VECMResult.normalized_alpha#

VECMResult.normalized_alpha(*, on=0)[source]#

Adjustment loadings rescaled to pair with normalized_beta().

Rescaling a cointegrating vector rescales its loading inversely, so the product and every test statistic are invariant and only the split between the two factors is convention. This returns the loadings that go with the readable basis.

Parameters:

on (int) – Index of the variable whose coefficient normalizes each vector.

Return type:

NDArray[double]

Returns:

A (k, r) array, empty when the rank is zero.

Raises:

SpecificationError – If the index is out of range or cannot carry the normalization.

Example

>>> import numpy as np
>>> rng = np.random.default_rng(0)
>>> common = np.cumsum(rng.standard_normal(300))
>>> y = np.column_stack([common, 2 * common + rng.standard_normal(300)])
>>> res = VECM(y, order=2, rank=1).fit()
>>> product = res.normalized_alpha() @ res.normalized_beta().T
>>> bool(np.allclose(product, res.cointegrating_matrix))
True
>>> res.normalized_alpha().ravel().round(2)
array([0.15, 2.63])
>>> res.normalized_alpha(on=2)
Traceback (most recent call last):
cultivars.exceptions.SpecificationError: on must index a variable, 0..1; got 2.