Peer effects on the United States Supreme Court

Holden, Richard; Keane, Michael P.; Lilley, Matthew Article Peer effects on the United States Supreme Court Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Holden, Richard; Keane, Michael P.; Lilley, Matthew (2021) : Peer effects on the United States Supreme Court, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 12, Iss. 3, pp. 981-1019, https://doi.org/10.3982/QE1296 This Version is available at: https://hdl.handle.net/10419/253582 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Quantitative Economics 12 (2021), 981–1019 1759-7331/20210981 Peer effects on the United States Supreme Court Richard Holden School of Economics, University of New South Wales Michael Keane School of Economics, University of New South Wales Matthew Lilley Department of Economics, Harvard University Using data on essentially every U.S. Supreme Court decision since 1946, we estimate a model of peer effects on the Court. We estimate the impact of justice ideology and justice votes on the votes of their peers. To identify the peer effects, we use two instruments that generate plausibly exogenous variation in the peer group itself, or in the votes of peers. The first instrument utilizes the fact that the composition of the Court varies from case to case due to recusals or absences for health reasons. The second utilizes the fact that many justices previously sat on Federal Circuit Courts, and justices are generally much less likely to overturn decisions in cases sourced from their former “home” court. We find large peer effects. For example, we can use our model to predict the impact of replacing Justice Ginsburg with Justice Barrett. Under the the assumption that Justice Barrett’s ideological position aligns closely with Justice Scalia, for whom she clerked, we predict that her influence on the Court will increase the Conservative vote propensity of the other justices by 47percentage points. That translates into 038 extra conservative votes per case on top of the impact of her own vote. In general, we find indirect effects are large relative to the direct mechanical effect of a justice’s own vote. Keywords. Peer effects, Supreme Court, voting, political economy. JEL classification. C31, C33, D72, K40. 1. Introduction Economists have long been interested in the impact of one’s social, educational, and workplace environment—and the characteristics of other agents in that environment— on one’s own behavior and outcomes.1The presence of positive spillovers, or peer effects, Richard Holden: [email protected] Michael Keane: [email protected] Matthew Lilley: [email protected] The editor and three referees provided very useful comments. We are grateful to Rosalind Dixon, John Friedman, Christine Jolls, Christopher Malloy, Emily Oster, Richard Posner, Jesse Shapiro, Andrei Shleifer, and Justin Wolfers for helpful discussions, and to seminar participants at Harvard, Harvard Law School, and MIT. 1In the context of education, the concept of peer effects dates to at least the “Coleman Report” (Coleman et al. (1966)). ©2021 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://qeconomics.org.https://doi.org/10.3982/QE1296 982 Holden, Keane, and Lilley Quantitative Economics 12 (2021) in such settings would suggest a range of policy interventions that may improve educational and labor-market outcomes. More generally, peer effects may be an important determinant of outcomes in many social settings. However, the problem of statistically identifying peer effects is formidable. As discussions in Manski (2000)andMoffitt (2001) make clear, plausible identification of peer effects requires (i) a clear definition of the peer group itself, (ii) exogenous variation in the behavior of peers (with the peer group held fixed or at least randomly assigned) and/or (iii) exogenous variation in the peer group itself (e.g., random assignment). Peers are often to some degree chosen, and thus in many contexts it is very difficult to find plausibly exogenous variation in the peer group.2Similarly, without exogenous variation in the peer group, it becomes very difficult to find interventions that exogenously shift the behavior of one or more peers while having no direct effect on other group members and while holding peer group composition fixed. The Supreme Court of the United States is a pertinent example of a context where peer effects may be of first-order importance. The Court issues decisions on important political, social, and constitutional questions. Accordingly, the question of how the composition of the Court and the interaction of its members affect justices’ individual votes and the majority decisions of the Court, is of intrinsic interest. Furthermore, the existence and magnitude of peer effects are important for understanding the cumulative effects of judicial appointments. In particular, the indirect effect of a new justice through the votes of existing justices may be large relative to the direct effect of their own vote. If peer effects are positive, this amplifies the stakes of judicial appointments.3This, in turn, speaks to the characteristics and design of legal institutions. As well as being of intrinsic interest, the Supreme Court is a context where, prima facie, peer effects appear difficult to isolate. Unlike some legal contexts where judges are plausibly randomly assigned to cases, the Supreme Court involves a panel of nine justices that evolves very slowly over time, and typically hear cases en banc, suggesting there is little variation in peer composition to exploit. Further, attempts to identify factors that exogenously shock a given justice’s vote in a case in order to analyze peer effects are subject to the common shock problem. Simply put, the most salient factors that affect any justice’s vote in a case are likely to directly affect all justices. Despite this challenge, we argue that careful consideration of the institutional environment of the Court enables peer effects to be identified. Several features of the Court are pertinent. First, the relevant peer group of a justice can be clearly defined as the group of eight other justices who sit on the same Court.4Second, even though the full 2Manski (1993) has particularly stressed the Reflection Problem—as people tend to chose peers who resemble themselves, there is typically a mechanical link between the characteristics of individuals and those of their peer group. This creates a great risk of falsely inferring that peer behavior affects own behavior, even if the causality actually runs the other way. See Manski (1993) and (2000). 3Given peer effects, the ideal appointment may not simply be the one that shifts the median justice closest to the view of the President. In general, the exact functional form by which peer effects operate may alter the optimal strategy of an administration in nominating justices. But as we restrict our attention to linear-in-means peer effects for tractability, we do not investigate this possibility, but note it as an avenue for future exploration. 4If we view the whole set of justices as the peer group, the group selection problem is largely irrelevant as justices have minimal choice over the identity of their peers (except via retirement decisions). Furthermore, Quantitative Economics 12 (2021) Peer effects on the United States Supreme Court 983 complement of peers is fixed (except in the infrequent instances when court composition changes) we will show there exists a highly plausibly source of case-to-case exogenous variation in any one justice’s peer group. A little known fact, at least outside the legal community, is that many Supreme Court cases are decided by less than the full complement of justices. That is, justices are frequently absent from particular cases due to illness, recusals, and other random factors. This stochastic process creates plausibly exogenous variation in justices’ peer groups on a case-to-case basis. Third and finally, the existence of “home court bias” generates a plausibly exogenous instrument that shifts the behavior of individual peer justices while the peer group is held exogenously fixed. Specifically, many Supreme Court justices previously sat on Federal Circuit Courts of Appeals. Epstein, Martin, Quinn, and Segal (2009) found strong evidence that justices are highly inclined to rule in favor of their respective home circuit court, even conditional on ideology and other factors, and we find the same effect. This provides us with a compelling instrument to identify peer effects. Together, these three facts mean that the question “How does the ideology or voting behavior of a justice’s peers affect his/her own vote?” is well-posed, as the peer group is well-defined, it is not self-selected, and it is subject to plausibly exogenous variation. And in addition, the behavior of peers is subject to plausibly exogenous variation induced by home court bias. Furthermore, this question is of policy relevance, because it