Contrast OLS, DiD, and PSM assumptions

For the shuttle impact problem, contrast OLS, DiD, and PSM rigorously. Do the following: 1) write the OLS and two-way fixed-effects DiD regression equations with all variables and fixed effects defined; 2) state the identifying assumptions for each (exogeneity, SUTVA, parallel trends, overlap/positivity) and the specific violations you expect here; 3) propose and justify a placebo test and a balance test, including metrics (e.g., standardized mean differences) and acceptance thresholds; 4) explain matching with vs. without replacement and the bias–variance–computational trade-offs; 5) after matching, show exactly how you would estimate effects (PSM+DiD) and compute ATT with valid standard errors; 6) discuss consequences of limited overlap and how you would diagnose and remedy it (trimming, calipers, alternative link).