Quantitative Economics | Macro Paper Warehouse

A method for solving and estimating heterogeneous agent macro models

This is a computational method for solving and estimating heterogeneous-agent macro models with aggregate shocks, built around one substitution: the infinite-dimensional cross-sectional distribution is replaced by a flexible parametric family, so its moments become finite-dimensional endogenous state variables, and the resulting system is then perturbed around its stationary equilibrium. The stated motivating gap is adoption rather than accuracy: of the many existing algorithms, “none are as general, efficient, or easy-to-use as the standard perturbation methods routinely employed to solve representative agent models using prepackaged toolboxes like Dynare. Due to this challenge, heterogeneous agent models have yet to reach widespread adoption, particularly among central banks and policy institutions.” The method has three steps — globally accurate projection approximations in the individual states (Chebyshev collocation for the firm’s value function, an exponential-polynomial density adapted from Algan, Allais and Den Haan for the distribution), computation of the stationary equilibrium without aggregate shocks, and a locally accurate Taylor expansion in the aggregate state, implemented directly in Dynare with a published code template. Demonstrated on the Khan and Thomas (2008) model of heterogeneous firms with fixed capital adjustment costs, a first-order solution takes 35 seconds for a degree-2 approximation of the distribution and 47 seconds for degree 4, most of that in the stationary equilibrium. Accuracy is reported by degree: degree 2 already gives steady-state aggregates “virtually indistinguishable” from a fine histogram (output 0.498 against 0.498, capital 1.006 against 1.007), while degree 1 is visibly wrong (capital 1.200), and higher degrees are needed only for the shape of the distribution, not the aggregates — cross-sectional covariance and variance of log capital do shift with the degree of approximation, though “these differences are small and do not generate differences in the dynamics of aggregate variables.” Business cycle moments are conventional: output volatility 2.14 percent, consumption 0.48 times as volatile as output, investment 3.86 times, all highly correlated because TFP is the only shock. Three generalizations establish the claimed breadth. A second-order approximation is a one-line Dynare change and shows quantitatively small sign- and state-dependence, consistent with Khan and Thomas; mass points from occasionally binding constraints are handled by adding the mass at the constraint as an extra distributional parameter, which is how the Krusell and Smith (1998) household model is solved; and approximate aggregation is shown not to be required, with a volatile-enough investment-specific shock breaking it. The efficiency claim is externally sourced: in Terry’s comparison project, an implementation of this method “solves and simulates the model in 0.098% of the time of the Krusell and Smith (1998) method.” That speed is what makes the paper’s estimation exercise possible: 10,000 Metropolis-Hastings draws in 44 minutes and 32 seconds, estimating the parameters of neutral and investment-specific productivity shocks conditional on three values of the fixed-cost upper bound. As frictions rise from 0.01 to 1, the estimated volatility of investment-specific shocks rises from 0.0056 to 0.0077 while the other parameters barely move, so “matching the aggregate investment data with larger adjustment frictions requires more volatile shocks.” The limits are stated rather than implied: the local approximation is “not well suited for capturing nonlinearities that are global in nature,” the parametric family fails for fat-tailed distributions, the estimation conditions on frictions rather than estimating them jointly, and the resulting variance decompositions differ only slightly across the friction values considered.

DeepHAM: A global solution method for heterogeneous agent models with aggregate shocks