helps to predict the impact of any potential Supreme Court appointment on the overall voting behavior of the Court. In this paper, we look at two types of peer effects: how peer ideology affects a justice’s voting behavior, and how actual peer votes affect a justice’s voting behavior. Following the literature, we will refer to these two types of peer effects as “exogenous” and “endogenous” peer effects, respectively (see, e.g., Moffitt (2001)). First, in Section 3, we consider a model of “exogenous” peer effects where justice voting behavior is determined both by their own ideology and the ideology of their peers. Specifically, we utilize a detailed coding of the votes in our data set as being either conservative (1) or liberal (0) in orientation, and then estimate a linear probability model of justice votes as a function of case characteristics, justice fixed effects (i.e., ideological positions), and mean peer ideology (constructed as the mean of peer justice fixed effects). Relying for identification primarily on changes in Court composition due to recusals and absences, we find clear evidence of ideology-based exogenous peer effects. In particular, we find that replacing a single justice with one who votes conservative 10 percentage points more frequently increases the probability that each other justice votes in the conservative direction by 14percentage points (on average). We then turn attention to investigating the possibility that peer effects may also be “endogenous,” meaning they operate through the actual votes cast by peer justices, not their ideology per se. In that case, identifying a true peer effect requires exogenous variation in voting propensity across justices, that is, a variable which directly affects how a given peer justice votes in a given case, but not the votes of other justices, except through this peer group is of policy interest, as it can be altered by a well-defined policy lever (i.e., presidential nomination and Senate confirmation). 984 Holden, Keane, and Lilley Quantitative Economics 12 (2021) the vote of the directly affected peer. We argue that the “home court bias” instrument described earlier has these properties. Properly investigating whether “endogenous” peer effects exist requires simultaneously testing for both “exogenous” and “endogenous” channels. This corresponds to the two terms in the most general “structural” model of peer effects discussed in Moffitt (2001), equation (16). As Moffitt (2001)andManski (1993) noted, in the reduced form of this structural model, own votes depend on own ideology, peer ideology, and the exogenous factor (i.e., home court bias) that shifts peer votes conditional on ideology. In the absence of endogenous peer effects, the exogenous factor that shifts peer votes drops out of the reduced form. Thus, testing for significance of the “home court” bias of peers in the reduced form is a simple test for existence of endogenous peer effects (a test that should not be too sensitive to the exact functional form through which peer votes operate). When we estimate this reduced form (Section 4.3), we find both peer ideology and home court bias of peers are significant, implying both exogenous and endogenous peer effects are present. Hence, in Section 4.5, we estimate structural models with both exogenous and endogenous peer effects, relying on both recusals and home court bias as sources of exogenous variation. In our preferred model, we find that a single peer shifting their vote from liberal to conservative increases each other justice’s conservative vote probability by roughly 11%. Thus, we find a strong causal impact of peer votes. Finally, we examine whether peer effects change pivotal votes, and hence case outcomes, or if they merely affect the size of the majority. If peer effects merely push a decision from 6–3to 5–4, or vice versa, then they are of limited practical interest.5To address this question, we aggregate votes at the case level, and consider how a single justice’s vote affects the collective voting behavior of their peers. We find strong evidence that peer effects can be pivotal. By affecting the votes of their peers, a single justice becoming 10% more likely to vote conservative increases the share of cases with a conservative outcome by 36percentage points—excluding the mechanical effect of that justice’s own vote—and reduces the share with a liberal outcome by 32percentage points. We are certainly not the first authors to consider the issues of judicial ideology and peer effects. Many political science and legal scholars have debated whether Supreme Court decision making is largely driven by justices’ own narrow policy preferences, or whether justices are also constrained by higher legal principles, such as deference to precedence and judicial restraint (Bailey and Maltzman (2011)), or political constraints, such as public opinion and executive discretion over compliance (Carrubba and Zorn (2010)). There is a significant empirical literature estimating the ideological position of judges and justices on measures that encapsulate both viewpoints. For instance, Martin and Quinn (2002) developed a dynamic item response model and estimate justice ideal points that can be time varying, and Martin, Quinn, and Epstein (2005) use the Martin– Quinn method to estimate the median Supreme Court justice on Courts dating from 5Of course, the credibility of the Court, and how political it looks, is an important issue, and is plausibly affected by the size of the majority in a case. 5–4decisions breaking along the lines of the party of the appointing President, for instance, may be seen as particularly political and this could be damaging to the image of the Court. Quantitative Economics 12 (2021) Peer effects on the United States Supreme Court 985 1937. If one thinks that peer effects operate through the characteristics of judges, then understanding judicial ideology is a necessary first step to study them, as well as being (arguably) of interest in its own right. Perhaps closer to our paper is the literature on panel effects on lower courts. A large literature considers peer effects (often referred to as “panel effects”) on U.S. Circuit Courts of Appeals.6Different authors emphasize different channels, such as: deliberation, group polarization, or aversion to dissent (Epstein, Landes, and Posner (2011)). Fischman (2015) argued that peer effects are best understood by reference to peers’ votes rather than characteristics, and reanalyzes 11 earlier papers on Circuit Court “panel” voting, as well as new data. 7He finds that—across the board—each judge’s vote increases the probability that a given judge votes in the same direction by approximately 40 percentage points. Boyd, Epstein, and Martin (2010) considered the impact of female judges and only finds strong effects for sex discrimination cases, suggesting an information channel is operative rather than alternative theories of influence (see also Peresie (2005)). Epstein and Jacobi (2008) argued the power of the median justice is due to bargaining power, not personality, and that ideological remoteness of the median justice makes them pivotal over a greater range of the ideological spectrum. Relative to this literature, we make several contributions: One, we focus on the United States Supreme Court rather than U.S. Circuit Courts of Appeals. Two, we analyze both the ideological channel and the vote channel using a novel identification strategy. And three, we focus on both peer effects and their impact in altering case outcomes. Methodologically, we also prevent a simple new method to estimate models where peer effects operate through fixed effects of peers. Once one is convinced that peer effects on the Supreme Court exist, a key question is what drives them. Perhaps the most most obvious channel is effects via persuasion. In the context of lower courts, several other possibilities have been raised, including: deliberation, group polarization, and aversion to dissent. In fact, our estimates would capture interdependence in justice votes due to any team-production based phenomena, including horsetrading (i.e., vote trading), dissent aversion, etc. Our paper is primarily about testing for existence of peer effects, not isolating the mechanisms through which they operate. But we touch on that question in our concluding remarks, where we also offer estimates of peer effects by issue area. The paper is organized as follows: Section 2describes the data. Section 3presents our analysis of the ideological channel for peer effects, while Section 4studies the ideological and voting channels jointly. Section 5examines peer effects on case outcomes, and Section 6concludes. 6Some notable papers in this literature are Revesz (1997), Miles and Sunstein (2006), and Posner (2008). 7He replaces the characteristics of panel colleagues with their votes, so the votes are endogenous, but colleague characteristics can be used as an instrument for colleague votes, assuming that they have no direct causal effect. 