Module 0: Getting Started

class: center, middle, inverse, title-slide

.title[
# Module 0: Getting Started
]
.subtitle[
## Part 2: Introduction to Causal Inference
]
.author[
### Ian McCarthy | Emory University
]
.date[
### Econ 470 & HLTH 470
]

---

<style type="text/css">
.remark-slide-content {
    font-size: 30px;
    padding: 1em 2em 1em 2em;    
}
.remark-code {
  font-size: 15px;
}
.remark-inline-code { 
    font-size: 20px;
}
</style>

# Why causal inference?

---
# Why causal inference?

Another example: **What price should we charge for a night in a hotel?**

.pull-left[
**Machine Learning**

- Focuses on prediction
- High prices are strongly correlated with higher sales
- Increase prices to attract more people?

]

.pull-right[
**Causal Inference**

- Focuses on **counterfactuals**
- What would sales look like if prices were higher?

]

---
# Goal of Causal Inference

- **Goal:** Estimate effect of some policy or program

- Key building block for causal inference is the idea of **potential outcomes**

---
# Some notation

**Treatment** `$D_{i}$`

`$$D_{i}=\begin{cases}
 1 \text{ with treatment} \\
 0 \text{ without treatment}
\end{cases}$$`

---
# Some notation

**Potential outcomes**

- `$Y_{1i}$` is the potential outcome for unit `$i$` with treatment
- `$Y_{0i}$` is the potential outcome for unit `$i$` without treatment

---
# Some notation

**Observed outcome**

`$$Y_{i}=Y_{1i} \times D_{i} + Y_{0i} \times (1-D_{i})$$`
or

`$$Y_{i}=\begin{cases}
Y_{1i} \text{ if } D_{i}=1 \\
Y_{0i} \text{ if } D_{i}=0
\end{cases}$$`

.footnote[
Assumes **SUTVA** (stable unit treatment value assumption)...no interference across units
]

---
# Example of "Potential Outcomes"

.pull-left[
![:scale 420px](pics/EmoryPicture.jpg)

`$Y_{1}$`= <span>&#36;</span>75,000

]

.pull-right[
![:scale 370px](pics/UNTPicture.jpg)

`$Y_{0}$`= <span>&#36;</span>60,000
]

---
# Example of "Potential Outcomes"
.pull-left[
![:scale 420px](pics/EmoryPicture.jpg)

`$Y_{1}$`= <span>&#36;</span>75,000

]

.pull-right[
![:scale 370px](pics/UNTPicture.jpg)

`$Y_{0}$`= <span>&#36;</span>60,000
]

Earnings due to Emory = `$Y_{1}-Y_{0}$` = <span>&#36;</span>15,000

---
# Example of "Potential Outcomes"

.pull-left[
![:scale 420px](pics/EmoryPicture.jpg)

`$Y_{1}$`= <span>&#36;</span>75,000

]

.pull-right[
![:scale 370px](pics/UNTPicture.jpg)

`$Y_{0}$`= ?
]

---
# Example of "Potential Outcomes"

.pull-left[
![:scale 420px](pics/EmoryPicture.jpg)

`$Y_{1}$`= <span>&#36;</span>75,000

]

.pull-right[
![:scale 370px](pics/UNTPicture.jpg)

`$Y_{0}$`= ?
]

Earnings due to Emory = `$Y_{1}-Y_{0}$` = ?

---
# Do we ever observe the potential outcomes?

.center[
  ![:scale 700px](https://media.giphy.com/media/zZeCRfPyXi9UI/giphy.gif)
]

--
Without a time machine...not possible to get *individual* effects.

---
# Fundamental Problem of Causal Inference

- We don't observe the counterfactual outcome...what would have happened if a treated unit was actually untreated.

- *ALL* attempts at causal inference represent some attempt at estimating the counterfactual outcome. We need an estimate for `$Y_{0}$` among those that were treated, and vice versa for `$Y_{1}$`.

