## Sign Restrictions with a New-Keyneysian Macro Model: Results From a “Quasi-Agnostic” Identification Procedure — Working Paper No. WP/2025/162

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---

### Introduction, background, and motivation
- Historical identification in SVAR:
  - Early SVAR literature used parametric (predominantly zero) contemporaneous restrictions on immediate responses of macro aggregates.
  - Justifications: variable sluggishness and observation delays.
- Move toward sign restrictions:
  - Sign restrictions impose only the direction of a shock’s impact on macro aggregates and are often called “agnostic” identification procedures.
- Empirical context and prior findings:
  - Prominent applications include Astveit et al. (2017), Canova and De Nicoló (2002), Elekdag and Han (2015), Faust (1998), and Uhlig (2005).
  - Uhlig (2005) finds contractionary monetary policy shocks do not exhibit a discernible effect on real GDP (monetary neutrality in his results).
  - Refinements (e.g., Arias et al. 2019) and external-instrument identification (Gertler and Karadi 2015) tend to reaffirm a short-term effect of monetary policy on GDP consistent with price rigidities à la Calvo (1983).

### Problem with agnostic sign-restriction procedures
- Low acceptance rates:
  - Agnostic sign-restriction procedures typically yield very low acceptance rates (proportion of simulated structural shocks that satisfy imposed sign restrictions).
  - Examples:
    - Fisher and Huh (2020) report acceptance rates of less than 1 percent using their algorithm.
    - Arias et al. (2018) also find low acceptance rates.
    - The algorithm of Ouliaris and Pagan (2016) yields very low acceptance rates when applied to Cho and Moreno (2003).

### Proposed contribution: “quasi-agnostic” identification procedure
- Core idea:
  - Iterative grid-search-like strategy that employs increasingly refined identification methods across iterations to identify plausible structural-parameter sets satisfying imposed sign restrictions.
  - Termed “quasi-agnostic” because each iteration narrows likely intervals of plausible structural parameters using information from earlier iterations.
- Key claim:
  - The strategy can significantly enhance acceptance rates, revealing a larger set of economic structures consistent with priors on the sign of responses of macro variables to structural shocks.

### Baseline model (three-variable SVAR)
- Data and variables:
  - Quarterly output gap (o), inflation (π), and average policy rate (r) for the US economy between 1982Q1 and 2000Q1.
- Structural form and normalization:
  - A0 Yt = A1 Yt−1 + A2 Yt−2 + B0 εt (equation (1)), with Y = [o, π, r]′.
  - A0 is lower triangular with ones on the diagonal.
  - B0 is diagonal with standard errors of shocks on the diagonal.
  - εt ∼ N(0, I3).
  - Normalized SVAR: standard deviations on diagonal of B0 and ones on diagonal of A0; unnormalized SVAR equivalent representation with ones on diagonal of B0 and parameters on diagonal of A0.
- Parameter counts and identification:
  - 27 parameters to be estimated.
  - Underlying VAR Yt = V1 Yt−1 + V2 Yt−2 + et (equation (2)) yields 24 parameters: 9 in V̂1, 9 in V̂2, and 6 in Σ̂ (symmetric).
  - Hence impose 3 zero contemporaneous restrictions on A0 to obtain exact identification.
- Estimation approach:
  - Exactly identified SVAR estimated via 2SLS using instrumental variables; A1 and A2 estimated via OLS.

