Does direction of causality between instrument and variable matter?

Question

The standard scheme of instrumental variable in terms of causality (->) is:

Z -> X -> Y

Where Z is an instrument, X an endogenous variable, and Y a response.

Is it possible, that following relations:

Z <- X ->Y

Z <-> X ->Y

are also valid?

While correlation between instrument and variable is satisfied, how may I think of exclusion restriction in such cases?

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NOTE: The notation <-> is not explicit and might lead to different understandings of the problem. Still, the answers highlight this issue and use it to show important aspects of the problem. When reading, please proceed with caution about this part of the question.

Answer 1

Yes, the direction matters. As pointed in this answer, to check whether $Z$ is an instrument for the causal effect of $X$ on $Y$ conditional on a set of covariates $S$, you have two simple graphical conditions:

1. $(Z \not\perp X|S)_{G}$
2. $(Z\perp Y|S)_{G_{\overline{X}}}$

The first condition requires $Z$ to be connected to $X$ in the original DAG. The second condition requires $Z$ to not be connected to $Y$ if we intervene on $X$ (represented by the DAG $G_{\overline{X}}$, where you remove the arrows pointing to $X$). Thus,

Z -> X -> Y : here Z is a valid instrument.

Z <-> X -> Y: here Z is a valid instrument (assuming that a bidirected edge represents an unobserved common cause, as it does in semi-Markovian models).

Z <- X -> Y : here Z is not a valid instrument.

PS: jsk's answer is not correct, let me show you how Z <-> X is a valid instrument.

Let the structural model be:

$$ Z = U_1 + U_z\\ X = U_1 + U_2 + U_x\\ Y = \beta X + U_{2} + U_y $$

Where all the $U$'s are unobserved mutually independent random variables. This corresponds to the DAG z <--> x -->y with also x<-->y. Thus,

$$ \frac{cov(Y, Z)}{cov(X, Z)} = \frac{\beta cov(X, Z)}{cov(X,Z)} = \beta $$

Answer 2

Yes, direction does matter.

According to Hernan and Robins' new causal inference book https://cdn1.sph.harvard.edu/wp-content/uploads/sites/1268/1268/20/hernanrobins_v2.17.21.pdf

the following three conditions must be met:

$i.$ $Z$ is associated with $X$.

$ii.$ $Z$ does not affect $Y$ except through its potential effect on $X$.

$iii.$ $Z$ and $Y$ do not share common causes.

Condition $(iii)$ rules out relations such as $X$ - > $Z$ or $X$ < - > $Z$ because $X$ cannot have a causal effect on both $Z$ and $Y$

Edit: whether or not $X<->Z$ is acceptable for an instrument depends on the definition of $X<->Z$. If it means that they are correlated because of a third variable, like in Carlos's example, then it's ok. If it suggests a feedback loop where a causal arrow can be drawn from X to Z as well then Z is not a valid instrument.