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---

### Key takeaways
- Between February and July 2022, measured high-frequency monetary stance shocks were persistently and statistically significantly higher than those implied by the filter.
- In the baseline specification, one can reject at the 5 percent confidence level a null that during February-July 2022 monetary policy was up to 25 percent less effective.
  - Interpretation offered: for every three interest rate hikes the Fed did during this period, they would have needed approximately one more to deliver the same degree of tightening as expected pre-COVID.
- Allowing the variance of other shocks to change to match the post-2021 covariance weakens the conclusion; the rejection at the 5 percent level then holds only in March and April 2022.
- Including policymakers’ speeches in the monetary shock series strengthens the finding, with the same effect holding at the 1 percent level.
- Central Bank information shocks: realized information shocks were larger than filter-implied ones throughout 2021; the data responded as if the Fed was communicating much more negatively than it did, suggesting Fed communication was less effective than usual during 2021.

### Introduction and empirical puzzle
- Context:
  - In March 2022 the Federal Reserve began a series of interest rate hikes; in the next 19 months the federal funds rate increased from zero to over five percent.
  - By comparison, the four preceding tightening cycles (starting in February 1994, June 1999, June 2004, and December 2015) took an average of 21 months to reach peaks typically 2.5 percentage points above their starting levels.
- Puzzle:
  - Output grew rapidly throughout 2022 and 2023, at more than double the rate of even the most optimistic forecasters, despite interest rates ending up much higher than expected.
  - Inflation returned to levels close to the Federal Reserve’s two percent target even during an aggressive tightening cycle.
- Research question: Was monetary policy transmission different during this episode, and if so, how?

### Methodological approach (three steps)
- Step 1 — Construct and estimate a semi-structural FAVAR:
  - Construct monetary policy shocks from high-frequency intra-day financial market data (in the spirit of Jarociński and Karadi [2020]) and use a pre-COVID sample to estimate a factor-augmented vector autoregression (FAVAR) that includes the monetary shocks as data.
  - Baseline VAR also includes: the fed funds rate, industrial production, private consumption, inflation, the excess bond premium, and the first five principal components of a large set of other macroeconomic time series.
  - Cholesky decomposition with the monetary shocks ordered first yields a semi-structural interpretation: monetary policy shocks identified; non-monetary shocks not identified.
- Step 2 — Recast the estimated FAVAR as a filter to infer shocks:
  - Invert the FAVAR as if monetary policy shocks were unknown to back out the most likely set of shocks from 2021 to the present under the null that the mapping from monetary policy to macro data has not changed.
  - The inferred monetary shocks depend on D_m and the reduced-form residual covariance ˆΩ_y only; they are invariant to alternative decompositions of non-monetary shocks that produce the same ˆΩ_y.
  - Extension: replace the estimated reduced-form covariance matrix with one estimated on post-2021 data to allow other shocks’ variance/covariance to change without specifying their exact nature.
- Step 3 — Compare distributions and test:
  - Compare the distribution of inferred shocks under the null to observed high-frequency monetary shocks since 2021.
  - Propose maximum likelihood tests to assess evidence that monetary transmission has changed and modifications permitting quantitative assessment of how much weaker monetary policy might have been.

### Formal VAR/filter setup and sample
- Data and sample:
  - Monthly data on y_t and v_t collected since 1990.
  - Estimation period: 1990-2019 with length N_est = 336.
  - Testing period: 2021-2023 with length N_test = 36.
- VAR specification:
  - Stack shocks and variables as [v_t; y_t] = sum_{j=1}^J A_j [v_{t-j}; y_{t-j}] + ε_t, ε_t ∼ N(0, Ω) with imposed zeros reflecting v_t i.i.d. and y_t depending only on lagged y_t.
  - Reduced-form errors ε_t related to structural shocks δ_t by ε_t = D δ_t, Cov(δ_t) ∼ N(0, I_{N+2}), DD′ = Ω.
  - Monetary shock columns are D_m; reduced-form covariance for y_t denoted ˆΩ_y = Eˆε_y_t ˆε_y_t′ = ˆD_y ˆD_y′.
- Filter inversion and inference:
  - Period-by-period solution for structural shocks given estimated parameters ˆθ = {ˆB_1,...,ˆB_J, ˆΩ}:
    - ˆδ_t|ˆθ = arg min_{δ} δ′δ  s.t. ˆD_y δ = ˆε_y_t  with solution ˆδ_t|ˆθ = ˆD_y′ (ˆΩ_y)^{-1} ˆε_y_t.
    - Monetary shock estimates: ˆδ_t^m|ˆθ = ˆD_m′ (ˆΩ_y)^{-1} ˆε_y_t.
    - Unbiasedness: E[ˆδ_t^m|ˆθ] = δ_t^m.
    - Mean square error: E_t[(ˆδ_t^m|ˆθ − δ_t^m)(ˆδ_t^m|ˆθ − δ_t^m)′] = I_K − ˆD_m′ ˆΩ_y^{-1} ˆD_m.
    - Conditional distribution: δ_t^m|t,ˆθ ∼ N(ˆD_m′ (ˆΩ_y)^{-1} ˆε_y_t, I_K − ˆD_m′ ˆΩ_y^{-1} ˆD_m).

### Main empirical findings and interpretation
- Pure monetary stance shocks:
  - From February to July 2022 measured high-frequency monetary stance shocks were persistently and statistically significantly higher than filter-implied shocks, both on average and in some individual months.
  - Baseline result: reject at the 5 percent confidence level the null that monetary policy was up to 25 percent less effective during February-July 2022.
  - When allowing other shocks’ variance to change to match the 2021-2023 covariance matrix, the rejection weakens and holds only for March and April 2022 at the same confidence level.
  - Including policymakers’ speeches in the shock series strengthens the result, making it significant at the 1 percent level.
- Central Bank information shocks:
  - Realized Central Bank information shocks were larger than filter-implied ones throughout 2021; macro data behaved as if the Fed was communicating more negatively than it actually did, consistent with reduced communication effectiveness in 2021.
- Impulse response features:
  - Monetary shock IRFs scaled to a 1 percentage point increase in the Federal Funds Rate; bootstrap with K=  1000 replications produces 68 and 90 percent confidence intervals and median bootstrap responses.
  - Baseline peak decline in industrial production per unit increase in peak short rates is similar in magnitude to Jarociński and Karadi [2020] but reached faster — around 7 to 8 months here versus around 10-20 months in Jarociński and Karadi [2020].
  - Baseline does not produce a hump-shaped response for short-term interest rates; excluding factors removes these differences.
  - Including interest rates at a variety of horizons helps control for correlated changes in the slope of the yield curve.