---
class: inverse, center, middle
name: ate

# Average Treatment Effects

---
# Different treatment effects

Tend to focus on **averages**<sup>1</sup>:

- **ATE**: `$\delta_{ATE} = E[ Y_{1} - Y_{0}]$`

- **ATT**: `$\delta_{ATT} = E[ Y_{1} - Y_{0} | D=1]$`

- **ATU**: `$\delta_{ATU} = E[ Y_{1} - Y_{0} | D=0]$`

.footnote[<sup>1</sup> or similar measures such as medians or quantiles]

---
# Average Treatment Effects

- **Estimand**: `$$\delta_{ATE} = E[Y_{1} - Y_{0}] = E[Y | D=1] - E[Y | D=0]$$`

- **Estimate**: `$$\hat{\delta}_{ATE} = \frac{1}{N_{1}} \sum_{D_{i}=1} Y_{i} - \frac{1}{N_{0}} \sum_{D_{i}=0} Y_{i},$$` where `$N_{1}$` is number of treated and `$N_{0}$` is number untreated (control)

- With random assignment and equal groups, inference/hypothesis testing with standard two-sample t-test

---
class: inverse, center, middle
name: selection

# Selection Bias

---
# Selection bias

- Assume (for simplicity) constant effects, `$Y_{1i}=Y_{0i} + \delta$`

- Since we don't observe `$Y_{0}$` and `$Y_{1}$`, we have to use the observed outcomes, `$Y_{i}$`

`$$\begin{align}
E[Y_{i} | D_{i}=1] &- E[Y_{i} | D_{i}=0] \\
=& E[Y_{1i} | D_{i}=1] - E[Y_{0i} | D_{i}=0] \\
=& \delta + E[Y_{0i} | D_{i}=1] - E[Y_{0i} | D_{i}=0] \\
=& \text{ATE } + \text{ Selection Bias}
\end{align}$$`

---
# Selection bias

- Selection bias means `$E[Y_{0i} | D_{i}=1] - E[Y_{0i} | D_{i}=0] \neq 0$`

- In words, the potential outcome without treatment, `$Y_{0i}$`, is different between those that ultimately did and did not receive treatment.

- e.g., treated group was going to be better on average even without treatment (higher wages, healthier, etc.)

---
# Selection bias

- How to "remove" selection bias?

- How about random assignment?

- In this case, treatment assignment doesn't tell us anything about `$Y_{0i}$`

`$$E[Y_{0i}|D_{i}=1] = E[Y_{0i}|D_{i}=0],$$` such that `$$E[Y_{i}|D_{i}=1] - E[Y_{i} | D_{i}=0] = \delta_{ATE} = \delta_{ATT} = \delta_{ATU}$$`

---
# Selection bias

- Without random assignment, there's a high probability that `$$E[Y_{0i}|D_{i}=1] \neq E[Y_{0i}|D_{i}=0]$$`

- i.e., outcomes without treatment are different for the treated group

---
# Omitted variables bias

- In a regression setting, selection bias is the same problem as omitted variables bias (OVB)

- Quick review: Goal of OLS is to find `$\hat{\beta}$` to "best fit" the linear equation `$y_{i} = \alpha + x_{i} \beta + \epsilon_{i}$`

---
# Regression review

`$$\begin{align}
\min_{\beta} & \sum_{i=1}^{N} \left(y_{i} - \alpha - x_{i} \beta\right)^{2}  = \min_{\beta} \sum_{i=1}^{N} \left(y_{i} - (\bar{y} - \bar{x}\beta) - x_{i} \beta\right)^{2}\\
0 &= \sum_{i=1}^{N} \left(y_{i} - \bar{y} - (x_{i} - \bar{x})\hat{\beta} \right)(x_{i} - \bar{x}) \\
0 &= \sum_{i=1}^{N} (y_{i} - \bar{y})(x_{i} - \bar{x}) - \hat{\beta} \sum_{i=1}^{N}(x_{i} - \bar{x})^{2} \\
\hat{\beta} &= \frac{\sum_{i=1}^{N} (y_{i} - \bar{y})(x_{i} - \bar{x})}{\sum_{i=1}^{N} (x_{i} - \bar{x})^{2}} = \frac{Cov(y,x)}{Var(x)}
\end{align}$$`

---
# Omitted variables bias

- Interested in estimate of the effect of schooling on wages

`$$Y_{i} = \alpha + \beta s_{i} + \gamma A_{i} + \epsilon_{i}$$`

- But we don't observe ability, `$A_{i}$`, so we estimate

`$$Y_{i} = \alpha + \beta s_{i} + u_{i}$$`

- What is our estimate of `$\beta$` from this regression?

