1 Introduction

1.1 Introduction: Matching Estimator

2 Identification

2.1 Matching
2.2 Identification
2.3 ATT and ATE

3 Estimation

3.1 Estimation Methods
3.2 Approach 1: Regression, or Analogue Approach
3.3 Nonparametric Estimation
3.4 Curse of dimensionality
3.5 Parametric Estimation, or going back to linear regression
3.6 Approach 2: $M-$ Nearest Neighborhood Matching
3.7 Approach 3: Propensity Score Matching

1 Introduction

1.1 Introduction: Matching Estimator

Idea: Compare individuals with the same characteristics $X$ across treatment and control groups
Key assumption: Treatment is random once we control for the observed characteristics.
Do you remember we already learnt a similar idea before?

2 Identification

2.1 Matching

Let $X_{i}$ denote the observed characteristics:
- age, income, education, race, etc..
Assumption 1: $D_{i}\perp(Y_{0i},Y_{1i})\left|X_{i}\right.$
- Conditional on $X_{i}$ , no selection bias.
- Selection on observables assumption / ignorability
Assumption 2: Overlap assumption $P(D_{i}=1|X_{i}=x)\in(0,1)\ \forall x$
- Given $x$ , we should be able to observe people from both control and treatment group.
- We call $P(D_{i}=1|X_{i}=x)$ propensity score.

2.2 Identification

The assumption implies that $\begin{aligned} E[Y_{1i}|D_{i} & =1,X_{i}]=E[Y_{1i}|D_{i}=0,X_{i}]=E[Y_{1i}|X_{i}]\\ E[Y_{0i}|D_{i} & =1,X_{i}]=E[Y_{0i}|D_{i}=0,X_{i}]=E[Y_{0i}|X_{i}]\end{aligned}$
The $ATT$ for $X_{i}=x$ is given by $\begin{aligned} E[Y_{1i}-Y_{0i}|D_{i}=1,X_{i}] & =E[Y_{1i}|D_{i}=1,X_{i}]-E[Y_{0i}|D_{i}=1,X_{i}]\\ & =E[Y_{i}|D_{i}=1,X_{i}]-E[Y_{0i}|D_{i}=0,X_{i}]\\ & =\underbrace{E[Y_{i}|D_{i}=1,X_{i}]}_{avg\ with\ X_{i}\ in\ treatment}-\underbrace{E[Y_{i}|D_{i}=0,X_{i}]}_{avg\ with\ X_{i}\ in\ control}\end{aligned}$
The components in the last line are identified (can be estimated).
Intuition: Comparing the outcome across control and treatment groups after conditioning on $X_{i}$

2.3 ATT and ATE

ATT is given by $\begin{aligned} ATT & =E[Y_{1i}-Y_{0i}|D_{i}=1]\\ & =\int E[Y_{1i}-Y_{0i}|D_{i}=1,X_{i}=x]f_{X_{i}}(x|D_{i}=1)dx\\ & =E[Y_{i}|D_{i}=1]-\int\left(E[Y_{i}|D_{i}=0,X_{i}=x]\right)f_{X_{i}}(x|D_{i}=1)\end{aligned}$
ATE is $\begin{aligned} ATE= & E[Y_{1i}-Y_{0i}]\\ = & \int E[Y_{1i}-Y_{0i}|X_{i}=x]f_{X_{i}}(x)dx\\ = & \int E[Y_{i}|D_{i}=1,X_{i}=x]f_{X_{i}}(x)dx\\ = & +\int E[Y_{i}|D_{i}=0,X_{i}=x]f_{X_{i}}(x)dx\end{aligned}$

3 Estimation

3.1 Estimation Methods

We need to estimate $E[Y_{i}|D_{i}=1,X_{i}=x]$ and $E[Y_{i}|D_{i}=0,X_{i}=x]$
Several ways to implement the above idea
1. Regression: Nonparametric and Parametric
2. Nearest neighborhood matching
3. Propensity Score Matching

3.2 Approach 1: Regression, or Analogue Approach

Let $\hat{\mu}_{k}(x)$ be an estimator of $\mu_{k}(x)=E[Y_{i}|D_{i}=k,X_{i}=x]$ for $k\in\{0,1\}$
The analog estimators are $\begin{aligned} \hat{ATE} & =\frac{1}{N}\sum_{i=1}^{N}\hat{\mu}_{1}(X_{i})-\hat{\mu}_{0}(X_{i})\\ \hat{ATT} & =\frac{N^{-1}\sum_{i=1}^{N}D_{i}(Y_{i}-\hat{\mu}_{0}(X_{i}))}{N^{-1}\sum_{i=1}^{N}D_{i}}\end{aligned}$
How to estimate $\mu_{k}(x)=E[Y_{i}|D_{i}=k,X_{i}=x]$ ?