### Uncertainty decomposition and parameter uncertainty
- Parameter uncertainty (Section 2.3):
  - The distribution in equation (5) conditions on a model and does not account for uncertainty about the true model; authors integrate over the distribution of the parameters via bootstrap using point estimates ˆθ.
  - Bootstrap simulation: simulate K draws of the estimation sample from the DGP using ˆθ; estimate parameters ˆθ^k on each simulated sample; compute mixture density f_t(δ_m) = 1/K ∑_{k=1}^K f_{t|ˆθ^k}(δ_m).
  - f_t(·) represents the distribution of shocks under the null while accounting for model uncertainty.
  - Uses bootstrapped draws to form confidence intervals for impulse responses and to decompose uncertainty:
    - f_{t,ˆθ}(·) captures irreducible uncertainty from colinearity of structural shocks.
    - f_t(·) adds model uncertainty, which is mitigated in longer samples.
- Decomposition results:
  - Uncertainty over model parameters accounts for a slim minority of total uncertainty.
  - Uncertainty due to colinearity of shocks (monetary vs other shocks producing similar impacts) is slightly more important than parameter uncertainty.
  - Inference uncertainty is often the main source of total uncertainty; model uncertainty can be substantial in periods with large reduced-form shocks (notably early 2021).

### Formal hypothesis testing and p-values
- Test definitions (three one-sided tests):
  - Test 1: H0: μi,t ≤ 0 for some t ∈ T vs H1: μi,t > 0 for all t ∈ T.
  - Test 2: H0: μi,t ≤ 0 for all t ∈ T vs H1: μi,t > 0 for some t ∈ T.
  - Test 3: H0: ̄μi ≤ 0 vs H1: ̄μi > 0, where ̄μi = (1/|T|) Σt∈T μi,t.
- Selected p-value evidence (Table 6; windows starting 2022-02-01 for monetary policy):
  - 2022-02-01 to 2022-02-01 |T|=1: Test 1 0.349, Test 2 0.346, Test 3 0.349.
  - 2022-02-01 to 2022-03-01 |T|=2: Test 1 0.349, Test 2 0.012 ∗∗, Test 3 0.017 ∗∗.
  - 2022-02-01 to 2022-04-01 |T|=3: Test 1 0.349, Test 2 0.021 ∗∗, Test 3 0.021 ∗∗.
  - 2022-02-01 to 2022-05-01 |T|=4: Test 1 0.349, Test 2 0.025 ∗∗, Test 3 0.013 ∗∗.
  - 2022-02-01 to 2022-06-01 |T|=5: Test 1 0.349, Test 2 0.036 ∗∗, Test 3 0.014 ∗∗.
  - 2022-02-01 to 2022-07-01 |T|=6: Test 1 0.468, Test 2 0.054 ∗, Test 3 0.021 ∗∗.
- Central Bank information p-values (windows starting 2021-05-01):
  - Example: 2021-05-01 to 2021-12-01 |T|=8: Test 1 0.383, Test 2 0.233, Test 3 0.017 ∗∗.
  - Longer windows into early 2022 show Test 3 p-values as low as 0.007, 0.008, indicating strong evidence that the average impact of the central bank information shock differed in the tested windows.
- Interpretation:
  - For monetary policy, Tests 2 and 3 provide evidence (around the 1 to 3 percent level) that monetary policy was weaker at least some of the time and on average in early 2022, while Test 1 generally cannot reject the null that transmission was different in all periods.
  - For central bank information, longer windows show very low p-values for Test 3, signaling substantial change in average impact during the second half of 2021 into early 2022.

### Quantifying changes in monetary transmission (Section 6)
- Nulls with scaled impacts:
  - Define modified impact ̃D(λ_1) = D_y × diag(λ_1, 1, ..., 1) for λ_1 ∈ (0,1), representing monetary transmission only λ_1 as strong as pre-COVID.
  - Under ̃D(λ_1) interest rates would need to increase by around 1/λ_1 percentage points to have the same effect as a 1pp increase pre-COVID.
- Empirical findings on λ_1:
  - Efficacy of monetary policy dropped considerably in March and April 2022.
  - During March–April 2022 one would struggle to reject at the 5 percent level any null where monetary policy was stronger than about 75 percent of its usual efficacy (i.e., λ_1 ≈ 0.75).
  - Translating: for every three interest rate hikes the Fed did during this period, they would have needed to do approximately one more to deliver the same degree of tightening as pre-COVID.
  - At the 1 percent significance level, monetary policy is concluded to be incrementally weaker only in April.
- Robustness to non-monetary shocks affecting covariance:
  - Because other shocks matter only via their effect on ˆΩ_y, these results hold identically no matter how non-monetary shocks changed so long as data covariance is unaffected.
  - Section 6.3 relaxes this assumption by replacing ˆΩ_y with the post-2021 residual covariance.

### Allowing for changing variances (Section 6.3) and robustness
- Motivation: post-COVID period likely changed the variance of other shocks during 2022.
- Evidence:
  - Figure 10 shows standard deviations of reduced form residuals for VAR data series in estimation and testing periods; every series shows an increase in standard deviation, in some cases more than doubling.
- Re-running tests with post-2021 ˆΩ_y:
  - Using the same D_y but the post-2021 residual covariance mitigates earlier findings somewhat; evidence of reductions in monetary impact is a little less strong.
  - Nevertheless, implication that policy was weaker than expected in at least March and April 2022 remains robust.
  - This experiment sets a relatively high threshold because the residual shock covariance itself is imprecisely estimated and including the full post-2021 period may not represent the population covariance during 2022.
- Including policymakers’ speeches:
  - Appendix B.3 and accompanying results show including speeches strengthens the findings.
  - Appendix Table A1 rejects unchanged monetary transmission even at the 1 percent level when speeches are included.
  - Figures A14–A15 imply monetary policy is statistically significantly weaker even if covariance of other shocks is different.

