Identification

Paul Schrimpf

2026-09-23

Reading

• Required: Song (2021) chapter 4 (which is the basis for these slides)
• Recommended: Lewbel (2019)
• Supplementary: Matzkin (2013), Molinari (2020) , Imbens (2020),

\[ \def\Er{{\mathrm{E}}} \def\cov{{\mathrm{Cov}}} \def\var{{\mathrm{Var}}} \def\R{{\mathbb{R}}} \]

Motivate by wanting to go from data to parameters …

Identification

The Identification Problem

• We have an economic model that describes how the world works. This model has parameters we care about (\(\theta_0\)).
• We have data, which we view as a sample from some probability distribution (\(P_X\)).

The Question of Identification

Can we learn the true value of the parameter \(\theta_0\) if we knew the true distribution of the data \(P_X\)?

The Identification Problem

• Identification is a conceptual problem about what is possible with infinite data.
• It is distinct from estimation, which is the problem of how to learn about \(\theta_0\) from a finite sample.
• Identification must come before estimation.

Identification

Definition

Let \(X\) be an observed random vector with distribution \(P_X\). Let \(\mathcal{P}\) be aprobability model — a collection of probabilities such that \(P_X \in \mathcal{P}\). Then \(\theta_0 \in \R^k\) is identified in \(\mathcal{P}\) if there exists a known \(\psi: \mathcal{P} \to \R^k\) s.t.

\[ \theta_0 = \psi(P_X) \]

The motivating idea here is we observe data. We do not know exactly the distribution of the data, \(P_X\), but we are willing to assume it comes from some collection of distributions, \(\mathcal{P}\). For example, suppose the data is the result of flipping a possibly biased coin \(n\) times. Then we could have \[ \mathcal{P} = \{ (\{0,1\}, 2^\{0,1\}, P(1) = p )^n : p \in [0,1] \} \] where the probability space to the \(n\) means the product space as constructed in the last question on problem set 1.

\(\theta_0\) is some parameter of interest related to \(P_X\). In the example, there is not much for \(\theta_0\) to be other than \(p\), but in general \(\theta_0\) could be some aspect of \(P_X\) that we especially care about instead of a complete description of it.

Examples

Example: Descriptive Statistics

• \(\theta_0 =\) mean of \(X\), then \(\theta_0\) is identified by \[ \psi_\mu(P) = \int x dP(x) \] in \(\mathcal{P} = \{P : \int x dP(x) < \infty \}\)

• Generally, descriptive statistics identified in a broad probability model with just regularity restrictions to ensure the statistics exist

In this example, the probability model, \(\mathcal{P}\), is a very large collection of distributions. This constrasts with the previous, biased-coin flipping, example where we could describe each element of \(\mathcal{P}\) with a single number.

Example: Linear Model

\[ Y = \alpha + \beta X + \epsilon \]

• \(\mathcal{P} = \{P_{X,Y}:\) \(Y=\alpha + \beta X + \epsilon\),

  1. \(| \mathrm{Cov}(X,Y) | < \infty\), \(0 < \mathrm{Var}(X) < \infty\)

  2. \(\mathrm{Cov}(X, \epsilon) = 0\) \(\}\)

• \(\beta\) identified as

\[ \beta = \frac{\int (x - \Er X) (y - \Er Y ) dP_{X,Y}(x,y)} {\int (x - \Er X)^2 dP_{X}(x)} = \frac{ \cov(X,Y) }{ \var(X) } \]

Assumptions

• Identification requires:
  1. Usually innocuous regularity conditions
  2. Substantive exogeneity restriction
• Evaluating plausibility of exogeneity restrictions requires a priori knowledge of data context and related economic theory

Examples of production function and returns to schooling here?

2 is more difficult to get right and more context dependent, so we will first tackle the easier problem of 1.

