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\newcommand{\blind}{0}

\newcommand{\tit}{Supplemental Appendix to ``Drawing Your Senator From a Jar: Term Length and
    Legislative Behavior''}

\if0\blind

{\title{\tit}

\begin{singlespace}   

  \author{ \Large Roc\'{i}o
    Titiunik\footnote{\href{mailto:titiunik@umich.edu}{\texttt{titiunik@umich.edu}},
      \mbox{\url{http://www.umich.edu/\~titiunik}}, Department of
      Political Science, University of Michigan, 5700 Haven Hall, 505 South State Street, 
      Ann Arbor, MI  48109-1045} \\
      Assistant Professor \\
     \large{Department of Political Science} \\ 
     \large{University of Michigan} }

%\date{ \begin{tabular} {ll}
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\tableofcontents
\if1\blind
\title{\bf \tit}
\maketitle
\fi

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\section{Overview} 

This supplemental appendix to the paper ``Drawing Your Senator From a
Jar: Term Length and Legislative Behavior'' is intended for online
publication only. Section 2 presents the
disaggregated data on term length that was used to produce Table 1 in
the main body of the paper. Section 3 provides the tables that were used
to produce Figures 4 through 7 in the main paper. Next, Section 4
shows power calculations based on a two-sample t-test and standard
deviations for various outcomes estimated from the data in each
session. Finally, in Section 5 I consider whether some of the
institutional features of the state senates considered in the main
paper might induce senators to engage in intertemporal strategic
behavior, which would induce a violation of the Stable Unit Treatment
Value Assumption (SUTVA) and preclude a simple policy recommendation
based on the effects reported in the paper.

\section{Term length in comparative perspective}
This section presents the information presented in Table 1 in the main body of the paper, disaggregated by country and chamber. The countries included are all countries with Polity score $\geq$ 8, obtained from the \cite{PolityIV} \citep[see also][]{MarshallGurrJaggers2013}, for which term length information was available from the \cite{IPU}. 

\input{TABLE-termlength-disaggregated-FINAL-part1.tex}
\input{TABLE-termlength-disaggregated-FINAL-part2.tex}
\clearpage

\section{Outcome tables and randomization inference balance tests}

Figures \ref{fig:balTX} and \ref{fig:balARIL} provide balance
tests analogous to those reported in Figures 1 and 2 in the main body
of the paper, but using exact (simulation-based) randomization
inference p-values instead of the $t$ distributional approximation
(see \cite{Fisher1935} and \cite{Rosenbaum2002} for details).

\begin{landscape}
\begin{figure}
\caption{Randomization Inference Balance Tests for Texas}
\label{fig:balTX}
\vspace{-0.4in}
\centering
\includegraphics[scale=0.8]{texas_balance_939503-RandInf.pdf}
\end{figure}
\end{landscape}

\begin{landscape}
\begin{figure}
\caption{Randomization Inference Balance Tests for Arkansas and Illinois}
\label{fig:balARIL}
\vspace{-0.4in}
\centering
\includegraphics[scale=0.8]{arkansas_illinois_balance_03-RandInf.pdf}
\end{figure}
\end{landscape}


Tables A2 through A5 below provide more details about the results
illustrated in Figures 4 through 7 in the main body of the paper.

\input{table-suppapp-AR.tex}
\input{table-suppapp-IL.tex}
\input{table-suppapp-TX1.tex}
\input{table-suppapp-TX2.tex}

\clearpage
\section{Power Calculations}

This section presents simple power calculations to provide some
information regarding the probability of detecting effects of
different size using the small-n experimental data analyzed in
the main body of the paper.

All calculations are illustrated graphically in Figures A3 through A11
below. All figures plot power (the probability of rejecting the null
hypothesis of no effect given that the alternative hypothesis of a
non-zero effect is true) for different values of the alternative
hypothesis. In particular, the power calculations assume a two-sample
t-test of difference means with a two-sided alternative and non-equal
sample sizes. There is one figure corresponding to each combination of
outcome and legislative session (with Texas' 1993 and 1995 analyzed
jointly as in the main paper). Each of these combinations uses the
sample sizes of the particular legislative session considered, and
estimates the standard deviation of the outcome from the data
corresponding to the particular outcome of the particular legislative
session considered.

The effect sizes in the x-axis are displayed in the natural units of
the outcomes (number of bills, proportion of abstentions, etc), but in
all figures the \textit{range} of the effect sizes goes from 0.1 to 1
standard deviations (SD) of the outcome, in increments of 0.05
SDs. The figures thus show the different effect sizes that are
needed for every combination of sample size to produced effects within
0.1 and one standard deviations.