### Diagnostics, impulse responses, and the price puzzle
- Model selection and diagnostics:
  - Lag selection criteria and likelihood-ratio tests suggest selecting lag order 2.
  - Wald statistics for Granger causality indicate variables treated as endogenous.
  - Stability: all eigenvalues lie in the unit circle.
- Key impulse response findings under recursive A0 (equation (3)):
  - Demand shock (one standard deviation):
    - Contemporaneous impact on output gap equal in magnitude to the shock; persistence about ten periods.
    - Modest positive contemporaneous effect on inflation.
    - Small immediate increase in policy rate; policy rate subsequently rises over time.
  - Supply shock (one standard deviation):
    - Zero contemporaneous impact on output gap (zero restriction) but increases to about ten basis points in about eight periods.
    - Positive contemporaneous response in inflation.
    - Contemporaneous increase in policy rate reaching about thirty basis points within eight periods before abating.
  - Monetary shock (one standard deviation):
    - Zero contemporaneous impact on output gap and inflation (zero restrictions).
    - Output gap response accelerates to about eighteen basis points in a few periods.
    - Inflation response is positive, producing a “price puzzle”.
- Standard corrective approaches to the price puzzle:
  - (i) Add endogenous variables (monetary aggregate, exchange rate, inflation expectations).
  - (ii) Redefine endogenous variables.
  - (iii) Employ non-recursive, less strict identification procedures.
  - (iv) Introduce latent variables (state-space, Kalman filter, MLE).
- Trade-offs:
  - Adding endogenous variables increases parameters and shocks and requires additional identifying restrictions; number of additional identifying restrictions required increases by (d(d−1)/2 + n d) where n is number of original endogenous variables and d is number added.
  - Latent-variable approaches require state-space and Kalman-filter estimation, beyond scope.

### Imposing agnostic sign restrictions (SRC procedure) — implementation and results
- SRC implementation:
  - Unrestricted structural matrix A0 = [1 a012 a013; a021 1 a023; a031 a032 1].
  - SRC draws random coefficients aij = θ/(1−abs(θ)) with θ ∼ U(−1,1).
  - Simulate 1,200 combinations of parameters a012, a013, a023.
  - For each draw: estimate B0 and remaining A0 parameters via 2SLS; estimate A1 and A2 via OLS.
  - Compare signs of contemporaneous IRF C0 = A−1 0 B0 with priors in Table 3; discard IRFs inconsistent with priors.
  - Acceptance rate = number of combinations consistent with priors / total draws.
- SRC results (sign restrictions at horizon zero):
  - Price puzzle resolved:
    - Demand shock: inflation impact between zero and twenty basis points; interest rate impact between thirty and seventy basis points.
    - Supply shock: inflation impact between twenty and seventy basis points; interest rate impact between zero and twenty-five basis points.
    - Monetary shock: negative impact on output gap between zero and thirty basis points; negative impact on inflation between zero and seventy basis points.
  - Low acceptance rate observed: Acceptance rate of 1.25 percent (1,200 simulations, sign restrictions at horizon 0).
  - Observations:
    - Sign restrictions solve structural identification but not model identification: each IRF corresponds to a unique set of structural parameters; summary IRF statistics do not resolve model identification.
    - Sign restrictions permit variation in standard deviations of structural shocks across models, affecting contemporaneous responses (can be normalized).

### From agnostic to “quasi-agnostic” sign restrictions — methodology and empirical behavior
- Motivation:
  - Completely agnostic identification (SRC) yields high standard deviation of possible structural parameters, causing many draws to fall outside plausible ranges and producing low acceptance rates.
- Quasi-agnostic approach:
  - Adaptive, grid-search-like refinement of identification procedures informed by ex-post results from earlier draws; progressively trim plausible parameter ranges to increase acceptance rates.
- Illustrative bivariate supply-demand example (no lags):
  - Model: Qt = α pt + es; Qt = β pt + ed (equation (4)).
  - SVAR A0 Yt = B0 εt with A0 = [1 a012; 1 a022] and B0 = diag(σs, σd) (equation (5)).
  - Priors matrix P = [ +  +;  + − ] (equation (6)): demand shock increases quantities and prices; supply shock increases quantities and decreases prices.
  - Contemporaneous IRF C0 = A−1 0 B0 yields conditions to satisfy P:
    - determinant of A0 negative: (â022 − â012) < 0,
    - â012 > 0,
    - â022 < 0.
  - Searching across (−∞,+∞) (SRC) leads to many implausible values; restricting search to implied sign intervals should increase acceptance.
- Empirical quasi-agnostic implementation (three-step refinement illustrated in Figure 5):
  - First pass: fully agnostic SRC draws; observation: all successful parameters positive; means order a012 smallest, then a023, then a013; a012 has very small standard deviation; a013 and a023 have larger, skewed distributions.
  - Second pass: abs(SRC) — take absolute values of SRC draws restricting parameters to (0, +∞); yields higher acceptance rates at all horizons.
  - Third pass: draws from Normal(0,1); broadly comparable to SRC.
  - Final pass: calibrate random draws exploiting means and standard deviations observed in earlier rounds; draw parameters concentrated around observed means — acceptance rate increases as draws concentrate around these means.
  - Observed behavior across horizons: larger plausible a013 parameters tend to disappear when IRFs are restricted at higher time horizons.
- Outcome:
  - Quasi-agnostic, progressively refined procedures can significantly enhance acceptance rates and reveal a larger set of economic structures consistent with sign priors.
  - Example numerical outcomes in figures:
    - SRC: AR = 1.25 at Horizon = 0 (1,200 simulations).
    - abs(SRC) and other refinements yield higher ARs (e.g., AR = 9.25 for abs(SRC) in a panel listing).
    - “Quasi agnostic” procedure: Acceptance rate of 8.58 percent with 1,200 simulations and sign restrictions imposed at horizon 2 (Figure 6).
- Limitations:
  - For higher-dimensional SVARs, using only ex-post information to restrict plausible structural parameters does not guarantee identification of all possible parameters consistent with sign priors.
  - The relative role of model misspecification versus identification-procedure limitations in producing low acceptance rates remains open.
  - For bivariate SVAR a closed-form solution can impose ex ante restrictions consistent ex post with sign priors; higher dimensions lack general ex ante characterizations.