986 Holden, Keane, and Lilley Quantitative Economics 12 (2021) 2. Data on Supreme Court votes The Supreme Court Database, developed by Spaeth and Epstein (2014), contains almost the entire universe of cases decided from the 1946 to 2013 terms.8It provides detailed information on each case, including the participants, the legal issue area, the court term when the case was heard and opinions issued, the winning party and the vote margin. The data includes the identity and vote of each justice, for each case in which they were involved. This allows us to model the votes of individual justices, and how they relate to the identity and voting decisions of the peers. For almost all cases, votes are categorized as ideologically liberal or conservative, following an explicit set of rules. Exceptions to this occur in cases without any clear ideological underpinning. We augment these data with biographical information on justices from the U.S. Supreme Court Justices Database developed by Epstein, Walker, Staudt, Hendrickson, and Roberts (2013). This provides information on which, if any, Circuit Court of Appeals a justice previously served on, and the length of their tenure on that court. This allows us to construct our “home court bias” instrument. In total, these data provide information on 116362 votes (including absences and recusals) involving 12981 legal provision-case pairs from 8561 cases.9Once we exclude absences, recusals and cases without any ideological direction, the data contain 110729 votes from 8420 cases (and 12779 legal provision-case pairs).10 One quarter of these cases involve a vote by less than the full panel of nine serving justices. Votes are ideologically balanced, with 48% issued in the conservative direction. In contrast, a slight majority (55%) of directional lower court decisions reviewed by the Supreme Court are in the conservative direction. There is a strong tendency towards overturning lower court decisions; 60% of Supreme Court decisions and 58% of individual justice votes are for reversal. The Supreme Court only reviews a small fraction of cases, so of course it tends to hear cases where several justices believe the lower court may have erred. Table 1breaks down vote directions by legal issue area. Of the 11 high-level issueareas in the database with a nontrivial number of cases, the conservative vote share over the 1946–2013 range of court terms varies from 29% conservative for Federal Taxation cases to 60% conservative in privacy cases. Grouping instead by the Circuit Court of Appeals that previously heard the case (for the ∼60% of cases that source from such a court) the conservative share of votes ranges from 43% for cases from the Seventh Circuit to 54% for Ninth Circuit cases. The variation in vote ideology is much greater across justices: the conservative vote share ranges from 22% for William O. Douglas to 72% for Clarence Thomas (see Table 2for details). Appendix C Figure 3shows that the conservative vote share has varied substantially over time. 8For example, per curiam decisions are not included unless the Court provided a summary or opinions were issued. 9Some cases involve separate votes on different points of law, or “provisions.” 10A small number of cases result in tied votes, following which the votes of individual justices are typically not made public. Provided that the case had a lower court decision with stated ideological direction, so that the case is known to have ideological relevance, the vote direction for each justice is coded as 05by convention. Quantitative Economics 12 (2021) Peer effects on the United States Supreme Court 987 Table 1. Descriptive statistics for votes and vote direction. Votes Provisions Vote Direction (Cons. %) Lower Court (Cons. %) Overturn (%) Total 110729 12779 4758 5503 5818 Legal Issue Area Criminal procedure 22549 2585 5212 6307 6023 Civil rights 18435 2112 4487 5347 5871 First amendment 9895 1140 4592 5666 5625 Due process 4975 577 4257 5365 5984 Privacy 1483 169 6035 3021 5738 Attorneys 1122 130 4323 5205 6034 Unions 4387 506 4525 5753 5587 Economic activity 21447 2500 4228 4882 5720 Judicial power 17041 1976 5832 5418 5833 Federalism 5805 670 4365 5666 5823 Federal taxation 3415 394 2949 5678 5271 Circuit Court Federal 937 107 4621 4300 6282 First 2125 246 4701 4082 5140 Second 8107 934 4835 5070 5485 Third 5008 575 5154 4984 5421 Fourth 4471 512 4596 6088 5540 Fifth 7907 914 4349 6512 6088 Sixth 5558 644 4759 5055 6017 Seventh 5523 645 4297 5907 5863 Eighth 4046 465 4530 4860 5794 Ninth 11835 1359 5430 3827 6280 Tenth 3153 367 5103 5122 6001 Eleventh 2203 247 4480 6768 5710 D.C. 6961 818 5215 5113 5946 3. Exogenous ideology-based peer effects First, we assume exogenous peer effects. That is, we assume peer effects work directly through ideological positions, with the votes of one justice directly influenced by the ideological positions of the other justices. In the terminology of Manski, this is a contextual peer effect as justice ideology is predetermined with respect to interactions with other justices. Under this mechanism, the voting decisions of a particular justice are influenced by the ideological positions of peer justices, regardless of how those peer justices actually vote in a particular case. 3.1 Model and estimation method We assume justices’ votes are influenced by their own ideology, the factual context of the case, and the ideology of peers. The ideological direction of the vote by justice jin case 988 Holden, Keane, and Lilley Quantitative Economics 12 (2021) Table 2. Justice ideology fixed effects and justice characteristics. Justice Ideology Estimate Segal–Cover Score Conservative Vote Share Party of President W. O. Douglas 01969 0730 02154 Democratic W. B. Rutledge 02042 1000 02336 Democratic F. M u r p h y 02137 1000 02424 Democratic T. Marshall 02435 1000 02802 Democratic W. J. Brennan 02728 1000 02930 Republican H. L. Black 02775 0875 02820 Democratic A. Fortas 02830 1000 03082 Democratic E. Warren 02896 0750 02703 Republican A. J. Goldberg 03092 0750 02404 Democratic J. P. Stevens 03976 0250 03889 Republican R. B. Ginsburg 04468 0680 03863 Democratic S. Sotomayor 04748 0780 03712 Democratic D. H. Souter 04749 0325 04183 Republican H. A. Blackmun 04755 0115 04790 Republican S. G. Breyer 04853 0475 04160 Democratic E. Kagan 05034 0730 03963 Democratic P. S t e w a r t 05179 0750 05046 Republican T. C. Clark 05241 0500 04764 Democratic B. R. White 05345 0500 05201 Democratic F. M . V i n s o n 05681 0750 05635 Democratic F. Frankfurter 05735 0665 05394 Democratic S. Minton 05974 0720 05688 Democratic S. F. Reed 05998 0725 05708 Democratic H. H. Burton 06030 0280 05669 Democratic L. F. Powell 06070 0165 06084 Republican C. E. Whittaker 06102 0500 05516 Republican R. H. Jackson 06192 1000 06157 Democratic J. Harlan II 06269 0875 05729 Republican W. E. Burger 06605 0115 06574 Republican S. D. O’Connor 06790 0415 06245 Republican A. M. Kennedy 06918 0365 06042 Republican J. G. Roberts 07374 0120 06126 Republican W. H. Rehnquist 07640 0045 07134 Republican A. Scalia 07813 0000 06793 Republican S. A. Alito 08020 0100 06653 Republican C. Thomas 08293 0160 07157 Republican Note: The ideology estimates are the fixed effects estimates from Model 1B, which includes justice, issue area, circuit court, and term fixed effects, as well as the mean active peer ideology measure. Note that only the relative values of the justice fixed effects are meaningful. So we normalize the mean of the ideology estimates to match the mean conservative vote share (net of the effects of the controls). N=110729 votes. c,whichwedenotebydjc, is either conservative (1) or liberal (0). We consider a linear probability model: p(djc =1|α β δ XcIjc)=αj+βp×1 |Ijc| i∈Ijc αi+X cβx+δt(c)(1) Quantitative Economics 12 (2021) Peer effects on the United States Supreme Court 995 Table 4. Exogenous peer effect models: peer ideology effects. (A) (B) (C) Model 1: Justice and term fixed effects All peer justices −0874 (1073) Active peer justices 1131 1244 (0319)(0432) Absent peer justices 0033 (0102) R-squared 01446 01454 01455 Model 2: Justice by issue, and term by issue FE All peer justices 0030 (0797) Active peer justices 1082 1199 (0238)(0315) Absent peer justices 0035 (0077) R-squared 02101 02111 02112 Model 3: Justice by issue by court, and term by issue FE All peer justices 2115 (1862) Active peer justices 1635∗∗∗ 1550∗∗∗ (0206)(0346) Absent peer justices −0027 (0082) R-squared 02378 02405 02406 Note:N=110729 votes. Supreme Court to overturn many decisions that it reviews (hence reversing the ideological direction of lower court decisions). Second, a consistent pattern of home court bias is evident. Consistent with results in Epstein et al. (2009), we find that justices who had previously served on a Circuit Court of Appeals (a justice’s home court) are less likely to overturn the lower court’s decision in a case sourced from that court. However, this