3.3 Nonparametric Estimation

Suppose that $X_{i}\in\{x_{1},\cdots,x_{K}\}$ is discrete with small $K$
- Ex: two demographic characteristics (male/female, white/non-white). $K=4$
Then, a nonparametric binning estimator is $\hat{\mu}_{k}(x)=\frac{\sum_{i=1}^{N}\mathbf{1}\{D_{i}=k,X_{i}=x\}Y_{i}}{\sum_{i=1}^{N}\mathbf{1}\{D_{i}=k,X_{i}=x\}}$
Here, I do not put any parametric assumption on $\mu_{k}(x)=E[Y_{i}|D_{i}=k,X_{i}=x]$ .

3.4 Curse of dimensionality

Issue: Poor performance if $K$ is large due to many covariates.
So many potential groups, too few observations for each group.
With $K$ variables, each of which takes $L$ values, $L^K$ possible groups (bins) in total.
This is known as curse of dimensionality.
Relatedly, if $X$ is a continuous random variable, can use kernel regression.

3.5 Parametric Estimation, or going back to linear regression

If you put parametric assumption such as $\begin{aligned} E[Y_{i}|D_{i}=0,X_{i}=x] & =\beta'x_{i}\\ E[Y_{i}|D_{i}=1,X_{i}=x] & =\beta'x_{i}+\tau_{0}\end{aligned}$ then, you will have a model $y_{i}=\beta'x_{i}+\tau D_{i}+\epsilon_{i}$
You can think the matching estimator as controlling for omitted variable bias by adding (many) covariates (control variables) $x_{i}$ .
This is one reason why matching estimator may not be preferred in empirical research.
- Remember: Controlling for those covariates is of course important. This can be combined with other empirical strategies (IV, DID, etc).

3.6 Approach 2: $M-$ Nearest Neighborhood Matching

Idea: Find the counterpart in other group that is close to me.
Define $\hat{y}_{i}(0)$ and $\hat{y}_{i}(1)$ be the estimator for (hypothetical) outcomes when treated and not treated. $\hat{y}_{i}(0)=\begin{cases} y_{i} & if\ D_{i}=0\\ \frac{1}{M}\sum_{j\in L_{M}(i)}y_{j} & if\ D_{i}=1 \end{cases}$
$L_{M}(i)$ is the set of $M$ individuals in the opposite group who are “close” to individual $i$
- Several ways to define the distance between $X_{i}$ and $X_{j}$ , such as $dist(X_{i},X_{j})=||X_{i}-X_{j}||^{2}$
Need to choose (1) $M$ and (2) the measure of distance
- R has several packages for this.

3.7 Approach 3: Propensity Score Matching

Use propensity score $P(D_{i}=1|X_{i}=x)$ as a distance to define who is the closest to me.
Implementation:
1. Estimate propensity score function by logit or probit using a flexible function of $X_i$ .
2. Calculate the propensity score for each observation. Use it to define the pair.

Program Evaluation (Causal Inference) 2: Matching

Instructor: Yuta Toyama

Last updated: 2020-06-22

1 Introduction

1.1 Introduction: Matching Estimator

2 Identification

2.1 Matching

2.2 Identification

2.3 ATT and ATE

3 Estimation

3.1 Estimation Methods

3.2 Approach 1: Regression, or Analogue Approach

3.3 Nonparametric Estimation

3.4 Curse of dimensionality

3.5 Parametric Estimation, or going back to linear regression

3.6 Approach 2: $M-$ Nearest Neighborhood Matching

3.7 Approach 3: Propensity Score Matching

Program Evaluation (Causal Inference) 2: Matching

Instructor: Yuta Toyama

Last updated: 2020-06-22

1 Introduction

1.1 Introduction: Matching Estimator

2 Identification

2.1 Matching

2.2 Identification

2.3 ATT and ATE

3 Estimation

3.1 Estimation Methods

3.2 Approach 1: Regression, or Analogue Approach

3.3 Nonparametric Estimation

3.4 Curse of dimensionality

3.5 Parametric Estimation, or going back to linear regression

3.6 Approach 2: M−M-Nearest Neighborhood Matching

3.7 Approach 3: Propensity Score Matching

3.6 Approach 2: $M-$ Nearest Neighborhood Matching