### Conclusions and avenues for future research
- Empirical findings:
  - Identified shocks exert durable influence on output gap and interest rates; effects on inflation are generally less persistent.
  - Recursive SVAR produces a price puzzle; imposing agnostic sign restrictions (SRC) at horizon zero resolves the price puzzle but yields low acceptance rates.
  - Quasi-agnostic refinements (progressive, data-informed narrowing of parameter draws) increase acceptance rates and uncover a wider set of plausible structures.
- Methodological implications:
  - Sign restrictions alone do not resolve model identification; normalization and additional restrictions or selection criteria are needed when choosing among plausible structural parameterizations.
  - Quasi-agnostic procedures exploit ex-post insights to improve identification procedure performance, but do not replace the need for ex ante restrictions, especially in higher dimensions.
- Recommendations for future research:
  - Identify ex ante restrictions on structural parameters for higher-dimensional SVARs to better link model and structural identification.
  - Quantify the relative importance of model misspecification versus identification-procedure limitations in determining acceptance rates.

### Appendix — selected exact numerical and display items preserved from figures and tables
- Parameter and simulation counts:
  - 27 parameters to estimate; 24 parameters identified from underlying VAR.
  - Number of simulations used in SRC and quasi-agnostic illustrations: 1,200.
- Acceptance rates explicitly reported:
  - Acceptance rate of 1.25 percent (SRC, horizon 0).
  - Acceptance rate of 8.58 percent (quasi-agnostic, horizon 2).
- Table 1: Lag Selection — horizon row entries (verbatim):
  - 0-3 78.58 10.899 10.902 10.941 10.999
  - 1-1 77.03 403.09 0.000 0.044 5.401 5.554 5.786
  - 2-1 48.47 57.12 *0.000 *0.025 *4.842 *5.110 *5.517
  - 3- -141.31 14.31 0.112 0.027 4.895 5.277 5.858
  - 4-134.00 14.63 0.102 0.029 4.943 5.440 6.196
  - 5-128.23 11.55 0.240 0.032 5.035 5.647 6.577
  - 6-124.40 7.65 0.569 0.037 5.183 5.910 7.014
  - 7-118.89 11.02 0.275 0.042 5.283 6.125 7.403
- Table 2: Granger causality (columns: EquationVariable, W, d.f., Pval — verbatim rows):
  - π1.010   2   0.604
  - o    r4.625   2   0.099
  - all8.148   2   0.086
  - π3.630   2   0.163
  - πr4.084   2   0.130
  - all11.360   2   0.023
  - π4.066   2   0.131
  - rr6.399   2   0.041
  - all8.088   2   0.088
- Table 3 (contemporaneous sign restrictions for positive structural shocks — as presented):
  - o+ − −
  - π++ − −
  - r +++
- Figure notes preserved:
  - Figure 1: Roots of the companion matrix (axes ticks at -1, -.5, 0, .5, 1).
  - Figure 2 and Figure 6: IRF panels with Horizon labels 0 5 10 15 20.
  - Figure 3: Acceptance rate of 1.25 percent, 1,200 simulations, sign restrictions at horizon 0.
  - Figure 6: Acceptance rate of 8.58 percent, 1,200 simulations, sign restrictions at horizon 2.
- Additional figure AR listings (verbatim entries from Figure 5 panel):
  - AR = 1.25, Horizon = 0, SRC
  - AR = 0.42, Horizon = 1
  - AR = 0.17, Horizon = 2
  - AR = 9.25, abs(SRC)
  - AR = 3.75
  - AR = 1.92
  - AR = 2.08
  - AR = 1.17
  - AR = 0.83
  - AR = 21.92, N(a,b)
  - AR = 13.75
  - AR = 8.58

