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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?

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.

cure
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2 Answers2

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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,

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

Carlos Cinelli
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  • I think this highlights the need to be very clear clear about what exactly $XZ$ actually means. In your revised example, I would argue that X and Z are driven by a third variable, which seems different than my understanding of the notation $XZ$. – jsk Jun 09 '19 at 22:40
  • @jsk this is standard notation for semi-Markovian models. – Carlos Cinelli Jun 09 '19 at 23:35
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    Not standard to everyone. Just read a paper by Pearl and Greenland in which they say that SOME authors use the notation in this way. There's nothing in the OP's question to suggest his interpretation of the notation, though he may very well agree with you. – jsk Jun 10 '19 at 00:19
  • What if $Y = \beta X + U_1 + U_y$? Would it not then be the case that $ZX$ but then Z would be correlated with the omitted variable and hence not be valid instrument? – Jesper for President Jun 10 '19 at 13:12
  • @JesperHybel If you have U1 in the structural equation of Y, this means the error terms of Z and Y are dependent. Thus you have an extra bidirected edge ZY and **no case works**, be it Z—>X or ZX. The graphical conditions are explicitly stated there. – Carlos Cinelli Jun 10 '19 at 14:25
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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.

jsk
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