If you are looking for a comprehensive research paper on this topic, the most recent and authoritative study is:
Banks often choose to pay high premiums (e.g., 13–44 basis points above the DW rate) to borrow from private markets rather than use the Fed, simply to avoid appearing financially weak. discount window treatment
Federal Reserve Bank of New York Staff Report No. 1137 (Revised March 2026) Key Findings: If you are looking for a comprehensive research
(2024): Explores why banks use the window outside of crises, noting that smaller banks with illiquid portfolios are the most frequent users. Stigma surfaced eight months before the 2023 regional
Stigma surfaced eight months before the 2023 regional banking crisis and has not significantly subsided since. Other Key Research Papers
Olivier Armantier, Marco Cipriani, and Asani Sarkar
(2025): Uses laboratory experiments to show that once a facility is stigmatized, it is extremely difficult to "cure" that perception, even with policy changes.