Dimension reduction provides a useful tool for analyzing high dimensional data. The recently developed Envelope method is a parsimonious version of the classical multivariate regression model through identifying a minimal reducing subspace of the responses. However, existing envelope methods assume an independent error structure in the model. While the assumption of independence is convenient, it does not address the additional complications associated with spatial or temporal correlations in the data. In this article, we introduce a Spatial Envelope method for dimension reduction in the presence of dependencies across space. We study the asymptotic properties of the proposed estimators and show that the asymptotic variance of the estimated regression coefficients under the spatial envelope model is smaller than that from the traditional maximum likelihood estimation. Furthermore, we present a computationally efficient approach for inference. The efficacy of the new approach is investigated through simulation studies and an analysis of an Air Quality Standard (AQS) dataset from the Environmental Protection Agency (EPA).

No abstract

No abstract

To explain the behavior of a whole system, we may need more than one multiple regression with its own set of explanatory variables. Seemingly Unrelated Regressions (SUR) is a way to model these equations simultaneously. In this approach, the cross‐correlation between equations will be accounted in the estimation of regression coefficients. SUR has been used in many different areas such as econometrics, environmetrics, and social science. In this article, we discuss SUR model and some of its extensions, along with some of its applications.