*Source: wpiea2025162-source-pdf*

### References10

### wpiea2025162-source-pdf - References10

### List of Figures and Tables
- Figures listed:
  - 1    Stability condition
  - 2    Structural impulse responses - recursive model.
  - 3    Contemporaneous sign restrictions on IRFs using the SRC procedure 1/.
  - 4    Ex ante restrictions on structural parameters.
  - 5    Distribution of restricted parameters using alternative identification procedures
  - 6    Results from “quasi agnostic” procedure /1.
- Tables listed:
  - 1    Lag Selection.
  - 2    Granger causality.
  - 3    Sign restrictions for positive structural shocks.
- Page references:
  - Figures and Tables span pages indicated (e.g., Figures on pages 12–15; Tables on page 12).

### Introduction: background and motivation
- Historical identification in SVAR:
  - Early SVAR literature identified policy and macroeconomic shocks via parametric (predominantly zero) contemporaneous restrictions on immediate responses of macro aggregates.
  - Justifications for contemporaneous zero restrictions include variable sluggishness and observation delays.
- Move toward sign restrictions:
  - Sign restrictions stipulate only the direction of a shock’s impact on macro aggregates, remaining agnostic about specific structural coefficients.
  - Procedures using sign restrictions are often called “agnostic” identification procedures.
- Empirical context and prior findings:
  - Prominent applications include Astveit et al. (2017), Canova and De Nicoló (2002), Elekdag and Han (2015), Faust (1998), and Uhlig (2005).
  - Uhlig (2005) finds contractionary monetary policy shocks do not exhibit a discernible effect on real GDP, consistent with monetary neutrality in his results.
  - Refinements (e.g., Arias et al. 2019) and external-instrument identification (Gertler and Karadi 2015) tend to reaffirm a short-term effect of monetary policy on GDP consistent with price rigidities à la Calvo (1983).

### Problem with agnostic sign-restriction procedures
- Low acceptance rates:
  - Agnostic sign-restriction procedures typically yield very low acceptance rates (proportion of simulated structural shocks that satisfy imposed sign restrictions).
  - Examples:
    - Fisher and Huh (2020) report acceptance rates of less than 1 percent using their algorithm.
    - Arias et al. (2018) also find low acceptance rates.
    - The algorithm of Ouliaris and Pagan (2016) yields very low acceptance rates when applied to Cho and Moreno (2003).

### Proposed contribution: “quasi-agnostic” identification procedure
- Core idea:
  - An iterative grid-search-like strategy that employs increasingly refined identification methods across iterations to identify the plausible set of structural parameters satisfying imposed sign restrictions.
  - Termed “quasi-agnostic” because each iteration narrows the likely interval of plausible structural parameters using information from earlier iterations.
- Key claim:
  - The strategy can significantly enhance acceptance rates, revealing a larger set of economic structures consistent with priors on the sign of responses of macro variables to structural shocks.

### Paper structure (as presented)
- Section I: lays out a three-variable recursive SVAR to estimate a conventional New-Keynesian small macro model.
- Section II: critiques reliance on a unique set of parametric restrictions (e.g., zero contemporaneous restrictions) and adopts the methodology proposed by Ouliaris and Pagan (2016) for imposing agnostic sign restrictions on impulse response functions at various time horizons.

