PacDev.tex

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\title{The Controlled Choice Design and Paternalism in Pawnshop Borrowing}
\author{Craig McIntosh\inst{1} \and Isaac Meza\inst{2} \and Joyce Sadka\inst{3} \and Enrique Seira\inst{4} \and Francis J. DiTraglia\inst{5} }
\institute[UTran]{\inst{1} UCSD, \inst{2} Harvard , \inst{3} ITAM , \inst{4} MSU , \inst{5} Oxford} 
                      
\date{PACDEV, March 2023}

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\begin{document}


\begin{frame}[c, noframenumbering]%{\phantom{title page}}
% The \phantom{title page} is a kludge to get the red bar on top
\titlepage
\end{frame}


\section{Motivation}



\begin{frame}{Motivation: Private paternalism}
\begin{itemize}
    \item Many institutions —firms, schools, financial contracts— restrict choice using built-in commitment mechanisms which help workers, students, borrowers overcome self-control problems
    \begin{itemize}
        \item Loans with fixed repayment schemes, homework due dates, etc.
    \end{itemize}
    \item At the same time these firms hide these forcing mechanisms and don’t market their commitment features, potentially because demand for them is low.
     \item  Laibson (2018) argues that clients that benefit from commitment may underestimate its value, and that in such cases private paternalism (i.e. allocating them to commitment) could be beneficial.
    \item  \textbf{We study the (a) benefits of imposing a structured repayment contract on financing cost, (b) whether there is demand for such structure, and (c) whether non-takers of such a commitment product would benefit from taking it.}
\end{itemize}

\end{frame}





\begin{frame}{Context}
\begin{itemize}
     \item Pawn loans involve borrowers leaving valuable liquid assets as collateral in exchange for an immediate cash loan
     \item The loan is overcollateralized (loan is 70\% of appraised value) and collateral is liquid.
    \begin{itemize}
        \item The lender approves loans in a few minutes without income or credit history check $\rightarrow$ used for emergencies.
    \end{itemize}
  \item Because the loan is overcollateralized and collateral is liquid, the lender may gain if the borrower defaults on the loan, especially if the borrower pays towards recovery on the way to default.
     \item Among those that lose their pawn (44\%), 48\% paid a positive amount towards its recovery and on average paid 42\% of loan.
     \item This happens in spite (or because?) of 74\% of borrowers reporting a 100\% subjective probability recovery ex-ante. 
    %\vfill   \item 13\% of borrowers are classified as present biased using the simple standard question.
        \item In such an environment, one may ask if putting more structure/committment in payments may help borrowers recover their pawn.
      \item Not only that: Will the structure/commitment be demanded by all (or most) people who benefit?
        

\end{itemize}
\end{frame}


\section{Context}


\begin{frame}{Context}
\begin{columns}
\begin{column}{.45\textwidth}
\begin{figure}[H]
    \begin{center}
    \caption{Pawnshop}
        \includegraphics[width=0.95\textwidth]{Figuras/empenio2_.png}
    \end{center}
    \end{figure}
\begin{figure}[H]
    \begin{center}
    \caption{Appraisers  inside a pawnshop}
        \includegraphics[width=0.95\textwidth]{Figuras/empenio9_.png}
    \end{center}
    \end{figure}    
    \end{column}
\begin{column}{.45\textwidth}
\begin{figure}[H]
    \begin{center}
    \caption{Waiting for appraisal}
        \includegraphics[width=0.95\textwidth]{Figuras/empenio11_.png}
    \end{center}
    \end{figure}
\begin{figure}[H]
    \begin{center}
    \caption{Lost pawns which are for sale}
        \includegraphics[width=0.95\textwidth]{Figuras/empenio3_.png}
    \end{center}
    \end{figure}    
    \end{column}    
    \end{columns}
\end{frame}



\section{Contribution}

\begin{frame}{Methodological Contribution}
\begin{itemize}
 %\vfill \item A context of particular interest to behavioral literature:  the demand for commitment in financial contracts (Laibson 1997, Bryan et al. 2010), and many more.
 
