Impulse Response Function

Definition

The Impulse Response Function (IRF) at horizon $s$ is the matrix of partial derivatives of $y_{t+s}$ with respect to a unit structural shock at time $t$ :

$\Phi_s = \frac{\partial y_{t+s}}{\partial \varepsilon_t'} \in \mathbb{R}^{n \times n}$

The $(i,j)$ element $\Phi_s^{(i,j)}$ is the response of variable $i$ to a one-standard-deviation shock to equation $j$ , $s$ periods after the shock.

Key Ideas

IRFs summarize the full dynamic structure of the system in one object.
They are defined only relative to a structural identification of $A_0$ ; reduced-form shocks have no unique economic interpretation.
Bayesian Error Bands around IRFs quantify estimation uncertainty.
IRFs from Dynamic Stochastic General Equilibrium (DSGE) models are routinely compared to vector autoregression (VAR) IRFs as a model-evaluation discipline.

How It Works

Recursive Computation

Let $B_\ell = A_0^{-1} A_\ell$ denote the reduced-form lag matrices. The impulse responses satisfy the recursion:

$\Phi_s = \Phi_{s-1} B_1 + \Phi_{s-2} B_2 + \cdots + \Phi_{s-p} B_p, \quad s \geq 1$

with boundary conditions:

$\Phi_0 = A_0^{-1}, \qquad \Phi_\nu = 0_{n \times n} \text{ for } \nu < 0$

This follows directly from the reduced-form moving-average (MA) representation. $\Phi_0 = A_0^{-1}$ encodes the contemporaneous response: the immediate effect of each structural shock on all variables, determined entirely by the identified $A_0$ .

MA(∞) Representation

The VAR has a moving-average representation:

$y_t = \mu + \sum_{s=0}^{\infty} \Phi_s\, \varepsilon_{t-s}$

where $\mu$ absorbs the deterministic terms. This converges if all eigenvalues of the companion matrix lie inside the unit circle (stationarity condition).

Long-Run Cumulative Response

$\Phi_\infty = \sum_{s=0}^{\infty} \Phi_s = \left(I_n - \sum_{\ell=1}^p B_\ell\right)^{-1} A_0^{-1}$

This matrix is used in long-run identification schemes (see Structural Identification).

Forecast Error Variance Decomposition

The $h$ -step forecast error variance of variable $i$ attributable to structural shock $j$ :

$\text{FEVD}_{ij}(h) = \frac{\sum_{s=0}^{h-1} [\Phi_s]_{ij}^2}{\sum_{s=0}^{h-1} \sum_{k=1}^n [\Phi_s]_{ik}^2}$

This complements IRFs by showing the relative importance of each shock at different horizons.

Affine Term Structure IRFs (Ang-Piazzesi 2001)

In an Affine Term Structure Model, the yield of maturity $n$ is $y_t^n = A_n + B_n' X_t$ , so the initial impulse response of yield $n$ to a unit shock in factor $k$ is simply the $k$ -th component of the loading vector $B_n$ . The full dynamic IRF combines these loadings with the VAR dynamics:

$\psi_i^n = \bar{B}_{n1}' P_i + \bar{B}_{n2}' P_{i-1} + \cdots$

where $P_i$ is the $i$ -step-ahead MA coefficient matrix of the companion VAR. Three features distinguish affine IRFs from standard VAR IRFs:

All maturities, analytically: because $B_n$ is a closed-form function of the no-arbitrage recursion parameters, IRFs are available for any maturity $n$ , not only the yields included in estimation. A standard macro-VAR can only compute IRFs for included yields.
No-arbitrage consistency: the yield responses at different maturities are automatically consistent with no-arbitrage — they are derived from a single pricing kernel rather than separate unconstrained equations.
Factor-shape interpretation: the pattern of $B_n$ across maturities $n$ labels factors: a flat $B_n$ profile = level factor; upward-sloping = slope factor; hump-shaped = curvature factor.

Empirical result (Ang-Piazzesi 2001): inflation shocks raise the yield curve at all maturities with a hump-shaped IRF peaking at ~1–2 years; real activity shocks have a smaller hump peaking at ~1 year. The short end of the curve responds more strongly than the long end to both macro shocks.

Permanent vs. Transitory Effects in Cointegrated Systems (Lütkepohl 1999)

For a stationary $I(0)$ system, $\Phi_s \to 0$ as $s \to \infty$ — all impulse responses are transitory. For a cointegrated $I(1)$ system with cointegrating rank $r$ , the limit $\Psi(1) = \sum_{s=0}^\infty \Phi_s$ has reduced rank:

$\mathrm{rank}\!\left(\sum_{s=0}^\infty \Phi_s\right) = n - r$

There are exactly $n - r$ structural shocks with permanent effects on the levels of $y_t$ , and $r$ shocks with transitory effects whose cumulative impact is zero (Engle and Granger 1987). This decomposition is the foundation of long-run identification schemes.

Long-run identification (Blanchard-Quah 1989). The Wold MA in first differences $\Delta y_t = \sum_{s=0}^\infty \Xi_s u_{t-s}$ can be orthogonalized as $\Delta y_t = \sum_{s=0}^\infty \Theta_s e_{t-s}$ where $e_t = P^{-1}u_t$ and $\Theta_s = \Phi_s P$ . The long-run effect of shock $j$ on variable $i$ is $(\sum_{s=0}^\infty \Theta_s)_{ij}$ . Imposing that a shock has no long-run effect on a variable is the restriction $e_i'(\sum \Theta_s)e_j = 0$ , where $e_i$ is a unit vector. For a cointegrated system with rank $r$ this provides $r(n-r)$ additional constraints on $P$ beyond $PP' = \Sigma_u$ , potentially achieving identification without contemporaneous zero restrictions.