---
# Omitted variables bias

`$$\begin{align}
\hat{\beta} &= \frac{Cov(Y_{i}, s_{i})}{Var(s_{i})} \\
 &= \frac{Cov(\alpha + \beta s_{i} + \gamma A_{i} + \epsilon_{i}, s_{i})}{Var(s_{i})} \\
 &= \frac{\beta Cov(s_{i}, s_{i}) + \gamma Cov(A_{i},s_{i}) + Cov(\epsilon_{i}, s_{i})}{Var(s_{i})}\\
 &= \beta \frac{Var(s_{i})}{Var(s_{i})} + \gamma \frac{Cov(A_{i},s_{i})}{Var(s_{i})} + 0\\
 &= \beta + \gamma \times \theta_{as}
 \end{align}$$`

---
# Removing selection bias without RCT

- The field of causal inference is all about different strategies to remove selection bias

- The first strategy (really, assumption) in this class: **selection on observables** or **conditional indpendence**

---
# Intuition

- Example: Does having health insurance, `$D_{i}=1$`, improve your health relative to someone without health insurance, `$D_{i}=0$`?

- `$Y_{1i}$` denotes health with insurance, and `$Y_{0i}$` health without insurance (these are **potential** outcomes)

- In raw data, `$[Y_{i} | D_{i}=1] > E[Y_{i} | D_{i}=0]$`, but is that causal?

---
# Intuition

Some assumptions:

- `$Y_{0i}=\alpha + \eta_{i}$`
- `$Y_{1i} - Y_{0i} = \delta$`
- There is some set of "controls", `$x_{i}$`, such that `$\eta_{i} = \beta x_{i} + u_{i}$` and `$E[u_{i} | x_{i}]=0$` (conditional independence assumption, or CIA)

`$$\begin{align}
Y_{i} &= Y_{1i} \times D_{i} + Y_{0i} \times (1-D_{i}) \\
&= \delta D_{i} + Y_{0i} D_{i} + Y_{0i} - Y_{0i} D_{i} \\
&= \delta D_{i} + \alpha + \eta_{i} \\
&= \delta D_{i} + \alpha + \beta x_{i} + u_{i}
\end{align}$$`

---
# ATEs versus regression coefficients

- Estimating the regression equation, `$$Y_{i} = \alpha + \delta D_{i} + \beta x_{i} + u_{i}$$` provides a causal estimate of the effect of `$D_{i}$` on `$Y_{i}$`

- But what does that really mean?

---
# ATEs vs regression coefficients

- *Ceteris paribus* ("with other conditions remaining the same"), a change in `$D_{i}$` will lead to a change in `$Y_{i}$` in the amount of `$\hat{\delta}$`

- But is *ceteris paribus* informative about policy?

---
# ATEs vs regression coefficients

- `$Y_{1i} = Y_{0i} + \delta_{i} D_{i}$` (allows for heterogeneous effects)

- `$Y_{i} = \alpha + \beta D_{i} + \gamma X_{i} + \epsilon_{i}$`, with `$Y_{0i}, Y_{1i} \perp\!\!\!\perp D_{i} | X_{i}$`

- Aronow and Samii, 2016, show that: `$$\hat{\beta} \rightarrow_{p} \frac{E[w_{i} \delta_{i}]}{E[w_{i}]},$$` where `$w_{i} = (D_{i} - E[D_{i} | X_{i}])^{2}$`

---
# ATEs vs regression coefficients

- Simplify to ATT and ATU
- `$Y_{1i} = Y_{0i} + \delta_{ATT} D_{i} + \delta_{ATU} (1-D_{i})$` 
- `$Y_{i} = \alpha + \beta D_{i} + \gamma X_{i} + \epsilon_{i}$`, with `$Y_{0i}, Y_{1i} \perp\!\!\!\perp D_{i} | X_{i}$`

`$$\begin{align}
\beta = & \frac{P(D_{i}=1) \times \pi (X_{i} | D_{i}=1) \times (1- \pi (X_{i} | D_{i}=1))}{\sum_{j=0,1} P(D_{i}=j) \times \pi (X_{i} | D_{i}=j) \times (1- \pi (X_{i} | D_{i}=j))} \delta_{ATU} \\
& + \frac{P(D_{i}=0) \times \pi (X_{i} | D_{i}=0) \times (1- \pi (X_{i} | D_{i}=0))}{\sum_{j=0,1} P(D_{i}=j) \times \pi (X_{i} | D_{i}=j) \times (1- \pi (X_{i} | D_{i}=j))} \delta_{ATT}
\end{align}$$`

---
# ATEs vs regression coefficients

What does this mean?

- OLS puts more weight on observations with treatment `$D_{i}$` "unexplained" by `$X_{i}$`

- "Reverse" weighting such that the proportion of treated units are used to weight the ATU while the proportion of untreated units enter the weights of the ATT

- This is *an* average effect, but probably not the average we want