  The Fay-Herriot (FH) model is widely used in small area estimation and uses
auxiliary information to reduce estimation variance at undersampled locations.
We extend the type of covariate information used in the FH model to include
functional covariates, such as social-media search loads or remote-sensing
images (e.g., in crop-yield surveys). The inclusion of these functional
covariates is facilitated through a two-stage dimension-reduction approach that
includes a Karhunen-Lo\`{e}ve expansion followed by stochastic search variable
selection. Additionally, the importance of modeling spatial autocorrelation has
recently been recognized in the FH model; our model utilizes the intrinsic
conditional autoregressive class of spatial models in addition to functional
covariates. We demonstrate the effectiveness of our approach through simulation
and analysis of data from the American Community Survey. We use Google Trends
searches over time as functional covariates to analyze relative changes in
rates of percent household Spanish-speaking in the eastern half of the United
States.
