Abstract: Via propensity score matching (PSM) and Rosenbaum Bounds (RB), this paper reports new evidence on the premiums charged by big 4 and the top 4 mid-tier (mid 4) auditors relative to their smaller counterparts in the private corporate market. The results demonstrate that big 4 and mid 4 premiums are in accord with theoretical predictions on auditor quality differences; and that these premiums are relatively insensitive to potential hidden bias when gauged by the RB method for appraising confounding variables under bounded uncertainty. Given the limitations of conventional methods, PSM is being increasingly adopted in accounting studies to estimate treatment effects. Employing paired and simultaneous multi-sample PSM premium estimates, we provide a comprehensive evaluation and illustration of the RB method, together with the advantages and limitations of PSM, on which RB is predicated, when compared to alternative estimators. We demonstrate that PSM, when coupled with RB, provide novel empirical evidence for premiums estimated across three different matched audit quality tiers, to those estimated in prior studies which employ Heckman methods, to hidden bias equivalents and to the sensitivity of the bounds parameters to the omission of covariates employed in the study. New evidence on premiums across size quartiles, and quantile regression estimates over audit fee percentiles, support the PSM findings.