LongRunSVAR#

class cultivars.multivariate.structural.zero_restrictions.LongRunSVAR(result, *, order=None)[source]#

Long-run recursive identification, Blanchard-Quah (1989).

The restriction lives at the infinite horizon: the cumulated response matrix is lower triangular in the declared ordering, so the first shock is the only one with a permanent effect on the first variable, and so on. In the bivariate Blanchard-Quah economy – output growth first, unemployment second – the first shock is supply, the only one that moves the level of output forever, and demand is whatever remains. Computed in closed form: with \(F = A(1)^{-1} M(1)\) the long-run impact of the innovations,

\[F \Sigma_u F' = \Theta \Theta', \qquad \Theta \text{ lower triangular}, \qquad B = F^{-1} \Theta,\]

the lower Cholesky factor of the long-run covariance is the long-run impact of the shocks, and the impact matrix is \(F^{-1}\) times it. The system must be stationary for \(F\) to exist, and nearly nonstationary systems make it ill-conditioned: the result is checked against the innovation covariance and refused when it fails to reproduce it.

Parameters:

  • result (ClosedSystemResult) – The fitted closed, stationary reduced-form result to identify.

  • order (Sequence[str] | None) – The ordering of permanence, most permanent first. None declares the variables as they stand.

Raises:

SpecificationError – If the result is not a closed system, is not stationary, or the ordering is not a permutation of its names.

Variables:

  • _source – The closed reduced-form result being identified.

  • _perm – The ordering as column indices into the variable names.

See also

References

Blanchard, O. J., & Quah, D. (1989). The dynamic effects of aggregate demand and supply disturbances. American Economic Review, 79(4), 655-673.

Example

A bivariate system built so that the long-run impact matrix of the structural shocks is lower triangular: only the first shock moves the first variable’s level permanently. The long-run factorization recovers the impact matrix – which is not triangular – and the first shock series:

>>> import numpy as np
>>> from cultivars.multivariate.reduced_form.vector_autoregression import VAR
>>> rng = np.random.default_rng(0)
>>> A = np.array([[0.5, 0.1], [0.0, 0.4]])
>>> F = np.array([[1.0, 0.0], [0.5, 1.0]])
>>> B = (np.eye(2) - A) @ F
>>> eps = rng.standard_normal((602, 2))
>>> y = np.zeros((602, 2))
>>> for t in range(1, 602):
...     y[t] = A @ y[t - 1] + B @ eps[t]
>>> res = VAR(y, order=1, names=("dy", "u")).fit()
>>> svar = LongRunSVAR(res).identify()
>>> svar.scheme, svar.shock_names, svar.is_complete
('long-run', ('dy', 'u'), True)
>>> svar.long_run_impact.round(2).tolist()
[[1.06, -0.0], [0.67, 0.94]]
>>> svar.impact.round(2).tolist(), B.round(2).tolist()
([[0.44, -0.11], [0.31, 0.58]], [[0.45, -0.1], [0.3, 0.6]])
>>> bool(np.corrcoef(svar.structural_shocks()[:, 0], eps[1:, 0])[0, 1] > 0.95)
True

Attributes#

k_endog

Number of variables in the source system.

names

Variable labels of the source system.

ordering

The declared ordering of permanence, most permanent first.

source

The fitted reduced-form result being identified.

Methods#

identify()

Factor the long-run covariance and map back to impact.