Lecture 6 (IV with heterogeneous treatment effects) Flashcards
With $Z_i$ the instrument status for individual $i$, in the potential outcome framework, we can think of the treatment $D_i$ for individual $i$ like the following:
D_i = D_i(0)+(D_i(1)-D_i(0))Z_i
That is, if $Z_i = 1$ we get $D_i = D_i(1)$ and if $Z_i = 0$ we get $D_i = D_i(0)$. This is equivalent to:
What kind of individuals do we regard in the experimental setting?
Express them in terms of D and Z
Different kinds of individuals:
In our analysis, we regard four types of individuals
Compliers:
takes the treatment only when they should
(D_i = 1 | Z_i=1) and (D_i = 0 | Z_i=0)
Defiers:
does the opposite of what they are told
(D_i = 1 | Z_i=0) and (D_i = 1 | Z_i=0)
Always-takers:
always take the treatment
(D_i = 1 | Z_i=1) and (D_i = 1 | Z_i=0)
Never-takers:
never take the treatment
(D_i = 0 | Z_i=1) and (D_i = 0 | Z_i=0)
How do the instrument effect the always-takers and never-takers?
The instrument has no predictive power for always-takers and never-takers since it does not affect their treatment status!
How do we deal with defiers?
We generally don’t think that we have a problem with defiers, and they are ruled out by the monotonicity assumption. However, in some settings, they can be a problem.
What type of effect do we estimate with IV when there is homogeneous treatment effects?
LATE = the average treatment effect among compliers.
With heterogeneity, we mainly answer questions in regard to internal validity, but if compliers are similar to the overall population, there is a strong claim for external validity. It is possible to recover the characteristics of the complier population.
Explain the 4. IV assumptions using potential outcomes in the heterogeneous case.
See notion.
Derive the LATE using potential outcome framework
See notion!
Can we check if we really have heterogeneous treatment effects?
The set of compilers is going to be a function of the instrument. If we use a different instrument, the set of compilers may be different. Our estimated LATE is thus only valid for our specific instrument.
Since we only identify local average treatment effects. With more than one instrument, you can use this insight to test whether treatment effects are homogeneous. Since the population of compliers is different across instruments, we can see if the estimated parameter is the same across different populations. If it changes, then we have heterogeneous treatment effects.
Who are the compilers?
To generalize our results to other populations we need to know this. We can not recover who they are, but the first stage regression coefficient gives us the total share of compliers in the population.
We can also study how covariates are distributed for compliers so we can recover their characteristics to see if they are like the broader population.
What is the LATE when we have multiple instruments?
The 2SLS is a weighted average of the underlying Wald estimates.
\beta_{2SLS} =\psi\beta_1+(1-\psi)\beta_2
The 2SLS produces a weighted average of $LATE(Z_1)$ and $LATE(Z_2)$.
What is the LATE when we have a single multivariat instrument?
We then get a weighted average LATE for every $j$ complier group. That is, a LATE for every value of $j.$
The weights are not, however, straightforward to interpret. We will also have
- overfitting and
- bias in small samples.
What is the LATE when we use covariates in our analysis?
In general, researchers do use covariates in a regression framework. We should then include the covariates in the first stage and the second stage.
Then, if we don’t saturate the model, including covariates will not in general identify a LATE! Saturation is hard to do it seems. It is thus better to use propensity score matching.
In the best case, using covariates 2SLS creates some kind of weighted average LATE of all the underlying covariate-specific LATE’s. This will rarely happen in a regression specification.
Write out the the IV Late estimator (the result from our derivation)
E[(Y_i(1)-Y_i(0)|D_i(1)-D_i(0)=1]
or equivivalently
E[(Y_i(1)-Y_i(0)|D_i(1)>D_i(0)] as Pischke writes.
Remember D_i(1)>D_i(0) are the compliers.
What happens if monotonicity is violated?
When the monotonicity assumption is violated, the LATE estimator is no longer identified. In this case, the compliers are no longer a well-defined group, and the instrumental variable cannot identify the treatment’s causal effect.
What do we estimate when then the control group do not have acces to the to the treatment?
If those in the control group cannot get the treatment, IV estimates the average treatment effect for the treated (ATT ).
Intuitively, the treated group can only consist of compliers, since “always-takers” are no longer present in $D_i = 1$ group Correcting the reduced-form (or intention-to-treat) effect for non-compliance (using the 1st stage) gives the ATT.This is sometimes called the Bloom estimator:
E[(Y_i(1)-Y_i(0))|D_i=1]
This is the wald estimator but with out the right term in the nämnare.
