EC5221 · Econometric Time Series Analysis
Impulse response functions in VAR models
A postgraduate note on stationary multivariate time series: how a vector autoregression is inverted into its moving-average form, how impulse response functions read off the dynamic effect of a shock, and why correlated reduced-form shocks must be orthogonalised — the Cholesky ordering — before the responses can be interpreted.
A vector autoregression has too many coefficients to interpret directly. The impulse response function is the tool that turns those coefficients into an economic story: the path of the whole system's response to a one-off shock. This note builds the IRF from the moving-average representation and confronts the identification problem that orthogonalisation solves.
The VAR and its moving-average form
A vector autoregression is the multivariate generalisation of an AR model: a \(k\times1\) vector \(Y_t\) in which each series depends on the lags of all the series. In lag-operator form,
with \(k\times k\) coefficient matrices \(\Phi_i\) and innovation vector \(\varepsilon_t\). If the VAR is stationary it can be inverted into a vector moving-average (Wold) representation — the current state as a weighted sum of present and past shocks:
Matching powers of \(L\) in \(I_k=\Theta(L)\Phi(L)\) gives the recursion for the MA matrices,
with \(\Phi_m=\mathbf 0\) for \(m>p\). These \(\Theta_s\) matrices are exactly what describe how a shock propagates through time.
Impulse responses: the AR(1) intuition
Take a stationary scalar AR(1), \(y_t=\phi y_{t-1}+\varepsilon_t=\sum_{i=0}^{\infty}\phi^i\varepsilon_{t-i}\). Perturb the period-\(t\) shock by one unit and trace the paths:
The response at horizon \(\ell\) is just the MA coefficient \(\phi^{\ell}\), and the whole sequence \(\{\partial y_{t+\ell}/\partial\varepsilon_t\}_{\ell\ge0}\) is the impulse response function. For a stationary process \(|\phi|\lt1\), so \(\phi^{\ell}\to0\): shocks die out geometrically and the system returns to its mean.
Impulse responses in the VAR
In the multivariate case a shock to one equation moves every variable's whole future path. Reading off the MA representation, the response of variable \(i\) at horizon \(s\) to a unit shock in variable \(j\) is the \((i,j)\) element of \(\Theta_s\):
Plotting \([\Theta_s]_{ij}\) against \(s\) gives the impulse response of \(i\) to \(j\). This is the workhorse output of applied VAR analysis.
A VAR(1) in two countries' GDP growth rates has an estimated residual covariance matrix such as \(\hat\Sigma=\begin{pmatrix}0.65 & 0.20\\ 0.20 & 0.81\end{pmatrix}\), implying a contemporaneous residual correlation of about \(0.20/\sqrt{0.65\times0.81}\approx0.28\). The two reduced-form shocks are correlated — which is precisely what makes the raw impulse responses hard to interpret.
The interpretation problem
A raw IRF shocks one equation's error while holding the others at zero. But the reduced-form shocks satisfy \(\mathbb{E}(\varepsilon_t\varepsilon_t')=\Sigma\), which is generally not diagonal: in the data, a shock to one equation typically arrives alongside shocks to the others. Setting the others to zero is therefore not a coherent experiment. We need a set of uncorrelated (orthogonal) shocks so that moving one in isolation is meaningful.
Orthogonalisation via the Cholesky decomposition
Pre-multiply the whole VAR by a constant matrix \(C\) to define a structural (orthogonalised) system whose disturbances \(u_t=C\varepsilon_t\) have a diagonal covariance \(\mathbb{E}(u_tu_t')\):
Many matrices \(C\) diagonalise \(\Sigma\); the standard choice is the Cholesky decomposition, \(\Sigma=PP'\) with \(P\) lower-triangular, taking \(C=P^{-1}\) so that \(\mathbb{E}(u_tu_t')=P^{-1}\Sigma(P^{-1})'=I\). Concretely, with a lower-triangular \(C\) the orthogonal shocks are built by sequential regression — \(u_{1t}=\varepsilon_{1t}\); \(u_{2t}\) is the part of \(\varepsilon_{2t}\) orthogonal to \(\varepsilon_{1t}\); \(u_{3t}\) the part of \(\varepsilon_{3t}\) orthogonal to both — a Gram–Schmidt construction.