Example: Multiple Regression

\[ Y = X'\beta + \epsilon \]

• \(\mathcal{P} = \{P: \Er [X \epsilon] = 0, \Er [X X'] \text{ invertible} \}\)

Example: Binary Choice

\[ Y = 1\{ \beta_0 + \beta_1 X > u \} \]

• \(\mathcal{P} = \{P: u \sim N(0,1), 0< \var(X) < \infty \}\)

• Is \(u \sim N(0,1)\) innocuous?

  • Suppose \(\theta_0 = \int \frac{\partial E[Y|X=x]}{dx} dP_X(x)\) (the average marginal effect or average derivative)

As, an aside, what the heck does \(\frac{\partial E[Y|X=x]}{dx}\) even mean? \(E[Y|X=x]\) is only unique almost everywhere, so we could pick a version that is not differentiable. One approach to just declare that we always work with the smoothest version of \(E[Y|X=x]\) and hope everything is okay. This mostly works in practice. But, the technically correct construction is that \(\frac{\partial E[Y|X=x]}{dx}\) is a weak derivative, i.e. it satisfies an integration by parts formula. Suppose \(f\) is differentiable in the usual sense with \(f(a)=f(b)=0\), then integration by parts says \[ \int_a^b E[Y|X=x] \frac{df}{dx}(x) dP_X(x) = \int_a^b \frac{\partial E[Y|X=x]}{dx} f(x) dP_X(x) \] We define \(\frac{\partial E[Y|X=x]}{dx}\) as the unique (almost everywhere) function that satisfies this equation for all such \(f\).

Anyway, some calculation will show that the average marginal effect is unaffected by change from \(N(0,1)\) to \(N(\mu, \sigma^2)\).

More generally, think about importance of statistical assumptions in terms of how they affect identifiable quantities and counterfactuals of interest.

Example: Potential Outcomes

• Data:

  • Treatment \(D_i\)
  • Potential outcomes \((Y_{i,0}, Y_{i,1})\), observed outcome \(Y_i = D_i Y_{i,1} + (1-D_i) Y_{i,0}\)
  • Covarites \(X_i\)

• Parameter: \(\theta_0 = \Er[Y_{i,1} - Y_{i,0}] =\) average treatment effect

• Assume:

  1. Unconfoundedness: \((Y_{i,0}, Y_{i,1})\) conditionally independent of \(D_i\) given \(X_i\)
  2. Overlap: \(\epsilon < P(D=1|X=x) < 1-\epsilon\) for some \(\epsilon > 0\) and all \(x\)

Example: Production Function Elasticity

• Data:
  • Revenue \(R_i = p_Y Y_i\)
  • Material expenditure \(p_M M_i\)
  • Capital \(K_i\), labor \(L_i\)
• Model:
  • Cobb-Douglas production: \(Y_i = A_i L_i^\alpha K_i^\beta M_i^\gamma\)
• Assume:
  1. Competitive, flexibly chosen \(M_i \in \mathrm{arg}\max_M \Er\left[p_Y A_i L_i^\alpha K_i^\beta M^\gamma - p_M M | \mathcal{I}_i \right]\)
  2. Other inputs and prices known when choosing \(M\), i.e. \(\sigma(L) \subset \mathcal{I}\), etc.
  3. \(\Er\left[\frac{p_Y Y}{p_M M}\right] < \infty\)

Example: Production Function Elasticity

What if production function is \(Y_i = A_i F(L_i, K_i, M_i)\), can the elasticity of output with respect to \(M\) be identified?

Reintrepretation of Estimators

Reintrepretation of Estimators

• Generalized and/or descriptive interpretation of population estimator
• Analyze familiar estimator under more general assumptions
  • Understand bias when exogeneity assumptions fail
  • Sometimes give more general interpretation of existing estimator

Example: Regression

• In linear model \(Y_i = X_i'\beta + \epsilon_i\), if just assume \(\Er[X X']\) invertible,
• Population regression \[ \begin{align*} \theta = & \Er[ X X']^{-1} \Er[ X Y] \\ = & \Er[X X']^{-1} \Er[X (X' \beta + \epsilon)] \\ = & \beta + \Er[X X']^{-1} \Er[X\epsilon] \end{align*} \]

