A Sufficient Statistics Approach for Macro Policy

The paper asks how a policymaker can tell whether current policy is optimal — and, if not, how to correct it — without knowing or estimating the full structural model of the economy, and shows that two directly observable objects are enough: the forecasts of outcomes conditional on current policy, and the impulse responses of those outcomes to a hypothetical policy perturbation. (Introduction, p. 2809). These two objects are the paper’s “sufficient statistics” — quantities policymakers such as central banks already produce and monitor (forecasts like the FOMC’s Summary of Economic Projections, and impulse responses from estimated VARs), so the approach avoids specifying preferences over, or estimating, every structural parameter of the economy.

In the paper’s “general linear environment” (Eq. 16, p. 2818) with a quadratic loss function (Eq. 15, p. 2817) and a unique rational-expectations equilibrium under both the baseline and any perturbed policy rule (Assumption 1), Proposition 1 (Eq. 22, p. 2820) shows that current policy is optimal if and only if R_y^0’ W E_t Y_t^0 = 0 — i.e., the loss-weighted inner product of the impulse-response matrix R_y^0 (outcomes’ response to a policy perturbation) and the baseline forecast E_t Y_t^0 (outcomes expected under current policy) equals zero. This holds because, in a linear economy with quadratic loss, the first-order condition for optimality is exactly the gradient of expected loss with respect to the policy perturbation evaluated at the baseline rule, and that gradient collapses to the product of the two sufficient statistics (Section II-III, pp. 2817-2820).

When the optimality condition fails, the authors define the “Optimal Policy Perturbation” δ_t = −(R_y^0’ W R_y^0)^{-1} R_y^0’ W E_t Y_t^0 (Eq. 24, p. 2821), a rescaled version of the gradient of the loss function with respect to the policy perturbation — formally analogous to a weighted-least-squares regression coefficient (the authors note the OPP formula “looks like the formula of a weighted least squares regression,” p. 2821), rather than a plain steepest-descent step.* Proposition 2 (p. 2821) shows δ_t* = 0 if and only if current policy is already optimal, and when it is nonzero, applying it once yields the exactly optimal policy path, E_t P_t^opt = E_t P_t^0 + R_p^0 δ_t* — a one-step property specific to the linear-quadratic setting (Appendix proof, p. 2843).

The paper decomposes the OPP (Eq. 26, p. 2822) as δ_t = (Γ_p^opt − Γ_p^0) S_t − R_p^0 ε_t^0, separating a correction for a systematically nonoptimal policy rule (when Γ_p^opt − Γ_p^0 ≠ 0) from a correction for purely exogenous policy shocks (ε_t^0 ≠ 0).* This distinction matters because a nonzero OPP driven by rule misspecification calls for a durable change in how policy responds to the state of the economy, whereas one driven by ε_t^0 reflects a one-off discretionary deviation.

Yes: the “subset OPP” (Eqs. 27-28, p. 2824) applies the same logic to a restricted set of instruments — for example, perturbing only the short rate δ_a while holding the yield-curve slope fixed — using δ_a,t = −(R_y,a^0’ W R_y,a^0)^{-1} R_y,a^0’ W E_t Y_t^0.* Corollary 1 (p. 2824) shows that if the full OPP is nonzero, the subset OPP is also nonzero whenever R_y,a^0 is a linear combination of the columns of the full R_y^0, but the subset OPP is (weakly) suboptimal relative to the full OPP: L_t(0,0) ≥ L_t(δ_a,t*, 0).

The “constrained OPP” (Eqs. 30-31, pp. 2825-2826) imposes a linear constraint C_a δ_t = 0 on the perturbation to respect a stated precommitment, illustrated using the September 2020 FOMC statement that rates would stay at the zero lower bound until inflation reached at least 0.5 percentage points above target for a year and unemployment fell to within 0.5 percentage points of its long-run level — thresholds the authors describe as chosen for illustration but state the results are robust to alternative reasonable choices (footnote 21, p. 2837). The paper notes this checks consistency with an existing precommitment, not whether the precommitment itself was a good idea (see Q9); it also proposes a related “time-consistent OPP” (Eq. 33, p. 2827) that augments the loss function to discipline future policy discretion, though this extension receives less empirical emphasis than the constrained OPP.

Forecasts of the inflation gap and unemployment gap are spliced from three sources across the 1990-2022 sample (pp. 2830-2831). For 1990-2006, the authors use the Fed’s semiannual Monetary Policy Report (MPR) — the SEP’s predecessor, which reports median/central-tendency/range FOMC forecasts only two years out — combined with real-time Greenbook estimates of long-run inflation and unemployment (positing linear convergence to those long-run values over five years). For 2007-2009, they use the FOMC’s Summary of Economic Projections (SEP), introduced in October 2007 and released four times a year with a three- to four-year horizon, but still combined with Greenbook long-run estimates, since the SEP itself did not begin asking members for long-run estimates until April 2009. From 2009 on, they use the SEP together with its own long-run estimates. Impulse responses come from a six-variable Bayesian VAR (inflation, unemployment, the Fed funds rate, the 10-year bond–Fed funds rate spread, and two monetary-surprise series ordered first) identified with a high-frequency instrument following Eberly, Stock, and Wright (2020), which builds on Kuttner’s (2001) Fed funds futures surprises, estimated over 1990-2018 (pp. 2831-2832). The evaluation uses a dual-mandate loss (equal weight, λ=1, on inflation and unemployment gaps), a five-year planning horizon, uniform discount weights, and covers the period 1990-2022; both the short-rate subset OPP and a yield-curve-slope subset OPP are computed separately (p. 2830).

Over the full 1990-2022 sample, the short-rate OPP averages approximately 25 basis points, while the slope OPP falls below -1 percentage point during 2009 and stays significantly different from zero through 2009-2013 (Figure 1, p. 2828). In four illustrative episodes: in April 2008 the short-rate OPP is -0.30, implying an additional ~25bp cut beyond the FOMC’s actual move would have reduced unemployment at the cost of a modest inflation overshoot roughly two years later, after the commodity-price surge subsided (Figure 2, p. 2836); in April 2010 the slope OPP is -0.90, implying nearly a full percentage point more accommodation at the long end in exchange for a small 2011 inflation overshoot (Figure 3, p. 2837); around the March 2021 FOMC meeting the unconstrained OPP calls for immediate liftoff with over 90 percent probability the funds rate is too low, but the constrained OPP — respecting the September 2020 precommitment — is exactly zero because unemployment was still substantially above target (Figure 5, p. 2838-2839); and by November 2021, with the labor market nearly fully recovered and inflation expected to remain above target for at least another year, the precommitment is no longer binding and both the constrained and unconstrained OPP call for an immediate rate rise and a steeper policy path (Figure 6, p. 2839-2840).

The optimality condition and the OPP are exact only in the paper’s “general linear environment” (Eq. 16); in nonlinear economies the OPP is valid only as a first-order local approximation. The OPP evaluates perturbations around the current policy path and does not assess whether a precommitment itself is optimal — the authors note that would require identifying separate “forward guidance” shocks at longer horizons (footnote 22, p. 2837). The empirical implementation inherits the exclusion restriction underlying the high-frequency-instrument identification of Eberly, Stock, and Wright (2020), and takes the FOMC’s SEP median projections as the relevant forecast without addressing any gap between those projections and a single structural model’s forecast. The application is restricted to monetary policy; extensions to fiscal, exchange-rate, or climate policy are discussed conceptually in the conclusion (pp. 2839-2840) but not implemented, and sensitivity of results to the baseline loss weight (λ=1) is not reported in detail in the main text.