It depends on which insurance company is asking you for the recorded statement. If it’s your insurance company, you have a contract with them.

You agree to pay them a certain amount of money every six months or every year, and they agree to provide you coverage. In your insurance policy somewhere it says something like: “If you’re in a wreck or if you need help, you will cooperate with us and give us a statement.” So, yes, if it’s your insurance company, you must allow them to record your statement.

Now, if it’s the insurance company of the at-fault party that caused the accident, if it’s that person’s insurance, you do not have an obligation to give a statement to them.