Structural VAR Identification by Short-Run Restrictions

From Reduced-Form Innovations to Structural Shocks

A reduced-form VAR summarizes how current variables relate to their lags, but its innovations may move together within a period. To interpret shocks as distinct economic disturbances, researchers need a mapping from those reduced-form innovations to structural shocks. Short-run identification supplies that mapping by restricting the immediate, or impact-period, relationships among variables and shocks.

The object to identify is the impact matrix. Its columns describe the immediate responses of the variables to the separately normalized structural shocks. Restrictions on that matrix can encode timing assumptions, but a mathematically identified matrix is not automatically a credible economic model: the restrictions must also be defensible for the application.

Definition: Write the reduced-form VAR innovations as \(u_t\), with \(E(u_t)=0\) and covariance matrix \(\Sigma_u=E(u_tu_t')\). A structural representation of these innovations is \(u_t=B\varepsilon_t\), where \(B\) is the structural impact matrix and the structural shocks satisfy \(E(\varepsilon_t)=0\) and \(E(\varepsilon_t\varepsilon_t')=I\). Thus \(\Sigma_u=BB'\). A short-run restriction constrains entries or other contemporaneous features of \(B\), or equivalently of a structural equation matrix that maps \(u_t\) to \(\varepsilon_t\).

The identity covariance normalization means the structural shocks have unit variance and are contemporaneously uncorrelated. This normalization fixes their scales, but not their economic names or signs. In an \(n\)-variable model, \(B\) has \(n^2\) entries, whereas the symmetric covariance matrix \(\Sigma_u\) supplies \(n(n+1)/2\) distinct equations. A further \(n(n-1)/2\) independent restrictions are therefore needed for exact identification under this normalization. This is a necessary counting condition, not a guarantee: restrictions must also be independent and yield an admissible solution.

Why the Covariance Matrix Is Not Enough

Even when \(\Sigma_u\) is known exactly, there are generally many matrices \(B\) that satisfy \(\Sigma_u=BB'\). For any orthogonal matrix \(Q\), meaning \(QQ'=I\), the matrix \(BQ\) produces the same covariance:

(BQ)(BQ)'=BQQ'B'=BB'=\Sigma_u.

Thus the reduced-form covariance identifies neither a unique set of structural shocks nor their unique impact effects. Short-run restrictions select a factorization from this observationally equivalent class. For exact identification, the restrictions must reduce the admissible set to a unique solution, apart from conventional sign or labeling choices. If multiple solutions remain, the model is not globally identified by those restrictions alone.

Worked Example: Counting Restrictions in a Three-Variable VAR

Suppose a three-variable reduced-form VAR has an innovation covariance matrix with six distinct entries: three variances and three covariances. The structural impact matrix \(B\) has nine entries, and the structural shocks are normalized to have covariance \(I\). How many independent short-run restrictions are required for exact identification?

The covariance equation \(\Sigma_u=BB'\) provides \(n(n+1)/2=3(4)/2=6\) distinct equations. The impact matrix has \(n^2=3^2=9\) unknown entries. The difference is

9-6=3 \qquad\text{or, in general,}\qquad \frac{n(n-1)}{2}=\frac{3(2)}{2}=3.

So three independent restrictions are needed by the parameter count. For example, three prespecified zero entries could potentially supply them. Merely writing down three zeros is not sufficient: if they are redundant, inconsistent with the covariance matrix, or leave multiple admissible solutions, exact identification has not been achieved.

Recursive Short-Run Restrictions

A recursive structure imposes a triangular pattern on \(B\). In a three-variable ordering, a lower-triangular impact matrix has the form

B= \begin{pmatrix} b_{11}&0&0\\ b_{21}&b_{22}&0\\ b_{31}&b_{32}&b_{33} \end{pmatrix}.

The zeros say that, within the period, the second and third structural shocks have no direct impact on the first variable, and the third shock has no direct impact on the second. They do not say that those variables can never respond to those shocks: lagged dynamics can transmit effects in later periods. The restrictions depend on the variable ordering, so changing that ordering changes the assumptions.

For a positive-definite \(\Sigma_u\), the Cholesky factorization gives a unique lower-triangular \(B\) with positive diagonal satisfying \(\Sigma_u=BB'\). The positive diagonal is a sign convention; without it, simultaneous sign changes of a column and the corresponding structural shock leave the representation unchanged. The triangular zeros provide \(n(n-1)/2\) restrictions, the number required by the count above.

Worked Example: Recovering a Two-Variable Recursive Impact Matrix

Suppose a hypothetical reduced-form VAR has innovation covariance matrix

\Sigma_u= \begin{pmatrix} 4&1\\ 1&2 \end{pmatrix}.

Assume a lower-triangular impact matrix and positive diagonal entries: \(B=\begin{pmatrix}b_{11}&0\\b_{21}&b_{22}\end{pmatrix}\). The structural shocks have covariance \(I\), so the required equation is \(BB'=\Sigma_u\). Multiplying gives

BB'= \begin{pmatrix} b_{11}^2 & b_{11}b_{21}\\ b_{11}b_{21} & b_{21}^2+b_{22}^2 \end{pmatrix} = \begin{pmatrix} 4&1\\ 1&2 \end{pmatrix}.

Equating entries and applying the positive-diagonal convention, \(b_{11}=\sqrt{4}=2\). Then \(b_{11}b_{21}=1\), so \(b_{21}=1/2=0.5000\). Finally, \(b_{21}^2+b_{22}^2=2\), giving \(b_{22}^2=2-(0.5000)^2=1.75\), and \(b_{22}=\sqrt{1.75}\). The exact identified factor is therefore

B= \begin{pmatrix} 2&0\\ 0.5000&\sqrt{1.75} \end{pmatrix}, \qquad BB'= \begin{pmatrix} 4&1\\ 1&2 \end{pmatrix}.

