# Idea Question: Do hospitals make people healthier? ![[s20220501_223359.png]] If we interpret these values at face value, the answer is no. Hospitals make people sicker: $3.21 -3.93 = -0.72$ Potential outcome $\begin{cases}\mathbf{Y}_{1 i} & \text { if } \mathbf{D}_{i}=1 \\ \mathbf{Y}_{0 i} & \text { if } \mathbf{D}_{i}=0\end{cases}$ Observed outcome ${Y}_{i}$ $\begin{aligned} \mathbf{Y}_{i} &= \begin{cases}\mathbf{Y}_{1 i} & \text { if } \mathbf{D}_{i}=1 \\ \mathbf{Y}_{0 i} & \text { if } \mathbf{D}_{i}=0\end{cases} \\ &=\mathbf{Y}_{0 i}+\left(\mathbf{Y}_{1 i}-\mathbf{Y}_{0 i}\right) \mathbf{D}_{i} \end{aligned}$ ${Y}_{1 i}-\mathbf{Y}_{0 i}$ is the causal effect of hospitalization for an individual, an estimate we never observe for each individual because we never see both potential outcomes for any one person. $\begin{aligned} \underbrace{E\left[\mathrm{Y}_{i} \mid \mathbf{D}_{i}=1\right]-E\left[\mathbf{Y}_{i} \mid \mathbf{D}_{i}=0\right]}_{\text {Observed difference in average health }} \ =\ & \underbrace{E\left[\mathrm{Y}_{1 i} \mid \mathbf{D}_{i}=1\right]-E\left[\mathrm{Y}_{0 i} \mid \mathbf{D}_{i}=1\right]}_{\text {Average treatment effect on the treated }} \\ &+\underbrace{E\left[\mathbf{Y}_{0 i} \mid \mathbf{D}_{i}=1\right]-E\left[\mathrm{Y}_{0 i} \mid \mathbf{D}_{i}=0\right]}_{\text {Selection bias }} . \end{aligned}$ Average causal effect of hospitalization on those who were hospitalized. $E\left[\mathrm{Y}_{1 i} \mid \mathrm{D}_{i}=1\right]-E\left[\mathrm{Y}_{0 i} \mid \mathrm{D}_{i}=1\right]=E\left[\mathrm{Y}_{1 i}-\mathrm{Y}_{0 i} \mid \mathrm{D}_{i}=1\right]$ # References - [[Angrist 2008 mostly harmless econometrics]]