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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 ...

This paper considers how ARCH effects may be handled in time series models formulated in terms of unobserved components. A general model is formulated, but this includes as special cases a random walk plus noise model with both disturbances subject to ARCH effects, an ARCH-M model with a time-varying parameter, and a latent factor model with ARCH effects in the factors. Although the model is not conditionally Gaussian, an approximate filter can be obtained and used as the basis for estimation. The performance of this method is examined on real data sets and Monte Carlo experiments are carried out. The method is extended to handle GARCH and a latent factor model based on the I-distribution.

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The Working Paper Series seeks to disseminate original research in economics and fi nance. All papers have been anonymously refereed. By publishing these papers, the Banco de España aims to contribute to economic analysis and, in particular, to knowledge of the Spanish economy and its international environment. The opinions and analyses in the Working Paper Series are the responsibility of the authors and, therefore, do not necessarily coincide with those of the Banco de España or the Eurosystem. The Banco de España disseminates its main reports and most of its publications via the INTERNET at the following website: http://www.bde.es. Reproduction for educational and non-commercial purposes is permitted provided that the source is acknowledged.

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The Working Paper Series seeks to disseminate original research in economics and fi nance. All papers have been anonymously refereed. By publishing these papers, the Banco de España aims to contribute to economic analysis and, in particular, to knowledge of the Spanish economy and its international environment.The opinions and analyses in the Working Paper Series are the responsibility of the authors and, therefore, do not necessarily coincide with those of the Banco de España or the Eurosystem.The Banco de España disseminates its main reports and most of its publications via the INTERNET at the following website: http://www.bde.es.Reproduction for educational and non-commercial purposes is permitted provided that the source is acknowledged.