Invariant Causal Prediction for Sequential Data

We investigate the problem of inferring the causal variables of a response $Y$ from a set of $d$ predictors $(X^1,\dots,X^d)$. Classical ordinary least squares regression includes all predictors that reduce the variance of $Y$. Using only the causal parents instead leads to models that have the advantage of remaining invariant under interventions, i.e., loosely speaking they lead to invariance across different "environments" or "heterogeneity patterns". More precisely, the conditional distribution of $Y$ given its causal variables remains constant for all observations. Recent work exploit such a stability to infer causal relations from data with different but known environments. We show here that even without having knowledge of the environments or heterogeneity pattern, inferring causal relations is possible for time-ordered (or any other type of sequentially ordered) data. In particular, this then allows to detect instantaneous causal relations in multivariate linear time series, in contrast to the concept of Granger causality. Besides novel methodology, we provide statistical confidence bounds and asymptotic detection results for inferring causal variables, and we present an application to monetary policy in macro economics.