   \vfill   \item Key quantity in debate about paternalism:  impacts on those who wouldn't elect to take the program versus impacts on those that do.   \begin{itemize}
       \item Oreopoulos 2006: effect of mandatory high school laws.
       \item Fowlie et al. 2021: defaulting customers into variable electricity pricing.
   \end{itemize} 
   
   
   
   
   %How do treatment effects relate to selection?  
    
    %Broadly speaking there are two approaches: the structural approach and the reduced form approach.

    %The structural approach combines instruments with a generalized Roy model and uses behavioral and statistical restrictions to extrapolate the causal effects for different sub-populations
    
    
    %The structural approach allows both selection on unobservables and selection on gains at the cost of modeling these channels. In contrast, the reduced form approach assumes that there is no selection on gains after conditioning on a set of covariates 


    \vfill \item  Large literature TE heterogeneity: \begin{itemize}
        \item LATE-and-reweight:  Aronow \& Carnegie (2013); Angrist \& Fernandez Val (2013). \alert{Assume no selection on gains}
        \item Structurally model selection:  Walters (2018) \alert{Need their parametric models to be correct}.
        %TUT and ATE implied by LATE-and-re-weight differ sharply from his model-based estimates
        \item MTE: Heckman \& Vytlacil MTE; Cornelissen et. al. (2018). \alert{Need instrumental variable with rich support, and separability of observed and unobserved determinants of TE.}
        %\item Brinch et. al. (2017) - discrete $Z$ but under some additivity restrictions on the MTE curve
        %\item Mogstad, Santos \& Torgovitsky (2018) - No parametric form assumption on MTE curve but only partial identification.
    \end{itemize}

    \vfill \item Our ``Controlled Choice'' design point identifies a number of relevant TE with mild assumptions
    \begin{itemize}
        \item Because we have two forcing-arms and a choice arm, we can use one IV to get at ToT, and a second IV to get at TuT.
    \end{itemize}
    
    
   \vfill  \item Consider winners and losers from paternalism.     

%   \vfill \pause  \item Three-arm design:  Control, Choice (voluntary takeup=ITT), Forcing (universal=ATE).  We illustrate how to use standard exclusion restrictions to point identify:
%     \begin{itemize}
%         \item Treatment on the Treated (ToT) and 
%         \item Treatment on the Untreated (TUT), also
% 	\item Average Selection on Gains, Average Selection Bias, and Average Selection on Levels
%     \end{itemize} 
    
  


\end{itemize}
\end{frame}





\begin{frame}{Treatment arms}
\label{treatment_arms}
   \begin{itemize}
    \vfill \item Randomization at the branch-day level.  Analysis at the pawn level.
    \begin{itemize}
        \vfill \item Control arm (1770 obs)
       \vfill \item Forced Commitment arm (1954 obs)
       \vfill \item Choice Commitment arm (2580 obs)
    \end{itemize}
    \vfill \item The existence of a choice arm allows us not only to measure if there is demand for such a contract, but who demands it, not only in demographic terms, but in terms of potential treatment effects (which include unobservables).
   \vfill \item  This design is innovative and critical for our purposes, as it enables us to explore whether or not forcing people into a structured payment contract could be more beneficial than allowing choice for a significant fraction of them.
\end{itemize}
\end{frame}






\begin{frame}{Data}
\label{data}
\begin{itemize}
    \item Administrative data: 1 month before and 8 months after the experiment ended
    \begin{itemize}
        \item Unique identifier for each client and each pawn.
        \item Value of the item, money loaned (70\%), date of pawning
        \item For all payments: date and amounts
        \item Fees incurred
        \item Whether the client lost the pawn, renewed the contract
    \end{itemize} 

