Counterfactual Analysis for Structural Dynamic Discrete Choice Models

Research Question. Discrete choice data identify only differences in agents’ utilities, not utility levels. In dynamic discrete choice (DDC) models this means many policy-relevant counterfactuals — those requiring knowledge of utility in levels — are not point-identified. Kalouptsidi, Kitamura, Lima, and Souza-Rodrigues ask: how much can researchers learn about counterfactual outcomes under mild, verifiable restrictions, without imposing the strong normalizations that are standard in applied work but often hard to justify and potentially sign-reversing in their effects?

Setting and Methodology. The paper works within a canonical infinite-horizon DDC framework where an agent chooses among a finite action set each period, with additively separable per-period payoffs and i.i.d. unobservables. The econometrician observes conditional choice probabilities (CCPs) and state transition functions from panel data, but the payoff vector is underidentified by X free parameters (one per state), which is the source of non-identification of many counterfactuals. The authors characterize the sharp identified set for counterfactual CCPs, for low-dimensional outcomes such as average welfare, and develop both identification theory and a feasible inference procedure.

Main Identification Results. The sharp identified set for counterfactual CCPs is a smooth, connected manifold whose dimension equals the rank of a specific matrix (CJ*QJ) that the econometrician can compute directly from the data. This rank is at most X minus the number of linearly independent equality restrictions imposed. Two classes of commonly used restrictions reduce the dimension further without requiring full point identification: (i) local counterfactuals — experiments affecting only a subset of the state-action space — reduce the dimension to at most the number of eigenvalues of the relevant transformation matrix that differ from one; (ii) parametric payoffs with ηγ free parameters reduce the dimension to at most ηγ. Combining both achieves the tightest bound. Point identification is the special case where the rank equals zero.

For scalar low-dimensional outcomes (e.g., average welfare), the identified set is a compact interval whose endpoints are obtained by solving constrained optimization programs implementable in standard nonlinear solvers (e.g., Knitro), feasible even when the state space is large.

Quantitative Illustration. In the firm entry/exit Monte Carlo with state space X = 4 and a counterfactual entry subsidy removal: under Restriction 1 alone (outside option = 0, non-negative costs, known variable profits), the identified set for the change in the long-run probability of being active is [-0.1235, 0.0000], correctly signed and containing the true value of -0.0638. Adding shape restrictions (Restrictions 1–2) tightens the upper bound to -0.0341; adding the scrap-value exclusion restriction (Restrictions 1–3) tightens it to -0.0421. Analogous patterns hold for consumer surplus (true: -0.0875; bounds narrowing from [-0.1735, 0.0000] to [-0.1735, -0.0573]) and firm value (true: 0.9513; bounds from [0.0000, 1.8229] to [0.6388, 1.8229]). Critically, the authors show that setting scrap values to zero — the standard identifying assumption — is rejected by the data under Restrictions 1 and 2, because that payoff vector does not lie in the identified set.

Empirical Application. Revisiting Das, Roberts, and Tybout (2007) on Colombian exporters, the paper re-examines the horserace among export revenue, fixed cost, and entry cost subsidies. The DRT ranking (revenue subsidies dominate, entry cost subsidies rank last) survives under weaker restrictions than originally imposed, but hinges on the assumption that scrap values do not vary across states. Without that restriction, entry cost subsidies can potentially outperform the other types, reversing the original conclusion.

Inference. The paper develops a subsampling-based inference procedure that is asymptotically uniformly valid (bootstrap fails here due to non-regularity of the set boundary). The confidence set is constructed by inverting a quadratic-form distance test statistic. The critical practical recommendation is subsample size hN = N^{2/3}. The procedure remains feasible in binary choice models with state spaces up to X = 240 (dimension of the optimization problem: 720), where standard moment-inequality approaches are computationally infeasible.

