Moving-Average Representations and Impulse Responses

From a VAR to a Moving-Average Representation

A VAR describes each variable using its own past, the past of the other variables, and a vector of new innovations. Its moving-average representation expresses the same process differently: each current value is written as the accumulated effect of current and past innovations. The representation makes propagation across variables and time explicit.

Consider a \(K\)-variable VAR(\(p\)) with a constant:

y_t=c+A_1y_{t-1}+\cdots+A_py_{t-p}+u_t,

where \(y_t\) is a \(K\times1\) vector, \(c\) is a \(K\times1\) vector, each \(A_j\) is a \(K\times K\) coefficient matrix, and \(u_t\) is the reduced-form innovation vector. Assume \(E(u_t)=0\), that innovations are serially uncorrelated with finite covariance matrix \(\Sigma_u\), and that the VAR is stable. For conditional-response interpretations below, also take the innovations to have conditional mean zero given information available before time \(t\).

Definition: A VAR is stable if all eigenvalues of its companion matrix have modulus strictly less than one. Equivalently, the roots of \(\det(I_K-A_1z-\cdots-A_pz^p)=0\) lie outside the unit circle. Stability ensures that the effects of past innovations decay sufficiently for a convergent moving-average representation.

Assume the VAR is in its stationary solution, and let \(\mu=E(y_t)\), its time-invariant unconditional mean. Taking expectations in the VAR gives \(\mu=c+(A_1+\cdots+A_p)\mu\), so \(\mu\) satisfies \((I_K-\sum_{j=1}^p A_j)\mu=c\). Define the centered vector \(x_t=y_t-\mu\). Then the dynamics are \(x_t=A_1x_{t-1}+\cdots+A_px_{t-p}+u_t\).

Definition: The stable VAR moving-average representation is

y_t=\mu+\sum_{h=0}^{\infty}\Psi_hu_{t-h},

where \(\Psi_h\) is a \(K\times K\) matrix, \(\Psi_0=I_K\), and the matrices \(\Psi_h\) describe how innovations propagate over time. The entry in row \(i\), column \(j\) of \(\Psi_h\) is the response of variable \(i\), at horizon \(h\), to a one-unit reduced-form innovation in equation \(j\), with the other innovation coordinates set to zero.

Deriving the Coefficient Recursion

The moving-average matrices are determined by the VAR coefficients, not estimated as an unrelated set of parameters. To see why, use the lag operator \(L\), where \(Lx_t=x_{t-1}\). The centered VAR becomes \((I_K-A_1L-\cdots-A_pL^p)x_t=u_t\). If the inverse operator has a power-series expansion \(\sum_{h=0}^{\infty}\Psi_hL^h\), its coefficients must satisfy the following recursion.

Result: Set \(\Psi_0=I_K\), and set \(\Psi_h=0\) for \(h<0\). For \(h\geq1\),

\Psi_h=\sum_{j=1}^{\min(p,h)}A_j\Psi_{h-j}.

For justification, multiply the proposed series by the VAR lag polynomial. The coefficient on \(L^0\) in the product is \(\Psi_0=I_K\). For any \(h\geq1\), the coefficient on \(L^h\) is \(\Psi_h-\sum_{j=1}^{\min(p,h)}A_j\Psi_{h-j}\). Setting this coefficient to zero yields precisely the recursion. Thus the product is \(I_K\), coefficient by coefficient. Stability ensures that this inverse has a convergent power-series representation, so applying it to \(u_t\) gives the stated moving-average form.

For a VAR(1), this reduces to \(\Psi_h=A_1^h\). For a VAR(2), the first terms are \(\Psi_0=I_K\), \(\Psi_1=A_1\), and \(\Psi_2=A_1^2+A_2\). Matrix multiplication order matters: the recursion uses \(A_j\Psi_{h-j}\), not \(\Psi_{h-j}A_j\).

Worked Example: Responses in a Stable VAR(1)

Suppose two hypothetical monthly economic indexes, both measured in index points, follow a centered VAR(1) with

x_t= \begin{pmatrix} 0.5&0.2\\ 0&0.4 \end{pmatrix}x_{t-1}+u_t.