bias diminishes with home court tenure, and justices with very long Circuit Court tenures (i.e., more than 10 years) are more likely to overturn cases sourced from their home court. We discuss this pattern further in Section 4.2. We now turn to our key results; the estimates of peer ideology effects. The results for Models 1 to 3, which use progressively richer controls for justice, issue and term effects, are reported in the three panels of Table 4. Columns A through C of each panel show results for different specifications of the peer ideology measure: all peers, active peers, or both. The results for Model 1, which includes term and justice fixed effects—so that each justice has a single invariant ideology estimate—are shown in the first panel. Column A reports results using the mean ideology of all peers to measure peer effects. As expected, 996 Holden, Keane, and Lilley Quantitative Economics 12 (2021) the estimate of the peer effect parameter βpis very imprecise, because Supreme Court panel rotation is infrequent and largely absorbed by term fixed effects, leaving little exogenous variation in peer ideology. Table 4, column B presents results using our preferred active peers measure, which exploits within-term variation in peers due to absences. This yields a substantial and tightly estimated active peer coefficient of 1131. This implies, for example, that replacing a justice with another who votes in the conservative direction 10 percentage points more frequently on average would increase the conservative vote probability of all other justices by 141 percentage points, generating a cumulative 0113 extra conservative votes by the peer justices per case (i.e., 00141 ×8=0113). Column C presents the placebo test where we include the absent peers measure. The estimate is small (0.033) and insignificant (SE =0102), providing no evidence that absent peer ideology is correlated with unobserved case characteristics that affect votes. Comparing columns B and C, we see that inclusion of the absent peers measure causes the coefficient on active peer ideology to increase very slightly to 1244.23 We take this as strong evidence supporting our assumption that absences/recusals can be taken as exogenous, and hence for the existence of peer effects. The results for Model 2, which contains both justice-by-issue-area and term-byissue-area fixed effects, are shown in the second panel of Table 4.24 As in Model 1, the all peers ideology effect is very imprecisely estimated. However, the active peer measure, which exploits within-year-and-issue-area variation in peer ideology due to justice absences, yields a positive and significant peer effect coefficient of 1082.IncolumnC, the coefficient on the placebo measure of absent justices is again small and completely insignificant—supporting our assumption that absences are as good as random—and the coefficient on active peers again increases only very slightly. Results for Model 3, which allows the ideology of each justice to change over time (by natural court) for each issue area, are displayed in the final panel of Table 4.InColumn B, we obtain a coefficient of 1635 on our preferred active peer ideology measure, with a standard error of 0206. In column C, the coefficient on placebo absent peer measure is again small and insignificant, consistent with our key identifying assumption of no selection into absence based on case unobservables. Collectively, the results in Table 4 provide strong evidence for existence of peer effects. This result is robust to alternative controls for justice, term, and issue area. Finally, the Appendix in the Online Supplementary Material (Holden, Keane, and Lilley (2021)) contains three robustness checks on these results suggested by referees: We consider specifications using median justice ideology, we consider only observations 23Notice that the standard errors on the active peer coefficients only increase by about one-third when the absent peer variable is included. This illustrates that the two variables are not highly collinearity (see Section 3.4). 24Once we introduce justice-by-issue-area fixed effects, changes in court composition induce differential changes in peer ideology by issue area. This source of variation may not be well distinguished from issuearea-specific ideological drift over time. Thus, Model 2 absorbs this source of variation via inclusion of term-by-issue-area fixed effects. Quantitative Economics 12 (2021) Peer effects on the United States Supreme Court 997 Table 5. Instrumenting for peer ideology using Segal–Cover scores. (A) (B) (C) Model 1: Justice and term fixed effects All peer justices −1847 (1518) Active peer justices 1129 1061 (0427)(0582) Absent peer justices −0009 (0136) First stage F-statistic 35706 111113 39333 Note: The mean Segal–Cover score of justices in each peer group (all, active or absent) is used to instrument peer ideology measure for that group. Column C reports the cluster robust Cragg–Donald F-statistic. from the post-Warren Court, and we consider only the subset of terms with no permanent changes in Court composition. Our results are little affected by these considerations. 3.5.1 Accounting for potential endogeneity of ideology We estimate justice ideology from each justice’s full voting record. Hence, a potential concern is that our ideology measures are not predetermined. That is, our ex post measure of a justice’s ideology may be influenced by the Supreme Court environment during his/her tenure (i.e., reverse causality from votes or interaction with peers to the ideology measures). However, we can deal with this concern by using a predetermined measure of ideology as an instrument. Segal and Cover (1989) developed estimates of justice ideology based on textual analysis of newspaper editorials between nomination by the President and Senate confirmation. These Segal–Cover scores predate a justice’s Supreme Court tenure, so they are predetermined with respect to voting behavior. As Table 2reveals, Segal–Cover scores are very imprecise compared to our vote-based ideology measures, but they are clearly highly correlated with our measures. Thus, we used Segal–Cover scores to instrument for justices’ ideologies when estimating Model 1.25 Table 5reports the results, which are very similar to the Model 1 estimates in Table 4top panel. In particular, the coefficient on active peer ideology in column B only moves slightly from 1131 to 1129. And the placebo test in column C again finds that ideology of absent justices is insignificant. So the IV results provide additional evidence of positive peer ideology effects. 4. A model with both exogenous and endogenous peer effects Here, we extend our analysis to allow for vote-based peer effects, in addition to exogenous ideology-based peer effects. If peers affect the votes of their colleagues through their own votes, then the votes of all justices are jointly determined on a case-by-case 25We discuss details of how to extend our nonlinear least squares estimator to the nonlinear 2SLS case in Section 4.5 and Appendix A.2. 998 Holden, Keane, and Lilley Quantitative Economics 12 (2021) basis, as in Fischman (2015). This fits within the framework of Manski’s endogenous peer effects (Manski (1993)). 4.1 Empirical specification and vote endogeneity We begin by extending the model in equation (1) to incorporate peer effects that operate through peer votes. Recall that djc isthevoteofjusticejin case c,equalto1for votes in a conservative direction and 0otherwise, and denote the set of peers who may affect justice jthrough their votes in case cas Vjc. Then we have the linear probability model: p(djc =1|θ d−jcXcVjcIjc) =αj+βv p×1 |Vjc| i∈Vjc dic +βid p×1 |Ijc| i∈Ijc αi+δt(c) +X cβx(3) This equation is identical to the specification in Model 1 of Section 3,exceptthatnow βid pcaptures the effect of the ideology of a justice’s peers, while βv pcaptures the effect of the votes of peers. Following Moffitt (2001), we refer to (3) as the “structural” model of peer effects, because it contains the endogenous peer vote variable. Later in Section 4.3 we will consider a reduced form of (3) where we substitute for peer votes using their exogenous determinants. We consider three alternative specifications of the relevant peer group of justices Vjc whose votes may affect the vote of justice j. First, we consider the votes of all other justices who vote in a case (active justices, in the language of Section 3). In this case, Vjc and Ijc are identical. Second, peers who have special expertise in a case may have a greater influence. Justices who previously served on the appellate court from which a case is sourced are plausibly more knowledgeable.26 Accordingly, we also consider the votes of home justices in home court cases. Third, as the impact of home peer votes may be stronger when they are more numerous, we also consider the net vote direction of the home justices (i.e., the number of home justices issuing conservative votes minus liberal votes), divided by the total number of all active peers.27 Obtaining consistent estimates of βv prequires the use of instrumental variables as votes are jointly determined. Of course, common unobservables that affect outcomes for both a person and their peers are a standard problem when estimating endogenous peer effects. Here, unobserved case characteristics are very important determinants of votes.28 In fact, 37% of the cases in our sample were decided unanimously, so the vote of 26Circuit courts tend to hear cases in certain areas, so a judge from such a court will tend to have more expertise in those areas. Second, a former circuit court judge may be more familiar with the legal reasoning of its judges. 