Higher-Order Perturbation in Sequence Space: the Certainty Correspondence

First-order sequence-space perturbation relies on certainty equivalence – the fact that, to first order in shock size, aggregate risk is neutral and future variables can be replaced by their expectations – which means a first-order solution has “no role for size- or history-dependent effects, nor is there any… precautionary response to risk,” making it “impossible to study how aggregate shocks, or policies responding to them, affect welfare” (Introduction, p. 2). The paper’s goal is to extend sequence-space methods, which sidestep the curse of dimensionality that state-space methods face in heterogeneous-agent models, to second and third order so that these questions become tractable.

It is the result that, given a nonlinear sequence-space solution 𝒳 expressing the economy’s path as a function of anticipated risk σ and the history of realized shocks, the higher-order terms of a Taylor expansion of 𝒳 around the deterministic steady state can be obtained from the derivatives of the perfect-foresight (“MIT shock”) solution alone (section 2.1-2.4, pp. 6-14; Proposition 3, p. 14). At second order, the “size-dependence” and “history-dependence” terms come directly from differentiating MIT-shock impulse responses with respect to shock size and shock timing (equations 13 and 15), while the pure “risk” term (𝒳σσ, which pins down the risky steady state) is obtained by perturbing steady-state expectations with a constant shift proportional to the size-dependence term and re-solving the steady state (equation 16-17) – so even the risk term, despite reflecting genuine uncertainty, can be computed without ever solving a stochastic problem directly.

Because Lan and Meyer-Gohde write the model with the expectation operator applied outside the equilibrium function, E_t[F(X_{t-1}, X_t, X_{t+1}, ϵ_t)] = 0, rather than with expectations only entering through E_t[X_{t+1}] inside F as in this paper’s equation (1); this difference in “where expectations are taken” means an additional term – reflecting the interaction of risk in X_{t+1} with the nonlinearity of F – survives in their framework and prevents the risk term from being recovered from perfect foresight alone (Introduction and section 2.4, pp. 4-5, 14; footnote and appendix A.6). The authors note that dynamic programming usually allows a model to be rewritten so expectations enter only through the mean, which is the reformulation that makes the certainty correspondence available.

It is the observation that each order of a general-equilibrium sequence-space solution can be obtained without solving any new nonlinear fixed point: evaluate the equilibrium system H nonlinearly on the previous order’s solution to obtain an “error” term at the next order (e.g., h_err,2 for the second-order error), then apply the single already-computed inverse steady-state Jacobian H_U^{-1} to that error to obtain the new order’s solution term (section 3.2, pp. 25-27, equations 33-36). This bypasses iterative nonlinear solving at each order and “dramatically cut[s] down on the time required to solve for the overall third-order perturbation” (p. 27), extending naturally from second to third order (appendix A.12).

In a calibrated heterogeneous-agent model with roughly 5,000 idiosyncratic grid points (200 asset points, 7 productivity states, 2 patience states, tracking both marginal value and distribution), a second-order sequence-space perturbation requires approximately 20,000 terms and a third-order perturbation approximately 300,000 terms, using a truncation horizon of T=300 (L=60 for second order, L=30 for third order); by contrast, a first-order state-space perturbation already has size 25 million, a second-order state-space perturbation is 125 billion, and a third-order state-space perturbation is 62 trillion – “infeasible because the resulting system could not even be stored on a computer” (section 3.3, Table 2, pp. 26-28).

The paper simulates the model under second- and third-order Volterra expansions and finds the transfer multiplier dY_t/dB_t is “about four times as large when a sequence of negative shocks has pushed output 3% below the deterministic steady state, compared to when a sequence of positive shocks has pushed output 2% above that steady state,” generating “a wide ergodic distribution of fiscal multipliers,” which the authors relate to the empirical literature on state-dependent fiscal multipliers (Auerbach and Gorodnichenko 2012), driven here by the association between recessions, weak balance sheets, and higher average marginal propensities to consume rather than by the zero lower bound (section 4.1, p. 31, Figure 10). The paper also finds that anticipated aggregate risk depresses output by less than 0.1% and interest rates by about 25 basis points in the risky steady state at the baseline calibration (σ = 5%) (p. 29, Figure 7).

Cost uncertainty widens firms’ inaction region and erodes risky-steady-state average markups, and – most centrally – the responsiveness of inflation to marginal cost shocks, dπ_t/dϵ_t, “steepens” at higher rates of current (trend) inflation, an effect “particularly pronounced in the third order solution, since then the relationship between average inflation and inflation responsiveness becomes convex,” echoing similar findings in other recent nonlinear-Phillips-curve studies (Blanco, Boar, Jones and Midrigan 2024) (section 4.2, pp. 32-35, Figures 11-14). The paper also finds that a fifth-order-accurate correction to second moments significantly raises the variance and lowers the autocovariance of inflation relative to the standard second-order-accurate approximation, and that the pruned second-order autocovariance can correct “in the wrong direction” relative to the more accurate estimate.

Comparing the partial-equilibrium heterogeneous-agent model’s third-order sequence-space solution to a global solution obtained via a Bellman approach, the paper finds the third-order solution “gets very close to the global solution,” and importantly that the third-order impulse response is closer to the global solution than the perfect-foresight solution is, “because the third-order solution captures the effect of aggregate precautionary savings on MPCs, which the perfect-foresight solution misses” (Introduction, p. 4; appendix C). For the part of the solution that ignores aggregate precautionary effects, the third-order solution is “extremely close” to the fully nonlinear perfect-foresight solution for shocks of plausible magnitude, persistence and distribution, and the paper separately verifies its third-order routines reproduce standard third-order state-space perturbation exactly on a neoclassical growth model.

The paper positions itself as complementary to both: Bhandari, Bourany, Evans and Golosov (2023) “provide a hybrid of state- and sequence-space approaches” and derive a second-order Volterra expansion, while Bilal (2023) and related work extend state-space perturbation to the heterogeneous-agent “Master Equation” context, which “can require significant model reduction” since a second-order state-space solution is cubic in the state dimension; this paper states that, “to our knowledge, our work is the first to point out the certainty correspondence result, and also to obtain a third-order solution, without model reduction, in the presence of substantial heterogeneity” (Introduction, p. 5). The paper also positions itself relative to the separate literature seeking fully “global” solutions to heterogeneous-agent models – including Kase, Melosi and Rottner (2022), Han and Yang et al. (2021, i.e. DeepHAM), Yang, Wang, Schaab and Moll (2025), and Lee (2025) – describing that literature as “highly complementary,” noting its own third-order solution performs well against a global partial-equilibrium benchmark while remarking that “the extent to which global solutions differ from our third-order solution for typical heterogeneous-agent models remains an intriguing open question” (Introduction, p. 5).