## _wp13257

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---

### III. Introduction — purpose and contribution
- Proposes a novel approach to elicit individual parameter priors using views on dynamic models’ properties — termed "system priors".
- Motivation:
  - Priors about many individual structural model parameters are difficult to formulate.
  - Independent marginal priors can imply infeasible or unintended prior distributions for model properties.
  - Direct specification of priors on model properties (impulse response functions, conditional correlations, sacrifice ratio) can be more transparent and intuitive.
- Main methodological claim:
  - Change the philosophy of formulating priors from bottom-up marginal parameter priors to top-down system priors that impose stochastic restrictions on combinations of parameters.
- Implementation note:
  - Technically a straightforward extension of standard Bayesian methods; translates to adding an evaluation of system priors on top of marginal priors and the likelihood in Monte Carlo estimation.

### III. Benefits and implications of system priors
- Benefits:
  - Introduce complex dependence among structural parameter priors.
  - Allow imposition of intuitive and economically meaningful priors about aggregate model behavior.
  - Avoid unintended consequences on model properties often associated with independent marginal priors.
  - Facilitate model comparison across differently structured models by using higher-level, more comparable priors.
  - Make explicit and testable many prior beliefs researchers already implicitly hold (e.g., horizon of monetary policy, conditional cross-correlations).
- Practical considerations:
  - Main cost is formulating relevant, transparent, and economically meaningful beliefs about model behavior.
  - Prior-predictive analysis is emphasized (following Geweke(2010)) to assess whether prior or data drive estimates of system properties.
  - Reporting priors for system properties (e.g., impulse response functions) is rare but recommended.

### II.A Motivation — problems with bottom-up priors and rationale for system priors
- Problems with standard priors:
  - Bottom-up independent priors can produce unintuitive aggregate properties due to nonlinear mapping from parameters to system properties.
  - Some individual parameters (e.g., Calvo price rigidity) may not map cleanly to comparable economic concepts across models, reducing transferability of parameter priors.
  - Marginal prior and posterior plots of structural coefficients are insufficient to reveal prior implications for system properties.
- Role of system priors:
  - Tie together parameter combinations that satisfy stochastic constraints defined by the analyst’s higher-level beliefs.
  - Allow direct expression of beliefs about such questions as the maximum length of inflation deviation after a demand shock.
  - Act as a non-dogmatic analog to imposing conditions like saddle-path (Blanchard-Kahn) stability, by introducing cross-dependence rather than hard truncation.

### II.B Candidates for system priors for DSGE models
- Any model-implied statistic can be used as a system prior. Selective candidates include:
  - Steady-state values of model variables (levels or growth rates).
  - Conditional or unconditional moments (e.g., cross-correlations of investment and private consumption after demand shocks).
  - Prominent policy scenarios (counter-factuals) such as:
    - (i) permanent disinflation, and
    - (ii) anticipated delayed response of monetary policy for several periods.
  - Characteristics of impulse-response functions (peak impacts, duration of recessions, horizon of policy effectiveness).
  - Frequency response function and spectral characteristics (signal-to-noise ratios, power of shocks at particular frequencies, coherence between macro variables).
- Construction guidance:
  - System priors can be constructed in relative or absolute terms but should be robust to trivial changes in scaling.
  - Working with a single summary statistic is often easier (e.g., sacrifice ratio), though vector forms (impulse-response paths) are possible.

### II.C Examples of system priors — focus on sacrifice-ratio priors
- Sacrifice ratio definition and motivation:
  - Measures cumulative loss of output after a permanent, unannounced 1% permanent reduction in the central bank’s target rate of inflation.
  - Policy relevance: a key statistic and "smell test" for monetary models and the role of monetary policy.
  - Practical value: disinflations are rare in typical sample data, so the likelihood is often uninformative on this property while historical episodes or expert judgment can inform a prior.
- Features of the sacrifice-ratio prior:
  - Highly non-linear function of important dynamic structural parameters; can be very responsive to parameter changes.
  - Using hyper-parameters for the system prior (distribution, mean, variance) penalizes parameterizations that deviate far from the prior.
  - From a global identification perspective, an informative sacrifice-ratio prior can be powerful:
    - It can select between parameterizations with similar likelihoods and dynamics but dramatically different policy implications, notably in open-economy models.

