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3 Y i X i Y i (1) X i = 1 Y i (0) X i = 0 Y i = X i Y i (1) + (1 X i )Y i (0) τ i Y i (1) Y i (0) τ E(τ i ) ( ) / 37

4 X i Y i = α + τx i + ϵ i α E(Y i (0)) ϵ i Y i (0) α + (τ i τ)x i (Y i (1), Y i (0)) X i E(X i ϵ i ) = 0 E(ˆτ) = τ ( ) / 37

5 K Z (k) i (k = 1, 2,..., K) ϵ i = k ζ (k) Z (k) i + ϵ i E(Z (k) i E(X i Z (k) i ) = 0 X i = 0) = E(Z (k) i X i = 1) ( ) / 37

6 X i X i ( ) / 37

7 (Fukumoto, Nda) i j X ij Y ij exp(ˆτ) = 1.5 (Hyde, 2007) i X i Y i ˆτ < 0 ( ) / 37

8 (Doherty, Gerber, and Green, 2006) i X i Y i ˆτ < 0 (Titiunik, 2016) i X i 2 4 Y i ˆτ > 0 ( ) / 37

9 27 ( ) / 37

10 i X i Y i (Fukumoto, Horiuchi, and Tanaka, Nd) (Fukumoto and Horiuchi, 2011) (Fukumoto and Horiuchi, 2016) (Fukumoto, Ndb) ( ) / 37

11 (Fukumoto and Miwa, 2018) i j X ij Y ij exp(ˆτ) = 1.69 (Elis, Malhotra, and Meredith, 2009) i X i Y i ˆτ > 0 ( ) / 37

12 (Gomez, Hansford, and Krause, 2007) X i Y i ˆτ < 0 (Healy and Malhotra, 2010) X i Y i ˆτ < 0 (Goodman, 2014) X i Y i ˆτ < 0 ( ) / 37

13 (Hirano, 2011) i X i Y i ˆτ < 0 (Cook, Jacobs, and Kim, 2010) i X i 1 2 Y i ˆτ > 0 i X i Y i ˆτ < 0 ( ) / 37

14 (Posner, 2004) i {, } X i Y i ˆτ > 0 (Montalvo, 2011) X i 3 Y i ˆτ < 0 ( ) / 37

15 (Krasno and Green, 2008) X i Y i ˆτ = 0 (Blattman, 2009) i X i Y i ˆτ > 0 ( ) / 37

16 (2013, 2017) ( ) / 37

17 ( ) / 37

18 (Lee, 2008) Y t X t Y t (1) Y t (0) Y t (1) Y t (0) ( ) / 37

19 V t 1 assignment/running/forcing variable X t = I(V t 1 c) c = 0 E(Y t (1) V t 1 = c) = lim v c E(Y t (1) V t 1 = v) y (1) E(Y t (0) V t 1 = c) = lim v c E(Y t (0) V t 1 = v) y (0). τ E(Y t (1) Y t (0) V t 1 = c) = lim v c E(Y t (1) V t 1 = v) lim v c E(Y t (0) V t 1 = v) = lim v c E(Y t V t 1 = v) lim v c E(Y t V t 1 = v) = y (1) y (0). ( ) / 37

20 f ( ) (x {0, 1}) E(Y t (x) V t 1 ) = y (x) + f (V t 1 x) f (c x) = 0. 4 f (v x) 4 d=1 γ (x) d (v c)d. Y t = α + τx t + f (V t 1 X t ) + ϵ t. (1) α y (0), E(ϵ t ) = 0. h h = 2% c h V t 1 c + h (1) ( ) / 37

21 c h V t 1 c + h V t 1 (Y t (1), Y t (0)) V t 1 (Y t (1), Y t (0)) X t (c h V t 1 < c, X t = 0) (c < V t 1 c + h, X t = 1) ϵ t X t E(X t ϵ t ) = 0 E(ˆτ) = τ ( ) / 37

22 (Gelman and King, 1990) (Ansolabehere, Snyder, and Stewart, 2000) (Hainmueller and Kern, 2008) ( ) / 37

23 (Ariga et al., 2016) (Ariga, 2015) ( ) / 37

24 K Z (k) t (k = 1, 2,..., K) c h V t 1 c + h ϵ t = g(z (1) t, Z (2) t,..., Z (K) t ) + ϵ t Z (k) t X t (c h V t 1 < c, X t = 0) (c < V t 1 c + h, X t = 1) Z (k) t E(Z (k) t c h V t 1 < c) = E(Z (k) t c < V t 1 c + h) ( ) / 37

25 Snyder (2005) Z (k) t = X t 1 (V t 2 ) E(X t 1 c h V t 1 < c) < E(X t 1 c < V t 1 c + h) Caughey and Sekhon (2011) h = 0.5% Z (k) t Lee (2008, 688, Fig. 4 (b)) (Z (k) t = V t 2 ) Grimmer et al. (2011) Z (k) t Pr(c h V t 1 < c) < Pr(c < V t 1 c + h) = ( ) / 37

26 ( ) / 37

27 Eggers et al. (2015) 40,000 9 NZ Ariga et al. (2016) Snyder, Folke, and Hirano (2015) Z (Dem) t V t 1 Pr(c h V t 1 < c) < Pr(c < V t 1 c + h) Z (k) t Z (Dem) t X t 1 V t 1 ( ) / 37