Q: Why are many counterfactuals not point-identified in DDC models, even after the model is estimated? A: Choice data identify only differences in value functions across actions, not utility levels. The identifying matrix M has rank AX, leaving X free payoff parameters undetermined. Counterfactuals that depend on utility levels — such as the welfare impact of an entry subsidy when scrap values are unknown — therefore cannot be recovered uniquely from the data, even with a fully estimated model.

Q: What is the key object the paper characterizes, and what does it look like geometrically? A: The paper characterizes the sharp identified set for the counterfactual CCP vector p̃. Proposition 1 establishes that this set is a smooth, connected manifold with boundary, whose interior dimension equals rank(CJ*QJ). Connectedness is important because it means the set has no gaps and boundary tracing is sufficient to characterize it.

Q: How does the dimension of the identified set depend on the type of model restrictions imposed? A: Equality restrictions (d of them) reduce the maximum possible dimension from X to X–d. Local counterfactuals (affecting L state-action pairs) reduce the dimension further to at most the number of eigenvalues of the payoff transformation H(L) that differ from one, which is at most L. Parametric payoffs with ηγ free parameters cap the dimension at ηγ. Combining local counterfactuals with parametric payoffs gives the tightest bound: at most the number of eigenvalues of a related matrix D that differ from one, which is at most min(L, ηγ).

Q: Under what conditions does the identified set for counterfactual behavior collapse to a point? A: When rank(CJ*QJ) = 0, every payoff vector in the identified set PI maps to the same counterfactual CCP — that is, p̃ is point-identified even though the structural payoff π may not be. This can occur through a combination of equality restrictions and specific structure of the counterfactual experiment, without requiring full identification of all model parameters.

Q: What properties does the identified set for a scalar low-dimensional outcome have, and how is it computed? A: Under continuity of the outcome function φ and boundedness of the payoff identified set, the identified set for a scalar outcome θ is a compact interval [θL, θU]. The endpoints are computed as the minimum and maximum of a constrained optimization program over the joint space of counterfactual CCPs and payoff vectors, subject to the model’s Bellman equations, model restrictions, and equality constraints linking observed to counterfactual behavior. These programs can be solved with standard nonlinear solvers.

Q: What do the Monte Carlo results show about the informativeness of the bounds? A: In the firm entry/exit example with X = 4, the identified sets under only mild restrictions (non-negative costs, known variable profits, zero outside option) are already informative and correctly signed. For the change in the probability of being active (true value: -0.0638), the set under Restriction 1 alone is [-0.1235, 0.0000], establishing that the probability does not increase. Adding shape restrictions and exclusion restrictions progressively tightens the interval. All intervals contain the true parameter value, confirming sharpness.

Q: What does the paper show about the assumption of zero scrap values, which is standard in the entry cost literature? A: The paper shows that setting scrap values to zero can be rejected by the data: in the firm entry/exit example, the payoff vector with s = 0 does not belong to the identified set PI under Restrictions 1 and 2. This is empirically important because Kalouptsidi, Scott, and Souza-Rodrigues (2021) had previously shown that mistakenly setting scrap values to zero not only biases estimated entry costs downward but can also reverse the sign of a subsidy’s predicted effect.

Q: What is the main finding of the empirical application to export subsidies? A: Revisiting Das, Roberts, and Tybout (2007), the paper finds that the DRT ranking — export revenue subsidies dominate, entry cost subsidies rank last — can be confirmed under restrictions weaker than those DRT originally imposed. However, the ranking is not robust to allowing scrap values to vary across states: under that generalization, entry cost subsidies can potentially outperform the other subsidy types, reversing the original policy conclusion.

Q: Why does the bootstrap fail for inference in this setting, and why does subsampling work? A: The test statistic ĴN(θ0) involves the minimum of a quadratic form over a non-regular (kinked), random, and possibly nonconvex set. Bootstrap critical values are not asymptotically uniformly valid in this non-regular setting. Subsampling with subsample size hN → ∞, hN/N → 0 (the paper recommends hN = N^{2/3}) delivers asymptotically uniformly valid critical values under weak conditions, because it does not require regularity of the constraint set boundary.

