IndependentNIW¶

The IndependentNIW model implements Bayesian VAR estimation with independent Normal-Inverse-Wishart priors. Unlike the natural-conjugate specification, the prior on the VAR coefficients is independent of \(\Sigma\), which allows a richer \((nk \times nk)\) full-system prior covariance matrix whose inverse encodes cross-variable shrinkage scaling by \(\sigma_i / \sigma_j\).

Because the prior is not conjugate, the posterior has no closed form and the model uses a Gibbs sampler that alternates between drawing \(\beta \mid \Sigma, Y\) and \(\Sigma \mid \beta, Y\). Burn-in draws are automatically discarded.

The public beta_point and sigma_point estimates come from the retained Gibbs draws, after burn-in. They are posterior means rather than posterior modes, and posterior_state_point references those same point arrays for forecasting with the full posterior state.

Note

IndependentNIW does not support marginal-likelihood optimisation, and its Gibbs sampler has no closed-form posterior point estimate, so optimisation_method="cross_validation" is not supported either (it requires refitting with point_only=True). Use optimisation_method="none" and set hyperparameters manually when constructing BVAR.

Constructor arguments (minnesota, soc, sur, covid, covid_dates) are documented on SamplingModel; soc and sur default to True.

Prior Structure¶

\[\beta \,\sim\, \mathcal{N}(\beta_0, V_\beta)\]

\[\Sigma \,\sim\, \mathcal{IW}(S_0, \nu_0)\]

where \(V_\beta\) is a full \((nk \times nk)\) covariance matrix. Its inverse is the prior precision used in the conditional posterior. The cross-variable off-diagonal blocks are scaled by a parameter c2 (Litterman, 1986), giving tighter shrinkage on cross-equation coefficients relative to own-equation coefficients. The Gibbs sampler iterates:

\[\beta \mid \Sigma, Y \;\sim\; \mathcal{N}\!\left(\tilde\beta,\; \tilde{V}\right), \qquad \tilde{V}^{-1} = V_\beta^{-1} + \Sigma^{-1} \otimes Z'Z\]

\[\Sigma \mid \beta, Y \;\sim\; \mathcal{IW}\!\left(S_0 + E'E,\; \nu_0 + T\right)\]

Hyperparameters¶

Parameter	Description	Default
`c2`	Cross-variable shrinkage	0.5
`c1`	Overall tightness	0.2
`c3`	Lag-decay exponent	2.0
`mu`	SOC tightness (if `soc=True`)	1.0
`theta`	SUR tightness (if `sur=True`)	1.0

Example¶

import bvar as bv

model = bv.IndependentNIW(
    c2=0.5,  # cross-variable shrinkage
    minnesota=True,
    soc=False,
    sur=False,
    covid=False,
)

# ML and cross-validation optimisation are not available; use "none"
bvar = bv.BVAR(n_lags=4, model=model, stationary=False, optimisation_method="none")
bvar.sample(data, N_draws=5000)

API¶

bvar.IndependentNIW ¶

IndependentNIW(c2: float = 0.5, minnesota: bool = True, soc: bool = True, sur: bool = True, covid: bool = False, covid_dates: Optional[list] = None)

BVAR with independent Normal-Inverse-Wishart priors.

The prior is β ~ N(β₀, V_β) independently of Σ ~ IW(S₀, ν₀), where V_β is a full (nk, nk) precision matrix encoding cross-variable scaling σ_i / σ_j.

PARAMETER	DESCRIPTION
`c2`	Cross-variable shrinkage (Litterman 1986 uses 0.5). TYPE: `float` DEFAULT: `0.5`
`minnesota`	Whether to use the Minnesota prior. TYPE: `bool` DEFAULT: `True`
`soc`	Whether to use the sum-of-coefficients prior. TYPE: `bool` DEFAULT: `True`
`sur`	Whether to use the single-unit-root prior. TYPE: `bool` DEFAULT: `True`
`covid`	Whether to include COVID dummy observations. TYPE: `bool` DEFAULT: `False`
`covid_dates`	Start and end dates for the COVID period. TYPE: `Optional[list]` DEFAULT: `None`

ATTRIBUTE	DESCRIPTION
`requires_burnin`	`True` — Gibbs draws require burn-in. TYPE: `bool`
`supports_ml`	`False` — no closed-form marginal likelihood. TYPE: `bool`
`supports_point_only`	`False` — the posterior has no closed-form point estimate, so `optimisation_method="cross_validation"` is not supported. Use `"none"` and set hyperparameters manually. TYPE: `bool`

set_priors ¶

set_priors(*, c2: float | None = None, **kwargs) -> None

Set prior hyperparameters including cross-variable shrinkage c2.

Also computes Gamma hyperprior parameters for c2.

fill_in_from_vector ¶

fill_in_from_vector(pars: ndarray) -> None

Vector layout: [c1, c3, c2, mu?, theta?].

sample ¶

sample(data: ndarray, n_lags: int, covid_indices: ndarray, vars_in_levels: ndarray, N_draws: int, point_only: bool = False, progressbar: bool = True, soc: Optional[bool] = None, sur: Optional[bool] = None, rng: Optional[Generator] = None) -> SamplingResult

Run the full independent-NIW Gibbs estimation pipeline.

PARAMETER	DESCRIPTION
`data`	Input data array. TYPE: `ndarray`
`n_lags`	Number of VAR lags. TYPE: `int`
`covid_indices`	Indices for COVID dummy observations. TYPE: `ndarray`
`vars_in_levels`	Indicators for variables in levels. TYPE: `ndarray`
`N_draws`	Number of posterior draws. TYPE: `int`
`point_only`	Whether to request a point estimate. TYPE: `bool` DEFAULT: `False`
`progressbar`	Whether to display a progress bar. TYPE: `bool` DEFAULT: `True`
`soc`	Effective sum-of-coefficients flag. TYPE: `Optional[bool]` DEFAULT: `None`
`sur`	Effective single-unit-root flag. TYPE: `Optional[bool]` DEFAULT: `None`
`rng`	Random number generator. TYPE: `Optional[Generator]` DEFAULT: `None`

RETURNS	DESCRIPTION
`SamplingResult`	Posterior draws and point estimates.

RAISES	DESCRIPTION
`ValueError`	If point_only is `True` (no closed-form posterior point estimate).

sample_posterior_state ¶

sample_posterior_state(Y: ndarray, Z: ndarray, current_state: PosteriorState, rng: Optional[Generator] = None) -> PosteriorState

Return the next Gibbs-sampled posterior state.

Performs one full Gibbs sweep (β|Σ then Σ|β) using the stored prior from the last call to :meth:sample. current_state.sigma seeds the sweep's covariance. The Gibbs kernel accepts current_state.beta for interface compatibility but ignores it; it samples β conditional on Σ within this sweep rather than carrying β forward.