Economics and/or politics?

 In recent decades, there have been a number of cases of "democratic erosion" or "backsliding"--elected leaders changing the rules to make it more difficult to oppose their policies or vote them out of office.  A paper by Eli Rau and Susan Stokes looks at democracies between 1995 and 2020 and finds that economic inequality was associated with a higher risk of backsliding.*  They considered per-capita GDP and the previous duration of democracy, which reduce the risk of other threats to democracy (like military coups and civil war) and find no clear connection to backsliding.  But they didn't consider another factor that I thought was potentially important:  presidential versus parliamentary systems of government.  First, it may be easier for outsiders to attain power in presidential systems, since it's not necessary to recruit and organize candidates for seats in parliament in order to win.  Second, presidential systems create the chance of conflict between the president and the legislature.  Third, it's more difficult to remove unpopular or unfit leaders in a presidential system:  impeachment takes time and requires a supermajority.  As a result, presidential systems may have a higher risk of backsliding.  

I began with a classification of forms of government from Wikipedia and reduced them to three types:  parliamentary, presidential, and an intermediate ("semi-presidential") category.  Semi-presidential systems have both a directly elected president and a prime minister who is responsible to the legislature--France is a prominent example.  There were also two countries in the system that didn't fit into the classification (Switzerland and Micronesia).  The table shows the risk of backsliding by form of government:

                                          Years at risk          episodes         Risk
Parliamentary                      1115                       4                   0.35%
Semi-Presidential                  516                       5                   0.97%
Presidential                            488                     14                   2.89%
                      
Other                                         78                      0                    0

For example, the United States had 22 years of being at risk (1995-2016), and in one of those (2016) began an episode of backsliding.  2017-20 are not considered because we were already in a state of backsliding.  I'll consider the "risk" of emerging from an episode later.  

So there is a substantial association between form of government and the risk of backsliding.  But what if we consider both inequality and form of government?

Constant    -7.52        -8.15          -8.51         -8.37
                   (1.04)     (1.18)          (1.46)       (1.21)

Gini            .078          .063           .081          .078
                   (.024)       (.027)        (.031)        (.026)

Form                            .59            
                                    (.28)         

Semi                                               1.34
                                                       (0.73)

Presidential                                    1.23
                                                       (0.59)

Non-Parliamentary                                             1.26  
                                                                           (0.56)

"Form" defines parliamentary systems as 1, semi-presidential as 2, and presidential as 3.  The third column compares each to the baseline of parliamentary systems.  The estimates for presidential and semi-presidential are similar, so the fourth column combines them into a single "non-parliamentary" group.  Statistical criteria don't permit a choice between the models in columns 2-4, but I think the arguments I gave apply about equally to presidential and semi-presidential forms, so I would go with #4 if I had to choose.  I thought that controlling for form of government might reduce or eliminate the estimated effect of inequality, but that wasn't the case.  

I also estimated models for the chance of leaving a state of backsliding.  Neither inequality nor the form of government had any clear effect, so I don't show the results.  

*The classification is from a working paper by Melis Laebens, who spoke of "executive aggrandizement," and suggested that it was a development of the post-Cold War period.  Until then, "aspiring autocrats would not be as concerned with maintaining a democratic facade."