*Source: wpiea2025162-source-pdf - References10*

### Section III explores different alternative “quasi-agnostic” identification procedures, proposing

### wpiea2025162-source-pdf - Section III explores different alternative “quasi-agnostic” identification procedures, proposing

### The baseline model
- Model: SVAR for a small macro model with three variables: quarterly output gap (o), inflation (π), and average policy rate (r) for the US economy between 1982Q1 and 2000Q1.
- Structural form:
  - A0 Yt = A1 Yt−1 + A2 Yt−2 + B0 εt (equation (1))
  - Y = [o, π, r]′
  - A0 is lower triangular with ones on the diagonal.
  - B0 is diagonal with standard errors of shocks on the diagonal.
  - εt ∼ N(0, I3)
- Parameter counts and identification:
  - There are 27 parameters to be estimated.
  - Only 24 parameters can be estimated from the underlying VAR Yt = V1 Yt−1 + V2 Yt−2 + et (equation (2)).
  - Underlying VAR yields 24 parameters: 9 in V̂1, 9 in V̂2, and 6 in Σ̂ (symmetric).
  - Hence 3 zero contemporaneous restrictions are imposed on A0 to obtain exact identification.
- Example normalized/unnormalized note:
  - A SVAR with standard deviations along the diagonal of B0 and ones along the diagonal of A0 is called normalized. A SVAR with ones along the diagonal of B0 and parameters to be estimated along the diagonal of A0 is called unnormalized. The two representations are equivalent.
- Estimation approach:
  - Because the SVAR defined with restrictions in A0 is exactly identified, estimation is performed via 2SLS using instrumental variables (chosen for ease when later contemporaneous restrictions differ from zero).

### Diagnostics, impulse responses, and the price puzzle
- Model selection and diagnostics:
  - Lag selection: Information criteria (FPE, AIC, BIC, HQIC) and likelihood-ratio tests suggest selecting lag order 2 to minimize overfitting.
  - Endogeneity: Wald statistics for Granger causality suggest variables can be treated as endogenous.
  - Stability: All eigenvalues lie in the unit circle.
- Impulse response function (IRF) findings under recursive A0 (equation (3)):
  - Demand shock (one standard deviation):
    - Contemporaneous impact on output gap equal in magnitude to the shock; persistence of about ten periods.
    - Modest positive contemporaneous effect on inflation.
    - Small immediate increase in policy rate; policy rate subsequently rises over time in response to expanding output gap.
  - Supply shock (one standard deviation):
    - Zero contemporaneous impact on output gap (due to zero restriction) but increases to about ten basis points in about eight periods.
    - Positive contemporaneous response in inflation.
    - Contemporaneous increase in policy rate reaching about thirty basis points within eight periods before abating.
  - Monetary shock (one standard deviation):
    - Zero contemporaneous impact on output gap and inflation (due to zero restrictions).
    - Output gap response accelerates to about eighteen basis points in few periods.
    - Inflation response is positive, yielding a “price puzzle”.
- Interpretation and standard corrective approaches to the price puzzle:
  - (i) Incorporate additional endogenous variables (e.g., a monetary aggregate, exchange rate, inflation expectations).
  - (ii) Redefine endogenous variables.
  - (iii) Employ non-recursive, less strict identification procedures.
  - (iv) Introduce latent variables (state-space, Kalman filter, MLE).
- Trade-offs noted:
  - Adding endogenous variables increases parameters and shocks, requiring additional identifying restrictions; number of additional identifying restrictions required increases by (d(d−1)/2 + n d) where n is number of original endogenous variables and d is number added.
  - Latent-variable approaches typically require state-space and Kalman-filter estimation, beyond scope.

