“Cross River is a state-chartered, F.D.I.C.-insured bank with robust regulatory standards,” said Phil Goldfeder, a bank spokesman. “Unlike other lenders who prioritized their own customers, Cross River addressed the S.B.A. call to action and did not limit the program to existing customers.”

Before the study was released, Blueacorn sent a letter Jay Hartzell, the president of the University of Texas at Austin, objecting to the researchers’ approach. Blueacorn said that by relying on interim data released by the Small Business Administration before the P.P.P. ended, the study counted loans that its lenders initially approved but later canceled because of suspicious traits. Nearly 157,000 applications — about 16 percent of all of the loans Blueacorn’s lenders approved — were canceled by the lenders before they were paid out.

“As we reviewed increasing volumes of loan applications, we learned, adapted, and enhanced our fraud detection capabilities and protocols,” Barry Calhoun, Blueacorn’s chief executive, said in a written statement. “Along the way, we partnered with the S.B.A. and other authorities to ensure the integrity of the P.P.P. while providing a traditionally overlooked population with access to the funds they needed and deserved.”

The researchers said they hoped their work would help inform the ongoing policy debate about the Paycheck Protection Program’s effectiveness.

“Our evidence, along with evidence that the P.P.P. saved relatively…

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