## 9.   The central bank should front-load tightening and then ease.

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**Canonical URL:** [9.   The central bank should front-load tightening and then ease.](https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023019-print-pdf.pdf)

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---

### Key results and mechanisms
- Three headline findings:
  - In a context of positive output gap and rising inflation, the central bank’s future interest rate path is steeper under adaptive learning (AL) expectations than under rational expectations (RE) because RE provides an additional anchor and self-enforcing tendency, whereas AL expectations are more inertial and harder to control once they start drifting away from the target.
  - Empirical estimation shows that inflation expectations move by more in Brazil than the USA when shocks hit the economy; the adaptive nature of inflation expectations likely propelled the Brazilian central bank to act earlier and more aggressively to anchor expectations.
  - Including a separate labor market in the model allows anchoring of inflation even with adaptive expectations and a positive output gap: a negative real wage gap can co-exist with a positive output gap, enabling inflation to be anchored without requiring a negative output gap.
- Expectation-formation mechanism:
  - Agents form expectations using a Perceived Law of Motion (PLM) based on a simple forecasting model (best-performing: univariate AR(2) with a constant and two lags).
  - Beliefs are updated each period via a Kalman filter. The AL Actual Law of Motion (ALM) is time-dependent: matrices 훼t, Tt and Rt depend on the evolving belief vector Bt.

### Data, estimation, and model fit
- Data sample:
  - quarterly macroeconomic data from 2000Q1 to 2019Q4 for Brazil and the USA.
- Variables used:
  - output gap, real wage gap, annualized quarterly price inflation deviation from target, and the policy rate.
  - Core PCE and the core IPCA are used for the US and Brazil, respectively.
- Construction and filters:
  - Composition-constant real wage: constructed for the USA using Howard, Rich, and Tracy(2022) method and analogously for Brazil.
  - Filter selection: linear filter chosen (Sun and Tsang(2019)) because the model has better out-of-sample forecast performance for wages and prices.
  - Neutral rate treatment: for Brazil, the policy rate deviation ˆit is calculated as deviation from its linear trend; an interest rate measurement equation is included for Brazil.
- Model fit (in-sample log marginal likelihood, Brazil):
  - Linear Filter: RE -173.9, AL -98.7
  - HP Filter: RE -169.04, AL -93.64
- Out-of-sample RMSE (selected highlights, Brazil; linear filter):
  - 1-quarter ahead RMSE:
    - Real Wage Gap RE 0.32 AL 0.29
    - Output Gap RE 0.78 AL 0.32
    - Policy Rate RE 0.06 AL 0.31
    - Inflation RE 0.33 AL 0.31
  - 4-quarter ahead RMSE:
    - Real Wage Gap RE 0.61 AL 0.72
    - Output Gap RE 2.47 AL 3.73
    - Policy Rate RE 0.24 AL 0.22
    - Inflation RE 0.39 AL 0.36
  - 8-quarter ahead RMSE:
    - Real Wage Gap RE 0.50 AL 0.52
    - Output Gap RE 2.60 AL 4.35
    - Policy Rate RE 0.19 AL 0.48
    - Inflation RE 0.22 AL 0.25
- Summary:
  - The AL model outperforms the RE model in-sample and often in short-term out-of-sample forecasts, in particular delivering the biggest gain in the short-term forecast of inflation under AL with a linear filter.
- Selected posterior estimates (posterior means, Table 2 examples):
  - ρy: RE 0.419, AL 0.533
  - β: RE 0.266, AL 0.517
  - κ: RE 0.084, AL 0.169
  - σϵx: RE 1.570, AL 0.706
  - σϵπ: RE 0.494, AL 0.317
  - σεπw: RE 0.851, AL 0.628

### Historical decomposition and cross-country differences
- Historical shocks (Brazil):
  - Large cost-push shocks in 2021 peaking in 2021Q4; offset by negative real wage shocks more recently.
  - Monetary policy: Brazil had diminishing negative contributions from interest rate shocks starting in 2021Q2 that became positive in 2021Q4.
- Historical shocks (USA):
  - Large real wage shocks explain about a third of recent inflation.
  - Monetary policy: the USA experienced increasing negative contributions from interest rate shocks in 2021.
- Expectations dynamics:
  - AL coefficients show that inflation and wage expectations in Brazil depend more on past outcomes than in the US (sum of AR(2) lag coefficients larger in Brazil).
  - Mean expected inflation was zero and stable before the pandemic in both countries; pandemic-era disruptions: mean expected inflation surged in Brazil in 2021Q1 and in the US by 2021Q3, with a larger surge in Brazil.

