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\begin{document}

\newpage

\section*{Appendix A: Incumbent Platforms Over Time}

In an effort to provide some empirical justification for our assumption that incumbent members of Congress adopt ideologically consistent platforms over short periods of time, we again used survey data from Project Vote Smart, this time to characterize the platform locations of incumbents. Specifically, we examined all incumbents who completed the questionnaire at least twice between 1996 and 2006. Using this criterion, we estimated platform locations for 228 unique legislators, which represents nearly one-third of all unique members (708) that served in the 105th-110th Congresses. Because some legislators completed the survey more than twice, we generated a total of 712 platform estimates for incumbents across the six election years. Furthermore, this enabled us to make additional comparisons between platforms that were chosen more than two years apart. In all, this provided a total of 1,153 comparisons between platforms chosen by the same candidate either two (497), four (312), six (191), eight (109), or ten (44) years apart.

Figure A.1 below shows the platform estimates for incumbents in elections \emph{t} and \emph{t+1}. The figure shows that the overwhelming majority of points lie along or very near to the 45 degree line, indicating no difference in platforms between subsequent elections. Indeed, the overall correlation between platform estimates across two elections is 0.97. Similarly, Poole and Rosenthal (1997, 72) show that in the post-World War II House, the correlation between a member's roll call voting record in Congress \emph{t} and \emph{t}+1 is 0.96. And, just as Poole and Rosenthal report modest declines in correlations over roll call records that are separated by greater amounts time, we find a slight dropoff in the correlations when comparing platforms across longer stretches of time:\ the correlations are 0.97, 0.97, 0.96, 0.94, and 0.92 for elections held two, four, six, eight, and ten years apart, respectively. Perhaps the more important point, though, is how strong the correlations remain even when elections are separated by as many as ten years. For the purposes of the repeated-elections analysis, however, we assume only that incumbent platforms are consistent across successive election cycles.

Using these scores, we also performed several multivariate analyses to identify whether there are any systematic predictors of platform consistency.  Specifically, we regressed the change in platform estimate from time 1 to time 2 on a battery of independent variables that could plausibly be related to changes in platform location, including the platform estimate from election 1, party affiliation, length of tenure in office, whether the member was a first-term incumbent, and the number of years that separated the two elections. We also included state fixed effects to account for systematic differences across states, and fixed effects for each pair of elections (e.g., whether we compared 1996 with 1998, 1996 with 2000, 1996 with 2002, and so forth). Across all of these variables, we found no evidence that platform consistency is correlated with any of these factors. The sole exception is that change was larger for members who completed the survey in 2000 and 2002$-$when redistricting occurred and incumbents likely had slightly different constituencies. This change also could have been affected by the aftermath of the 9/11 attacks and the newly initiated war on terror.

On the basis of these two sets of findings, we find the evidence persuasive that incumbents adopt ideologically consistent platforms across successive elections. 


\begin{figure}[H]
\begin{center}
Figure A.1: Ideological Consistency in Incumbent Platforms
\includegraphics[width=6in]{platformcompare.pdf}
\end{center}
\vspace{-0.25in}
\begin{footnotesize}
The x-axis is the incumbents' estimated platforms in an initial election, and the y-axis represents the platform estimate in a subsequent election. The 45 degree line represents the locations of platforms that are identical across two election cycles. The correlation between the two measures is 0.97.
\end{footnotesize}
\end{figure}



