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Equation Syntax Reference

Source: vignettes/koma-equations.Rmd
koma-equations.Rmd

Overview

This vignette describes the syntax for specifying stochastic and identity equations, priors, and lags in the koma package.

1. Stochastic Equations

Stochastic (regression) equations model a dependent variable with an error term. An intercept is included by default.

# With default intercept:
consumption ~ gdp + consumption.L(1)

# Without intercept:
consumption ~ gdp + consumption.L(1) - 1

# Explicit intercept:
consumption ~ 1 + gdp + consumption.L(1)

2. Identity Equations

Identity equations enforce exact relationships.

# Identity equations with explicitly defined weights:
# To aggregate the component growth rates into a growth rate for GDP we need to define weights.
# This is done by specifying the weights in the equation. 
# You can, e.g. use the nominal level weights of the last observed period.
gdp == 0.7*consumption + 0.2*investment + 0.2*government - 0.1*net_exports

3. Injected Parameters 

# Ratios computed from data:
gdp == (nom_cons/nom_gdp) * cons

4. Lag Notation

Lags are specified with L() or lag() notation. Ranges and combinations are supported.

# Single lag:
x.L(1)
lag(x, 1)

# Range of lags:
x.L(1:4)
lag(x, 1:4)

# Mix range and specific lags:
x.L(1:3, 5)

5. Priors

There are two kinds of priors in koma equations:

  • coefficient priors, written in front of a term
  • the error-term prior, written at the end of the equation

Coefficient priors are written in front of the term they belong to:

{mean, variance} variable

For example:

consumption ~ {0.4, 0.1} gdp + consumption.L(1)

This sets a prior with mean 0.4 and variance 0.1 on the coefficient of gdp.

You can use priors on:

  • the intercept, written as 1 or constant
  • exogenous variables
  • lagged variables
  • contemporaneous endogenous variables
consumption ~
  {0, 1000} 1 +
  {0.4, 0.1} gdp +
  {0.9, 10} consumption.L(1) +
  {0.2, 0.5} service

The error-term prior is different. It is written as a final prior with no variable name:

consumption ~ gdp + consumption.L(1) + {3, 0.001}

In the error-term prior, the two values specify:

  • degrees of freedom
  • scale

Some valid examples for priors are:

# Prior on the intercept
consumption ~ {0, 1000} 1 + gdp

# Prior on a lagged term
consumption ~ gdp + {0.9, 10} consumption.L(1)

# Prior on an endogenous regressor
consumption ~ {0.2, 0.5} service + gdp

# Error-term prior: {df, scale}
consumption ~ gdp + consumption.L(1) + {3, 0.001}

Rules:

  • Priors are only supported in stochastic equations.
  • Coefficient priors have the form {mean, variance} variable.
  • The error-term prior has the form {df, scale} and appears without a variable name.
  • The dependent variable cannot have a prior.
  • The error-term prior must be the last prior in the equation.

6. Equation-specific Tau

You can override the default tau in your gibbs_settings for a single equation by appending [tau = value] after its equation. If the acceptance rate falls outside 30%-60 %, a warning is emitted.

"consumption ~ constant + gdp + consumption.L(1) + consumption.L(2) [tau = 1.2]"