### Scenario analysis: cost-push shock and RE vs AL outcomes
- Scenario assumptions:
  - Both scenarios assume Brazil faces an unexpected cost-push shock that takes actual inflation to observed 2022Q2 levels with a half-life of 6.5 quarters.
  - Output shock dynamics unwind according to the estimated AR(1) process in equation (4).
- Scenario outcomes:
  - Under RE expectations:
    - Inflation returns to target by early 2024. Inflation expectations peak in 2022Q2 but are lower than realized inflation, aiding the decisive fall in inflation and interest rates over 2024.
  - Under estimated AL expectations:
    - Inflation remains well above Brazil’s central bank target for the next three years; inflation remains at an annual rate of 4.8 percent by 2024Q4 despite tighter monetary policy and a negative output gap from 2024Q1.
    - The persistence is driven by sharp responses in inflation expectations: households start to believe future inflation will run much higher than the target, keeping inflation above target despite negative output gap.
- Mechanism insight:
  - A price Phillips curve that includes only the output gap (no labor sector) would imply that lowering inflation under AL requires a negative output gap. With an explicit labor market, a negative real wage gap can offset cost-push shocks and help anchor inflation even when the output gap is positive.

### Optimal monetary policy implications and recommendations
- Welfare-minimizing policy problem (equation preserved):
  - Central bank chooses ˆit to minimize Et Σβt [0.75(ˆit − ˆit−1) + ˆy2t + ˆπ2t], with equal weights on output gap and inflation deviations and a role for interest rate smoothing.
  - Simulations assume the central bank knows current shocks, all future shocks, and how policy actions impact expectations (expectations still follow AL).
- Optimal policy prescription under AL (derived implications):
  - When inflationary shocks are present and the output gap is positive, it is optimal for monetary policy to:
    - Respond sooner,
    - Respond more strongly (front-load tightening),
    - Then ease subsequently.
  - Rationale: aggressive early action reduces the risk that high inflation becomes entrenched via adaptive expectations; these actions influence inflation through three channels:
    - Demand channel: tighter policy lowers output gap and inflation.
    - Expectations channel 1: lowering current inflation reduces next-period inflation expectations through the AR(2) formation.
    - Expectations channel 2: by producing outcomes below agents’ forecasts, the central bank can alter households’ learning (the AR(2) coefficients), reducing the backward-looking persistence in expectations.
- Policy tradeoffs:
  - More aggressive policy is warranted in economies where inflation expectations have a larger backward-looking component (e.g., emerging markets like Brazil), consistent with empirical results that advanced economies have better-anchored long-term inflation expectations.
  - When inflation is well-anchored, monetary policy should not respond as strongly to inflation fluctuations.

### The optimal (policy path and interpretation)
- Optimal policy prescription (quantified):
  - The optimal path has policy tighter by 35 basis points in the 2022Q2 but then looser over the next two years and a half.
  - With that policy, the output gap does not fall as much and inflation is a bit higher than in the estimated monetary policy reaction function.
  - It takes time for the deviation in monetary policy to influence inflation and the difference gets higher over time.
- Model, expectations, and mechanism:
  - The paper uses a New Keynesian model with wage and price Phillips curves and a backward-looking AL expectation formation process in which households learn from previous forecasting mistakes.
  - The AL mechanism creates a channel through which central banks can affect inflation by influencing households’ learning process.
  - A DSGE model with AL expectations outperforms the model with RE in in-sample and out-of-sample forecasting performance.
- Role of labor market and real wage gap:
  - Including labor market developments is important when discussing expectation formation effects.
  - In a model with labor markets, a negative real wage gap can act as an anchor to inflation even with fully adaptive expectations.
- Explaining central bank behavior and simulations:
  - Results rationalize why Emerging Market central banks tightened monetary policy earlier than Advanced Economies in response to adverse inflation news: if inflation expectations are likely to drift, aggressive monetary policy reactions are warranted.
  - The learning model captures a recent example in Brazil and rationalizes an early monetary policy response by the central bank.
- Key takeaways for policy:
  - Front-loading tightening (e.g., tighter by 35 basis points in 2022Q2) followed by easing can limit the decline in the output gap while accepting a bit higher inflation relative to the estimated reaction function.
  - Central banks should consider how policy actions influence adaptive learning and expectations formation, not just contemporaneous macroeconomic trade-offs.
  - Aggressive and timely policy responses can be warranted when expectations risk becoming unanchored, especially in Emerging Markets.

