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Replication — Uhlig (2005)

The Ramey-Zubairy page targets a published number. This one targets a published shape of uncertainty: Uhlig (2005) identified a contractionary monetary policy shock purely from sign restrictions and reported that the data then refuse to say what the shock does to output — the 68% band on the real-GDP response straddles zero at essentially every horizon, while the price puzzle of recursive VARs disappears. The width of the band is the finding, which makes this the natural flagship for sign_restricted_svar: a replication that must reproduce an ambiguity, quantitatively.

.venv/bin/python docs/examples/replication_uhlig_monetary.py

The data is the paper's own: 468 months (1965:1–2003:12) of six US series — real GDP (interpolated to monthly), the GDP deflator, a commodity price index, the federal funds rate, nonborrowed reserves, total reserves — all in 100·log except the funds rate, exactly as distributed in the RATS replication archive and re-distributed by the R package VARsignR as uhligdata. It is committed at fixtures/uhlig2005.csv, so the replication runs fully offline.

The identification

A contractionary monetary policy shock is defined as any structural shock under which, for months \(k = 0, \dots, 5\) after impact (Uhlig's \(K = 5\)),

  • the GDP deflator is not positive,
  • the commodity price index is not positive,
  • nonborrowed reserves are not positive,
  • the federal funds rate is not negative,

and — the agnostic part — real GDP and total reserves are left unrestricted, because the output response is the question. In tsecon's (variable, shock, horizon, sign) convention that is 24 tuples:

restr = []
for h in range(6):                      # horizons 0..5 = Uhlig's K = 5
    restr += [(1, 3, h, "-"),           # GDP deflator       not positive
              (2, 3, h, "-"),           # commodity prices   not positive
              (4, 3, h, "-"),           # nonborrowed res.   not positive
              (3, 3, h, "+")]           # federal funds rate not negative

r = tsecon.sign_restricted_svar(data, restr, lags=12, horizon=60,
                                n_draws=2000, max_tries=1000,
                                seed=0, lambda1=10.0)

VAR(12) on the monthly levels, 60-month impulse responses, 2000 reduced-form posterior draws. The run accepts 2000 of 2000 posterior draws in 34,106 Haar rotations (acceptance rate 5.9%) and takes about 3 seconds through the Rust core.

The result

Pointwise 16/50/84% posterior quantiles (Uhlig's Fig. 6 plots exactly these three), seed 0. Responses in percent (the data are 100·log), funds rate in percentage points:

h (months) real GDP 16% 50% 84% deflator 16% 50% 84% ffr 50% commod. 84% NBR 84%
0 −0.007 +0.097 +0.190 −0.063 −0.036 −0.014 +0.197 −0.375 −0.394
3 +0.001 +0.145 +0.277 −0.100 −0.059 −0.027 +0.221 −0.447 −0.339
6 −0.072 +0.073 +0.209 −0.118 −0.067 −0.029 +0.114 −0.690 −0.181
12 −0.041 +0.091 +0.212 −0.207 −0.136 −0.071 +0.033 −0.905 −0.184
24 −0.110 +0.036 +0.165 −0.366 −0.232 −0.111 −0.051 −1.052 −0.028
36 −0.119 +0.039 +0.190 −0.522 −0.325 −0.162 −0.085 −0.778 +0.202
48 −0.128 +0.041 +0.212 −0.633 −0.392 −0.184 −0.089 −0.465 +0.372
60 −0.142 +0.035 +0.214 −0.707 −0.427 −0.191 −0.072 −0.197 +0.448

Against the paper's three headline claims:

(a) No price puzzle. Uhlig's abstract: the GDP price deflator "falls only slowly" after a contractionary shock. Here the deflator's 84% quantile is negative at every one of the 61 horizons (its maximum is −0.014, at impact) — the restrictions only impose this for months 0–5; that it stays negative for five years is the replicated finding. The median declines gradually to −0.43% by month 60: falling, and only slowly. The commodity price index shows the same pattern (84% quantile negative through month 60).