For example, Figure A3 (which corresponds to the scenario in Arkansas
for abstention rates) shows that when there are 17 and 18 observations
and the SD is equal to 0.03, the power for an effect of 0.1 SD is well
below 0.2 (first point in the graph, corresponding to an absolute-value difference in
abstention rates below 0.005), but for an effect of 1
SD power is above 0.80 (last point in the graph, roughly an absolute-value difference
in abstention rates equal to 0.030). All other figures can be
interpreted analogously.

\begin{figure}
\caption{Power Calculations Based on Abstention Rates in Arkansas}
\centering
\includegraphics[scale=0.8]{../../joint-analysis/output/power-Arkansas-AbstentionRateLV-effect.pdf}
\end{figure}


\begin{figure}
\caption{Power Calculations Based on Abstention Rates in Texas}
\centering
\includegraphics[scale=0.8]{../../joint-analysis/output/power-Texas-AbstentionRateLV-effect.pdf}
\end{figure}

\begin{figure}
\caption{Power Calculations Based on Abstention Rates in Illinois}
\centering
\includegraphics[scale=0.8]{../../joint-analysis/output/power-Illinois-AbstentionRateLV-effect.pdf}
\end{figure}

\begin{figure}
\caption{Power Calculations Based on Bills Introduced in Arkansas}
\centering
\includegraphics[scale=0.8]{../../joint-analysis/output/power-Arkansas-NumberBillsIntroducedLV-effect.pdf}
\end{figure}


\begin{figure}
\caption{Power Calculations Based on Bills Introduced in Texas}
\centering
\includegraphics[scale=0.8]{../../joint-analysis/output/power-Texas-NumberBillsIntroducedLV-effect.pdf}
\end{figure}

\begin{figure}
\caption{Power Calculations Based on Bills Introduced in Illinois}
\centering
\includegraphics[scale=0.8]{../../joint-analysis/output/power-Illinois-NumberBillsIntroducedLV-effect.pdf}
\end{figure}

\begin{figure}
\caption{Power Calculations Based on NOMINATE Scores in Arkansas}
\centering
\includegraphics[scale=0.8]{../../joint-analysis/output/power-Arkansas-NOMINATEScoreLV-effect.pdf}
\end{figure}


\begin{figure}
\caption{Power Calculations Based on NOMINATE Scores in Texas}
\centering
\includegraphics[scale=0.8]{../../joint-analysis/output/power-Texas-NOMINATEScoreLV-effect.pdf}
\end{figure}

\begin{figure}
\caption{Power Calculations Based on NOMINATE Scores in Illinois}
\centering
\includegraphics[scale=0.8]{../../joint-analysis/output/power-Illinois-NOMINATEScoreLV-effect.pdf}
\end{figure}

\begin{comment}
%%%%%%%% In terms of  SD
\begin{figure}
\caption{Power Calculations Based on Abstention Rates in Arkansas}
\centering
\includegraphics[scale=0.8]{../../joint-analysis/output/power-Arkansas-AbstentionRateSD-effect.pdf}
\end{figure}


\begin{figure}
\caption{Power Calculations Based on Abstention Rates in Texas}
\centering
\includegraphics[scale=0.8]{../../joint-analysis/output/power-Texas-AbstentionRateSD-effect.pdf}
\end{figure}

\begin{figure}
\caption{Power Calculations Based on Abstention Rates in Illinois}
\centering
\includegraphics[scale=0.8]{../../joint-analysis/output/power-Illinois-AbstentionRateSD-effect.pdf}
\end{figure}

\begin{figure}
\caption{Power Calculations Based on Bills Introduced in Arkansas}
\centering
\includegraphics[scale=0.8]{../../joint-analysis/output/power-Arkansas-NumberBillsIntroducedSD-effect.pdf}
\end{figure}


\begin{figure}
\caption{Power Calculations Based on Bills Introduced in Texas}
\centering
\includegraphics[scale=0.8]{../../joint-analysis/output/power-Texas-NumberBillsIntroducedSD-effect.pdf}
\end{figure}

\begin{figure}
\caption{Power Calculations Based on Bills Introduced in Illinois}
\centering
\includegraphics[scale=0.8]{../../joint-analysis/output/power-Illinois-NumberBillsIntroducedSD-effect.pdf}
\end{figure}

