VARMA._recursive_innovations#

VARMA._recursive_innovations(deterministic, ar, ma)[source]#

Innovations implied by a parameter draw, computed forward in time.

Pre-sample innovations are set to zero; under an invertible moving-average block their influence decays geometrically, which is exactly why the caller checks invertibility before trusting this.

Parameters:

  • deterministic (NDArray[double]) – Deterministic coefficients, one row per term.

  • ar (NDArray[double]) – (p, k, k) autoregressive stack.

  • ma (NDArray[double]) – (q, k, k) moving-average stack.

Return type:

NDArray[double]

Returns:

An (nobs, k) array, zero for the first p rows.

Example

At the true parameters the recursion recovers the innovations that generated the sample almost exactly:

>>> import numpy as np
>>> rng = np.random.default_rng(0)
>>> u = rng.standard_normal((302, 2))
>>> y = np.zeros((302, 2))
>>> for t in range(1, 302):
...     y[t] = 0.5 * y[t - 1] + u[t] + 0.4 * u[t - 1]
>>> model = VARMA(y[2:], order=(1, 1))
>>> ar, ma = 0.5 * np.eye(2)[None], 0.4 * np.eye(2)[None]
>>> recovered = model._recursive_innovations(np.zeros((1, 2)), ar, ma)
>>> recovered.shape, bool(np.allclose(recovered[0], 0.0))
((300, 2), True)
>>> bool(np.corrcoef(recovered[50:, 0], u[52:, 0])[0, 1] > 0.99)
True