Causal inference uses observational data to infer the causal structure of the data generating system. We study a class of restricted Structural Equation Models for time series that we call Time Series Models with Independent Noise (TiMINo). These models require independent residual time series, whereas traditional methods like Granger causality exploit the variance of residuals. This work contains two main contributions: (1) Theoretical: By restricting the model class (e.g. to additive noise) we provide more general identifiability results than existing ones. The results cover lagged and instantaneous effects that can be nonlinear and unfaithful, and non-instantaneous feedbacks between the time series. (2) Practical: If there are no feedback loops between time series, we propose an algorithm based on non-linear independence tests of time series. When the data are causally insufficient, or the data generating process does not satisfy the model assumptions, this algorithm may still give partial results, but mostly avoids incorrect answers. The Structural Equation Model point of view allows us to extend both the theoretical and the algorithmic part to situations in which the time series have been measured with different time delays (as may happen for fMRI data, for example). TiMINo outperforms existing methods on artificial and real data. Code is provided.
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