No abstract

This paper develops a generalized autoregressive conditional correlation~GARCC! model when the standardized residuals follow a random coefficient vector autoregressive process+ As a multivariate generalization of the Tsay~1987, Journal of the American Statistical Association 82, 590-604! random coefficient autoregressive~RCA! model, the GARCC model provides a motivation for the conditional correlations to be time varying+ GARCC is also more general than the Engle~2002, Journal of Business & Economic Statistics 20, 339-350! dynamic conditional correlation~DCC! and the Tse and Tsui~2002, Journal of Business & Economic Statistics 20, 351-362! varying conditional correlation~VCC! models and does not impose unduly restrictive conditions on the parameters of the DCC model+ The structural properties of the GARCC model, specifically, the analytical forms of the regularity conditions, are derived, and the asymptotic theory is established+ The Baba, Engle, Kraft, and Kroner~BEKK! model of Engle and Kroner~1995, Econometric Theory 11, 122-150! is demonstrated to be a special case of a multivariate RCA process+ A likelihood ratio test is proposed for several special casesThe authors thank the co-editor, Bruce Hansen, and three referees for insightful suggestions and Manabu Asai,

Typical multivariate economic time series may exhibit co-behavior patterns not only in the conditional means, but also in the conditional variances. In this paper we give two new de®nitions of variance noncausality in a multivariate setting; a Granger-type noncausality and a linear Granger noncausality through projections on Hilbert spaces. Both de®nitions are related to a previous second-order noncausality concept de®ned by Granger et al. in a bivariate setting. The implications of secondorder noncausality on multivariate ARMA processes with GARCH-type errors are investigated. We derive exact testable restrictions on the parameters of the processes considered, implied by this type of noncausality. Conditions for the ®niteness of the fourth-order moment of the multivariate GARCH process are derived and related to earlier results in the univariate framework. We include an illustration of second-order noncausality in a trivariate model of daily ®nancial returns.Keywords. BEKK; GARCH; second-order noncausality.autoregressive conditional heteroskedasticity (ARCH) model and its various subsidiaries. See, for instance, Engle (1982), Bollerslev (1986), Nelson (1991), Bollerslev et al. (1992 and the many references therein, Bollerslev and Engle (1993) and Bollerslev et al. (1994).While it is clear that, at least for volatile data, causality testing in variance processes is just as important as causality testing for mean processes, so far only limited attempts in a bivariate setting have been made to characterize second-order noncausality. These include Granger et al.'s (1986) bivariate de®nition of second-order noncausality, and Cheung and Ng's (1996) development of causality in variance test, again in a bivariate setting.The purpose of this paper is to characterize variance noncausality in a multivariate framework. Two new de®nitions of variance noncausality are given and are related to the previous second-order noncausality concept introduced by Granger et al. (1986). The ®rst is a Granger-type noncausality and the second is a linear Granger noncausality through projections on Hilbert spaces. It is stressed that other types of noncausality notions described in the statistics literature, and in particular those covered by Holland (1986), are not dealt with here. The de®nitions provided are consistent with the mainstream econometric de®nitions of Granger's noncausality in mean or in the density function. We show that second-order noncausality leads to exact testable restrictions on the parameters of the general class of VARMA models with GARCH-type errors.The plan for the paper is as follows. In Section 2 we review the basic notions of Granger and linear noncausalities in mean and give two de®nitions of variance noncausality. Connections between mean, second-order and variance noncausality concepts are drawn. We show that in VARMA models with martingale difference innovations possessing a constant conditional variance matrix, there is always second-order noncausality. In Section 3 we give suf®cient conditions for seco...

International audienceAn agent is asked to assess a real-valued variable Yp based on certain characteristics Xp = (Xp-super-1, ..., Xp-super-m), and on a database consisting of Xi-super-1, ... Xi-super-m, Yi) for i = 1, ..., n. A possible approach to combine past observations of X and Y with the current values of X to generate an assessment of Y is similarity-weighted averaging. It suggests that the predicted value of Y, Ȳp-super-s, be the weighted average of all previously observed values Yi, where the weight of Yi for every i = 1, ..., n, is the similarity between the vector Xp-super-1, ..., Xp-super-m, associated with Yp, and the previously observed vector, Xi-super-1, ..., Xi-super-m. We axiomatize this rule. We assume that, given every database, a predictor has a ranking over possible values, and we show that certain reasonable conditions on these rankings imply that they are determined by the proximity to a similarity-weighted average for a certain similarity function. The axiomatization does not suggest a particular similarity function, or even a particular form of this function. We therefore proceed to suggest that the similarity function be estimated from past observations.We develop tools of statistical inference for parametric estimation of the similarity function, for the case of a continuous as well as a discrete variable. Finally, we discuss the relationship of the proposed method to other methods of estimation and prediction. Copyright by the President and Fellows of Harvard College and the Massachusetts Institute of Technology

No abstract