# Process of establishing a causal relationship

I am trying to clarify my understanding of establishing a causal effect, as mentioned in textbooks/online resources that I am reading.

Is the following a correct understanding?

To identify a causal effect of X on Y, we need to justify:

1. The direction of causality, i.e. that it's indeed X->Y
2. A ceteris paribus effect to give us the magnitude of the relationship, i.e. the "b" from Y = a + bX + u

The causal direction of an econometric model is an a priori component of the model. This causal direction is described by the "structural model", which is a statement (written as an equation) that claims some relationship exists (supported via theory for how the researcher believes the world works). The next task is to extract a ceteris paribus relationship from the aforementioned structural model, through regression analysis, etc. Once a ceteris paribus relationship is found, this can be described as the causal effect of X on Y, assuming the structural model is correct.

A ceteris paribus effect is not sufficient to make a causal claim, because one can take some data and extract a statistical relationship that satisfies E(u|X) = 0, however this doesn't guarantee it's a causal relationship.

• Is it possible to provide links to your references? or the book titles? Commented Mar 30 at 11:28

It depends on the definition of causality. In the statistics/econometrics world the most common definition uses the counterfactual model which, in a more philosophical setting, is closely related to the theory of possible worlds by Lewis. Say you have a treatment $$A$$ and you can either assign it ($$1$$) or not ($$0$$) and are interested in an outcome $$Y$$. The outcome you observe is $$Y^1$$ if you give the treatment and $$Y^0$$ if you do not. The outcome, might be influenced by a myriad of things but what you're interested in the effect the treatment has on the outcome. Here is where ceteris paribus comes in. In possible worlds jargon, you ask yourself "what would have happened in the closest world?" The closest world is the world where everything else "before the treatment" stays the same except the treatment and all the variables it causally influences. If I had some inter dimensional portal and was able to compare the two then I would be able to decide what the treatment does. Is it better? Is it worse? Now, from a less sci-fi point of view, you have your model (structural equations, DAG,...) and you intervene on the treatment. You force the value of treatment you'd like to compare and look what happens to the outcome. I personally find the SWIG framework proposed by Robins to be particularly intuitive. You replace the node of your DAG with a split in half node, the arrows coming directly into the node do not habe an effect anymore. This is the same idea as closest world! In general keep in mind that structural equations $$Y=\beta_0+\beta_a\cdot A\dots$$ are not to be interpreted as 'equalities' but more as causes(one the right) and outcomes (on the left).
Now, the first step you described, deciding which way the arrow points $$X\rightarrow Y$$ or $$X\leftarrow Y$$ is extremely hard! Some people like to believe in 'automatic structure discovery' approaches like the PC algorithm which decide from the data which way arrows point but you often need experts to tell you which way your arrows point, which even then the structure may be questionable.
• As a comment: the PC algorithm is unable to distinguish between a particular arrow's direction if no $v$ structures are created or destroyed by reversing that arrow's direction. This is Theorem 1.2.8 in Pearl's Causality, 2nd Ed. Essentially, the PC algorithm can detect the skeleton and the $v$-structures, but likely not the final, correct causal graph. Commented Apr 1 at 15:24