Open Access

Kevin Debeire, Andreas Gerhardus, Renée Bichler, Jakob Runge, Veronika Eyring

• 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 theIn 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. In a numerical experiment, we compare the uncertainty ranges of the proposed asymptotic-based approach with four bootstrap-based methods. Finally, we apply our methods to investigate the effects of economic activities on air pollution in Northern Italy, demonstrating their ability to reveal local effects. Our novel approach provides a comprehensive framework for analyzing the impacts of perturbations on both large- and small-scale variables, thereby enhancing our understanding of complex systems. Our research has implications for various disciplines where the study of perturbation effects is crucial for understanding and predicting systems’ behavior.…