Optimal Monetary Policy When the Economy Must Be Learned

Introduction (Section 1)

The paper measures learning at four levels and conditions its conclusions about optimal policy on all four: the economy’s agents, the econometrician, a correction to the model’s residuals, and the policymaker.

The page follows three agents against the realized federal funds rate: the baseline agent, estimated and trained on the full sample (Section 8.1); the frozen agent, estimated and trained on data through 2015:Q4 and held fixed (Section 8.2); and the rolling agent, re-estimated and retrained each quarter from 2006:Q1 on the data then available (Section 8.3).

The Real-Time Data (Section 2)

With three exceptions, the neutral-rate path, core PCE inflation, and the term premium, every observation the model reads at a decision point is one the FOMC could have read before its meeting.

The data come from 30 series, each held with every release as a separate vintage, drawn from the Archival Federal Reserve Economic Data (ALFRED), the Real-Time Data Set for Macroeconomists (RTDSM), the Survey of Professional Forecasters (SPF), the Federal Reserve Board staff’s Tealbook projections, the neutral-rate estimates of Holston, Laubach, and Williams, and the term premiums of Adrian, Crump, and Moench. The neutral-rate path is the ex post Holston–Laubach–Williams (HLW) estimate, except for the rolling agent, which reads the real-time HLW vintages from 2016:Q1. Core PCE inflation is read from the price index’s vintage of 26 August 2026, and the estimator’s Kalman filter at a decision point reads the decision quarter’s own inflation, a quarter whose three months of the price index had been published at none of the 118 meetings. At the 103 decision points whose real-time vintages cover the four preceding quarters, four-quarter inflation in the vintage of 26 August 2026 differs from the value those vintages give by 0.28 pp root mean square and most, at 2009:Q2, by 0.89 pp. Section 8.1 reports how the baseline agent’s mean difference changes with that inflation withheld from the filter and with the preceding quarters’ inflation read from the vintage available at each decision point. The term premium is read from one release. A quarter is at the ELB where the lower end of the target range for the federal funds rate is zero at the decision point; there the realized federal funds rate is set to the model’s lower bound of 0.125 percent. At the decision point of 2020:Q1, the meeting of 29 January 2020, the realized federal funds rate is the midpoint of the target range of 1.50 to 1.75 percent in force at the meeting.

Optimal Policy, Rules, and Learners (Section 3)

The optimal policy of Adrian, Laxton, and Obstfeld’s volume minimizes a quadratic loss in inflation, the output gap, and changes in the policy rate; equation (1) is that loss, its sum taken from one quarter ahead, the lower limit the volume’s equation (12.3) prints.

π⁴ is four-quarter core PCE inflation, the superscript marking the four-quarter average; π* is the inflation target of 2 percent; y is the output gap in percent of potential output; i_t+i is the policy rate i quarters ahead; the weights α and β are one and γ one half. The volume is Tobias Adrian, Douglas Laxton, and Maurice Obstfeld, eds., Advancing the Frontiers of Monetary Policy (International Monetary Fund, 2018), chapter 12. The volume’s optimal policy, which the volume labels DM1 (its dual-mandate loss at these weights), is a path, re-optimized each quarter; the paper’s agent is a rule, of the inflation-forecast-based (IFB) form.

The volume’s experiment on the learned rule (Section 3.1)

From the volume’s 2014:Q4 state, in the paper’s reproduction of the volume’s model with no shocks, the volume’s IFB rule lifts off in 2015:Q1, the DM1 path holds the rate at the ELB through 2015:Q4, its rate in 2016:Q1 standing 0.5 bp above the ELB, and the baseline agent holds the rate at the ELB for five quarters and lifts off in 2016:Q2.

Each loss is the loss of equation (1) over the sixteen quarters 2015:Q1 through 2018:Q4 plus the loss along the policy’s perfect-foresight path from the state of 2018:Q4; the reproduction carries the IS block in the paper’s calibration. The baseline agent’s path overshoots: four-quarter inflation ends at 2.22 percent, above the target, and the output gap is positive from 2016:Q4 on. The baseline agent’s loss lies between the DM1 path’s and the volume’s IFB rule’s. The learned rule of the section’s title is the baseline agent.

