QVAR#

class cultivars.multivariate.nonlinear.quantile.QVAR(endog, *, order, trend='c', names=None)[source]

Bases: _QuantileVectorAutoRegressionModel[QVARResult]

Quantile vector autoregression, after Koenker and Bassett, equation by equation.

Models the conditional \(\tau\)-quantile of every variable as linear in the system’s lags, with its own coefficient stack at every requested level,

\[Q_\tau(y_{i,t} \mid y_{t-1}, \ldots, y_{t-p}) = c_i(\tau)' d_t + \sum_{l=1}^{p} a_{i,l}(\tau)' y_{t-l},\]

so the tails get their own dynamics rather than the mean’s dynamics plus a symmetric error. Each equation at each level is one exact check-loss linear program, solved by HiGHS in primal form: the residual is split into its positive and negative parts, priced at \(\tau\) and \(1 - \tau\), under \(X\beta + u - v = y\). Equations and levels separate because the check loss is additive across both, so a fit is \(k \times Q\) independent programs over one design. No prior surface is offered: the base’s shrinkage prior is a Gaussian belief about conditional-mean coefficients and has no meaning for a check loss.

Variables:

  • _endog – The observed panel, from the base model.

  • _order – The autoregressive order p, from the base model.

  • _trend – The deterministic specification, from the base model.

  • _names – The variable labels, from the base model.

  • _prior – The base slot, held at _NoPrior because no prior is offered.

Parameters:

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

  • order (int) – Autoregressive order; zero fits the unconditional quantiles with deterministics only.

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

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

Raises:

See also

  • QVARResult – the record fit returns.

  • VAR – the conditional-mean system on the same design.

  • GARCH – a moving tail modelled through the variance instead.

References

Koenker, R., & Bassett, G. (1978). Regression quantiles. Econometrica, 46(1), 33-50.

White, H., Kim, T.-H., & Manganelli, S. (2015). VAR for VaR: Measuring tail dependence using multivariate regression quantiles. Journal of Econometrics, 187(1), 169-188.

Adrian, T., Boyarchenko, N., & Giannone, D. (2019). Vulnerable growth. American Economic Review, 109(4), 1263-1289.

Example

On a Gaussian VAR(1) the three levels share the same lag coefficients and differ in their intercepts, which order with the level:

>>> import numpy as np
>>> rng = np.random.default_rng(0)
>>> y = np.zeros((200, 2))
>>> for t in range(1, 200):
...     y[t] = 0.5 * y[t - 1] + rng.standard_normal(2)
>>> res = QVAR(y, order=1).fit(quantiles=(0.25, 0.5, 0.75))
>>> res.coefficients(0.5).shape
(1, 2, 2)
>>> res.quantile_path("y1", 0.75).shape
(199,)
>>> spread = res.coefficient_stacks.max(axis=0) - res.coefficient_stacks.min(axis=0)
>>> bool(spread.max() < 0.3)
True
>>> intercepts = res.deterministics[:, 0, 0]
>>> bool(intercepts[0] < intercepts[1] < intercepts[2])
True
>>> QVAR(y, order=1, names=("a", "b")).fit(quantiles=(0.5,)).names
('a', 'b')