---
name: brycewang-stanford/identification-proofs
source: https://app.decimal.ai/s/brycewang-stanford-identification-proofs@1/SKILL.md
source_sha256: 5b46e75438fe
---

# Identification Proofs

Reference for writing formal and informal identification arguments: from stating the target parameter precisely, through deriving the identification result, to connecting it to a feasible estimator.

**Detail files** (load on demand):
- `references/derivation-tools.md` — IFT approach, completeness, worked proofs for LATE/RDD/DiD/BLP
- `references/proof-template.md` — LaTeX and plain-language templates for identification propositions
- `references/regularity-and-partial-id.md` — Regularity conditions checklist and partial identification methods

## When to Use This Skill

Use when the user is:
- Writing a formal identification proposition for a paper or theory appendix
- Deriving whether a structural or causal parameter is point identified
- Stating and verifying regularity conditions for an identification result
- Working through rank or order conditions for GMM moment conditions
- Arguing identification for IV, DiD, RDD, or structural models
- Checking whether two models are observationally equivalent
- Characterizing an identified set under partial identification

Skip when:
- The task is implementing a causal estimator (use `causal-inference` skill)
- The task is structural model estimation code (use `structural-modeling` skill)
- The user needs only informal intuition, not a formal argument

---

## What Identification Means

**Core definition.** A parameter $\theta_0$ is *identified* if the map from the true parameter value to the distribution of observables is injective: $P_{\theta_1} = P_{\theta_2} \implies \theta_1 = \theta_2$.

**Key distinctions:**
- **Local vs global**: Local identification holds in a neighborhood of $\theta_0$ (Rothenberg 1971). Global identification requires uniqueness over the entire parameter space. Estimation needs global identification for a well-defined probability limit.
- **Point vs set**: Under point identification, data uniquely determine $\theta_0$. Under partial identification (Manski 1990), data are consistent with an identified set $\Theta^* \supseteq \{\theta_0\}$.
- **Observational equivalence**: Identification fails when two distinct parameter values generate the same observable distribution.

**Why identification precedes estimation.** A parameter can only be consistently estimated if it is identified. Code runs and produces output even when parameters are unidentified — checking identification before estimation prevents hard-to-diagnose failures.

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## The 7-Step Canonical Structure

Every formal identification argument follows this architecture. Work through all seven steps before claiming identification.

### Step 1 — Target Parameter
State precisely *what* $\theta$ you want to identify — not "the causal effect" but the exact functional or structural parameter (e.g., coefficient $\beta$ under endogeneity, the ASF $g(x) = E[Y(x)]$, taste parameters in BLP). **Common mistake:** conflating the target with the estimand (LATE is not ATE; ATT from DiD is not ATE).

### Step 2 — Model Primitives
State observables $(Y, X, Z)$, latent variables ($\varepsilon$, unobserved heterogeneity), structural equations, error restrictions (independence, mean independence), functional form (parametric vs nonparametric), and equilibrium concept if applicable.

### Step 3 — Source of Variation
State what observable variation provides identification leverage: instrument variation (IV), policy changes across groups (DiD), proximity to a cutoff (RDD), cost shifters entering supply but not demand (structural). The source must be distinct from functional form assumptions.

### Step 4 — Key Assumptions
Enumerate identifying assumptions explicitly (label A1, A2, ...). Common categories: exclusion restrictions, rank/order conditions, support conditions, independence, monotonicity (LATE), continuity (RDD), parallel trends (DiD). Each must be statable in population terms and either testable or defended substantively.

### Step 5 — Identification Result
Derive identification via one of three strategies:
1. **Explicit formula**: $\theta_0 = h(P_{\theta_0})$ — strongest form, gives both identification and an estimator
2. **Implicit function theorem**: moment conditions $E[m(X;\theta)] = 0$ have $\theta_0$ as unique solution (Jacobian has full rank) — see `references/derivation-tools.md`
3. **Injectivity argument**: show $P_{\theta_1} = P_{\theta_2} \implies \theta_1 = \theta_2$ directly

### Step 6 — Regularity Conditions
State conditions under which the result holds: support, rank, order, compactness, continuity, integrability, unique zero, monotonicity, no anticipation, overlap. Full checklist in `references/regularity-and-partial-id.md`.