The World Model (Section 4)

The world model, the paper’s estimated model of the economy, is a reduced form of the volume’s semi-structural economy with an IS equation, a hybrid Phillips curve, and an IFB policy equation with an ELB, estimated by maximum likelihood; the baseline estimate is the deepest optimum the multi-start search reaches with the inflation response normalized.

Along the inflation response of the estimated policy equation, offsetting moves of its intercept and of its shock’s standard deviation keep much of the fit, so the likelihood determines the equation’s scale only weakly, and the paper normalizes the response at the only local minimum of the profile of the objective at which no parameter lies at a bound; at an inflation response of 8, where the objective is lowest along the profile and the log-likelihood is higher by about 8 than at the normalized value, the persistence of the policy shock lies at its bound. The baseline estimate is reached from two of four non-random starting values, optima of the likelihood found outside the random search, and from none of 40 random starting values. Table 2 has five lines: the output gap one quarter ahead, at 79 decision points through 2019:Q3, and inflation one quarter ahead, at 84, each from the IS and Phillips equations alone and with the residual components added; and four-quarter inflation four quarters ahead, at 83 decision points. The staff’s RMSE is the lower on every line. A loss differential is the model’s squared forecast error less the staff’s. On none of the four one-quarter-ahead lines does the ratio of the mean loss differential to its standard error exceed the critical value of 2.36; on the four-quarter line, read over its 75 consecutive quarters through 2018:Q3, the ratio is 2.71, which exceeds that value, and over the eight quarters after the line’s missing quarter, for which the estimator’s rule gives one term, it is 0.82 against 12.71. The comparison does not separate a difference in information from a difference in model.

Model-Consistent and Alternative Expectations (Section 5)

Each of three families of departures from model-consistent expectations, an anchor, cognitive discounting, and an anchor learned from experience, is estimated from the baseline and the alternative estimates as starting values, and each family’s deeper optimum lies at the model-consistent value, a parameter value of one (Table 3).

The learned-anchor family makes the anchor’s level a constant-gain mean of past realized inflation, the learned level. The long-run survey expectation, the Survey of Professional Forecasters’ ten-year expectation of consumer price index (CPI) inflation expressed in core PCE units (the survey less the sample’s mean wedge between headline CPI and core PCE inflation), is taken as an observation of the learned level, and the gain so estimated has a half-life of about 23 quarters. The survey’s measurement error about the learned level has an estimated standard deviation of 0.30 pp, larger than the survey’s own standard deviation of 0.21 pp over the sample; the survey lies more than 0.21 pp below the learned level in 43 of the 122 quarters, all of them before 2010:Q1 or from 2023:Q1 on. The survey varies less than the learned level, whose standard deviation is 0.32 pp, although from 2020:Q3 through 2022:Q4 the survey rises by 0.92 pp, against 0.68 pp for the learned level; the expectations that enter the IS and Phillips equations place no weight on the learned level and remain model-consistent. The learned-anchor family is estimated on 958 observation rows, the 836 of Section 2 and the 122 of the ten-year survey. In the paper’s Monte Carlo study, on five simulated samples with an ELB episode and the staff’s output-gap observations present in every quarter, the estimator of the anchored family returns anchoring weights of 0.769 to 0.809 against a true value of 0.80; on simulated economies with an ELB episode, the estimator of cognitive discounting returns discounts of 0.828 to 0.931 against a true value of 0.85, and that of the learned anchor returns gains of 0.0483 to 0.0534 against a true value of 0.05 (Appendix A).

Residual Components and a Null Distribution with an ELB Episode (Section 6)

Against a null distribution simulated with an ELB episode, the IS component’s leave-one-out (LOO) log-score excess exceeds all 25 simulated statistics of its null, and the Phillips component’s lies at the 48th percentile of its null, which is no evidence of signal.