The coefficient matrix is triangular, so its eigenvalues are its diagonal entries, \(0.5\) and \(0.4\). Both have modulus below one; the VAR is stable. Therefore \(\Psi_0=I_2\), and \(\Psi_h=A_1^h\). At horizon one,

\Psi_1= \begin{pmatrix} 0.5&0.2\\ 0&0.4 \end{pmatrix}.

At horizon two, multiply the matrices:

\Psi_2=A_1^2= \begin{pmatrix} 0.5(0.5)+0.2(0)&0.5(0.2)+0.2(0.4)\\ 0(0.5)+0.4(0)&0(0.2)+0.4(0.4) \end{pmatrix} = \begin{pmatrix} 0.25&0.18\\ 0&0.16 \end{pmatrix}.

A one-index-point innovation in equation 2 has response vector \(\Psi_0(0,1)'=(0,1)'\) at horizon zero, \(\Psi_1(0,1)'=(0.2,0.4)'\) at horizon one, and \(\Psi_2(0,1)'=(0.18,0.16)'\) at horizon two. Thus the first index responds through the cross-variable coefficient, while the second index’s response decays from 1 to 0.4 to 0.16 index points. These are responses to reduced-form innovations, not by themselves causal effects.

Impulse Responses from a VAR(2)

An impulse response traces the effect of an innovation at a specified date while holding other innovations fixed at zero for the calculation. In a VAR with conditional-mean-zero future innovations, it also equals the change in the conditional expected value at a future horizon caused by changing the current innovation vector, with the same starting conditions. In matrix form, the response at horizon \(h\) to a one-unit innovation in coordinate \(j\) is column \(j\) of \(\Psi_h\).

Worked Example: Building VAR(2) Responses

Suppose the centered dynamics of two hypothetical monthly outcomes, each measured in its own recorded units, are

x_t=A_1x_{t-1}+A_2x_{t-2}+u_t,\qquad A_1= \begin{pmatrix} 0.4&0.1\\ 0.1&0.3 \end{pmatrix}, \quad A_2= \begin{pmatrix} 0.08&0\\ 0&0 \end{pmatrix}.

First check stability rather than assuming it. The VAR(2) companion matrix is \(\begin{pmatrix}A_1&A_2\\I_2&0\end{pmatrix}\). Its characteristic polynomial is \(\lambda(\lambda^3-0.7\lambda^2+0.03\lambda+0.024)\); its eigenvalues are approximately \(0.575,\ 0.276,\ -0.151,\) and \(0\). Each has modulus below one, so the VAR is stable.

Now calculate the moving-average matrices. The first two are \(\Psi_0=I_2\) and \(\Psi_1=A_1\). For horizon two,

\Psi_2=A_1\Psi_1+A_2 =A_1^2+A_2 = \begin{pmatrix} 0.17&0.07\\ 0.07&0.10 \end{pmatrix} + \begin{pmatrix} 0.08&0\\ 0&0 \end{pmatrix} = \begin{pmatrix} 0.25&0.07\\ 0.07&0.10 \end{pmatrix}.

At horizon three, both lag matrices contribute:

\begin{aligned} A_1\Psi_2&= \begin{pmatrix} 0.107&0.038\\ 0.046&0.037 \end{pmatrix}, & A_2\Psi_1&= \begin{pmatrix} 0.032&0.008\\ 0&0 \end{pmatrix},\\ \Psi_3&=A_1\Psi_2+A_2\Psi_1 = \begin{pmatrix} 0.139&0.046\\ 0.046&0.037 \end{pmatrix}. \end{aligned}

For example, the second coordinate’s response to a one-unit innovation in equation 1 at horizon three is the entry in row 2, column 1: \(0.1(0.25)+0.3(0.07)+0=0.046\). The zero term comes from the second row of \(A_2\Psi_1\), which is zero. The response vectors to a unit innovation in equation 1 at horizons zero through three are therefore \((1,0)'\), \((0.4,0.1)'\), \((0.25,0.07)'\), and \((0.139,0.046)'\), respectively.

