Identification, Asymptotics, and Axiomatic Foundations (ecta-identification)

When to trigger

• An estimator is defined and shown consistent, but its limiting distribution is missing
• Identification is asserted ("the parameter is identified") without a proof or counterexample analysis
• A theory model posits behavior but the axioms are not isolated, or existence/uniqueness is unproven
• Inference is proposed (standard errors, tests) without the asymptotic theory that justifies it

This is the formal spine. Econometrica referees check it first; a gap here sinks the paper.

Re-slant for Econometrica. Identification here is not primarily "do I have a credible research design for a causal estimate" (that framing belongs to AER / QJE / JPE / REStud). Econometrica's core is identification and estimator validity inside structural and econometric models — is the structural parameter / functional a one-to-one image of the data distribution, and does the proposed estimator have a derived limiting distribution that licenses its inference? Credible-design content (Branch D) is still in scope for the journal's applied/structural submissions, but the methodological object — completeness, rank, support, the asymptotic law of your estimator — is what carries the paper. Lineage: GMM identification and asymptotics (Hansen 1982), nested fixed-point identification of a dynamic discrete-choice model (Rust 1987), selection-model identification (Heckman 1979).

Branch A — Econometric theory: identification

1. Define the parameter / object as a functional of the data-generating process, separate from any estimator. Identification is a property of the population, not the sample.
2. State the identification conditions as numbered assumptions (rank / completeness / support / exclusion / monotonicity, as relevant). For each, say what fails without it.
3. Prove identification: show the map from distribution to parameter is one-to-one on the admissible class. Where identification can fail, give the explicit failure (partial identification, set identification, point identification under added conditions).
4. Distinguish point vs. partial identification. If only set identification holds, define the identified set and characterize it; do not silently assume point identification.

Branch B — Econometric theory: asymptotic distribution theory

1. Consistency under stated conditions (which sample sizes / sequences; i.i.d., dependent, or panel asymptotics — be explicit about the regime).
2. Rate of convergence — root-n or nonstandard (n^{1/3}, boundary, super-consistent). A nonstandard rate must be derived, not assumed.
3. Limiting distribution — derive it; state the asymptotic variance and a consistent estimator of it. If the limit is non-normal (e.g., from a boundary, a non-differentiable moment, or a unit root), characterize it and justify inference accordingly.
4. Uniformity — is the asymptotics pointwise or uniform over the parameter space? Modern referees ask for uniform validity (weak-identification-robust, boundary-robust) where the pointwise theory is known to mislead.
5. Regularity conditions — smoothness, moment, and bandwidth/tuning conditions stated precisely; primitive where possible rather than high-level.

Branch C — Micro / game / decision theory: axioms and existence/uniqueness

1. Isolate the axioms on the primitive (preference relation, choice function, payoff structure). Each axiom should be behaviorally interpretable and stated independently.
2. Existence — prove an equilibrium / representation / solution exists (fixed-point, topological, or constructive argument), with the topology and continuity conditions made explicit.
3. Uniqueness (or characterization of the set) — prove uniqueness or characterize multiplicity; a representation theorem should pin the functional form up to its known degrees of freedom (e.g., affine transformations of a utility index).
4. Independence / tightness of axioms — show no axiom is redundant (each is necessary) and, ideally, that the axiom set is tight (relaxing any one breaks the representation).
5. Behavioral payoff — translate the formal result into a statement about observable behavior or comparative statics.

Branch D — Structural / empirical (and credible-design applied)

1. State the model's microfoundations and the identifying restrictions explicitly.
2. Argue identification of the structural parameters from the available variation (functional form, exclusion, support, instruments) — separate what is identified nonparametrically from what relies on parametric assumptions.
3. Provide the estimator's asymptotics or a justified inference procedure; if you use a known estimator off the shelf, cite the precise theorem that licenses your standard errors.
4. Counterfactuals must be objects the identification argument actually delivers.
5. Credible-design content still belongs here for applied/structural submissions (a DID, RDD, or IV used inside the paper). But at Econometrica the design alone is not the contribution — the methodological or identification argument is. If the design is off-the-shelf and the estimand is the whole point, the paper is general-interest-applied, not Econometrica (see ecta-topic-selection).

Execution bridge (StatsPAI / Stata MCP)

Estimate and audit the design, don't only describe it. Full map: execution-with-mcp. Econometrica publishes econometric theory and applied micro; the chain below serves its applied/empirical papers (weak-IV-robust and modern-DiD reporting expected) — pure theory uses its own apparatus.

• detect_design → recommend → fit with as_handle=true → audit_result.
• Observational causal claims: staggered DiD (callaway_santanna / sun_abraham + bacon_decomposition + honest_did_from_result); IV (effective_f_test + anderson_rubin_ci); RDD (rdrobust + mccrary_test).
• Experiments: randomization-based inference + romano_wolf for many-outcome control.
• Sensitivity: oster_delta / sensemakr for observational claims.

Report the magnitude in interpretable units; route the full battery to the appendix. A run end-to-end (synthetic data, real returns) is in the JF execution walkthrough.

Checklist

• Parameter/object defined as a population functional, separate from the estimator
• Identification conditions numbered; each shown to bind (counterexample if dropped)
• Point vs. partial identification stated honestly
• Rate of convergence derived (not assumed), including any nonstandard rate
• Limiting distribution derived; asymptotic variance + consistent estimator given
• Pointwise vs. uniform asymptotics addressed; weak-identification robustness considered
• Regularity conditions primitive and minimal
• (Theory) axioms isolated, independent, behaviorally interpretable; existence + uniqueness proven

Anti-patterns

• "The parameter is clearly identified" with no argument and no failure analysis
• Reporting standard errors with no asymptotic theory justifying them
• Assuming root-n / normality when the moment condition is non-differentiable or on a boundary
• Pointwise asymptotics in a setting (weak IV, near-boundary) where they are known to fail
• High-level "regularity conditions" that quietly assume the hard part
• Axioms that overlap or include a redundant one; existence claimed without a fixed-point argument
• A representation theorem that does not pin the functional form (uniqueness left open)

Output format

【Branch】identification / asymptotics / axioms / structural
【Object/parameter】... (population functional or representation)
【Identification】point / partial — argument: ...
【Rate & limit】rate: ...; limiting distribution: ...; variance estimator: ...
【Uniformity】pointwise / uniform / weak-id-robust
【Regularity conditions】[...]  (gaps: [...])
【Next step】ecta-theory-model