GraphicalVAR#

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

Bases: _SparseVectorAutoRegressionModel[GraphicalVARResult]

Graphical VAR, Barigozzi-Brownlees: sparse dynamics, sparse precision.

Two stages. The dynamics are a lasso VAR,

\[\min_{A}\ \tfrac{1}{2}\sum_t \lVert y_t - c - \textstyle\sum_i A_i y_{t-i} \rVert^2 + \lambda \sum_i \lVert A_i \rVert_1,\]

with \(\lambda\) fixed by the caller or chosen by rolling-origin forecast error along a path; the contemporaneous network is then estimated on the residuals by Meinshausen-Buhlmann nodewise lasso regressions – each residual on all the others – with each node’s penalty chosen by K-fold cross-validation, which is legitimate in the second stage because residual rows carry no serial ordering once the dynamics are removed. Deterministic terms are never penalized.

Variables:

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

  • _order – The autoregressive order.

  • _trend – The deterministic specification.

  • _names – One label per variable.

Parameters:

  • endog (ArrayLike) – The observed panel, shape (nobs, k).

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

  • trend (Literal['n', 'c', 'ct']) – Deterministic terms, "n", "c" or "ct", never penalized.

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

Raises:

  • SpecificationError – If order is below 1, trend is not one of the three names, or names does not have one entry per column.

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

See also

  • GraphicalVARResult – the record fit returns.

  • SparseVAR – the first stage on its own, with the elastic-net and group penalties.

  • VAR – the unpenalized model, for systems small enough to afford it.

References

Barigozzi, M., & Brownlees, C. (2019). NETS: Network estimation for time series. Journal of Applied Econometrics, 34(3), 347-364.

Meinshausen, N., & Buhlmann, P. (2006). High-dimensional graphs and variable selection with the lasso. Annals of Statistics, 34(3), 1436-1462.

Example

>>> import numpy as np
>>> rng = np.random.default_rng(0)
>>> y = np.zeros((300, 5))
>>> for t in range(1, 300):
...     y[t] = 0.5 * y[t - 1] + rng.standard_normal(5)
>>> res = GraphicalVAR(y, order=1).fit(seed=0)
>>> res.precision.shape
(5, 5)
>>> res.contemporaneous_adjacency().dtype
dtype('bool')
>>> bool(res.lambda_path.size) and bool(res.cv_errors.size)
True
>>> fixed = GraphicalVAR(y, order=1).fit(lam=0.05, seed=0)
>>> fixed.lam, fixed.lambda_path.shape, fixed.lam_nodes.shape
(0.05, (0,), (5,))