model our treatment allocation, $Z_1$, as $$\label{chap3:ztime1}
    logit[P(Z_1=1|V)]=\gamma_0+\gamma_1V.$$ For the potential outcome at $t=1$, $Y_{Z_1}$, we model it as $$\label{chap3:ytime1}
    Y_{Z_1}=\beta_0+\beta_ZI\{Z_1=1\}+\beta_VV+\beta_{VZ}VI\{Z_1=1\}+e,$$ where $e\sim N(0,1)$.

We model the potential survival status at $t=2$, $S_{Z_1}$ as $$\begin{aligned}
    \label{chap3:stime2}
    logit(P[S_{Z_1}=1|V,Y_{Z_1}])&=\alpha_0+\alpha_{Y_1}Y_1I\{Z_1=1\}+\alpha_{Y_0}Y_0[1-I\{Z_1=1\}] \nonumber\\
    &\quad+\alpha_ZI\{Z_1=1\}+\alpha_VV+\alpha_{VZ}VI\{Z_1=1\}.\end{aligned}$$ Monotonicity is imposed by setting $S_0=1$ if $S_1=1$. Because a negative wealth shock is an absorbing state, if $Z_1=1$, then $Z_2=1$. So when $Z_1=0$, we have $$\label{chap3:ztime2}
    logit(P[Z_2=1|V,Y_0])=\gamma_0+\gamma_{Y_0,2}Y_0+\gamma_2V.$$ We model the potential outcome at $t=2$, $Y_{Z_1,Z_2}$ as $$\begin{aligned}
    \label{chap3:ytime2}
    Y_{Z_1,Z_2}&=\beta_0+\beta_{Z_{01}}I\{Z_1=0,Z_2=1\}+\beta_{Z_{11}}I\{Z_1=1,Z_2=1\}\nonumber\\
    &\quad+\beta_{Y_0Z_{00}}Y_0I\{Z_1=0,Z_2=0\}+\beta_{Y_0Z_{01}}Y_0I\{Z_1=0,Z_2=1\}\nonumber\\
    &\quad+\beta_{Y_1Z_{11}}Y_1I\{Z_1=1,Z_2=1\}+\beta_VV+\beta_{VZ_{01}}VI\{Z_1=0,Z_2=1\}\nonumber\\
    &\quad+\beta_{VZ_{11}}VI\{Z_1=1,Z_2=1\}+e,\end{aligned}$$ where $e\sim N(0,1)$.

For the potential survival status at $t=3$, $S_{Z_1,Z_2}$, if $S_{Z_1}=0$, then $S_{Z_1,Z_2}=0$. When $S_{Z_1}=1$, we have $$\begin{aligned}
    \label{chap3:stime3}
    logit(P[S_{Z_1,Z_2}=1|X,Y_{Z_1,Z_2},S_{Z_1}=1])&=\alpha_0+\alpha_{Z_{01}}I\{Z_1=0,Z_2=1\}\nonumber\\
    &\quad+\alpha_{Z_{11}}I\{Z_1=1,Z_2=1\}\nonumber\\
    &\quad+\alpha_{Y_{00}Z_{00}}Y_{00}I\{Z_1=0,Z_2=0\}\nonumber\\
    &\quad+\alpha_{Y_{01}Z_{01}}Y_{01}I\{Z_1=0,Z_2=1\}\nonumber\\
    &\quad+\alpha_{Y_{11}Z_{11}}Y_{11}I\{Z_1=1,Z_2=1\}\nonumber\\
    &\quad+\alpha_VV+\alpha_{VZ_{01}}VI\{Z_1=0,Z_2=1\}\nonumber\\
    &\quad+\alpha_{VZ_{11}}VI\{Z_1=1,Z_2=1\}.\end{aligned}$$ Again, we impose monotonicity by setting $S_{00}=S_{01}=1$ if $S_{11}=1$ and $S_{00}=1$ if $S_{01}=1$. For the treatment allocation at $t=3$, $Z_3$, if $Z_1=Z_2=0$, we have $$\label{chap3:ztime3}
    logit(P[Z_3=1|X,Y_{00}])=\gamma_0+\gamma_{Y_{00}}Y_{00}+\gamma_{Y_{0,3}}Y_0+\gamma_3V.$$ For the potential outcome at $t=3$, $Y_{Z_1,Z_2,Z_3}$, we have $$\begin{aligned}
    \label{chap3:ytime3}
    Y_{Z_1,Z_2,Z_3}&=\beta_0+\beta_{Z_{001}}I\{Z_1=0,Z_2=0,Z_3=1\}+\beta_{Z_{011}}I\{Z_1=0,Z_2=1,Z_3=1\}\nonumber\\
    &\quad+\beta_{Z_{111}}I\{Z_1=1,Z_2=1,Z_3=1\}+\beta_{Y_{00}Z_{000}}Y_{00}I\{Z_1=0,Z_2=0,Z_3=0\}\nonumber\\
    &\quad+\beta_{Y_{00}Z_{001}}Y_{00}I\{Z_1=0,Z_2=0,Z_3=1\}\nonumber\\
    &\quad+\beta_{Y_{01}Z_{011}}Y_{01}I\{Z_1=0,Z_2=1,Z_3=1\}\nonumber\\
    &\quad+\beta_{Y_{11}Z_{111}}Y_{11}I\{Z_1=1,Z_2=1,Z_3=1\}+\beta_{Y_0Z_0}Y_0I\{Z_1=0\}\nonumber\\
    &\quad+\beta_{Y_1Z_1}Y_1I\{Z_1=1\}+\beta_VV+\beta_{VZ_{001}}VI\{Z_1=0,Z_2=0,Z_3=1\}\nonumber\\
    &\quad+\beta_{VZ_{011}}VI\{Z_1=0,Z_2=1,Z_3=1\}\nonumber\\
    &\quad+\beta_{VZ_{111}}VI\{Z_1=1,Z_2=1,Z_3=1\}+e.\end{aligned}$$

Table \[chap3:sim\_para1\] shows the parameters we used to achieve the three different simulation scenarios. Scenario 1 is achieved by setting $\gamma_1$, $\alpha_Z$, $\gamma_2$, $\gamma_{Y_0,2}$, $\alpha_{Z_{01}}$, $\alpha_{Z_{11}}$, $\gamma_3$, $\gamma_{Y_0,3}$, and $\gamma_{Y_{00}}$ to be about 10 times smaller than the values in Scenarios 2 and 3. The 