    \vfill \item Survey data
    \begin{itemize}
        \item During experiment, we asked clients to complete a 5-minute survey before going to the teller window to appraise their piece and before treatment status was known to them.
        \item Demographics, proxies for income/wealth, education, present-biased preferences, experience pawning, if family or friends commonly asked for money, cost of going to branch, the subjective probability of recovering, the subjective value, etc.
    \end{itemize}
\end{itemize}    
\end{frame}





\begin{frame}{Main outcomes: financial cost and default}
\label{fc_outcome}
\begin{itemize}
    \item We are interested in measuring the financial cost of borrowing, which very saliently includes the cost of defaulting on the loan and losing the pawn.
    \item We will measure loan default using an indicator $\mathds{1}(\text{Default}_i)$, and the cost in pesos using the following definition that capture borrower outlays:
\end{itemize}

\begin{align*}
    \text{Financial Cost}_i =&  \sum_t P^I_{it} +\sum_t P^F_{it}  
     + \mathds{1}(\text{Default}_i) \times \left(\text{Value}_i-\text{Loan}_i + \sum_t P^c_{it}\right)
\end{align*}

\vspace{.2in}
\: \: \: \: APR: equivalent yearly interest generated by sum of payments:
   \begin{equation*}
    (\text{APR})_i =\left( 1 + \frac{\frac{\text{Financial Cost}_i}{\text{Loan Value}_i}}{\text{loan term}_i}\right)^{\text{loan term}_i}-1 
\end{equation*}

   \vspace{.2in}
\begin{itemize}
    \item We \alert{will not talk about welfare}, only financial cost.
    \begin{itemize}
        \item Welfare would depend on reduced liquidity, anxiety from monthly payments, cost of going to branch, etc. We don't observe these.
    \end{itemize}
\end{itemize}
\end{frame}










\begin{frame}{Experimental integrity}
\label{experimental_integrity}
    \begin{table}[H]
\caption{Attrition table}
\label{attrition_table}
\begin{center}
\small{\input{./Tables/Attrition.tex}}
\end{center}
\end{table}
\end{frame}








\section{Main Results}
\begin{frame}{Main results: ITT}
\label{main_results}


\begin{itemize}
    \item  Takeup of commitment in Choice arm is only 11\%.
    \item Financial cost reduced by \$204 pesos (20\% of mean). That is: charging fee + structure \textit{decreased} cost.
    \item Default decreases by 6.5pp (15\% of mean), APR decreases by 11 points, from mean of 57\%.
\end{itemize}
\vspace{.3in}
\begin{table}[H]
\begin{center}
\resizebox{0.95\textwidth}{!}{
\small{\input{./Tables/decomposition_main_te.tex}}
}
\end{center}
\end{table}

   \vfill 
  %\hyperlink{several_def_cost}{\beamerbutton{More}}

\end{frame}








\begin{frame}{Intermediate outcomes (1)}
\label{mechanism_appendix}

\begin{table}[H]
\caption{Intermediate outcomes}
\begin{center}
\scriptsize{\input{./Tables/mechanism_pres1.tex}}
\end{center}
\end{table}
\vfill
%\hyperlink{intermediate_outcomes}{\beamerbutton{Back}}
 \begin{itemize}
     \item \textbf{Speed of payment:} The first payment of the forced commitment contract  occurs 14 days earlier, increasing the fraction who recover on the first visit by 7.9 pp. Conditional on recovering, they recover 28 days faster.
\end{itemize}
\end{frame}




\begin{frame}{Intermediate outcomes (2)}

\begin{table}[H]
\caption{Intermediate outcomes}
\begin{center}
\footnotesize{\input{./Tables/mechanism_pres2.tex}}
\end{center}
\end{table}