Example: Regression

• If relevant moments exist (no linear model required) population regression solves \[ \Er[ X X']^{-1} \Er[ X Y] \in \mathrm{arg}\min_b \Er[ (X'b - \Er[Y|X])^2 ] \]

Caution

• Regression being a linear approximation to \(\Er[Y|X]\) does not mean \(\beta = \Er[X X']^{-1} \Er[X Y]\) necessarily has the sign you want

• In example below, \(\Er[Y|x_1=1, x_2] > \Er[Y|x_1=0,x_2]\), but \(\beta_1 < 0\)

using LinearAlgebra
n = 100_000
EYX(x) = x[2]>0 ? -1.0 + 1.0*x[1] : (1.0 + 0.1*x[1])
x2 = randn(n)
x1 = (x2*3 + randn(n) .> 0)
X = hcat(x1,x2)
y = EYX.(eachrow(X)) + randn(n)
X = hcat(ones(n), X)
β = (X'*X) \ (X'*y)

3-element Vector{Float64}:
  0.5226655090702952
 -0.13343797382338
 -0.3513147652124974

Example: Potential Outcomes

• Matching initially studied with a linear regression model, e.g. Cochran (1953) \[ Y_i = \alpha D_i + X_i' \beta + \epsilon_i \]
• Implies constant treatment effect \(Y_{i,1} - Y_{i,0} = \alpha\)

The LATE interpretation of IV developed along similar lines – replace a constant treatment effect linear model with a heterogenous effect potential outcomes framework, and see what IV does.

Similarly, the recent literature on difference in differences with staggered treatment timing arises from moving from a model with a constant treatment effect over time to one where treatment effects vary with time.

Observational Equivalence

Observational Equivalence

• Identification can be defined without explicit mapping from data to parameters, e.g. Hsiao (1983), Matzkin (2007) by using observational equivalence
• Especially useful for showing non-identification

Structures

• Be explicit about how distribution of data relates to an underlying economic model
• Set of model structures \(S\)
• Mapping from structure to distribution of observed data \(P(\cdot;s)\) \[ \mathcal{P} = \{ P(\cdot; s): s \in S \} \]

Observational Equivalence

Definition: Observationally Equivalent

• Let \(\mathcal{P} = \{ P(\cdot; s) : s \in S \}\), two structures \(s\) and \(\tilde{s}\) in \(S\) are observationally equivalent if they imply the same distribution for the observed data, i.e. \[ P(B;s) = P(B; \tilde{s}) \] for all \(B \in \sigma(X)\).

Observationally Equivalent Parameters

Definition: Observationally Equivalent Parameters

• Let \(\lambda: S \to \R^k\), \(\theta\) is observationally equivalent to \(\tilde{\theta}\) if \(\exists s, \tilde{s} \in S\) that are observationally equivalent and \(\theta = \lambda(s)\) and \(\tilde{\theta} = \lambda(\tilde{s})\)
  • Let \(\Gamma(\theta, S) = \{P(\dot; s) | s \in S, \theta = \lambda(s) \}\), then \(\theta\) and \(\tilde{\theta}\) are observationally equivalent iff \(\Gamma(\theta,S) \cap \Gamma(\tilde{\theta}, S) \neq \emptyset\)

Non-constructive Identification

Definition: (Non-Constructive) Identification

• \(s_0 \in S\) is identified if there is no \(s\) that is observationally equivalent to \(s_0\)

• \(\theta_0\) is identified (in \(S\)) if there is no observationally equivalent \(\theta \neq \theta_0\)

  • i.e. \(\Gamma(\theta_0, S) \cap \Gamma(\theta, S) = \emptyset\) \(\forall \theta \neq \theta_0\)

• Compared to constructive definition with \(\theta_0 = \psi(P)\):
  • Less clear how to use identification to estimate
  • Easier to show non-identification
  • Set of observationally equivalent structures can be of interest

