## _wp16231

## Source details

**Canonical URL:** [_wp16231](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp16231.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp16231.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp16231.pdf.json)

---

### 1. Parameter regions for different priors
- Content unit provided: heading only — "1. Parameter regions for different priors".
- Page reference: 13.
- No substantive findings, figures, tables, or policy recommendations included for this section in the supplied content.

### 2. Model properties for admissible regions

#### I. INTRODUCTION
- Introduces “system priors” as a tool to incorporate prior knowledge into an economic model; originally devised by Andrle and Benes (2013).
- Key properties of system priors:
  - Are priors about high-level model properties (functions of individual parameters) rather than individual parameters themselves.
  - Impose restrictions on the joint prior distribution of individual model coefficients because many features of a model are non-trivial functions of its parameters.
  - Can express beliefs about impulse-response characteristics, frequency-domain properties, or other computable model properties.
  - Apply beyond DSGE models and extend to state-space models, Bayesian VARs, linear and non-linear models.
  - Implementation updates initial marginal priors using both the conventional likelihood and the information contained in system priors, producing a composite prior.
  - Can be interpreted as an additional penalty in a penalized-likelihood objective (analogy to “dummy observation” priors).
  - Do not require linearity, Gaussian shocks, or conjugate priors; computational cost is higher but tractable with modern parallel architectures.

#### II. SYSTEM PRIORS — formal structure and computation
- Bayesian formulation (exact expression preserved):
  - Start with marginal priors p_m(θ).
  - Define a statistic r = h(θ; M).
  - Introduce a system prior p_s(h(θ); M).
  - Posterior kernel with system priors:
    - p(θ|Y; M) ∝ L(Y|θ; M) × [ p_s(h(θ); M) × p_m(θ) ].
- Interpretation and implementation notes:
  - System prior acts as an “artificial likelihood” for r, yielding a composite prior after updating p_m with p_s, then updating with the conventional likelihood.
  - From a non-Bayesian view, equivalent to a penalized-likelihood with two penalties (marginal prior and system-prior penalties).
  - Existing Bayesian code requires minimal changes: for each draw θ_j evaluate three components instead of two, with the system-prior component added.
  - The inverse Hessian at the posterior mode can be used for (non-Bayesian) inference or as input to MCMC.
  - Prior predictive analysis (evaluating composite prior without the conventional likelihood) is recommended to check intended model properties.
  - System priors need not have closed-form solutions and can be incorporated via simulation-based methods.

#### III. EXAMPLE – SYSTEM PRIORS FOR AN AR(2) PROCESS
- Model setup (exact):
  - Zero-mean AR(2) process: y_t = φ1 y_{t-1} + φ2 y_{t-2} + ε_t,  ε_t ~ N(0, σ_ε^2).
- Problems with independent marginal priors:
  - Independent Gaussian marginals φ1 ~ N(0, σ_{φ1}^2), φ2 ~ N(0, σ_{φ2}^2) can permit implausible dynamics (wild oscillations, non-stationarity) and do not efficiently encode cyclical beliefs.
  - Polar re-parameterization (amplitude A and periodicity τ) can be infeasible for complex models and still insufficiently specific.
- Spectral-based system-prior construction (exact expressions preserved):
  - Spectral density of y_t:
    - S_y(w) = σ_ε^2 / [2π(1 + φ1^2 + φ2^2 + 2φ1(φ2 - 1) cos(w) - 2φ2 cos(2w))],  where w ∈ [0, π).
    - Note: σ_ε^2 sets spectrum level but not shape.
  - Business-cycle variance ratio r (exact definition):
    - r = ∫_{a}^{b} S_y(w) dw  /  ∫ S_y(w) dw,  where (a,b) are business-cycle frequencies.
    - r ∈ [0,1]; independent of σ_ε^2.
  - Two illustrative system-prior specifications:
    - Hard constraint: at least 60% of variance of y_t originates from business-cycle frequencies (i.e., r ≥ 60%).
    - Statistical prior: r ∼ Be(15,5).
- Stationarity conditions for AR(2) (exact):
  - φ1 + φ2 < 1;  φ2 − φ1 < 1;  |φ2| < 1.
- Computational approach in the example:
  - Characterize the composite prior by simulating with the conventional likelihood omitted (prior-predictive analysis).
  - Implementation methods:
    - Rejection sampling with normally distributed marginal priors as proposal for the hard constraint (r ≥ 60%).
    - Sequential Monte Carlo (SMC) sampling for the Beta system prior r ∼ Be(15,5); SMC preferred for higher-dimensional models over Metropolis-Hastings.
  - Empirical/graphical findings (as reported):
    - Stationarity alone does not meaningfully restrict cyclical behavior.
    - The spectral-based system prior yields parameter regions producing economically-plausible cyclical spectra and impulse responses.
- Practical observations:
  - The spectral system prior is informative and leads to non-Normal joint parameter distributions.
  - System priors are flexible: one could instead impose priors directly on impulse-response functions.
  - Implementation requires users to provide their own prior-restriction functions; once interfaced, these plug into existing Bayesian toolkits.