28 Erikson and Rader (2017) (X t 1 = I(V t 2 c) ) de la Cuesta and Imai (2016) E(Z (k) t c h V t 1 < c) = E(Z (k) t c < V t 1 c + h) Z (k) t = α + τx t + f (V t 1 X t ) + ϵ t. ( ) / 37

29 ( ) / 37

30 Z (Dem) t (X t 1 = I(V t 2 c)) ( ) / 37

31 I Ansolabehere, Stephen, James M. Snyder, and Charles Stewart Old Voters, New Voters, and the Personal Vote: Using Redistricting to Measure the Incumbency Advantage. American Journal of Political Science 44(1): 17. Ariga, Kenichi Incumbency Disadvantage under Electoral Rules with Intraparty Competition: Evidence from Japan. Journal of Politics 77(3): Ariga, Kenichi, Yusaku Horiuchi, Roland Mansilla, and Michio Umeda No sorting, No advantage: Regression discontinuity estimates of incumbency advantage in Japan. Electoral Studies 43: Blattman, Christopher From violence to voting: War and political participation in uganda. American Political Science Review 103(2): ( ) / 37

32 II Caughey, Devin, and Jasjeet S. Sekhon Elections and the regression discontinuity design: Lessons from close U.S. House Races, Political Analysis 19(4): Cook, Fay Lomax, Lawrence R. Jacobs, and Dukhong Kim Trusting what you know: Information, knowledge, and confidence in social security. Journal of Politics 72(2): de la Cuesta, Brandon, and Kosuke Imai Misunderstandings About the Regression Discontinuity Design in the Study of Close Elections. Annual Review of Political Science 19(1): Doherty, Daniel, Alan S. Gerber, and Donald P. Green Personal income and attitudes toward redistribution: A study of lottery winners. Political Psychology 27(3): ( ) / 37

33 III Eggers, Andrew C., Anthony Fowler, Jens Hainmueller, Andrew B. Hall, and James M. Snyder On the Validity of the Regression Discontinuity Design for Estimating Electoral Effects: New Evidence from Over 40,000 Close Races. American Journal of Political Science 59(1): Elis, Roy, Neil Malhotra, and Marc Meredith Apportionment cycles as natural experiments. Political Analysis 17(4): Erikson, Robert S., and Kelly Rader Much ado about nothing: RDD and the incumbency advantage. Political Analysis 25(2): Fukumoto, Kentaro. Nda. The Effect of Candidate List Position on Vote Share: Improving Internal and External Validity. Unpublished manuscript. Fukumoto, Kentaro. Ndb. Political Ticket-Fixing. Unpublished manuscript. ( ) / 37

34 IV Fukumoto, Kentaro, and Hirofumi Miwa Share the Name, Share the Vote: A Natural Experiment of Name Recognition. Journal of Politics 80(2): Fukumoto, Kentaro, and Yusaku Horiuchi Making outsiders votes count: Detecting electoral fraud through a natural experiment. American Political Science Review 105(3): Fukumoto, Kentaro, and Yusaku Horiuchi Identifying the effect of mobilization on voter turnout through a natural experiment. Electoral Studies 44: Fukumoto, Kentaro, Yusaku Horiuchi, and Shoichiro Tanaka. Nd. Treated Politicians, Treated Voters: A Natural Experiment of Political Budget Cycles. Unpublished manuscript. Gelman, Andrew, and Gary King Estimating Incumbency Advantage without Bias. American Journal of Political Science 34(4): ( ) / 37

35 V Gomez, Brad T., Thomas G. Hansford, and George A. Krause The Republicans should pray for rain: Weather, turnout, and voting in U.S. presidential elections. Journal of Politics 69(3): Goodman, Joshua Flaking Out: Student Absences and Snow Days as Disruptions of Instructional Time. Working Paper National Bureau of Economic Research. Grimmer, Justin, Eitan Hersh, Brian Feinstein, and Daniel Carpenter Are Close Elections Random? Unpublished manuscript pp Hainmueller, Jens, and Holger Lutz Kern Incumbency as a source of spillover effects in mixed electoral systems: Evidence from a regression-discontinuity design. Electoral Studies 27(2): Healy, Andrew, and Neil Malhotra Random Events, Economic Losses, and Retrospective Voting: Implications for Democratic Competence. Quarterly Journal of Political Science 5(2): ( ) / 37

36 VI Hirano, Shigeo Do individual representatives influence government transfers? Evidence from Japan. Journal of Politics 73(4): Hyde, Susan D The Observer Effect in International Politics: Evidence from a Natural Experiment. World Politics 60(01): Krasno, Jonathan S., and Donald P. Green Do televised presidential ads increase voter turnout? Evidence from a natural experiment. Journal of Politics 70(1): Lee, David S Randomized experiments from non-random selection in U.S. House elections. Journal of Econometrics 142(2): Montalvo, José G Voting after the Bombings: A Natural Experiment on the Effect of Terrorist Attacks on Democratic Elections. Review of Economics and Statistics 93(4): Posner, Daniel N Political Salience Are Allies of in Cultural Zambia Difference : Why Adversaries Chewas in Malawi. American Political Science Review 98(4): ( ) / 37

37 VII Snyder, James M., Olle Folke, and Shigeo Hirano Partisan Imbalance in Regression Discontinuity Studies Based on Electoral Thresholds. Political Science Research and Methods 3(02): Snyder, Jason Detecting Manipulation in U.S. House Elections. Unpublished manuscript. Titiunik, Rocío Drawing Your Senator from a Jar: Term Length and Legislative Behavior. Political Science Research and Methods 4(02): , In, ed.. pp , (10): ( ) / 37

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