Subvector conditional likelihood-ratio distribution under multiple endogenous regressors

Establish that, in the instrumental variables model with instruments Z, endogenous regressors X (of interest) and W (nuisance) as defined in Model 1, under the null hypothesis β = β0, the subvector conditional likelihood-ratio statistic LR(β) is asymptotically bounded above by the distribution Γ(k − m, m_x, \tilde s_min(β0)), where k is the number of instruments and m = m_x + m_w. Define λ1 and λ2 as the smallest and second-smallest eigenvalues of (n − k) \big[ [X \ W \ y]^T M_Z [X \ W \ y] \big]^{-1} [X \ W \ y]^T P_Z [X \ W \ y], and define μ(β) = (n − k) · λ_min\{ \big[ [W \ y − Xβ]^T M_Z [W \ y − Xβ] \big]^{-1} [W \ y − Xβ]^T P_Z [W \ y − Xβ] \}. Let \tilde s_min(β) := λ1 + λ2 − μ(β). Show that LR(β0) ≤ Γ(k − m, m_x, \tilde s_min(β0)) under both strong and weak instrument asymptotics.

Background

The conditional likelihood-ratio (CLR) test of Moreira (2003) provides weak-instrument-robust inference when testing a scalar parameter (m_w = 0). However, in many applications partial inference for components of a vector causal parameter is needed when multiple endogenous variables are present (m_w > 0). The classical CLR result does not directly apply in this subvector setting.

Kleibergen (2021) proposed a subvector extension and conjectured a distributional upper bound for the LR statistic that conditions on an instrument-strength measure built from eigenvalues of matrices involving P_Z and M_Z. Establishing this bound would justify a practical, robust testing procedure for individual coefficients in overidentified models.