27For example, if there are three home peers, of which two vote liberal and the other conservative, the variable is −1 8. If there are two home peers, and both vote liberal, it equals −2 8. 28The observed case characteristics are legal issue area, the term the case is heard, the lower court decision, and the Circuit Court (if any) the case stems from. Conditioning of these variables leaves much of the variation in case vote outcomes unexplained, implying that unobserved case characteristics are very important determinants of votes. Quantitative Economics 12 (2021) Peer effects on the United States Supreme Court 999 Figure 1. Home court bias in overturn rate of lower court decisions. a single justice has substantial predictive power for how other justices vote, irrespective of the existence of peer effects. To identify true peer vote effects, it is necessary to find an instrument that generates exogenous variation in voting propensity across justices, unrelated to unobserved case characteristics. For this purpose, we use our “home court” instrument, which we explain in the next section. 4.2 Constructing instruments for peer votes Epstein et al. (2009) found that justices who had previously served on a Circuit Court of Appeals, their home court in our terminology, are ceteris paribus less likely to overturn decisions in cases sourced from their home court. Figure 1documents this pattern by plotting the rate at which justices overturn decisions in cases from their home court relative to all other cases, against the duration of home court tenure, for each of the 19 justices who previously served on a Circuit Court of Appeals. All but 4justices lie above the x-axis, indicating deference to home court decisions. Our results in Section 3.5,Table3also indicate that justices with previous service on a circuit court are less likely to overturn decisions sourced from that “home court”— consistent with Epstein et al. (2009). But we find that this home court bias diminishes with longer lower court tenure.29 A possible explanation for this pattern is that circuit courts often handle cases in particular legal areas, so circuit court judges develop expertise in those areas. A Supreme Court justice with relatively short tenure on a circuit court may give deference to decisions of his former colleagues because he/she recognizes their expertise in the issue areas the circuit court deals with. On the other hand, a 29In particular, Justices Kennedy and Berger both exhibited bias against their home courts, and both had long tenures: Kennedy served on the 9th Circuit for 12 years, and Chief Justice Burger served on the D.C. Circuit Court for 13 years, often clashing with colleagues during his tenure; see Greenhouse (2007). 1000 Holden, Keane, and Lilley Quantitative Economics 12 (2021) justice with long tenure on a circuit would have developed expertise in those issue areas him/herself, perhaps leading to less deference to the lower court judgement. We argue that variables capturing home court bias are plausible instruments for peer votes. A valid instrument should affect how a justice votes in a given case only through its effect on a peer’s vote (see Moffitt (2001)). The home court instrument satisfies this condition, as there is no plausible reason that the mere presence of a peer justice from a lower court would affect how another justice votes on a case sourced from that court. Any plausible effect must operate through how the home peer justice actually votes. To form the instruments, let Ia cdenote the set of justices in case cwho previously served on lower court a,andletya jdenote justice j’s tenure on lower court a. Our instruments are the share of other justices at home 1 N−1j=iI(j ∈Ia c)and the average length of home court tenure per justice 1 N−1j=i(I(j ∈Ia c)×ya j)in the case.30 Both are interacted with the lower court decision direction to convert effects on overturn propensity into effects on ideological disposition. To negate any possibility that the instruments are contaminated by selection into absence, we construct them in two ways. First, using only the justices active in a case, second using all justices on the Supreme Court (regardless of whether they are active). If there are no selection effects the former specification is more intuitive, as the endogenous peer vote variable in equation (3) is only based on active peer votes. Finally, recall that all our models contain circuit court fixed effects (i.e., indicators for the circuit court if any, from which each case is sourced). This controls for the possibility that some circuit courts are overturned more frequently than others. It should also allay any concern that we may confound home court effects with circuit court effects. 4.3 A simple reduced-form test for endogenous peer effects Before estimating the structural equation in (3), we first present a simple reduced-form test for whether endogenous peer effects exist. To obtain the reduced form of (3), we substitute out for peer votes using their exogenous determinants. In practice, this is equivalent to simply adding the “home bias” instruments directly to equation (1). If the home court variables are jointly significant in the reduced form voting model, we take it as evidence that endogenous peer effects exist. Table 6reports coefficients on the key variables of interest in the reduced form voting model. We report results for Model 1 that includes justice and term fixed effects. The peer home court measures are jointly highly significant, as shown by the F-statistics. The sign pattern is consistent with the idea that votes to uphold by home court peers reduce the propensity of the Supreme Court to overturn lower court verdicts, as the justices show some deference to peers from the lower court. It is implausible the home court variables would affect justices’ votes directly, rather than indirectly through peer votes, so this is strong evidence that endogenous peer effects exist. Note that active peer ideology is also significant in the reduced form. Its coefficient is little changed from that in the first panel of Table 4. The significance of peer ideology 30Note that we take the average over both home and away justices. Quantitative Economics 12 (2021) Peer effects on the United States Supreme Court 1001 Table 6. Reduced form voting model. (A) (B) (C) All peer justices −0873 (1060) Active peer justices 1135 1258 (0319)(0432) Absent peer justices 0036 (0101) Share of peers on lower court ×Conservative decision 0121 0129 0131 (0170)(0170)(0170) ×Liberal decision −0477 −0488 −0489 (0178)(0178)(0178) Peer mean tenure at lower court ×Conservative decision −0032 −0032 −0033 (0027)(0027)(0027) ×Liberal decision 0070 0071 0071 (0021)(0021)(0021) R-squared 01454 01463 01463 Home court variables: F-statistic 4320 4343 4362 Home court variables: P-value 00017 00016 00016 Note: All regressions include the same set of controls as in Model 1 in Table 3.N=110729 votes. in the reduced form may arise either because exogenous peer effects exist, or because peer ideology affects a justice’s own vote through its effect on peer votes (i.e., an endogenous peer effect). Thus, our reduced form results may be consistent with a structural model that contains both exogenous and endogenous peer effects, or a model that only contains the latter. Next we estimate the structural equation in (3)tosortoutthesetwo explanations. 4.4 A nonlinear 2SLS estimation algorithm If the peer ideology variable ¯α-jc =1 |Icj |i∈Icj αiwere observed, we could estimate (3)by 2SLS. However, as ¯α-jc is unobserved we instead implement the nonlinear 2SLS estimator that solves: min θ∈Θ cj Z jcrjc(θ) cj Z jcZjc−1 cj Z jcrjc(θ)(4) where we have defined residuals rjc(θ) =djc −p(djc =1|θ d−jcXcVjcIcj )based on equation (3), and we have also defined an instrument vector Zjc that includes all the exogenous variables (justice dummies, court dummies, case characteristics) as well as our home court instruments Hjc.31 The instruments must satisfy the exogeneity and rank 31Recall our home court instruments Hjc are the share of peer justices at home 1 N−1j=iI(j ∈Ia c),their average length of home court tenure 1 N−1j=i(I(j ∈Ia c)×ya j), and both interacted with the lower court decision direction. 