\section*{Appendix B: Robustness Checks}



\begin{table}[H]
\begin{center}
Table B.1: Ideological Extremity and Challenger Vote Shares
\begin{tabular}{lD{.}{.}{-1.2}D{.}{.}{-1.2}}
 \toprule
 Independent Variables  & \multicolumn{0}{c}{(1)} & \multicolumn{0}{c}{(2)}\\
 \midrule
Challenger extremity   & -0.14 & -0.11\\
 & (0.49) & (0.47) \\ [\defaultaddspace]
Incumbent extremity &  & 8.80\\
 & & (1.90) \\ [\defaultaddspace]
Quality challenger  & 2.02 & 2.23\\
 & (0.64) & (0.64) \\ [\defaultaddspace]
Presidential vote share  & -0.40 & -0.47\\
 & (0.03) & (0.03) \\ [\defaultaddspace]
Spending difference  & 0.85 & 0.81\\
 & (0.10) & (0.09) \\ [\defaultaddspace]
In-party & 1.47 & 0.21\\
 & (0.53) & (0.57) \\ [\defaultaddspace]
First term incumbent & 3.35 & 3.05\\
 & (0.67) & (0.68) \\ [\defaultaddspace]
1998  & -0.57 & -0.74\\
 & (0.53) & (0.52)\\ [\defaultaddspace]
2000  & -2.17 & -2.40\\
 & (0.66) & (0.66) \\ [\defaultaddspace]
2002  & -2.54 & -3.26\\
 & (0.72) & (0.72)\\ [\defaultaddspace]
2004  & 0.34 & -0.27\\
 & (0.55) & (0.58)\\ [\defaultaddspace]
2006 & 1.82 & 1.17\\
) & (0.73) & (0.73)\\ [\defaultaddspace]
Intercept  & 58.46 & 59.49\\
 & (2.37) & (2.28)\\ [\defaultaddspace]
\midrule
N  & \multicolumn{0}{c}{725} & \multicolumn{0}{c}{725}  \\
Clusters  & \multicolumn{0}{c}{177} & \multicolumn{0}{c}{177} \\
%$\hat{\sigma}_{year}$ & 3.67 & 2.33 & 2.38 & 2.35 & 1.47 & 1.43 & 1.28\\
MSE & 5.36 & 5.25\\ 
\bottomrule
\end{tabular}
\label{tab:cross}
\end{center}
\vspace{-.1in}
\begin{footnotesize}
Entries are linear regression coefficients and standard errors (clustered by congressional district).  The dependent variable is the challenger's vote share (in percentage points)}.
\end{footnotesize}
\end{table}

\newpage

\begin{table}[H]
\begin{center}
Table B.2: The Effect of Challenger Positioning on Vote Share -- Partisan Differences\\
\scalebox{1}{
\begin{tabular}{lD{.}{.}{-1.2}D{.}{.}{-1.2}D{.}{.}{-1.2}D{.}{.}{-1.2}}
 \toprule
 & \multicolumn{2}{c}{Democrats} & \multicolumn{2}{c}{Republicans} \\
 & \multicolumn{1}{c}{(1)} & \multicolumn{1}{c}{(2)}  & \multicolumn{1}{c}{(1)} & \multicolumn{1}{c}{(2)} \\
  \midrule
Moderation & -0.71 & -0.77 & -0.26 & 0.07\\
 & (0.57) & (0.57) & (0.56) & (0.53)\\ [\defaultaddspace]
Change in spending & 0.05 & 0.05 & 0.07 & 0.06\\
\hspace{0.1in} (ten thousands) & (0.01) & (0.01) & (0.01) & (0.01) \\ [\defaultaddspace]
One term incumbent & -1.24 & -1.36 & -3.18 & -2.04\\
 & (1.16) & (1.18) & (1.27) & (1.13)\\ [\defaultaddspace]
Incumbent extremity & -3.07 & -3.61 & 4.91 & 0.99\\
 & (3.69) & (3.75) & (2.11) & (1.94)\\ [\defaultaddspace]
Vote share,  & -0.21 & -0.20 & -0.50 & -0.44\\
\hspace{0.1in} election 1 & (0.04) & (0.04) & (0.08) & (0.09) \\ [\defaultaddspace]
Change in quality & 2.07 & 2.03 & 0.75 & 0.69 \\
 & (1.17) & (1.14) & (0.76) & (0.66) \\ [\defaultaddspace]
(Intercept)  & 6.16 & 5.16 & 16.07 & 13.54\\
 & (2.55) & (2.62) & (3.46) & (3.72)\\ [\defaultaddspace]
 \midrule
N & \multicolumn{0}{c}{203} & \multicolumn{0}{c}{203} & \multicolumn{0}{c}{241} & \multicolumn{0}{c}{241} \\
Clusters & \multicolumn{0}{c}{116} & \multicolumn{0}{c}{116} & \multicolumn{0}{c}{129} & \multicolumn{0}{c}{129} \\
MSE & 4.91 & 4.92 & 5.72 & 5.32\\ 
\midrule
Year fixed effects & \multicolumn{0}{c}{No} & \multicolumn{0}{c}{Yes} & \multicolumn{0}{c}{No} & \multicolumn{0}{c}{Yes}
\bottomrule
\end{tabular}}
\label{tab:main2}
\end{center}
\vspace{-.1in}
\begin{footnotesize}
Entries are linear regression coefficients, standard errors (clustered by congressional district).  The dependent variable is the change in challenger vote share (in percentage points) between elections \emph{t} and \emph{t}+1, where positive values indicate that the challenger in election \emph{t}+1 received a larger share of the vote than the challenger in election \emph{t}. Year fixed effects were also estimated where indicated but are not shown. 
\end{footnotesize}
\end{table}

\end{document}