### Implementation overview and operational relation to existing literature
- Operationalization:
  - Express higher-level system priors as stochastic restrictions on parameters; combine with marginal priors to act as a penalty on the likelihood.
  - The composite prior typically not analytically tractable but obtainable via standard sampling techniques (Monte Carlo).
- Relation to literature:
  - Builds on Bayesian estimation literature for dynamic models (e.g., An and Schorfheide(2007)).
  - Extends emphasis on prior and posterior predictive analysis (e.g., Geweke(2010), Faust and Gupta(2012), Faust(2009)).
  - Relates operationally to Del Negro and Schorfheide(2008) approaches of priors based on pre-sample data and long-run stylized facts; philosophy differs by shaping parameter priors around model-implied properties.
  - More general than Jarociński and Marcet(2010) approach of using a prior on output growth expectations at the beginning of the sample.

### 2. Delayed policy response — effects and duration priors
- Effects of delayed policy action:
  - Scenario: a demand shock followed by keeping the policy rate unchanged for k periods.
  - Key mechanism: the longer markets expect no change in the policy instrument, the more accommodative the policy is.
  - Statistic of interest: how much more inflation increases compared to a standard reaction of the policy (for example, when policy follows a variant of a Taylor rule).
  - Nature of the statistic: a very nonlinear function of parameters and a statistic that goes beyond the properties of the model that can be directly observed in the data.
  - Identification benefit: the simulation entails anticipated actions and thus uses both stable and unstable dynamics of the model, allowing for richer identification options than the likelihood.
- Duration priors:
  - Motivation: duration system priors proved useful in the model estimation to prevent a priori unreasonable dynamics.
  - Example prior: a prior suggesting that no significant deviation of inflation from its target is to be found after the fourth year would seem like a rather consensual prior and may work as an upper bound on the response of inflation.
  - Scale-invariant duration priors (implementation):
    - Compute the integral of the particular variable’s response in absolute value up to horizon T and T+ h and express the prior about the relative contribution of the h-period horizon as a ratio of these two numbers.
    - Interpretation: no dynamics between periods T and T+h implies that the statistic of interest is one.
    - Property: such a definition of dynamics is invariant to the size of the shock in a linear model; in non-linear models a variety of modifications are possible.
    - Extension: if needed, the measure of persistence of the inflation response can be related to the persistence of the output expansion.

### 4. Frequency-domain Priors
- Frequency-domain rationale and applications:
  - System priors based on statistics computed in the frequency domain offer flexibility and control over estimation and are intuitive when the contribution of trends, cycles, or measurement errors is of concern.
  - Equivalence between time domain and frequency domain allows choice of tool based on purpose; frequency-domain is natural for restricting contribution of processes at particular periodicities.
  - Examples: priors on spectral properties, signal-to-noise ratios, and frequency response functions (Wiener-Kolmogorov or Kalman filter implied by the structural model).
- Measurement-error priors:
  - For uncorrelated measurement error: spectral density is flat; contribution to estimated variance of observed endogenous variable can be constrained a priori.
  - For correlated measurement errors: a priori information about up to which frequencies measurement error can contribute is implementable.
  - For each parametrization, contribution of measurement error to population or sample variance can be evaluated and penalized for deviations from prior-implied statistic.
- Priors on trends, cycles, and frequency responses:
  - System priors can be associated with the frequency-domain representation of the model, or the frequency response function of the filter implied by the structural model.
  - Practical prior choices include the mid-point of the filter’s implied gain function for a variable-to-variable relationship, or priors on signal-to-noise ratio to guard against the pile-up problem.
  - In complex models with endogenous trends, system priors on implied signal-to-noise ratio and frequency response function provide an effective way to express a priori views even without closed-form solutions.

### Implementing system priors — Bayesian formulation and composite prior
- Bayesian update (general):
  - p(θ|Yo,M) ∝ L(Yo|θ,M) × ̃p(θ|M).
- Typical starting prior in DSGE work:
  - Independent marginal priors: p(θ|M) = p(θ1) × ··· × p(θk).
- Introduce model properties Z = h(θ) with distribution Z ∼ D(Zs) (Zs describes mean and variance of prior properties distribution).
- System-prior likelihood for non-sample information: pS(Zs|θ,h,M).
- Composite joint prior:
  - ̃p(θ|Zs,h,M) ∝ pS(Zs|θ,h,M) × p(θ|M).
- Posterior combining sample and system-prior information:
  - p(θ|Yo,Zs,M) ∝ L(Yo|θ,M) × pS(Zs|θ,h,M) × p(θ|M) ∝ L(Yo|θ,M) × ̃p(θ|Zs,h,M).

