Identifiability of Gaussian structural equation models with same error variances

We consider structural equation models in which variables can be written as a function of their parents and noise terms, the latter are assumed to be jointly independent. Corresponding to each structural equation model, there is a directed acyclic graph describing the relationships between the variables. In Gaussian structural equation models with linear functions, the graph can be identified from the joint distribution only up to Markov equivalence classes assuming faithfulness. However, this constitutes an exceptional case. For linear functions and non-Gaussian noise, the directed acyclic graph becomes identifiable. Apart from few exceptions, the same is true for non-linear functions and arbitrarily distributed additive noise. In this work, we prove identifiability for a third modification: if we require all noise variables to have the same variances the directed acyclic graph can be recovered from the joint Gaussian distribution. Our result has direct implications for causal inference: if the data follow a Gaussian structural equation model with same error variances and assuming that all variables are observed, the causal structure can be inferred from observational data only. We propose a statistical method and a corresponding algorithm that exploit our theoretical findings.