From compound-science
This skill covers formal identification arguments and proofs in structural and reduced-form econometrics. Use when the user needs to prove or formalize that a parameter is identified — including writing identification propositions, stating regularity conditions, deriving rank conditions, or showing observational equivalence fails. Triggers on "identification proof", "identification argument", "identify the parameter", "show identification", "identification condition", "exclusion restriction proof", "rank condition", "order condition", "identification strategy formal", "nonparametric identification", "parametric identification", "local identification", "global identification", "observational equivalence", "identification at infinity", "completeness condition", "regularity conditions", "Rothenberg", "proof of identification", "identification result", "identified parameter", "point identified", "set identified", "partial identification".
npx claudepluginhub james-traina/science-plugins --plugin compound-scienceThis skill uses the workspace's default tool permissions.
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.
Provides UI/UX resources: 50+ styles, color palettes, font pairings, guidelines, charts for web/mobile across React, Next.js, Vue, Svelte, Tailwind, React Native, Flutter. Aids planning, building, reviewing interfaces.
Fetches up-to-date documentation from Context7 for libraries and frameworks like React, Next.js, Prisma. Use for setup questions, API references, and code examples.
Guides Payload CMS config (payload.config.ts), collections, fields, hooks, access control, APIs. Debugs validation errors, security, relationships, queries, transactions, hook behavior.
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/BLPreferences/proof-template.md — LaTeX and plain-language templates for identification propositionsreferences/regularity-and-partial-id.md — Regularity conditions checklist and partial identification methodsUse when the user is:
Skip when:
causal-inference skill)structural-modeling skill)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:
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.
Every formal identification argument follows this architecture. Work through all seven steps before claiming identification.
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).
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.
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.
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.
Derive identification via one of three strategies:
references/derivation-tools.mdState 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.
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.
| 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 | $E[Y(0){t=1}-Y(0){t=0} | D=1] = E[Y_{t=1}-Y_{t=0} | D=0]$ |
| Sharp RDD | Continuity at cutoff | $E[Y(0) | X=x]$ continuous at $c$ | Manipulation |
| 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 |
Full derivations for each method: references/derivation-tools.md
Formal proof required when:
Informal argument sufficient when:
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}
If identification fails, characterize the identified set $\Theta^*$. Key approaches:
Full treatment: references/regularity-and-partial-id.md. For empirical sensitivity exercises, see sensitivity-analysis.md in the empirical-playbook skill.
identification-critic agent — Reviews a completed identification argument (assumption completeness, rank conditions, support)mathematical-prover agent — Verifies proof steps: fixed-point arguments, rank conditions, uniquenessidentification-critic agent — Interactive review of a completed identification argumentcausal-inference skill — Method-specific implementation after identification is establishedstructural-modeling skill — BLP and dynamic DC implementation detailsempirical-playbook skill → sensitivity-analysis.md — Oster bounds, specification curves, breakdown frontiers