People say that central bank with autonomy is needed to avoid political business cycle but then how do we explain the existence of central bank in arab oil exporting countries like saudi arabia where government has no election to run periodically? Why can't the government do the work of central bank in such countries ?
Government (meaning here executive branch of government since in most countries central bank is already government institution just independent like courts) could do that, in fact in many dictatorships/monarchies central banks are not properly independent so governments are de facto doing it if not de jure. In many such places central bank as an institution exists only nominally the same way as many dictatorships hold elections or have parliaments.
This being said historical experience shows governments are very bad at managing money supply because there ultimately complicated monetary decisions are being made made by people who don’t understand them at all. Historically, politically independent central banks perform much better at keeping prices stable and also actually better job at stimulating economy exactly when it’s needed. Hence to the extend that dictators care about having good monetary policy they would opt to have independent central bank. Although let me again reemphasize that in most dictatorships/monarchies central banks are not really independent and they are de facto controlled by dictator/monarch.