<p>I am confused about how to draw a Directed Acyclic Graph (DAG) for a causal chain with an instrumental variable for a setting in which the endogenous variable is the sum of the instrumental variable plus something else. Depending on how I represent it, I get that my instrument affects the outcome through more than one channel or not. I would like to figure out and understand what is the right way of drawing the DAG for my example. Below are pictures for the 2 options I thought of.</p>
<p>Let Y be my outcome, X be an edogenous variable such that X = Z + XmZ (XmZ stands for X - Z) and Z be an instrument. Also, Z impacts X both mechanically through its own presence in X, and through XmZ. Since my idea is to instrument X with Z, I draw the DAG shown below:</p>
<p><a href="https://i.sstatic.net/YF03M4Dx.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/YF03M4Dx.png" alt="DAG1" /></a></p>
<p>Note that there is no direct effect of Z on Y.</p>
<p>The other way I have thought of drawing this DAG is shown below. In this case, because I don't include X directly (I include Z and XmZ separately instead), we have that Z impacts Y directly, as well as through XmZ.</p>
<p><a href="https://i.sstatic.net/Ox2yW718.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Ox2yW718.png" alt="DAG2" /></a></p>
<p>I feel that the right DAG is the first one, because my regression really has X as its endgenous variable. However, it feels weird that the first and second DAGs have different implications in terms of if the instrument impacts the outcome directly. I am confused if this means I am approaching this problem incorrectly. Thanks for your help!</p>
<p>UPDATE: Following a comment I am clarifying what my empirical setting is to explain why I include X, XmZ and Z in my DAG. I am trying to run a regression where Y are crop yields and X is percentage of forest around a farm. Because forest cover is not random, I plan on instrumenting X (percentage of forest around a farm) with Z (percentage of public forests around a farm). On a nutshell, public forests are not under the control of farmers, so they could be a good instrument. Because forests can only be public or non-public, XmZ (which is X - Z) represents the percentage of non-public forests around a farm. My intuition tells me that changes in the percentage of public forests around a farm impact the percentage of forests (both public and non-public) around a farm in two ways: (i) mechanically, because Z is part of X and (ii) through the response of farmers in terms of their forest holdings (XmZ). The latter channel could be, for instance, because farmers provide environmental services to crops, so if public lands around you now have less farm, you and your neighbours might want to increase forest in your properties to make up for the lost environmental services. Finally, the confounder C here just highlights potential confounders related to manipulation of forest cover by farmers.</p>
<p>UPDATE 2: I realized I forgot to include a confounder to argue for the need of an instrument so I am adding it now to my DAGs. I have also excluded the mediator M, since it is not needed for the question I am asking here.</p>
<p>UPDATE 3: Following a comment, I have a third possible case for my DAG shown below:</p>
<p><a href="https://i.sstatic.net/3ADanolD.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/3ADanolD.png" alt="DAG3" /></a></p>
<p>The confounder C could, for instance, a new unobserved farming practice adopted by some farmers that both affects yields positively and creates incentives for farmers to change the amount of forest they hold in their land.</p>

CC BY-SA 4.0
2024-11-11T10:58:14.857Z
The term XmZ seems meaningless. If you intend 'm' to be minus you should write up the structural with an actual '-'. DAGs are representations of non-parametric structural models. What are you hoping to represent with the equation $X = Z + XmZ$? So X is causally affected by $Z$ - which is fine - but also by $X-Z$ - this seems to open up the possibility that $X$ affects $X$ which introduces a cycle in a DAG and is therefore inconsistent. It might be more helpful if you describe in normal language what problem you are trying to solve.
Jesper Hybel
CC BY-SA 4.0
2024-11-12T20:54:51.497Z
@JesperHybel, I have just updated my question to reflect your points. Update 1 describes the empirical setting I have in mind to help with intuition. Update 2 I did to make stuff clearer. On your comment about m being minus, indeed $XmZ$ is just $X - Z$. I wrote m instead of - because R was giving an error when I tried to write - in my DAG. But is is really the same thing. Let me know if it is not clear whay I mean to represent with $X = Z + XmZ$ and if there is a cycle in $X$ in my DAG now and how can I correctly represent my setting in terms of a DAG.
RobertoAS
CC BY-SA 4.0
2024-11-13T13:29:55.583Z
Ok, that clarifies matters ... not completely but for the better. Your second DAG matches your story partially. However, your instrument Z cannot directly affect the dependent variable. The instrument changes Y only through X. You are interested in the effect of X on Y. But C confounds X. So to nullify the confounder you only focus on variation in X that is due to variation in Z. So I would change XmZ to simply X and remove the arrow from Z to Y. And then work on improving the story ... How, the confounding occurs and what C is is currently unclear.
Jesper Hybel
CC BY-SA 4.0
2024-11-14T05:37:07.003Z
@JesperHybel, thanks for your comment. I have further updated my question to reflect your comments. So the correct DAG would be the third one (the one shown in update 3)? If that's the case, it would be helpful if you could develop a little bit more why it is fine not including $XmZ$ in the DAG. As for your comment that the  instrument $Z$ cannot directly affect the dependent variable, I am aware of that and my question was motivated by my concern that maybe $Z$ is not a good instrument due to affecting $Y$ directly (that would be the case I feel if the second DAG was the correct one).
RobertoAS
<econometrics><instrumental-variable><causality>
Directed Acyclic Graph (DAG) endogenous variable is the sum of the instrumental variable plus something else
RobertoAS
# Directed Acyclic Graph (DAG) endogenous variable is the sum of the instrumental variable plus something else

(Asked by: username_0 on 2024-11-11)

I am confused about how to draw a Directed Acyclic Graph (DAG) for a causal chain with an instrumental variable for a setting in which the endogenous variable is the sum of the instrumental variable plus something else. Depending on how I represent it, I get that my instrument affects the outcome through more than one channel or not. I would like to figure out and understand what is the right way of drawing the DAG for my example. Below are pictures for the 2 options I thought of.

