and leptokurtosis. The autocorrelation of the squared process of the model generates the long memory property following a trend commonly observed in real data. Our theory of misspeci cation shows that IGARCH could be the result of missing a relevant persistent covariate in the GARCH(1,1) process. Moreover, the kurtosis of the model is randomly bigger than the kurtosis of the GARCH(1,1) model. This version: December 2007 JEL classi cation: C22, C50, G12 Keywords; GARCH, persistent covariate, quasi-maximum likelihood estimator, asymptotic distribution theory, long memory property, IGARCH, leptokurtosis 1 I would like to thank Joon Y. Park and conference participants at the 18th EC meeting on advances in econometric time series analysis (Faro), the third symposium on econometric theory and applications (SETA2007, Hong Kong) and 2007 Korean econometric society meeting (Seoul) for helpful comments and suggestions. I also thank Je¤ Fleming for providing the data for rms in the MMI. The research for this paper was supported by the NUS Risk Management Institute. Address correspondence: Department of Economics, National University of Singapore, 1 Arts Link, Singapore 117570; phone: (65) 6516-6258; fax: (65) 6775-2646; e-mail: ecshhj@nus.edu. sg 2 1 Introduction ARCH type models have been widely used to model the volatility of economic and nancial time series since the seminal work by Engle (1982) and the extension made by Bollerslev (1986). However, since most ARCH type models have been univariate, relating the volatility of time series only to the information contained in its own past history, they shed little light on an economic source of volatility.2 One of the simplest ways to consider an economic source of volatility is to include economic variables in volatility models. Thus there have been works that consider ARCH type models with macroeconomic variables as covariates. Since the GARCH(1,1) model has been popular and many macroeconomic variables are persistent, these works use the GARCH(1,1) model with a persistent covariate as following; t = ! + y 2 t 1 + 2 t 1 + jxt 1j (or xt 1); where (yt) is a demeaned time series and t is its variance conditional on the information available at time t 1: As a covariate (xt), Glosten et al (1993) and Engle and Patton (2001) used three-month U.S. Treasury bill rates for stock return volatility, and Gray (1996) used the level of interest rates in his generalized regime-switching model of short-term interest rates. Likewise, forward-spot spreads and interest rate di¤erentials between countries were used as covariates respectively by Hodrick (1989) and Hagiwara and Herce (1999) for exchange rate return volatility. Note that these covariates are persistent; interest rates and interest rate di¤erentials between countries may well be modeled as time series with a unit root or a near unit root. Like most empirical applications of ARCH type models, the quasi-maximum likelihood estimation procedure was used in the literature to estimate the GARCH(1,1) model with a persistent covariate, which has either a unit root or a near unit root. However, there is no existing literature that provides the asymptotic distribution theory of the quasi-maximum likelihood estimator (QMLE) in such a model. The asymptotic distribution theory of the QMLE in the GARCH(1,1) model, stationary GARCH(1,1) as well as IGARCH(1,1), was formally established by Lumsdaine (1996) and Lee and Hansen (1994). Their theories depend on covariance-stationarity or strict stationarity (in case of IGARCH(1,1)) of the GARCH(1,1) process. However, their theories are not valid for the GARCH(1,1) process with a persistent covariate because the process is not stationary due to a nonstationary covariate. Recently, Jensen and Rahbek (2004) established the asymptotic distribution theory of the QMLE in the GARCH(1,1) model with the entire parameter region, including nonstationary behavior. However, their theory also cannot be applied in the GARCH(1,1) model with a persistent covariate. Since there is no existing literature which investigates the statistical properties of the GARCH(1,1) model with an additional persistent covariate, this paper attempts to ll this As noted by Engle and Rangel (2005), "After more than 25 years of research on volatility, the central unsolved problem is the relation between the state of the economy and aggregate nancial volatility. The number of models that have been developed to predict volatility based on time series information is astronomical, but the models that incorporate economic variables are hard to nd. "