Each component is a GP with a squared-exponential kernel on four features of the filtered state. An ELB episode makes the filtered shocks depend on the state even at the true parameters, so a null distribution simulated without one would count that dependence as signal. The paper states the evidence of signal in the IS residual as a limitation of the IS equation (Section 9); the baseline agent’s training environment does not include the components. Each component is capped where it enters a simulation, the cap bounding its signal standard deviation; the capped components, judged against simulated economies whose components are capped in the same way, give the same two results: the capped IS component’s excess, 0.092, exceeds every simulated statistic, the largest being 0.064, and the capped Phillips component’s lies at the 48th percentile. The null’s 25 simulated economies with an ELB episode differ from the sample in three respects: each has 96 quarters where the sample has 122; their shocks are filtered at the parameters that generated them, where the sample’s are filtered at the estimate; and their parameters are the volume’s calibration with an anchoring weight of 0.80, not the baseline estimate. They differ in a fourth respect, the length of the ELB episode, where the paper measures it. Of them, five are the Monte Carlo study’s samples of condition O16 of Appendix A (the anchored specification with an ELB episode and the sample’s pattern of missing observation rows) and have 7 to 17 quarters at the ELB, against the 28 quarters of the sample’s first episode, which falls in the sample’s first 96 quarters; the paper does not measure the number for the other 20.

The Policy Agent (Section 7)

The baseline agent sets the policy rate by the IFB rule of the paper’s equation (4), whose four coefficients, the smoothing $\rho$, the inflation response $c_{\pi }$, the gap response $c_y$, and the intercept $c_0$, are trained in the model at the baseline estimate on draws around the estimate and selected among eight training runs by the loss on a validation set of starting states, from which training episodes also start, and of draws that enter no training episode.

The intercept c₀ is −0.580 pp. A draw around the estimate is a draw of four coefficients of the IS and Phillips equations, each from an independent normal distribution centered on the coefficient’s estimate with its standard error; the draws are not draws from the sampling distribution of the estimator. The loss of this comparison is the loss of equation (1) summed over 24 quarters from the 2014:Q4 state with no shocks, in the model at each draw, and differs from the loss of Section 3.1. An agent trained with the residual components in its environment has a higher loss than the IFB rule’s on every one of those draws, by 3.72 on average, in the estimated model without the components (Section 7.2). With the components in the model, at the one state and estimate evaluated, the agent trained with them has the lower loss of the two agents, 13.296 against 17.755. The selected checkpoint, a checkpoint being the rule’s coefficients at one step of a training run, comes from a run that ranks eighth of eight on the test set, a second such set of starting states and draws, which the selection of the reported checkpoint does not use. The Spearman rank correlation of the checkpoints’ validation and test losses is −0.98; the paper reads the coefficients as one selection among rules of similar loss.

The baseline agent and the realized federal funds rate (Section 8.1)

Over the 118 decision points from 1997:Q1 through 2026:Q2, the baseline agent prescribes rates above the realized federal funds rate on average in every period of Table 4; in the two ELB episodes the difference cannot be negative, and over the first its mean is +1.3 bp.

• Realized federal funds rate
• Baseline agent
• Baseline agent’s band, its 10th through 90th percentiles across 20 draws around the estimate
• ELB episodes

Ticks on the horizontal axis mark the first quarter of the year. The shaded quarters are the two ELB episodes, in which the realized federal funds rate is set to the model’s lower bound of 0.125 percent. The band varies the four coefficients of a draw around the estimate and holds the filtered state; with 20 draws, each percentile rests on few observations. Over all quarters the mean difference is +1.483 pp for the Taylor (1993) rule, +1.603 pp for the balanced-approach rule, +0.223 pp for the first-difference rule, and +0.405 pp for the volume’s IFB rule, against the baseline agent’s +0.723 pp (Table 5); before the first ELB episode the agent’s mean difference, +0.659 pp, exceeds the Taylor (1993) and balanced-approach rules’ +0.130 pp and +0.285 pp. The Taylor (1993), balanced-approach, and first-difference rules are three of the rules presented in the Board of Governors’ Monetary Policy Report of February 2019, which writes them with the unemployment gap; the paper writes them in output-gap form. The first two take a neutral real rate of 2 percent, where the model takes the neutral-rate path of Section 2. The first-difference rule’s mean difference is below the agent’s in every period of Table 5 except the two ELB episodes. With the decision quarter’s inflation withheld from the filter at each decision point, the baseline agent’s mean difference changes by −0.3 bp over all quarters and by −6.5 bp from 2021:Q1. With the preceding quarters’ inflation read from the vintage available at each decision point as well, it changes by −5.6 bp over all quarters and by −16.3 bp from 2021:Q1. From 2021:Q1 the ratios of the two changes to their standard errors, 3.59 and 4.33, exceed the critical value of 3.18; over all quarters the ratios, 0.24 and 1.66, do not exceed the critical value of 2.26.

