Treatment Effect

definition: The term ‘treatment effect’ refers to the causal effect of a binary (0–1) variable on an outcome variable of scientific or policy interest.

The term ‘treatment effect’ originates in a medical literature concerned with the causal effects of binary, yes-or-no ‘treatments’, such as an experimental drug or a new surgical procedure.
Economics examples include the effects of government programmes and policies, such as those that subsidize training for disadvantaged workers, and the effects of individual choices like college attendance.

Given a data-set describing the labour market circumstances of trainees and a non-trainee comparison group, we can compare the earnings of those who did participate in the programme and those who did not. Any empirical study of treatment effects would typically start with such simple comparisons.

In general, omitted variables bias (also known as selection bias) is the most serious econometric concern that arises in the estimation of treatment effects. The link between omitted variables bias, causality, and treatment effects can be seen most clearly using the potential-outcomes
In general, omitted variables bias (also known as selection bias) is the most serious econometric concern that arises in the estimation of treatment effects. The link between omitted variables bias, causality, and treatment effects can be seen most clearly using the potential-outcomes
framework.

Model

First assume a binary treatment. For each population unit, two possible outcomes: $y(0)$ (the outcome without treatment) and $y(1)$ (the outcome with treatment). The binary “treatment” indicator is $D$, where $D=1$ denotes “treatment”, The nature of $y(0)$ and $y(1)$ – discrete,
continuous, some mix -is ok.

The gain from treatment is:

$y(1)-y(0)$

For a particular unit $i$, the gain from treatment is
$y_{i}(1)-y_{i}(0)$

Problem: For each unit i, only one of $y(0)$ and $y(1)$ is observed.
In effect, we have a missing data problem (even though we will
eventually assume a random sample of units)

Two parameters are of primary interest: ATE(average treatment
effect) and ATT(average treatment effect on the treated )

ATE
$\tau_{ATE}=E[y(1)-y(0)]$
The expected gain for a randomly selected unit from the population. This is sometimes called the average causal effect.
ATT
$\tau_{ATT}=E[y(1)-y(0)|D=1]$
With heterogeneous treatment effects, (2) and (3) can be very different. ATE averages across gain from units that might never be subject to treatment.
how we estimate $\tau_{ATT}$ and $\tau_{ATE}$ depends on what we
assume about treatment assignment.
the problem in estimating is that sample is limited, which means we only know $y(1|D=1)$ and $y(0|D=0)$we can not know the $y(1|D=0)$

Sampling Assumptions
Assume independent, identically distributed observations from the underlying population. The data we would like to have is $(y_{i}(0),y_{i}(1)) : i = 1, . . . ,N$

But we only observe $D_{i}$ and

$y_{i}=(1-D_{i})y_{i}(0)+D_{i}y_{i}(1)=y_{i}(0)+D_{i}{y_{i}(1)-y_{i}(0)}$

in other word( each individual can only be observed in one of the possible treatments)

Random sampling rules out treatment of one unit having an effect on
other units
Estimation under Random Assignment

Strongest form of random assignment, $[y(0),y(1)]$ is independent of $D$ $E(y|D=1)-E(y|D=0)=E(Y(1))-E(y(0))=\tau_{ATE}=\tau_{ATT}$

under mean independence and the means on the left hand side can be
estimated by using sample averages on the two subsamples

To start with, we consider a linear structural outcome equation, and homogeneous effects:

$y=\beta d+\epsilon$

DID (different in different method)

Definition

Two groups:
- D = 1 Treated units
- D = 0 Control units
Two periods:
- T = 0 Pre-Treatment period
- T = 1 Post-Treatment period
Potential outcome $Y_{d}(t )$
- $Y_{1i}(t )$ outcome unit i attains in period t when treatment between t and t − 1
- $Y_{0i}(t )$ outcome unit i attains in period t with control between t and t − 1

ESITMATE ATET
Focus on estimating the average effect of the treatment on the
treated

$\alpha_{ATT}=E[Y_{1}(1)-Y_{0}(1)|D=1]$

Tables	Post-period(t=1)	Pre-period(t=0)
Treated D=1	$E[Y_{1}(1);D=1]$	$E[Y_{0}(0);D=1]$
Control D=0	$E[Y_{0}(1);D=0]$	$E[Y_{0}(0);D=0]$

Problem
Missing potential outcome: $E[Y_{0}(1)|D = 1]$, i.e., what is the average post-period outcome for the treated in the absence of the treatment?

solving strategy

Before vs. After
- Use: $E [Y (1)|D = 1] − E [Y (0)|D = 1]$
- Assumes: $E [Y_{0}(1)|D = 1]= E [Y_{0}(0)|D = 1]$
Treated vs. Control in Post-Period
- Use: $E [Y (1)|D = 1] − E [Y (1)|D = 0]$
- Assumes: $E [Y_{0}(1)|D = 1]= E [Y_{0}(1)|D = 0]$
Difference-in-Differences
Use:
- $\{E [Y (1)|D = 1] − E [Y (1)|D = 0] \}- \{ E [Y (0)|D = 1] − E [Y (0)|D = 0]\}$
- Assumes: $E[Y_{0}(1) − Y_{0}(0)|D = 1] = E[Y_{0}(1) − Y_{0}(0)|D = 0]$

The rest of the content is in the slides
[resource link]