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Identify a monetary policy shock with sign restrictions

A Cholesky ordering is a point identification you usually cannot defend. Sign restrictions ask for less: state only the signs you are confident about — "a contractionary policy shock raises the policy rate and lowers prices for a couple of quarters" — and let the data report the set of responses consistent with them. The width of the resulting band is the finding.

The recipe

import numpy as np, tsecon

rng = np.random.default_rng(2)                       # columns: 0 output, 1 prices, 2 rate
A0 = np.array([[0.6, 0.2, -0.3],                     # impact of [demand, supply, policy]
               [0.4, -0.5, -0.2],
               [0.3, 0.1, 0.5]])
B = np.diag([0.6, 0.5, 0.7])
Y = np.zeros((300, 3))
for t in range(1, 300):
    Y[t] = B @ Y[t - 1] + A0 @ rng.standard_normal(3)

R = [(2, 2, 0, "+"), (2, 2, 1, "+"),                 # policy shock raises the rate, h=0,1
     (1, 2, 0, "-"), (1, 2, 1, "-")]                 # and lowers prices, h=0,1
out = tsecon.sign_restricted_svar(Y, restrictions=R, lags=2, horizon=8,
                                  n_draws=1000, seed=0)

q = np.array(out["quantiles"])                       # [h][variable][shock][prob]
for h in (0, 2, 4, 6, 8):                            # probs = 5, 16, 50, 84, 95%
    print(f"h={h}   output {q[h,0,2,2]:+.3f}   68% [{q[h,0,2,1]:+.3f}, {q[h,0,2,3]:+.3f}]")
print("acceptance rate:", round(out["diagnostics"]["acceptance_rate"], 4))


h=0   output -0.167   68% [-0.559, +0.264]
h=2   output -0.092   68% [-0.201, +0.056]
h=4   output -0.036   68% [-0.081, +0.010]
h=6   output -0.010   68% [-0.034, +0.006]
h=8   output -0.003   68% [-0.014, +0.004]

acceptance rate: 0.4237

Reading it

A restriction is a (variable, shock, horizon, sign) tuple. Here shock 2 is the policy shock, and we constrain only variables 2 (the rate) and 1 (prices). Output is deliberately left unrestricted — its response is the question, and its band is the answer.

That answer, honestly reported: output is centred negative at every horizon but the 68% band straddles zero throughout. In this sample, sign restrictions on the rate and prices are not enough to sign the output response. That is a real result, and a point-identified scheme would have hidden it behind a single confident line.

quantiles is [horizon][variable][shock][prob] with probs = [0.05, 0.16, 0.50, 0.84, 0.95], so index 2 is the median and 1/3 bracket the 68% band. The median is not "the" IRF — it mixes different rotations at different horizons, so no single structural model produces that path.

The acceptance rate is a diagnostic

lo, hi = np.array(out["set_min"]), np.array(out["set_max"])
print(f"identified set for output at h=0: [{lo[0,0,2]:+.3f}, {hi[0,0,2]:+.3f}]")
print("rotations tried:", out["diagnostics"]["rotations_tried"],
      " accepted:", out["diagnostics"]["accepted"])

identified set for output at h=0: [-0.832, +0.649]
rotations tried: 2360  accepted: 1000

set_min / set_max are the raw identified-set envelope — every value a surviving rotation produced, with no probability attached. Compare that [-0.83, +0.65] to the 68% quantile band [-0.56, +0.26]: the gap between them is entirely the Haar prior on rotations, not evidence.

An acceptance rate of 0.42 is healthy. Rates near 1e-5 mean your "posterior" is a handful of draws and your restrictions are close to mutually inconsistent — watch it every time you add a restriction, because acceptance decays roughly exponentially in their number.

Gotchas

  • The Haar prior is not uninformative about the responses you care about (Baumeister-Hamilton 2015). Part of any sign-restricted band is prior. For prior-robust bounds use tsecon.robust_svar_bounds; for the single coherent draw closest to the median path, tsecon.fry_pagan_svar.
  • Do not stack restrictions to narrow the band without watching acceptance. Narrowing by assumption is not evidence.
  • seed and n_draws belong in your write-up. Both change the numbers.
  • Need a zero restriction too ("this shock has no contemporaneous effect on output")? That is tsecon.zero_sign_svar. Have an external instrument? Use tsecon.proxy_svar instead — it point-identifies.

See also