### Validation, diagnostics, and historical context
- Filter validation on estimation period (1990-2019):
  - Regression of actual shocks on inferred shocks yields coefficients near 1 with intercepts near 0:
    - Monetary Policy: Estimated shock coefficient 1.011 ∗∗∗ (0.187); Constant 0.005 (0.051); Observations 356; R2 0.076; Adjusted R2 0.074; Residual Std. Error (df = 354) 0.966.
    - Central Bank Information: Estimated shock coefficient 1.005 ∗∗∗ (0.122); Constant −0.025 (0.049); Observations 356; R2 0.161; Adjusted R2 0.158; Residual Std. Error (df = 354) 0.921.
  - Coverage ratios (mixture of normals percentiles):
    - Monetary Policy: 68 percent: 80.9; 90 percent: 92.1; 95 percent: 94.7.
    - Central Bank Information: 68 percent: 83.7; 90 percent: 93.0; 95 percent: 94.7.
  - Excess kurtoses around 13; inner confidence intervals tend to be over-covered and outer ones under-covered due to fat tails.
- Historical persistence:
  - Table 5 reports long consecutive streaks of realized shocks above/below filter-implied estimates; such long departures are rare and historically comparable only to extreme stress episodes like the Global Financial Crisis and the onset of the COVID-19 pandemic.

### Conclusions and implications
- Method: presents a general method for assessing whether a dynamic data generating process changed when externally-identified shocks are available (estimate FAVAR, invert to filter, compare realized high-frequency shocks to filter-implied distributions, conduct hypothesis tests).
- Application: to US monetary policy transmission during the 2022 tightening cycle.
- Main conclusion:
  - During 2022 the macro data and shocks were not consistent with unchanged monetary transmission.
  - Monetary policy was probably around 25 percent less effective than would have been expected pre-COVID.
- Robustness:
  - Method accounts for the full set of shocks and is robust to concerns that other shocks may mask monetary policy impact.
  - Allowing for changing shock variance weakens but does not overturn findings.
  - Including policymakers’ speeches strengthens findings.
- Potential applications: method can be applied broadly where well-identified shocks exist to assess whether propagation of those shocks has changed (example: state-dependence of public spending multipliers using narrative fiscal shocks).

*Source: wpiea2024129-print-pdf (2022) — content as provided.*

### 2022. These differences imply that monetary transmission was around 25 percent weaker than normal. Our

### wpiea2024129-print-pdf - 2022. These differences imply that monetary transmission was around 25 percent weaker than normal. Our

### Key takeaways
- Between February and July 2022, measured high-frequency monetary stance shocks were persistently and statistically significantly higher than those implied by the filter.
- In the baseline specification, one can reject at the 5 percent confidence level a null that during February-July 2022 monetary policy was up to 25 percent less effective.
  - Interpretation offered: for every three interest rate hikes the Fed did during this period, they would have needed approximately one more to deliver the same degree of tightening as expected pre-COVID.
- Allowing the variance of other shocks to change to match the post-2021 covariance weakens the conclusion; the rejection at the 5 percent level then holds only in March and April 2022.
- Including policymakers’ speeches in the monetary shock series strengthens the finding, with the same effect holding at the 1 percent level.
- Central Bank information shocks: realized information shocks were larger than filter-implied ones throughout 2021; the data responded as if the Fed was communicating much more negatively than it did, suggesting Fed communication was less effective than usual during 2021.

### Introduction and empirical puzzle
- Context:
  - In March 2022 the Federal Reserve began a series of interest rate hikes; in the next 19 months the federal funds rate increased from zero to over five percent.
  - By comparison, the four preceding tightening cycles (starting in February 1994, June 1999, June 2004, and December 2015) took an average of 21 months to reach peaks typically 2.5 percentage points above their starting levels.
- Puzzle:
  - Output grew rapidly throughout 2022 and 2023, at more than double the rate of even the most optimistic forecasters, despite interest rates ending up much higher than expected.
  - Inflation returned to levels close to the Federal Reserve’s two percent target even during an aggressive tightening cycle.
- Research question: Was monetary policy transmission different during this episode, and if so, how?

### Methodological approach (three steps)
- Step 1 — Construct and estimate a semi-structural FAVAR:
  - Construct monetary policy shocks from high-frequency intra-day financial market data (in the spirit of Jarociński and Karadi [2020]) and use a pre-COVID sample to estimate a factor-augmented vector autoregression (FAVAR) that includes the monetary shocks as data.
  - Baseline VAR also includes: the fed funds rate, industrial production, private consumption, inflation, the excess bond premium, and the first five principal components of a large set of other macroeconomic time series.
  - Cholesky decomposition with the monetary shocks ordered first yields a semi-structural interpretation: monetary policy shocks identified; non-monetary shocks not identified.
- Step 2 — Recast the estimated FAVAR as a filter to infer shocks:
  - Invert the FAVAR as if monetary policy shocks were unknown to back out the most likely set of shocks from 2021 to the present under the null that the mapping from monetary policy to macro data has not changed.
  - The inferred monetary shocks depend on D_m and the reduced-form residual covariance ˆΩ_y only; they are invariant to alternative decompositions of non-monetary shocks that produce the same ˆΩ_y.
  - Extension: replace the estimated reduced-form covariance matrix with one estimated on post-2021 data to allow other shocks’ variance/covariance to change without specifying their exact nature.
- Step 3 — Compare distributions and test:
  - Compare the distribution of inferred shocks under the null to observed high-frequency monetary shocks since 2021.
  - Propose maximum likelihood tests to assess evidence that monetary transmission has changed and modifications permitting quantitative assessment of how much weaker monetary policy might have been.

### Formal VAR/filter setup and sample
- Data and sample:
  - Monthly data on y_t and v_t collected since 1990.
  - Estimation period: 1990-2019 with length N_est = 336.
  - Testing period: 2021-2023 with length N_test = 36.
- VAR specification:
  - Stack shocks and variables as [v_t; y_t] = sum_{j=1}^J A_j [v_{t-j}; y_{t-j}] + ε_t, ε_t ∼ N(0, Ω) with imposed zeros reflecting v_t i.i.d. and y_t depending only on lagged y_t.
  - Reduced-form errors ε_t related to structural shocks δ_t by ε_t = D δ_t, Cov(δ_t) ∼ N(0, I_{N+2}), DD′ = Ω.
  - Monetary shock columns are D_m; reduced-form covariance for y_t denoted ˆΩ_y = Eˆε_y_t ˆε_y_t′ = ˆD_y ˆD_y′.
- Filter inversion and inference:
  - Period-by-period solution for structural shocks given estimated parameters ˆθ = {ˆB_1,...,ˆB_J, ˆΩ}:
    - ˆδ_t|ˆθ = arg min_{δ} δ′δ  s.t. ˆD_y δ = ˆε_y_t  with solution ˆδ_t|ˆθ = ˆD_y′ (ˆΩ_y)^{-1} ˆε_y_t.
  - Monetary shock estimates: ˆδ_t^m|ˆθ = ˆD_m′ (ˆΩ_y)^{-1} ˆε_y_t.
  - Unbiasedness: E[ˆδ_t^m|ˆθ] = δ_t^m.
  - Mean square error: E_t[(ˆδ_t^m|ˆθ − δ_t^m)(ˆδ_t^m|ˆθ − δ_t^m)′] = I_K − ˆD_m′ ˆΩ_y^{-1} ˆD_m.
  - Conditional distribution: δ_t^m|t,ˆθ ∼ N(ˆD_m′ (ˆΩ_y)^{-1} ˆε_y_t, I_K − ˆD_m′ ˆΩ_y^{-1} ˆD_m).

