Long-Run and Sign Restrictions in Structural VARs

Why Structural Restrictions Matter

A reduced-form VAR describes how observed variables respond over time to their own past values and to reduced-form innovations. Those innovations can be correlated across equations, so their separate components do not automatically represent distinct economic shocks. A structural VAR adds a model for mutually uncorrelated shocks and asks how each shock affects each variable. The reduced-form data alone generally do not determine that decomposition uniquely; restrictions on long-run effects or response signs can narrow the possibilities.

The distinction between a unique answer and a set of admissible answers is central. A long-run zero restriction can, in some systems, determine a rotation up to sign conventions. Sign restrictions often leave a continuum of rotations consistent with the stated economic predictions. In that case the response is set-identified, not point-identified.

Structural Shocks and Long-Run Effects

Consider a stable reduced-form VAR for a $k$-vector $y_t$, with reduced-form innovation $u_t$. Suppose its innovation covariance matrix is $\Sigma_u$, which is positive definite. A structural representation of the innovations is

u_t=B\varepsilon_t,\qquad E(\varepsilon_t)=0,\qquad E(\varepsilon_t\varepsilon_t')=I_k.

Here $B$ is the impact matrix, and the components of $\varepsilon_t$ are standardized structural shocks that are uncorrelated contemporaneously. Thus $\Sigma_u=BB'$. Let the reduced-form moving-average representation be $y_t=\text{deterministic terms}+\sum_{h=0}^{\infty}\Psi_h u_{t-h}$, where $\Psi_0=I_k$. The structural response matrix at horizon $h$ is $\Theta_h=\Psi_h B$: its $j$th column gives the responses to a unit structural shock $j$.

Definition: For a stable VAR, the structural long-run impact matrix is $$ \Theta(1)=\sum_{h=0}^{\infty}\Theta_h =\left(\sum_{h=0}^{\infty}\Psi_h\right)B. $$ Its entry in row $i$, column $j$ is the cumulative response of variable $i$ to a unit shock $j$, summed over all horizons. In a stable VAR with autoregressive matrices $A_1,\ldots,A_p$, $\sum_{h=0}^{\infty}\Psi_h=(I_k-\sum_{\ell=1}^p A_\ell)^{-1}$.

Stability ensures that the infinite sum converges. In a cointegrated or otherwise nonstationary system, a finite stable-VAR long-run multiplier may not exist in this form; long-run restrictions must then be formulated for the appropriate common-trend effects. The calculations here use the stable VAR setting.

Rotations and the Identification Problem

Choose any matrix $P$ satisfying $PP'=\Sigma_u$, such as a Cholesky factor. Every impact matrix of the form $B=PQ$, where $Q$ is orthogonal ($QQ'=Q'Q=I_k$), produces the same reduced-form innovation covariance:

BB'=PQQ'P'=PP'=\Sigma_u.

Conversely, any impact matrix $B$ satisfying $BB'=\Sigma_u$ can be written this way when $\Sigma_u$ is positive definite. Indeed, set $Q=P^{-1}B$. Then $QQ'=P^{-1}\Sigma_u(P^{-1})'=I_k$, so $Q$ is orthogonal. The reduced-form covariance therefore does not select one rotation; structural restrictions are needed to distinguish among the observationally equivalent candidates.

If $C=\sum_{h=0}^{\infty}\Psi_h$, the structural long-run matrix for a candidate rotation is $CPQ$. A long-run restriction sets a specified entry, or other specified function of entries, of this matrix to a required value, commonly zero. It is a restriction on the cumulative response, not necessarily on the impact response or on the response at any one intermediate horizon.

Worked Example: A Long-Run Zero Restriction

Suppose a hypothetical stable two-variable reduced form has $P=I_2$, and its long-run reduced-form multiplier is $C=\begin{pmatrix}\sqrt{0.91}&1.3\\1.3&5\end{pmatrix}$. This is compatible with stability: $C$ is positive definite and its eigenvalues exceed $0.5$, so the implied VAR(1) matrix $I_2-C^{-1}$ has eigenvalues strictly between $-1$ and $1$. Impose the restriction that structural shock 1 has zero cumulative effect on variable 1, namely $(CQ)_{11}=0$.