Worked Example: Combining and Scaling Innovations

Use the stable VAR(2) above. Suppose the reduced-form innovation vector at time \(t\) is \((0.5,-0.2)'\), in the units of the two outcomes, and subsequent innovations are set to zero. The response at horizon \(h\) is \(\Psi_h(0.5,-0.2)'\). At horizon two, direct multiplication gives

\Psi_2 \begin{pmatrix}0.5\\-0.2\end{pmatrix} = \begin{pmatrix} 0.25(0.5)+0.07(-0.2)\\ 0.07(0.5)+0.10(-0.2) \end{pmatrix} = \begin{pmatrix}0.111\\0.015\end{pmatrix}.

At horizon three, the same calculation gives

\Psi_3 \begin{pmatrix}0.5\\-0.2\end{pmatrix} = \begin{pmatrix} 0.139(0.5)+0.046(-0.2)\\ 0.046(0.5)+0.037(-0.2) \end{pmatrix} = \begin{pmatrix}0.0603\\0.0156\end{pmatrix}.

These are changes in the two outcomes’ conditional expectations relative to the no-innovation path, at the corresponding horizons. The calculation uses linearity: the response to a vector of innovations is the weighted sum of the responses to its coordinates. The numerical units of each component are the units of its corresponding outcome.

Interpreting Reduced-Form Responses

A response matrix contains one response path per column. Row \(i\), column \(j\) of \(\Psi_h\) concerns outcome \(i\), an innovation in equation \(j\), and horizon \(h\). Horizon zero is always \(I_K\): an innovation enters its own equation immediately, while the other outcomes have zero contemporaneous response under this reduced-form accounting. At later horizons, off-diagonal entries show propagation across equations.

A reduced-form innovation is not automatically an independently manipulable shock. The coordinates of \(u_t\) can be contemporaneously correlated, as represented by \(\Sigma_u\). Consequently, setting one coordinate to one and all others to zero defines an algebraic response, but may not describe a historically or economically isolated disturbance. The matrix \(\Psi_h\) alone does not establish a causal interpretation. That interpretation requires additional assumptions about what the innovations represent.

The moving-average form matters because it turns a collection of lag coefficients into a horizon-by-horizon account of dynamic transmission. It also provides the building blocks for studying forecast uncertainty and the contribution of shocks to movements in the system. Stability is important in this interpretation: when the VAR is stable, responses eventually die away rather than grow without bound.

Common Mistakes and Checks

Using only \(A_1\) at every horizon. That works for horizon one, not generally for later horizons. Apply the full recursion; in a VAR(2), for instance, \(\Psi_3=A_1\Psi_2+A_2\Psi_1\).
Reversing matrix multiplication. The stated recursion is \(A_j\Psi_{h-j}\). Since matrices generally do not commute, changing the order can change the answer.
Calling a reduced-form response causal. A column of \(\Psi_h\) describes the effect of a specified coordinate perturbation in the reduced-form system. Correlated innovations mean that additional identification assumptions are needed for a structural causal claim.
Ignoring units or horizon. Report which outcome responds, which equation’s innovation is perturbed, the horizon, and the units of the response.
Assuming stability from small-looking coefficients. Check the eigenvalues of the companion matrix, or the equivalent VAR characteristic roots; inspecting individual entries is not a stability test.

Key takeaway: For a stable VAR, start with \(\Psi_0=I_K\) and compute later matrices from \(\Psi_h=\sum_{j=1}^{\min(p,h)}A_j\Psi_{h-j}\). The resulting columns trace the propagation of reduced-form innovations, with interpretation tied to the outcome, equation, horizon, and units.

Check Your Understanding

Use the definitions and calculations above to answer these questions.

For a stable VAR(1), why is \(\Psi_h=A_1^h\), and what condition on the eigenvalues makes the process stable?
For a VAR(2), write the recursion for \(\Psi_3\). Which matrices must be multiplied, and in what order?
In an impulse-response matrix, what does the entry in row 2, column 1 at horizon 4 represent?
Why does a reduced-form response to one coordinate of \(u_t\) not, by itself, establish a causal effect?
If an innovation vector is multiplied by a scalar \(a\), how does linearity change its response at each horizon?