# Inference and Linear Regression when dependent and independent variables are political party vote shares

Imagine there are two parties that compete with each other for votes. If Party A wins votes in a certain town, it's most often from winning over voters from Party B.

In my scenario, Party A is a large mainstream conservative party and Party B is a smaller "niche" right-wing party. In my project overall, I am trying to qualitatively assess the extent to which Party B has pushed Party A to the right. In the 2006 election, Party B was quite small, but in the 2010 election it had a major breakthrough and rode an extreme right-wing platform. The period from 2006-2010 also coincided with Party B's efforts to improve its organization, mobilize voters at the grassroots, and develop a media network. Since 2010, Party A and Party B have been directly competing for the conservative vote.

I have data for four parliamentary elections (2006, 2010, 2014, and 2018), showing the breakdown of support for each party in each town in each election (so an observation is an election-town). If I plot Party A's vote share change from one election to the next against Party B's vote share change from the same election to the next (i.e. change between 2010 and 2014, 2014 and 2018, etc), I get a strong negative correlation, as would be expected since the parties are competing directly for the same voters. Party A's win is Party B's loss and vice versa.

Party A has moved far to the right, largely in an effort to capture Party B's votes (as I claim qualitatively), but I want to provide some statistical support for this claim. I found a pretty striking positive correlation between Party A's vote share change from 2014-2018 (the period Party A shifted far to the right and had lots of electoral success) and Party B's change in vote share from 2006 to 2010 (which represents how popular Party B's right wing message was in a certain town).

I want to be able to say that this provides evidence that Party B's right-wing messaging early on likely made voters more receptive to Party A's hard-right turn in the 2018 election. (Disclaimer: This is political science research; I want descriptive evidence to support my argument rather than a causally robust natural experiment). But I am concerned that this relationship is just an arithmetical fact. If Party A and Party B compete directly for votes, then places where Party B gained a lot of votes in 2010 are areas that Party A had a lot of potential voters that it could "flip" toward itself in 2018. So it would make sense that the change in vote share would be higher in these towns.

So I was wondering:

a) Are my concerns about inferring anything from the data I've discussed valid or am I excessively concerned? b) Might a linear regression strategy be possible? Imagine I regress Party A's change in vote share from 2014-2018 and regress this Party B's change in vote share from 2006-2010 and Party A's vote share in 2010. That way, would I be controlling for bias that could emerge from Party A having an initially low vote share in the 2010 elections in towns that they could then "flip" in the 2018 elections? I worry about collinearity but VIFs in these regressions are generally around 1.