%\hyperlink{intermediate_outcomes}{\beamerbutton{Back}}
   \vfill \item  \textbf{Separating those who will not pay:} Commitment contract decreases fraction of borrowers making a payment and \textit{not} recovering by 7pp.
    \begin{itemize}
        \item Actually, among those that default, those in the forced commitment arm are 14pp less likely to have paid towards recovery. They also visit the branch less: as if the commitment contract makes them realize they will end up losing pawn anyway, and stop paying early on.
    \end{itemize}
\end{frame}







%\begin{frame}{Intermediate outcomes: forced-commitment}
%\label{intermediate_outcomes}
% \begin{itemize}
 %    \item \vfill \textbf{Speed of payment:} The first payment of the forced commitment contract  occurs 13 days earlier, increasing the fraction who recover on the first visit by 7.9 pp. Conditional on recovering, they recover 28 days faster.
     
  %  \item \vfill \textbf{Separating those who will not pay:} Commitment contract decreases fraction of borrowers making a payment and \textit{not} recovering by 7pp.
   % \begin{itemize}
    %    \item Actually, among those that default, those in the forced commitment arm are 14pp less likely to have paid towards recovery. They also visit the branch less: as if the commitment contract makes them realize they will end up losing pawn anyway, and stop paying early on.
    %\end{itemize}
    %\end{itemize}

  % \vfill
   
%   \hyperlink{mechanism_appendix}{\beamerbutton{Details}}
%\end{frame}









 
\begin{frame}{Using a Double Exclusion Restriction}

Our 3-armed experiment + dual exclusion restriction allows us to estimate: 
\begin {itemize}
 \item Treatment Effect on the Treated (ToT), as with standard IV on one-sided non-compliance.
 \item Treatment on the Untreated (TUT), symmetric instrumentation for \textit{not} complying as compared to forcing arm.
\end{itemize}
\vspace{.2in}


Assumptions:

\begin{itemize}
    \item $Z \independent (Y_0,Y_1,C)$, achieved by randomization
    \item $D = \mathbbm{1}(Z_i \neq 2) Z + \mathbbm{1}(Z_i=2) C$, i.e. \alert{being assigned to a contract has same effect as choosing it} (e.g. used in compulsory school attendance and returns to schooling lit).
\end{itemize}

\vspace{.2in}

Prop:
\begin{itemize}
    \item ToT := $\mathbbm{E}(Y_{i1} - Y_{i0} | C_i = 1)$ is point-identified, and equals $\frac{\mathbbm{E}(Y_i|Z_i=2) - \mathbbm{E}(Y_i|Z_i =0)}{\mathbbm{E}(D_i|Z_i=2)} $
     \item TuT := $\mathbbm{E}(Y_{i1} - Y_{i0} | C_i = 0)$ is point-identified, and equals $\frac{\mathbbm{E}(Y_i|Z_i=1) - \mathbbm{E}(Y_i|Z_i =2)}{1-\mathbbm{E}(D_i|Z_i=2)} $
     \item Also identified: 
     \begin{itemize}
         \item Average Selection on Gains $ASG:=ToT-TuT$
         \item Average Selection Bias $ASB:= \E[Y_0 | C=1]-\E[Y_0 | C=0]$
         \item Average Selection on Levels $ASL:= \E[Y_1 | C=1]-\E[Y_1 | C=0]$
     \end{itemize}
\end{itemize}
\end{frame}





\begin{frame}{The ``Controlled Choice'' Design}
\label{cc_design}
\begin{itemize}
    \item Recall that under one-sided non-compliance (no always-takers), LATE=ToT.
\end{itemize}
\vspace{.2in}
\begin{figure}[H]
    \begin{center}
        \centering
        \includegraphics[width=1.0\textwidth]{Figuras/tot_tut_intuition.png}
    \end{center}
\end{figure}   
 \vfill
%\hyperlink{identification_randomized_choice}{\beamerbutton{Identification}}
\end{frame}




\begin{frame}{On avg. those that do not choose would benefit: TuT $>$ 0}
\begin{itemize}
    \vfill \item Commitment increases financial benefit even for borrowers who don't choose  it.
    \vfill \item APR, FC ToT>TuT, but cannot reject equality due to large s.e. in ToT.
\end{itemize}
\vspace{.2in}
\begin{table}[H]
%those who are most likely to benefit from it and those whose outcomes are most adverse under the status quo.
\label{tot_tut}
\begin{center}
\small{\input{./Tables/tot_tut.tex}}
\end{center}
\end{table}
\end{frame}

