📖 Introduction to structural estimation

Historical overview¶

Stage 1: early foundations (1930s–1950s)¶

Leonid Hurwicz, Jacob Marschak, Trygve Haavelmo, Tjalling Koopmans, and the Cowles Commission

Key contributions:

Introduced the structural vs reduced form distinction.
Formalized simultaneous equations, endogeneity, identification.
Haavelmo (1944): probability approach to econometrics → modern estimation foundations.
Marschak: structure needed for policy analysis.
Hurwicz: identification theory (order/rank conditions).

Legacy: SEM becomes the dominant approach to causal inference in economics.

Stage 2: the Lucas critique and microfoundations (1970s)¶

Robert E. Lucas Jr. and the rational expectations revolution

Key points:

Structural parameters must be policy invariant.
Ad hoc SEM (with “behavioral equations”) fail under policy changes.
Led to microfounded models derived from optimization and equilibrium conditions.
Birth of DSGE models as dynamic, expectations-driven SEM descendants.

Legacy: SEM concepts survive but become embedded inside microfounded dynamic systems.

Stage 3: individual-level structural modelling (1980s–1990s)¶

John Rust (1987): dynamic discrete choice V. Joseph Hotz and Robert A. Miller (1993): CCP inversion, reduced-form based identification Steven Berry, James Levinsohn and Ariel Pakes (1995): random-coefficients demand estimation

Key innovations:

Microfoundations at the agent level (Bellman equations).
Structural estimation using NFXP and GMM.
Discrete choice demand becomes the IO workhorse.

Legacy: structural microeconometrics becomes a major field.

Stage 4: modern structural IO (1990s–2020s)¶

Richard Ericson and Ariel Pakes (1995) $\rightarrow$ dynamic games Victor Aguirregabiria and Pedro Mira, Ariel Pakes et al., and more recent computational IO

Advances:

Multi-agent dynamic games, heterogeneous firms, entry/exit, investment.
GMM, simulation, and high-dimensional methods.
Close integration with industrial organization and antitrust practice.

Legacy: modern IO is a fully microfounded, dynamic descendant of SEM.

Flow diagram: classical SEM leads to DSGE models, then to dynamic discrete choice models, then to structural IO

Modern structural econometric models — DSGE, dynamic discrete choice, and structural IO — are dynamic, microfounded generalizations of classical simultaneous equations models (SEM).

Structural and reduced form econometrics¶

Aspect	Structural econometrics	Reduced form econometrics
Essence	Estimation of parameters of economic models derived from theory	Estimation of relationships directly from the data
Purpose	Policy analysis, counterfactuals, understanding theoretical mechanisms	Prediction, local causal inference
Model	Based on economic theory, optimization behavior	Statistical relationships without explicit economic model
Assumptions	About details of economic behavior	About statistical properties of data
Identification	Exclusion restrictions, instruments, functional form	Often relies on natural experiments, IV, regression discontinuity
Estimation methods	MLE, GMM, simulated methods	OLS, IV, matching, regression discontinuity
Data requirements	Often requires detailed microdata	Generally less detailed or aggregate data
Applications	Structural models of demand, dynamic programming, games	Reduced form impact evaluations, treatment effects

Structural econometrics focuses on estimating parameters of economic models derived from theory, allowing for counterfactual analysis and policy simulations.
Reduced form econometrics focuses on estimating relationships directly from data without explicit reference to underlying economic models, often used for prediction or causal inference without structural interpretation.

Is economics a falsifiable science according to Popper (1934)?
How can we estimate the effect of a large new and unique policy?
Can a model be useful without being realistic?
Evidence-based (experimental) vs. model-based (structural) policy analysis -- in medicine?

Why structural econometrics?¶

Four things a structural approach buys you, each of them a consequence of committing to an explicit model.

Internal consistency. Rational agents facing constraints; uncertainty stated as an explicit probability distribution; well-defined equilibrium concepts (competitive, Nash, and so on); an explicit data-generating process; estimation grounded in the LLN and the CLT.

Elegance and transparency. Every step can be independently verified, and there is little room for researcher discretion — though the numerical implementation can still hide problems.

Causality. A model-based concept of causality, resting on assumptions that are stated rather than implied.

Counterfactuals. Generated by the model itself. They are valid only within the maintained structure, and taking them outside it requires further external validity assumptions.

Components of a structural estimation project¶

Economic model derived from theory — optimizing agents, possibly with bounded rationality, dynamics, observed and unobserved heterogeneity, and either equilibrium conditions or strategic interaction between agents.
Data generated by that model — cross-section, time series or panel; individual or aggregate; often censored, truncated or incomplete.
Preliminary data analysis — cleaning, descriptive statistics, visualization, and reduced form estimates that feed back into how the model is built.
Estimation method — maximum likelihood (full or limited information), GMM, simulated methods (simulated MLE, method of simulated moments), or Bayesian.
Identification strategy — exclusion restrictions, functional form assumptions, instruments for endogenous variables, policy invariance.
Counterfactual simulations — the estimated model as a synthetic laboratory for welfare effects, market outcomes and policy analysis.

Prototype dynamic discrete choice model¶

Choices¶

Periods: $t = 1, \dots, T$ , possibly $T = \infty$
Actions: $j = 1, \dots, J$
Indicators: $d_{jt} \in \{0,1\}$

\sum_{j=1}^J d_{jt} = 1, \; \forall t

Mutual exclusivity is not restrictive: combinations can be redefined as distinct actions.

States and transitions¶

Let the state be $z_t \in \mathcal{Z}$ . This is all the information that is relevant for the decision at time $t$ .

Transition probabilities when action $j$ is chosen at period $t$

f_{jt}(z_{t+1} \mid z_t)

State spaces may be large but are often sparse.