\begin{figure}
\caption{Power Calculations Based on NOMINATE Scores in Arkansas}
\centering
\includegraphics[scale=0.8]{../../joint-analysis/output/power-Arkansas-NOMINATEScoreSD-effect.pdf}
\end{figure}


\begin{figure}
\caption{Power Calculations Based on NOMINATE Scores in Texas}
\centering
\includegraphics[scale=0.8]{../../joint-analysis/output/power-Texas-NOMINATEScoreSD-effect.pdf}
\end{figure}

\begin{figure}
\caption{Power Calculations Based on NOMINATE Scores in Illinois}
\centering
\includegraphics[scale=0.8]{../../joint-analysis/output/power-Illinois-NOMINATEScoreSD-effect.pdf}
\end{figure}
\end{comment}

\clearpage
\section{SUTVA and the possible limitations imposed by strategic interactions}
If senators interacted strategically with each other, the Stable
Treatment Unit Value Assumption (SUTVA) might be violated. If this
happens, the policy implications of the experimental results presented
in the main body of the paper would need to be revised. When SUTVA
holds, units do not interfere with each other, in the sense that the
outcome of every unit is solely affected by the treatment received by
that unit, regardless of the treatment status assigned to the rest of
the units participating in the experiment. Formally, letting
$\mathbf{T} = \left(T_1,T_2,\cdots,T_N\right)$ and $\mathbf{T^\prime}
= \left(T_1^\prime,T_2^\prime,\cdots,T_N^\prime\right)$ be two
$N$-dimensional vectors that contain the treatment assignment of each
of the $N$ units in the experiment, SUTVA is defined as
$Y_i^\mathbf{T} = Y_i^\mathbf{T^\prime}$ if $T_i = T_i^\prime$
\citep{Rubin1990}.  In the experiments considered in the main paper,
SUTVA requires that a senator assigned a four-year term behave
identically whether all other senators are assigned two-year terms,
all other senators are assigned four-year terms, or any other possible
arrangement of treatment assignment among the remaining senators.

Recent theoretical results developed by \citet{MuthooShepsle2010},
\citet{ShepsleVanHouwelingDickson2004} and
\citet{ShepsleVanHouwelingAbramsHanson2009}, suggest that the
staggered structure of terms might induce strategic intertemporal
interactions between legislators. The authors study a
divide-the-dollar game in a staggered legislature, where each
legislator serves a term of three sessions and a third of the members
faces reelection at the end of each session. The model shows that in a
staggered legislature, an equilibrium can be sustained in which the
different generations of legislators engage in an intertemporal deal
according to which in every period the amount to redistribute is
shared \emph{only} among those senators who are facing reelection in
that period. In this equilibrium, legislators agree to forgo resources
early in their terms to aid the generations that are seeking
reelection, a behavior that is later reciprocated by their fellow
legislators (given a credible punishment regime) and hence allows them
to concentrate or ``backload'' all benefits in their last period, just
before they face reelection.

This theoretical argument is of direct relevance to the term
experiments considered in the main body of the paper. If senators
engaged in an intertemporal distributional agreement of this sort, the
behavior of four-year term senators would not be the behavior that
they would have exhibited if the \emph{entire} senate had been elected
for four years. The estimated difference between two-year and
four-year senators would reflect not only the effect of serving a
shorter term, but would also include the compensating effect of the
four-year senators, who would be ``allowing'' two-year senators to
behave in the observed way, under the agreement that they would be
reciprocated in future sessions.  In this case, one could no longer
define a \emph{pair} of potential outcomes
$\left(Y_{i0},Y_{i1}\right)$ for each senator since individual
outcomes would depend on the entire vector of treatment
assignments. That is, $Y_i^\mathbf{T} \neq Y_i^\mathbf{T^\prime} \: \:
\mbox{with} \: \: T_i = T_i^\prime$ and SUTVA would be violated. On
the other hand, if these intertemporal deals were not occurring, the
results of the experiment would be directly informative of the
possible effects decreasing term length for all members of the
legislature.