Ordering is an identifying assumption. A lower-triangular \(C\) imposes a recursive contemporaneous ordering: the first-ordered variable responds to no other within the period, the last responds to all. Because \(C\) multiplies the entire VAR, the lag matrices \(A_j=C\Phi_j\) change too, not just the shocks. Different orderings give different orthogonalised impulse responses, so the ordering should be justified by economic theory, not chosen arbitrarily.
Application: world shocks and the UK economy
A published central-bank study asks how global shocks have transmitted into the UK since the financial crisis. It estimates a VAR in two blocks — a world block (world GDP growth, world price growth, a US interest-rate spread, a volatility index) and a UK block (UK GDP growth, CPI inflation, the policy rate) — ordered world before UK. That ordering encodes the small-open-economy assumption: UK variables do not affect world variables within the quarter, but world variables affect the UK. Orthogonalising by Cholesky, the structural UK shocks are the parts of the UK reduced-form shocks not explained by the world (and, recursively, by higher-ordered UK) variables. Switching off the domestic structural shocks and feeding only the world shocks back through the estimated system builds a counterfactual path that isolates the contribution of world events to UK output, inflation and interest rates.
Self-check questions
What is the impulse response at horizon ℓ for an AR(1), and why does it vanish for a stationary process?
From \(y_t=\sum_{i\ge0}\phi^i\varepsilon_{t-i}\), the response of \(y_{t+\ell}\) to a unit shock at \(t\) is \(\partial y_{t+\ell}/\partial\varepsilon_t=\phi^{\ell}\). Stationarity of an AR(1) requires \(|\phi|\lt1\), so \(\phi^{\ell}\to0\) as \(\ell\to\infty\): the effect decays geometrically and the series returns to its mean. This is the scalar version of the VAR result \(\partial y_{i,t+s}/\partial\varepsilon_{j,t}=[\Theta_s]_{ij}\).
Given a non-diagonal Σ, why can't we read a "shock to variable 1 only" off the raw IRF, and what does Cholesky do?
A non-diagonal \(\Sigma\) means the reduced-form shocks are contemporaneously correlated, so empirically a shock to variable 1 tends to coincide with shocks to the others. Setting the others to zero — what a raw IRF does — is not a realistic experiment. The Cholesky decomposition \(\Sigma=PP'\) defines orthogonal shocks \(u_t=P^{-1}\varepsilon_t\) with identity covariance. With variable 1 ordered first, its structural shock is \(\varepsilon_{1t}\) itself, while the others' shocks are the parts orthogonal to it, so shocking \(u_{1t}\) coherently carries its typical contemporaneous spillover into the rest of the system.
Why is the Cholesky ordering "not innocuous"?
Cholesky forces a lower-triangular \(C\), i.e. a recursive contemporaneous ordering: the first variable reacts to no other within the period, the last reacts to all. Reversing the order changes which variable may react contemporaneously, changing \(C\), the structural shocks \(u_t=C\varepsilon_t\), the transformed lag matrices \(A_j=C\Phi_j\), and therefore the entire set of orthogonalised impulse responses. The remedy is to justify the ordering with economic theory — for instance world-before-UK in a small-open-economy model, or placing a monetary-policy rate last so it responds to everything within the period.
Related notes
- EC5221 lecture slides and problem sets — ARMA and stationarity, estimation and inference, and the bivariate VAR(1) problem set with impulse responses and Granger causality.
- Time series econometrics guide · AR and ARMA processes explained.
- Econometrics explained · All teaching materials.
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