fix readme

README.Rmd

@@ -40,10 +40,7 @@ devtools::install_github("thomaswiemann/civ", dependencies = TRUE)
 To illustrate ``civ`` on a simple example, consider the data generating process from the simulation of Wiemann (2023): For $i = 1, \ldots, n$, the data generating process is given by $$
     Y_i = D_i \pi_{0}(X_i) + X_i\beta_0 + U_i,\qquad \text{and} \qquad
     D_i= m_0(Z_i) + X_i\gamma_0 + V_i,
-$$
-where $(U_i, V_i)\sim \mathcal{N}(0, \left[\begin{smallmatrix}
-1 & 0.6 \\
-0.6 & \sigma_V^2 \end{smallmatrix}\right])$, $D_i$ is a scalar-valued endogenous variable, $X_i\sim\textrm{Bernoulli}(0.5)$ is a binary covariate and $\beta_0 = \gamma_0 = 0$, and $Z_i$ is the categorical instrument taking values in ${1, \ldots, 40\}$ with equal probability. To introduce correlation between $Z_i$ and $X_i$, I further set $\Pr(Z_i \text{ is odd}\vert X_i = 0) = \Pr(Z_i \text{ is even}\vert X_i = 1) = 0$. The optimal instrument $m_0$ is constructed by first partitioning the support of $Z_i$ into two equal subsets and then assigning either $0$ or $C$ as values. I choose the scalars $\sigma_V^2$ and $C$ such that the variance of the first stage variable is fixed to 1 and the concentration parameter for $n=800$ is $\mu^2 = 180$. The data generating process allows for individual treatment effects $\pi_0(X_i)$ to differ with covariates. Here, $\pi_0(X_i) = 1 + 0.5(1 - 2X_i)$ so that the expected treatment effect is simply $E\pi_0(X) = 1.$
+$$ where $(U_i, V_i)$ are mean-zero multivariate normal with $\sigma_U^2 = 1$, $\sigma_V^2 =\sqrt{0.81}$, and $\sigma_{UV} = 0.6$. $D_i$ is a scalar-valued endogenous variable, $X_i\sim\textrm{Bernoulli}(0.5)$ is a binary covariate and $\beta_0 = \gamma_0 = 0$, and $Z_i$ is the categorical instrument taking values in $\{1, \ldots, 40\}$ with equal probability. To introduce correlation between $Z_i$ and $X_i$, I further set $\Pr(Z_i \text{ is odd}\vert X_i = 0) = \Pr(Z_i \text{ is even}\vert X_i = 1) = 0$. The optimal instrument $m_0$ is constructed by first partitioning the support of $Z_i$ into two equal subsets and then assigning either $0$ or $C$ as values. The scalars $\sigma_V^2$ and $C$ are chosen such that the variance of the first stage variable is fixed to 1 and the concentration parameter for $n=800$ is $\mu^2 = 180$. The data generating process allows for individual treatment effects $\pi_0(X_i)$ to differ with covariates. Here, $\pi_0(X_i) = 1 + 0.5(1 - 2X_i)$ so that the expected treatment effect is simply $E\pi_0(X) = 1.$
 
 The code snippet below draws $n=800$ observations from this data generating
 process.

README.md

@@ -42,17 +42,17 @@ process from the simulation of Wiemann (2023): For $i = 1, \ldots, n$,
 the data generating process is given by $$
     Y_i = D_i \pi_{0}(X_i) + X_i\beta_0 + U_i,\qquad \text{and} \qquad
     D_i= m_0(Z_i) + X_i\gamma_0 + V_i,
-$$ where
-$(U_i, V_i)\sim \mathcal{N}(0, \left[\begin{smallmatrix} 1 & 0.6 \\ 0.6 & \sigma_V^2 \end{smallmatrix}\right])$,
+$$ where $(U_i, V_i)$ are mean-zero multivariate normal with
+$\sigma_U^2 = 1$, $\sigma_V^2 =\sqrt{0.81}$, and $\sigma_{UV} = 0.6$.
 $D_i$ is a scalar-valued endogenous variable,
 $X_i\sim\textrm{Bernoulli}(0.5)$ is a binary covariate and
 $\beta_0 = \gamma_0 = 0$, and $Z_i$ is the categorical instrument taking
-values in ${1, \ldots, 40\}$ with equal probability. To introduce
+values in $\{1, \ldots, 40\}$ with equal probability. To introduce
 correlation between $Z_i$ and $X_i$, I further set
 $\Pr(Z_i \text{ is odd}\vert X_i = 0) = \Pr(Z_i \text{ is even}\vert X_i = 1) = 0$.
 The optimal instrument $m_0$ is constructed by first partitioning the
 support of $Z_i$ into two equal subsets and then assigning either $0$ or
-$C$ as values. I choose the scalars $\sigma_V^2$ and $C$ such that the
+$C$ as values. The scalars $\sigma_V^2$ and $C$ are chosen such that the
 variance of the first stage variable is fixed to 1 and the concentration
 parameter for $n=800$ is $\mu^2 = 180$. The data generating process
 allows for individual treatment effects $\pi_0(X_i)$ to differ with