Introduction to Causal Inference

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# Introduction to Causal Inference
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### Ian McCarthy | Emory University
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### Econ 771, Fall 2022
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# Outline

1. [Motivation](#motivation)

2. [Potential Outcomes](#notation)

3. [Canonical Research Designs](#designs)

4. [Note on Regression Coefficients](#interpretation)

---
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name: motivation

# Why care about causal inference?

---
# Why causal inference?

---
# Why causal inference?

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

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**Machine Learning**

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

]

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**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**

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name: notation

# Introduction to Potential Outcomes

---
# Potential outcomes

- Framework for causal inference
- Introduced by Neyman (1923), pushed by Rubin in 1970s and 1980s
- Requires counterfactual reasoning. These counterfactuals are thought experiments. They are not observed.

---
# Treatment

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

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

---
# Potential outcomes

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

--
<br>
- Potential outcomes are **NOT** realized. They are purely hypothetical.

---
# Potential vs observed (switching equation)

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

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# Do we ever observe the potential outcomes?

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

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# 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 `$Y_{1}$`.
    - Explicit estimation as in nearest neighbor matching or synthetic control
    - Implicit as in RD or IV

---
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name: ate

# Average Treatment Effects

---
# Different treatment effects

Since `$Y_{1i}$` and `$Y_{0i}$` are not observed for the same unity, we 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

---
# 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

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

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# Canonical Research Designs

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# 1. Selection on Observables

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

- One strategy: Impose **selection on observables** or **conditional independence**

- Otherwise known as "identification purely by assumption"

---
# 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)

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

---
# Selection on observables in practice

- Matching or reweighting
- Regression
- Both! (doubly-robust)

---
# 2. Instrumental variables

- Easy!
    - Exogenous variation from "outside" variable `$z$`, related to endogenous variable `$x$` but not otherwise directly related to outcome `$y$`
    - Incorporate `$z$` as instrument in IV, 2SLS, LIML, etc.
    
- Hard!
    - Finding an instrument
    - Weak instruments
    - Many vs single instrument
    - Many endogenous variables
    - Nonlinear models

---
# 3. Regression discontinuity

- Easy!
    - Exploit discontinuity in rule assigning people to one group or another based on relatively arbitrary threshold
    - Jump in linear regression below and above threshold
    
- Hard!
    - Manipulation of running variable
    - Selecting bandwidth and bin width
    - Sparsity of data around threshold

---
# 4. Difference in differences

- Easy!
    - Compare difference in trends before and after treatment among treated and control group
    - 2x2 difference in means
    
- Hard!
    - Assumption of parallel trends (i.e., treatment group would have followed the same trend as control group in the absence of treatment), tests for "parallel trends", and appropriateness of the control group
    - Staggered treatment timing
    - Heterogenous treatment effects

---
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name: interpretation

# What do we really "get" from a regression?

---
# 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