### Step 7 — Estimation Link
Connect identification to a feasible estimator: explicit formula yields plug-in estimator $\hat\theta = h(P_n)$; moment conditions yield GMM; likelihood yields MLE. State the consistency result.

---

## Identification Arguments by Method

| Method | Key Assumption | Formal Statement | Common Failure | Test |
|--------|---------------|-----------------|----------------|------|
| IV/2SLS | Exclusion | $Z \perp \varepsilon$ | Direct effect of $Z$ on $Y$ | Overid test; substantive argument |
| LATE | Exclusion + monotonicity | $D(1) \geq D(0)$ a.s.; $Z \perp (Y(0),Y(1),D(0),D(1))$ | Defiers; exclusion violated | Monotonicity untestable; falsification |
| DiD | Parallel trends | $E[Y(0)_{t=1}-Y(0)_{t=0}|D=1] = E[Y_{t=1}-Y_{t=0}|D=0]$ | Differential anticipation/trends | Pre-trends event study; Rambachan-Roth |
| Sharp RDD | Continuity at cutoff | $E[Y(0)|X=x]$ continuous at $c$ | Manipulation | McCrary/rddensity; covariate smoothness |
| Fuzzy RDD | Continuity + first stage | Cutoff shifts $D$ discontinuously | Compound discontinuity | Placebo outcomes; covariate balance |
| Structural (BLP) | Rank on instruments | $\mathrm{rank}(E[Z'X]) = K$ | Weak instruments | First-stage F; Cragg-Donald |
| Structural (dynamic) | Exclusion in Bellman | State captures payoff-relevant history | Omitted state variable | Residual correlation test |
| Nonparametric IV | Completeness | $E[\phi(X)|Z]=0 \implies \phi=0$ a.s. | Discrete instrument | Support conditions (not directly testable) |

Full derivations for each method: `references/derivation-tools.md`

---

## Formal vs Informal Arguments

**Formal proof required when:**
- Proposing a new estimator or identification strategy
- Identification relies on non-standard assumptions (partial ID, extrapolation)
- Model involves equilibrium/fixed-point arguments where uniqueness is non-obvious
- A reviewer raised an identification concern
- Paper is methodological

**Informal argument sufficient when:**
- Applying a well-known method with established identification results
- Assumptions are standard for the method and context
- Paper's contribution is empirical, not methodological

For formal proofs, use the LaTeX and prose templates in `references/proof-template.md`. Minimal skeleton:

```latex
\begin{assumption}[Model restrictions]\label{ass:model}
  (i) Structural equation; (ii) Exogeneity; (iii) Relevance/rank;
  (iv) Support; (v) Compactness; (vi) Continuity.
\end{assumption}

\begin{proposition}[Identification of $\theta_0$]\label{prop:id}
  Under Assumption~\ref{ass:model}, $\theta_0$ is the unique element of
  $\Theta$ satisfying $\mathbb{E}[m(X_i;\theta_0)] = 0$.
\end{proposition}

\begin{proof}
  Step 1: Observational implications (model $\to$ moments).
  Step 2: Rank condition $\implies$ local injectivity (IFT).
  Step 3: Global uniqueness argument. \hfill$\square$
\end{proof}
```

---

## When Point Identification Fails

If identification fails, characterize the identified set $\Theta^*$. Key approaches:
- **Manski sharp bounds** — worst-case bounds without distributional assumptions
- **Intersection bounds** — tighten via multiple assumption-driven bounds (Chernozhukov, Lee, Rosen 2013)
- **Sensitivity analysis** — Oster (2019) bounds relate to partial ID under proportional selection

Full treatment: `references/regularity-and-partial-id.md`. For empirical sensitivity exercises, see `sensitivity-analysis.md` in the `empirical-playbook` skill.

---

## Integration with compound-science

- **`identification-critic` agent** — Reviews a completed identification argument (assumption completeness, rank conditions, support)
- **`mathematical-prover` agent** — Verifies proof steps: fixed-point arguments, rank conditions, uniqueness
- **`identification-critic` agent** — Interactive review of a completed identification argument
- **`causal-inference` skill** — Method-specific implementation after identification is established
- **`structural-modeling` skill** — BLP and dynamic DC implementation details
- **`empirical-playbook` skill → `sensitivity-analysis.md`** — Oster bounds, specification curves, breakdown frontiers