Sign Restriction Identification

Definition

Sign restriction identification is a set-identification strategy for structural VARs that recovers structural shocks by requiring their impulse responses to satisfy sign constraints — e.g., a contractionary monetary policy shock must raise the federal funds rate and lower prices — without imposing any zero restrictions on contemporaneous responses. Because sign restrictions typically do not point-identify the model, inference is over a set of admissible structural parameterizations.

Key Ideas

Avoids zero restrictions: Classical Cholesky (recursive) identification imposes that certain variables do not respond at all to certain shocks on impact. Sign restrictions replace these sharp exclusions with softer inequality constraints that are more easily defended on economic grounds.
Leaves variables of interest unrestricted: The "agnostic" variant (Uhlig 2004) deliberately omits sign restrictions on the variable being studied (e.g., output), so that the identification does not prejudge the answer to the research question.
Produces set-identified impulse responses: Rather than a unique structural decomposition, sign restrictions admit a continuum of admissible impulse vectors. Posterior inference integrates over this set (or optimizes over it), yielding bands rather than point estimates.
Convexity of the admissible set: The set $\mathcal{A}(B, \Sigma, K)$ of impulse vectors satisfying sign constraints at horizons $k=0,\ldots,K$ is the intersection of half-spaces in the $\alpha$ -parameterization and is therefore convex.
Horizon parameter K: Sign restrictions are imposed for $K$ periods after the shock. Longer $K$ tightens identification but can paradoxically alter substantive conclusions (Uhlig 2004 finds longer K pushes GDP bands upward).

How It Works

Impulse Vector Parameterization

Any structural impulse vector consistent with the reduced-form covariance $\Sigma$ can be written:

$a = \tilde{A}\alpha, \qquad \alpha \in S^m$

where $\tilde{A}$ is the lower Cholesky factor of $\Sigma$ and $\alpha$ is a unit-length vector on the $m$ -sphere. The full set of structural decompositions corresponds to all orthogonal rotations $Q$ with $\Sigma = \tilde{A}QQ'\tilde{A}'$ — parameterizing by $\alpha$ selects one column of $Q$ .

Sign Restriction Set

For a shock with impulse vector $a$ and restricted variables indexed by $j \in \mathcal{J}$ with signs $s_j \in \{-1, +1\}$ , the admissible set is:

$\mathcal{A}(B, \Sigma, K) = \left\{ a = \tilde{A}\alpha : s_j \cdot \iota_j' r_a(k) \geq 0,\; \forall j \in \mathcal{J},\; k = 0, \ldots, K \right\}$

where $r_a(k)$ is the impulse response vector at horizon $k$ and $\iota_j$ is the $j$ -th unit basis vector.

Pure Sign Restriction Approach (Uhlig 2004)

Draw $(B, \Sigma)$ from the unrestricted Normal-Wishart posterior (diffuse prior: $N_0=0$ , $\nu_0=0$ ).
Draw $\alpha$ uniformly from $S^m$ .
Compute $a = \tilde{A}\alpha$ and check sign constraints at all $k \leq K$ .
Retain accepted draws; discard the rest.
Report quantiles of retained impulse responses as posterior credible bands.

Penalty Function Approach (Uhlig 2004)

Instead of accepting/rejecting, minimize an asymmetric penalty for each posterior draw:

$\hat{\alpha} = \arg\min_{\alpha \in S^m} \Psi(a(\alpha)), \qquad \Psi(a) = \sum_{j \in \mathcal{J}} \sum_{k=0}^{K} f\!\left(\frac{s_j \iota_j' r_a(k)}{\sigma_j}\right)$

where $f(x) = -x$ for $x \geq 0$ (reward correct sign) and $f(x) = 100 \cdot (-x)$ for $x < 0$ (heavily penalize violation). This yields a point-like solution per draw and produces sharper bands. It also permits weighting — the reward component can favor large correctly-signed responses.

Why It Matters

Sign restrictions became a dominant identification strategy in structural macroeconomics after Uhlig (2004) and Canova-de Nicoló (2002) because they:

Respect agnosticism: Identifying assumptions can be restricted to variables whose responses are uncontroversial, leaving the variable of interest (output, inflation) free.
Avoid the price puzzle by construction: Contractionary shocks are defined to lower prices; no ad hoc commodity price controls are needed.
Reveal the role of implicit assumptions: Uhlig shows that Cholesky's zero-impact restriction on output is what drives conventional large-negative-GDP results — relaxing it makes results ambiguous, revealing that the data are less informative than point-identified methods suggest.

Open Questions

Lack of point identification: Wide posterior bands are the cost of agnosticism — the method may be too weak to distinguish competing theories.
Multiplicity of admissible models: Without additional structure, many economically different models satisfy the same sign restrictions; inference averages over all of them.
K-choice sensitivity: Results can be sensitive to the restriction horizon K in non-obvious ways (e.g., longer K weakening the conventional monetary contraction result in Uhlig 2004).
Extensions to narrative and external instruments: Recent work combines sign restrictions with external instrument (proxy SVAR) or narrative restrictions to tighten identification while maintaining partial agnosticism.