brycewang-stanford/identification-proofs

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

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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.

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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

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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:

\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}

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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.

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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