Q: How does the inference approach handle the high dimensionality of DDC settings? A: The paper develops a computational algorithm specifically tailored to the structure of DDC models, exploiting the linear Bellman equation constraints to reduce the effective dimensionality of the optimization problem. In a binary choice model with X = 90, the joint optimization is over a 270-dimensional space; with X = 240 (as in Blundell, Gowrisankaran, and Langer, 2020), the dimension is 720. Standard moment-inequality inference methods (Kaido, Molinari, Stoye, 2019; Bugni, Canay, Shi, 2017) are computationally infeasible at these scales; the authors’ algorithm remains tractable.

Q: How does the paper relate to Norets and Tang (2014), the closest alternative approach? A: Norets and Tang (2014) partially identify structural parameters and high-dimensional counterfactual CCPs by relaxing the assumed distribution of idiosyncratic shocks, focusing on binary choice models and using a pointwise-valid Bayesian approach. The present paper instead targets low-dimensional policy outcomes (nonlinear functions of payoffs and counterfactual CCPs), accommodates multinomial choice, provides asymptotically uniformly valid frequentist inference via subsampling, and restricts the source of underidentification to the payoff function rather than the error distribution. The two contributions are non-nested and complementary.

Q: What is the practical workflow the paper enables for applied researchers? A: A researcher can (i) select any combination of model restrictions (equality or inequality, parametric or shape), (ii) specify any counterfactual experiment via an affine payoff transformation (H, g), and (iii) define any low-dimensional outcome of interest φ, then directly compute the identified set and a valid confidence interval by solving two constrained optimization programs — without deriving new analytical identification results for each specification. The rank condition for checking the dimension of the identified set is computable from the data.

Dynamic Discrete Choice (DDC) Model. A discrete-time infinite-horizon model where agents choose among a finite action set each period, with per-period utilities additively separable into an observed payoff function π and an i.i.d. unobservable shock, and agents maximize expected discounted lifetime utility. The model is parameterized by payoffs π, transition function F, discount factor β, and shock distribution G.

Conditional Choice Probability (CCP). The probability that an agent selects a given action in a given state, integrating out the unobservable shocks. CCPs and state transitions are directly identifiable from panel data and serve as the sufficient statistics for the identified set, in place of the unidentified payoff vector.

Sharp Identified Set for Counterfactual CCPs. The set PĨ(p, F) of all counterfactual CCP vectors p̃ that are consistent with the observed data (p, F) and the imposed model restrictions, given the specified counterfactual transformation. Characterized as a smooth connected manifold with dimension equal to rank(CJ*QJ).

Local Counterfactual. A counterfactual experiment in which the payoff transformation H modifies only a subset L of the state-action pairs, leaving the rest unchanged. Local counterfactuals reduce the dimension of the identified set relative to global experiments, because only the payoffs in the affected subset matter for the unidentified component of the counterfactual response.

Partial Identification / Identified Set for Outcomes. Rather than seeking a unique estimate of a counterfactual outcome θ, partial identification recovers the set ΘI of all values of θ consistent with the data and restrictions. For scalar outcomes this is a compact interval [θL, θU] whose endpoints solve constrained optimization problems over payoff and counterfactual CCP spaces.

Subsampling Inference. A procedure for constructing asymptotically uniformly valid confidence sets by repeatedly computing the test statistic on subsamples of size hN < N, approximating the sampling distribution of ĴN(θ0) without requiring regularity (smoothness) of the boundary of the constraint set — a requirement that fails here due to the kinked, nonconvex nature of the identified set.

Rank Condition for Dimension. The dimension of the identified set for counterfactual CCPs is determined by the rank of the matrix CJ*QJ, which depends on the counterfactual transformation H, the model restrictions, and the observed data. The econometrician can compute this rank from observables to assess, before imposing any strong assumptions, how many dimensions of freedom remain in the identified set.