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Suppose Z t is the square of a time series Y t whose conditional mean is zero. We do not specify a model for Y t, but assume that there exists a p×1 parameter vector Φ such that the conditional distribution of Z t| Z t−1 is the same as that of [ABSTRACT FROM AUTHOR]

In this article, we propose a new generalized index to recover relationships between two sets of random vectors by finding the vector projections that minimize an distance between each projected vector and an unknown function of the other. The unknown functions are estimated using the Nadaraya-Watson smoother. Extensions to multiple sets and groups of multiple sets are also discussed, and a bootstrap procedure is developed to detect the number of significant relationships. All the proposed methods are assessed through extensive simulations and real data analyses. In particular, for environmental data from Los Angeles County, we apply our multiple-set methodology to study relationships between mortality, weather, and pollutants vectors. Here, we detect existence of both linear and nonlinear relationships between the dimension-reduced vectors, which are then used to build nonlinear time-series regression models for the dimension-reduced mortality vector. These findings also illustrate potential use of our method in many other applications. A comprehensive assessment of our methodologies along with their theoretical properties are given in a Web Appendix. [ABSTRACT FROM AUTHOR]

We propose a notion of central mean dimension reduction subspace for time series {x1} which does not require specification of a model but seeks to find a p x d matrix Φd, d ≤ p, so that the d x 1 vector where ΦdTXt-1 where Xt-1= (xt-1,…Xt-p)T for some p ≥ 1, includes all the information about xt that is available from E(xt∣Xt-1). For known p and d, we estimate the mean central subspace through the Nadaraya-Watson kernel smoother and establish the strong consistency of our estimator. In addition, we propose estimation of d and p using a modified Schwarz Bayesian criterion, if either of d and p is unknown. Finally, we examine the performance of all the estimators extensively through a variety of simulations and provide a new analysis of the well-known Canadian lynx data. Supplemental materials for this article are available online. [ABSTRACT FROM AUTHOR]