## References

Abadie, Alberto. 2005. Semiparametric difference-in-differences estimators.
*The Review of Economic Studies*
72(1):1–19.

Abadie, Alberto, Diamond, Alexis, and Hainmueller, Jens. 2010. Synthetic control methods for comparative case studies: Estimating the effect of California’s tobacco control program.
*Journal of the American Statistical Association*
105(490):493–505.

Abadie, Alberto, Diamond, Alexis, and Hainmueller, Jens. 2015. Comparative politics and the synthetic control method.
*American Journal of Political Science*
59(2):495–510.

Acemoglu, Daron, Johnson, Simon, Kermani, Amir, Kwak, James, and Mitton, Todd. 2016. The value of connections in turbulent times: Evidence from the United States.
*Journal of Financial Economics*
121(2):368–391.

Alvarez, R. Michael, Ansolabehere, Stephen, and Wilson, Catherine H.. 2002. Election day voter registration in the United States: How one-step voting can change the composition of the American electorate. Working Paper, Caltech/MIT Voting Technology Project.

Angrist, Joshua D., Jord, Scar, and Kuersteiner, Guido. 2013. Semiparametric estimates of monetary policy effects: String theory revisited. NBER Working Paper No. 19355.

Bai, Jushan. 2003. Theory for factor models of large dimensions.
*Econometrica*
71(1):135–137.

Bai, Jushan. 2009. Panel data models with interactive fixed effects.
*Econometrica*
77:1229–1279.

Beck, Nathaniel, and Katz, Jonathan N.. 1995. What to do (and not to do) with time-series cross-section data.
*American Political Science Review*
89(3):634–647.

Blackwell, Matthew, and Glynn, Adam. 2015. How to make causal inferences with time-series cross-sectional data. Harvard University. Mimeo.

Brians, Craig Leonard, and Grofman, Bernard. 2001. Election day registration’s effect on US voter turnout.
*Social Science Quarterly*
82(1):170–183.

Brodersen, Kay H., Gallusser, Fabian, Koehler, Jim, Remy, Nicolas, and Scott, Steven L.. 2014. Inferring causal impact using Bayesian structural time-series models.
*Annals of Applied Statistics*
9(1):247–274.

Burden, Barry C., Canon, David T., Mayer, Kenneth R., and Moynihan, Donald P.. 2009. The effects and costs of early voting, election day registration, and same day registration in the 2008 elections. University of Wisconsin-Madison. Mimeo.

Burnham, Walter Dean. 1980. The appearance and disappearance of the American voter. In
*Electoral participation: A comparative analysis*
, ed. Rose, Richard. Beverly Hills, CA: Sage Publications.

Cain, Bruce E., Donovan, Todd, and Tolbert, Caroline J.. 2011.
*Democracy in the states: Experiments in election reform*
. Washington, DC: Brookings Institution Press.

Campbell, John Y., Lo, Andrew W., and Craig MacKinlay, A.. 1997.
*The econometrics of financial markets*
. Princeton, NJ: Princeton University Press.

Dube, Arindrajit, and Zipperer, Ben. 2015. Pooling multiple case studies using synthetic controls: An application to minimum wage policies. IZA Discussion Paper No. 8944.

Efron, Brad. 2012. The estimation of prediction error.
*Journal of the American Statistical Association*
99(467):619–632.

Efron, Brad, and Tibshirani, Rob. 1993.
*An introduction to the bootstrap*
. New York: Chapman & Hall.

Fenster, Mark J.
1994. The impact of allowing day of registration voting on turnout in US elections from 1960 to 1992 A research note.
*American Politics Research*
22(1):74–87.

Gaibulloev, Khusrav, Sandler, Todd, and Sul, Donggyu. 2014. Dynamic panel analysis under cross-sectional dependence.
*Political Analysis*
22(2):258–273.

Glynn, Adam N. Glynn, and Quinn, Kevin M.. 2011. Why process matters for causal inference.
*Political Analysis*
19:273–286.

Gobillon, Laurent, and Magnac, Thierry. 2016. Regional policy evaluation: Interactive fixed effects and synthetic controls.
*The Review of Economics and Statistics*
98(3):535–551.

Hanmer, Michael J.
2009.
*Discount voting: Voter registration reforms and their effects*
. New York: Cambridge University Press.

Highton, Benjamin. 1997. Easy registration and voter turnout.
*The Journal of Politics*
59(2):565–575.

Highton, Benjamin. 2004. Voter registration and turnout in the United States.
*Perspectives on Politics*
2(3):507–515.

Holland, Paul W.
1986. Statistics and causal inference.
*Journal of American Statistical Association*
81(8):945–960.

Hsiao, Cheng, Ching, Steve H., and Wan, Shui Ki. 2012. A panel data approach for program evaluation: Measuring the benefits of political and economic integration of Hong Kong with Mainland China.
*Journal of Applied Econometrics*
27(5):705–740.