1002 Holden, Keane, and Lilley Quantitative Economics 12 (2021) Table 7. First stage IV for peer votes: home court instruments. Active Peer Votes Home Peer Votes Net Home Peer Votes (1) (2) (3) (4) (5) (6) Share of peers at home ×Conservative decision 0211 0406 0197 (0169)(0211)(0112) ×Liberal decision −0581 −1343 −0258 (0176)(0228)(0118) Peer mean years at home ×Conservative decision −0039 −0088 −0023 (0026)(0030)(0009) ×Liberal decision 0079 0212 0052 (0020)(0025)(0008) Shareofactivepeersathome ×Conservative decision 0289 0495 0263 (0173)(0236)(0136) ×Liberal decision −0578 −1375 −0228 (0177)(0255)(0142) Active peer mean years at home ×Conservative decision −0054 −0090 −0022 (0026)(0031)(0009) ×Liberal decision 0070 0227 0059 (0021)(0027)(0008) R-squared 06886 06886 05908 05927 00844 00899 Home court variables: F-statistic 5368 5285 24485 25878 20018 22020 Home court variables: P-value 00003 00003 00000 00000 00000 00000 Note: All regressions include a control for the mean ideology of active peers, plus the same controls as in Model 1 in Table 3. N=110729 votes. conditions E[Z jcrjc(θ0)]=0and Rank E[Z jc∇θrjc(θ0)]=Pwhere θ∈Θ⊂RP.Therequirements for consistency are essentially identical to those for NLLS discussed earlier, as uniform (in θ) convergence in probability of the sample objective function in (4)toits population analogue is again the key point.32 Like the NLLS problem in (2), the nonlinear 2SLS problem in (4) involves function minimization over a large number of fixed effects and other parameters. As before, we use a simple iterative algorithm to construct the estimator. We discuss the algorithm— which is a simple extension of the NLLS algorithm—in detail in Appendix A.2. 4.5 Structural model results 4.5.1 First stage Here, we present nonlinear IV estimates of equation (3). Our first stage results from regressing peer votes on the instruments are shown in Table 7.Column(1) presents our first specification, where the endogenous variable in equation (3)isthe mean vote of all active peers, which is regressed on the home court instruments based on all home justices. The instruments are highly significant determinants of peer votes (F=537,p=0000). The point estimates indicate that as the share of home peers in32See Wooldridge (2010), pages 525–526 and 530–531 for details. Quantitative Economics 12 (2021) Peer effects on the United States Supreme Court 1003 creases, there are fewer peer votes to overturn the lower court decision.33 This effect is diminished if the home peers had longer tenure on the lower court. In column (2), we see the results are little changed if we base the instruments on only active peers rather than all peers. Columns (3) and (4) report first stage results for our second specification where the endogenous variable in equation (3) is the mean vote of home peers only. Unsurprisingly, our home court instruments are more highly significant in this model (F=2449), as they are better predictors of home peer votes than of all peer votes. Finally, columns (5) and (6) report results for our third version of the endogenous variable, the net vote of home peers. The results are similar. The Stock–Yogo weak instrument F-test critical values in the case of one endogenous variable and four instruments are 532,1023,and1672, respectively, for asymptotic bias of 2SLS relative to OLS being less than 30%,10%,or5%, respectively (see Skeels and Windmeijer (2018)). Thus, our home court instruments are—not surprisingly—much stronger when used to predict home court justice votes as opposed to all active peer votes. Still, we are not concerned about a weak instrument problem even in the case of the all peers measure, for two reasons: First, the very high F-test values in columns (3)–(6), as well the prior literature and our own results in Tables 3and 6, make clear that “home court bias” is a very real and quantitatively important phenomenon, not just a weak effect we pick up because we have a very large sample. The lower F-test values in columns (1)–(2) are directly attributable to the fact that only a small subset of justices are from the home courts, and only they are directly affected by the home court instruments (but for them the effects are large). Second, as we shall see below, these instruments still deliver precise estimates of the coefficient on the endogenous active peer vote variable in the second stage. 4.5.2 Second stage We present our instrumental variable estimates of the full structural model of equation (3)inTable8. The IV estimates document substantial positive peer effects. In column 1 the coefficient on the active peer vote variable is 0894 (SE =0037). In the typical full panel case (with 8peer justices), a single peer shifting their vote from liberal to conservative increases the active peer measure by 1/8. According to our estimates, this increases each other justice’s conservative vote probability by (1/8)(0894)=11%. Thus, we find a very strong causal impact of peer votes. The only difference between columns (1) and (2) is whether we base our four home court instruments on the total number of home peers in a case or only on the number of active home peers in a case. As we see in Table 8, this makes almost no difference. We examine the influence of home peer justices in columns (3) and (4). In a full panel case with one home peer, a shift in the peer’s vote from liberal to conservative increases our home peer vote measure by 10.According to our estimates, this increases the conservative vote probability of the other justices by 30% to 34%. This is three times greater than the 11% effect of a generic justice vote, highlighting the very strong influence of home peer votes in cases sourced from their home circuit court. We argue this large 33From the sign pattern of coefficients in Table 7column (1), we see that if the lower court decision was in the conservative (liberal) direction, the peer vote share in the conservative (liberal) direction increases. 1004 Holden, Keane, and Lilley Quantitative Economics 12 (2021) Table 8. Structural model of endogenous and exogenous peer effects. (1) (2) (3) (4) (5) (6) Active peer votes 0894 0877 (0037)(0041) Home peer votes 0342 0303 (0068)(0064) Net home peer votes 1366 1071 (0282)(0254) Active peer ideology −0505 −0471 1111 1113 1095 1102 (0087)(0097)(0310)(0311)(0309)(0311) Home peer instruments All Active All Active All Active First stage F-statistic 5368 5285 24485 25878 20018 22020 Note: All regressions include the same controls as Model 1 in Table 3.N=110729 votes. effect is plausible, as home justices are perceived as having greater expertise in cases originating from their home court. In fact, our estimate is smaller then the effect that Fischman (2015) finds for peer votes on circuit courts. The final two columns consider our third measure of peer votes, the net vote direction of active peers. In a full panel case with one home peer, a shift in the peer’s vote from liberal to conservative increases this measure from −1/8to 1/8. Our estimates imply this would increase the conservative vote probability of the other justices by 27% to 34%. An additional home justice voting conservative would increase this measure by an additional one-eighth, raising the conservative vote probability of other justices by afurther14% to 17%. These results again illustrate the strong influence of home peer justices. The structural models in Table 8also provide estimates of exogenous peer effects, operating through mean active peer ideology. Notice that in our main model in column (1) the effect of active peer ideology is negative. But care must be taken in interpreting this result: It should be interpreted as the effect of a change in peer ideology, holding peer votes constant. This has an intuitive interpretation: holding the number of peers who vote in the conservative direction fixed, a more conservative peer group makes that vote less convincing. Conversely, the more liberal is the peer group of justices who vote conservatively, the more persuasive that vote is.34 In summary, our results provide strong evidence for the existence of endogenous peer effects operating through peer votes. In the next section, we turn to the question of whether peer effects actually change case outcomes. 