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:
- 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.
- 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).
- 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).
- 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
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.
| Assumption |
Primitive or high-level? |
Necessary or convenience? |
Holds in a leading example? |
| Moment / tail |
state which |
for which CLT/LLN |
verify in one DGP (minimal exponent?) |
| Dependence (mixing/NED) |
primitive preferred |
controls the variance term |
e.g., a stable VAR |
| Smoothness / tuning |
bandwidth/penalty rate |
bias-variance trade |
concrete kernel/penalty |
| Identification / rank |
primitive on the model |
for consistency |
a structural example |
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
Source: brycewang-stanford/Awesome-Journal-Skills → Econometric-Theory-Skills/skills/ectheory-identification-strategy/SKILL.md
1---2name: ectheory-identification-strategy3description: Use when the assumptions, regularity conditions, and asymptotic results are the bottleneck for an Econometric Theory (ET) theorem-proof paper — adapt "identification" to mean stating defensible assumptions, proving the limit theory, and establishing generality, not causal design.4---5
6
7# Assumptions, Results & Proof Plan (ectheory-identification-strategy)
8
9## When to trigger
10
11- The estimator/test is stated but the **regularity conditions** under which it works are not pinned down
12- A proof has a gap, or an assumption is doing suspicious load-bearing work
13- You need to decide what limit theory applies (stationary vs nonstationary, fixed vs growing dimension)
14- You are unsure your result clears ET's rigor bar before drafting theorems
15
16## The ET rigor bar (theory journal — adapt "identification" accordingly)
17
18At ET the analogue of an "identification strategy" is a **complete, defensible assumption set plus a
19correct, general proof**. The most common ET referee objection is an unstated or implausibly strong
20regularity condition. Treat the assumption-result-proof triple as the spine:
21
221. **Assumptions.** State each one explicitly and minimally. For every assumption ask: is it
23 necessary, can it be weakened, and is it satisfied by a leading example DGP? Distinguish
24 primitive conditions from high-level ones; if you use high-level conditions, show they hold in a
25 concrete case.
262. **Results.** State theorems precisely — the object, the mode of convergence (in probability, in
27 distribution, almost surely, uniformly), the rate, and the limiting law (normal, mixed-normal,
28 functional of Brownian motion, non-standard).
293. **Proof exposition.** Give a readable proof: a roadmap up front, key lemmas isolated, and the
30 probabilistic machinery named (LLN/CLT, triangular-array CLT, FCLT/weak convergence,
31 empirical-process bounds, mixing / near-epoch dependence, concentration inequalities).
324. **Generality.** Show the result is not an artifact of a special case — handle dependent data,
33 non-standard limits, or growing dimension where relevant.
34
35## Branch paths by environment
36
37- **Stationary, fixed dimension** — standard LLN/CLT; verify moment and dependence conditions; give
38 the asymptotic variance explicitly and a consistent estimator of it.
39- **Nonstationary / unit-root / cointegration** — FCLT and continuous-mapping arguments; limits as
40 functionals of Brownian motion; care with normalizing rates (e.g., super-consistency).
41- **High-dimensional / many regressors** — dimension growing with n; concentration inequalities,
42 sparsity or regularization conditions; uniformity over the parameter space.
43- **Non-standard inference / partial identification** — characterize the (possibly non-normal) limit,
44 size control under the least-favorable configuration, and robustness of the inference.
45- **Semiparametric / nonparametric** — empirical-process / stochastic-equicontinuity arguments;
46 bandwidth/tuning conditions; bias-variance trade-off made explicit.
47
48## Checklist
49
50- [ ] Every assumption stated explicitly, minimally, and motivated by a leading example
51- [ ] High-level conditions verified in at least one primitive case
52- [ ] Mode of convergence, rate, and limiting law stated precisely in each theorem
53- [ ] Proof has a roadmap; lemmas isolated; probabilistic tools named
54- [ ] Asymptotic variance / limiting functional given, with a consistent estimator where relevant
55- [ ] Generality shown (dependence, nonstationarity, or dimension handled, as applicable)
56- [ ] No claim exceeds what the assumptions support
57
58## Anti-patterns
59
60- An assumption that secretly assumes the conclusion (e.g., directly assuming asymptotic normality)
61- High-level conditions that are never shown to hold in any concrete DGP
62- A proof sketch that hides the hard step (uniformity, the non-standard limit, the edge case)
63- Plain CLT machinery applied to nonstationary or high-dimensional data without justification
64- Stating a rate or limit law without the supporting convergence argument
65
66## The assumption-audit table ET referees run first
67
68The single most common Econometric Theory objection is that a regularity condition is too strong or not
69primitive. Audit each assumption against the columns below before drafting theorems.
70
71| Assumption | Primitive or high-level? | Necessary or convenience? | Holds in a leading example? |
72| --- | --- | --- | --- |
73| Moment / tail | state which | for which CLT/LLN | verify in one DGP (minimal exponent?) |
74| Dependence (mixing/NED) | primitive preferred | controls the variance term | e.g., a stable VAR |
75| Smoothness / tuning | bandwidth/penalty rate | bias-variance trade | concrete kernel/penalty |
76| Identification / rank | primitive on the model | for consistency | a structural example |
77
78A high-level condition with no concrete DGP satisfying it is a classic desk-reject flag.
79
80## Worked vignette and the primitive-condition fixes
81
82For beta-hat in a cointegrating regression with a near-integrated regressor (root rho = 1 + c/n): assume a
83martingale-difference innovation array with finite fourth moments, and prove n(beta-hat - beta) converges to
84a ratio of stochastic integrals against an Ornstein-Uhlenbeck process via an FCLT plus continuous-mapping,
85with a separate lemma making the bias o_p(1) uniformly in c — the delicate step. The fixes: "conditions too
86strong / not primitive" → swap a high-level condition for a primitive moment-plus-dependence pair; "rate
87without distribution theory" → supply the limiting law (mode, normalizer, functional); "uniformity not
88established" → isolate it as a named lemma.
89
90## Output format
91
92```
93【Environment】stationary / nonstationary / high-dimensional / non-standard / semiparametric
94【Assumptions】listed + each justified by a leading example? [Y/N]
95【Result】object, mode of convergence, rate, limiting law
96【Proof plan】roadmap + key lemmas + named tools
97【Generality】what is handled beyond the base case
98【Gaps】[...]
99【Next step】ectheory-contribution-framing
100```
101
102---
103
104**Source:** [`brycewang-stanford/Awesome-Journal-Skills`](https://github.com/brycewang-stanford/Awesome-Journal-Skills) → `Econometric-Theory-Skills/skills/ectheory-identification-strategy/SKILL.md`
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