Assumptions, Results & Proof Plan (ectheory-identification-strategy)

When to trigger

• The estimator/test is stated but the regularity conditions under which it works are not pinned down
• A proof has a gap, or an assumption is doing suspicious load-bearing work
• You need to decide what limit theory applies (stationary vs nonstationary, fixed vs growing dimension)
• You are unsure your result clears ET's rigor bar before drafting theorems

The ET rigor bar (theory journal — adapt "identification" accordingly)

At ET the analogue of an "identification strategy" is a complete, defensible assumption set plus a correct, general proof. The most common ET referee objection is an unstated or implausibly strong regularity condition. Treat the assumption-result-proof triple as the spine:

1. Assumptions. State each one explicitly and minimally. For every assumption ask: is it necessary, can it be weakened, and is it satisfied by a leading example DGP? Distinguish primitive conditions from high-level ones; if you use high-level conditions, show they hold in a concrete case.
2. Results. State theorems precisely — the object, the mode of convergence (in probability, in distribution, almost surely, uniformly), the rate, and the limiting law (normal, mixed-normal, functional of Brownian motion, non-standard).
3. Proof exposition. Give a readable proof: a roadmap up front, key lemmas isolated, and the probabilistic machinery named (LLN/CLT, triangular-array CLT, FCLT/weak convergence, empirical-process bounds, mixing / near-epoch dependence, concentration inequalities).
4. Generality. Show the result is not an artifact of a special case — handle dependent data, non-standard limits, or growing dimension where relevant.

Branch paths by environment

• Stationary, fixed dimension — standard LLN/CLT; verify moment and dependence conditions; give the asymptotic variance explicitly and a consistent estimator of it.
• Nonstationary / unit-root / cointegration — FCLT and continuous-mapping arguments; limits as functionals of Brownian motion; care with normalizing rates (e.g., super-consistency).
• High-dimensional / many regressors — dimension growing with n; concentration inequalities, sparsity or regularization conditions; uniformity over the parameter space.
• Non-standard inference / partial identification — characterize the (possibly non-normal) limit, size control under the least-favorable configuration, and robustness of the inference.
• Semiparametric / nonparametric — empirical-process / stochastic-equicontinuity arguments; bandwidth/tuning conditions; bias-variance trade-off made explicit.

Checklist

• Every assumption stated explicitly, minimally, and motivated by a leading example
• High-level conditions verified in at least one primitive case
• Mode of convergence, rate, and limiting law stated precisely in each theorem
• Proof has a roadmap; lemmas isolated; probabilistic tools named
• Asymptotic variance / limiting functional given, with a consistent estimator where relevant
• Generality shown (dependence, nonstationarity, or dimension handled, as applicable)
• No claim exceeds what the assumptions support

Anti-patterns

• An assumption that secretly assumes the conclusion (e.g., directly assuming asymptotic normality)
• High-level conditions that are never shown to hold in any concrete DGP
• A proof sketch that hides the hard step (uniformity, the non-standard limit, the edge case)
• Plain CLT machinery applied to nonstationary or high-dimensional data without justification
• Stating a rate or limit law without the supporting convergence argument

The assumption-audit table ET referees run first

The single most common Econometric Theory objection is that a regularity condition is too strong or not primitive. Audit each assumption against the columns below before drafting theorems.

A high-level condition with no concrete DGP satisfying it is a classic desk-reject flag.

Worked vignette and the primitive-condition fixes

For beta-hat in a cointegrating regression with a near-integrated regressor (root rho = 1 + c/n): assume a martingale-difference innovation array with finite fourth moments, and prove n(beta-hat - beta) converges to a ratio of stochastic integrals against an Ornstein-Uhlenbeck process via an FCLT plus continuous-mapping, with a separate lemma making the bias o_p(1) uniformly in c — the delicate step. The fixes: "conditions too strong / not primitive" → swap a high-level condition for a primitive moment-plus-dependence pair; "rate without distribution theory" → supply the limiting law (mode, normalizer, functional); "uniformity not established" → isolate it as a named lemma.

Output format

【Environment】stationary / nonstationary / high-dimensional / non-standard / semiparametric
【Assumptions】listed + each justified by a leading example? [Y/N]
【Result】object, mode of convergence, rate, limiting law
【Proof plan】roadmap + key lemmas + named tools
【Generality】what is handled beyond the base case
【Gaps】[...]
【Next step】ectheory-contribution-framing