CLR bound in models with included exogenous regressors

Establish that, in the extended instrumental variables model with included exogenous regressors C and D (and endogenous W treated as nuisance), under the null (β, δ) = (β0, δ0), the conditional likelihood-ratio statistic satisfies LR(β0, δ0) ≤ Q_{m_d} + Γ(k − m_x − m_w, m_x, \tilde s_min(β0)), where Q_{r} denotes a χ^2(r) random variable and \tilde s_min(β0) is the smallest generalized eigenvalue defined via matrices involving M_{[Z, D]} and P_{[Z, D]}. In the special case m_x = 0, show that LR(δ0) ≤ Q_{m_d} + max{ Q_{k − m_w} − \tilde s_min(δ0), 0 }.

Background

When exogenous covariates are explicitly included (as both instruments and regressors), extending weak-instrument-robust inference is nontrivial because some matrices in the CLR derivation become singular. The paper derives one bounding distribution (Q_{m_d} + Γ(k − m_x, m_x, s_min)) and then suggests a potentially tighter bound that subtracts the nuisance dimension m_w.

Validating a stronger bound would improve power for testing exogenous components (e.g., race effects) while maintaining correct size under weak instruments, and clarify the precise conditioning needed when D is included as both instrument and regressor.