### Imposing agnostic sign restrictions (SRC procedure)
- SRC setup:
  - Unrestricted structural matrix A0 = [1 a012 a013; a021 1 a023; a031 a032 1].
  - SRC draws random coefficients aij = θ/(1−abs(θ)) with θ ∼ U(−1,1).
  - Simulate 1,200 combinations of parameters a012, a013, a023.
  - For each draw: estimate B0 and remaining A0 parameters via 2SLS; estimate A1 and A2 via OLS.
  - Compare signs of contemporaneous IRF (C0 = A−1 0 B0) with priors on signs reported in Table 3; discard IRFs inconsistent with priors.
  - Acceptance rate = number of combinations consistent with priors / total draws.
- SRC results (sign restrictions at horizon zero):
  - Price puzzle resolved:
    - Demand shock: inflation impact between zero and twenty basis points; interest rate impact between thirty and seventy basis points.
    - Supply shock: inflation impact between twenty and seventy basis points; interest rate impact between zero and twenty-five basis points.
    - Monetary shock: negative impact on output gap between zero and thirty basis points; negative impact on inflation between zero and seventy basis points.
  - Low acceptance rate observed.
  - Sign restrictions solve structural identification but not model identification: each IRF corresponds to a unique set of structural parameters; summary statistics (median, percentiles, average IRFs) do not solve the model identification problem.
  - Sign restrictions permit variation in standard deviations of structural shocks across models, affecting contemporaneous responses (can be normalized).

### From agnostic to “quasi-agnostic” sign restrictions
- Motivation:
  - Completely agnostic identification (SRC) yields high standard deviation of possible structural parameters, causing many draws to fall outside plausible ranges and producing low acceptance rates.
- Quasi-agnostic concept:
  - Adaptive, grid-search-like refinement of identification procedures informed by ex-post results from earlier draws; progressively trim plausible parameter ranges to increase acceptance rates.
- Illustrative bivariate supply-demand example (no lags):
  - Model equations: Qt = α pt + es; Qt = β pt + ed (equation (4)).
  - SVAR A0 Yt = B0 εt with A0 = [1 a012; 1 a022] and B0 = diag(σs, σd) (equation (5)).
  - Priors matrix P = [ +  +;  + − ] (equation (6)): demand shock increases quantities and prices; supply shock increases quantities and decreases prices.
  - Contemporaneous IRF C0 = A−1 0 B0 yields expressions (equation (7)); to satisfy P three restrictions implied on structural parameters:
    - determinant of A0 negative: (â022 − â012) < 0,
    - â012 > 0,
    - â022 < 0.
  - Searching across (−∞,+∞) for parameters (as SRC does) leads to many implausible values; restricting search to implied sign intervals should increase acceptance.
- Empirical quasi-agnostic implementation (three-step refinement illustrated in Figure 5):
  - First pass: fully agnostic SRC draws; observation: all successful parameters positive; means in order a012 smallest, then a023, then a013; a012 has very small standard deviation; a013 and a023 have larger, skewed distributions.
  - Second pass: abs(SRC) — take absolute values of SRC draws restricting parameters to (0, +∞); yields higher acceptance rates at all horizons.
  - Third pass: draws from Normal(0,1); broadly comparable to SRC.
  - Final pass: calibrate random draws exploiting means and standard deviations observed in earlier rounds; draw parameters concentrated around observed means — acceptance rate increases the more parameters are drawn around these means.
  - Observed behavior across horizons: larger plausible a013 parameters tend to disappear when IRFs are restricted at higher time horizons.
- Outcome:
  - Quasi-agnostic, progressively refined procedures can significantly enhance acceptance rates and reveal a larger set of economic structures consistent with sign priors.
- Limitations:
  - For higher-dimensional SVARs, using only ex-post information to restrict plausible structural parameters does not guarantee identification of all possible parameters consistent with sign priors.
  - The relative role of model misspecification versus identification-procedure limitations in producing low acceptance rates remains open.
  - For bivariate SVAR a closed-form solution can impose ex ante restrictions consistent ex post with sign priors; higher dimensions lack such general ex ante characterizations.

