Instrument Variables and Exclusion Restriction from a Mediation perspective

Question

I'm having trouble making sense of the exclusion restriction in instrumental variables.

I understand that the unbiased treatment effect is $B = \frac{Cov(Y, Z)}{Cov(S, Z)}$, where $Y$ is the outcome, $S$ is the treatment, and $Z$ is the instrument. In other words, $B = \frac{ITT} {\text{Compliance Rate}}$.

However, if I think about this in a mediation framework, and apply the exclusion restriction, this makes less and less sense.

In a mediation framework, ITT = the total effect, or the $Cov(S,Z)\cdot Cov(Y,S) + Cov(Y,Z|S)$. So, the unbiased treatment effect is:

$\frac{(Cov(S,Z)\cdot Cov(Y,S) + Cov(Y,Z|S))}{Cov(S,Z)}$, which reduces to:

$Cov(Y,S) + \frac{Cov(Y, Z|S)}{Cov(S, Z)}$,

so the unbiased causal estimate is the effect of the biased treatment + the effect of the instrument ($\frac{controlling for the treatment} { compliance rate}$.

However, with the exclusion restriction, there is not supposed to be an effect of the instrument once we control for the treatment.

An example, from Gelman's Sesame Street example. First, obtaining the unbiased treatment effect via 2SLS:

fit.2s <- lm(regular ~ encour, data = df)
watched.hat <- fit.2s$fitted
fit.2b <- lm(postlet ~ watched.hat, data = df)
summary(fit.2b)

which gives the answer, 7.934.

And now, within an SEM framework:

library(foreign)
library(lavaan)
mod  <-
'
regular ~ a*encour
postlet ~ b*regular + c*encour
ind := a*b
total := a*b + c
'
fit <- sem(mod, data = df)
summary(fit)

 Regressions:
               Estimate  Std.Err  Z-value  P(>|z|)
  regular ~                                           
encour     (a)    0.362    0.051    7.134    0.000

  postlet ~                                           
  regular    (b)   13.698    2.079    6.589    0.000
  encour     (c)   -2.089    1.802   -1.160    0.246


  Defined Parameters:
               Estimate  Std.Err  Z-value  P(>|z|)
  ind               4.965    1.026    4.840    0.000
  total             2.876    1.778    1.617    0.106

13.698 - 2.089/.362 = 7.92

So, the only reason that the unbiased treatment effect is not just the biased treatment effect is the there is still an effect of the instrument when controlling for the treatment, which, according to the exclusion restriction should not happen.

Am I missing something here?

Answer 1

You correctly state that under the LATE-style IV assumptions with a causal effect of the IV Z on the treatment S, exogenous instrument, and no direct effect on the outcome Y, your treatment effect B of S on Y is identified as

$Cov(Y,Z)/Cov(S,Z) = ITT/Compliance Rate$

So clearly,

$ITT = Cov(Y, Z)$,

and not $Cov(S,Z)⋅Cov(Y,S)+Cov(Y,Z)$, as you mistakenly state. This is also intuitively clear because if the instrument is exogenous with respect to the treatment and the outcome, its causal effects are identified and (under linearity) are simply the correlation between the instrument and the outcome.

You further seem to imply that the IV and Y should be unrelated in a regression of Y on the treatment and the IV. This is not the case if treatment S is actually endogenous. Then, it is a collider, as it is caused by the IV and the unobserved error term of Y. Conditioning on the treatment makes the IV and the error term dependent, and therefore also the IV and Y. So you get a non-zero regression coefficient even if the exclusion restriction is valid. This should be very clear if you draw the causal graph.

If it was not the case, we could actually test the exclusion restriction, but of course we know we usually can't test it! (At least not that easily).