34In contrast, in Table 8, columns 3 to 4, where we estimate the effects of home peer votes in home court cases, the coefficients on active peer ideology are positive and roughly 111. This is very similar to the peer ideology effects reported for Model 1B in Section 3. Controlling for the vote share of home peers in home court cases has only a minor effect on the ideology coefficient, as justices are home rather infrequently. The same logic applies to columns 5 and 6. Quantitative Economics 12 (2021) Peer effects on the United States Supreme Court 1011 effects variables. Thus, we already obtain on iteration 2the vector of fixed effects αthat solve equation (7). Thus, on iteration 3we will already have ¯α3 -jc =1 |Icj |i∈Icj α3 i.Sothe iteration 3estimates of the parameter vector (α β δ) will also solve equation (7). A.1 Properties of the second iteration estimates The first iteration of our algorithm estimates justice ideology fixed effects under the assumption of no peer effects. Hence, our first iteration justice ideology measures are contaminated by the peer effects coming from other justices. This in turn causes our peer ideology measures ( ¯α-jc) to be contaminated by a justice’s own ideology. However, this contamination is washed out by the fixed effects in our second iteration regression. As a result, our second iteration (i) generates consistent estimates of justice ideologies that solve the NLLS minimization problem in equation (7), and (ii) gives a consistent estimate of βp(N−1 N−1−βp), the true peer coefficient βptimes a scale factor that depends on the number of justices N. We now show these results formally. To clarify the key idea that drives the results, first consider a simplified version of equation (1) where votes are determined by the linear probability model: djc =αj+β¯α-jc +εjc and a simple data generating process where court composition is unchanged during the tenure of each justice j, and where the full panel of judges hears all cases. Then ¯α-jc is a constant, which we denote by ¯α-j. So if we estimate the (misspecified) equation djc =αp j+ξjc that ignores peer effects, we will obtain, in large samples, the proxy ideology measures αp j=αj+β¯α-j. Thus, our initial ideology measures, obtained from a model that ignores peer effects, are contaminated by those peer effects. Suppose we nevertheless use them to construct an initial estimate of the peer ideology variable, which we denote by ¯αp −j: ¯αp -j=1 N−1 k=j αp k=1 N−1 k=j (αk+β¯α-k) =1 N−1 k=j αk+β1 N−1 k=j ¯α-k =¯α-j+β N−11 N−1(α2+···+αj+···+αN) +(α1+α3+···+αj+···+αN)+··· +(α1+···+αj+···+αN−1) =(1+β)¯α-j+β N−1(αj−¯α-j)=1+N−2 N−1β¯α-j+β N−1αj Thus the proxy peer effect measure consists of a scaled version of the true peer variable ¯α-j, plus a “contamination” due to the justice’s own ideology (the β N−1αjterm). 1012 Holden, Keane, and Lilley Quantitative Economics 12 (2021) Now consider the more realistic data generating process where justice jis observed sitting on a number of different courts g=1G, each with a different (but typically overlapping) group of N−1other justices who are concurrently appointed to the court. This allows the exposure of a particular justice to another to vary across cases and across justice pairs, while within a group g, with membership denoted by the set Sg, composition of the court may still vary by case due to absences. The true model is now djc =αj+β¯α-jcg +εjc If one instead estimates djc =αp jg +ξjc then, given a large number of cases alongside each peer, the Khintchine law of large numbers implies that one obtains the initial ideology measures: αp jg =αj+βπj g1α1+πj g2α2+···+πj gnαn where πj gk is the exposure weight of justice jto justice k∈Sg,withiπj gi =1and zero exposure to self, πj gj =0.Letusnowconstruct ¯αp -jcg, the mean estimated ideology that justice jfaces in a case cwith cohort g. ¯αp -jcg =1 Nc−1 k=jk∈Sg αp kgIc k =1 Nc−1 k=jk∈Sg αkIc k+β Nc−1 k=jk∈Sgπk g1α1+πk g2α2+···+πk gnαnIc k where Ic kis an indicator for the presence of justice k∈Sgin a case c,andNc−1is the number of active peers in case c. The exposure of peers of jto the ideology of jcan be separated out, ¯αp -jcg =1 Nc−1 k=jk∈Sg αkIc k+β Nc−1 i∈Sg k=jk∈Sg Ic kπk giαi =1 Nc−1 k=jk∈Sg αkIc k+β Nc−1 i=ji∈Sg k=jk∈Sg Ic kπk giαi +β Nc−1 k=jk∈Sg Ic kπk gj αj =1 Nc−1 k=jk∈Sg αkIc k+β i=ji∈Sg Ic iπi gk+β Nc−1 k=jk∈Sg Ic kπk gj αj where the final term captures the exposure of the peers jto the ideology of j. Now suppose we use these contaminated ideology measures in the fixed effects regression: djcg =γjg +θ¯αp -jcg +ωjc(8) Quantitative Economics 12 (2021) Peer effects on the United States Supreme Court 1013 where γjg are justice-by-group fixed effects and θis the key estimated parameter that captures peer effects. For example, gmay categorize the intersection of issue area and natural court. To implement the estimation with justice-by-group fixed effects, we must de-mean the ¯αp -jcg over the Tcases cwithin each group gby justice jpair. The mean is ¯αp -jcg =1 T c1 Nc−1 k=jk∈Sg αkIc k +1 T cβ Nc−1 k=jk∈Sg  i=ji∈Sg Ic iπi gkαk +1 T c β Nc−1 k=jk∈Sg Ic kπk gj αj To make this tractable, assume that justice absences are independent and equally likely within g. Then each justice k∈Sgis equally exposed to each other justice k∈Sg(that is, πk gi =1 N−1∀k= i k ∈Sg)andwehave ¯αp -jcg =1 Nc−1 k=jk∈Sg αkIc k+1 Nc−1 k=jk∈Sg αkβNc−1−Ic k N−1 +β Nc−1Nc−1 N−1αj =1 Nc−1 k=jk∈Sg αkIc k+β N−1 k=jk∈Sg αk −β N−11 Nc−1 k=jk∈Sg αkIc k+β N−1αj =1−β N−1¯α-jcg +βα-jg +β N−1αj Averaging over cases, we obtain ¯αp -jcg =1−β N−1¯α-jcg +βα-jg +β N−1αj Observe that the αjterm collapses to β N−1αj, which is constant across cases within g, and thus drops out upon demeaning. It follows that ¯αp -jcg − ¯αp -jcg =1−β N−1(¯α-jcg − ¯α-jcg) Observe that the justice fixed effects αjdrop out of this equation as claimed. 1014 Holden, Keane, and Lilley Quantitative Economics 12 (2021) This leaves us with the fixed-effects regression: djcg −djcg =θ1−β N−1(¯α-jcg − ¯α-jcg)+(ωjc −¯ωjc) (9) where the first parenthetical term on the right is the attenuation factor and the second is the “correct” regressor. If we estimate the fixed effects model in (8) using the within transform in (9), the rescaling of the peer variable by the attenuation factor has no impact on the estimates of the fixed effects. So the estimated fixed effects that we obtain on iteration 2minimize the NLLS objective function in (7), and our search algorithm converges for other parameters on iteration 3.43 We emphasize that this is a numerical property rather than an asymptotic result. As for asymptotic properties, in large samples our iteration 2estimate of θconverges to θ=β/1−β N−1 Therefore, θis consistent for βif β=0,itisattenuated if β<0and it is inflated if β>0.44 In our case N=9, so if, for example, β=1131, then plimn→∞ of the iteration 2estimate of θis β( 8 8−1131 )=1317. So our iteration 2estimate of the peer effect parameter is slightly inflated. Finally, consider using the estimates of equation (9) to back out iteration 2estimates of the justice ideology fixed effects. In large samples, we obtain consistent estimates of the true fixed effects, as the scaling of the peer variable has no impact on the estimated fixed effects. It follows that the iteration 3estimates are consistent for all parameters (which is apparent as they correspond to the NLLS estimates). A.2 Extension to nonlinear 2SLS Just as with the NLLS problem in (2), the nonlinear 2SLS problem in (4) involves function minimization over a large number of fixed effects and other parameters. As before, we use a simple iterative algorithm to construct the estimator. The meta-algorithm is identical to the NLLS algorithm, except we replace the OLS step with a 2SLS step: Start with a guess for the vector of peer effect variables (¯α-jc), and then solve for the parameter vector θby 2SLS. Then refine our guess of the peer effects variables until the guess coincides with the fixed effects we estimate by 2SLS. At which point, we have a solution of the nonlinear 2SLS minimization problem in (4). We now describe the algorithm in more detail: The first stage equation of 2SLS is 1 |Vjc| i∈Vjc dic =κj+λi p×1 |Ijc| i∈Ijc αi+μt(c) +X cλ+H cγ+ζjc(10) 43Note that Model 1 in the main text contains justice and term fixed effects, while Models 2 and 3 have justice-by-issue and term-by-issue fixed effects. In each case, the fixed effects wash out the contamination of the ideology measures by a justice’s own ideology on iteration 2(as described above). 