### Numerical implementation and sampling methods
- Rely on numerical specification and Monte Carlo integration, fitting standard DSGE estimation workflows.
- Posterior analysis via Monte Carlo:
  - Random-Walk Metropolis variants feasible; proposal example N(θ(i) − θ(i−1), cΣ), where Σ can be derived from the Hessian of the posterior mode θ? and c is a scaling parameter.
- Additional step to evaluate pS(.). For most system priors model solution need not be re-solved.
- If analytical pS(.) unavailable, consider Rejection Sampling or Importance Sampling for smaller problems; Importance Sampling may be preferred in complex problems for parallelization and exploration.
- Preferred practical approach: Random-Walk Metropolis applied to posterior with likelihood ‘switched-off’ (set to constant) to analyze composite prior — introduces necessity of two sampling steps to evaluate posterior and prior separately.

### Effects on prior dependence, diagnostics, and identification
- System priors induce complex interdependence among parameters (global restrictions across θ support), sharpening identification and dissecting parameter space into feasible regions.
- Marginal prior/posterior plots for individual parameters are insufficient to reveal joint implications; prior predictive analysis and prior distribution of model system properties (e.g., impulse-response functions) are crucial.
- System priors can affect posterior distributions of purely unidentified or weakly identified parameters because priors regularize the likelihood.
- Decomposition of posterior-mode Hessian H(θ?) into components:
  - H(θ?) = HY(θ?) + HS(θ?) + HP(θ?)
  - HY from log-likelihood, HS from system properties priors, HP from independent marginal priors (HP diagonal).
- System priors may act as a transparent mode-selection device; e.g., a sacrifice-ratio prior can navigate estimation toward a particular likelihood mode in a bi-modal likelihood.
- Global identification can be aided by system priors based on solutions featuring anticipated events (forward expansion), which depend on unstable eigenvalues and may identify parameters not present in standard recursive solutions.

### Illustration: Simple monetary policy model (setup)
- Model: New-Keynesian closed economy with price rigidities.
- Key equations (as numbered in source):
  - Output cycle: ˆyt = α1 y t+1|t + α2 y t−1 + α3 (rrt − rr t) + εy t (equation (6))
  - Inflation: πct = λ1 πct+1|t + (1 − λ1) πct−1 + λ2 ˆyt + επ t (equation (7))
  - Policy rule (interest rate): it = γ1 it−1 + (1 − γ1) × [ (rrt + πt) + γ2 (πy/y t+3|t − πt+3) + γ3 yt ] + εi t (equation (8))
  - Real rate: rrt = it − πt+1|t (equation (9))
  - Inflation target: πt = ρπ πt−1 + επ t (equation (10))
- Parameter vector: θ = {α1,2,3, λ1,2, ρi, γ1,2,3} (eight parameters) plus four standard deviations for structural shocks.
- Assumption: inflation target random-walk with ρπ = 1.
- Model derivation based on GHH preferences, habit formation, flexible wages, labor as single input, one-sector economy.

### Computational experiment: sacrifice-ratio system prior — setup and findings
- Sacrifice ratio defined as cumulative loss in output after a permanent decrease of central bank’s inflation target by 1 percentage point.
- System prior specification:
  - Sacrifice ratio prior: N(-0.8,0.05).
  - Additionally truncate parameter space to Blanchard-Kahn stable regions (dogmatic system prior).
- Marginal independent priors for individual parameters chosen as rather diffuse truncated Normal distributions with example moments cited in figures:
  - α1 prior: normal μ=0.6 σ=0.1
  - α2 prior: normal μ=0.4 σ=0.1
  - α3 prior: normal μ=0.05 σ=0.02
  - λ1 prior: normal μ=0.6 σ=0.2
  - λ2 prior: normal μ=0.05 σ=0.2
  - γ1 prior: normal μ=0.75 σ=0.1
  - γ2 prior: normal μ=1.75 σ=0.2
  - γ3 prior: normal μ=0.2 σ=0.1
- Key computational findings:
  - Applying the sacrifice-ratio system prior converts a diffuse multivariate Normal prior of parameters into a composite prior that implies a much tighter prior on the sacrifice ratio.
  - The marginal prior mode implied by independent marginal priors for the sacrifice ratio is -0.5.
  - Independent marginal priors map nonlinearly to a highly non-Gaussian prior distribution for the sacrifice ratio before applying the system prior.
  - After enforcing the system prior N(-0.8,0.05), the composite prior for the sacrifice ratio becomes concentrated around -0.8 with variance implied by 0.05.
  - Figure diagnostics summary:
    - Figure 1: marginal independent priors (blue dashed) vs composite system priors (red solid) for individual parameters; output and inflation equation coefficients are most affected, monetary policy rule coefficients are least identified.
    - Figure 2: left panel shows prior predictive distribution of sacrifice ratio under independent marginal priors (mode at -0.5); right panel shows composite prior after system prior applied.
  - Both graphs in Figure 2 are based on 15000 draws from the relevant prior distributions.