### Main empirical findings and interpretation
- Pure monetary stance shocks:
  - From February to July 2022 measured high-frequency monetary stance shocks were persistently and statistically significantly higher than filter-implied shocks, both on average and in some individual months.
  - Baseline result: reject at the 5 percent confidence level the null that monetary policy was up to 25 percent less effective during February-July 2022.
  - When allowing other shocks’ variance to change to match the 2021-2023 covariance matrix, the rejection weakens and holds only for March and April 2022 at the same confidence level.
  - Including policymakers’ speeches in the shock series strengthens the result, making it significant at the 1 percent level.
- Central Bank information shocks:
  - Realized Central Bank information shocks were larger than filter-implied ones throughout 2021; macro data behaved as if the Fed was communicating more negatively than it actually did, consistent with reduced communication effectiveness in 2021.

### Uncertainty decomposition
- The paper decomposes sources of uncertainty in the filter estimates:
  - Uncertainty over model parameters accounts for a slim minority of total uncertainty.
  - Uncertainty due to colinearity of shocks (the degree to which monetary and other shocks have similar impacts on the data and are thus hard to disentangle) is slightly more important than parameter uncertainty.

### Relation to prior literature
- Builds on high-frequency identification of monetary shocks (Kuttner [2001], Gürkaynak et al. [2004], Jarociński and Karadi [2020], Swanson [2023], Swanson and Jayawickrema [2023]) and extends event set to include speeches by all FOMC Board members.
- Connects to methodological literature using high-frequency shocks as identified inputs in VARs (Stock and Watson [2012], Mertens and Ravn [2013], Gertler and Karadi [2015]); the paper explicitly includes shocks in the VAR and then inverts the VAR as a filter.
- Contributes to literature on monetary policy response to COVID-19 and aftermath (English et al. [2024], Ball et al. [2022], Stedman and Pollard [2023], Cohen [2023], D’Amico and King [2023], Berger et al. [2021], Eichenbaum et al. [2022]) by providing a statistical test for weakening of monetary transmission relative to pre-pandemic.

### Paper organization (as presented)
- Section 2: Method description.
- Section 3: Construction of monetary policy shocks.
- Section 4: Description of the estimated FAVAR and validation on the pre-COVID sample.
- Section 5: Main results comparing filter-implied shocks with high-frequency measured shocks.
- Section 6: Alternative interpretations.
- Section 7: Conclusion.

### JEL and keywords; author contacts
- JEL Classification Numbers: C32, E43, E52
- Keywords: Monetary Policy; Semi-structural Identification; VAR; Filtering
- Author’s E-Mail Address: PBarrett@imf.org; JPlatzer@imf.org

*Source: wpiea2024129-print-pdf (2022) — content as provided.*

### 2.3    Parameter Uncertainty

### 2.3    Parameter Uncertainty

### Concept and motivation
- The distribution in equation (5) conditions on a model and does not account for uncertainty about the true model.
- To account for model uncertainty, the authors integrate over the distribution of the parameters via bootstrap using the point estimates ˆθ = {ˆB1,...,ˆBJ, ˆΩ}.

### Bootstrap simulation procedure
- Simulate K draws of the estimation sample {(vk1, yk1), ..., (vkNest, ykNest)}_{k=1}^K from the data generating process in equation (1) using the point estimates ˆθ.
- On each simulated sample estimate parameters ˆθ^k = {ˆB^k_1,...,ˆB^k_J, ˆΩ^k}.
- Compute the distribution for δ_{t|t}, the structural shocks conditional only on the data, as a mixture of normals:
  - f_t(δ_m) = 1/K ∑_{k=1}^K f_{t|ˆθ^k}(δ_m)   (equation (6))

### Interpretation and use of the resulting density f_t(·)
- f_t(·) represents the distribution of shocks under the null that the relationship between the data and the shocks remains unchanged, while accounting for uncertainty around that relationship (i.e., model uncertainty).
- This mixture density can be used to assess hypothesis tests of the null that the data–shock relationship is unchanged.

### Additional benefits of the bootstrap parameter draws
- Compute bootstrapped confidence intervals for impulse responses using the simulated parameter draws.
- Compare, in each testing period t, the distributions f_{t,ˆθ}(·) and f_t(·) to decompose sources of uncertainty:
  - f_{t,ˆθ}(·) captures the irreducible uncertainty coming from the colinearity of the structural shocks.
  - f_t(·) adds the effect of model uncertainty, which is mitigated in longer samples.
- The authors note they return to this point later in Section 5.4.

*wpiea2024129-print-pdf - 2.3    Parameter Uncertainty*

### 3. CPI index (100*log) 4. Excess bond premium (percent) 5. Employment−Population Ratio (percent)

### wpiea2024129-print-pdf - 3. CPI index (100*log) 4. Excess bond premium (percent) 5. Employment−Population Ratio (percent)

### Impulse responses: Monetary policy shock
- Figure 5 presents the responses of headline variables to a monetary policy shock scaled to a 1 percentage point increase in the Federal Funds Rate.
- Estimated VAR includes five main variables, both shocks, and five factors.
- Presentation details preserved:
  - Solid lines: point estimates.
  - Shaded regions: 68 and 90 percent confidence intervals from a bootstrap with K=  1000  replications.
  - Dashed lines: median responses from the bootstrap.
  - Data is monthly and the estimation sample is January 1990-December 2019.
- Key comparative findings:
  - Compared with Jarociński and Karadi [2020], the baseline peak decline in industrial production per unit increase in peak short rates is similar in magnitude but reached faster — around 7 to 8 months in this study versus around 10-20 months in Jarociński and Karadi [2020].
  - Baseline results do not produce a hump-shaped response for short-term interest rates, unlike Jarociński and Karadi [2020]; excluding the factors removes these differences.
- Role of factors:
  - Removing the factors causes interest rates to stay persistently higher and output and employment to be slower to respond.
  - Including interest rates at a variety of horizons helps control for correlated changes in the slope of the yield curve.

