What is semi-presidentialism?

Semi-presidentialism is where a constitution includes a popularly elected fixed-term president and a prime minister and cabinet who are collectively responsible to the legislature

The assumption

A very varied set of countries have semi-presidential constitutions. We should not use semi-presidentialism as an explanatory variable. We should distinguish between different sub-types of semi-presidentialism and explore the effects of each

Sub-types of semi-presidentalism

I like the distinction between premier-presidential  and president-parliamentary forms of semi-presidentialism. This distinction was first made by Matthew Shugart and John Carey in Presidents and Assemblies. Constitutional Design and Electoral Dynamics, Cambridge: Cambridge University Press (1992)