Let Y be my outcome, X be an edogenous variable such that X = Z + XmZ (XmZ stands for X - Z) and Z be an instrument. Also, Z impacts X both mechanically through its own presence in X, and through XmZ. Since my idea is to instrument X with Z, I draw the DAG shown below:

[![DAG1](https://i.sstatic.net/YF03M4Dx.png)](https://i.sstatic.net/YF03M4Dx.png)

Note that there is no direct effect of Z on Y.

The other way I have thought of drawing this DAG is shown below. In this case, because I don't include X directly (I include Z and XmZ separately instead), we have that Z impacts Y directly, as well as through XmZ.

[![DAG2](https://i.sstatic.net/Ox2yW718.png)](https://i.sstatic.net/Ox2yW718.png)

I feel that the right DAG is the first one, because my regression really has X as its endgenous variable. However, it feels weird that the first and second DAGs have different implications in terms of if the instrument impacts the outcome directly. I am confused if this means I am approaching this problem incorrectly. Thanks for your help!

UPDATE: Following a comment I am clarifying what my empirical setting is to explain why I include X, XmZ and Z in my DAG. I am trying to run a regression where Y are crop yields and X is percentage of forest around a farm. Because forest cover is not random, I plan on instrumenting X (percentage of forest around a farm) with Z (percentage of public forests around a farm). On a nutshell, public forests are not under the control of farmers, so they could be a good instrument. Because forests can only be public or non-public, XmZ (which is X - Z) represents the percentage of non-public forests around a farm. My intuition tells me that changes in the percentage of public forests around a farm impact the percentage of forests (both public and non-public) around a farm in two ways: (i) mechanically, because Z is part of X and (ii) through the response of farmers in terms of their forest holdings (XmZ). The latter channel could be, for instance, because farmers provide environmental services to crops, so if public lands around you now have less farm, you and your neighbours might want to increase forest in your properties to make up for the lost environmental services. Finally, the confounder C here just highlights potential confounders related to manipulation of forest cover by farmers.

UPDATE 2: I realized I forgot to include a confounder to argue for the need of an instrument so I am adding it now to my DAGs. I have also excluded the mediator M, since it is not needed for the question I am asking here.

UPDATE 3: Following a comment, I have a third possible case for my DAG shown below:

[![DAG3](https://i.sstatic.net/3ADanolD.png)](https://i.sstatic.net/3ADanolD.png)

The confounder C could, for instance, a new unobserved farming practice adopted by some farmers that both affects yields positively and creates incentives for farmers to change the amount of forest they hold in their land.

**Tags:** econometrics, instrumental-variable, causality

> **Comments:**
> **username_1**: The term XmZ seems meaningless. If you intend 'm' to be minus you should write up the structural with an actual '-'. DAGs are representations of non-parametric structural models. What are you hoping to represent with the equation $X = Z + XmZ$? So X is causally affected by $Z$ - which is fine - but also by $X-Z$ - this seems to open up the possibility that $X$ affects $X$ which introduces a cycle in a DAG and is therefore inconsistent. It might be more helpful if you describe in normal language what problem you are trying to solve.  
> **username_0**: @JesperHybel, I have just updated my question to reflect your points. Update 1 describes the empirical setting I have in mind to help with intuition. Update 2 I did to make stuff clearer. On your comment about m being minus, indeed $XmZ$ is just $X - Z$. I wrote m instead of - because R was giving an error when I tried to write - in my DAG. But is is really the same thing. Let me know if it is not clear whay I mean to represent with $X = Z + XmZ$ and if there is a cycle in $X$ in my DAG now and how can I correctly represent my setting in terms of a DAG.  
> **username_1**: Ok, that clarifies matters ... not completely but for the better. Your second DAG matches your story partially. However, your instrument Z cannot directly affect the dependent variable. The instrument changes Y only through X. You are interested in the effect of X on Y. But C confounds X. So to nullify the confounder you only focus on variation in X that is due to variation in Z. So I would change XmZ to simply X and remove the arrow from Z to Y. And then work on improving the story ... How, the confounding occurs and what C is is currently unclear.  
> **username_0**: @JesperHybel, thanks for your comment. I have further updated my question to reflect your comments. So the correct DAG would be the third one (the one shown in update 3)? If that's the case, it would be helpful if you could develop a little bit more why it is fine not including $XmZ$ in the DAG. As for your comment that the  instrument $Z$ cannot directly affect the dependent variable, I am aware of that and my question was motivated by my concern that maybe $Z$ is not a good instrument due to affecting $Y$ directly (that would be the case I feel if the second DAG was the correct one).  

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