Feasibility testing for linear programs with estimated coefficients

Develop a statistically valid and detailed methodology to test the feasibility of the linear programs with estimated coefficients introduced for Synthetic Parallel Trends inference—both with and without nonnegativity constraints—including deriving appropriate profiled test statistics, establishing asymptotic validity under the sampling frameworks considered, and addressing settings with noncompact constraint sets and potentially rank-deficient coefficient matrices.

Background

The paper proposes an inference approach based on profiling out nuisance weights ω and reformulating the identification problem as minimizing a quadratic criterion over a simplex. It outlines how similar profiling-based statistics could be used to test whether the associated linear programs (LPs) possess feasible solutions, thereby making the SPT assumption refutable.

The remark points out that extending this to LPs without nonnegativity constraints or to testing existence of certain affine transformations introduces technical complications (e.g., noncompact constraint sets, convergence issues), and notes that a comprehensive treatment of feasibility testing for LPs with estimated coefficients is left for future work.