Identification: When Observational Data Is Enough

You have a causal graph encoding your assumptions, and observational data. The question that organises this entire field is:

Can the causal effect $P(Y \mid do(X))$ be written as a formula involving only quantities you can estimate from observed data?

If yes, the effect is identifiable, and the formula tells you exactly what to compute. Statistical estimation then proceeds normally.

If no, then no amount of data helps. This is the point most often misunderstood. Non-identifiability is not a small-sample problem or a modelling weakness. It means that two different causal structures produce the identical observational distribution, so no procedure operating on that distribution can distinguish them. A billion rows will not.

Note the order of operations, which is the opposite of common practice. Identification comes first, from the graph, before any data is touched. It is a question about structure. Only once you know a quantity is identifiable and by what formula does estimation become relevant.

Running a regression and interpreting a coefficient causally skips this step entirely, which is why so much applied work makes claims its design cannot support.

The workhorse result tells you when adjusting for a set of variables suffices.

A backdoor path from X to Y is any path that starts with an arrow pointing into X. These are the non-causal routes through which X and Y become associated, and the arrow into X is what marks a path as carrying influence from a common cause rather than from X itself.

A set of variables Z satisfies the backdoor criterion if two conditions hold. Z blocks every backdoor path from X to Y. And Z contains no descendant of X.

The second condition is easy to overlook and matters. A descendant of X is affected by the treatment, so conditioning on one blocks part of the causal effect you are trying to measure, or opens a collider path.

When the criterion holds, the effect is identified by the adjustment formula:

$P(Y \mid do(X{=}x)) = \sum_z P(Y \mid X{=}x, Z{=}z)\, P(Z{=}z)$

Read the structure carefully, because it is the whole point. You compute the conditional distribution within each stratum of Z, then average using the marginal distribution of Z rather than its distribution given X.

That re-weighting is what simulates the intervention: it asks what the outcome would be if the whole population, not just those who received the treatment, had received it.

The formula is exact and, applied naively, unusable. With several continuous confounders there are no populated strata to average over.

The practical estimators all approximate the same quantity by different routes.

Outcome regression. Model $E[Y \mid X, Z]$ with any regression method, predict for every unit under treatment and under control, and average the difference. Machine learning models are perfectly acceptable here, since the model's job is prediction within strata.

Propensity scores. Model the probability of treatment given covariates, $e(Z) = P(X{=}1 \mid Z)$. A key result is that adjusting for this single number suffices when adjusting for all of Z would, which collapses a high-dimensional problem to one dimension. Use it by matching treated to control units with similar scores, or by inverse-probability weighting.

Doubly robust estimators. Combine both. The attractive property is that the estimate remains consistent if either the outcome model or the propensity model is correctly specified, giving two chances rather than one.

A warning applies to all of them. Any of these fits happily even when the backdoor criterion does not hold. The estimator does not know whether Z is a valid adjustment set, and it will return a confidently wrong number.

Backdoor adjustment requires measuring the confounders. What if the confounder is unmeasured, and unmeasurable?

Surprisingly, the effect can sometimes still be identified, through a mediator.

The structure required is specific. X affects Y only through a mediator M. There is no unblocked backdoor path from X to M. And every backdoor path from M to Y is blocked by X.

When this holds, the effect is identified even with an unmeasured confounder of X and Y, by chaining two adjustments: the effect of X on M, and the effect of M on Y, adjusting for X.

The intuition is that you decompose the causal path into two segments, each of which is unconfounded even though the whole is not. The unmeasured confounder affects X and Y, but not the X-to-M link nor, conditional on X, the M-to-Y link.

The classic illustration is smoking and cancer with an unmeasured genetic confounder, where tar deposits serve as the mediator.

In practice front-door identification is used far less than backdoor adjustment, because its assumptions are demanding: all of the effect must flow through the observed mediator, and no unmeasured factor may affect both mediator and outcome. Both are strong and frequently implausible.

The strategies are not interchangeable. Each requires a different structure, and the correct move is to determine which structure your problem plausibly has before choosing an estimator.

The third major strategy comes from econometrics and is the most widely used when confounding is unmeasured.

An instrument Z is a variable satisfying three conditions.

Relevance. Z affects X. This is testable in your data.

Exclusion. Z affects Y only through X. There is no direct arrow from Z to Y.

Independence. Z shares no common cause with Y.

Given these, the logic is that variation in X driven by Z is as-if random with respect to the confounders, because Z is unrelated to them. So you use only that portion of X's variation.

The standard estimator implements exactly that in two stages: regress X on Z to obtain the predicted values, then regress Y on those predicted values.

The difficulty is that the two crucial conditions are untestable. Exclusion and independence are assumptions about arrows that are absent from the graph, and absence of an arrow cannot be verified from data.

Two failure modes are common enough to name. A weak instrument, one only slightly related to X, produces estimates with enormous variance and can be badly biased. And a violated exclusion restriction biases the estimate arbitrarily, with no diagnostic that reliably detects it.

Sometimes no strategy applies, and the professional response is not to proceed anyway.

There are principled things to do.

Report bounds. Even without point identification, the effect can often be bounded. The range may be wide and it is honest, and sometimes a bound excluding zero is all the decision requires.

Sensitivity analysis. Ask how strong an unmeasured confounder would have to be to explain away your estimate. If it would need to be implausibly strong, that is real evidence. If a modest one suffices, your estimate was never informative.

Report the association as an association. A correlation honestly labelled is useful. The same number labelled as a causal effect is a false claim.

Change the design. Find a natural experiment, a policy discontinuity, or a subpopulation where assignment was as good as random. This is usually a better investment than a more elaborate estimator.

The failure to avoid is the common one: fitting a flexible model with many covariates and interpreting a coefficient causally. Model flexibility does not create identification. A neural network fitted to confounded data estimates a confounded quantity with great precision.

What each approach requires and where each breaks.

Two points to carry forward.

Estimation is the easy part. Modern machine learning gives excellent tools for fitting outcome and propensity models, and none of that capability helps if the identification argument is wrong. The hard work is upstream and it is about structure.

Untestable assumptions should be stated, not hidden. Every non-experimental strategy rests on assumptions the data cannot check. Naming them lets a reader disagree with your reasoning rather than only with your numbers, and that is the honest form of an observational claim.

All of this assumed you already have the graph. The next lesson asks whether you can learn it.