% Despite substantial treatment effect heterogeneity, most borrowers would experience higher financial benefits under a commitment contract : $F_{\operatorname{TUT}}^{-1}(0.72)>0$









\begin{frame}{If commitment works, why don't people choose it?}
    \begin{itemize}
         \onslide<1>{\item Need to learn about it}
         \onslide<2>{\item Discounting}
        \onslide<3>{\item Present-Bias}
    \end{itemize}

\vfill


\only<1>{  
 

    \textbf{Do people learn from Forced exposure to the program that they benefit from it? } 
    \vspace{.1in}
    \begin{itemize}
    \item  Small sample (small power): focus on those that, after being either forced to commitment contract or to status quo, come back within a few months and are subject to choice arm.
    \vspace{.1in}
    \item  \textbf{No learning?} We don't find that being forced to the commitment contract makes them more likely to voluntarily chose commitment later.
    %\vfill \item Are people learning? \hyperlink{learning_table}{\beamerbutton{Table}}
 \end{itemize}  
 \vspace{.2in}
\begin{table}[H]
\begin{center}
\small{\input{./Tables/learning_exp.tex}}
\end{center}
\end{table}
 %\hyperlink{learning}{\beamerbutton{Back}}
}




    
\vfill
\only<2>{  

\textbf{Can impatience explain why not take up a contract that decreases overall cost?}

\begin{figure}[H]
        \caption{Financial cost for different discount rates}
    \label{fc_discount_rates}
    \begin{center}
        \centering
        \includegraphics[width=0.50\textwidth]{Figuras/discount_effect_tut.pdf}
    \end{center}
\end{figure}   
    \begin{itemize}
    \item   Commitment contract imposes up-front costs for later benefits (collateral recovery).  
   
    \item  Requires an interest rate > 4,000\% to make NPV TuT insignificant. 
    \end{itemize}
}

\only<3>{  
  

 \textbf{Standard behavioral angle:}  compliers are sophisticated time-inconsistent, non-compliers are a mix of naifs and the time consistent (who don't need commitment).  

\begin{figure}[H]
    \begin{center}
        \centering
        \includegraphics[width=0.65\textwidth]{Figuras/hyperbolicity_strata.png}
    \end{center}
\end{figure}   
    \begin{itemize}
    \item We have a survey measure of time inconsistency taken at baseline.
    \item  Effect of forcing commitment should be entirely among the time-inconsistent.  Is this true?  No, but PB measure may not be good.
    \end{itemize}
}



\end{frame}



\begin{frame}{Possible Behavioral explanations : TuT \& ToT by groups}

\begin{itemize}
    \item The TuT effect is concentrated on those that say they have 100\% of recovery. This is what we would expect if the Sure-confident don't demand because they think they don't need it (but are wrong).
    \item Also on those with no PB (as measured by us).
\end{itemize}
\begin{figure}[H]
    \caption{TuT}
        \includegraphics[width=0.5\textwidth]{Figuras/tut_beh_partition.pdf}
\end{figure}
\end{frame}












\begin{frame}{Understanding Heterogeneity with a Causal Forest}
\label{HTE}
\begin{itemize}

\item We use (Athey et.al. 2019) GRF to estimate CATE (causal RF), CTUT and CTOT (instrumental RF).
\end{itemize}
    
    
        \begin{figure}
       % \label{choose_wrong}
        \caption{Mistakes in choice arm}
        \centering
        \includegraphics[width=0.65\textwidth]{Figuras/line_cw_apr_tot_tut.pdf}
        \end{figure}

\begin{itemize}
    \item   Use Random Forest to estimate ToT, TUT conditional on covariates.
    \item   Those making either choice make mistakes if benefit of other choice is higher.
    \item $>$ 70\% of non-choosers would have seen some benefit from assignment to commitment.
    \item  $<$ 20\% of choosers made a mistake.
\end{itemize}        


\end{frame}




\begin{frame}{How well can we target paternalism?}

\begin{itemize}
    \item   Why be paternalistic to everyone if we have a strong ability to predict those who lose and we can assign them to control?
    
