Testing Inequalities Linear in Nuisance Parameters

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This paper proposes a new test for inequalities that are linear in possibly partially identi ed nuisance parameters, called the generalized conditional chi-squared (GCC) test. It extends the subvector conditional chi-squared (sCC) test in Cox and Shi (2023, CS23) to a setting where the nuisance parameter is pre-multiplied by an unknown and estimable matrix of coefficients. Properly accounting for the estimation noise in this matrix while maintaining the simplicity of the sCC test is the main innovation of this paper. As such, the paper provides a simple solution to a broad set of problems including subvector inference for models represented by linear programs, nonparametric instrumental variable models with discrete regressor and instruments, and linear unconditional moment inequality models. We also derive a simplified formula for computing the critical value that makes the computation of the GCC test elementary.