C68 | Macro Paper Warehouse

Determinacy and Large-Scale Solutions in the Sequence Space

This paper studies the mathematical structure of sequence-space Jacobians – the derivative operators, mapping perfect-foresight paths of shocks to paths of aggregate outcomes, that underlie the increasingly popular “sequence-space” approach to solving macroeconomic models with rich heterogeneity. The authors prove that under general conditions these Jacobians are “quasi-Toeplitz operators”: a Toeplitz operator (whose matrix has constant diagonals, reflecting time-invariant responses to well-anticipated shocks) plus a compact correction that captures the extra effect of a shock’s not being anticipated before the initial date, and that vanishes for shocks announced sufficiently far in advance. They establish two structure theorems – that the Jacobian of any stationary heterogeneous-agent block is quasi-Toeplitz, and, more generally, that the solution operator of any expectational linear difference equation satisfying standard stability conditions is quasi-Toeplitz – implying that quasi-Toeplitz structure is close to universal in sequence-space macroeconomics. The authors exploit this structure in three ways. First, they derive a “winding number” test, building on Onatski (2006), that determines whether a sequence-space system has a unique solution, suffers from indeterminacy, or has no solution at all, by counting how many times a related complex-valued “symbol” function winds around the origin; they show this test agrees with the classic Blanchard-Kahn root-counting criterion when applicable, but extends to a much broader class of models, including heterogeneous-agent models with no finite-dimensional canonical form, and they show the test holds “generically” for quasi-Toeplitz operators, addressing a genericity critique previously raised by Sims (2007) against Onatski’s original test. Second, they show that quasi-Toeplitz structure can be exploited computationally to sharply reduce the cost of avoiding truncation error, either by using the (cheap-to-compute) Toeplitz part of a Jacobian’s inverse as a preconditioner for iterative solvers such as GMRES, or by representing the compact correction term with a low-rank approximation. Third, and most strikingly, they apply these methods to solve a heterogeneous-agent, multi-country fiscal policy model in which 190 countries trade according to a realistic, asymmetric bilateral trade network – a sequence-space system with roughly 190,000 unknowns at each of 1,000 time periods, far too large to solve by direct matrix inversion – in just 12 iterations and under three seconds on a laptop, versus an extrapolated multi-year cost for a comparable state-space solution method. Throughout, the paper’s applications center on stationary models (technically, Jacobians mapping into the space of square-summable sequences), explicitly excluding representative-agent models with a unit root in consumption, which the authors flag as a limitation and direction for future work.

Exploiting MIT shocks in heterogeneous-agent economies: the impulse response as a numerical derivative

This paper proposes a new way to compute the equilibrium of heterogeneous-agent models with aggregate uncertainty: rather than building a recursive, linearized law of motion for a high-dimensional state such as the cross-sectional wealth distribution – the strategy behind existing linearization methods including Reiter (2009, 2010), Childers (2017), and Ahn, Kaplan, Moll, Winberry, and Wolf (2017) – the authors solve, using entirely standard nonlinear methods, a single deterministic perfect-foresight transition path that follows a one-time, unexpected (“MIT”) shock away from the model’s non-stochastic steady state. Under the working assumption that the true stochastic equilibrium is well approximated by a linear system, that one nonlinear transition path is literally the model’s numerical derivative at each future time horizon, so the response to any sequence of aggregate shocks – and to any number of independent shocks, with computation time rising only linearly in the number of shocks – can be recovered by scaling and summing copies of this single impulse response, with no analytical differentiation and no explicit treatment of the distribution as a linearized state. The only nontrivial numerical tool the method requires is value-function iteration: once to solve the model’s non-stochastic steady state, and again, backward over a finite horizon, to solve for the transition path, with the passage of calendar time serving as the sole state variable added along the way. Applied to a standard Aiyagari-style economy with valued leisure and two aggregate technology shocks, the method reproduces conditional-moment results close to Dynare’s linearization- and second-order-perturbation-based solutions in a representative-agent benchmark, delivers accurate impulse responses (including for distributional statistics like the Gini coefficient and the hand-to-mouth share) in the heterogeneous-agent case, and, in an extension with a consumption externality and deficit-financed transfers, shows that Ricardian equivalence breaks down and fiscal transfers have real stabilizing effects – all without added numerical difficulty. The paper reports that its own scalability and additivity checks are satisfied “with flying colors” in this application, but is explicit that the whole approach rests on the assumption that linearization is in fact a good approximation for the model at hand, offers no formal stability or determinacy characterization of the transition path it computes, and, unlike recursive methods, provides no separate goodness-of-fit metric for that assumption.

Solving heterogeneous-agent models by projection and perturbation

This paper proposes a numerical method for solving stochastic general-equilibrium models with incomplete markets and a continuum of heterogeneous agents – a class of problems where, as the paper puts it, “the state vector includes the whole cross-sectional distribution of wealth,” an infinite-dimensional object in principle. The dominant approach at the time, pioneered by Krusell and Smith (1998), represents that distribution with only a small number of statistics (typically the mean) and works very well for the models it was built for; but the paper argues this cannot serve as a general solution, since in models where the shape of the distribution itself drives the dynamics – such as (S,s) pricing or inventory models, or the redistributive-shock example this paper constructs – a low-dimensional summary can miss essential dynamics. The proposed method instead computes a solution that is fully nonlinear in the idiosyncratic (individual) shocks but only linear in the aggregate shocks: it first solves precisely for the steady-state cross-sectional distribution and consumption function (with no aggregate shocks but the full idiosyncratic shock process), using cubic splines for the consumption function and a fine histogram (up to 1000, and as a robustness check 5000, intervals) for the wealth distribution; it then computes a first-order perturbation of that whole high-dimensional representation with respect to small aggregate shocks, using Sims (2001)’s solver for linear rational-expectations systems. Applied to a test model of household saving with uninsurable income risk, liquidity constraints, an aggregate technology shock, and an i.i.d. redistributive capital-tax shock, the method reproduces the Krusell-Smith “approximate aggregation” finding when only the technology shock is active (a one-moment forecast of future aggregate capital is nearly exact), but shows that this breaks down once the tax shock is introduced, in which case even a four-moment forecast leaves sizable error while the paper’s high-dimensional, spline-based solution remains accurate to roughly 10^-6 in absolute forecast error. The method is explicitly a linear approximation in the aggregate dimension – suited to cases “where individual shocks are much bigger than aggregate shocks” – and the paper proposes it as a first step that can be combined with state-space reduction (via spline or principal-component bases) before, in future work, attempting higher-order perturbations in a reduced aggregate state space.

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.