SpectralDensityResult.phase#

SpectralDensityResult.phase(effect, cause)[source]#

The phase of the cross-spectrum f[effect, cause], in radians.

\(\arg f_{ij}(\omega)\) with \(i\) the effect and \(j\) the cause. Under the transfer convention \(e^{-i\omega l}\) used throughout the package, a negative phase at frequency \(\omega\) means cause leads effect by \(-\phi(\omega) / \omega\) observations at that frequency: for effect_t = cause_{t-1} exactly, the phase is \(-\omega\) on the whole grid. Half the literature states the mirror convention, so check signs against a known lead before trusting a plot.

Parameters:

  • effect (str) – The variable being explained.

  • cause (str) – The variable explaining it.

Return type:

NDArray[double]

Returns:

A real (n,) array in (-pi, pi].

Raises:

SpecificationError – If either label is unknown.

Example

A one-period lead recovers the phase \(-\omega\) exactly:

>>> import numpy as np
>>> from cultivars.multivariate.reduced_form.vector_autoregression import VAR
>>> rng = np.random.default_rng(1)
>>> x = rng.standard_normal(2000)
>>> y = np.zeros((2000, 2))
>>> y[:, 1] = x
>>> y[1:, 0] = x[:-1] + 0.05 * rng.standard_normal(1999)
>>> res = VAR(y, order=1, names=("gdp", "rate")).fit()
>>> density = SpectralDensity(res, n_frequencies=16).compute()
>>> lead = density.phase("gdp", "rate")[1:] / density.frequencies[1:]
>>> bool(np.allclose(lead, -1.0, atol=0.01))
True