QVARResult.pseudo_r_squared

QVARResult.pseudo_r_squared(quantile)

Koenker-Machado goodness of fit per equation at one level.

1 - loss / loss_location: the share of check loss the regression removes relative to the unconditional quantile,

\[R^1(\tau) = 1 - \frac{\sum_t \rho_\tau(y_t - \hat Q_\tau(y_t \mid x_t))} {\sum_t \rho_\tau(y_t - \hat q_\tau)}.\]

Local to its level by construction – an R1 at the median says nothing about the tails – and not comparable to a least-squares R^2.

Parameters:

quantile (float) – An estimated level.

Return type:

dict[str, float]

Returns:

A mapping from equation name to its R1.

Raises:

SpecificationError – If the level was not estimated.

Example

A persistent system explains more check loss than white noise at every level, and every R1 lies in [0, 1]:

>>> import numpy as np
>>> rng = np.random.default_rng(0)
>>> y = np.zeros((300, 2))
>>> for t in range(1, 300):
...     y[t] = 0.7 * y[t - 1] + rng.standard_normal(2)
>>> res = QVAR(y, order=1).fit(quantiles=(0.1, 0.5))
>>> fit = res.pseudo_r_squared(0.5)
>>> sorted(fit), bool(all(0.0 <= v <= 1.0 for v in fit.values()))
(['y1', 'y2'], True)
>>> noise = QVAR(rng.standard_normal((300, 2)), order=1).fit(quantiles=(0.5,))
>>> bool(fit["y1"] > noise.pseudo_r_squared(0.5)["y1"])
True