REMOTE EFFECTS SPATIAL PROCESS MODELS be generalized, the definitions restrict our use of the RESP model to working in the perfect prognosis downscaling setting described at the end of Section 1. The responses Y ( s , t ) and Y ( s , t $^{′}$) for t = t ′ are independent given covariates and sufficiently separated time points, like successive winters (e.g., winter 1991, winter 1992, etc.). The remote covariates in the teleconnection term (4) naturally induce temporal non-stationarity in the response’s variance; extensions to accommodate serial dependence are discussed in Section 4. The remote covariance C$_{α}$ also induces a separable structure for the remote coefficients α ( s , r ), which constrains the spatial variability of teleconnection effect fields and simultaneously constrains the teleconnection effects { α ( s , r ) : r ∈ D$_{Z}$} and { α ( s $^{′}$, r ) : r ∈ D$_{Z}$} to be similar for nearby locations s , s ′ ∈ D$_{Y}$ . Simpler covariance structures for the teleconnection effects α ( s , r ) may not capture these physical properties of teleconnection as directly. Similarly, although climate data are often available as gridded data products, we choose to work with geostatistical covariance models (or their discrete approximations, e.g., Lindgren, Rue, & Lindstr¨ om, 2011) instead of neighborhood-based spatial models so that we may avoid inducing potentially counterintuitive covariance structures (Assun¸c˜ ao & Krainski, 2009; Wall, 2004). To apply the RESP model (3), additional constraints need to be imposed due to the potential multicollinearity in the covariates. Remote covariates z ( r , t ) in teleconnection applications will often consist of data that measure ocean properties at high spatial resolution, like sea surface temperature or sea level pressure. This raises concerns for estimating the remote coefficients α ( s , r ) in (4) as the main trends in the remote covariates z ( r , t ) are highly collinear over D$_{Z}$ . Physically, however, this suggests the remote coefficients should be highly correlated as well. We use predictive processes to mitigate multicollinearity in the remote covariates, which is an alternative motivation for predictive processes. Banerjee, Gelfand, Finley, and Sang (2008) originally introduce predictive processes so that parameters of geo- 2.2 Reduced rank approximation 9