*Source: wpiea2023019-print-pdf — “9.   The central bank should front-load tightening and then ease.” (PDF chapter/section).*

### References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  26

### wpiea2023019-print-pdf - References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  26

### References
- References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  26

### Tables
- 1.   Prior Distribution of the Estimated Parameters. . . . . . . . . . . . . . . .  14
- 2.   Posterior Estimates for RE and AL Models. . . . . . . . . . . . . . . . . .  15
- 3.   In-Sample Forecast Performance for RE and AL Models. . . . . . . . . . . .  15
- 4.   Out-of-Sample Forecast Performance for RE and AL Models. . . . . . . . .  16

### Figures
- 1.   Brazil: Low wage workers suffered larger employment losses. . . . . . . .  13
- 2.   Real composition-constant wages did not increase as much. . . . . . . .  14
- 3.   Brazil: Shock decomposition. . . . . . . . . . . . . . . . . . . . . . . . . .  18
- 4.   USA: Shock decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . .  19
- 5.   Inflation and wages expectations respond to past values by more in Brazil. . .  20
- 6.   Impulse response functions for specific shocks. . . . . . . . . . . . . . . . .  20
- 7.   Impulse response functions for specific shocks. . . . . . . . . . . . . . . . .  21
- 8.   Inflation is stickier when expectations are adaptive learning. . . . . . . .  22

*Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023019-print-pdf.pdf*

### 9.   The central bank should front-load tightening and then ease. . . . . . . . . . . . .  24

### 9.   The central bank should front-load tightening and then ease.

### Key results and mechanisms
- Three headline findings:
  - In a context of positive output gap and rising inflation, the central bank’s future interest rate path is steeper under adaptive learning (AL) expectations than under rational expectations (RE) because RE provides an additional anchor and self-enforcing tendency, whereas AL expectations are more inertial and harder to control once they start drifting away from the target.
  - Empirical estimation shows that inflation expectations move by more in Brazil than the USA when shocks hit the economy; the adaptive nature of inflation expectations likely propelled the Brazilian central bank to act earlier and more aggressively to anchor expectations.
  - Including a separate labor market in the model allows anchoring of inflation even with adaptive expectations and a positive output gap: a negative real wage gap can co-exist with a positive output gap, enabling inflation to be anchored without requiring a negative output gap.

- Expectation-formation mechanism:
  - Agents form expectations using a Perceived Law of Motion (PLM) based on a simple forecasting model (best-performing: univariate AR(2) with a constant and two lags).
  - Beliefs are updated each period via a Kalman filter. The AL Actual Law of Motion (ALM) is time-dependent: matrices 훼t, Tt and Rt depend on the evolving belief vector Bt.

### Data, estimation, and model fit
- Data sample: quarterly macroeconomic data from 2000Q1 to 2019Q4 for Brazil and the USA.
- Variables used: output gap, real wage gap, annualized quarterly price inflation deviation from target, and the policy rate. Core PCE and the core IPCA are used for the US and Brazil, respectively.
- Composition-constant real wage: constructed for the USA using Howard, Rich, and Tracy(2022) method and analogously for Brazil to mitigate workforce composition effects.
- Filter selection: linear filter chosen (Sun and Tsang(2019)) because the model has better out-of-sample forecast performance for wages and prices.
- Neutral rate treatment: for Brazil, the policy rate deviation ˆit is calculated as deviation from its linear trend; an interest rate measurement equation is included for Brazil.