### Conclusions and avenues for future research
- Empirical findings:
  - Identified shocks exert durable influence on output gap and interest rates; effects on inflation are generally less persistent.
  - Recursive SVAR produces a price puzzle; imposing agnostic sign restrictions (SRC) at horizon zero resolves the price puzzle but yields low acceptance rates.
  - Quasi-agnostic refinements (progressive, data-informed narrowing of parameter draws) increase acceptance rates and uncover a wider set of plausible structures.
- Methodological implications:
  - Sign restrictions alone do not resolve model identification; normalization and additional restrictions or selection criteria are needed when choosing among plausible structural parameterizations.
  - Quasi-agnostic procedures exploit ex-post insights to improve identification procedure performance, but do not replace the need for ex ante restrictions, especially in higher dimensions.
- Recommendations for future research:
  - Identify ex ante restrictions on structural parameters for higher-dimensional SVARs to better link model and structural identification.
  - Quantify the relative importance of model misspecification versus identification-procedure limitations in determining acceptance rates.

*Source: wpiea2025162-source-pdf - Section III explores different alternative “quasi-agnostic” identification procedures, proposing*

### References

### Sign Restrictions with a New-Keynesian Macro Model: Results From a “Quasi-Agnnostic” Identification Procedure — References and Appendix

### References
- Arias, J.A., J.F. Rubio-Ramírez, and D.F. Waggoner (2018). “Inference Based on Structural Vector Autoregressions Identified With Sign and Zero Restrictions: Theory and Applications”. In: Econometrica. Vol. 86.2, pp. 685–720. doi:https://doi.org/10.3982/ECTA14468.
- Arias, J.A., D. Caldara, and J.F. Rubio-Ramírez (2019). “The systematic component of monetary policy in SVARs: An agnostic identification procedure”. In: Journal of Monetary Economics. Vol. 101, pp. 1–13. doi:https://doi.org/10.1016/j.jmoneco.2018.07.011.
- Astveit, K.A., G.J. Natvik, and S. Sola (2017). “Economic Uncertainty and the Effectiveness of Monetary Policy”. In: Journal of International Money Finance. Vol. 101, pp. 50–67. doi:https://doi.org/10.1016/j.jimonfin.2017.05.003.
- Calvo, G.A. (1983). “Staggered prices in a utility-maximizing framework”. In: Journal of Monetary Economics. Vol. 12.3, pp. 383–398. doi:https://doi.org/10.1016/0304-3932(83)90060-0.
- Canova, F. and G. De Nicoló (2002). “Monetary disturbances matter for business fluctuations in the G-7”. In: Journal of Monetary Economics. Vol. 49.6, pp. 1131–1159. doi:https://doi.org/10.1016/S0304-3932(02)00145-9.
- Cho, S. and A. Moreno (2003). “A Small-Sample Study of the New-Keynesian Macro Model”. In: Journal of Money, Credit and Banking. Vol. 38.6, pp. 1461–1481. doi:https://doi.org/10.1353/mcb.2006.0078.
- Elekdag, S. and F. Han (2015). “What Drives Credit Growth in Emerging Asia?” In: Journal of Asian Economics. Vol. 38, pp. 1–13. doi:https://doi.org/10.1016/j.asieco.2015.03.001.
- Faust, J. (1998). “The robustness of identified VAR conclusions about money”. In: Carnegie-Rochester Conference Series on Public Policy. Vol. 49.1, pp. 207–244. doi:https://doi.org/10.1016/S0167-2231(99)00009-3.
- Fisher, L.A. and H.S. Huh (2020). “Combining sign and parametric restrictions in SVARs by utilising Givens rotations”. In: Studies in Nonlinear Dynamics & Econometrics. Vol. 24.3, pp. 1–19. doi:https://doi.org/10.1515/snde-2018-0104.
- Fry, R. and A. Pagan (2011). “Sign Restrictions in Structural Vector Autoregressions: A Critical Review”. In: Journal of Economic Literature. Vol. 49.4, pp. 938–960.
- Gertler, M. and P. Karadi (2015). “Monetary Policy Surprises, Credit Costs, and Economic Activity”. In: American Economic Journal: Macroeconomics. Vol. 7.1, pp. 44–76. doi:https://doi.org/10.1257/mac.20130329.
- Ouliaris, S. and A. Pagan (2016). “A Method for Working with Sign Restrictions in Structural Equation Modelling”. In: Oxford Bulletin of Economics and Statistics. Vol. 79.5, pp. 605–622. doi:https://doi.org/10.1111/obes.12137.
- Uhlig, H. (2005). “What are the effects of monetary policy on output? Results from an agnostic identification procedure”. In: Journal of Monetary Economics. Vol. 52.2, pp. 381–419. doi:https://doi.org/10.1016/j.jmoneco.2004.05.007.

