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

The paper addresses the computational challenge that equilibrium in heterogeneous-agent models involves a time-varying, high-dimensional distribution of agents, by linearizing the model’s equilibrium conditions in the “sequence space” of perfect-foresight aggregate paths rather than in the state space of the distribution itself, so that the size of the resulting linear system “is independent of the size of the state space” (Introduction, pp. 1-2). Like Reiter (2009), the authors perturb the model to first order in aggregates while preserving full nonlinearity with respect to idiosyncratic shocks; unlike Reiter, whose linear system grows (and becomes costly to solve, scaling roughly with the cube of the state-space size) with the dimension of the discretized distribution, their sequence-space system’s size depends only on the number of aggregate variables and the time horizon T, “making it feasible to solve and estimate models that feature very rich heterogeneity” (pp. 1-2, 5).

A sequence-space Jacobian is the derivative, evaluated around the steady state, of a mapping from a time path of aggregate inputs (e.g., real interest rates) to a time path of aggregate outputs (e.g., aggregate consumption) implied by heterogeneous households’ or firms’ optimizing behavior; the paper calls it a sufficient statistic because “to know the aggregate effect of r on C, all we need to know is [the Jacobian]” – the underlying heterogeneous household responses and distributional dynamics do not need to be recomputed once the Jacobian is known (Introduction, p. 2). The general representation the paper works with (equations 10-12, Section 3.1) expresses any such heterogeneous-agent problem via three linked functions describing (i) a value-function-like object’s backward recursion, (ii) the distribution’s forward law of motion, and (iii) the aggregation of individual outcomes into aggregate outputs.

The fake news algorithm (Section 3.2, Proposition 1) computes the full sequence-space Jacobian using only a single backward iteration – to obtain, for a shock at each possible date s, the resulting date-0 output response and date-1 distribution response – combined with a single set of forward-iterated “expectation vectors” computed once from the steady state, whereas the “direct” approach requires a full separate backward-then-forward iteration for a shock at each of the T dates (Section 3.2, pp. 11-14). The name reflects the intuition behind the recursion: the date-1 distributional response to a shock announced for some future date s behaves like a piece of “fake news” that is essentially retracted by date 1 if that shock does not occur, so its later effects can be reconstructed from steady-state “expectation vectors” (Lemma 3) rather than resimulated. This lowers the computational cost “by a factor of about T relative to the direct approach,” and the paper reports that computing all heterogeneous-agent Jacobians for each of its three example models takes well under 11 seconds on a laptop (Introduction, pp. 2-4; Table 1).

General equilibrium is written as a nonlinear system F(X, Z) = 0 in endogenous variables X and shocks Z, whose first-order solution dX = -F_X^{-1} F_Z dZ requires the Jacobians F_X and F_Z, formed by combining the heterogeneous-agent Jacobians (from the fake news algorithm) with the Jacobians of the model’s other equilibrium conditions via the chain rule (Section 4.1, pp. 19-21). The authors implement this by representing the model as a directed acyclic graph (DAG) of “blocks” – as illustrated for the Krusell-Smith economy in their Figure 3 – and composing block Jacobians along the graph via “forward accumulation,” a procedure they say “can be automated, and it usually only takes a few milliseconds” once the component Jacobians are known (p. 2, p. 21).

The authors benchmark their sequence-space Jacobian (SSJ) method against the Reiter method, both implemented with automatic differentiation to eliminate derivative-approximation error, and find that on models small enough for Reiter to remain feasible, the two methods “deliver exactly the same solution,” with impulse responses differing by less than 10^-9 and the recovered state-space law-of-motion matrices differing by less than 10^-8 (Section 4.2, pp. 21-23). They further note that SSJ implemented with numerical rather than automatic differentiation produces “larger but still relatively minor errors,” and discuss how the truncation horizon T affects accuracy.

Truncated linear impulse responses form a moving-average, MA(T-1), representation of the model (following Boppart, Krusell and Mitman 2018), from which the authors compute autocovariances analytically via a standard formula rather than by simulation, recover a state-space law of motion, and apply the Kalman filter to evaluate the exact Gaussian likelihood of an observed time series – with the crucial efficiency gain that heterogeneous-agent Jacobians, once computed, can be reused across the many likelihood evaluations required by posterior-mode search and Markov Chain Monte Carlo (Sections 5.1-5.4, pp. 25-32). Quantitatively, they report that a single likelihood evaluation takes a fraction of a second even for the two-asset HANK model, that finding the posterior mode takes under nine minutes for every model considered, and that a 200,000-draw Random Walk Metropolis-Hastings run over the two-asset model’s parameters takes under twelve hours – versus an estimated “about 8 full days” just to re-evaluate the heterogeneous-agent Jacobian 200,000 times using their own fake news algorithm, and “about 6 years” using the direct algorithm, underscoring that Jacobian reuse, not merely Jacobian speed, drives the estimation gains (Section 5.4, pp. 32-33).

The paper solves the model’s nonlinear perfect-foresight transition equations with a quasi-Newton iteration that updates a guess for the path of unknowns using the fixed steady-state Jacobian at every step, rather than recomputing the true Jacobian at each guess, and shows this “falls in the class of quasi-Newton methods” yet converges rapidly (Section 6, pp. 33-34). For a large -5 percentage-point monetary policy shock in the two-asset HANK model, the algorithm converges to a residual below 10^-8 in 13 iterations, versus 5 iterations for a smaller -1 percentage-point shock; comparing the linear approximation to the true nonlinear response, the authors find them “almost identical” for the small shock and only modestly different even for the large one (0.81% versus 0.86% on impact), leading them to conclude that “nonlinearities in the household model… do not play an important role for plausibly-sized shocks” in this application (p. 34). The same procedure, applied with the terminal steady-state Jacobian, computes the nonlinear transition following a permanent 1% TFP shock in 7 iterations (p. 34).

The paper applies its methods to three canonical heterogeneous-agent models of rising complexity – a Krusell-Smith neoclassical model (and a “high-dimensional” version with 250,000 idiosyncratic states), a one-asset New Keynesian HANK model in the spirit of McKay, Nakamura and Steinsson (2016), and a two-asset New Keynesian HANK model in the spirit of Kaplan, Moll and Violante (2018) (Section 1, Table 1, pp. 3-4). Reported laptop computing times (Table 1) show heterogeneous-agent Jacobians computed in under 11 seconds and full sets of general-equilibrium impulse responses (matrix G) in a fraction of a second for all models, with the heaviest cost – full MCMC estimation of the two-asset model – still completing in under twelve hours, which the authors describe as faster than what “the leading computational techniques existing today” can achieve for models of comparable richness (p. 4).

The authors note that, like all methods that linearize with respect to aggregates, their approach “does not generate risk premia,” leaves “portfolio choice… indeterminate,” and leaves “optimal Ramsey policy… ill-defined,” so that applications requiring these objects need higher-order perturbation or global solution methods instead (Introduction, footnote 7, p. 5). They also flag a structural limitation of their core framework (equations 10-12): the value function vt “is not allowed to depend on” the distribution Dt, which “prevents us from applying the fake news algorithm when the behavior of heterogeneous agents depends on the anticipated future distribution through the value function, in a way that cannot be intermediated via aggregates… in general equilibrium” – examples they name include OLG models with endogenous bequest distributions and labor- or money-search models with wage posting or distribution-dependent bargaining (Section 3.3, “Scope and limitations,” p. 19). For other extensions – entry and exit, nonlinear functionals of the distribution, sub-period timing – the paper’s appendix generalizes Proposition 1 (its Propositions 2 and 3) so the fake news algorithm continues to apply (p. 19).