Prediction-powered inference for time series across space

The following motif is common in spatiotemporal settings: we have a labeled dataset consisting of a sequence of covariate and response pairs observed for a short, recent time period. We have access to unlabeled covariates over a longer time period. Data is observed over many spatial locations. For instance, crop yield might be observed over a large geographical area for recent years, but weather data (which is informative about crop yield) is available for a much longer historical period. The goal is to estimate, at each spatial location, the expected response (e.g. crop yield) in the future and provide a valid confidence interval for this value. The observed time period alone is too short for reliable estimates. Imputing missing labels with machine learning can cause substantial bias. Prediction-powered inference (PPI) can correct for this bias, but it relies on an i.i.d. assumption that breaks under our expected temporal dependencies. Heteroskedasticity and autocorrelation consistent (HAC) procedures account for temporal correlation, but have not been adapted to cases where some responses are imputed. In the present work, we provide reliable point estimates and confidence intervals for the problem outlined above. We show our method outperforms natural alternatives.