Solving heterogeneous-agent (HA) models with aggregate shocks efficiently, reliably, interpretably, and generally has proven difficult: the Krusell-Smith (KS) method approximates the distribution with a small number of moments and is efficient for simple models but suffers a curse of dimensionality with multiple shocks or endogenous states, while the local-perturbation method of Reiter (2009) handles complex models but is unreliable wherever aggregate shocks generate nonlinear or nonlocal effects (a zero lower bound, large shocks, or a risky steady state that departs from the deterministic one). This paper’s method, DeepHAM, is designed to satisfy all four requirements – efficiency, reliability, interpretability, and generality – at once. It represents each agent’s value and policy functions with deep neural networks, and, rather than feeding these networks the entire cross-sectional distribution, first extracts a small number of “generalized moments”: neural-network-determined, permutation-invariant summary statistics of the distribution that play a role analogous to classical moments (such as the first moment of wealth) but are automatically optimized rather than fixed a priori. The networks are trained by directly optimizing the model’s objective over simulated economic paths, sidestepping the fixed-point iterations that both KS-style and Reiter-style methods require. In a calibrated Krusell-Smith benchmark, DeepHAM using only the first moment already reduces the Bellman equation error by 27.2% relative to the classic KS solution, and DeepHAM with one algorithmically-optimized generalized moment reduces the error by 40.3%; the resulting generalized moment reveals that the mapping from individual wealth to the moment is concave, implying that a purely redistributive, unanticipated policy shock changes the welfare of “middle” households who are not part of the redistribution program – a channel invisible to the standard KS solution, in which welfare depends only on the first moment. DeepHAM also efficiently solves more complex environments, including a model with a Brunnermeier-Sannikov-style financial sector, without suffering the curse of dimensionality that afflicts moment-matching or discretized state-space methods as the number of shocks or endogenous states grows. Finally, because DeepHAM’s neural networks are trained directly against a stated objective rather than derived from a decentralized equilibrium concept, the same framework solves the planner’s constrained-efficiency problem as easily as the competitive equilibrium, which the authors note “opens up new possibilities for studying optimal monetary and fiscal policies in heterogeneous agent models with aggregate shocks.” The paper’s stated scope excludes models in which aggregate variables are determined recursively as a function of expected future aggregate variables (such as inflation in a forward-looking New Keynesian Phillips curve), which the authors flag as requiring an additional price function and leave for companion work.

Solving discrete time heterogeneous agent models with aggregate risk and many idiosyncratic states by perturbation

This is a solution method for discrete-time heterogeneous-agent models with aggregate risk. It extends the perturbation approach of Reiter (2002, 2009) and complements the continuous-time work of Ahn, Kaplan, Moll, Winberry and Wolf (2017) by placing the dimensionality reduction at a new point in the pipeline: after the stationary equilibrium without aggregate risk has been solved, but before the nonlinear difference equation is linearized. Two reductions do the work. Value (or policy) functions are written as sparse expansions around their stationary-equilibrium counterparts using the discrete cosine transform, with only the largest coefficients allowed to move and the rest held at stationary values — the authors’ analogy is lossy video compression against a lightly compressed reference frame. The joint distribution over idiosyncratic states is factored into its marginal histograms, which vary freely, and a copula, which is held fixed at its stationary value. Because the deviation being zero exactly reproduces the stationary value function, the compression introduces no approximation error in the stationary equilibrium “irrespective of the degree of sparseness that is used in the calculation of the model dynamics.” The problem being solved is concrete: for a household problem with two assets and idiosyncratic income at 50 × 50 × 9 grid points, the distribution and value function are each vectors of 22,500 entries, the Jacobian blocks exceed 45,000 × 45,000, and each would occupy more than 7 GB stored densely. On the Krusell-Smith (1998) benchmark with the JEDC comparison-project calibration, the reduced method’s simulated log capital stock differs from the original Krusell-Smith algorithm’s by 0.0324% on average, exactly as the unreduced Reiter method does, and from the unreduced Reiter solution by 0.0003% on average; its Den Haan error is 0.0100% mean and 0.0191% max, against 0.0051% and 0.0131% for the Krusell-Smith algorithm, which remains the more accurate of the two. Run time is 0.38 seconds versus 91.61 for Krusell-Smith and 1.19 unreduced — “more than 240 times faster,” or 13 times faster once the 7.05 seconds for the stationary equilibrium are included. The scalability claim rests on a two-asset HANK model with 120,000 states and 240,000 controls that “is infeasible to solve for the aggregate dynamics… on the full histogram”: the fixed copula cuts states to 236 and DCT compression at 99.9999% energy cuts controls to 1427, giving a 5-minute solve plus 22 minutes for the stationary equilibrium on a laptop, with Den Haan errors of 0.033% mean and 0.092% max for capital. The costs are stated openly. The selection of retained coefficients is a heuristic, and the coefficients dropped “are only unimportant in the stationary equilibrium,” so robustness must be checked with Den Haan’s test rather than guaranteed. And the fixed copula is not innocuous for every shock: under TFP shocks the Jensen-Shannon distance between the reduced and unreduced joint distributions is a negligible 0.0005, but under idiosyncratic income-uncertainty shocks — which hit the joint distribution directly — the difference “attains a significant order of magnitude,” and recovering accuracy requires perturbing the copula’s own largest DCT coefficients (41 out of a possible 2100 in their example).