Test: Part-time Workers Effect on Unemployment Rate
This is mostly related to 2):

Not only is there reversed causality, but also potential for omitted variable bias. Identifying causality instead of correlation is something that no econometric method can do for you, you will have to work for that yourself.

For example, you can


Find some proxy that is going to move part-time employment, but not any other variable that could be somehow linked to the unemployment rate
Find exogenous (random?) variation in part-time employment. The most common tool here are (quasi-) natural experiments.