Huang, Chi, and Shields, Todd G.. 2000. Interpretation of interaction effects in logit and probit analyses reconsidering the relationship between registration laws, education, and voter turnout.
*American Politics Research*
28(1):80–95.

Imai, Kosuke, and Kim, In Song. 2016. When should we use linear fixed effects regression models for causal inference with panel data. Princeton University. Mimeo.

Keele, L., and Minozzi, W.. 2013. How much is Minnesota like Wisconsin? Assumptions and counterfactuals in causal inference with observational data.
*Political Analysis*
21(2):193–216.

Kim, Dukpa, and Oka, Tatsushi. 2014. Divorce law reforms and divorce rates in the USA: An interactive fixed-effects approach.
*Journal of Applied Econometrics*
29(2):231–245.

King, James D., and Wambeam, Rodney A.. 1995. Impact of election day registration on voter turnout: A quasi-experimental analysis.
*Policy Studies Review*
14(3):263–278.

Knack, Stephen. 2001. Election-day registration The second wave.
*American Politics Research*
29(1):65–78.

Knack, Stephen, and White, James. 2000. Election-day registration and turnout inequality.
*Political Behavior*
22(1):29–44.

Leighley, Jan E., and Nagler, Jonathan. 2013.
*Who votes now? Demographics, issues, inequality, and turnout in the United States*
. Princeton, NJ: Princeton University Press.

Mitchell, Glenn E., and Wlezien, Christopher. 1995. The impact of legal constraints on voter registration, turnout, and the composition of the American electorate.
*Political Behavior*
17(2):179–202.

Moon, Hyungsik Roger, and Weidner, Martin. 2015. Dynamic linear panel regression models with interactive fixed effects.
*Econometric Theory*
33(1):158–195.

Mora, Ricardo, and Reggio, Ilina. 2012. Treatment effect identification using alternative parallel assumptions. Universidad Carlos III de Madrid. Mimeo.

Neiheisel, J. R., and Burden, B. C.. 2012. The impact of election day registration on voter turnout and election outcomes.
*American Politics Research*
40(4):636–664.

Neyman, Jerzy. 1923. On the application of probability theory to agricultural experiments: Essay on principles.
*Statistical Science*
5:465–480, Section 9 (translated in 1990).

Pang, Xun. 2010. Modeling heterogeneity and serial correlation in binary time-series cross-sectional data: A Bayesian multilevel model with AR(p) errors.
*Political Analysis*
18(4):470–498.

Pang, Xun. 2014. Varying responses to common shocks and complex cross-sectional dependence: Dynamic multilevel modeling with multifactor error structures for time-series cross-sectional data.
*Political Analysis*
22(4):464–496.

Park, Jong Hee. 2010. Structural change in US presidents’ use of force.
*American Journal of Political Science*
54(3):766–782.

Park, Jong Hee. 2012. A unified method for dynamic and cross-sectional heterogeneity: Introducing hidden Markov panel models.
*American Journal of Political Science*
56(4):1040–1054.

Rhine, Staci L.
1992. An analysis of the impact of registration factors on turnout in 1992.
*Political Behavior*
18(2):171–185.

Rubin, Donald B.
1974. Estimating causal effects of treatments in randomized and nonrandomized studies.
*Journal of Educational Psychology*
5(66):688–701.

Springer, Melanie Jean. 2014.
*How the states shaped the nation: American electoral institutions and voter turnout, 1920–2000*
. Chicago, IL: University of Chicago Press.

Stewart, Brandon. 2014. Latent factor regressions for the social sciences. Princeton University. Mimeo.

Teixeira, Ruy A.
2011.
*The disappearing American voter*
. Washington, DC: Brookings Institution Press.

Timpone, Richard J.
1998. Structure, behavior, and voter turnout in the United States.
*The American Political Science Review*
92(1):145.

Timpone, Richard J.
2002. Estimating aggregate policy reform effects: New baselines for registration, participation, and representation.
*Political Analysis*
10(2):154–177.

Wolfinger, Raymond E., and Rosenstone, Steven J.. 1980.
*Who votes?*
New Haven, CT: Yale University Press.

Xu, Yiqing. 2016. Replication data for: Generalized synthetic control method: Causal inference with interactive fixed effects models. doi:10.7910/DVN/8AKACJ, Harvard Dataverse. 2 To gauge the uncertainty of the estimated treatment effect, the synthetic control method compares the estimated treatment effect with the “effects” estimated from placebo tests in which the treatment is randomly assigned to a control unit.

5 When the treatment effect is heterogeneous (as it is almost always the case), an IFE model that imposes a constant treatment effect assumption gives biased estimates of the average treatment effect because the estimation of the factor space is affected by the heterogeneity in the treatment effect.