     \vspace{.2in}
    
    \item   Standard targeting analysis, except need to consider covariates carefully given context.
    \item   Private sector lender, competitive market, part of attraction of pawning is no application!  
    \begin{itemize}
        \item Only use readily observable and non-falsifiable covariates.
    \item   Cannot use any behavioral covariates for which truthful reporting becomes non-incentive-compatible if incentivized.
    \item   Cannot use choice and then compel compliance among non-choosers in a commercial context.  
    \end{itemize}
    
     \vspace{.2in}
    
    \item Which leaves us with:
    \begin{itemize}
    \item   Age
    \item   Gender
    \item   HS Education or above
    \item   Ever pawned before 
    \end{itemize}
    
     \vspace{.2in}
    
    \item  We call these our `narrow' targeting criteria, while the full set of covariates used in the RF are `wide'.
\end{itemize}        


\end{frame}





\begin{frame}{Comparison of Targeting Performance:  Hit-Miss Table}


\begin{table}[H]
\caption{Type I \& II errors using targeting narrow rules}
\label{hit_miss_rule}
\begin{center}
\resizebox{0.95\textwidth}{!}{
\small{\input{./Tables/hit_miss_rule.tex}}
}
\end{center}
\end{table} 
     

\begin{itemize}
    \item   As seen above, even RF with `narrow' variables results in very modest increase in targeting efficiency relative to universal paternalism.
    \item  Logit performs very poorly because it makes symmetric Type I and Type II errors and hence assigns many to control who would have been better off with commitment.
    \item  \textbf{Takeaway:}  In this context, with these variables, targeting paternalism relatively ineffective.
\end{itemize}        


\end{frame}

\section{Conclusion}
\begin{frame}{Conclusion}
    \begin{itemize}
     \vfill \item Laibson has spoken of `veiled paternalism' in contexts where principals desire reliability; here we have a case of `veiled non-paternalism' where features encouraging default might be embedded.
     \vfill \item Novel design to study essential heterogeneity: ``Controlled Choice''
        %\begin{itemize}
         %   \item Combine ``Controlled Choice'' with MTE: Can we extend results of Mogstad et al (2017) to use our ATE, TUT, and TOT as additional inequality restrictions that constrain partial identification bounds for MTEs?
        %\end{itemize}
        \vfill \item  APR reduced by 19\% in the commitment arm
        \vfill \item Mystery of low take-up combined with large TuT seems best explained by sure-confidence among pawnshop customers.
        \vfill\item  Ability to target paternalism using objective covariates is poor; even using RF without cross-validation gains from targeting are small and with logit actually makes things worse.
        \vfill \item Suggests mandated commitment-based contract structures in payday/pawnshop loans as a form of pro-poor regulation?
    \end{itemize}  
\end{frame}

\appendix



\section{Appendix}


\begin{frame}{Appendix}
\vfill \centering APPENDIX
\end{frame}




\begin{frame}{Difference number in pawns}
    \[Pawns \: per \: day_{jt} = \alpha_j + \gamma f(t) + \beta_b \mathbbm{1}(t \in MB)_{t} +\beta_a \mathbbm{1}(t \in MA)_{t}\]
    
    \begin{table}[H]
\caption{Number of pawns balance before and after the experiment}
\label{num_pawns_bal}
\begin{center}
\small{\input{./Tables/num_pawns_bal.tex}}
\end{center}
\end{table}