The agent frozen at the end of the first ELB episode (Section 8.2)

The frozen agent prescribes rates nearer the realized federal funds rate after 2015:Q4 than the baseline agent does, in mean and in root-mean-square difference.

• Realized federal funds rate
• Baseline agent
• Rolling agent
• Frozen agent, estimated and trained on data through 2015:Q4
• ELB episodes

Ticks on the horizontal axis mark the first quarter of the year, and the shaded quarters are the ELB episodes. The frozen agent’s rates begin at 2016:Q1. Over the 42 decision points from 2016:Q1 through 2026:Q2, the ratio of the frozen agent’s mean difference to its standard error, 1.07, is below the critical value of 2.78, and the baseline agent’s, 3.64, exceeds it. The chart draws the rolling agent of Section 8.3 beside the two. The frozen agent’s prescription less the baseline agent’s has a mean of −1.057 pp over the 42 decision points and a standard error of 0.449 pp, a ratio of 2.35 against 2.78, which it does not exceed; through 2020:Q4 the ratio is 1.28 against 4.30, which it does not exceed, and from 2021:Q1 it is 7.03 against 3.18, which it exceeds. A division is a quarter at which the 42 decision points are divided into an earlier and a later period. At the state of the paper’s Section 2, which sets the decision quarter’s IS and Phillips shocks at zero, the ratio does not exceed its critical value over the earlier period and exceeds it over the later at each of the 24 divisions from 2019:Q3 through 2025:Q2, and at none before, the later period resting on one term of the estimator from 2023:Q3; a change cannot be dated at 2020:Q4 from these ratios, and the paper reports no test of the change between the two periods’ means. The frozen estimate’s Phillips slope is 0.0172 (standard error 0.0037) against the baseline estimate’s 0.0596 (standard error 0.0166). The two estimates also differ in the weight on expected inflation (0.693 against 0.946), in the IS weights on the lagged and the expected gap (0.883 and 0.087 against 0.771 and 0.199), in the gap response of the estimated policy equation (1.31 against 0.84), in its smoothing and intercept, in the persistence and standard deviation of its shock (Table 6), and in the residual components and measurement-error moments. Exchanged between the two estimates, both in the rule’s training and in the model with which its rate is computed, the Phillips slope does not account for the difference between the two agents’ prescriptions. Over the 42 decision points, a rule trained on the baseline estimate with the frozen estimate’s slope has a mean difference of +1.421 pp, and a rule trained on the frozen estimate with the baseline estimate’s slope one of +0.117 pp, so that neither exchange moves the mean difference toward the other agent’s. The first of the two rules stands at a checkpoint within the final 200 steps of its training runs of 4,000 steps (the procedure of Section 7.3 extends runs of 2,000 steps once to that length when their selected checkpoint falls in their final 200), and whether longer runs would select a rule with another mean difference is not established. From 2021:Q1 the difference between the two agents’ prescriptions is accounted for by the output gap each estimate’s filter infers. No observation of the gap enters either estimate’s filter after 2020:Q4, and from 2021:Q3 the baseline estimate’s filtered gap exceeds the frozen estimate’s by 2.76 to 4.60 percentage points of potential output. Averaged over the six orders in which the rule, the model’s parameters, and the filtered gap can be exchanged between the two agents, replacing the baseline estimate’s filtered gap by the frozen estimate’s lowers the mean difference over 2021:Q1 through 2026:Q2 by 2.572 pp, against 1.840 pp between the two agents’ mean differences over those quarters, while replacing the rule raises it by 0.206 pp and replacing the parameters by 0.526 pp. The exchange is an accounting of that distance and not a model that either estimate describes, and the paper does not separate which of the two estimates’ differences, in the parameters (the slope among them) and in the error moments and the survey’s wedge that their filters read, produce the divergence of the two filtered gaps. A search on the frozen estimate’s sample, the data through 2015:Q4, from the baseline and the alternative estimates as starting values reaches no deeper optimum. The root-mean-square difference from the realized federal funds rate over the 42 decision points is 0.547 pp for the frozen agent and 1.501 pp for the baseline agent (Table 7).

