We propose a new method to estimate structural parameters in multi-way networks while controlling for rich structures of fixed effects. The method is based on a series of classification tasks and is agnostic to both the number and structure of fixed effects. In contrast to full maximum likelihood approaches, our estimator does not suffer from the incidental parameter problem. For sparsely connected networks, it is also computationally faster than PPML. We provide empirical evidence that our estimator yields more reliable confidence intervals than PPML and its bias-correction strategies. These improvements hold even under model misspecification and are more pronounced in sparse settings. While PPML remains competitive in dense, low-dimensional data, our approach offers a robust alternative for multi-way models that scales efficiently with sparsity. The method is applied to study the causal effect of a policy reform on spatial accessibility to health care in France.

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