7 Cases in which the treatment switches on and off (or “multiple-treatment-time”) can be easily incorporated in this framework as long as we impose assumptions on how the treatment affects current and future outcomes. For example, one can assume that the treatment only affect the current outcome but not future outcomes (no carryover effect), as fixed effects models often do. In this paper, we do not impose such assumptions. See Imai and Kim (Reference Imai and Kim2016) for a thorough discussion.

8
$\unicode[STIX]{x1D6FD}$
is assumed to be constant across space and time mainly for the purpose of fast computation in the frequentist framework. It is a limitation compared with more flexible and increasingly popular random coefficient models in Bayesian multi-level analysis.

9 For this reason, additive unit and time fixed effects are not explicitly assumed in the model. An extended model that directly imposes additive two-way fixed effects is discussed in the next section.

12 For a clear and detailed explanation of quantities of interest in TSCS analysis, see Blackwell and Glynn (Reference Blackwell and Glynn2015). Using their terminology, this paper intends to estimate the Average Treatment History Effect on the Treated given two specific treatment histories:
$\mathbb{E}[Y_{it}(\text{}\underline{a}_{t}^{1})-Y_{it}(\text{}\underline{a}_{t}^{0})|\text{}\underline{D}_{i,t-1}=\text{}\underline{a}_{t-1}^{1}]$
in which
$\text{}\underline{a}_{t}^{0}=(0,\ldots ,0)$
,
$\text{}\underline{a}_{t}^{1}=(0,\ldots ,0,1,\ldots ,1)$
with
$T_{0}$
zeros and
$(t-T_{0})$
ones indicate the histories of treatment statuses. We keep the current notation for simplicity.

13 We attempt to make inference about the ATT in the sample we draw, not the ATT of the population. In other words, we do not incorporate uncertainty of the treatment effects
$\unicode[STIX]{x1D6FF}_{it}$
.

20 In the Online Appendix, we list the years during which EDR laws were enacted and first took effect in presidential elections.

21 See Wolfinger and Rosenstone (Reference Wolfinger and Rosenstone1980), Mitchell and Wlezien (Reference Mitchell and Wlezien1995), Rhine (Reference Rhine1992), Highton (Reference Highton1997), Timpone (Reference Timpone1998), Timpone (Reference Timpone2002), Huang and Shields (Reference Huang and Shields2000), Alvarez, Ansolabehere, and Wilson (Reference Alvarez, Ansolabehere and Wilson2002), Brians and Grofman (Reference Brians and Grofman2001), Hanmer (Reference Hanmer2009), Burden *et al.* (Reference Burden, Canon, Mayer and Moynihan2009), Cain, Donovan, and Tolbert (Reference Cain, Donovan and Tolbert2011), Teixeira (Reference Teixeira2011) for examples. The results are especially consistent for the three early adopters, Maine, Minnesota, and Wisconsin.

24 We do not use the voting-eligible population (VEP) as the denominator because they are not available in early years.

25 As is shown in the figure and has been pointed out by many, turnout rates are in general higher in states that have EDR laws than states that have not, but this does not necessarily imply a causal relationship between EDR laws and voter turnout.

26 Note that although the estimated ATT of EDR on voter turnout is presented in the same row as the coefficient of EDR using the FE model, the GSC method does not assume the treatment effect to be constant. In fact, it allows the treatment effect to be different both across states and over time. Predicted counterfactuals and individual treatment effect for each of the nine treated states are shown in the Online Appendix.

27 The results are similar if additive state and year fixed effects are not directly imposed, though not surprisingly, the algorithm includes an additional factor.

28 Although it is not guaranteed, this is not surprising since the GSC method uses information of all past outcomes and minimizes gaps between actual and predicted turnout rates in pretreatment periods.

29 The results are essentially the same with or without controlling for the other two registration reforms.

30 Although we can control for indicators of Jim Crow laws in the model, such indicators may not be able to capture the heterogeneous impacts of these laws on voter turnout in each state.

31 In the Online Appendix, we show that the treatment effects are positive (and relatively large) for all three early adopting states, Maine, Minnesota, and Wisconsin. Using a fuzzy regression discontinuity design, Keele and Minozzi (Reference Keele and Minozzi2013) show that EDR has almost no effect on the turnout in Wisconsin. The discrepancy with this paper could be mainly due to the difference in the estimands. Two biggest cities in Wisconsin, Milwaukee and Madison constitute a major part of Wisconsin’s constituency but have neglectable influence to their local estimates. One advantage of Keele and Minozzi (Reference Keele and Minozzi2013)’s approach over ours is the use of fine-grained municipal level data.

32 Glynn and Quinn (Reference Glynn and Quinn2011) argue that traditional cross-sectional methods in general overestimate the effect of EDR laws on voter turnout and suggest that EDR laws are likely to have minimum effect on turnout in non-EDR states (the ATC). In this paper, we focus on the effect of EDR in EDR states (the ATT) instead.