Using the Sequence-Space Jacobian to Solve and Estimate Heterogeneous-Agent Models

This paper proposes a general and highly efficient method for solving and estimating general-equilibrium heterogeneous-agent models with aggregate shocks in discrete time. Building on Reiter (2009)’s idea of perturbing a heterogeneous-agent model to first order in aggregates, the authors write the linearized equilibrium conditions not in the state space (as Reiter does) but in the “sequence space” – as a system relating perfect-foresight paths of aggregate variables – so that the size of the resulting linear system no longer depends on the size of the underlying distributional state space. The paper’s central objects are sequence-space Jacobians: derivatives of the mapping from aggregate input sequences (such as interest rates or wages) to aggregate output sequences (such as consumption or investment), which the authors show are “sufficient statistics” summarizing everything about household or firm heterogeneity relevant for general equilibrium. Their main technical contribution is a “fake news” algorithm (Proposition 1) that computes these Jacobians using a single backward iteration and a single set of forward-iterated expectation vectors, rather than the costly direct approach of repeating a full backward-then-forward solve separately for a shock at each date – lowering the computational cost by a factor of roughly T, the number of periods considered, which is typically 300 to 1,000 in practice. These heterogeneous-agent Jacobians are then combined with the Jacobians of the model’s other equilibrium conditions – represented as a directed acyclic graph of blocks – via the chain rule, to obtain full general-equilibrium impulse responses essentially instantaneously. The authors verify the method’s accuracy by showing it reproduces the Reiter method’s solutions, using automatic differentiation in both methods, to within machine precision on models small enough for Reiter to remain feasible. They then develop two applications that this speed makes newly practical: full-information, likelihood-based Bayesian estimation of heterogeneous-agent models (by recovering an MA representation, computing autocovariances analytically, and applying the Kalman filter, while reusing Jacobians across repeated likelihood evaluations), and the computation of nonlinear perfect-foresight transitions via a quasi-Newton method that reuses the steady-state Jacobian at every iteration. Applied to three canonical models of increasing complexity – a Krusell-Smith neoclassical model, a one-asset New Keynesian HANK model, and a two-asset New Keynesian HANK model – the methods compute all heterogeneous-agent Jacobians in under 11 seconds, obtain posterior-mode estimates in under nine minutes, and trace out full posterior distributions via Markov Chain Monte Carlo with 200,000 draws in under twelve hours even for the most complex two-asset model – estimation exercises the authors describe as previously out of reach for the literature.

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Using the Sequence-Space Jacobian to Solve and Estimate Heterogeneous-Agent Models