### Conclusions
- System priors are priors directly related to the structural system properties of the model, that allow one to back out implied complex joint prior distribution for individual parameters.
- System priors allow for eliciting transparent and economically meaningful priors for the estimation of structural models, without unintentional consequences as is often the case with independent marginal parameter priors.
- System priors are transparent and easy to communicate; they foster debate and require researchers to explain subjective priors.
- System priors can serve both as a complement and substitute for individual independent marginal parameter priors.
- Although direct priors on structural parameters remain first best in many cases, system priors may impose further structure and regularize estimation.

### Appendix: Behavioral foundations of a simple monetary model — main assumptions and log-linearized form
- Production and preferences:
  - Yt(j) = Zt Lt(j)^γ (equation (11))
  - Consumers maximize: max ∑_{t=s}^∞ β^t ((Ct(i) − h Ct−1) − χ L_{t}^{1+η}/(1 + η))^{1+σ} × 1/(1 + σ) (equation (13))
  - Period budget constraint: Pt Ct(i) + Bt(i) = R^B_{t−1} B_{t−1}(i) + Wt Lt(i) + Πt(i) + Pt Tt(i) (equation (14))
- Pricing and Phillips Curve:
  - Calvo(1983) pricing with parameter θ; Phillips Curve:
    - π^y_t = β/(1 + β) π^y_{t+1|t} + 1/(1 + β) π^y_{t−1} + ((1−θ)(1−θβ))/(θ(1 + β)) ˆrmc_t + ε^μ_t (equation (12)), where ˆrmc = ˆw_t − ˆp_t − ˆz_t.
- Banking, interest rates, and monetary policy:
  - Banks refinance at the central bank and face shocks φ_t, so R^B_t = R_t φ^B_t.
  - Monetary authority sets nominal policy rate:
    - ˆR_t = ρ ˆR_{t−1} + (1−ρ) [ γ_π (π^y/y_{t+3} − ̄π_{t+3}) + γ_y ˆy ] + ε^i_t (equation (15))
- Log-linearized model equations include:
  - ˆy_t = 1/(1 + h) y_{t+1|t} + h/(1 + h)(1 + ι_L) ˆy_{t−1} − (1−h)/(1 + h) ζ_L/σ (ˆR_t − π_{t+1}) + ˆφ^B_t (equation (16))
  - π^y_t = β/(1 + β) π^y_{t+1} + 1/(1 + β) π^y_{t−1} + (((1−θ)(1−θβ))/(θ(1 + β))) η/γ (ˆy_t − ˆz_t) + ε^μ_t (equation (17))
  - ˆR_t = ρ ˆR_{t−1} + (1−ρ) [ γ_π (π^y/y_{t+3} − ̄π_{t+3}) + γ_y ˆy ] + ε^i_t (equation (18))
  - Definitions: ζ_L ≡ (1 − ι_L/[(1 + η)(1 − h)]) and ι_L ≡ χ L^η (L/C)/(1 + η).

*Source: Excerpt from _wp13257 - References .  .  .  .  .  .  .  .  .  .  .  .  .  . .*

### References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### _wp13257 - References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### III. Introduction — purpose and contribution
- Proposes a novel approach to elicit individual parameter priors using views on dynamic models’ properties — termed "system priors".
- Motivation:
  - Priors about many individual structural model parameters are difficult to formulate.
  - Independent marginal priors can imply infeasible or unintended prior distributions for model properties.
  - Direct specification of priors on model properties (impulse response functions, conditional correlations, sacrifice ratio) can be more transparent and intuitive.
- Main methodological claim:
  - Change the philosophy of formulating priors from bottom-up marginal parameter priors to top-down system priors that impose stochastic restrictions on combinations of parameters.
- Implementation note:
  - Technically a straightforward extension of standard Bayesian methods; translates to adding an evaluation of system priors on top of marginal priors and the likelihood in Monte Carlo estimation.

### III. Benefits and implications of system priors
- Benefits:
  - Introduce complex dependence among structural parameter priors.
  - Allow imposition of intuitive and economically meaningful priors about aggregate model behavior.
  - Avoid unintended consequences on model properties often associated with independent marginal priors.
  - Facilitate model comparison across differently structured models by using higher-level, more comparable priors.
  - Make explicit and testable many prior beliefs researchers already implicitly hold (e.g., horizon of monetary policy, conditional cross-correlations).
- Practical considerations:
  - Main cost is formulating relevant, transparent, and economically meaningful beliefs about model behavior.
  - Prior-predictive analysis is emphasized (following Geweke(2010)) to assess whether prior or data drive estimates of system properties.
  - Reporting priors for system properties (e.g., impulse response functions) is rare but recommended.

