Disclaimer. Don't rely on these old notes in lieu of reading the literature, but they can jog your memory. As a grad student long ago, my peers and I collaborated to write and exchange summaries of political science research. I posted them to a wiki-style website. "Wikisum" is now dead but archived here. I cannot vouch for these notes' accuracy, nor can I say who wrote them.
Gerber and Morton. 1998. Primary election systems and representation. Journal of Law, Economics, and Organization 14:304-324.
Research question: How does variation in primary election rules (open, closed primaries) affect representation?
It's self-evident why closed primaries produce the most extreme candidates. However, it takes some explanation to understand why semi-closed primaries produce the most moderate candidates. In any election, voters might cast ballots either sincerely or strategically. In primary elections, strategic voting takes the form of voting for a radical in the other party's primary so that your party's nominee will have a better chance of winning the general election; sincere voting involves casting a primary vote for the candidate that you most hope will win the general election.
Strategic voting is very costly in a closed primary, since you have to actually join the other party to do it. As such, closed primaries lead to sincere voting, resulting in extreme candidates winning the primary.
However, strategic voting is not costly enough in an open primary; it's too easy to meddle with the other party's primary election. If members of both parties strategically voted for extreme candidates in the other party's primary, this would push the general election candidates toward the extremes.
Thus, semi-closed primaries have the best balance between incentives for sincere and strategic voting. Fierce partisans (who are most likely to want to be strategic) would have high costs of being strategic (i.e. they have to join the other party or quit their own), but independents (who are mostly likely to want to be sincere) would have low costs of being sincere. The independents, then, gain great sway over the primary election result in a semi-closed primary.
The empirical test assumes that the representativeness of the winner reflects the representativeness of the two nominees. It does not actual test whether the nominees are moderates, only whether the general election winner is.
Looks at US congressional elections (by district) from 1982 and 1990.
Footnote 29 (p 320): If you re-run the analysis with NOMINATE (generally considered a better measure of Congressional representativeness), the significance goes away. That's a problem.
Spatial autocorrelation risk: This is district-level data. Over the four presidential years included, California's 50 districts end up constituting 200 of the N=868 (Table 4). Not only is this almost a quarter of all cases, this might be close to half of the "closed primary" districts. How much does that throw things off? Moreover, California was strange during this period. It had a Republican governor and voted for Reagan (a Californian). Thus, California looked more conservative than usual during these years (if you measure each district's ideology based on its vote for president). And since California contributes such a huge portion of the "closed primary" cases, it might make closed primaries look much more representative than they really are--which would bias the "open" and "semi" dummies (which use "closed" as the baseline) toward zero. In other words, the real effects might be stronger than the study suggests.
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