Thanks, coauthor!

]]>Part (b) isn’t saying that exclusion is violated because the noncompliers have a different treatment effect than compliers. It’s saying that exclusion is violated because they have a non-zero effect of the instrument. Setting the effect to half that of the compliers was just a way of making that non-zero effect concrete. But I agree that the wording is sloppy (particularly use of the word treatment). I have a better version that I used in my causal inference course last year that I’ll be adding to the new book this week! That one has been beta-tested by students so should be more clear.

]]>This is especially straightforward in the case of a binary instrument and binary treatment variable, as there are then only 4 principal strata to worry about: the well-known Compliers, Defiers, Never Takers and Always Takers. To simulate from these groups, we just need to assign class probabilities to these four principal strata. At this point, we can start adding in model assumptions in terms of how we choose these probabilities. For instance, we can make the monotonicity assumption by assigning a probability of 0 to the Defier class. Then, we can think about a model for the conditional distribution of the potential outcomes within each of these principal strata. For instance, we may model Y(1) | Complier ~ N(mu_{C1}, sigma^2_{C1}) and Y(0) | Complier ~ N(mu_{C0}, sigma^2_{C0}). This tells us that the Complier Average Causal Effect is mu_{C1} – mu_{C0}. Here again we can add in assumptions through our choice of parameters. For instance, the exclusion restriction says that the instrument has no effect on the outcome except through the treatment actually received. In other words, the distribution of Y(1) | Never Taker is the same as the distribution of Y(0) | Never Taker, and similarly for the Always Takers.

By the way, this paper by Imbens and Rubin is a fantastic explanation of the binary treatment/binary instrument IV model, and shows how to approach IV models in terms of distributions of potential outcomes conditional on principal strata: https://projecteuclid.org/download/pdf_1/euclid.aos/1034276631

Here’s a short R script demonstrating this approach for the binary instrument/binary treatment IV model under the exclusion restriction:

###

n <- 100

# we'll assume there are only Never Takers and Compliers, with 80% of the population being Compliers

complier <- rbinom(n, 1, .8)

# we now generate potential outcomes Y(0) and Y(1) for each person in our sample.

# we'll use a normal model where for simplicity all variances are equal to 1.

# we'll also generate Y(0) and Y(1) separately.

# correlation between potential outcomes can be incorporated by simulating

# from a bivariate normal distribution.

Y0 <- Y1 <- rep(NA, n)

# simulating potential outcomes for compliers.

# we'll make the CACE equal to 0.5.

Y0[complier == 1] <- rnorm(sum(complier), mean = 1, sd = 1)

Y1[complier == 1] <- rnorm(sum(complier), mean = 1.5, sd = 1)

# simulating potential outcomes for never takers.

# under the exclusion restriction, Y(0) and Y(1) have the same distribution for never takers

Y0[complier == 0] <- rnorm(n – sum(complier), mean = 1, sd = 1)

Y1[complier == 0] <- rnorm(n – sum(complier), mean = 1, sd = 1)

# simulate a randomized instrument

Z <- rbinom(n, 1, .5)

# construct observed data

# observed treatment is 0 if Z = 0.

# if Z = 1, observed treatment is 0 for never takers and 1 for compliers

treatment_observed <- 0 + Z*complier

Y_observed <- (1 – Z)*Y0 + Z*Y1

###