44Note that tests for the existence of peer effects will still be consistent in this case, as β=0under the null (see Wooldridge (2010, pp. 158–160), where in his notation, G=0so 2SLS standard errors and test statistics are valid). Quantitative Economics 12 (2021) Peer effects on the United States Supreme Court 1015 Here, the dependent variable 1 |Vjc|i∈Vjc dic is the endogenous peer vote measure that appears in equation (3)ofSection4.1.Thetermκjis a fixed effect for justice j, while μt(c) is a fixed effect for the court term t(c),andζjc is an idiosyncratic error. As before, the term ¯α-jc =1 |Icj |i∈Icj αiis the mean ideology of peers active in case c, and this regressor is unobserved and must be updated as decsribed below. The second stage of 2SLS is simply to estimate equation (3) by OLS (or fixed effects) after substituting the predicted peer vote variable from the first stage. As before, let ¯αt -jc denote the assumed value of the peer effect variable on iteration t,andletαtdenote the vector of fixed effects estimated from equation (3) on iteration t(conditional on ¯αt -jc). The algorithm is as follows: In the first iteration, estimate the first stage equation (10) assuming that ¯α1 -jc =0(which also means λi pis not estimated). Substitute the fitted values of the peer vote variable obtained from (10) into the second stage equation (3). Then estimate (3) by fixed effects, still assuming ¯α1 -jc =0(so βid pin (3) is not estimated). In the second iteration, construct the peer effect variable using the estimated fixed effects from the first iteration, ¯α2 -jc =1 |Icj |i∈Icj α1 i. Then reestimate equations (10)and (3), using this updated value of the peer variable. Now the parameters λi pin (10)andβid p in (3) are estimated. In each subsequent iteration, construct ¯αt -jc using the fixed effects αt−1 iestimated from the prior iteration, and reestimate (10)and(3). Repeat until the estimates of the fixed effects converge to a desired degree of tolerance. This procedure again converges in three iterations: In the first iteration, we estimate the model by 2SLS ignoring the peer ideology variable. This causes the ideology fixed effects to be contaminated by the omitted peer effects. But for exactly the same reason we discussed in Appendix A.1, the fixed effects (within) transformation wipes out this contamination, up to scale. Hence, for the same reason, we obtain the ideology fixed effects that minimize the nonlinear 2SLS objective function on iteration 2. And this, in turn, allows us to obtain the optimized values of all other parameters on iteration 3. Appendix B: Endogenous case selection As we noted in Section 2, the justices select which cases the Supreme Court will hear. It is possible that the characteristics of chosen cases may depend on justice ideology. For example, a majority coalition of justices with similar ideology may seek to enshrine it’s own preferences in precedent. Winning cases thus becomes an instrumental goal. The appointment of a new justice that strengthens such a coalition may make it more willing to take on cases that are more ideological (in their favored direction), and thus offer a greater prospect of setting important precedent. But these more ideological cases are also relatively hard for such a grouping to win, that is, the more ideological a case is, the more likely is any given justice to vote in the opposite ideological direction.45 Thus, if such endogenous case selection exists, a change in the Court’s ideological composition in one direction will change the distribution of cases heard, moving the average vote of 45A corollary of this idea is that if a majority wins all cases by too large a margin, they could have chosen harder targets and still been successful. 1016 Holden, Keane, and Lilley Quantitative Economics 12 (2021) Figure 2. Endogenous case ideology selection. continuing justices in the opposite direction. This, in turn, may bias estimates of peer effects downwards. In order to shed light on whether this case selection mechanism is important, we consider the relationship between the mean Segal–Cover score of justices sitting on the court and case characteristics that are known to be viewed as particularly conservative or liberal. If case selection effects exist, then reviewing a larger number of conservative lower court decisions is behavior that would intuitively be consistent with a comparatively liberal Court. Figure 2reveals a strong relationship as hypothesized, with more liberal Supreme Court cohorts (high average Segal–Cover scores) mostly reviewing conservative lower court opinions, and vice versa. This analysis reveals an important reason to control for term fixed effects in the models in Sections 3.5 and 3.5.1. To the extent that case selection is governed by the justices jointly, case selection effects will be common (at least by issue area) within a natural court. Term dummies capture this effect, so the peer effect coefficients we report in Tables 4and 5would not be biased by endogenous case selection. Appendix C: Conservative vote share Figure 3shows that the conservative vote share has varied substantially over time. Particularly notable is the high liberal vote share during the Warren Court era from 1953-69. In the Appendix in the Online Supplementary Material (Holden, Keane, and Lilley (2021)) we re-estimate our models using only data from the post-Warren Court era. We find our results are little affected. Quantitative Economics 12 (2021) Peer effects on the United States Supreme Court 1017 Figure 3. Evolution of conservative vote share by term. References Arcidiacono, P., G. Foster, N. Goodpaster, and J. Kinsler (2012), “Estimating spillovers using panel data, with an application to the classroom.” Quantitative Economics, 421– 470. [1009] Bailey, M. A. and F. Maltzman (2011), The Constrained Court: Law, Politics, and the Decisions Justices Make. Princeton University Press. [984] Black, R. and L. Epstein (2005), “Recusals and an equally divided Supreme Court.” The Journal of Appellate Practice and Process, 7, 75–100. [992,993] Boyd, C. L., L. Epstein, and A. D. Martin (2010), “Untangling the causal effects of sex on judging.” American Journal of Political Science, 54, 389–411. [985] Carrubba, C. J. and C. Zorn (2010), “Executive discretion, judicial decision making, and separation of powers in the United States.” The Journal of Politics, 72, 812–824. [984] Coleman, J. S., E. Q. Campbell, C. J. Hobson, J. McPartland, A. M. Mood, F. D. Weinfeld, and R. L. York (1966), Equality of Educational Opportunity. U.S. Government Printing Office, Washington D.C. [981] Epstein, L. and T. Jacobi (2008), “Super medians.” Stanford Law Review, 61, 37–100. [985] Epstein, L., W. M. Landes, and R. A. Posner (2011), “Why (and when) judges dissent: A theoretical and empirical analysis.” Journal of Legal Analysis, 3, 101–137. [985] Epstein, L., A. D. Martin, K. M. Quinn, and J. A. Segal (2009), “Circuit effects: How the norm of federal judicial experience biases the Supreme Court.” University of Pennsylvania Law Review, 157, 833–880. [983,995,999,1008] Epstein, L., A. D. Martin, J. A. Segal, and C. Westerland (2007), “The judicial common space.” Journal of Law, Economics and Organization, 23, 303–325. [1008] 1018 Holden, Keane, and Lilley Quantitative Economics 12 (2021) Epstein, L., T. G. Walker, N. Staudt, S. Hendrickson, and J. Roberts (2013), “The U.S. Supreme Court justices database.” Available at http://epstein.wustl.edu/research/ justicesdata.html.[986] Fischman, J. B. (2015), “Interpreting circuit court voting patterns: A social interactions framework.” Journal of Law, Economics and Organization, 31, 808–842. [985,998,1004, 1006] Greenhouse, L. (2007), “Becoming Justice Blackmun: Harry Blackmun’s Supreme Court Journey.” Macmillan. [999] Holden, R., M. Keane, M. Lilley (2021), “Supplement to ‘Peer effects on the United States Supreme Court’.” Quantitative Economics Supplemental Material, 12, https://doi.org/ 10.3982/QE1296.[996,1016] Hume, R. J. (2014), “Deciding not to decide: The politics of recusals on the U.S. Supreme Court.” Law and Society Review, 48, 621–655. [993] Manski, C. F. (1993), “Identification of endogenous social effects: The reflection problem.” Review of Economic Studies, 60. [982,984,998] Manski, C. F. (2000), “Economic analysis of social interaction.” Journal of Economic Perspectives 14. [982] Martin, A. D. and K. M. Quinn (2002), “Dynamic ideal point estimation via Markov chain Monte Carlo for the U.S. Supreme Court, 1953–1999.” Political Analysis, 10 (2), 134–153. [984] Martin, A. D., K. M. Quinn, and L. Epstein (2005), “The median justice on the United States Supreme Court.” North Carolina Law Review, 1275–1322. [984] Miles, T. J. and C. R. Sunstein (2006), “Do judges make regulatory policy? An empirical investigation of Chevron.” University of Chicago Law Review, 73. [985] Moffitt, R. A. (2001), “Policy interventions, low-level equilibria, and social interactions.” In Social Dynamics (S. Durlauf and H. Young, eds.), 45–82, MIT Press, Cambirdge, MA. [982,983,984,998,1000] Peresie, J. L. (2005), “Female judges matter: Gender and collegial decision making in the federal appelate courts.” Yale Law Journal, 114, 1759–1790. [985] Posner, R. A. (2008), How Judges Think. Harvard University Press, Cambridge, MA. [985, 1007] Revesz, R. L. (1997), “Environmental regulation, ideology, and the D.C. circuit.” Virginia Law Review, 83, 1717–1772. [985] Segal, J. A. and A. D. Cover (1989), “Ideological values and the votes of U.S. Supreme Court justices.” American Political Science Review., 83, 557–565. [997] Skeels, C. and F. Windmeijer (2018), “On the Stock–Yogo tables.” Econometrics, 6, 1–23. [1003] Quantitative Economics 12 (2021) Peer effects on the United States Supreme Court 1019 Spaeth, H. J. and L. Epstein (2014), “Supreme Court database, version 2014.” Available at http://supremecourtdatabase.org.[986] Wooldridge, J. (2010), Econometric Analysis of Cross Section and Panel Data. MIT Press. [989,990,1002,1014] Co-editor Christopher Taber handled this manuscript. Manuscript received 18 February, 2019; final version accepted 13 January, 2021; available online 28 January, 2021.