#### IV. CONCLUSION — implications and recommendations
- System priors provide a top-down, flexible approach to elicit priors about computable, high-level model characteristics (impulse-responses, frequency-response functions, long-run features).
- They enable informative and economically-meaningful priors that restrict parameter estimation to regions consistent with substantive prior beliefs (illustrated by imposing that more than 60% of dynamics come from business-cycle frequencies).
- System priors:
  - Are general and extend to state-space models, Bayesian VARs, and other econometric models.
  - Are straightforward to incorporate into existing Bayesian toolkits with modest computational overhead.
  - Allow transparent, interpretable prior specifications that analysts and audiences can evaluate and contest.
- Recommended practice:
  - Conduct prior-predictive analysis of model properties to detect unintended consequences of chosen system priors.
  - Prefer simulation-based algorithms (e.g., SMC) for complex models where system priors are applied.

### Appendix: Pseudo code for the posterior kernel
- Restated formula (exact):
  - p(θ|Y;Μ) ∝ L(Y|θ;Μ) ×[ p_s(h(θ);M)×  p_m(θ)].
- Implementation notes and required user inputs:
  - Crucial input: user-defined function logsprior_user_fun that evaluates the system priors for a given coefficient vector θ.
  - Function object can be a callable that “remembers” data or previous state.
  - Efficient evaluation recommends solving the model once per θ or evaluating components in parallel.
  - Switches permit:
    - Bayesian estimation with System Priors,
    - Bayesian estimation without system priors,
    - maximum likelihood estimation with no explicit priors,
    - investigation of the compound prior by switching off the likelihood component.
- Exact pseudo-code structure (preserved lines and structure):
  - [crit]  = function(theta, Model, Data, logsprior_user_fun, ... ) 
  - BEGIN
  - IF (do_mprior == TRUE)
    - Log_mprior = evalMarginalPriors(theta, hyperParameters);
    - ELSE Log_mprior = 0;
  - IF (do_sprior == TRUE)
    - Log_sprior = call(@logsprior_user_fun(theta, Model, Data);
    - ELSE Log_sprior = 0;
  - IF (do_loglik == TRUE)
    - Log_lik  = evalLoglikelihood(theta, Data, Model);
    - ELSE Log_lik  = 0;
  - crit = Log_lik + Log_sprior + Log_mprior
  - END

*Source: _wp16231 (pages 13–14)*

### 1. Parameter regions for different priors _______________________________________ 13

### 1. Parameter regions for different priors

### Overview
- Content unit provided: heading only — "1. Parameter regions for different priors".
- Page reference in supplied content: 13.
- No substantive findings, figures, tables, or policy recommendations were included in the supplied content for this section.

*Source: _wp16231 - 1. Parameter regions for different priors (page 13)*

### 2. Model properties for admissible regions _____________________________________ 14

### _wp16231 - 2. Model properties for admissible regions _____________________________________ 14

### I. INTRODUCTION
- Introduces “system priors” as a tool to incorporate prior knowledge into an economic model; originally devised by Andrle and Benes (2013).
- System priors:
  - Are priors about high-level model properties (functions of individual parameters) rather than individual parameters themselves.
  - Impose restrictions on the joint prior distribution of individual model coefficients because many features of a model are non-trivial functions of its parameters.
  - Can express beliefs about impulse-response characteristics, frequency-domain properties, or other computable model properties.
- Key conceptual points:
  - System priors are applicable beyond DSGE models and extend to state-space models, Bayesian VARs, linear and non-linear models.
  - Implementation updates initial marginal priors using both the conventional likelihood and the information contained in system priors, producing a composite prior.
  - System priors can be interpreted as an additional penalty in a penalized-likelihood objective (analogy to “dummy observation” priors).
  - They do not require linearity, Gaussian shocks, or conjugate priors; computational cost is higher but tractable with modern parallel architectures.