The agent re-estimated and retrained each quarter (Section 8.3)

The rolling agent prescribes rates above the realized federal funds rate on average from 2016:Q1, and the ratio of its mean difference to its standard error exceeds its critical value over 2016:Q1 through 2026:Q2, over 2022:Q2 through 2026:Q2, and over 2021:Q1 through 2026:Q2.

The chart under Section 8.2 draws the rolling agent’s rates beside the other two agents’. Of the 40 decision points from 2006:Q1 through 2015:Q4, 28 are at the ELB, where the difference cannot be negative, and the paper reports no standard error for that span. Over the 42 decision points from 2016:Q1 through 2026:Q2, the rolling agent’s mean difference has a standard error of 0.474 pp, by the equal-weighted cosine estimator of the long-run variance, and its ratio to that standard error, 2.88, exceeds the critical value of 2.78. A paired difference is one agent’s prescription less another’s at one decision point. The rolling agent’s mean difference lies 0.097 pp above the baseline agent’s, a paired difference with a standard error of 0.144 pp and a ratio of 0.67, which does not exceed that critical value. Of the rolling agent’s mean differences over 2016:Q1 through 2020:Q1 and over 2021:Q1 through 2023:Q4, neither period’s ratio, 1.58 and 2.70, exceeds the critical value of 4.30, and the paper reports no test of the change between those two means. Over 2022:Q2 through 2026:Q2 and over 2021:Q1 through 2026:Q2 its ratios are 8.12 against 4.30 and 4.52 against 3.18.

Counterfactual outcomes (Section 8.4)

With the structural residuals recovered from the realized data fed back as shocks, four-quarter core PCE inflation over 2021:Q1 through 2023:Q4 averages below its realized level under the baseline agent, in the model at the baseline estimate, and above it under the frozen agent, in the model at the frozen estimate.

The paths simulated under the two agents’ rules are outcomes in the estimated models, not outcomes the FOMC could have chosen. The persistent disturbance of the estimated policy equation, the FOMC’s deviation from its estimated reaction function, is not fed back; over 2021:Q1 through 2023:Q4 it averages −5.14 pp at the baseline estimate, the realized rate standing below the estimated policy equation’s prescription. Three conditions bound the comparison: the residuals are treated as invariant to the rule, the condition the Lucas critique questions; expectations are model-consistent under the simulated rule; and the output gap on the realized path is each model’s filtered gap, since the true gap is latent. That filtered gap averages 2.72 percent of potential output over 2021:Q1 through 2023:Q4 at the baseline estimate and −0.87 percent at the frozen estimate. The rates on a simulated path differ from the prescriptions at the realized states: over the same quarters the baseline agent’s rate averages 2.78 percent on its simulated path against the realized 2.21 percent, while its prescriptions at the realized states have a mean difference of +1.898 pp from the realized rate over those quarters (Tables 4 and 9). Against the path simulated under the estimated policy equation, the baseline agent’s loss is the lower from 2016:Q1, 38.8 against 332.7, and the higher before, 198.4 against 186.2; the frozen agent’s loss, against the path simulated under the frozen estimate’s policy equation, is 158.4 against 158.4 from 2016:Q1, the two alike at Table 9’s precision, and 172.3 against 222.6 before. These comparisons are made under each model’s filtered gap, and the two agents’ losses are not ranked across the two models. In the model with the weight on expected inflation held at the volume’s 0.70 (Table 9), and under the Congressional Budget Office’s measure of the output gap (Table C.3), the baseline agent’s loss is lower in both periods than that of the path simulated under the corresponding estimated policy equation; under that measure the frozen agent’s loss is higher in both than that of the path simulated under the frozen estimate’s policy equation.