\hyperlink{consort}{\beamerbutton{Back}}
\end{frame}



\begin{frame}{Histogram of payments}

\begin{columns}
\begin{column}{.33\textwidth}
    \begin{figure}[H]
    \caption{Status-quo}
    \begin{center}
        \includegraphics[width=\textwidth]{Figuras/hist_payments_sq.pdf}
    \end{center}
\end{figure}
\end{column}

\begin{column}{.33\textwidth}
    \begin{figure}[H]
    \caption{Forced commitment}
    \begin{center}
        \includegraphics[width=\textwidth]{Figuras/hist_payments_fc.pdf}
    \end{center}
\end{figure}
\end{column}

\begin{column}{.33\textwidth}
    \begin{figure}[H]
    \caption{Choice commitment}
    \begin{center}
        \includegraphics[width=\textwidth]{Figuras/hist_payments_cc.pdf}
    \end{center}
\end{figure}
\end{column}
\end{columns}


    
    \hyperlink{treatment_arms}{\beamerbutton{Back}}
\end{frame}





\begin{frame}{Costs fall from Forcing by multiple measures}
\label{several_def_cost}
\begin{table}[H]
\caption{Effects on several definitions of cost}
\label{table_robustness_fc}
\begin{center}
\resizebox{0.95\textwidth}{!}{
\small{\input{./Tables/fc_robustness.tex}}
}
\end{center}
 \scriptsize 
 
%\textit{Do file: } \texttt{fc\_robustness.do}
\end{table}
   % \hyperlink{main_results}{\beamerbutton{Back}}
 \begin{itemize}
     \item \vfill Significantly reduced costs even when we include:
     \begin{itemize}
         \item Subjective rather than objective value of pawn.
         \item Travel costs.
         \item Valuing lost liquidity using the interest rate.
         \item All three at once.
     \end{itemize}
\end{itemize}
\end{frame}



\begin{frame}{Do borrowers value Forcing as measured by repeat business?}


\vspace{.2in}
\begin{table}[H]
\begin{center}
\footnotesize{\input{./Tables/repeat_loans.tex}}
\end{center}
\end{table}    
\begin{itemize}
    \item Client forced into commitment is 20\% more likely to come back again and pawn.
    \item Not mechanically from having recovered the pawned, since also pawn other pieces.
    \item Not within the life of the loan, but after. So unlikely that is a liquidity story.
    \item Conditional on recovery (endogenous), ``effect'' twice as big.
\end{itemize}

\end{frame}




\begin{frame}{Distribution of financial cost (\$MXN)}
    \begin{figure}
     \centering
        \includegraphics[width=.8\textwidth]{Figuras/hist_fc.pdf}
    \end{figure}
    
    \hyperlink{fc_outcome}{\beamerbutton{Back}}

\end{frame}






\begin{frame}{Survival Graph}
   \label{interpolation_censoring_imp}

\begin{columns}
\begin{column}{.45\textwidth}

\begin{figure}[H]
    \begin{center}
    \caption{Ended}
        \includegraphics[width=1.1\textwidth]{Figuras/survival_graph_ended.pdf}
    \end{center}
    \end{figure}

    
    \end{column}
 
    
\begin{column}{.45\textwidth}

\begin{figure}[H]
    \begin{center}
    \caption{Recovery}
        \includegraphics[width=1.1\textwidth]{Figuras/survival_graph_unpledge.pdf}
    \end{center}
    \end{figure}
 
    
    \end{column}    
    \end{columns}
\end{frame}


\begin{frame}{Censoring}
\label{censoring}
    \begin{table}[H]
\caption{Bounding censoring}
\label{bounding_censoring}
\begin{center}
\resizebox{0.9\textwidth}{!}{
\footnotesize{\input{./Tables/censoring_imp.tex}}
}
\end{center}
 \scriptsize 
%\textit{Do file: } \texttt{censoring_imp.do, censoring_imp_pr.do}
\end{table}


