You must give a statement to your insurance company, but not the other side's.
No, you don't have to give a recorded statement to the insurance company. We need to make a couple of distinctions here between a recorded statement and a deposition. A recorded statement is an informal event. A deposition is a formal event where you would be asked questions under oath. That occurs later in the case after litigation is started.
As to answer the question, specifically, do you have to give a recorded statement to your own insurance company? The answer is, yes, you generally have to cooperate with your own insurance company. You never under any circumstances have to give a voluntary recorded statement to the insurance company for the other party involved in an automobile accident.