Sensitivity and Limitations (Section 9)

Under four variants of the loss weights, each mean difference from the realized federal funds rate lies within 0.19 pp of the baseline agent’s +0.723 pp, and across the five trainings at the baseline weights the mean differences span 3.2 bp.

The paper states these limitations:

1. the IS residual carries evidence of signal beyond the null distribution with an ELB episode, and the baseline agent’s training environment does not include the residual components;
2. the baseline estimate’s Phillips equation is nearly forward-looking, with ${\lambda }_1$ at 0.946; the counterfactual reduction in inflation is nearly the same with ${\lambda }_1$ held at the volume’s 0.70, where the baseline rule lowers 2021:Q1 through 2023:Q4 inflation by 1.93 pp against the path simulated under that model’s estimated policy equation, beside 1.95 pp at the baseline estimate; the reduction is measured at these two values of ${\lambda }_1$ alone, and the counterfactual does not feed back the persistent policy disturbance;
3. the loss has the target-deviation form of the volume’s chapter 12, without the history dependence of the framework the FOMC adopted in 2020, an aim of inflation that averages 2 percent over time, which the updated Statement on Longer-Run Goals and Monetary Policy Strategy that the FOMC approved on 22 August 2025, inside the sample, omits; the agent’s prescriptions over 2021 through 2023 are therefore a statement about the loss as much as about the agent;
4. the Monte Carlo study’s ELB episodes last 7 to 17 quarters under the anchored specification and 5 to 12 under cognitive discounting, shorter than the sample’s first episode of 28 quarters; under the learned anchor they last 14 to 43 quarters; the study’s estimates of each specification’s own parameter are for episodes of those lengths, four of the five estimates of the cognitive discount lie above its true value, and the study does not show the recovery of the sum of the two IS weights or of the estimated policy equation’s inflation and gap responses;
5. the agent’s rate at each state is computed under expectations consistent with the agent’s rule, and the estimated policy equation is the FOMC’s reaction function as estimated, so the comparison holds the private sector’s way of forming expectations, model-consistent under each policy, fixed across the two policies, the condition of Section 8.4;
6. the scale of the estimated policy equation, which the likelihood determines only weakly, is fixed by normalizing its inflation response to 3.0, not estimated; along the profile the log-likelihood is higher by about 8 at an inflation response of 8 than at the normalized value, and the paper does not report how the agents’ prescriptions change with the normalized value;
7. the neutral-rate path is the ex post estimate except for the rolling agent from 2016:Q1, and the final rolling window, on the same data as the baseline estimate except the real-time neutral-rate path, fits better than the baseline estimate by 9.15 log-likelihood units;
8. the selected checkpoint is the best of eight training runs on a validation set of 32 draws and eight starting states; across the eleven searches of the paper’s Section 7.3, each a selection among eight training runs, the Spearman correlation of the runs’ validation and test rankings is positive in none; in 32 of the 82 rolling windows of the paper’s Section 8.3, each also a selection among eight training runs, the selected run ranks in the better half on a fresh set of starting states and draws, against the 41 a random choice among the eight gives on average; and the repetitions of the training (Table 11) measure how the prescriptions vary with the training’s random-number streams;
9. no observation of the output gap enters the model after 2020:Q4, and the prescriptions of every agent after 2020, and the difference between the baseline and the frozen agent’s prescriptions there, depend on the gap the model infers, as do the published rules’ readings and the agent’s band after 2020 and the rolling agent’s larger mean difference after 2020; the paper does not estimate the model with an observation of the gap in those quarters;
10. the standard deviations of the IS and Phillips shocks are assumed, the paper does not vary them on the sample, and the estimates, the likelihood comparisons, and the standard errors are conditional on the two values. The moments of the staff’s errors that the filter reads, those of its estimates and projections of the output gap and of its projection of inflation, are also inputs fixed before the estimation: at the baseline estimate, in the rolling windows from 2021:Q1 (from 2022:Q2 for the inflation projection’s), and in the simulated economies of the paper’s Sections 4.2 and 6, these moments include quarters whose errors are computed from the staff’s estimates and projections of 2020:Q4, which the data carry to later decision points because the staff’s projections are released with a five-year lag; the paper does not estimate the model, or simulate those economies, with the moments of the other quarters alone;
11. core PCE inflation is read from one vintage of the price index, the filter reads the decision quarter’s own inflation, and the term premium is read from one release, so the states, the estimates, and the frozen and rolling agents carry later information in those inputs as well as in the neutral-rate path (the rolling agent’s only before 2016:Q1). From 2021:Q1 the changes in the baseline agent’s mean difference when the decision quarter’s inflation is withheld and when the preceding quarters’ inflation is read from the vintage available at each decision point as well, −6.5 and −16.3 bp, have ratios to their standard errors that exceed their critical values, and over all quarters the changes, −0.3 and −5.6 bp, do not, and beyond what the paper’s Sections 8.1 and 8.2 and its Table C.1 report under the two treatments, nothing is computed under either;
12. the standard errors of the mean differences come from one estimator of the long-run variance, whose rule for the number of terms is derived for series less persistent than the differences over all quarters, and a ratio over 12 to 22 quarters rests on two or three terms of the estimator, and one over fewer than 12 quarters on one; ratios stand near their critical values, among them, on four terms over 42 quarters, the rolling agent’s from 2016:Q1, 2.88 against 2.78, and three ratios of the frozen agent’s paired difference with the baseline agent at the divisions of Section 8.2; and
13. the agent is trained inside a model whose forecasts have a higher root-mean-square error than the Federal Reserve Board staff’s on every line of Table 2; the comparison does not separate a difference in information from a difference in model, and the paper does not measure how the agent’s prescriptions depend on the model’s forecast accuracy.