### II.A Motivation — problems with bottom-up priors and rationale for system priors
- Problems with standard priors:
  - Bottom-up independent priors can produce unintuitive aggregate properties due to nonlinear mapping from parameters to system properties.
  - Some individual parameters (e.g., Calvo price rigidity) may not map cleanly to comparable economic concepts across models, reducing transferability of parameter priors.
  - Marginal prior and posterior plots of structural coefficients are insufficient to reveal prior implications for system properties.
- Role of system priors:
  - Tie together parameter combinations that satisfy stochastic constraints defined by the analyst’s higher-level beliefs.
  - Allow direct expression of beliefs about such questions as the maximum length of inflation deviation after a demand shock.
  - Act as a non-dogmatic analog to imposing conditions like saddle-path (Blanchard-Kahn) stability, by introducing cross-dependence rather than hard truncation.

### II.B Candidates for system priors for DSGE models
- Any model-implied statistic can be used as a system prior. Selective candidates include:
  - Steady-state values of model variables (levels or growth rates).
  - Conditional or unconditional moments (e.g., cross-correlations of investment and private consumption after demand shocks).
  - Prominent policy scenarios (counter-factuals) such as:
    - (i) permanent disinflation, and
    - (ii) anticipated delayed response of monetary policy for several periods.
  - Characteristics of impulse-response functions (peak impacts, duration of recessions, horizon of policy effectiveness).
  - Frequency response function and spectral characteristics (signal-to-noise ratios, power of shocks at particular frequencies, coherence between macro variables).
- Construction guidance:
  - System priors can be constructed in relative or absolute terms but should be robust to trivial changes in scaling.
  - Working with a single summary statistic is often easier (e.g., sacrifice ratio), though vector forms (impulse-response paths) are possible.

### II.C Examples of system priors — focus on sacrifice-ratio priors
- Sacrifice ratio definition and motivation:
  - Measures cumulative loss of output after a permanent, unannounced 1% permanent reduction in the central bank’s target rate of inflation.
  - Policy relevance: a key statistic and "smell test" for monetary models and the role of monetary policy.
  - Practical value: disinflations are rare in typical sample data, so the likelihood is often uninformative on this property while historical episodes or expert judgment can inform a prior.
- Features of the sacrifice-ratio prior:
  - Highly non-linear function of important dynamic structural parameters; can be very responsive to parameter changes.
  - Using hyper-parameters for the system prior (distribution, mean, variance) penalizes parameterizations that deviate far from the prior.
  - From a global identification perspective, an informative sacrifice-ratio prior can be powerful:
    - It can select between parameterizations with similar likelihoods and dynamics but dramatically different policy implications, notably in open-economy models.

### Implementation overview and operational relation to existing literature
- Operationalization:
  - Express higher-level system priors as stochastic restrictions on parameters; combine with marginal priors to act as a penalty on the likelihood.
  - The composite prior typically not analytically tractable but obtainable via standard sampling techniques (Monte Carlo).
- Relation to literature:
  - Builds on Bayesian estimation literature for dynamic models (e.g., An and Schorfheide(2007)).
  - Extends emphasis on prior and posterior predictive analysis (e.g., Geweke(2010), Faust and Gupta(2012), Faust(2009)).
  - Relates operationally to Del Negro and Schorfheide(2008) approaches of priors based on pre-sample data and long-run stylized facts; philosophy differs by shaping parameter priors around model-implied properties.
  - More general than Jarociński and Marcet(2010) approach of using a prior on output growth expectations at the beginning of the sample.

*Source: Excerpt from _wp13257 - References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .*

### 2.   Delayed policy response

### 2.   Delayed policy response

### Effects of delayed policy action
- Scenario: a demand shock followed by keeping the policy rate unchanged forkperiods.
- Key mechanism: the longer markets expect no change in the policy instrument, the more accommodative the policy is.
- Statistic of interest: how much more inflation increases compared to a standard reaction of the policy (for example, when policy follows a variant of a Taylor rule).
- Nature of the statistic: a very nonlinear function of parameters and a statistic that goes beyond the properties of the model that can be directly observed in the data.
- Identification benefit: the simulation entails anticipated actions and thus uses both stable and unstable dynamics of the model, allowing for richer identification options than the likelihood.

