Orthogonalized and Generalized Impulse Responses

Why Two Impulse-Response Conventions?

A reduced-form VAR innovation can move several variables at once. Its covariance matrix records this contemporaneous co-movement, so a response to one component of the innovation is not automatically a response to an isolated shock in that variable. Two commonly used responses address the correlation in different ways: Cholesky-orthogonalized impulse responses transform the innovations into uncorrelated components, while generalized impulse responses condition on an unusually large innovation in one component and account for the expected movements in the others.

Both methods start with the moving-average representation. For a stable VAR, write the centered variables as \(y_t-\mu=\sum_{h=0}^{\infty}\Phi_h u_{t-h}\), where \(u_t\) is the reduced-form innovation vector, \(E(u_t)=0\), and \(E(u_tu_t')=\Sigma_u\). The matrix \(\Phi_h\) describes the effect of an innovation at one date on the variables \(h\) periods later. The focus here is how to combine \(\Phi_h\) with \(\Sigma_u\) when innovations are contemporaneously correlated.

The Cholesky response describes the effect of a one-unit shock to one orthogonalized innovation, with the other orthogonalized innovations set to zero. Since the orthogonalized innovations have identity covariance, one unit is one standard deviation. The generalized response describes the change in the conditional mean path associated with observing a one-standard-deviation innovation in component \(j\), while allowing correlated components of the same innovation vector to move as their conditional means imply. This is a conditional association, not by itself a causal effect.

Cholesky-Orthogonalized Responses

Suppose \(\Sigma_u\) is positive definite, so it has a unique lower-triangular Cholesky factor \(P\) with positive diagonal satisfying \(\Sigma_u=PP'\). Define \(\varepsilon_t=P^{-1}u_t\). Then \(u_t=P\varepsilon_t\), and \(E(\varepsilon_t\varepsilon_t')=P^{-1}\Sigma_u(P^{-1})'=I\). Substituting into the moving-average representation gives \(y_t-\mu=\sum_{h=0}^{\infty}\Phi_hP\varepsilon_{t-h}\). Therefore, a one-unit increase in orthogonalized shock \(i\) at date \(t\), with the other orthogonalized shocks held at zero, changes the path \(h\) periods later by the \(i\)th column of \(\Phi_hP\).

This transformation resolves correlation algebraically, but it allocates the contemporaneous covariance according to the chosen Cholesky ordering. In a lower-triangular \(P\), the first orthogonalized shock can affect all variables on impact; the second can affect the second variable and those after it, but not the first on impact. These zero restrictions follow from the factorization and ordering. They are not supplied by the reduced-form VAR alone.

Generalized Responses and Conditional Innovation Means

A generalized response does not first rotate the entire innovation vector into orthogonal components. Instead, it considers the conditional expectation of the reduced-form innovation vector when component \(j\) takes the value \(\sqrt{\sigma_{jj}}\), its own standard deviation. Under a multivariate normal distribution with mean zero and covariance \(\Sigma_u\), \[ E(u_t\mid u_{j,t}=\sqrt{\sigma_{jj}}) =\frac{\Sigma_u e_j}{\sqrt{\sigma_{jj}}}. \] The \(j\)th component of this conditional mean is \(\sqrt{\sigma_{jj}}\); the other components generally have nonzero conditional means when they are correlated with component \(j\).

For the impulse-response formula to represent a change in the full conditional mean path, assumptions about time as well as the same-date covariance are needed. Here, assume the innovations are independent and identically distributed over time, jointly Gaussian with mean zero and covariance \(\Sigma_u\), and independent of the information available just before date \(t\). Compare the conditional expected path given \(u_{j,t}=\sqrt{\sigma_{jj}}\) with the path given the same past information but without that restriction on \(u_t\). Past innovations then have the same conditional means in both paths, and future innovations have conditional mean zero in both. Only the terms involving \(u_t\) differ, yielding the generalized response below.

The independence-over-time assumption matters: without it, conditioning on the current innovation could change the conditional means of past or future innovations, so the displayed difference would not follow from \(\Sigma_u\) alone. The Gaussian assumption makes the conditional-mean formula exact. With non-Gaussian innovations, the same expression is often used as a linear-projection-based generalized response, but it need not equal the full conditional-mean response.

Why Ordering Changes Cholesky Results

Cholesky-orthogonalized responses are not invariant to reordering the variables. To see the source of this dependence, reorder the same covariance matrix so that \(z_t\) comes before \(x_t\): \[ \begin{pmatrix}1&1.2\\1.2&4\end{pmatrix}. \] Its lower-triangular Cholesky factor is \[ \begin{pmatrix}1&0\\1.2&1.6\end{pmatrix}, \] since \(1.2^2+1.6^2=4\). The first orthogonalized shock now has impact vector \((1,1.2)'\) in the reordered coordinates. In the original \((x,z)\) coordinates, that is \((1.2,1)'\): the impact pattern previously associated with the generalized response to a one-standard-deviation \(z\) innovation.

Under this reversed ordering, the shock to \(x\) is second. Its impact vector in reordered coordinates is \((0,1.6)'\), or \((1.6,0)'\) when written in the original \((x,z)\) coordinates. This differs from the original-order Cholesky response to shock 1, \((2,0.6)'\). The reduced-form covariance matrix has not changed in substance; the allocation of its contemporaneous covariance across orthogonalized shocks has changed because the factorization order changed.

Generalized responses do not depend on a Cholesky ordering. Reordering the variables simply reorders the entries of the covariance matrix, coefficient matrices, and response vectors; the response associated with a given named variable is the same after translating back to the original coordinates. This invariance does not mean generalized responses isolate independent shocks. They retain the conditional movements in correlated innovations, as the calculation for innovation 2 demonstrated.

Common Pitfall and Practical Interpretation

A frequent mistake is to treat the generalized response as the effect of changing \(u_{j,t}\) while holding all other components of \(u_t\) at zero. That reasoning would produce an impact vector with only its \(j\)th entry nonzero, which generally is not \(\Sigma_u e_j/\sqrt{\sigma_{jj}}\). The generalized response instead conditions on the specified value of \(u_{j,t}\); correlated components have the conditional means implied by \(\Sigma_u\). Conversely, a Cholesky response does set the other orthogonalized innovations to zero, but this is not the same as setting the other reduced-form innovations to zero.

A second pitfall is to interpret the triangular zeros in a Cholesky impact matrix as empirical proof that one variable cannot respond immediately to another. Those zeros are consequences of the chosen ordering and decomposition. The ordering may be motivated by substantive knowledge, but the reduced-form covariance matrix alone does not select it. When reporting either response, specify the variable order, the shock normalization, and whether the response is orthogonalized or generalized. For generalized responses interpreted as conditional means, state the distributional and time-independence assumptions that support that interpretation.

The methods therefore answer related but distinct descriptive questions. Cholesky orthogonalization is useful when an explicitly ordered decomposition of correlated innovations is desired. Generalized responses provide an ordering-invariant way to summarize the effects associated with a component-specific innovation while retaining correlated co-movements. Their differences are informative: they reveal how much the reported path depends on allocating contemporaneous correlation to separate shocks.