Not only does effective central bank policy lead to good inflation outcomes, but it is the primary contributor to monetary credibility. Sound policy and credibility are tightly, and perhaps inextricably, intertwined and cannot be separately measured with current statistical techniques. Am I bothered by the paucity of direct academic evidence regarding the specific role of monetary credibility? Not at all.
The research I have briefly described makes clear that a commitment to fighting inflation, low inflation expectations, and central bank independence all contribute to restraining inflation. Thus, it would be surprising if monetary credibility did not play a role, even if, because it cannot be directly measured, its specific contribution is hard to gauge.