For every voting precinct in the United States, a researcher can easily observe the count of registered voters who voted in the 2004, 2008, 2012 and 2016 Presidential elections and the count of the precinct who voted for each party's candidate (Democrat, Republican, other).
Facebook knows each of its users home address and geographic location. These Big Data can be aggregated up so that Facebook could give the Congress data by year/state/voting precinct on the total count of Facebook users in each voting precinct and how much time they spend each day on Facebook (so this would proxy for quantity and quality of FB exposure).
Using just these data (and keep in mind that the unit of analysis is a state/county/voting precinct/year), use linear regression methods and run the following regression;
% vote republican on precinct fixed effects , county time trend + B1*% of Precinct adults with active Facebook Accounts + B2*average minutes spent on Facebook + U
By interacting the Facebook right hand side variables with a 2016 election year dummy, we can test whether B1 and B2 > 0. This is the start of a statistical test of whether "Facebook caused Trump".
Now Step #2
Use the Facebook Big Data to identify the voting precincts where the Russian fake accounts targeted Facebook members. Facebook would need to use their machine learning algorithms here. Why? The Russians buy the ads knowing that the Facebook ML algorithm will target "the right people". Facebook knows who was exposed to those ads. Calculate a state/year/voting precinct new variable called "Russian Targets". So if this variable = 9% this indicates that 9% of Facebook users in a voting precinct received the Russian Ads. Include this variable in the regression above;
% vote republican on precinct fixed effects , county time trend + B1*% of Precinct adults with active Facebook Accounts + B2*average minutes spent on Facebook + B3*Russian Targets + U
By interacting the Facebook right hand side variables with a 2016 election year dummy, we can test whether B1 and B2 and B3 > 0. This is a good non-experimental statistical test of whether "Facebook caused Trump". This regression can easily be run.
I realize that it is an ecological regression but voting precincts are quite small.
My test is a "conservative test" because it assumes away a contagion effect. If Michael is targeted with Trump ads in his voting precinct and he talks to Jane in another precinct about what he has learned on Facebook, my procedure does not capture the impact of FB on Jane.
Facebook could provide more data on the precinct to precinct "friends network" and the researcher could then construct a "Russian Target adjacency" variable. So to repeat this point, if everyone in precinct 34 is buddies with someone in precinct 89 then by treating precinct 34 with Russian ads the Russians have also partially treated precinct 89. If FB provided the micro level friends network, the researcher could explicitly test for this "contagion effect".
Facebook could provide more data on the precinct to precinct "friends network" and the researcher could then construct a "Russian Target adjacency" variable. So to repeat this point, if everyone in precinct 34 is buddies with someone in precinct 89 then by treating precinct 34 with Russian ads the Russians have also partially treated precinct 89. If FB provided the micro level friends network, the researcher could explicitly test for this "contagion effect".