### Impulse responses: Central Bank Information shock
- Figure 6 shows responses to a Central Bank Information shock (also scaled to a 1 percentage point increase in the Federal Funds Rate).
- Same estimation setup and presentation conventions as Figure 5 (five main variables, both shocks, five factors; bootstrap with K=  1000; monthly data; estimation sample January 1990-December 2019).
- Robustness checks:
  - Extending the shock series to include speeches since 2008 (per Swanson [2023]) yields broadly similar impulse responses, with a slightly more delayed response of industrial production and employment (Appendix Figures A11 and A12).

### Validation of the filter and model
- Filter check: applied to estimation period (1990-2019), backing out two implied policy shocks from the data.
- Regression of actual shocks on inferred shocks (Table 3) — expected slope of one and intercept zero if unbiased:
  - Monetary Policy (column 1): Estimated shock coefficient 1.011 ∗∗∗ (0.187); Constant 0.005 (0.051); Observations 356; R2 0.076; Adjusted R2 0.074; Residual Std. Error (df = 354) 0.966.
  - Central Bank Information (column 2): Estimated shock coefficient 1.005 ∗∗∗ (0.122); Constant −0.025 (0.049); Observations 356; R2 0.161; Adjusted R2 0.158; Residual Std. Error (df = 354) 0.921.
  - Note: ∗ p<0.1; ∗∗ p<0.05; ∗∗∗ p<0.01.
- Interpretation:
  - Regression slopes close to one and intercepts close to zero for both shocks, consistent with an unbiased filter.
  - R2 is low for both shocks (especially the pure monetary policy shock), indicating much of the variation in realized shocks is not captured by the VAR — possibly model mis-specification or fundamentally limited information in macro data.
- Coverage ratios for the realized shocks using percentile ranges from the mixture of normals (Table 4):
  - Monetary Policy: Shock coverage — 68 percent: 80.9; 90 percent: 92.1; 95 percent: 94.7.
  - Central Bank Information: Shock coverage — 68 percent: 83.7; 90 percent: 93.0; 95 percent: 94.7.
  - Excess kurtoses of around 13 noted; inner confidence intervals tend to be over-covered and outer ones under-covered due to fat tails.

### Results from the Filter (post-2021 application)
- Method: Apply VAR-based filter to post-2021 data to compute distributions of monetary and other shocks under null that data generating process is unchanged; compare realized high-frequency shocks to these distributions (out-of-sample test).
- Figure 7 summary:
  - Shaded regions: 68 and 90 percent confidence intervals from a bootstrap with K= 1000 replications.
  - Blue line with large dots: actual shock from high frequency data.
  - Observations:
    - Measured monetary policy shock exceeds the filter point estimate throughout January to July 2022 — interpreted as the FOMC repeatedly moving financial markets consistent with tightening while macro data behaved as if loosening (i.e., transmission was looser).
    - Fed information shock exceeds the point estimate from April 2021 until April 2022 — markets interpreted Fed announcements as revealing positive information about the state of the economy but monthly macro data did not confirm that subsequently.
  - Statistical confidence month-by-month is typically low (blue dots usually inside center of distribution), but the persistence of deviations across months motivates pooling evidence across periods.

### Informal assessment of persistence and historical context
- Table 5 reports longest streaks where observed high-frequency shocks are consistently above or below filter-implied point estimates (full sample 1990-2023). Examples listed as terminal months and consecutive periods:
  - Monetary Policy:
    - 2007-03-01: Consecutive periods above 6
    - 2008-04-01: Consecutive periods below 70
    - 2020-06-01: Consecutive periods above 109
    - 2022-07-01: Consecutive periods below 70
  - Central Bank Information:
    - 2010-09-01: Consecutive periods above 90
    - 2014-12-01: Consecutive periods above 90
    - 2019-04-01: Consecutive periods above 100
    - 2022-04-01: Consecutive periods above 130
- Interpretation: Such long consecutive departures are rare and historically comparable only to extreme stress episodes like the Global Financial Crisis and the onset of the COVID-19 pandemic.