(b) The GDP response is ambiguous. The paper's quantitative statement: "with a two-thirds probability, a typical shock will move real GDP by up to 0.2 percent" — in either direction, "consistent with the conventional view, but also consistent with monetary neutrality". Here the 16–84% band on real GDP straddles zero at all 55 horizons from month 6 to month 60, and over those horizons stays within [−0.14, +0.22]% — the ~0.2-percent magnitude of the text. (In months 0–5 the median is positive, up to +0.15 — the short-lived "output puzzle" tilt visible in Uhlig's Fig. 6 — and the 16% quantile grazes zero at h = 3; the ambiguity claim lives at h ≥ 6, and is pinned there.)

(c) The shock looks like monetary policy. The funds-rate median rises about +0.2 percentage points on impact and decays within roughly two years; nonborrowed reserves fall on impact and recover. Those magnitudes are our run's, not digitized from the paper's figures — the claims pinned against the paper are the sign/straddle facts and the ~0.2% band magnitude its text states.

The regression test (test_replication_uhlig.py) pins all of (a)–(c) at 300 draws with a fixed seed — across seeds the 16/50/84 quantiles at these horizons move by only ~0.02–0.04, so the pins hold with generous margins.

What matches Uhlig's procedure, and what does not

This is Uhlig's pure-sign-restriction (rejection) approach — the paper's benchmark, Fig. 6 — not its penalty-function variant (which trades the hard accept/reject for a sign-violation penalty and is not implemented here; tsecon's fry_pagan_svar addresses the "median mixes models" concern the penalty function also targets). The correspondence and the honest deviations:

  • Impulse-vector distribution — matches. Uhlig draws an impulse vector uniformly on the unit sphere per posterior draw; tsecon draws a full Haar rotation and reads off the restricted column. A column of a Haar-distributed orthogonal matrix is uniform on the sphere, so the candidate distribution is identical. The per-shock sign flip (try \(-\alpha\) when \(\alpha\) fails) matches the reference implementation's accept logic (VARsignR uhlig.reject).
  • Weighting across posterior draws — differs, second order. tsecon keeps the first accepted rotation per posterior draw (up to max_tries), so every reduced-form draw that yields any acceptance is weighted equally. Uhlig kept every accepted (draw, sub-draw) pair, which weights reduced-form draws in proportion to the Haar volume of their accepted set. With acceptance rates in the 5–6% range and stable across draws, the difference is second order here.
  • Reduced-form prior — differs, made negligible. Uhlig used a flat Normal-inverse-Wishart prior on a VAR with no constant. tsecon's sampler draws from a Minnesota-NIW posterior with an intercept; the replication sets lambda1 = 10.0, which loosens the Minnesota shrinkage to the point where the likelihood dominates (lambda1 = 10 and lambda1 = 2 give bands identical to ~0.01). The script re-runs at the library default lambda1 = 0.2 and prints the comparison: band edges move by a few hundredths of a percent and no conclusion changes.
  • Strict vs weak inequalities — measure zero. Uhlig's restrictions are weak ("not positive"); tsecon's checker requires strict signs. For continuous posterior draws the difference has probability zero.
  • Draw counts. Uhlig's published bands use all accepted draws from a large joint sample; ours use 2000 accepted draws, which is past the point where the 16/50/84 quantiles are stable (seed-to-seed movement ~0.02).

What is deliberately not claimed: agreement with figure pixels. The paper's figures were not digitized; every number pinned here comes from the paper's text and the procedure's own output.

Citation. Uhlig, H. (2005), "What are the effects of monetary policy on output? Results from an agnostic identification procedure," Journal of Monetary Economics 52(2):381–419. Data redistributed from the VARsignR package (Danne, 2015, CRAN, GPL ≥ 3), originally the paper's RATS replication archive; the underlying series are US-government statistics (FRED/BEA) plus a Global Financial Data commodity index. Please cite the paper if you use the data.

See also. sign_restricted_svar model card · guide: causal identification, which walks the same punchline on synthetic data · robust_svar_bounds for the Baumeister-Hamilton caveat that part of any sign-restriction band is prior, not evidence — a caveat Uhlig's ambiguity finding is robust to, since removing the Haar prior only widens the set around zero.