Combining Zero and Sign Restrictions#

ZeroSignRestriction imposes exact zeros on the impact matrix alongside sign restrictions on impulse responses. Use it when part of your identification rests on a timing or exclusion argument you can defend outright, and the rest on the direction a response should take.

A worked example#

Take a three-variable system: a temperature anomaly, a measure of economic activity, and emissions. Economic activity moves emissions immediately, and emissions eventually move temperature, but the thermal inertia of the ocean means an activity shock cannot register in this year’s temperature anomaly. That is a zero, not a sign.

from impulso.identification import ZeroSignRestriction

scheme = ZeroSignRestriction(
    shock_names=["climate", "activity", "emissions"],
    zero_restrictions={
        # An activity shock has no contemporaneous effect on temperature:
        # the physical lag between emissions and warming is far longer
        # than the sampling frequency.
        "temperature": ["activity"],
    },
    sign_restrictions={
        "temperature": {"climate": "+"},
        "activity":    {"climate": "-", "activity": "+"},
        "emissions":   {"activity": "+", "emissions": "+"},
    },
    random_seed=42,
)

identified = fitted.set_identification_strategy(scheme)
irf = identified.impulse_response(horizon=20)

Specifying restrictions#

  • shock_names fixes the column order of the returned structural matrix. Name fewer shocks than you have variables and the rest are labelled unidentified_1, unidentified_2, … — those columns carry no restrictions and are rotation-arbitrary, so fevd() masks their shares.

  • zero_restrictions maps a variable to the shocks that do not move it on impact. Zeros bind at horizon 0 only; long-run zeros are not supported.

  • sign_restrictions uses the same format as SignRestriction: variable → shock → "+" or "-".

  • restriction_horizon=H imposes the signs at horizons 0..H. The zeros stay at impact.

A cell cannot be restricted to zero and to a sign at once — that is a contradiction at horizon 0, and construction fails with a ValueError.

How many zeros are admissible#

Sort the shocks by how many zeros they carry, most first. The shock in position j may carry at most n - j zeros. Break that and identification is impossible for any orthogonal matrix, so identify() raises before sampling starts rather than burning through rotations. At the limit — n - 1, n - 2, …, 0 — the zeros exactly identify the system and reproduce the Cholesky factor.

When draws fail#

A draw fails when no candidate satisfies the sign restrictions within n_rotations attempts.

Failed draws become NaN, not Cholesky

ZeroSignRestriction fills failed draws with NaN and warns once with the count and fraction. It deliberately does not fall back to the unrotated factor the way SignRestriction does: that fallback would silently break the zero restrictions, which are the whole point of the scheme.

NaN draws propagate into impulse-response and variance-decomposition summaries and are rejected by the scenario methods, so treat a non-trivial failure fraction as a result to act on, not a warning to suppress. Raise n_rotations, relax the signs, or set on_failure="raise" to stop at the first failure.

Tuning#

  • Each unpinned column sign is effectively a coin flip, so a scheme with k sign-restricted shocks accepts roughly one candidate in 2**k even when the restrictions are otherwise easy. Budget n_rotations accordingly.

  • Read the acceptance rate off the shock matrix:

    attrs = identified.shock_matrix().attrs
attrs["zero_sign_acceptance_rate"]     # fraction of draws identified
attrs["zero_sign_mean_attempts"]       # candidates drawn per draw
attrs["zero_sign_max_zero_violation"]  # largest |zero cell|, expect ~1e-14

  • Set random_seed for reproducibility.

  • Naming a shock without giving it any sign restriction leaves its column sign unidentified, so posterior summaries average over both directions. Impulso warns when this happens.

Draws are unweighted

No importance weight corrects for the volume element of the zero-restricted manifold, so set-identified results are not the uniform-conditional prior of Arias, Rubio-Ramírez and Waggoner (2018). See the identification explanation for what this does and does not affect.