### Formal assessment: Hypothesis testing
- Define ηt = vt − ̄δt (difference between observed high-frequency shocks and filter mean); μt = Eηt. Tests consider whether μt > 0 in different senses over a window T:
  - Test 1: H0: μi,t ≤ 0 for some t ∈ T vs H1: μi,t > 0 for all t ∈ T.
  - Test 2: H0: μi,t ≤ 0 for all t ∈ T vs H1: μi,t > 0 for some t ∈ T.
  - Test 3: H0: ̄μi ≤ 0 vs H1: ̄μi > 0, where ̄μi = (1/|T|) Σt∈T μi,t.
- Notes on tests:
  - All tests are one-sided.
  - Test 1 is stringent; power does not grow with |T| for Tests 1 and 2 because new parameters enter with each observation; Test 3 benefits from central limit theorem and power grows with |T|.
- p-values (Table 6) — selected entries for monetary policy (windows starting 2022-02-01) and central bank information (windows starting 2021-05-01):
  - Monetary Policy (rows show TStart, TEnd, |T|, Test 1, Test 2, Test 3):
    - 2022-02-01 to 2022-02-01 |T|=1: Test 1 0.349, Test 2 0.346, Test 3 0.349.
    - 2022-02-01 to 2022-03-01 |T|=2: Test 1 0.349, Test 2 0.012 ∗∗, Test 3 0.017 ∗∗.
    - 2022-02-01 to 2022-04-01 |T|=3: Test 1 0.349, Test 2 0.021 ∗∗, Test 3 0.021 ∗∗.
    - 2022-02-01 to 2022-05-01 |T|=4: Test 1 0.349, Test 2 0.025 ∗∗, Test 3 0.013 ∗∗.
    - 2022-02-01 to 2022-06-01 |T|=5: Test 1 0.349, Test 2 0.036 ∗∗, Test 3 0.014 ∗∗.
    - 2022-02-01 to 2022-07-01 |T|=6: Test 1 0.468, Test 2 0.054 ∗, Test 3 0.021 ∗∗.
  - Central Bank Information:
    - 2021-05-01 to 2021-05-01 |T|=1: Test 1 0.383, Test 2 0.382, Test 3 0.383.
    - 2021-05-01 to 2021-06-01 |T|=2: Test 1 0.383, Test 2 0.537, Test 3 0.316.
    - 2021-05-01 to 2021-07-01 |T|=3: Test 1 0.383, Test 2 0.346, Test 3 0.149.
    - 2021-05-01 to 2021-08-01 |T|=4: Test 1 0.383, Test 2 0.363, Test 3 0.103.
    - 2021-05-01 to 2021-09-01 |T|=5: Test 1 0.383, Test 2 0.379, Test 3 0.074 ∗.
    - 2021-05-01 to 2021-10-01 |T|=6: Test 1 0.383, Test 2 0.442, Test 3 0.072 ∗.
    - 2021-05-01 to 2021-11-01 |T|=7: Test 1 0.383, Test 2 0.407, Test 3 0.046 ∗∗.
    - 2021-05-01 to 2021-12-01 |T|=8: Test 1 0.383, Test 2 0.233, Test 3 0.017 ∗∗.
    - 2021-05-01 to 2022-01-01 |T|=9: Test 1 0.383, Test 2 0.260, Test 3 0.014 ∗∗.
    - 2021-05-01 to 2022-02-01 |T|=10: Test 1 0.383, Test 2 0.289, Test 3 0.013 ∗∗.
    - 2021-05-01 to 2022-03-01 |T|=11: Test 1 0.383, Test 2 0.270, Test 3 0.008 ∗∗∗.
    - 2021-05-01 to 2022-04-01 |T|=12: Test 1 0.383, Test 2 0.300, Test 3 0.007 ∗∗∗.
  - Note: ∗ p <0.1, ∗∗ p <0.05, ∗∗∗ p <0.01.
- Inference from p-values:
  - For monetary policy, Tests 2 and 3 provide evidence (around the 1 to 3 percent level) that monetary policy was weaker at least some of the time and on average in early 2022, while Test 1 generally cannot reject the null that transmission was different in all periods.
  - For central bank information, longer windows show very low p-values for Test 3 (e.g., 0.007, 0.008), indicating strong confidence that the average impact of the central bank information shock was different during the tested windows (second half of 2021 into early 2022).

### Role of uncertainty and decomposition
- Figure 8 decomposes the 90 percent confidence intervals from Figure 7 into contributions from:
  - Model uncertainty (parameter uncertainty).
  - Inference uncertainty (uncertainty from the filter due to missing information).
  - Interaction term.
- Key points:
  - Inference uncertainty is constant (reflects lack of knowledge over shocks due to missing information) and is most of the time the main source of uncertainty because shocks are fundamentally hard to disentangle.
  - Model uncertainty can be substantial in periods when reduced-form shocks are largest (notably early 2021), because parameter uncertainty is amplified by large shocks.
  - In periods with smaller shocks, model uncertainty contribution is lower.

### Interpretation and open questions
- Rejecting the tests indicates the relationship between high-frequency identified shocks and monthly macro data differs from pre-COVID norms, but does not by itself quantify how transmission differs or identify alternative explanations.
- The paper poses three interpretation questions to address subsequently:
  - Quantitatively, how much weaker would transmission need to be to reconcile observed high-frequency shocks with the macro data?
  - Could changing impacts of other shocks explain the results?
  - What implications do the results have for standard structural macro models?

*Source: wpiea2024129-print-pdf (https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024129-print-pdf.pdf)*

### 6.1    Set up

### 6.1    Set up

### Mapping data into implied structural shocks
- Key ingredients of the filter that maps data into implied structural shocks are the matrices ˆD_y and ˆΩ_y, which together define the mean and covariance of the filter estimate conditional on the parameters, δ_{t|t, ˆθ}.
- Modified ˆD_y matrices can be used to consider alternate null transmissions by multiplying by a diagonal matrix Λ, partitioned as:
  - Λ = ["Λ_m" 0_{K×N}; 0_{N×K} "Λ_m"]
- The modified causal impact matrix is:
  - ̃D_y = D_y Λ = [D_m Λ_m  D_x Λ_m]
- Interpretation: ̃D_y represents the statistical model where the impact of each shock has been scaled by the corresponding entry of Λ. Example: if the top-left first entry of Λ = 0.9 and all other entries = 1, the pure monetary shock is uniformly ten percent less effective — a given monetary shock moves all outcomes by ten percent less than in the baseline model.
- Procedure: run the filter on the same data but replace D_y with ̃D_y to produce the distribution for high frequency indicators under the new null (e.g., monetary shocks ten percent weaker). Then perform regular hypothesis tests using the modified model to test hypotheses such as monetary policy being at least ten percent weaker in a sample.

### Choosing Λ and handling missing variation
- Altering a column of D without offsetting changes to other shocks will generally change the data-generating covariance.
- Because other shocks matter only insofar as they affect the covariance of the data, ˆΩ_y, one need only be explicit about assumptions on other shocks up to how they affect ˆΩ_y.
- Shortcut: if ̃D_y modifies only the monetary shock but leaves ˆΩ_y unchanged, computing the filter with unchanged ˆΩ_y will recover the distribution of monetary shocks under the assumption that unidentified shocks change in whatever ways are needed to keep the estimated data-generating process covariance the same.
  - Cost: the time series for the other, unidentified, shocks will not be correct — but they are not the focus.
- If one believes the variance of the data changed during the test period, ˆΩ_y can be replaced with an alternative; this is discussed further in Section 6.3.