\end{frame}


\begin{frame}{Interpolation on bounding censoring}
    
\begin{figure}[H]
        \caption{Significance area for Default}
  
    \begin{center}
        \includegraphics[width=0.75\textwidth]{Figuras/frontera_sig_def_imp.pdf}
    \end{center}
\end{figure}
 \hyperlink{main_results}{\beamerbutton{Back}}
\end{frame}



% \begin{frame}{Contract terms}

% \begin{figure}[H]
%      \caption{Contract Terms Summary}
%     \label{PaperSlip}
%     \begin{center}
%         \includegraphics[width=0.65\textwidth]{Figuras/TicketLenderP.png}

%     \end{center}

% \end{figure}
% \end{frame}

 
 
\begin{frame}{Identification of treatment parameters}
\label{identification_randomized_choice}
\begin{equation*}
    Y_i = \mathbbm{1}(Z_i =0) Y_{i0} + \mathbbm{1}(Z_i = 1)  Y_{i1}  + \mathbbm{1}(Z_i = 2) \left[(1 - C_i) Y_{i0} + C_i Y_{i1} \right].
\label{eq:potentialOutcomes}
\end{equation*}

    Viewing $Z_i$ as an instrumental variable, the randomized choice design can be interpreted as a \emph{pair} of RCTs, each subject to one-sided non-compliance. \\
    \begin{itemize}
        \item The first of these compares $Z_i=0$ to $Z_i = 2$.  This setting is identical to a ``randomized encouragement'' design in which treatment is only available to those who are encouraged: $Z_i = 2$. Under this interpretation, those with $C_i = 1$ are ``the compliers'' and it follows that 
\begin{equation*}
\frac{\mathbbm{E}(Y_i|Z_i=2) - \mathbbm{E}(Y_i|Z_i =0)}{\mathbbm{E}(D_i|Z_i=2)-\mathbbm{E}(D_i|Z_i=0)}  = \mathbbm{E}(Y_{i1} - Y_{i0}|C_i = 1)
\label{eq:ToT}
\end{equation*}

\item The second considers $Z_i = 1$ to be the ``encouragement'' and compare the outcomes for these individuals to those with $Z_i = 2$.
\begin{equation*}
\frac{\mathbbm{E}(Y_i|Z_i=1) - \mathbbm{E}(Y_i|Z_i =2)}{\mathbbm{E}(D_i|Z_i=1)-\mathbbm{E}(D_i|Z_i=2)}  = \mathbbm{E}(Y_{i1} - Y_{i0} | C_i = 0)
\label{eq:TuT}
\end{equation*}


\item Because non-compliance is one-sided only, LATE=ToT in one case (since treated=compliers+always takers, but there are no always takers, as they cannot take without being assigned) and LATE=TuT in the other.

\end{itemize}
\hyperlink{cc_design}{\beamerbutton{Back}}


\end{frame}




\begin{frame}{Honest Causal Tree construction}
\label{honest_causal_tree}
Given a dataset with an outcome $Y$, covariates $X$, and a randomized condition $T$ that takes on the value of 0 for control and 1 for treatment:

\begin{enumerate}
\item Split the data into subsample $I$ and subsample $J$

\item Train a decision tree on subsample $I$ predicting $Y$ from $X$, with the requirement that each terminal node has at least $k$ observations from each condition in subsample $J$

\item Apply the decision tree constructed on subsample $I$ to subsample $J$

\item At each terminal node, get the mean of predictions for the $T = 1$ cases from subsample $J$ and subtract the mean of predictions for the $T = 0$ cases from subsample $J$; the resulting difference is the estimated treatment effect.

\end{enumerate}

This is then applied to a causal random forest by using bagging or bootstrapping.
\hyperlink{HTE}{\beamerbutton{Back}}\\

\end{frame}


\end{document}