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Put confidence bands on a VAR impulse response

tsecon.var_irf gives you the point path and nothing else — deliberately, so that the cheap call stays cheap. The moment you want to say whether a response is distinguishable from zero, switch to tsecon.var_irf_bands, which takes the same (data, lags, horizon, orth) arguments and returns a dict with point, se, lower, and upper.

The recipe

import numpy as np, tsecon

rng = np.random.default_rng(0)                      # a toy 2-variable macro system
A = np.array([[0.55, 0.10], [0.35, 0.60]])          # VAR(1) coefficients
L = np.array([[1.00, 0.00], [0.50, 0.80]])          # correlated innovations
y = np.zeros((240, 2))
for t in range(1, 240):
    y[t] = A @ y[t - 1] + L @ rng.standard_normal(2)

b = tsecon.var_irf_bands(y, lags=2, horizon=10, method="bootstrap",
                         n_boot=2000, seed=0, alpha=0.10)   # 90% percentile bands
point, lower, upper = (np.array(b[k])[:, 1, 0] for k in ("point", "lower", "upper"))
for h in (0, 2, 4, 6, 8, 10):
    print(f"h={h:2d}   {point[h]:+.3f}   [{lower[h]:+.3f}, {upper[h]:+.3f}]")

h= 0   +0.511   [+0.406, +0.607]
h= 2   +0.680   [+0.533, +0.797]
h= 4   +0.471   [+0.293, +0.594]
h= 6   +0.253   [+0.095, +0.372]
h= 8   +0.118   [+0.003, +0.227]
h=10   +0.049   [-0.024, +0.139]

Reading it

Every array is indexed [horizon][response][shock], exactly like var_irf, so [:, 1, 0] is "variable 1's response to a shock in variable 0". The response is hump-shaped, peaks at h=2, still excludes zero at h=8, and no longer does by h=10 — that last line is the finding, not a footnote. Slicing the band arrays the same way you slice var_irf means you can drop banded responses into an existing plot without reshaping anything.

Two knobs decide what the band means:

  • alpha is the non-coverage: alpha=0.10 is a 90% band, alpha=0.32 the 68% band macro papers often plot.
  • seed makes the bootstrap reproducible. Report it. A bootstrap band without a seed and an n_boot is not a replicable number.

The asymptotic alternative

method="asymptotic" — the default — gives Lütkepohl delta-method standard errors and Wald bands (point ± z·se) with no resampling. It is instant, and it hands you a usable se array:

a = tsecon.var_irf_bands(y, lags=2, horizon=10, method="asymptotic", alpha=0.10)
print("method:", a["method"], " n_boot:", a["n_boot"])
for h in (0, 4, 8):
    print(f"h={h:2d}   se {np.array(a['se'])[h, 1, 0]:.3f}   "
          f"[{np.array(a['lower'])[h, 1, 0]:+.3f}, {np.array(a['upper'])[h, 1, 0]:+.3f}]")

method: asymptotic  n_boot: None
h= 0   se 0.058   [+0.415, +0.606]
h= 4   se 0.091   [+0.322, +0.620]
h= 8   se 0.072   [-0.001, +0.236]

Note that the two methods disagree about h=8: the bootstrap band excludes zero, the asymptotic one just barely includes it. That disagreement is information — delta-method bands are symmetric by construction and are known to undercover at longer horizons in short samples, which is why method="bootstrap" is the one to reach for when the answer matters. n_boot comes back None on the asymptotic branch so a logged result always records which method ran.

Gotchas

  • orth=True is the default and it is an identification assumption. The bands describe sampling uncertainty about a Cholesky-ordered response; they say nothing about whether the ordering is defensible. For a causal reading, go to sign restrictions or a local projection.
  • Check stability first. tsecon.var_fit(y, lags=2)["is_stable"] must be True; near-unit-root systems make long-horizon bands unreliable under either method.
  • bias_correct=True applies the Kilian (1998) bootstrap bias correction on the bootstrap branch — worth trying when the VAR is persistent.
  • Want cumulative (level) responses? Pass cumulative=True; it means the same thing here as in var_irf.

See also