Limited-Information Estimation of Heterogeneous Agent Models

Abstract: We develop a method for estimating and testing a single block of a macroeconomic model with heterogeneous agents, without placing assumptions on the structure of the rest of the economy. In a large class of models, individual agents' decisions depend on the macroeconomy only through their expectations of the evolution of a finite-dimensional vector of "sufficient statistics" (e.g., asset returns or aggregate earnings). Our estimator selects the structural parameters that provide the best model-consistent fit between empirical impulse responses with respect to identified macro shocks of (a) cross-sectional moments of agent choices (e.g., moments of consumption) and (b) the vector of sufficient statistics. In a simulation illustration, we estimate a two-asset heterogeneous household model block without restricting production, firm investment, financial intermediation, monetary policy, trade, etc.

• The paper develops a limited-information spectral GMM estimator that identifies household-block parameters by matching impulse responses of cross-sectional moments to sequence-space Jacobian predictions, without specifying firms, financial intermediaries, or policy rules.
• The method is consistent and asymptotically normal under weak dependence assumptions, but simulations with 120 quarters show that bootstrap confidence intervals provide near-nominal coverage of 0.857–0.947 while analytical intervals under-cover substantially.
• The framework enables researchers to estimate and test individual blocks of heterogeneous-agent models using identified shocks and macroeconomic sufficient statistics, although validity depends on instrument relevance, correct block specification, and reliable bootstrap inference.

Motivation and contribution

Structural estimation of heterogeneous agent (HA) macroeconomic models has historically required specifying the full general equilibrium: production, financial intermediation, monetary and fiscal policy, and so on. This creates a familiar fragility—misspecification of one block contaminates estimation of parameters belonging to other blocks. The paper by Liu, Plagborg-Møller, and Tan develops a limited-information alternative for HA models, analogous in spirit to GMM estimation of a single Euler equation or Phillips curve in representative agent settings (2608.13953). The key observation is that in a large class of HA models, individual decisions depend on the macroeconomy only through a finite-dimensional vector of aggregate "sufficient statistics"—typically prices such as asset returns or after-tax earnings. The estimator matches empirical impulse responses of cross-sectional moments of agent choices to model-implied responses constructed from sequence-space Jacobians (SSJs) and empirical impulse responses of the sufficient statistics, without imposing any process on their dynamics.

Moment conditions from shocks as instruments

The starting point is the linearized sequence-space representation of aggregated household outputs $y_t$ around a deterministic steady state:

$$y_t - y^{ss} \approx E_t\left[\sum_{k=0}^\infty J_k(\theta)(x_{t+k}-x^{ss})\right] + \xi_{t-1}(\theta),$$

where $\{J_k(\theta)\}$ are the SSJs mapping expected future paths of sufficient statistics $x_t$ into aggregated choices, and $\xi_{t-1}(\theta)$ captures the pre-determined cross-sectional distribution of idiosyncratic states. Two obstacles prevent direct nonlinear least squares: expectations at all horizons are unobserved, and $\xi_{t-1}(\theta)$ is an unobservable function of the state distribution that is correlated with $x_t$.

The authors resolve both using identified economic shocks as instruments $z_t$ satisfying two conditions: orthogonality to past shocks (hence to $\xi_{t-1}$), and inability to predict households' forecast errors at any horizon—implied by rational expectations plus instruments being in the information set. This yields moment conditions

$\cov(y_t,z_t) = \sum_{k=0}^\infty J_k(\theta_0)\cov(x_{t+k},z_t).$

Notably, instruments need not have unique economic interpretations; composites of several shocks (e.g., high-frequency monetary surprises that mix interest rate and information shocks) remain valid. The moment conditions also tolerate general measurement error processes in $$y_t - y^{ss} \approx E_t\left[\sum_{k=0}^\infty J_k(\theta)(x_{t+k}-x^{ss})\right] + \xi_{t-1}(\theta),$$0, $$y_t - y^{ss} \approx E_t\left[\sum_{k=0}^\infty J_k(\theta)(x_{t+k}-x^{ss})\right] + \xi_{t-1}(\theta),$$1, and $$y_t - y^{ss} \approx E_t\left[\sum_{k=0}^\infty J_k(\theta)(x_{t+k}-x^{ss})\right] + \xi_{t-1}(\theta),$$2 under dynamic uncorrelatedness restrictions.