Bootstrap Confidence Intervals for IRFs (Lütkepohl 1999)

Analytical asymptotic variances of IRF coefficients follow from the delta method:

$\sqrt{T}(\hat\phi_{ij,s} - \phi_{ij,s}) \xrightarrow{d} \mathcal{N}(0, \sigma^2_{ij,s}), \qquad \sigma^2_{ij,s} = \frac{\partial\phi_{ij,s}}{\partial a'} \Sigma_a \frac{\partial\phi_{ij,s}}{\partial a}$

where $a$ collects the VAR coefficients. Bootstrap methods are popular because (1) the analytical expressions are complicated, and (2) bootstrapping can improve finite-sample coverage.

However, bootstrap confidence intervals (CIs) cannot fix the degenerate case: when $\sigma^2_{ij,s} = 0$ (the gradient is zero, e.g., a restricted subset VAR where $\phi_{ij,s} = 0$ is an exact consequence of the restrictions), neither the asymptotic normal approximation nor the bootstrap provides a valid confidence interval. The bootstrap failure in this case is not a numerical issue but a fundamental one — there is no distributional uncertainty to quantify when the parameter is constrained to zero.

DSGE Model IRF vs. VAR IRF

A DSGE model with state-space $(A,B,C,D)$ has its own economic model IRF, read directly from the state-space matrices without any VAR estimation:

$d_0 = D, \qquad d_j = CA^{j-1}B, \quad j \geq 1 \tag{5}$

The VAR IRF, by contrast, is computed from the innovations representation $(A,K,C,\Sigma)$ produced by the Kalman filter:

$c_0 = G, \qquad c_j = CA^{j-1}KG, \quad j \geq 1 \tag{15}$

where $G$ is the identification matrix for the Wold innovations $\varepsilon_t$ .

Condition for equality: $d_j = c_j$ for all $j \geq 0$ if and only if the Invertibility Problem eigenvalue condition holds — all eigenvalues of $A - BD^{-1}C$ strictly less than 1 in modulus. Under this condition:

$K = BD^{-1}, \qquad \Sigma = 0, \qquad G = D$

so that $c_j = CA^{j-1}(BD^{-1})D = CA^{j-1}B = d_j$ . The structural identification matrix $G = D$ is then known directly from the model, not from an arbitrary sign or zero restriction.

When invertibility fails, the VAR innovation covariance is $GG' = C\Sigma C' + DD' > DD'$ (strictly), and no identification of $G$ can equate the VAR IRF to the model IRF. This means DSGE-VAR comparisons based on impulse-response matching are fundamentally compromised unless the invertibility pre-condition is verified first.

Persistence Measure for Stationary Processes (Fiorentini-Sentana 1998)

Standard long-run persistence measures (the sum of MA coefficients $\Psi(1) = \sum_{j=0}^\infty \psi_j$ ) are zero for any I(0) process, making them uninformative about the degree of persistence within the stationary class. Fiorentini and Sentana (1998) propose an IRF-based measure that quantifies the total impulse-response energy:

$P_\infty(x_t|\epsilon_t) = \sum_{j=0}^\infty \psi_j^2 = \frac{V(x_t)}{V(\epsilon_t)}$

This equals the ratio of the process variance to the innovation variance, or equivalently $2\pi$ times the spectral density at frequency zero normalised by $V(\epsilon_t)$ . For an AR(1): $P_\infty = 1/(1-\phi^2)$ ; for ARMA(1,1): $P_\infty = [1 + (\phi-\theta)^2/(1-\phi^2)]$ .

The $k$ -period interim persistence is $P_k(x_t|\epsilon_t) = \sum_{j=0}^k \psi_j^2$ , equal to the ratio of the $(k+1)$ -step-ahead forecast error variance to the innovation variance.

Key identity: For the same innovation $\epsilon_t$ , the persistence of the conditional mean $\mu_{t+1} = E_t(x_{t+1})$ and the persistence of $x_t$ satisfy:

$P_\infty(\mu_{t+1}|\epsilon_t) = P_\infty(x_t|\epsilon_t) - 1$

This follows from the fact that $x_t = \mu_t + u_t$ where $u_t$ is the one-step innovation, so $V(x_t) = V(\mu_t) + V(u_t)$ . In the univariate case, white noise for $x_t$ ( $P_\infty = 1$ ) forces constant $\mu_t$ ( $P_\infty = 0$ ) — observable returns are unpredictable if and only if their conditional mean is constant. The multivariate case allows white-noise returns with a highly persistent conditional mean, provided innovations to the two series are nearly perfectly negatively correlated (see Risk-Return Tradeoff).

Why It Matters

IRFs are the primary output of structural VAR analysis. They are used to:

Characterize monetary transmission mechanisms (e.g., response of output and inflation to a policy rate shock).
Test theoretical predictions of DSGE models.
Calibrate structural models via impulse-response matching.

Open Questions

IRFs are estimated with substantial uncertainty, especially at long horizons; error bands (see Error Bands) can be very wide.
In near-integrated systems, the long-run cumulative IRF $\Phi_\infty$ is highly sensitive to lag specification.