- Model fit and forecast performance:
  - In-sample log marginal likelihood (Brazil):
    - Linear Filter: RE -173.9, AL -98.7
    - HP Filter: RE -169.04, AL -93.64
  - Out-of-sample RMSE (selected highlights, Brazil; linear filter):
    - 1-quarter ahead RMSE: Real Wage Gap RE 0.32 AL 0.29; Output Gap RE 0.78 AL 0.32; Policy Rate RE 0.06 AL 0.31; Inflation RE 0.33 AL 0.31
    - 4-quarter ahead RMSE: Real Wage Gap RE 0.61 AL 0.72; Output Gap RE 2.47 AL 3.73; Policy Rate RE 0.24 AL 0.22; Inflation RE 0.39 AL 0.36
    - 8-quarter ahead RMSE: Real Wage Gap RE 0.50 AL 0.52; Output Gap RE 2.60 AL 4.35; Policy Rate RE 0.19 AL 0.48; Inflation RE 0.22 AL 0.25
  - Summary: The AL model outperforms the RE model in-sample and often in short-term out-of-sample forecasts, in particular delivering the biggest gain in the short-term forecast of inflation under AL with a linear filter.

- Selected posterior estimates (comparison RE vs AL; real model table headers preserved):
  - Examples from Table 2 (posterior means):
    - ρy: RE 0.419, AL 0.533
    - β: RE 0.266, AL 0.517
    - κ: RE 0.084, AL 0.169
    - σϵx: RE 1.570, AL 0.706
    - σϵπ: RE 0.494, AL 0.317
    - σεπw: RE 0.851, AL 0.628
  - (The table includes standard deviations for each estimate; the overlay preserves the numeric values exactly as reported.)

### Historical decomposition and cross-country differences
- Historical shocks (Brazil):
  - Large cost-push shocks in 2021 peaking in 2021Q4; offset by negative real wage shocks more recently.
  - Monetary policy: Brazil had diminishing negative contributions from interest rate shocks starting in 2021Q2 that became positive in 2021Q4.
- Historical shocks (USA):
  - Large real wage shocks explain about a third of recent inflation.
  - Monetary policy: the USA experienced increasing negative contributions from interest rate shocks in 2021.
- Expectations dynamics:
  - AL coefficients show that inflation and wage expectations in Brazil depend more on past outcomes than in the US (sum of AR(2) lag coefficients larger in Brazil).
  - Mean expected inflation was zero and stable before the pandemic in both countries; pandemic-era disruptions: mean expected inflation surged in Brazil in 2021Q1 and in the US by 2021Q3, with a larger surge in Brazil.

### Scenario analysis: cost-push shock and consequences under RE vs AL
- Scenario assumptions:
  - Both scenarios assume Brazil faces an unexpected cost-push shock that takes actual inflation to observed 2022Q2 levels with a half-life of 6.5 quarters.
  - Output shock dynamics unwind according to the estimated AR(1) process in equation (4).
- Scenario outcomes:
  - Under RE expectations:
    - Inflation returns to target by early 2024 (blue line in figure 8). Inflation expectations peak in 2022Q2 but are lower than realized inflation, aiding the decisive fall in inflation and interest rates over 2024.
  - Under estimated AL expectations:
    - Inflation remains well above Brazil’s central bank target for the next three years; inflation remains at an annual rate of 4.8 percent by 2024Q4 despite tighter monetary policy and a negative output gap from 2024Q1.
    - The persistence is driven by sharp responses in inflation expectations: households start to believe future inflation will run much higher than the target, keeping inflation above target despite negative output gap.
- Mechanism insight:
  - A price Phillips curve that includes only the output gap (no labor sector) would imply that lowering inflation under AL requires a negative output gap. With an explicit labor market, a negative real wage gap can offset cost-push shocks and help anchor inflation even when the output gap is positive.

