| name | ecta-theory-model |
| description | Use when the central theorem of an Econometrica manuscript must be stated cleanly and proven completely — assumptions, generality, and proof strategy. Builds and audits the theorem-and-proof core; it does not derive the identification/asymptotic conditions (use ecta-identification) or run simulations (use ecta-robustness). |
Theorem and Proof Strategy (ecta-theory-model)
When to trigger
- The central result is stated informally or its conditions are scattered through the text
- The proof has gaps: "it is easy to see," "by a standard argument," or a missing step
- Regularity conditions are invoked mid-proof but never stated as assumptions
- You are unsure whether the result is as general as claimed, or whether the proof secretly uses more
This is the heart of an Econometrica paper. The product is a complete, correct proof of a
general, cleanly stated theorem. Econometrica is the field's theorem-proof journal: the
referee pool (routed by the handling co-editor) reads proofs line by line, and a single
genuine gap can sink an otherwise strong paper — this is the most common rejection cause
here, in contrast to applied siblings (AER / QJE / JPE / REStud) where a credible empirical
narrative can carry the day. Plan around the 45-page main-text limit (incl. references
and appendices): the body conveys the theorem, its interpretation, and the architecture of
the proof, while the complete, formal proofs live in the Supplemental Material (the
≤25-page Supplemental Appendix or the unrestricted online Supplemental Material), which does
not count against the body's page budget. "Proof omitted to save space" is not acceptable —
the proof must exist somewhere complete.
Structure: definitions → assumptions → theorems → proofs
Organize the formal content in this order, with consistent theorem numbering:
- Definitions — every object used in the theorem (spaces, operators, parameter set,
solution concept) defined before it appears.
- Assumptions — numbered, each stated once, each used. Group standing assumptions
separately from those invoked only for specific results.
- Theorems / Propositions / Lemmas — the headline result first or clearly flagged;
supporting lemmas factored out so the main proof reads cleanly.
- Proofs — in-text for short proofs; the full proof in the Supplemental Material
(Econometrica's term for the online appendix; ≤25-page Supplemental Appendix or
unrestricted online Supplemental Material) when length would break the exposition or
exceed the 45-page body cap. Either way the proof must be complete and self-contained.
Stating the theorem
- State the theorem so it is self-contained: a reader should be able to verify the
hypotheses and conclusion without hunting through the text.
- Reference assumptions by number in the hypothesis ("Under Assumptions 1–3, ...").
- Make the conclusion a precise mathematical statement (existence, the limiting law, the
representation, the bound), not a verbal summary.
- If the result is sharp (assumptions are necessary, the rate is optimal), say so and prove it
or cite the matching lower bound.
Proof strategy toolkit
Pick the architecture before writing line-by-line:
| Result type | Typical machinery |
|---|
| Existence of equilibrium / solution | Fixed-point (Brouwer / Kakutani / Banach / Schauder), Berge maximum theorem |
| Representation theorem | Separation / Hahn–Banach, mixture-space, biseparable arguments |
| Consistency | Uniform law of large numbers, argmax / M-estimation continuity, identification + compactness |
| Limiting distribution | CLT (for arrays / dependent data), delta method, empirical-process / Donsker arguments, stochastic equicontinuity |
| Uniqueness / comparative statics | Contraction, monotone-comparative-statics (lattice / single-crossing), index theory |
| Bounds / minimax | Le Cam two-point / Fano, coupling |
Write the proof sketch first (the architecture and the one or two hard steps), then expand
every step. Flag the genuinely novel step — referees want to see where the work is.
Generality audit
- For each assumption: is it used? Locate the line of the proof that needs it. Unused
assumptions must go.
- Is each assumption as weak as the proof allows? If the proof only needs continuity, do
not assume differentiability.
- Does the result hold on the largest natural class, or is it tied to a parametric example?
If the latter, either generalize or reframe honestly as an example.
- Are there edge cases (boundary of the parameter space, degenerate distributions, ties,
non-uniqueness) the statement quietly excludes? State the exclusions.
Checklist
Anti-patterns
- "It is easy to see that ..." standing in for the load-bearing step
- "By standard arguments" where the argument is not standard in this setting
- A regularity condition that appears only inside the proof, never in the assumptions
- Differentiability / compactness assumed for convenience but never genuinely needed
- A theorem that holds only for one functional form, presented as general
- Existence proven, uniqueness silently assumed
- Citing a CLT / fixed-point theorem whose hypotheses your setting does not actually satisfy
Output format
【Central theorem】... (one-line statement + assumption numbers)
【Proof architecture】fixed-point / ULLN+M-estimation / empirical process / ...
【Hard step】...
【Assumptions audit】all used? [yes/no — list unused]; minimal? [...]
【Generality】largest class: ...; excluded edge cases: ...
【Full proof location】in-text / Supplemental Material §...
【Next step】ecta-robustness
