No problem at all if the IV is properly controlled for by the said M.. It shows how strong the mediator variable is in determining the effect of treatment variable on the response variable(outcome). In the world of medicine. You take drugs with water to fight illness. But at same time you are asked to eat enough fruits. The fruits or water plays a role in the effectiveness of the drug for you to get well. The effectiveness might not be held down only to the drug alone, but to the exposure of the patient to fruits and water used in taking the drug.

@Atilla Onkulu

It's not at all problematic. In fact, in a mediation model, better are the chances for stronger indirect effect i.e., product of paths X>M and M>Y respectively.

Hello Atilla,

1) I do not quite understand your problem. What do you mean with "correlation between M and Y"? The zero-order-correlation or the error-correlation? If it is the zero-order correlation you would assume a strong correlation due to the effect of M on Y and perhaps confounding (that you perhaps have accounted for)?

2) Your model is not clear. When you say you use an instrument Z: What do you instrument with Z? X?

3) When you say that you have the exclusion criteria that X-->M and X-->Y, then M is a partial mediator or in other words: X has a indirect effect on Y mediated by M AND a direct effect. If this is the case, X is not valid instrument for the M-Y effect. In this case, you need an instrument for M.

The optimal situation for mediation is

a) in case you have a full mediation (no effect of X on Y): In this situation, X qualifies as an instrument for the M-Y effect (assumed a strong effect). This allows to test for endogeneity (by adding an error covariance on the M-Y link) and if this is significant, leave the error covariance estimated. This identifies the M-Y effect. For the X-M effect to be identified, you need an instrument for X that is conditionally independent of M (controlling for X) or in other words: has no direct effect on M.

b) In case you have partial mediation (where X as also a direct effect on Y): Here, X is no instrument for M (unless you can select a further mediator (M2) that turns X into a "conditional instrument"--that is, it is valid/exclusive when you control for M2. If not, you need an instrument for X and an instrument for M.

Hope that helps

--Holger

Busari Yusuf and Imran Anwar thank you for the answers!

Holger Steinmetz thank you for the answer and sorry for not being clear enough in the question. Let me explain a little further.

(1) By "correlation between M and Y", I mean ordinary pairwise correlation between observed variables, not between unobserved error terms. And it is this confounding that I worry whether I account for or not.

(2) & (3) I use a single instrument (M) for X in both M(x) and Y(x). With the help of two (well-known) exclusion restrictions between X and both M(x) and Y(x), Z as an instrument for X can enable the causal effect of X on M as well as X on Y. The exclusion restriction that applies to the mediator M can also apply to outcome Y.

The model explained in Dippel et al (2019) brings a third (novel) exclusion restriction between Z and Y(m) (conditional on X). This implies that instrumental variable Z can be used to evaluate the causal relation of M on Y if (and only if) conditioned on X.

In line with what you discuss in (a), I have a full mediation (no significant direct effect of X on Y). So, I was hoping that my single IV (Z) accounts for the endogeneity in both of the relationships in indirect effect.

I hope I am not making something already complicated even more complicated by trying to explain it.

Thank you all again for your interest in my question.

Hi Attilla,

Ad 1: As I said you would expect a strong correlation

Ad 2: I do not quite understand how you can use M (that is an endogenous mediator) as an instrument for X. I do not understand the notation M(x) and Y(x), Looks like potential outcome notation with which I am not very familiar.

As I said: If you have a direct mediation with a hopefully strong X --> M effect, X

qualifies as an instrument for M-Y and Z can be used as an instrument for X.

The part of your post, however is unclear: You say you have a full mediation but you need and instrument for the X-Y effect. In a full mediation model, there is not X-Y effect (conditionally on M).

Best,

--Holger

Dear Holger Steinmetz , thank you again for your comments. I'll need to rethink about the underlying assumptions behind ivmediate command and how they fit to my theoretical model. Using a single IV with the help of 3 separate 2SLSs is a novel approach and I'll need to dig deeper to grasp how this approach resolves multiple (in)dependence problem.

Regards,

Atilla

Hi perhaps, switching to SEM with the lavaan package (www.lavaan.org) in R would be an easy way.

In such a model the code would look like this

model