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.