Estimator and inference

The estimator is formulated as spectral GMM: a minimum distance procedure matching cross-periodograms $$y_t - y^{ss} \approx E_t\left[\sum_{k=0}^\infty J_k(\theta)(x_{t+k}-x^{ss})\right] + \xi_{t-1}(\theta),$$3 against $$y_t - y^{ss} \approx E_t\left[\sum_{k=0}^\infty J_k(\theta)(x_{t+k}-x^{ss})\right] + \xi_{t-1}(\theta),$$4 across Fourier frequencies, where $$y_t - y^{ss} \approx E_t\left[\sum_{k=0}^\infty J_k(\theta)(x_{t+k}-x^{ss})\right] + \xi_{t-1}(\theta),$$5. The frequency-domain formulation facilitates handling the infinite decision horizon and motivates a Gaussian multiplier bootstrap that resamples periodograms and re-runs the nonlinear optimization. Standard errors use a HAC estimator applied to an approximate score process, and over-identification follows the conventional minimum distance statistic with asymptotic $$y_t - y^{ss} \approx E_t\left[\sum_{k=0}^\infty J_k(\theta)(x_{t+k}-x^{ss})\right] + \xi_{t-1}(\theta),$$6 distribution.

On theory, the paper proves consistency and asymptotic normality under weak assumptions: strict stationarity with summable second- and fourth-order cumulants, compact parameter space, and uniform summability of the SSJs and their derivatives. A technical contribution is that stochastic equicontinuity of the sample moments holds under these same fourth-moment conditions—whereas existing results (Dahlhaus 1988) require moments of all orders—because the moment function is bilinear in the periodogram and the SSJs. Identification requires the map $$y_t - y^{ss} \approx E_t\left[\sum_{k=0}^\infty J_k(\theta)(x_{t+k}-x^{ss})\right] + \xi_{t-1}(\theta),$$7 to be injective; intuitively, instruments must induce sufficiently rich dynamic variation in the sufficient statistics, particularly at long horizons when identifying forward-looking parameters such as the discount factor.

Simulation evidence

The method is illustrated on data simulated from the two-asset HANK model of Auclert, Bardóczy, Rognlie, and Straub (ABRS), estimating four household-block parameters—the EIS $$y_t - y^{ss} \approx E_t\left[\sum_{k=0}^\infty J_k(\theta)(x_{t+k}-x^{ss})\right] + \xi_{t-1}(\theta),$$8, discount factor $$y_t - y^{ss} \approx E_t\left[\sum_{k=0}^\infty J_k(\theta)(x_{t+k}-x^{ss})\right] + \xi_{t-1}(\theta),$$9, and adjustment cost parameters $\{J_k(\theta)\}$0—using nine output series (moments of log consumption and wealth-bracket asset holdings), three sufficient statistics, and two instruments (monetary and government spending shock innovations), with 20% measurement error added to each output series. Only the household block's structure is used; nothing about firms, intermediaries, or policy rules enters the estimation.

With $\{J_k(\theta)\}$1 quarters across 512 Monte Carlo replications, biases and standard deviations are modest relative to parameter scales. The analytical confidence intervals under-cover substantially for three of four parameters (coverage as low as 0.467 for $\{J_k(\theta)\}$2 against a nominal 0.90), but bootstrap intervals achieve near-nominal coverage (0.857–0.947). The analytical over-identification test over-rejects severely (rejection rate 0.949 at nominal size 0.10), while the bootstrap version is near-nominal (0.115). These results imply that practitioners should rely on the bootstrap rather than delta-method inference, at least in samples of this size.

Limitations and open questions

Several limitations are acknowledged explicitly. First-order linearization ignores aggregate risk effects beyond precautionary behavior embedded in the steady state; incorporating them would require higher-order expansions. The method cannot estimate counterfactuals involving parameters outside the estimated block, though some objects (e.g., steady-state wealth Lorenz curves in some HANK models) depend only on household-block parameters. Exploitation of micro panel data is limited to moments of at most two consecutive periods of individual states and choices; extending SSJ computation to longer panels remains open. Bootstrap validity is not formally proven, and the non-Gaussian correction via the Multivariate Frequency-domain Hybrid Bootstrap relies on a conjectured extension of Meyer and Paparoditis (2023) to data-dependent spectral means. The efficient weight matrix may behave poorly when the number of moments is large relative to $\{J_k(\theta)\}$3, and the authors recommend application-specific Monte Carlo comparisons. Finally, identification depends on instrument relevance at appropriate horizons—an easily testable but substantive requirement—and the correct specification of the block itself is assumed, though testable via over-identification.

Conclusion

The paper provides a rigorous limited-information framework for estimating and testing single blocks of heterogeneous agent models, requiring only time series on macro sufficient statistics and identified shocks, and leaving all other equilibrium structure unrestricted. It combines the robustness logic of classic GMM estimation of optimality conditions with modern sequence-space computational tools, delivers consistency and asymptotic normality under weak dependence assumptions, and demonstrates through calibrated simulations that bootstrap-based inference is reliable even in moderate samples.