Write a two-dimensional rotation as $Q=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$. The first entry of $CQ$ is $\sqrt{0.91}\cos\theta+1.3\sin\theta$. Setting it to zero and choosing $\cos\theta>0$ gives $\cos\theta=1.3/\sqrt{2.60}$ and $\sin\theta=-\sqrt{0.91}/\sqrt{2.60}$, since $1.3^2+0.91=2.60$.

Numerically, $\cos\theta\approx0.8062$ and $\sin\theta\approx-0.5916$. Substitution verifies the restriction: $1.3(-0.5916)+0.9539(0.8062)\approx-0.7691+0.7690\approx0$, with the small remaining difference due to rounding. In exact form, the two terms are $-1.3\sqrt{0.91}/\sqrt{2.60}$ and $1.3\sqrt{0.91}/\sqrt{2.60}$, so they cancel exactly. The sign convention $\cos\theta>0$ fixes the otherwise arbitrary sign of shock 1; without it, the opposite column also satisfies the zero restriction.

Sign Restrictions on Impulse Responses

A sign restriction specifies whether selected responses must be positive or negative, often over a stated set of horizons. For example, an economic interpretation might require that one shock raise output and lower prices for the first few periods. Such restrictions are inequalities on entries of $\Theta_h=\Psi_hPQ$. The variables, shock orientation, and horizons must be stated: a claim about the impact response is not the same as a claim that the response keeps its sign for several periods.

Definition: A rotation is admissible under a given set of sign restrictions if every specified response entry has the required sign at every specified horizon. The identified set is the collection of structural response matrices generated by all admissible rotations. It is point-identified only if the restrictions leave a unique response matrix after the chosen normalizations.

Signs do not by themselves fix the scale or orientation of a shock. Here shock variance is normalized to one, and the sign restrictions help orient the columns. If a column and its negative both satisfy the restrictions, that shock's sign remains ambiguous. In practice, sign-restriction procedures explore rotations and retain those satisfying all inequalities; a retained rotation is an admissible model, not evidence that it is the uniquely correct one.

Worked Example: Checking Sign Restrictions Across Horizons

Let a hypothetical reduced-form VAR(1) have $\Phi=\operatorname{diag}(0.5,0.2)$ and $\Sigma_u=I_2$. Thus $P=I_2$, $\Psi_h=\Phi^h$, and candidate structural impact matrices are rotations $B=Q$. Require shock 1 to raise variable 1 and lower variable 2, and shock 2 to raise both variables, at horizons $h=0$ and $h=1$.

For the rotation $Q=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$, its impact columns are $(\cos\theta,\sin\theta)'$ and $(-\sin\theta,\cos\theta)'$. If $-\pi/2<\theta<0$, then $\cos\theta>0$ and $\sin\theta<0$. Shock 1 therefore has the required positive and negative impact signs; shock 2 has two positive impact responses. At horizon 1, the responses are multiplied by $0.5$ for variable 1 and $0.2$ for variable 2, so none of those signs changes. Every rotation in this open interval is admissible.

For instance, at $\theta=-\pi/4$, both sine and cosine have magnitude $1/\sqrt{2}\approx0.7071$. Shock 1's impact responses are approximately $(0.7071,-0.7071)'$, and its horizon-1 responses are $(0.3536,-0.1414)'$. Shock 2's corresponding responses are $(0.7071,0.7071)'$ and $(0.3536,0.1414)'$. All four required signs hold at both horizons.

Set-Valued Responses and Their Interpretation

The preceding example shows why sign restrictions usually do not produce a single impulse response. The admissible angles include $-\pi/4$ and $-\pi/6$, among infinitely many others. At $-\pi/6$, shock 1's impact responses are $(0.8660,-0.5000)'$, and its horizon-1 responses are $(0.4330,-0.1000)'$; the same signs hold, but the magnitudes differ from the responses at $-\pi/4$. Both candidates have the same reduced-form covariance and dynamics.

