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[IrisToolbox] for Macroeconomic Modeling

Intro to estimation and calibration using system priors

jaromir.benes@iris-toolbox.com

Priors in bayesian estimation

  • Traditionally, priors on individual parameters

  • However, more often than not we simply wish to control some properties of the model as a whole system: system properties

  • And not so much the individual parameters

Examples of system properties

  • Model-implied correlation between output and inflation

  • Model-implied sacrifice ratio

  • Frequency response function from output to potential output (band of periodicities ascribed to potential output)

  • Suppress secondary cycles in shock responses (e.g. more than 90% of a shock response has to occur within the first 10 quarter)

  • Make sure a type 2 policy error is costly (delays in the policy response to inflationary shocks calls for a larger reaction later)

  • Anything...

  • Even qualitative properties (e.g. sign restrictions) can be expressed as system priors

System priors formally

Posterior density

\[ \underbrace{ \ p\left(\theta \mid Y, m \right) \ }_{ \text{Posterior}} \propto \underbrace{\ p\left(Y \mid \theta, m\right)\ }_{\text{Data likelihood}} \times \underbrace{\ p\left(\theta \mid m\right) \ }_{\text{Prior}} \]

Prior density typically consists of independent marginal priors

\[ p\left(\theta \mid m \right) = p_1\left(\theta_1 \mid m \right) \times p_2\left(\theta_2 \mid m \right) \times \cdots \times p_n\left(\theta_n \mid m \right) \]

Complement or replace with density involving a property of the model as a whole, \(h\left(\theta\right)\)

\[ p\left(\theta \mid m \right) = p_1\left(\theta_1 \mid m \right) \times \cdots \times p_n\left(\theta_n \mid m \right) \times {\color{highlight} q_1\bigl(h(\theta)\mid m\bigr) \times \cdots \times q_k\bigl(h(\theta)\mid m\bigr) } \]

Benefits of system priors in estimation

  • A relatively low number of system priors can push parameter estimates into a region where the properties of the model as a whole make sense and are well-behaved…

  • …without enforcing a tighter prior structure on individual parameters

Non-bayesian interpretation of priors: Penalty/shrinkage

  • Shrinkage (or penalty) function

  • Keep the parameters close to our “preferred” values

  • ”Close” is defined by the shape/curvature of the shrinkage/penalty function

  • Example: Normal priors are equivalent to quadratic shrinkage/penalty

Priors in calibration: Maximize prior mode

  • Exclude/disregard data likelihood

  • Only maximize prior mode

  • Case 1: only independent priors on individual parameters
    \(\Rightarrow\) modes of marginals

\[ p\left(\theta \mid m \right) = p_1\left(\theta_1 \mid m \right) \times \cdots \times p_n\left(\theta_n \mid m \right) \times \cdots \]
  • Case 2: only a small number of system priors
    \(\Rightarrow\) very likely underdetermined (singular)
\[ p\left(\theta \mid m \right) = q_1\bigl(h(\theta)\mid m\bigr) \times \cdots \times q_k\bigl(h(\theta)\mid m\bigr) \]
  • Case 3: Combination of priors on individual parameters and system priors
    \(\Rightarrow\) deviate as little as possible from the "preferred" values of parameters while delivering sensible system properties $$ p\left(\theta \mid m \right) = p_1\left(\theta_1 \mid m \right) \times \cdots \times p_n\left(\theta_n \mid m \right) \times q_1\bigl(h(\theta)\mid m\bigr) \times \cdots \times q_k\bigl(h(\theta)\mid m\bigr) $$

Implemenation in IrisT

The following @Model class functions (methods) can be used to construct a @SystemProperty object for efficient evaluation of system properties

Function Description
simulate Any kind of simulation, including complex simulation design
acf Autocovariance and autocorrelation functions
xsf Power spectrum and spectral density functions
ffrf Filter frequency response function