Mackay said he could look back at his time at Cardiff with "immense pride".
"I leave with my head held high having gained a level of experience that, upon reflection, I suspect would have been difficult to find anywhere else in British football," he said in a statement issued on his behalf by the League Managers' Association.
"The foundations laid at Cardiff City FC will provide a solid platform for my successor. And for the sake of the club, I hope that all of the progress that has been made so far will not be impaired in any way by today's enforced changes."
Mackay's removal follows a public row with Tan, who wrote to to the manager on 16 December asking him to
resign or be sacked.
Tan also accused Mackay of overspending in the transfer market and announced the manager would have nothing to spend in the January window.
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