This paper develops exact ﬁnite sample and asymptotic distributions for structural equation tests based on partially restricted reduced form estimates. Particular attention is given to models with large numbers of instruments, wherein the use of partially restricted reduced form estimates is shown to be especially advantageous in statistical testing even in cases of uniformly weak instruments and reduced forms. Comparisons are made with methods based on unrestricted reduced forms, and numerical computations showing ﬁnite sample performance of the tests are reported. Some new results are obtained on inequalities between noncentral chi-squared distributions with diﬀerent degrees of freedom that assist in analytic power comparisons.
Phillips, Peter C.B. and Gao, Wayne Yuan, "Structural Inference from Reduced Forms with Many Instruments" (2016). Cowles Foundation Discussion Papers. 2524.