  Dimension reduction provides a useful tool for analyzing high dimensional
data. The recently developed \textit{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 \textit{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).
