Allen turned down an offer of $12 million for two years to sign for $9 million over three years with Miami.
"I didn't know one way or the other," Celtics coach Doc Rivers said. "I just knew financially that we were in a better position, but he had his reasons for leaving.
"No, I didn't agree, but that doesn't matter. It doesn't matter what I think. I just wish we could have found a way, and we didn't. Always put it on us. It's easier.
"He talked to me. It was a good talk. You know how I am. A guy makes his mind up, I'm not going to try and change it. I respect him for all he did for us, and I thought he should have stayed with us. Actually, you just have to let it go."
There was talk Allen was swayed by the Heat's presentation and the vigor with which the club went after him.
"We recruited," Rivers said. "We couldn't fly him to Miami because he was in Boston, but other than that, we recruited him just as hard.
"I thought we (courted Allen). Danny (Ainge)in particular did exactly what he should have done. Kevin Garnett was our focal point, and he should have been. If that got anyone ruffled, then that's probably too bad. We did everything we were supposed to do.
"At the end of the day, you just follow the money trail, and in this case, for whatever reason, it didn't work. He had his reasons. He probably got bent sideways a bit by us courting Kevin. I really don't know, honestly."
The Celtic coach then struck a conciliatory note.
"At the end of the day, he was great for us here, and that's all you can say," Rivers said. "Clearly we would have liked him to stay with us, but I respect Ray, and that will never go away.
"Our relationship won't change. Just because I disagree or I'm disappointed in Ray's decision doesn't change the way I feel about him."