GibbsBVAR#

class cultivars.multivariate.large_dim.gibbs.GibbsBVAR(endog, *, order, prior=None, trend='c', names=None)[source]

Bases: _GibbsBayesianVectorAutoRegressionModel[GibbsBVARResult]

Non-conjugate Bayesian VAR: independent Normal-Wishart, sampled by Gibbs.

One model, two families of priors the conjugate path refuses. Static moment priors with any diagonal variance – the default IndependentNormalWishartPrior, which is Litterman’s full prior with the cross-equation weight kept, or a MinnesotaPrior passed directly – and the adaptive shrinkage hierarchies: HorseshoePrior, SpikeAndSlabPrior, DirichletLaplacePrior, NormalGammaPrior. The prior is \(\beta \sim N(\underline\beta, \underline V)\) independent of \(\Sigma \sim IW(\underline S, \underline\nu)\), so the coefficient conditional is the GLS posterior

\[\bar V = \bigl(\underline V^{-1} + \Sigma^{-1} \otimes X'X\bigr)^{-1}, \qquad \bar\beta = \bar V \bigl(\underline V^{-1} \underline\beta + (\Sigma^{-1} \otimes X')\, y\bigr),\]

one \((k^2 p + k n_{det})\)-dimensional Gaussian per sweep, and the covariance conditional is inverse-Wishart on the residual scatter. Static moment priors may be composed with + (a SumOfCoefficientsPrior or DummyInitialObservationPrior adds artificial rows the sampler treats as data); adaptive priors stand alone.

Variables:

  • _endog – The validated (nobs_total, k) panel.

  • _order – The autoregressive order.

  • _trend – The deterministic specification.

  • _names – One label per variable.

  • _prior – The validated prior.

Parameters:

  • endog (ArrayLike) – The observed panel, shape (nobs, k). A proper prior makes wide systems admissible, though each Gibbs sweep costs a Cholesky of the joint coefficient precision, so very large k belongs to the conjugate model.

  • order (int) – Autoregressive order, at least 1.

  • prior (_Prior | None) – Any finite-variance prior, static or adaptive. Defaults to IndependentNormalWishartPrior(). Adaptive priors do not compose with +.

  • trend (Literal['n', 'c', 'ct']) – Deterministic terms, "n", "c" or "ct".

  • names (Sequence[str] | None) – One label per variable. Defaults to y1 ... yk.

Raises:

  • SpecificationError – If the prior is improper (NoPrior), trend is not one of the three names, or names does not have one entry per column. An adaptive prior inside a composition is accepted here and refused at fit.

  • DimensionError – If the panel is not a time-down-the-rows matrix or too short for the order.

See also

  • GibbsBVARResult – the posterior record fit returns.

  • BVAR – the conjugate model: exact posterior, closed-form evidence, any dimension, but a Kronecker-restricted prior.

  • BVARSV – the same sampler with stochastic volatility on the innovations.

References

Koop, G., & Korobilis, D. (2010). Bayesian multivariate time series methods for empirical macroeconomics. Foundations and Trends in Econometrics, 3(4), 267-358.

George, E. I., Sun, D., & Ni, S. (2008). Bayesian stochastic search for VAR model restrictions. Journal of Econometrics, 142(1), 553-580.

Example

>>> import numpy as np
>>> from cultivars.bayes.priors import SpikeAndSlabPrior
>>> rng = np.random.default_rng(0)
>>> y = np.zeros((200, 3))
>>> for t in range(1, 200):
...     y[t] = 0.6 * y[t - 1] + rng.standard_normal(3)
>>> model = GibbsBVAR(y, order=1, prior=SpikeAndSlabPrior())
>>> res = model.fit(n_draws=300, n_burn=100, seed=0)
>>> res.inclusion_probabilities().shape
(1, 3, 3)
>>> bool(np.all(np.diag(res.inclusion_probabilities()[0]) > 0.5))
True
>>> from cultivars.bayes.priors import HorseshoePrior, SumOfCoefficientsPrior
>>> composed = GibbsBVAR(y, order=1, prior=HorseshoePrior() + SumOfCoefficientsPrior())
>>> composed.fit(n_draws=60, n_burn=20, seed=0)
Traceback (most recent call last):
cultivars.exceptions.SpecificationError: adaptive shrinkage priors do not compose: ...