# 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:

• $$y_{i0}$$ - the outcome if the entity was not treated

• $$y_{i1}$$ - the outcome if the entity was treated

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)

$$ATE=E[y_{i1}-y_{i0}|D_i=1]$$

$$ATE=E[y_{i1}|D_i=1]-E[y_{i0}|D_i=1]$$

### 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 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: $$E[y_{0i}|\mathbf x_{i}, D_i]=\mathbf x_i \theta$$.

## 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

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