Preferences and expected utility¶

Flow/current/instantaneous utility at time period $t$ when action $j$ is chosen

u_{jt}(z_t)

Discount factor

\beta \in (0,1)

Expected utility

\E\left\{\left. \sum_{t=1}^T \sum_{j=1}^J \beta^{t-1} d_{jt} u_{jt}(z_t) \right | z_1 \right\}

Value functions and Bellman equation¶

Define the optimal policy $d_t^\star(z_t)$ as a vector of zeros and one, indicating the most desirable action.

The value function conditions on optimal behavior in all future periods; it is the maximal attainable expected utility from period $t$ on

V_t(z_t) = \E\left\{\left. \sum_{\tau=t}^T \sum_{j=1}^J \beta^{\tau-t} d_{j\tau}^\star(z_\tau) u_{j\tau}(z_\tau) \right| z_t \right\}

Bellman equation:

V_t(z_t) = \sum_{j=1}^J d_{jt}^\star \left[ u_{jt}(z_t) + \beta \sum_{z'} V_{t+1}(z') f_{jt}(z' \mid z_t) \right]

We will see later how the Bellman equation can be solved and value functions computed numerically.

Define the choice-specific value:

v_{jt}(z_t) = u_{jt}(z_t) + \beta \sum_{z'} V_{t+1}(z') f_{jt}(z' \mid z_t)

By definition the optimal choice is:

d_{jt}^\star(z_t) = \mathbf{1} \left\{ v_{jt}(z_t) \ge v_{kt}(z_t)\, \;\forall k \right\}

Why unobserved heterogeneity is needed¶

If agents with identical observed states are observed in the data to choose differently

the model implies indifference between actions
all actions appear optimal
the model loses empirical content!

Therefore fully observed heterogeneity is useless for data analysis.

Unobserved heterogeneity framework¶

Decompose the state:

z_t = (x_t, e_t)

$x_t$ : observed by both agents and econometrician
$e_t$ : unobserved by econometrician, but observed by agents

The objective becomes predicting choice probabilities, not individual choices.

Data generating process¶

Observed data are states and corresponding choices:

(x_1, d_1, \dots, x_T, d_T),

with the individual observations given by

(x_1^{(n)}, d_1^{(n)}, \dots, x_T^{(n)}, d_T^{(n)}), \; n = 1, \dots, N

The likelihood integrates out unobservables:

\Pr(d_1, x_2, \dots, d_T \mid x_1) = \int \cdots \int \prod_t \sum_{j=1}^J d_{jt} \Pr(d_t \mid x_t, e_t) \Pr(x_{t+1} \mid x_t, d_t) \, de_1 \cdots de_T

A huge multidimensional integral in the general case!
We will see how Rust’s assumptions simplify this drastically

Maximum likelihood estimation¶

Let $\theta$ index utilities, transitions, and $\beta$ .

\hat{\theta}_{ML} = \arg\max_\theta \frac{1}{N} \sum_{n=1}^N \log \Pr(\text{data}_n \mid x_{1n}; \theta)

Early applications include Miller (1984) and Wolpin (1984).

Other estimation approaches:

Two-step methods based on conditional choice probabilities estimated directly from the data (CCP methods) (Hotz–Miller, Aguirregabiria–Mira)
Method of simulated moments (MSM)
Calibration (no standard errors)

Multiple decision makers $\rightarrow$ equilibrium models¶

Macro style models with aggregate states¶

Infinitely many agents
Individual actions do not affect aggregate states
Aggregate states affect individual payoffs and transitions
Aggregate states evolve according to the collective behavior of all agents

Dynamic Markov games¶

Finite number of agents
Individual actions affect payoffs and transitions of all other agents
Joint individual actions affect payoffs and transitions
Equilibrium defined by mutual best responses
- Nash
- Bayesian Nash
- Markov perfect equilibrium (MPE)
- Oblivious equilibrium, etc.

References¶

Gillingham, K., Iskhakov, F., Munk-Nielsen, A., Rust, J., & Schjerning, B. (2022). Equilibrium Trade in Automobiles. Journal of Political Economy, 130(10), 2534–2593. 10.1086/720463
Keane, M. (2010). Structural vs. atheoretic approaches to econometrics. Journal of Econometrics, 156(1), 3–20. https://econpapers.repec.org/article/eeeeconom/v_3a156_3ay_3a2010_3ai_3a1_3ap_3a3-20.htm
Wolpin, K. I. (2013). The Limits of Inference without Theory. The MIT Press. 10.7551/mitpress/9258.001.0001
Rust, J. (2014). The Limits of Inference with Theory: A Review of Wolpin (2013). Journal of Economic Literature, 52(3), 820–850. 10.1257/jel.52.3.820
Sargent, T. J. (2024). Critique and consequence. Journal of Monetary Economics, 141, 2–13. 10.1016/j.jmoneco.2023.10.001

📖 Introduction to structural estimation

Historical overview¶

Stage 1: early foundations (1930s–1950s)¶

Stage 2: the Lucas critique and microfoundations (1970s)¶

Stage 3: individual-level structural modelling (1980s–1990s)¶

Stage 4: modern structural IO (1990s–2020s)¶

Structural and reduced form econometrics¶

Why structural econometrics?¶

Components of a structural estimation project¶

Prototype dynamic discrete choice model¶

Choices¶

States and transitions¶

Preferences and expected utility¶

Value functions and Bellman equation¶

Why unobserved heterogeneity is needed¶

Unobserved heterogeneity framework¶

Data generating process¶

Maximum likelihood estimation¶

Multiple decision makers →\rightarrow→ equilibrium models¶

Macro style models with aggregate states¶

Dynamic Markov games¶

Multiple decision makers $\rightarrow$ equilibrium models¶