# Homogeneous treatment effects

## Introduction

### Treatment data

#### Recap

With multilevel data with fixed coefficients we have:

\(y_{ij}=\mathbf x_{ij}\theta +m_j + \epsilon_{ij}\)

We can estimate \(m_j\) using fixed effects or similar methods.

#### Treatment data

If the data is grouped by whether an entity was treated then will have:

However we only observe \(y_i\) and \(D_i\).

\(y_i=y_{i0}+D_i(y_{i1}-y_{10})\)

### Average Treatment Effects (ATE, ATET, ATEUT)

#### Average Treatment Effect (ATE)

\(ATE=E[y_{i1}-y_{i0}]\)

#### Average Treatment Effect on the Treated (ATET)

\(ATE=E[y_{i1}-y_{i0}|D_i=1]\)

\(ATE=E[y_{i1}|D_i=1]-E[y_{i0}|D_i=1]\)

#### Average Treatment Effect on the Untreated (ATEUT)

### Conditional Average Treatment Effect (CATE)

\(E[y_{i1}-y_{i0}|\mathbf x_i]\)

## Exogenous treatment

### Randomly Controlled Trials (RCTs)

If the model is:

\(y_i=D_i\theta +g(X) +\epsilon_i\)

And \(D\) is randomly assigned, then we can estimate

\(y_i=D_i\theta +\epsilon_i\)

To get an estimate for \(\theta \) without collecting data on \(X\).

### Calculating CATEs in RCTs with interaction terms

### Calculating CATEs in RCTs with subgroup analysis

## Calculating treatment effects without estimating missing data

### Regression

We can simply regress outcomes on variables, including treatment.

This assumes treatment effects are constant.

This also assumes that outcomes \(y_{1i}\) and \(y_{0i}\) are independent of \(D_i\), conditional on \(X\).

If we are missing variables in \(X\) then we will have biased estimates.

This also assumes the effects of \(X\) are linear.

We assume: \(E[y_{0i}|\mathbf x_{i}, D_i]=\mathbf x_i \theta\).

### Instrumental Variables and natural experiments

### Regression discontinuity

### Synthetic controls

## Calculating treatment effects by estimating missing data

### Matching

Matching is similar to regression. We assume that effects are constant, and the effect of treatment on \(y_{0i}\) and \(y_{1i}\) are independent of treatment, once controlling for \(X\).

Again, this is biased if this is not the case.

We however do not have to assume a linear form for \(X\).

We assume: \(E[y_{ji}|\mathbf x_{i}, D_i]=E[y_{ji}|\mathbf x_{i}]\)

For each entity, find a near entity which had the opposite treatment.

### Propensity score matching

Match on the chance of getting treatment, given covariates.

### Matrix completion

\(E[y_{i1}-y_{i0}|\mathbf x_i]\)

## Using semi-parametric

## Other

### Estimating ATE using MCMC