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SKILL.md
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.
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:
Explicit formula: $\theta_0 = h(P_{\theta_0})$ — strongest form, gives both identification and an estimator
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
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.
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.