Example: Multiple Regression

\[ Y = X'\beta + \epsilon \]

• Assume \(\Er[X\epsilon] = 0\)

• If \(\beta\) and \(\tilde{\beta}\) observationally equivalent, then \(0 = \Er[X(Y - X'\beta)] = \Er[X(Y-X'\tilde{\beta})]\), so \(0 = \Er[XX'](\beta - \tilde{\beta})\)

• \(X = [X_1\, X_2]'\), if rank \(\Er X X' = 1\), then \(\beta_1, \beta_2\) is observationally equivalent to any \(\tilde{\beta}_1, \tilde{\beta}_2\) s.t. \[ \tilde{\beta}_1 + \tilde{\beta}_2 \frac{\cov(X_1, X_2)}{\var(X_1)} = \beta_1 + \beta_2 \frac{\cov(X_1, X_2)}{\var(X_1)} \]

• \(\theta_0 = \lambda( \beta ) = \beta_1 + \beta_2\frac{\cov(X_1, X_2)}{\var(X_1)}\) is identified if rank \(\Er [X X'] \geq 1\)

Example: Random Coefficients Logit

• \(Y_i = 1\{\beta_0 + \beta_i X_i \geq U_i \}\)

  • \(U\) independent \(X_i,\beta_i\),
  • \(\beta_i\) indepedent \(X_i\),
  • \(F_u(z) = \frac{e^z}{1+e^z}\)

• \(\Er[Y|X] = \int \frac{e^{\beta_0 + \beta X_i}} {1+e^{\beta_0 + \beta X_i}} dF_\beta(\beta)\)

• Non-constructive and constructive identification of \(F_\beta\) in Fox et al. (2012)

Hard to give a brief example where non-constructive argument is required. Non constructive identification proofs typically leverage some high level mathematical result.

Christensen (2015) is another example of non-constructive identification.

References

Christensen, Timothy M. 2015. “Nonparametric Identification of Positive Eigenfunctions.” Econometric Theory 31 (6): 1310–30.

Cochran, William G. 1953. “Matching in Analytical Studies.” American Journal of Public Health and the Nations Health 43 (6_Pt_1): 684–91.

Fox, Jeremy T, Kyoo il Kim, Stephen P Ryan, and Patrick Bajari. 2012. “The Random Coefficients Logit Model Is Identified.” Journal of Econometrics 166 (2): 204–12.

Hsiao, Cheng. 1983. Chapter 4 Identification. Vol. 1. Handbook of Econometrics. Elsevier. https://doi.org/https://doi.org/10.1016/S1573-4412(83)01008-9.

Imbens, Guido W. 2020. “Potential Outcome and Directed Acyclic Graph Approaches to Causality: Relevance for Empirical Practice in Economics.” Journal of Economic Literature 58 (4): 1129–79. https://doi.org/10.1257/jel.20191597.

Lewbel, Arthur. 2019. “The Identification Zoo: Meanings of Identification in Econometrics.” Journal of Economic Literature 57 (4): 835–903. https://doi.org/10.1257/jel.20181361.

Matzkin, Rosa L. 2007. Chapter 73 Nonparametric Identification. Edited by James J. Heckman and Edward E. Leamer. Vol. 6. Handbook of Econometrics. Elsevier. https://doi.org/https://doi.org/10.1016/S1573-4412(07)06073-4.

Matzkin, Rosa L. 2013. “Nonparametric Identification in Structural Economic Models.” Annu. Rev. Econ. 5 (1): 457–86.

Molinari, Francesca. 2020. “Chapter 5 - Microeconometrics with Partial Identification.” In Handbook of Econometrics, Volume 7A, edited by Steven N. Durlauf, Lars Peter Hansen, James J. Heckman, and Rosa L. Matzkin, vol. 7. Handbook of Econometrics. Elsevier. https://doi.org/https://doi.org/10.1016/bs.hoe.2020.05.002.

Song, Kyunchul. 2021. Introduction to Econometrics.