ggplot(data = (dartmouth.data %>% filter(Year==2015)),
mapping = aes(x = Expenditures, y = Total_Mortality)) +
geom_point(size = 1) + theme_bw() + scale_x_continuous(label = comma) +
geom_smooth(method="lm", se=FALSE, color="blue", size=1/2) +
labs(x = "Spending Per Capita ($US)",
y = "Mortality Rate",
title = "Mortality and Health Care Spending")
Another example: What price should we charge for a night in a hotel?
Machine Learning
Causal Inference
Treatment \(D_{i}\)
\[D_{i}=\begin{cases} 1 \text{ with treatment} \\ 0 \text{ without treatment} \end{cases}\]
Potential outcomes
Observed outcome
\[Y_{i}=Y_{1i} \times D_{i} + Y_{0i} \times (1-D_{i})\] or
\[Y_{i}=\begin{cases} Y_{1i} \text{ if } D_{i}=1 \\ Y_{0i} \text{ if } D_{i}=0 \end{cases}\]
\(Y_{1}\)= $75,000
\(Y_{0}\)= $60,000
Single-year earnings due to Emory = \(Y_{1}-Y_{0}\) = $15,000
\(Y_{1}\)= $75,000
\(Y_{0}\)= ?
Earnings due to Emory = \(Y_{1}-Y_{0}\) = ?
Without a time machine…not possible to get individual effects.
Tend to focus on averages:
\[\begin{align} E[Y_{i} | D_{i}=1] &- E[Y_{i} | D_{i}=0] \\ =& E[Y_{1i} | D_{i}=1] - E[Y_{0i} | D_{i}=0] \\ =& \delta + E[Y_{0i} | D_{i}=1] - E[Y_{0i} | D_{i}=0] \\ =& \text{ATE } + \text{ Selection Bias} \end{align}\]
\[\begin{align} \min_{\beta} & \sum_{i=1}^{N} \left(y_{i} - \alpha - x_{i} \beta\right)^{2} = \min_{\beta} \sum_{i=1}^{N} \left(y_{i} - (\bar{y} - \bar{x}\beta) - x_{i} \beta\right)^{2}\\ 0 &= \sum_{i=1}^{N} \left(y_{i} - \bar{y} - (x_{i} - \bar{x})\hat{\beta} \right)(x_{i} - \bar{x}) \\ 0 &= \sum_{i=1}^{N} (y_{i} - \bar{y})(x_{i} - \bar{x}) - \hat{\beta} \sum_{i=1}^{N}(x_{i} - \bar{x})^{2} \\ \hat{\beta} &= \frac{\sum_{i=1}^{N} (y_{i} - \bar{y})(x_{i} - \bar{x})}{\sum_{i=1}^{N} (x_{i} - \bar{x})^{2}} = \frac{Cov(y,x)}{Var(x)} \end{align}\]
\[\begin{align} \hat{\beta} &= \frac{Cov(Y_{i}, s_{i})}{Var(s_{i})} \\ &= \frac{Cov(\alpha + \beta s_{i} + \gamma A_{i} + \epsilon_{i}, s_{i})}{Var(s_{i})} \\ &= \frac{\beta Cov(s_{i}, s_{i}) + \gamma Cov(A_{i},s_{i}) + Cov(\epsilon_{i}, s_{i})}{Var(s_{i})}\\ &= \beta \frac{Var(s_{i})}{Var(s_{i})} + \gamma \frac{Cov(A_{i},s_{i})}{Var(s_{i})} + 0\\ &= \beta + \gamma \times \theta_{as} \end{align}\]
Some assumptions:
\[\begin{align} Y_{i} &= Y_{1i} \times D_{i} + Y_{0i} \times (1-D_{i}) \\ &= \delta D_{i} + Y_{0i} D_{i} + Y_{0i} - Y_{0i} D_{i} \\ &= \delta D_{i} + \alpha + \eta_{i} \\ &= \delta D_{i} + \alpha + \beta x_{i} + u_{i} \end{align}\]
\[\begin{align} \beta = & \frac{P(D_{i}=1) \times \pi (X_{i} | D_{i}=1) \times (1- \pi (X_{i} | D_{i}=1))}{\sum_{j=0,1} P(D_{i}=j) \times \pi (X_{i} | D_{i}=j) \times (1- \pi (X_{i} | D_{i}=j))} \delta_{ATU} \\ & + \frac{P(D_{i}=0) \times \pi (X_{i} | D_{i}=0) \times (1- \pi (X_{i} | D_{i}=0))}{\sum_{j=0,1} P(D_{i}=j) \times \pi (X_{i} | D_{i}=j) \times (1- \pi (X_{i} | D_{i}=j))} \delta_{ATT} \end{align}\]
What does this mean?