### Optimal monetary policy implications and recommendations
- Welfare-minimizing policy problem (equation preserved):
  - Central bank chooses ˆit to minimize Et Σβt [0.75(ˆit − ˆit−1) + ˆy2t + ˆπ2t], with equal weights on output gap and inflation deviations and a role for interest rate smoothing.
  - Simulations assume the central bank knows current shocks, all future shocks, and how policy actions impact expectations (expectations still follow AL).
- Optimal policy prescription under AL (derived implications):
  - When inflationary shocks are present and the output gap is positive, it is optimal for monetary policy to:
    - Respond sooner,
    - Respond more strongly (front-load tightening),
    - Then ease subsequently.
  - Rationale: aggressive early action reduces the risk that high inflation becomes entrenched via adaptive expectations; these actions influence inflation through three channels:
    - Demand channel: tighter policy lowers output gap and inflation.
    - Expectations channel 1: lowering current inflation reduces next-period inflation expectations through the AR(2) formation.
    - Expectations channel 2: by producing outcomes below agents’ forecasts, the central bank can alter households’ learning (the AR(2) coefficients), reducing the backward-looking persistence in expectations.
- Policy tradeoffs:
  - More aggressive policy is warranted in economies where inflation expectations have a larger backward-looking component (e.g., emerging markets like Brazil), consistent with empirical results that advanced economies have better-anchored long-term inflation expectations.
  - When inflation is well-anchored, monetary policy should not respond as strongly to inflation fluctuations.

*Source: wpiea2023019-print-pdf — “9.   The central bank should front-load tightening and then ease.” (PDF chapter/section).*

### 9. The optimal

### 9. The optimal

### Optimal policy prescription
- The optimal policy prescribes front-loading the interest rate tightening and then easing compared to the estimated monetary policy reaction function.
- The optimal path has policy tighter by 35 basis points in the 2022Q2 but then looser over the next two years and a half.
- With that policy, the output gap does not fall as much and inflation is a bit higher than in the estimated monetary policy reaction function.
- It takes time for the deviation in monetary policy to influence inflation and the difference gets higher over time.
- The more aggressive tightening of the estimated monetary policy reaction function is also a result of not incorporating the full path of inflation shocks in the interest rate decisions.
- The blue line in figure 9 shows the optimal interest rate path; the secondary axis shows the difference between the optimal path and the estimated reaction function.

### Model, expectations, and mechanism
- The paper introduces a standard New Keynesian model that includes wage and price Phillip’s curves and departs from rational expectations by including a "limited rationality" backward-looking expectation formation process in which households learn from previous forecasting mistakes.
- This adaptive learning (AL) mechanism creates a channel through which central banks can affect inflation by influencing households’ learning process.
- A standard DSGE model with AL expectations outperforms the model with rational expectations (RE) in terms of in-sample and out-of-sample forecasting performance.

### Role of labor market and real wage gap
- Including labor market developments is important when discussing expectation formation effects.
- A price Phillip’s curve that includes only the output gap (and not a labor sector) would predict that under AL expectations the only way to lower inflation is with a negative output gap.
- In a model with labor markets, a negative real wage gap can act as an anchor to inflation even with fully adaptive expectations.

### Explaining central bank behavior and simulations
- The results rationalize why Emerging Market central banks tightened monetary policy earlier than Advanced Economies in response to adverse inflation news: if inflation expectations are likely to drift, aggressive monetary policy reactions are warranted.
- The learning model captures a recent example in Brazil and rationalizes an early monetary policy response by the central bank.
- Simulations representing the current situation with a positive output gap and rising inflation show:
  - It is optimal for monetary policy to respond sooner and more strongly under AL expectations than under RE expectations.
  - The self-enforcing tendency to the target in the RE expectations model provides an additional anchor.
  - The optimal monetary policy response seeks to influence the learning process and avoid that beliefs of higher inflation lead to higher costs of disinflation.

### Key takeaways for policy
- Front-loading tightening (e.g., tighter by 35 basis points in 2022Q2) followed by easing can limit the decline in the output gap while accepting a bit higher inflation relative to the estimated reaction function.
- Central banks should consider how policy actions influence adaptive learning and expectations formation, not just contemporaneous macroeconomic trade-offs.
- Aggressive and timely policy responses can be warranted when expectations risk becoming unanchored, especially in Emerging Markets.

*The Distributional Impacts of Worker Reallocation: Evidence from Europe — Working Paper No. WP/23/19*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023019-print-pdf.pdf_