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 even say who wrote them. If you have more recent summaries to add to this collection, send them my way I guess. Sorry for the ads; they cover the costs of keeping this online.
Ansolabehere and Konisky. 1994. The introduction of voter registration and its effect on turnout. Political Analysis 14.
Previous studies have found that registration laws have large, significant effects on turnout: Imposing registration laws could lower turnout by 5-10%. But these studies have suffered from methodological flaws that have inflated the findings. This study uses improved methods. It does find a statistically significant effect for registration laws, but the effect is only half as large.
Uses county-level data on turnout and registration laws in New York and Ohio from the 1950s through the 1990s. In the 1970s (separate years), New York and Ohio changed their law to impose a uniform registration law on all of counties in the state. Prior to this, only some counties had a registration law. However, the counties that already had registration laws different in important ways from the ones that had it thrust on them: They were more populous and more urban. Thus, the authors use time-series data with several controls to estimate the actual effect of the new registration laws. Also, recognizing that new laws can cause some confusion among election workers (and therfore depressed turnout), they compare the first election after the reform with the first several after the reform.
The lit just did cross-sectional, state-level studies. These have two problems. First, they don't control for the fact that places that have registration laws differ from places that don't. Because the authors' controls and county-level analysis shows that registration laws have half the effect that previous studies have found the authors conclude that county-level variables explain the other half of observed effects.
The second problem with these cross-sectional studies is their low statistical power, so they can be easily distorted by a few states. The county-level time-series has far more statistical power.
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