This discussion illustrates that even when the treatment of interest
is randomly assigned, understanding the incentives imposed by the
institutional structure in which individuals interact is crucial to
the correct interpretation of the parameters under study. To
approximately test whether senators are behaving according to this
model, I take advantage of the ``placebo'' experiment that occurs in the
Texas Senate. As explained in the paper, in 1993, Texas senators were randomly assigned to
two-year and four-year terms in the \emph{middle} of the legislative
session and, as a consequence, during the first half of the session
they ignored who would be running first for reelection. These periods
during which \emph{all} senators ignored when their reelection would
occur can be used as a baseline to estimate their behavior when
intertemporal deals are \emph{not} occurring --under the assumption
that when senators ignore whose reelection is coming first an
intertemporal deal is impossible to reach. This is a plausible
assumption, since before terms are assigned there is no way to know
who should be receiving the benefits first, and so ``backloading'' is
undefined.

\begin{comment}
For the purposes of evaluating the plausibility of the
model, the most important piece is the punishment mechanism. The
theoretical results show that a backloading equilibrium \emph{can} be
sustained, but this is not the only possible equilibrium. This
equilibrium will occur only if the appropriate punishment regime can be
implemented. This punishment regime must essentially punish all
off-the-equilibrium-path behavior; for example, if a coalition of
legislators manages to pass a proposal that give them resources before
their reelection session, the punishment regime must be such that this
behavior is punished in the \emph{future} sessions. But note that this
assumes that legislators do not retire or lose elections. If this
assumption is relaxed and the turnover rate is allowed to be greater
than zero, implementing the punishment mechanism becomes a very
demanding requirement, since the agreement must supersede individual
legislators and be applied to their seats.  Thus, one necessary
condition for the achievement of this equilibrium appears to be an
institution that can ensure the punishment regime even under positive
turnover.
\end{comment}


During these ``blind'' periods when senators ignore when they will run
for reelection, their behavior will not be contaminated by
intertemporal deals. Thus, one can observe whether after terms have
been assigned both groups deviate from their previous behavior in the
blind period. If one observed that, for example, abstention rates
increased for the two-year group and simultaneously decreased for the
four-year group relative to the blind period abstention rates, then
this would provide evidence in favor of the existence of intertemporal
logrolling.
\begin{comment}
\footnote{Note that observing a difference for both groups
  between the blind period and the period when all reelection dates
  are known could be interpreted not only as evidence of an
  intertemporal deal, but also as an uncertainty effect. Indeed, if
  the optimal behavior under a 2-year term is to set $y=y_2$ and the
  optimal behavior under a 4-year term is to set $y=y_4$, rational
  senators would incorporate the uncertainty during the blind period
  and have $y=1/2 y_2 + 1/2 y_4$ (assuming equal probability of being
  assigned to either group).  After the uncertainty resolves, we would
  observe 2-year senators having $y=y_2$ and 4-year senators having
  $y=y_4$. Both groups would alter their behavior after terms are
  revealed, but this change would not be due to an intertemporal deal
  being put in place, but rather to the resolution of previous
  uncertainty. I thank Bob Powell for this observation.} 
\end{comment}
If, on the other hand, the abstention rates of the two-year group
increased but the abstention rates of the other group remained at the
same blind-period levels, this would be evidence against the existence
of intertemporal deals. In general, finding no difference between the
two periods for \emph{only one} of the groups and not the other is the
strongest piece of evidence against intertemporal deals, since these
deals are by definition zero-sum and require that one group give up
resources in favor of the other. 

Remarkably, as shown in Table \ref{tab:suppappTXPL}, average
abstention rates among senators assigned 4-year terms are almost
exactly identical before and after the assignment (2.62\% versus
2.69\%). In contrast, senators assigned 2-year terms more than double their
abstention rates after learning that they will only serve for
two years. In other words, senators do not change their behavior if
they receive long terms, but they do if they receive short terms. This
limited piece of evidence is not at all compatible with the kind of
SUTVA violation discussed above, according to which both groups should
change their behavior to implement an intertemporal deal.

\input{table-suppapp-TXPL.tex}

\begin{comment}
Taking
advantage of a placebo test that occurs in the Texas Senate in 1993, I
provide evidence, to the extent possible, that these intertemporal
deals do not seem to be a major factor in these legislatures and thus
SUTVA seems to hold.  This is reassuring, since it means that the
results here can help understand what would happen if terms changed
for \emph{all} members of a legislature simultaneously. Nevertheless,
the discussion in this section should make clear that SUTVA is not a
minor assumption, and that an experiment is not a substitute for a
careful analysis of the institutional conditions surrounding the
individuals under study.
\end{comment}

\clearpage 
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\pdfbookmark[1]{References}{sec:references}
\bibliography{./causalinf.bib,./redistricting.bib,./lots.bib,./lots2.bib}
%\bibliography{../../../bibliography/main.bib}
%\end{singlespace} 

\end{document}
