Quadratic ARCH Models

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Oxford University Press and The Review of Economic Studies, Ltd. are collaborating with JSTOR to digitize, preserve and extend access to The Review of Economic Studies.We introduce a new model for time-varying conditional variances as the most general quadratic version possible within the ARCH class. Hence, it encompasses all the existing restricted quadratic variance functions. Its properties are very similar to those of GARCH models, but avoids some of their criticisms. In univariate applications to daily U.S. and monthly U.K. stock market returns, QARCH adequately represents volatility and risk premia. QARCH is easy to incorporate in multivariate models to capture dynamic asymmetries that GARCH rules out. Such asymmetries are found in an empirical application of a conditional factor model to 26 U.K. sectorial stock returns.(or QARCH) model. As we shall see, it combines many attractive features of several existing ARCH models while ensuring that estimated conditional variances are nonnegative.In particular, the QARCH model nests the recently proposed Augmented ARCH (AARCH) model of Bera and Lee (1990), which in turn encompasses Engle's (1982) original ARCH model. It also nests the linear "standard deviation" model discussed by Robinson (1991), and the asymmetric ARCH model in Engle (1990). This nesting has at least three non-trivial advantages. First, many theoretical results derived for these models still hold with minor modifications for the QARCH model, including estimation and testing, stationarity conditions and persistence properties, autocorrelation structure, temporal and contemporaneous aggregation, forecasting, etc. Second, QARCH conditional variances can also be easily integrated in economic models, just as linear time-series models for the conditional mean are (see Campbell and Hentschel (1992) for an application to a stock returns model). Third, the QARCH model is capable of improving the empirical success of ARCH models, since it avoids some of its criticisms without departing significantly from the standard specification. In particular, it provides a very simple way of calibrating and testing for dynamic asymmetries in the conditional variance function of the kind postulated for some financial series (see e.g. Black (1976) and Nelson (1991)).The QARCH formulation can also be interpreted as a second-order Taylor approximation to the unknown function &2( ), or, alternatively, as the quadratic projection of the square innovation on the information set. Actually, this projection coincides with the projection of the true conditional variance a2 on a "quadratic information set", so that we can also understand the QARCH model as producing smooth filtered "estimates" of ...

Persistence and Kurtosis in GARCH and Stochastic Volatility Models

“…Notice that the covariance stationarity of y t does not depend on the parameter g that measures the asymmetry. Sentana (1995) shows that the properties of the GARCH(1,1) and QGARCH(1,1) models are very similar. In fact, both have the same unconditional mean and variance equal to zero and s 2 y ¼ v 1 p , respectively.…”

“…In this section we consider the simplest specifications directly related with the GARCH(1,1) and ARSV(1) models analyzed before. In particular, we consider the QGARCH model proposed by Sentana (1995), the EGARCH model of Nelson (1991), and the asymmetric ARSV model proposed by Harvey and Shephard (1996).…”

“…In this article we introduce this asymmetry in the GARCH model using the specification proposed by Sentana (1995), known as the quadratic GARCH (QGARCH) model, because its statistical properties are easy to derive. On the other hand, we consider the asymmetric ARSV model proposed by Harvey and Shephard (1996).…”

The Intertemporal Relation between Expected Return and Risk on Currency

“…10 Chan, Karolyi, and Stulz (1992) employ a bivariate GARCH-in-mean model to estimate the conditional variance, and they also fail to obtain a significant coefficient estimate for the United States. Campbell and Hentchel (1992) use the quadratic GARCH (QGARCH) model of Sentana (1995) to determine the existence of a risk-return tradeoff within an asymmetric GARCH-in-mean framework. Their estimate is positive for one sample period and negative for another sample period, but neither is statistically significant.…”

An Introduction to Univariate GARCH Models

“…Several extensions of the GARCH model aim at accommodating the asymmetry in the response. These include the GJR-GARCH model of Glosten, Jagannathan and Runkle (1993), the asymmetric GARCH models of Engle and Ng (1993) and the quadratic GARCH of Sentana (1995). The GJR-GARCH model has the form (2) ; where…”