HEWITT, HOETING, DONE, AND TOWLER 3.2 RESP model and prior specification In the RESP model (3), we specify a linear relationship between the local covariate T and response P so that β in (3) has intercept β$_{0}$ and slope β$_{T}$ components β = ( β$_{0}$, β$_{T}$ ) $^{T}$. While the RESP model as described in Section 2.1 uses a stationary covariance model and precipi- tation is non-stationary in space, stationary models have comparable predictive performance in Colorado (Paciorek & Schervish, 2006). For the RESP model’s remote coefficients, knots are placed at 93 locations that are roughly evenly spaced across the Pacific Ocean and along coastal locations (Supplement A, ?? ). While knot selection can be problematic, Baner- jee et al. (2008) find that reasonably dense, regularly spaced grids can yield good results. Since the ENSO teleconnection is scientifically meaningful, we will interpret the transformed teleconnection effects α $^{′}$( s , 1) from (12), which are associated with ENSO through its connec- tion to the first empirical orthogonal function (EOF) of sea surface temperature anomalies ψ$_{1}$ : D$_{Y}$ → $_{R}$. We adopt a combination of weakly informative and non-informative prior distributions. A dispersed normal prior is used for the fixed effects β ∼ N ( 0 , 10 I ). We use σ 2 w ∼ IG (2 , 1), σ 2 ε ∼ IG (2 , 1), ρ$_{w}$ ∼ U (1 , 600), and ρ$_{α}$ ∼ U (1 , 2000). The Mat´ ern covariance smoothness parameters (7) are fixed at ν$_{w}$ = ν$_{α}$ = . 5, which correspond to the smoothest well-defined Mat´ ern covariances for Gaussian processes on spheres (Gneiting, 2013). In exploratory anal- ysis, variograms for the local and remote data fit this parameterization well. The prior for σ 2 α is informative to increase the identifiability of this parameter and the remote range ρ$_{α}$ (Zhang, 2004). The prior σ 2 α ∼ IG (6 , 10) keeps the model from exploring parameter combi- nations that would imply very large teleconnection influence relative to the scale of the data Y ( s , t ). We demonstrate the benefit of remote covariates by comparing the RESP model to RE and SP submodels that, respectively, exclude local and remote covariates. We also show improve- 3.3 Comparison models 18