Effective lower bound (ELB) data augmentation¶

What is the ELB constraint?¶

For some interest-rate series, observations are effectively censored at a lower bound (e.g. 0 or slightly negative). When the ELB binds, the observed series no longer behaves like an unconstrained Gaussian variable.

The basic modelling idea¶

In an ELB/shadow-rate setting, the observed rate \(i_t\) is treated as a censored version of an underlying latent series \(s_t\) (a shadow rate). One simple relationship is:

\[ i_t = \max\{\mathrm{ELB},\, s_t\}. \]

Data augmentation in MCMC¶

A standard way to fit these models is data augmentation:

treat ELB-bound observations as latent,
sample latent values conditional on the VAR parameters and the constraint \(s_t \le \mathrm{ELB}\).

In practice, this becomes sampling from a (possibly univariate) truncated normal conditional distribution for each constrained time point.

In this toolkit¶

When ELB support is enabled via ElbSpec, fit() uses Gibbs steps that alternate between:

sampling VAR parameters conditional on the current latent series, and
sampling latent shadow values for ELB observations conditional on the VAR parameters.

With stochastic volatility enabled, the conditional distribution accounts for time-varying variance through the current log-volatility state.