### Duration priors
- Motivation: duration system priors proved useful in the model estimation to prevent a priori unreasonable dynamics.
- Practical intuition: economists often hold strong priors about the duration or timing of economic phenomena (e.g., duration of business cycles, adjustment speed to shocks). Inflation-targeting central banks commonly operate within a “monetary policy horizon” during which inflation should return to its target after the initial shock.
- Example prior: a prior suggesting that no significant deviation of inflation from its target is to be found after the fourth year would seem like a rather consensual prior and may work as an upper bound on the response of inflation.
- Scale-invariant duration priors (implementation):
  - Compute the integral of the particular variable’s response in absolute value up to horizon T and T+ h and express the prior about the relative contribution of the h-period horizon as a ratio of these two numbers.
  - Interpretation: no dynamics between periods TandT+h implies that the statistic of interest is one.
  - Property: such a definition of dynamics is invariant to the size of the shock in a linear model; in non-linear models a variety of modifications are possible.
  - Extension: if needed, the measure of persistence of the inflation response can be related to the persistence of the output expansion.

*Source: _wp13257 - 2.   Delayed policy response*

### 4.   Frequency-domain Priors

### 4.   Frequency-domain Priors

### Frequency-domain rationale and applications
- System priors based on statistics computed in the frequency domain offer flexibility and control over estimation and are intuitive when the contribution of trends, cycles, or measurement errors is of concern.
- Equivalence between time domain and frequency domain allows choice of tool based on purpose; frequency-domain is natural for restricting contribution of processes at particular periodicities.
- Examples of system priors in the frequency domain include priors on spectral properties, signal-to-noise ratios, and frequency response functions (Wiener-Kolmogorov or Kalman filter implied by the structural model).

### Measurement-error priors
- Standard priors on measurement error variance lack sophisticated restrictions on the role of measurement errors across frequencies.
- For uncorrelated measurement error:
  - Spectral density is flat; contribution to estimated variance of observed endogenous variable can be constrained a priori.
- For correlated measurement errors or general processes:
  - A priori information about up to which frequencies measurement error can contribute is implementable.
- For each parametrization, contribution of measurement error to population or sample variance can be evaluated and penalized for deviations from prior-implied statistic.

### Priors on trends, cycles, and frequency responses
- System priors can be associated with:
  - Frequency-domain representation of the model, or
  - Frequency response function of the filter implied by the structural model.
- Properties of the implied filter are determined by model parameterization and hyper-parameters handling initial conditions.
- Practical prior choices include the mid-point of the filter’s implied gain function for a variable-to-variable relationship, or priors on signal-to-noise ratio to guard against the pile-up problem.
- In complex models with endogenous trends, or trend-cycle models, system priors on implied signal-to-noise ratio and frequency response function provide an effective way to express a priori views even without closed-form solutions.

### Implementing system priors — Bayesian formulation and composite prior
- Bayesian update (general):
  - p(θ|Yo,M) ∝ L(Yo|θ,M) × ̃p(θ|M).
- Typical starting prior in DSGE work:
  - Independent marginal priors: p(θ|M) = p(θ1) × ··· × p(θk).
- Introduce model properties Z = h(θ) with distribution Z ∼ D(Zs) (Zs describes mean and variance of prior properties distribution).
- System-prior likelihood for non-sample information: pS(Zs|θ,h,M).
- Composite joint prior:
  - ̃p(θ|Zs,h,M) ∝ pS(Zs|θ,h,M) × p(θ|M).
- Posterior combining sample and system-prior information:
  - p(θ|Yo,Zs,M) ∝ L(Yo|θ,M) × pS(Zs|θ,h,M) × p(θ|M) ∝ L(Yo|θ,M) × ̃p(θ|Zs,h,M).

### Numerical implementation and sampling methods
- Full reliance on numerical specification and Monte Carlo integration is proposed, fitting standard DSGE estimation workflows.
- Posterior analysis via Monte Carlo:
  - Random-Walk Metropolis variants are feasible; proposal distribution example: N(θ(i) − θ(i−1), cΣ), where Σ can be derived from the Hessian of the posterior mode θ? and c is a scaling parameter.
- Evaluation of p(.) and L(.) unchanged; additional step to evaluate pS(.). For most system priors model solution need not be re-solved.
- If analytical pS(.) unavailable, consider Rejection Sampling or Importance Sampling for smaller problems. Importance Sampling may be preferred in complex problems for parallelization and exploration of composite prior support.
- Preferred practical approach: Random-Walk Metropolis applied to posterior with likelihood ‘switched-off’ (set to constant) to analyze composite prior — introduces necessity of two sampling steps to evaluate posterior and prior separately.

### Effects on prior dependence and diagnostics
- System priors induce complex interdependence among parameters (global restrictions across θ support), sharpening identification and dissecting parameter space into feasible regions.
- Marginal prior/posterior plots for individual parameters are insufficient to reveal joint implications; prior predictive analysis and prior distribution of model system properties (e.g., impulse-response functions) are crucial.
- Even Normal marginal priors are complex because mapping from parameters to model solution/properties is highly nonlinear.

