Graphical Identification

A causal graph turns assumptions about direction and common causes into a visible structure. The main question is not which model predicts best. It is whether the effect of an intervention can be recovered from the available observations under the proposed graph.

Backdoor adjustment

Let $X$ be an action or exposure, $Y$ the outcome, and $Z$ a set of observed common causes. For a discrete $Z$ , the adjustment formula is

\mathbb{E}[Y\mid \operatorname{do}(X=x)] =\sum_z \mathbb{E}[Y\mid X=x,Z=z]\Pr(Z=z).

The expression on the left is the average outcome if $X$ were set to $x$ . The inner expectation is the observed average outcome among cases with $X=x$ and $Z=z$ . The probability $\Pr(Z=z)$ weights that average by how often each value of $Z$ occurs. The formula is valid only when the stated graph makes $Z$ a sufficient adjustment set.

The arrow from $X$ to $Y$ represents the effect of interest. The path through $Z$ creates a second association, so $Z$ must be included in the adjustment under this graph.

Frontdoor identification

Sometimes no observed set blocks the backdoor paths from $X$ to $Y$ , but an observed mediator $M$ can still identify the effect. For discrete variables, the frontdoor formula is

\Pr(y\mid \operatorname{do}(x)) =\sum_m \Pr(m\mid x) \sum_{x'} \Pr(y\mid m,x')\Pr(x').

Here, $M$ lies on the causal route from $X$ to $Y$ , $m$ indexes its possible values, and $x'$ indexes the observed values of $X$ used in the second adjustment. This formula needs stronger structural conditions than the equation alone can show. In particular, the graph must support identification of both the effect of $X$ on $M$ and the effect of $M$ on $Y$ .

What the graph permits

Structure	Question answered	Main condition to examine
Backdoor adjustment	Which observed variables should be adjusted for?	All noncausal paths from $X$ to $Y$ are blocked
Frontdoor identification	Can a mediator recover the effect?	The required paths into and out of the mediator satisfy the frontdoor conditions
Instrumental variable	Can outside variation isolate part of the effect?	The instrument changes $X$ but has no separate route to $Y$

A graph can make assumptions precise and reveal contradictions. It cannot prove that its own arrows are correct from observational data alone. Subject knowledge and study design remain essential.

Comparing routes to causal identification

The backdoor experiment repeats the analysis while increasing the influence of a common cause. The known effect is fixed, so the error of each estimate can be measured directly.

Average signed error and root mean squared error for adjusted and unadjusted estimates as common-cause strength increases. — The unadjusted estimate develops substantial error as the common cause becomes stronger. The adjusted estimate stays near the known effect in these generated samples.

The frontdoor experiment compares the full distribution of estimates over 200 generated samples.

Distributions of frontdoor and ordinary regression estimates over 200 generated samples, with the known effect marked. — The frontdoor estimates gather around the known effect. Ordinary regression remains displaced because the treatment and outcome share an unmeasured common cause in this experiment.

The instrumental-variable experiment then changes the strength of the first-stage relationship.

First-stage strength and estimated effects at three levels of instrument relevance. — As instrument relevance weakens, the first-stage statistic falls and the instrumental estimate becomes unstable. The warning line at 10 is a practical check, not proof that an instrument is valid.

Reading the results

These experiments show what follows from each stated graph and set of assumptions. They do not tell us whether those assumptions describe a particular market; that judgment still depends on subject knowledge and study design.

Notebooks

All three notebooks are public and have passed full execution checks.


1. Compare safe and unsafe adjustment choices — Effect estimates and graph checks
2. Recover an effect through a mediator — Identification diagnostics
3. Examine the effect of weak instruments — Stability figures and summary tables