Worked Example: Reporting an Admissible Response Range

Continue with the diagonal VAR and the sign requirements in the previous example. For shock 1, the impact response of variable 1 is $\cos\theta$. Across admissible angles $-\pi/2<\theta<0$, this response is strictly between 0 and 1; its closure is the interval $[0,1]$. At horizon 1, the response is $0.5\cos\theta$, so its admissible values lie strictly between 0 and $0.5$, with closure $[0,0.5]$.

For variable 2, shock 1's impact response is $\sin\theta$, which lies strictly between $-1$ and 0, and its horizon-1 response is $0.2\sin\theta$, strictly between $-0.2$ and 0. These bounds describe separate response entries. They do not imply that every arbitrary combination of values inside the four ranges is feasible: all entries must come from the same angle $\theta$, so they are linked.

A report should therefore convey that multiple response paths meet the restrictions and describe the resulting range or envelope, rather than presenting one retained rotation as uniquely identified. Strict sign inequalities exclude zero at the endpoints; reporting the closure is a convenient way to summarize limiting bounds.

Why the Choice of Restrictions Matters

Long-run and sign restrictions encode different economic claims. A long-run zero says that a particular shock has no cumulative effect on a particular variable; it does not say that the variable never responds along the way. A sign restriction concerns directions of response at specified horizons, but ordinarily says little about their exact magnitudes. Neither restriction follows from the reduced-form VAR alone. Their credibility depends on the economic argument for imposing them.

The restrictions also differ in how much they identify. In a two-shock system, one scalar long-run zero can impose an equation on the rotation angle and may determine the angle up to sign conventions. A finite list of strict sign inequalities usually accepts a range of angles, as in the example. With more shocks, even several restrictions may leave multiple rotations. Always assess whether the restrictions deliver a unique structural interpretation or only a set.

Specify the structural setup.
State the shock normalization, the reduced-form covariance factor $P$, and the structural response formula $\Theta_h=\Psi_hPQ$.

State the restrictions precisely.
For long-run restrictions, name the cumulative effect being constrained. For sign restrictions, name each variable, shock, direction, and horizon.

Check admissible rotations.
Verify every equality or inequality using the same candidate $Q$, and respect shock sign and scale normalizations.

Describe what is identified.
Distinguish a unique response from a collection or range of responses consistent with the imposed restrictions.

Common Pitfalls

Calling a sign-restricted response uniquely identified. One admissible rotation is not necessarily the only one. Check whether other rotations satisfy all the restrictions.
Confusing a long-run zero with a zero impact effect. A cumulative response can equal zero even when intermediate responses are nonzero.
Ignoring shock orientation. Reversing a structural shock reverses its entire response column. State the sign normalization or restrictions that orient it.
Mixing up horizons or variables. A sign requirement must identify exactly which response entries it constrains; signs at impact need not persist later.
Treating restrictions as data-derived facts. They are maintained economic assumptions used to identify the structural shocks, not consequences of the reduced-form covariance alone.

Key takeaway: Structural long-run and sign restrictions select among rotations that fit the same reduced-form VAR. Long-run zeros constrain cumulative effects; sign restrictions constrain directions over specified horizons. Always report whether the resulting structural responses are point-identified or remain set-valued.

Check Your Understanding

Answer each question using the definitions and calculations in this tutorial.

Why do all orthogonal rotations $PQ$ have the same reduced-form innovation covariance?
In a stable VAR, what does entry $(i,j)$ of the structural long-run impact matrix represent?
Why does a zero long-run effect not require the impact response to be zero?
For the sign restrictions in the diagonal VAR example, why does every angle strictly between $-\pi/2$ and 0 satisfy the required signs at horizons 0 and 1?
Why are the response bounds for separate entries not necessarily independently combinable into one response path?
What must be established before describing a structural response as point-identified rather than set-identified?