Uncertainty bounds for long-term causal effects of perturbations in spatiotemporal systems

Abstract

In time-dependent systems, autoregressive models are frequently employed to investigate the interactions between variables of interest in fields such as climate science, macroeconomics, and neuroscience. Typically, these variables are aggregated from smaller-scale variables into large-scale variables, for instance, representing modes of climate variability in climate science. A key aspect of these models is estimating the long-term effects of external perturbations, once the system stabilizes. Our primary contribution is an explicit formula for quantifying these long-term effects on small-scale variables, which is directly estimable from the model’s linear coefficients and aggregation weights. This improves traditional autoregressive models by providing a localized understanding of the system behavior. We conduct a series of numerical experiments to evaluate the performance of various methods to estimate perturbation effects from data. Our second contribution is the derivation of the asymptotic properties of these estimators under suitable assumptions. These asymptotic properties can be leveraged for uncertainty quantification.