### II. SYSTEM PRIORS — formal structure and computation
- Bayesian formulation with system priors:
  - Start with marginal priors p_m(θ) (often independent marginals).
  - Define a statistic r = h(θ; M) (a computable function of parameters describing system properties).
  - Introduce a system prior p_s(h(θ); M).
  - Posterior kernel with system priors:
    - p(θ|Y; M) ∝ L(Y|θ; M) × [ p_s(h(θ); M) × p_m(θ) ].
- Interpretation:
  - System prior acts as an “artificial likelihood” for r, yielding a composite prior after updating p_m with p_s, then updating with the conventional likelihood.
  - From a non-Bayesian view, this is a penalized-likelihood with two penalties (marginal prior and system-prior penalties).
- Practical computation:
  - Existing Bayesian code requires minimal changes: for each draw θ_j evaluate three components instead of two, with the system-prior component added.
  - The inverse Hessian at the posterior mode can be used for (non-Bayesian) inference or as input to MCMC.
  - Prior predictive analysis (evaluating composite prior without the conventional likelihood) is recommended to check intended model properties.
  - System priors need not have closed-form solutions and can be incorporated via simulation-based methods.

### III. EXAMPLE – SYSTEM PRIORS FOR AN AR(2) PROCESS
- Model setup:
  - Zero-mean AR(2) process: y_t = φ1 y_{t-1} + φ2 y_{t-2} + ε_t,  ε_t ~ N(0, σ_ε^2).
- Problem with independent marginal priors:
  - Independent Gaussian marginals φ1 ~ N(0, σ_{φ1}^2), φ2 ~ N(0, σ_{φ2}^2) can permit implausible dynamics (wild oscillations, non-stationarity) and do not efficiently encode cyclical beliefs.
  - Polar re-parameterization (amplitude A and periodicity τ) is one approach but not generally feasible for complex models and can still be insufficiently specific.
- System-prior application via spectral restriction:
  - Spectral density of y_t:
    - S_y(w) = σ_ε^2 / [2π(1 + φ1^2 + φ2^2 + 2φ1(φ2 - 1) cos(w) - 2φ2 cos(2w))],  where w ∈ [0, π).
    - Note: σ_ε^2 sets spectrum level but not shape.
  - Define business-cycle variance ratio r:
    - r = ∫_{a}^{b} S_y(w) dw  /  ∫ S_y(w) dw,  where (a,b) are business-cycle frequencies.
    - r ∈ [0,1]; independent of σ_ε^2.
  - Two illustrative system-prior specifications:
    - Hard constraint: at least 60% of variance of y_t originates from business-cycle frequencies (i.e., r ≥ 60%).
    - Statistical prior: r ∼ Be(15,5) (Beta distribution on [0,1] placing mass on high business-cycle shares).
- Stationarity conditions for AR(2):
  - φ1 + φ2 < 1;  φ2 − φ1 < 1;  |φ2| < 1.
- Computational approach in the example:
  - To characterize the composite prior (marginal priors updated by system priors) simulate with the conventional likelihood omitted.
  - Implementation methods:
    - Rejection sampling with normally distributed marginal priors as proposal for the hard constraint (r ≥ 60%).
    - Sequential Monte Carlo (SMC) sampling for the Beta system prior r ∼ Be(15,5); SMC preferred for higher-dimensional models over Metropolis-Hastings.
  - Kernel-density depictions and region plots (Figure 1) show:
    - Upper-left: φ1, φ2 ~ N(0,2) independent marginal priors.
    - Upper-right: same marginals restricted to stationarity region.
    - Bottom-left: stationarity + r ≥ 60%.
    - Bottom-right: stationarity + r ∼ Be(15,5).
  - Prior-implied model behavior (Figure 2):
    - Stationarity alone does not meaningfully restrict cyclical behavior.
    - The spectral-based system prior yields parameter regions producing economically-plausible cyclical spectra and impulse responses.
- Practical observations:
  - The spectral system prior is informative and leads to non-Normal joint parameter distributions.
  - System priors are flexible: one could instead impose priors directly on impulse-response functions.
  - Implementation requires users to provide their own prior-restriction functions; once interfaced, these plug into existing Bayesian toolkits.

### IV. CONCLUSION — implications and recommendations
- System priors provide a top-down, flexible approach to elicit priors about computable, high-level model characteristics (impulse-responses, frequency-response functions, long-run features).
- They enable informative and economically-meaningful priors that restrict parameter estimation to regions consistent with substantive prior beliefs (illustrated by imposing that more than 60% of dynamics come from business-cycle frequencies).
- System priors:
  - Are general and extend to state-space models, Bayesian VARs, and other econometric models.
  - Are straightforward to incorporate into existing Bayesian toolkits with modest computational overhead.
  - Allow transparent, interpretable prior specifications that analysts and audiences can evaluate and contest.
- Recommended practice:
  - Conduct prior-predictive analysis of model properties to detect unintended consequences of chosen system priors.
  - Prefer simulation-based algorithms (e.g., SMC) for complex models where system priors are applied.