The variants halve and raise by half the output-gap weight β (to 0.5 and 1.5) and the rate-change weight γ (to 0.25 and 0.75), the inflation weight α held at one; for each variant the ratio of the mean difference to its standard error exceeds its critical value, the smallest ratio being 2.98. Table 10 reports the variants and Table 11 the repetitions.

Conclusion (Section 10)

In the paper, optimal monetary policy when the economy must be learned is a rule whose optimality is conditional on the four learners’ estimates: the expectations specification, the estimated model, the residual components, and the selected checkpoint.

That the baseline agent’s prescribed rates lie above the realized federal funds rate on average rests on the expectations specification at which the likelihood is highest, model-consistent expectations (the paper’s Section 5), and on the limitations of the paper’s Section 9, among them:

• a loss without the history dependence of the framework the FOMC adopted in 2020; and
• a scale of the estimated policy equation that a normalization fixes, the likelihood determining it only weakly.

Those rates are a property of this conditional rule and not an evaluation of monetary policy as the FOMC conducted it. Over the 42 decision points from 2016:Q1 through 2026:Q2, the frozen agent’s mean difference is +0.212 pp against the baseline agent’s +1.269 pp. The ratio of the mean of the frozen agent’s prescription less the baseline agent’s to its standard error does not exceed its critical value over those decision points or through 2020:Q4 and exceeds it from 2021:Q1; the same pair of results holds wherever those decision points are divided from 2019:Q3 through 2025:Q2, so that a change cannot be dated at 2020:Q4, and the paper reports no test of the change. From 2021:Q1 the difference between the two agents’ prescriptions is accounted for, in the paper’s exchange of rule, parameters, and filtered gap, by the output gaps the two estimates infer in quarters in which no observation of the gap enters either filter.