Numerically, \(\sqrt{1.75}\approx1.323\), so the impact matrix is approximately \(\begin{pmatrix}2&0\\0.5000&1.323\end{pmatrix}\). The exact factorization above uses the exact square root; the rounded value is not an exact factor. Under this ordering, the first variable can respond immediately to shock 1 but not shock 2. The second variable can respond immediately to either shock.

Nonrecursive Short-Run Restrictions

Not every economically motivated timing structure is triangular. A nonrecursive structure can allow contemporaneous feedback while imposing a different pattern of zeros. One convenient representation is

A u_t=\varepsilon_t,\qquad E(\varepsilon_t\varepsilon_t')=I.

If \(A\) is invertible, then \(B=A^{-1}\) and \(\Sigma_u=A^{-1}A^{-T}\). Equivalently, the precision matrix, the inverse covariance matrix, satisfies \(\Sigma_u^{-1}=A'A\). Restrictions on \(A\), often including a normalization of its diagonal, can therefore be assessed through the covariance implied by the reduced form. Unlike the recursive case, a nonrecursive pattern does not generally have a simple triangular factorization; the restrictions must be solved together and checked for a unique admissible solution.

Worked Example: Identifying a Nonrecursive Three-Variable Structure

Consider a hypothetical three-variable system with normalized diagonal entries in \(A\) and the nonrecursive pattern

A= \begin{pmatrix} 1&x&0\\ 0&1&y\\ z&0&1 \end{pmatrix}.

This pattern allows a cycle of contemporaneous relationships rather than the one-directional ordering of a recursive system. It imposes three zeros and fixes the diagonal to one. Suppose the reduced-form covariance is positive definite and its inverse is

\Sigma_u^{-1}= \begin{pmatrix} 1.16&-0.20&-0.40\\ -0.20&1.04&-0.30\\ -0.40&-0.30&1.09 \end{pmatrix}.

Under the stated structural model, \(\Sigma_u^{-1}=A'A\). The columns of \(A\) are \((1,0,z)'\), \((x,1,0)'\), and \((0,y,1)'\). Their pairwise products give

A'A= \begin{pmatrix} 1+z^2&x&z\\ x&1+x^2&y\\ z&y&1+y^2 \end{pmatrix}.

Matching the off-diagonal entries identifies \(x=-0.20\), \(z=-0.40\), and \(y=-0.30\). The diagonal entries check the restrictions: \(1+z^2=1+0.16=1.16\), \(1+x^2=1+0.04=1.04\), and \(1+y^2=1+0.09=1.09\). Thus the solution satisfies all six covariance equations, not only the three off-diagonal ones. The matrix is invertible because its determinant is \(1+xyz=1+(-0.20)(-0.30)(-0.40)=0.976\ne0\). The resulting impact matrix is \(B=A^{-1}\); it maps each unit-variance structural shock into contemporaneous changes in the three variables. The cycle makes this system nonrecursive even though the restrictions identify its parameters in this example.

Identification Checks and Interpretation

A restriction count is a first diagnostic, not a proof of identification. In an applied model, the researcher must check that the restrictions are economically justified, that the covariance equations and restrictions can be satisfied jointly, and that the resulting impact matrix is nonsingular. For nonlinear restrictions, multiple solutions may satisfy the same equations even when the number of restrictions equals the parameter-count difference. Restrictions in excess of the minimum make the model overidentified; whether the additional restrictions are compatible with the data is a separate question.

Short-run restrictions concern contemporaneous impacts only. They should not be confused with restrictions on how effects accumulate over time, nor with restrictions on the signs of responses. The immediate interpretation is read from \(B\): element \(b_{ij}\) is the impact-period response of variable \(i\) to a one-unit structural shock \(j\), under the unit-variance normalization. Subsequent responses also depend on the VAR's lag dynamics.

Common pitfall: “The reduced-form covariance matrix is known, so its structural shocks are uniquely determined.” This is false: orthogonal rotations of a valid factor give the same covariance. The correction is to state the shock normalization, impose economically justified short-run restrictions, and verify that those restrictions identify a unique admissible impact matrix. A Cholesky factor is one answer only after a recursive pattern and variable ordering have been specified.

Specify the normalization.
State the covariance and scale convention for the structural shocks, such as \(E(\varepsilon_t\varepsilon_t')=I\).

Count and motivate restrictions.
Compare the number of unknown impact parameters with the distinct covariance equations, then justify the additional restrictions substantively.

Solve and verify.
Recover the candidate matrix and check the covariance equation, the restriction pattern, and nonsingularity. Investigate uniqueness rather than treating the count as a guarantee.

Check Your Understanding

Use the normalization and definitions in this tutorial to answer the following questions.

For a four-variable VAR with unit-variance structural shocks, how many additional independent restrictions are needed by the parameter count for exact identification?
Why does multiplying a valid impact matrix on the right by an orthogonal matrix leave the reduced-form innovation covariance unchanged?
In a lower-triangular recursive impact matrix, what does a zero above the diagonal say about the impact-period response? Does it rule out later responses?
For a structural equation representation \(A u_t=\varepsilon_t\) with shock covariance \(I\), express the reduced-form covariance and its inverse in terms of \(A\).
Why does having the required number of restrictions not by itself establish identification?