Higher-Order Perturbation in Sequence Space: the Certainty Correspondence

Sequence-space methods have made first-order solutions of heterogeneous-agent models with aggregate shocks fast, by exploiting “certainty equivalence”: to first order, aggregate risk is neutral, so the response to a one-time, perfect-foresight (“MIT”) shock is the same as the true stochastic impulse response. But that restriction to first order rules out any role for precautionary behavior, welfare effects of risk, or history- and size-dependent responses – exactly the questions a growing literature wants to ask of these models. This paper shows how to go beyond first order in the sequence space by establishing a “certainty correspondence”: nearly all the terms in the second- and third-order Taylor expansion of the model’s full nonlinear sequence-space solution – including the risky steady state and the interaction between shock size and shock history – can be computed purely from perfect-foresight (“MIT shock”) impulse responses, differentiated with respect to shock size and shock timing, without ever manipulating the derivatives of the underlying equilibrium system directly. Combined with a “one-shot principle” that lets each order of the general-equilibrium solution be obtained by evaluating equilibrium conditions on the previous order’s solution and applying a single already-computed inverse Jacobian, this makes third-order solutions of large heterogeneous-agent models computationally practical: an unreduced HANK model with roughly 5,000 idiosyncratic grid points needs only about 300,000 terms at third order in the sequence space, versus a state-space alternative that would require roughly 62 trillion terms and could not be stored on a computer. Applying the method to a quantitative HANK model, the paper finds the fiscal (transfer) multiplier is about four times larger when a sequence of shocks has pushed output several percent below steady state than when output is above steady state, consistent with empirical evidence on state-dependent fiscal multipliers. Applying it to a menu-cost model with strategic complementarity, the paper finds the responsiveness of inflation to nominal marginal cost is significantly steeper when trend inflation is already high, echoing recent findings on nonlinear Phillips curves. Extensive accuracy checks – against third-order state-space perturbation on a small model, and against a global Bellman solution in partial equilibrium – show the third-order sequence-space solution tracks both closely, and in fact tracks the global solution more closely than the perfect-foresight solution does, because it captures the effect of aggregate precautionary saving on marginal propensities to consume that a purely perfect-foresight calculation misses.

Rodolfo Rigato | Macro Paper Warehouse

Higher-Order Perturbation in Sequence Space: the Certainty Correspondence