
The first study is Presidents with Prime Ministers: Do Direct Elections Matter? From the title you can probably tell that the author did not set out to study constitutional monarchies. In fact they are dealt with after the fact in a footnote. The end conclusion is that presidential republics suffer more voter fatigue and a drop in votes for legislative elections of about 5-7% than either constitutional monarchies or legislative republics. (UPDATE: While the original post uses 'presidential' and 'legislative' descriptors it should in fact be 'directly-elected' and 'indirectly-elected' respectively. While most legislative republics use indirect election via a legislature, some do not.)