Causal inference on time series data is a challenging problem, especially in the presence of unobserved confounders. This work focuses on estimating the causal effect between two time series, which are confounded by a third, unobserved time series. Assuming spectral sparsity of the confounder, we show how in the frequency domain this problem can be framed as an adversarial outlier problem. We introduce Deconfounding by Robust regression (DecoR), a novel approach that estimates the causal effect using robust linear regression in the frequency domain. Considering two different robust regression techniques, we first improve existing bounds on the estimation error for such techniques. Crucially, our results do not require distributional assumptions on the covariates. We can therefore use them in time series settings. Applying these results to DecoR, we prove, under suitable assumptions, upper bounds for the estimation error of DecoR that imply consistency. We show DecoR's effectiveness through experiments on synthetic data. Our experiments furthermore suggest that our method is robust with respect to model misspecification.
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