### Identification issues
- System priors can affect posterior distributions of purely unidentified or weakly identified parameters because priors regularize the likelihood.
- System priors provide prior parameter interdependence similar to Blanchard-Kahn stability requirement, affecting global identification.
- System priors may act as a transparent mode-selection device; e.g., a sacrifice-ratio prior can navigate estimation toward a particular likelihood mode in a bi-modal likelihood.
- Local identification tools:
  - Decomposition of posterior-mode Hessian H(θ?) into components:
    - H(θ?) = HY(θ?) + HS(θ?) + HP(θ?)
    - HY from log-likelihood, HS from system properties priors, HP from independent marginal priors (HP diagonal).
  - HP regularizes HY (akin to ridge), HS typically non-diagonal and reveals identification patterns consistent with system priors.
- Global identification can be aided by system priors based on solutions featuring anticipated events (forward expansion), which depend on unstable eigenvalues and may identify parameters not present in standard recursive solutions.

### Illustration: Simple monetary policy model (setup)
- Model: New-Keynesian closed economy with price rigidities.
- Key variables and relations include (equations numbered in source):
  - Output cycle: ˆyt = α1 y t+1|t + α2 y t−1 + α3 (rrt − rr t) + εy t (equation (6))
  - Inflation: πct = λ1 πct+1|t + (1 − λ1) πct−1 + λ2 ˆyt + επ t (equation (7))
  - Policy rule (interest rate): it = γ1 it−1 + (1 − γ1) × [ (rrt + πt) + γ2 (πy/y t+3|t − πt+3) + γ3 yt ] + εi t (equation (8))
  - Real rate: rrt = it − πt+1|t (equation (9))
  - Inflation target: πt = ρπ πt−1 + επ t (equation (10))
- Parameter vector: θ = {α1,2,3, λ1,2, ρi, γ1,2,3} (eight parameters) plus four standard deviations for structural shocks.
- Assumption: inflation target random-walk with ρπ = 1.
- Model derivation based on GHH preferences, habit formation, flexible wages, labor as single input, one-sector economy (appendix derivation in source).

### Computational experiment: sacrifice-ratio system prior
- Goal: show how a single system prior (sacrifice ratio) induces changes in individual parameter priors starting from diffuse marginal priors.
- Sacrifice ratio defined as cumulative loss in output after a permanent decrease of central bank’s inflation target by 1 percentage point.
  - Example: in the linear model, disinflation from 3% to 2% produces same loss as from 15% to 14%.
- System prior specification:
  - Sacrifice ratio prior: N(-0.8,0.05).
  - Additionally truncate parameter space to Blanchard-Kahn stable regions (dogmatic system prior).
- Marginal independent priors for individual parameters chosen as rather diffuse truncated Normal distributions (specific marginal prior moments cited in figures: e.g., alpha1 prior: normal μ=0.6 σ=0.1; alpha2 prior: normal μ=0.4 σ=0.1; alpha3 prior: normal μ=0.05 σ=0.02; lambda1 prior: normal μ=0.6 σ=0.2; lambda2 prior: normal μ=0.05 σ=0.2; gamma1 prior: normal μ=0.75 σ=0.1; gamma2 prior: normal μ=1.75 σ=0.2; gamma3 prior: normal μ=0.2 σ=0.1).
- Key computational findings:
  - Applying the sacrifice-ratio system prior converts a diffuse multivariate Normal prior of parameters into a composite prior that implies a much tighter prior on the sacrifice ratio.
  - The marginal prior mode implied by independent marginal priors for the sacrifice ratio is -0.5 (red vertical line in figure panels).
  - The marginal independent priors map nonlinearly to a highly non-Gaussian prior distribution for the sacrifice ratio before applying the system prior.
  - After enforcing the system prior N(-0.8,0.05), the composite prior for the sacrifice ratio becomes concentrated around -0.8 with variance implied by 0.05.
  - Figure diagnostics (summary from source):
    - Figure 1: shows marginal independent priors (blue dashed) vs composite system priors (red solid) for individual parameters; output and inflation equation coefficients are most affected, monetary policy rule coefficients are least identified.
    - Figure 2: left panel shows prior predictive distribution of sacrifice ratio under independent marginal priors (mode at -0.5); right panel shows composite prior after system prior applied.
  - Both graphs in Figure 2 are based on 15000 draws from the relevant prior distributions.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp13257.pdf*

### CONCLUSIONS

### CONCLUSIONS

### System priors: definition and advantages
- System priors are priors directly related to the structural system properties of the model, that allow one to back out implied complex joint prior distribution for individual parameters.
- System priors allow for eliciting transparent and economically meaningful priors for the estimation of structural models, without unintentional consequences as is often the case with independent marginal parameter priors.
- System priors – no matter what they are – are transparent and easy to communicate.
- With system priors, it is perfectly acceptable to have economic priors and it is likewise acceptable to have quite informative priors.
- System priors foster transparency and it is their transparency and economically meaningful nature that allow the priors to be discussed and debated.
- These are subjective priors that need to be explained.

