Federal Reserve Chairman Kevin Warsh couldn't have asked for a better bond market reaction after his debut address at the Jackson Hole economic conference.

Long-dated government bond yields fell and short-term notes' yields rose after Warsh doubled down on the central bank's inflation-fighting mind-set. The Treasury market's response suggests his remarks might have helped restore some of the Fed's credibility in its battle to bring inflation back to the bank's 2% target. But the market hasn't fully bought into the hawkish stance yet.

"Here is my standard: We must be confident that underlying inflation is moving to our objective, clearly and at sufficient speed," he said during Friday's speech at the Fed's annual economic policy symposium in Jackson Hole, Wyoming. "Otherwise, we have work to do. That's our job . . . our mandate . . . and our charge to keep."

Such assurance that the central bank is ready and willing to raise interest rates to restore price stability received a vote of confidence from the bond market. Bonds trade based on new information, and shorter-dated yields-which are sensitive to the interest rates set by the Fed-rose, indicating that bond traders see a rate increase coming soon. Bond yields and prices move inversely.

Not since the Aug. 25, 2023 symposium has the market seen a similar move after a Fed chair's Jackson Hole speech, with short-bond yields rising and long-bond yields declining.

Warsh "gave the clearest signal yet that tighter policy will be forthcoming if the Committee isn't soon convinced that inflation is on a path toward 2.0%," wrote Vail Hartman, U.S. rates strategist at BMO.

Just last week, 30-year yields had shot to their highest levels in about 19 years, partly because of concerns about the inflationary impact from the Iran war and rising oil prices. That made Friday's fall in yields welcome news, even though the decline was brief and the magnitude-a 3.4 basis points decline-was nothing to write home about. (One basis point equals 0.01%).

Warsh's speech came after some of his peers, including Kansas City Fed President Jeff Schmid and Cleveland Fed President Beth Hammack, communicated their support for higher rates.

"I think it's appropriate for us to put some restraint there to help bring inflation back down to target," Hammack told CNBC on Thursday.

But the Fed needs to prove that this isn't just all rhetoric if it wants Treasuries to sustain the latest moves-and if Warsh wants to gain the market's trust. The fear for some Fed watchers is that the hawkish tone could be a head fake, and that the central bank is only creating the illusion of a rate increase being on the table-rather than truly committing to action as inflation remains sticky for longer.

Darius Dale, founder of macro-research firm 42 Macro, said Friday that he isn't convinced by the Fed's hawkish commentary, and he actually expects the Fed to lower interest rates in 2027 after a hike this year.

The Fed is trying to regain the credibility on a transitory basis with the bond market with the hawkish speeches, Dale said. He expects the Fed to cite findings from the newly created five task forces as a reason to cut rates later on. If they tried to go from today's sticky inflation problem to cutting rates tomorrow, "you're going to blow up the bond market," he said. "They have to walk this delicate tight rope."

The Fed declined to comment Friday.

BofA strategists also want Warsh and his peers to walk the walk.

"We were encouraged by Warsh's speech. But talk is cheap," Aditya Bhave, head of U.S. economics research for BofA Global Research, and team wrote Friday. "The onus is now on him to deliver a hike in September (unless the August jobs and inflation data are very soft). Else he will probably lose the credibility he gained today."

Warsh has limited Fed communications because he wants the market to make its own judgments about inflation and the economy. And, for now, the bond market is saying the Fed needs to hike.