Inductive Identification Across Multiple Decision Times

From Two Decisions to a Sequence

With several treatment decisions, a regime-specific mean is not obtained by adjusting for one collection of covariates in a single step. At each decision time, the covariate distribution depends on the treatment history that preceded it. Identification therefore proceeds through a sequence: identify the final conditional outcome mean, average it over the covariate distribution after the preceding treatment, and continue backward until only baseline covariates remain.

The two-time identification formula is the first instance of this pattern. Here the aim is to establish the pattern for any finite number of decisions, and to show why its recursive form follows from consistency and sequential exchangeability.

Histories, Regimes, and Assumptions

Suppose treatment is assigned at times \(t=0,\ldots,K\), where \(K\) is finite. Let \(L_0\) denote baseline covariates. For \(t=0,\ldots,K\), let \(A_t\) be treatment at time \(t\), and let \(L_{t+1}\) be the covariates measured after \(A_t\) when \(t<K\). The outcome \(Y\) is measured after the final treatment \(A_K\). Write \(\bar A_t=(A_0,\ldots,A_t)\) and \(\bar L_t=(L_0,\ldots,L_t)\). The history just before treatment at time \(t\) is \(H_t=(\bar L_t,\bar A_{t-1})\), with \(H_0=L_0\).

Fix a treatment regime \(\bar a=(a_0,\ldots,a_K)\). Write \(L_{t+1}^{\bar a_t}\) for the covariate that would follow treatment history \(\bar a_t=(a_0,\ldots,a_t)\), and \(Y^{\bar a}\) for the outcome under the full regime. The counterfactual history before decision \(t\) is \(H_t^{\bar a}=(L_0,L_1^{a_0},\ldots,L_t^{\bar a_{t-1}},\bar a_{t-1})\).

These assumptions have distinct jobs. Consistency connects observed quantities to counterfactuals when the observed treatment matches the regime. Exchangeability lets the observed treatment-specific conditional distributions stand in for counterfactual distributions. Support ensures those observed conditional distributions exist where the regime needs them. The treatment versions represented by each \(a_t\) must also be well defined, and the outcome must have a finite mean for the mean formula below to be finite.

The Backward Recursion

The formula is easiest to state recursively. At the final decision, define the conditional outcome mean

For \(t=K-1,\ldots,0\), define

The argument \((h_t,a_t,l_{t+1})\) is the next history with treatment set to its regime value. When covariates are discrete, the integral is a sum. The final identified mean averages the resulting baseline-specific quantity over the baseline distribution:

Proof. Begin at the final decision. Given a counterfactual history \(H_K^{\bar a}\), sequential exchangeability allows the conditional mean of \(Y^{\bar a}\) to be represented by the mean among people with that observed history who receive \(A_K=a_K\). Consistency identifies their observed outcome with \(Y^{\bar a}\). Thus the conditional counterfactual outcome mean is \(\mu_K(H_K^{\bar a})\).

Now suppose the conditional counterfactual mean from the next decision onward is represented by \(\mu_{t+1}\). By the law of total expectation,

Under the induction hypothesis, the inner conditional mean is \(\mu_{t+1}(H_{t+1}^{\bar a})\). Sequential exchangeability at time \(t\), together with consistency, identifies the conditional distribution of \(L_{t+1}^{\bar a_t}\) given the regime history with the observed distribution \(F(L_{t+1}\mid H_t,A_t=a_t)\). Therefore the outer expectation is exactly the recursion defining \(\mu_t(H_t^{\bar a})\). This proves the recursion backward through all decision times. Finally, \(H_0^{\bar a}=L_0\), whose distribution is the observed baseline distribution; averaging \(\mu_0(L_0)\) over that distribution gives \(E[Y^{\bar a}]\). The finite number of steps and the stated integrability condition make each conditional expectation well defined. \(\square\)

The same argument also explains the equivalent integral form: each recursive averaging step inserts one conditional covariate distribution. No single joint distribution of future covariates has to be guessed; the regime mean is built from the baseline law and the successive treatment-specific transition laws.

Worked Examples

Why the Induction Matters

A tempting shortcut is to calculate the outcome mean among everyone whose observed treatment history exactly equals the regime. That generally does not reproduce the regime-specific mean. Treatment assignment can depend on past covariates, so selecting people who happened to follow the entire regime can change the distribution of their histories. The recursive formula instead averages each conditional distribution using the treatment-specific transition law at that decision, then integrates over the preceding history.

This formula is an identification result, not a guarantee that an estimate will be reliable. In practice, estimating it often requires models for the conditional outcome mean and for each covariate transition. Poorly specified models can yield poor estimates even when the identifying assumptions are plausible. Support failures also matter: if the regime treatment is never observed for some histories that arise under the regime, the relevant conditional distribution cannot be learned directly from the observed data.

These approaches are alternatives for estimation, not different definitions of the causal target. The recursion is especially useful for seeing exactly which conditional distributions identify that target and where a required distribution is unsupported.

Common Mistake and Diagnostics

• Wrong reasoning: “The people observed to follow the regime are the people who would follow it under intervention, so their outcome average is the regime mean.” This ignores selection into observed treatment histories.
• Correction: Use the regime-specific conditional outcome and covariate distributions in the backward recursion, and integrate over the resulting histories rather than simply restricting to observed adherers.
• Warning sign: The estimated regime mean changes sharply when a conditional model is modestly altered. This can indicate model dependence, sparse histories, or extrapolation.
• Warning sign: Some treatment-history strata needed by the regime have no observations. The corresponding conditional law is not empirically identified without additional assumptions.
• Limitation: The derivation cannot repair unmeasured confounding, ill-defined treatment versions, or unsupported treatment choices. Those are failures of the assumptions, not algebraic problems with the recursion.