While SE CE, it is not generally true thatSE = CE. Consider now an intervention onX0. In the example, this only happens when we set the weights, means and variances to very particular values. Here we present a slightly adapted version of Invariant Causal Prediction [27]. Under this approach, the complexity oftesting asingle setofpredictors ofsize k is the cost of performing a least-squares regression and computing the residuals (O(k2N)) and the cost of performing the t-test and F-test over each split of thee environments (O(eN)).

One of the most common mistakes made when performing data analysis is attributing causal meaning to regression coefficients. Formally, a causal effect can only be computed if it is identifiable from a combination of observational data and structural knowledge about the domain under investigation (Pearl, 2000, Ch. 5). Building on the literature of instrumental variables (IVs), a plethora of methods has been developed to identify causal effects in linear systems. Almost invariably, however, the most powerful such methods rely on exponential-time procedures. In this paper, we investigate graphical conditions to allow efficient identification in arbitrary linear structural causal models (SCMs). In particular, we develop a method to efficiently find unconditioned instrumental subsets, which are generalizations of IVs that can be used to tame the complexity of many canonical algorithms found in the literature. Further, we prove that determining whether an effect can be identified with TSID (Weihs et al., 2017), a method more powerful than unconditioned instrumental sets and other efficient identification algorithms, is NP-Complete. Finally, building on the idea of flow constraints, we introduce a new and efficient criterion called Instrumental Cutsets (IC), which is able to solve for parameters missed by all other existing polynomial-time algorithms.

Learning the unknown causal parameters of a linear structural causal model is a fundamental task in causal analysis. The task, known as the problem of identification, asks to estimate the parameters of the model from acombination of assumptions on the graphical structure of the model and observational data, represented as a non-causal covariance matrix.In this paper, we give a new sound and complete algorithm for generic identification which runs in polynomial space. By a standard simulation result, namely \mathsf{PSPACE} \subseteq \mathsf{EXP},this algorithm has exponential running time which vastly improves the state-of-the-art double exponential time method using a Gröbner basis approach. The paper also presents evidence that parameter identification is computationally hard in general. In particular, we prove, that the taskasking whether, for a given feasible correlation matrix, there are exactly one or two or more parameter sets explaining the observed matrix, is hard for \forall \mathbb{R}, the co-class of the existential theory of the reals.

UPDATE: Thank you for the thoughtful response, those changes should improve the things that were unclear to me. There is a rich recent literature on identification criteria for linear structural causal models, but most of the recently proposed criteria largely ignore the question of efficient computability. This paper answers important questions in this area by given efficient algorithms for some criteria, while showing others to be NP-complete. The paper is original and generally clear and of high quality. Minor comments: l100: double "a" l104: the equation you refer to is in the supplement, which should be mentioned here.

The paper proposes a method to efficiently find instrumental subsets for identification in linear acyclic SCMs. The reviewers think that the method is interesting and relevant. An improvement to its evaluation would be the addition of an experimental section -- the authors indicated that they will add it in the revised version of the paper.

One of the most common mistakes made when performing data analysis is attributing causal meaning to regression coefficients. Formally, a causal effect can only be computed if it is identifiable from a combination of observational data and structural knowledge about the domain under investigation (Pearl, 2000, Ch. 5). Building on the literature of instrumental variables (IVs), a plethora of methods has been developed to identify causal effects in linear systems. Almost invariably, however, the most powerful such methods rely on exponential-time procedures. In this paper, we investigate graphical conditions to allow efficient identification in arbitrary linear structural causal models (SCMs).

One of the most common mistakes made when performing data analysis is attributing causal meaning to regression coefficients. Formally, a causal effect can only be computed if it is identifiable from a combination of observational data and structural knowledge about the domain under investigation (Pearl, 2000, Ch. 5). Building on the literature of instrumental variables (IVs), a plethora of methods has been developed to identify causal effects in linear systems. Almost invariably, however, the most powerful such methods rely on exponential-time procedures. In this paper, we investigate graphical conditions to allow efficient identification in arbitrary linear structural causal models (SCMs).