### Transparency, specification, and researcher behavior
- System priors free analysts from ‘prior specification fishing,’ chasing for the combination of marginal independent priors that roughly convey their more often vague ideas about model behavior, if they have any.
- System priors force researchers to carefully articulate their priors.
- Although several examples of system priors were discussed, these are just the tip of the iceberg.

### Relationship with individual parameter priors
- System priors can serve both as a complement and substitute for individual independent marginal parameter priors.
- There is no doubt that having priors directly about structural parameters is the first best approach in many cases.
- System priors are not mutually exclusive with individual parameter priors in principle; better, system priors may impose further structure on those, as is clear from the computational procedure suggested.

### Appendix: Behavioral foundations of a simple monetary model — main assumptions and log-linearized form
- The standard New Keynesian Model, see e.g. Gali (2008), amounts to a system similar to our simple model, with the restriction α2 = 0.
- Production and preferences:
  - Yt(j) = Zt Lt(j)^γ, where Zt is aggregate technology shock. (equation (11) in source)
  - Consumers maximize:
    - max ∑_{t=s}^∞ β^t ((Ct(i) − h Ct−1) − χ L_{t}^{1+η}/(1 + η))^{1+σ} × 1/(1 + σ). (equation (13) in source)
  - Period budget constraint:
    - Pt Ct(i) + Bt(i) = R^B_{t−1} B_{t−1}(i) + Wt Lt(i) + Πt(i) + Pt Tt(i). (equation (14) in source)
- Pricing and Phillips Curve:
  - Calvo(1983) pricing with parameter θ determining the probability of changing the price in the current period; non-reoptimizing firms index prices by previous period change of the average price level.
  - Phillips Curve form:
    - π^y_t = β/(1 + β) π^y_{t+1|t} + 1/(1 + β) π^y_{t−1} + ((1−θ)(1−θβ))/(θ(1 + β)) ˆrmc_t + ε^μ_t, where ˆrmc = ˆw_t − ˆp_t − ˆz_t. (equation (12) in source)
  - The ‘cost-push shock’ ε^μ_t follows from a time variation in the elasticity of substitution among intermediate goods.
- Banking, interest rates, and monetary policy:
  - Consumers use a deposit account at a perfectly competitive bank with rate R^B_t; banks refinance at the central bank and are subject to stochastic shocks φ_t, so R^B_t = R_t φ^B_t.
  - Cashless-limit economy; monetary authority operates under flexible inflation targeting and sets the nominal policy rate according to:
    - ˆR_t = ρ ˆR_{t−1} + (1−ρ) [ γ_π (π^y/y_{t+3} − ̄π_{t+3}) + γ_y ˆy ] + ε^i_t. (equation (15) in source)
  - ˆR_t denotes annualized deviation of the nominal policy rate from its trend (equilibrium) value ̄R^S_t.
- Log-linearized model equations:
  - ˆy_t = 1/(1 + h) y_{t+1|t} + h/(1 + h)(1 + ι_L) ˆy_{t−1} − (1−h)/(1 + h) ζ_L/σ (ˆR_t − π_{t+1}) + ˆφ^B_t. (equation (16) in source)
  - π^y_t = β/(1 + β) π^y_{t+1} + 1/(1 + β) π^y_{t−1} + (((1−θ)(1−θβ))/(θ(1 + β))) η/γ (ˆy_t − ˆz_t) + ε^μ_t. (equation (17) in source)
  - ˆR_t = ρ ˆR_{t−1} + (1−ρ) [ γ_π (π^y/y_{t+3} − ̄π_{t+3}) + γ_y ˆy ] + ε^i_t. (equation (18) in source)
  - Definitions: ζ_L ≡ (1 − ι_L/[(1 + η)(1 − h)]) and ι_L ≡ χ L^η (L/C)/(1 + η).
- Closing remark:
  - It is clear that the simple structural model is a rather restricted version of the semi-structural model estimated in the main body of the paper.

*Source: CONCLUSIONS and Appendix (Behavioral foundations) from _wp13257 - CONCLUSIONS*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp13257.pdf_