### 6.2    The changing impact of monetary policy
- Goal: quantify change in strength of monetary policy transmission, initially assuming post-2022 residual covariance ˆΩ_y is unchanged.
- Consider modified structural impact matrices:
  - ̃D(λ_1) = D_y × diag(λ_1, 1, ..., 1) for λ_1 ∈ (0,1).
  - ̃D(λ_1) embodies the idea that monetary transmission is only λ_1 as strong as in the pre-COVID baseline.
  - Interpretation: interest rates would need to increase by around 1/λ_1 percentage points to have the same effect as a 1pp increase pre-COVID.
- For each λ_1 re-run hypothesis tests replacing D_y with ̃D(λ_1). Figure 9 plots resulting p-values for six monthly windows T starting in February 2022. Values at λ_1 = 1 match corresponding columns of Table 6.
- Empirical suggestion:
  - Efficacy of monetary policy dropped considerably in March and April 2022.
  - During March–April 2022 one would struggle to reject at the 5 percent level any null where monetary policy was stronger than about 75 percent of its usual efficacy.
  - Translating: for every three interest rate hikes the Fed did during this period, they would have needed to do approximately one more to deliver the same degree of tightening as pre-COVID.
  - Later in the year, this statement continues to hold on average.
  - At the 1 percent significance level, monetary policy is concluded to be incrementally weaker only in April.
- Robustness note: because other shocks only matter via their effect on ˆΩ_y, these results hold identically no matter how non-monetary shocks changed so long as data covariance is unaffected. Section 6.3 relaxes this assumption.

### 6.3    Allowing for changing variances
- Motivation: post-COVID period likely changed the nature (variance) of other shocks during 2022.
- Advantage of approach: can capture impact of such changes on filter-implied monetary shocks without specifying exactly how other shocks changed, by describing solely how changes affect reduced form residual covariance matrix ˆΩ_y.
- Caveat: if reduced form residual covariance changed during test period, this undermines interpreting results as a change in efficacy of monetary policy.
  - Example: the filter expects tighter monetary policy to, at the margin, produce lower inflation (see Figure 5). Persistence of inflation despite high interest rates will be interpreted as weaker monetary policy unless offsetting shocks are accounted for correctly. If variance of offsetting shocks changed, the filter will account for them incorrectly.
- Evidence on residual variances:
  - Figure 10 shows standard deviations of reduced form residuals for VAR data series in estimation and testing periods; every series shows an increase in standard deviation, in some cases more than doubling.
- Re-run using post-2021 residual covariance for ˆΩ_y (Figure 11):
  - Using the same D_y but the post-2021 residual covariance, tests for changes in monetary impact are repeated allowing excess volatility to be explained by changing variances of other shocks.
  - Results mitigate earlier findings somewhat: evidence of reductions in monetary impact is a little less strong.
  - Nevertheless, implication that policy was weaker than expected in at least March and April 2022 remains robust.
  - This experiment sets a relatively high threshold for concluding a change in monetary impact, since the residual shock covariance itself is imprecisely estimated and including the full post-2021 period may not represent the population covariance during 2022.
- Additional robustness:
  - Appendix B.3 repeats tests including policymakers’ speeches and shows they strengthen results.
  - Appendix Figure A13 shows post-2022 outturns vs filter-inferred shocks; differences larger than baseline.
  - Appendix Table A1 rejects unchanged monetary transmission even at the 1 percent level when speeches included.
  - Figures A14 and A15 imply monetary policy is statistically significantly weaker even if covariance of other shocks is different.

### Conclusions (from Section 7 as summarized here)
- Method: general method for assessing whether a dynamic data generating process changed when externally-identified shocks are available.
- Application: to whether transmission of US monetary policy was different during the 2022 tightening cycle.
- Main conclusion:
  - During 2022 the macro data and shocks were not consistent with unchanged monetary transmission.
  - Monetary policy was probably around 25 percent less effective than would have been expected pre-COVID.
- Robustness features:
  - Method accounts for the full set of shocks and is robust to concerns that other shocks may mask monetary policy impact.
  - Allowing for changing shock variance weakens but does not overturn findings.
  - Including policymakers’ speeches strengthens findings.
- Potential: method can be applied broadly where well-identified shocks exist to assess whether propagation of those shocks has changed (example: state-dependence of public spending multipliers using narrative fiscal shocks).

*Italic: Source: wpiea2024129-print-pdf - 6.1    Set up*

### 5. Employment−Population Ratio (percent)Factor #1Factor #2

### 5. Employment−Population Ratio (percent)Factor #1Factor #2

### Impulse responses: overview
- Figure A4: Response of headline variables to a monetary policy shock, scaled to a 1 percentage point increase in the Federal Funds Rate. Solid lines are point estimates; shaded regions are the 68 and 90 percent confidence intervals from a bootstrap with K= 1000 replications. Dashed lines show median responses from the bootstrap.
- Figures A5–A10: Point estimates of factor-augmented VAR responses to:
  - Monetary policy shock and Central Bank information shock, each scaled to a 1 percentage point increase in the interest rate.
  - Variations across:
    - different interest rate series (Baseline = Federal Funds Rate; alternatives: One−year government bond, Three−month treasury);
    - different lag structures (Baseline = 2 lags chosen by AIC; alternatives: 3 Lags, 4 Lags, 6 Lags);
    - different number of factors (Baseline = five factors; alternatives: 1 Factor, 3 Factor, 8 Factors, No factors).
- All specifications include five factors; responses of the factors themselves are not shown.

### Robustness checks: interest rate series, lag length, and factor count
- Interest rate series:
  - Figures A5 and A6 compare Baseline (Fed Funds Rate), One−year government bond, and Three−month treasury for monetary policy and Central Bank information shocks, respectively.
  - All IRFs are scaled to a 1 percentage point increase in the interest rate.
- Lag structures:
  - Figures A7 and A8 show IRFs under Baseline (2 lags), 3 Lags, 4 Lags, and 6 Lags for monetary policy and Central Bank information shocks, respectively.
- Number of factors:
  - Figures A9 and A10 show IRFs under Baseline (5 factors), 1 Factor, 3 Factors, 8 Factors, and No factors for monetary policy and Central Bank information shocks, respectively.

### Including policymakers’ speeches
- Figures A11 and A12:
  - Compare Baseline against specifications "Including speeches in shocks" for monetary policy and Central Bank information shocks.
  - Both figures: point estimates of factor-augmented VAR responses, scaled to a 1 percentage point increase in the interest rate. All specifications include five factors.
- Figure A13: Inferred and realized shocks for 2021-2023, including speeches in shocks.
  - Solid lines: point estimates of inferred shocks under the null that the data generating process remains unchanged.
  - Shaded regions: 68 and 90 percent confidence intervals from a bootstrap with K= 1000 replications.
  - Blue line with large dots: actual shock computed from high frequency data. This version uses shocks including policymakers’ speeches.

