Uhlig (2004) What Are the Effects of Monetary Policy on Output? Results from an Agnostic Identification Procedure

Summary

Uhlig proposes an agnostic identification strategy for monetary policy shocks in a vector autoregression (VAR): impose sign restrictions only on variables other than output (prices, nonborrowed reserves, federal funds rate must respond in the expected direction for $K$ months), and leave the output response completely unconstrained. Applying this to a six-variable monthly VAR (1965–2003), he finds that monetary policy shocks have an ambiguous effect on real GDP — within $\pm$0.2% with two-thirds probability — consistent with both the conventional negative view and near-neutrality. The key insight is that Cholesky identification's implicit zero-impact restriction on GDP is what drives conventional large-negative results, not the data itself.

Key Claims

• A contractionary monetary policy shock defined only by sign restrictions on prices, nonborrowed reserves, and the federal funds rate leaves the GDP response statistically indeterminate.
• With $K=5$, two-thirds of posterior probability mass for the peak GDP impulse response lies within $\pm 0.2\%$; the median is near zero.
• Monetary policy shocks explain roughly 5–10% (median) of GDP forecast-error variance at a 5-year horizon; the 84th percentile is around 25%.
• Cholesky identification implies monetary policy shocks account for ~50% of GDP variance — Uhlig argues this is implausibly large and results from the zero-contemporaneous-restriction on GDP, not from the data.
• Longer restriction horizons $K$ shift GDP bands upward (less negative), suggesting tighter sign restrictions do not strengthen the conventional view.
• The penalty-function approach and the pure sign-restriction approach agree on the core result: GDP ambiguity is robust across methods.
• There is no price puzzle under sign restriction identification, by construction — the identifying restrictions rule out price increases directly.

Model Specification

VAR: $y_t = B_1 y_{t-1} + \cdots + B_\ell y_{t-\ell} + u_t$, $E[u_t u_t'] = \Sigma$, with $m=6$ variables (real GDP, GDP deflator, commodity price index, total reserves, nonborrowed reserves, federal funds rate), $\ell=12$ lags, monthly data January 1965–December 2003, variables in log levels (except the federal funds rate (FFR)).

Impulse vector: Any impulse vector can be written $a = \tilde{A}\alpha$ where $\tilde{A}$ is the lower Cholesky factor of $\Sigma$ and $\alpha \in S^m$ is a unit-length vector. All structural shocks consistent with a given reduced-form are parameterized this way.

Sign restriction set: $\mathcal{A}(B, \Sigma, K)$ = set of admissible impulse vectors satisfying, for each restricted variable $j$ and horizon $k=0,\ldots,K$:

$$\iota_j' r_a(k) \geq 0 \quad (\text{or} \leq 0)$$

where $r_a(k)$ is the impulse response at horizon $k$ and $\iota_j$ selects variable $j$. This set is the intersection of half-spaces in $\alpha$-space and is convex.

Pure sign restriction approach: Use a Normal-Wishart prior with $N_0=0$, $\nu_0=0$ (diffuse). Draw $(B, \Sigma)$ from posterior, draw $\alpha$ uniformly from $S^m$, retain draws where $a = \tilde{A}\alpha \in \mathcal{A}(B,\Sigma,K)$. Posterior bands are the quantiles of retained draws.

Penalty function approach: For each posterior draw of $(B,\Sigma)$, find:

$$\hat{\alpha} = \arg\min_{\alpha \in S^m} \Psi(a(\alpha)), \quad \Psi(a) = \sum_j \sum_{k=0}^{K} f\!\left(\frac{\iota_j r_a(k)}{\sigma_j}\right)$$

where $f(x)$ is asymmetric: slope 1 for $x \geq 0$ (reward), slope 100 for $x < 0$ (penalty). This produces sharper bands that closely mimic point identification.

Variance share: $\phi_{a,j,k} = (r_{a,j}(k))^2 \big/ \sum_{i=1}^{m}(r_{i,j}(k))^2$, the fraction of $j$'s forecast variance at horizon $k$ attributable to shock $a$.

Concepts Introduced or Extended

• Sign Restriction Identification
• Structural Identification
• Monetary Policy Shocks
• Impulse Response Function
• Bayesian VAR
• Price Puzzle
• Monetary Neutrality

Entities Mentioned

• Harald Uhlig
• Ben Bernanke
• Ilian Mihov

Quotes

"The agnostic procedure leaves the response of output to a monetary policy shock unrestricted. Essentially, I find that this agnostic procedure results in an ambiguous response of output."

"The Cholesky decomposition implicitly imposes that output does not respond contemporaneously to monetary policy shocks. This restriction has considerable bite and drives the output results."

My Take

This paper is methodologically important: it shows that a widely cited finding (monetary contractions depress output) rests on an implicit zero-restriction that has nothing to do with economics. The sign restriction approach is elegant precisely because it avoids imposing restrictions on the object of interest. The main limitation is that "agnosticism" produces wide, nearly uninformative bands — the result is less "monetary policy is neutral" and more "we cannot tell." The K-sensitivity finding is underappreciated: requiring sign restrictions to hold for longer paradoxically weakens the negative-output evidence, which ought to concern users who treat K as a robustness check. The penalty function approach rescues point-like inference but at the cost of optimizing over a somewhat arbitrary loss function.