### Appendix: Tables and Figures — Key items and exact numeric values
- Table 1: Lag Selection
  - Horizon row entries (preserve formatting):
    - 0-3 78.58 10.899 10.902 10.941 10.999
    - 1-1 77.03 403.09 0.000 0.044 5.401 5.554 5.786
    - 2-1 48.47 57.12 *0.000 *0.025 *4.842 *5.110 *5.517
    - 3- -141.31 14.31 0.112 0.027 4.895 5.277 5.858
    - 4-134.00 14.63 0.102 0.029 4.943 5.440 6.196
    - 5-128.23 11.55 0.240 0.032 5.035 5.647 6.577
    - 6-124.40 7.65 0.569 0.037 5.183 5.910 7.014
    - 7-118.89 11.02 0.275 0.042 5.283 6.125 7.403
  - Source: Authors’ calculations.

- Table 2: Granger causality (columns: EquationVariable, W, d.f., Pval)
  - π1.010   2   0.604
  - o    r4.625   2   0.099
  - all8.148   2   0.086
  - π3.630   2   0.163
  - πr4.084   2   0.130
  - all11.360   2   0.023
  - π4.066   2   0.131
  - rr6.399   2   0.041
  - all8.088   2   0.088
  - Source: Authors’ calculations.

- Figure 1: Stability condition
  - Plot axes labeled Imaginary and Real with ticks at -1, -.5, 0, .5, 1 and Real axis showing -1 -.5 0.5 1; caption: Roots of the companion matrix.
  - Source: Authors’ calculations.

- Figure 2: Structural impulse responses - recursive model
  - Panels shown for Output Gap, Demand Shock, Supply Shock, Monetary Shock, Inflation, Interest Rate with Horizon labels: 0 5 10 15 20.
  - Source: Authors’ calculations.

- Table 3: Sign restrictions for positive structural shocks (contemporaneous signs)
  - Demand Shock: o +
  - Supply Shock: π + +
  - Monetary Shock: r + ++
  - (As presented: o+ − − ; π++ − − ; r +++)

- Figure 3: Contemporaneous sign restrictions on IRFs using the SRC procedure
  - Note: Acceptance rate of 1.25 percent, 1,200 simulations, and sign restrictions imposed at horizon 0.
  - Source: Authors’ calculations.

- Figure 4: Ex ante restrictions on structural parameters
  - (Figure presented without numeric table entries in source.)

- Figure 5: Distribution of restricted parameters using alternative identification procedures
  - AR values and labels preserved as presented:
    - AR = 1.25, Horizon = 0, SRC
    - AR = 0.42, Horizon = 1
    - AR = 0.17, Horizon = 2
    - AR = 9.25, abs(SRC)
    - AR = 3.75
    - AR = 1.92
    - AR = 2.08
    - AR = 1.17
    - AR = 0.83
    - AR = 21.92, N(a,b)
    - AR = 13.75
    - AR = 8.58
  - Values of A[1,2], A[1,3], Values of A[2,3] plotted across panels.
  - Source: Authors’ calculations.

- Figure 6: Results from “quasi agnostic” procedure
  - Note: Acceptance rate of 8.58 percent, 1,200 simulations, and sign restrictions imposed at horizon 2.
  - Panels include Output Gap, Demand Shock, Supply Shock, Monetary Shock, Inflation, Interest Rate with Horizon labels: 0 5 10 15 20.
  - Source: Authors’ calculations.

- Additional numerical and methodological details preserved exactly from figure notes:
  - Acceptance rate values: 1.25 percent and 8.58 percent.
  - Number of simulations: 1,200.
  - Sign restriction horizons explicitly stated: horizon 0 and horizon 2.

*Sign Restrictions with a New-Keynesian Macro Model: Results From a “Quasi-Agnostic” Identification Procedure — Working Paper No. WP/2025/162*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025162-source-pdf.pdf_