### External instruments versus direct inclusion
- Figure A16: Monetary policy shock
  - Line labelled “External instrument”: point estimate of response to a one-standard-deviation shock computed using the high-frequency monetary policy series as an external instrument.
  - Line labelled “Direct inclusion in VAR”: baseline IRFs, rescaled to match the same initial impulse as for the external instrument.
  - Five factors included in estimation; factor responses omitted for clarity.
- Figure A17: Central Bank information shock — same comparison and notes as Figure A16.

### Hypothesis tests: setup and procedures (Section C)
- Definitions and normalized variables:
  - ηt = vt − ̄δt where ̄δt = R δ δ f t (δ) dδ.
  - (σ i t )2 = R δ (δ − ̄δt)2 f t (δ) dδ are the mean and variance of the filter-implied distribution for the ith structural shock.
  - μt = Et ηt is the mean of ηt.
  - Normalized shock: xt = η i t /σ i t, assumed known root mean square error.
  - xt ∼i.i.d. N(μt,1) ∀t∈T.
- Three tests of interest (for fixed i and sample t = 1,...,T):
  - Test 1: H0: μ i t ≤ 0 for some t ∈ T vs. H1: μ i t > 0 for all t ∈ T.
  - Test 2: H0: μ i t ≤ 0 for all t ∈ T vs. H1: μ i t > 0 for some t ∈ T.
  - Test 3: H0: ̄μ i ≤ 0 vs. H1: ̄μ i > 0, where ̄μ i = 1/|T| Σ t∈T μ i t.
- Test 3 (sample mean) construction:
  - ̄xT = 1/|T| Σ t∈T xt.
  - E ̄xT = ̄μ i and ̄xT ∼ N(̄μ i ,1/√T).
  - One-sided p-value: p3(̄xT) = 1 − Φ(̄xT / √T).
- Tests 1 and 2:
  - Framed as likelihood ratio tests with parameter restriction sets:
    - Θ1 0 = {μ ∈ R T | μ t ≤ 0 ∀ t ∈ T}.
    - Θ2 0 = {μ ∈ R T | μ t ≤ 0 for some t ∈ T}.
  - LR statistic λ(x) = supΘ0 L(θ|x) / supΘ L(θ|x).
  - Rejection regions defined by Euclidean distance from observed data to nearest point inside the constrained set: dj(x) = inf μ∈Θ j 0 ||x − μ||.
  - For Test 1 a closed-form p-value exists: p1(x) = 1 − Φ(min t∈T xt).
  - For Test 2: μ∗ = 0 (the origin) maximizes the probability over the null; p-values computed by numerical integration drawing 100,000 points from the mean-zero unit-variance multivariate normal and computing the fraction of observations in R2(x).

### Empirical p-value results (including speeches in shocks)
- Table A1: p-values for joint hypothesis tests, including speeches in shocks. Each line considers three tests that the observed high frequency shocks are drawn from a distribution with a higher mean than expected based on pre-COVID transmission of monetary policy, over a period of consecutive months. Significance notation: ∗ p <0.1, ∗∗ p <0.05, ∗∗∗ p <0.01. This version uses shocks including policymakers’ speeches.
- Selected rows (as presented in Table A1):
  - Monetary Policy
    - 2022-02-01 2022-02-01 |T| 1 : 0.385 0.382 0.385
    - 2022-02-01 2022-03-01 |T| 2 : 0.385 0.011 ∗∗ 0.019 ∗∗
    - 2022-02-01 2022-04-01 |T| 3 : 0.385 0.008 ∗∗∗ 0.006 ∗∗∗
    - 2022-02-01 2022-05-01 |T| 4 : 0.385 0.013 ∗∗ 0.006 ∗∗∗
    - 2022-02-01 2022-06-01 |T| 5 : 0.385 0.002 ∗∗∗ 0.001 ∗∗∗
    - 2022-02-01 2022-07-01 |T| 6 : 0.933 0.003 ∗∗∗ 0.008 ∗∗∗
  - Central Bank Information
    - 2021-05-01 2021-05-01 |T| 1 : 0.281 0.281 0.281
    - 2021-05-01 2021-06-01 |T| 2 : 0.281 0.311 0.162
    - 2021-05-01 2021-07-01 |T| 3 : 0.281 0.192 0.061 ∗
    - 2021-05-01 2021-08-01 |T| 4 : 0.281 0.162 0.029 ∗∗
    - 2021-05-01 2021-09-01 |T| 5 : 0.281 0.182 0.022 ∗∗
    - 2021-05-01 2021-10-01 |T| 6 : 0.399 0.234 0.026 ∗∗
    - 2021-05-01 2021-11-01 |T| 7 : 0.399 0.185 0.013 ∗∗
    - 2021-05-01 2021-12-01 |T| 8 : 0.399 0.070 ∗ 0.003 ∗∗∗
    - 2021-05-01 2022-01-01 |T| 9 : 0.399 0.070 ∗ 0.002 ∗∗∗
    - 2021-05-01 2022-02-01 |T| 10 : 0.507 0.091 ∗ 0.003 ∗∗∗
    - 2021-05-01 2022-03-01 |T| 11 : 0.507 0.012 ∗∗ 0.000 ∗∗∗
    - 2021-05-01 2022-04-01 |T| 12 : 0.507 0.016 ∗∗ 0.000 ∗∗∗

### Sensitivity of hypothesis tests to shock strength and residual covariance
- Figures A14 and A15: p-values for hypothesis tests (Tests 2 and 3) as the strength of the monetary policy shock varies under nulls indexed by λ1.
  - Horizontal lines show 1, 5, and 10 percent confidence levels.
  - Interpretation: λ1 represents a null where monetary policy is only λ1 as effective as pre-COVID.
  - Each panel computes the tests on a window starting in February 2022 and ending in the titular month. Values when λ1 = 1 match corresponding columns in Table 6.
  - Figure A15 repeats these p-value exercises using the post-2021 residual covariance for ˆΩ y and shocks including policymakers’ speeches.

### Analytical derivations and numerical implementation details
- Likelihood ratio tests follow the formulation in Casella and Berger [2024] (chapter 8 referenced in source).
- Rejection regions built from Euclidean distance to constrained parameter sets (Figures A18 and A19 illustrate two-period cases).
- For Test 2 p-values, numerical integration implemented with 100,000 draws from the mean-zero unit-variance multivariate normal to estimate the fraction of observations in the rejection region R2(x).

*Source: wpiea2024129-print-pdf - 5. Employment−Population Ratio (percent)Factor #1Factor #2 (figures, tables, and C. Deriving the hypothesis tests as provided).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024129-print-pdf.pdf_