Causal Inference by Identification of Vector Autoregressive Processes   with Hidden Components
  A widely applied approach to causal inference from a non-experimental time series $X$, often referred to as "(linear) Granger causal analysis", is to regress present on past and interpret the regression matrix $\hat{B}$ causally. However, if there is an unmeasured time series $Z$ that influences $X$, then this approach can lead to wrong causal conclusions, i.e., distinct from those one would draw if one had additional information such as $Z$. In this paper we take a different approach: We assume that $X$ together with some hidden $Z$ forms a first order vector autoregressive (VAR) process with transition matrix $A$, and argue why it is more valid to interpret $A$ causally instead of $\hat{B}$. Then we examine under which conditions the most important parts of $A$ are identifiable or almost identifiable from only $X$. Essentially, sufficient conditions are (1) non-Gaussian, independent noise or (2) no influence from $X$ to $Z$. We present two estimation algorithms that are tailored towards conditions (1) and (2), respectively, and evaluate them on synthetic and real-world data. We discuss how to check the model using $X$. 
Philipp Geiger, Kun Zhang, Mingming Gong, Dominik Janzing, Bernhard   Sch\"olkopf
stat.ML