*Source: _wp16231 - 2. Model properties for admissible regions*

### REFERENCES

### _wp16231 - REFERENCES

### References list
- Andrle, M., and Benes, J. (2013): “System Priors: Formulating Priors about DSGE Models System  Properties,” Working   Paper   WP/13/257,   International   Monetary   Fund, Washington DC.
- Andrle, M., and Plašil, M. (2016): “System Priors for VARs,” mimeo.
- Clark, P. (1987): “The Cyclical Comovement of U.S. Economic Activity,” Quarterly Journal of Economics, 102, pp. 797-814.
- Geweke, J. (2010): Complete and Incomplete Econometric Models, Princeton University Press.
- Giannone, D., M. Lenza, and G.E. Primiceri (2014): “Priors for the Long Run”, mimeo
- Harvey, A.C., and Koopman, S.J. (1997): Multivariate structural time series models, In: Heij, C., et al. (Eds.), System Dynamics in Economic and Financial Models. Wiley, Chichester.
- Herbst, E., and Schorfheide, F. (2014): “Sequential Monte Carlo Sampling for DSGE Models,” Journal of Applied Econometrics, Vol. 29, Issue 7, pp.1073-1098.
- Kuttner, K. (1994): “Estimating potential output as a latent variable,” Journal of Business and Economic Statistics, 12, 3, 361-368.
- Planas, C., Rossi, A., and Fiorentini, G. (2008): “Bayesian Analysis of the Output Gap,” Journal of Business & Economic Statistics, 26 (1), pp. 18-32.
- Sims, Ch. (2005): “Conjugate Dummy Observations for VARs,” Princeton University mimeo.
- Sims, Ch. And T. Zha (1998): “Bayesian Methods for Dynamic Multivariate Models”, International Economic Review, 39, 949—968
- Theil, H., and Goldberger, A. (1961): “On Pure and Mixed Statistical Estimation in Economics,” International Economic Review, Vol. 2, No. 1, pp. 65-78.
- Villani, M. (2009): “Steady-State Priors for Vector Autoregressions”, Journal of Applied Econometrics, Vol 24, 630—650
- Watson, M. (1986): “Univariate detrending methods with stochastic trends,” Journal of Monetary Economics, 18, pp. 49-75.

### Appendix: Pseudo code for the posterior kernel
- Restated formula (exact):
  - p(θ|Y;Μ) ∝ L(Y|θ;Μ) ×[ p
s
(h(θ);M)×  p
m
(θ)].
- Purpose:
  - Provide simplified pseudo-code to implement computations to evaluate formula (2) in the main text.
  - Function evaluates all three components of p(θ|Y;Μ) given inputs: vector of coefficients θ, Model (already solved for θ or to be solved for θ), observed Data required for evaluation of the log-likelihood and possibly for system priors.
- Crucial input:
  - User-defined function logsprior_user_fun that evaluates the system priors for a given coefficient vector θ.
  - The function handle or function object needs to follow a pre-specified application programming interface (API).
  - Note: footnote 13 references function objects; exact text: "Function object is an object that can be called like a function, yet can do more, for instance “remember” a lot of data, its previous state, etc."
- Efficiency note:
  - Evaluation can be efficient by solving the model with a new vector of coefficients only once or evaluating all three components in parallel.
- Switches:
  - Allow switching between:
    - Bayesian estimation with System Priors,
    - Bayesian estimation without system priors,
    - maximum likelihood estimation with no explicit priors,
    - investigation of the compound prior by switching off the likelihood component.
  - “do_xx” switches are included in the function (or function object) though not shown explicitly as inputs.

### Pseudo-code (exact structure and lines)
- Function signature and begin:
  - [crit]  = function(theta, Model, Data, logsprior_user_fun, ... ) 
  - BEGIN
- Marginal priors p_m(θ):
  - IF (do_mprior == TRUE)
    - Log_mprior = evalMarginalPriors(theta, hyperParameters);
  - ELSE
    - Log_mprior = 0;
  - END
- SYSTEM priors p_s(h(θ);M):
  - IF (do_sprior == TRUE)
    - Log_sprior = call(@logsprior_user_fun(theta, Model, Data);
  - ELSE
    - Log_sprior = 0;
  - END
- Likelihood L(Y|θ;Μ):
  - IF (do_loglik == TRUE)
    - Log_lik  = evalLoglikelihood(theta, Data, Model);
  - ELSE
    - Log_lik  = 0;
  - END
- Assemble and return posterior value:
  - crit = Log_lik + Log_sprior + Log_mprior
  - END

*Source: _wp16231 - REFERENCES*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp16231.pdf_