  Counterfactual estimation using synthetic controls is one of the most
successful recent methodological developments in causal inference. Despite its
popularity, the current description only considers time series aligned across
units and synthetic controls expressed as linear combinations of observed
control units. We propose a continuous-time alternative that models the latent
counterfactual path explicitly using the formalism of controlled differential
equations. This model is directly applicable to the general setting of
irregularly-aligned multivariate time series and may be optimized in rich
function spaces -- thereby improving on some limitations of existing
approaches.
