Impulse Response Functions (IRFs)

Macroeconomics (M8674)

February 2026


Vivaldo Mendes, ISCTE
vivaldo.mendes@iscte-iul.pt

1. Main goal

What are IRFs?


  • Impulse response functions represent the response of the endogenous variables of a given system, when one (or more than one) of these variables is hit by an exogenous shock.
  • The nature of the shock can be:
    • Temporary
    • Permanent
    • Systematic
  • Linear systems. The magnitude of the shock does not change the stability properties of the system.
  • Nonlinear systems. In this case, the magnitude of the shock is of great importance and it can change the stability of the system under consideration.

An example

  • Consider the simplest case, an AR(1):

\[y_{t+1}= {\color{blue}a} y_{t}+\varepsilon_{t+1}, \quad \varepsilon_{t} \sim N(0,1) \tag{2}\]

  • Assume that for \(t \in(1,n)\): \[y_{1}=0 \ ; \ \varepsilon_{2}=1 \ ; \ \varepsilon_{t}=0 \ , \ \forall t\neq2\]
  • This implies that at \(t=2 \Rightarrow y_2 = 1\). But what happens next, if there are no more shocks?
  • The IRF of \(y\) provides the answer.
  • The dynamics of \(y\) will depend crucially on the value of \(\color{blue}a\). Six examples: \[\color{blue} a = \{0,0.5,0.9, 0.99, 1, 1.01\}\]

The IRFs of the AR(1) Process


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Another example


  • Consider a more sophisticated case, an AR(2):

\[y_{t+1}= {\color{blue}a} y_{t}+{\color{red}b} y_{t-1} + \varepsilon_{t+1}, \quad \varepsilon_{t} \sim N(0,1)\] * Assume that for \(t \in(1,n)\): \[y_{1}=0 \ ; \ y_2=0 \ ; \ \varepsilon_{3}=1 \ ; \ \varepsilon_{t}=0 \ , \ \forall t\neq 3\]

  • This implies that at
    \[t=3 \ , \ \varepsilon_{3}=1 \ \Rightarrow \ y_3 = 1.\]

  • What happens next, if there are no more shocks? The IRF of \(y\) provides the answer.

  • The dynamics of \(y\) will depend on the values of \(\color{blue}a\) and \(\color{red}b\). For simplicity consider: \[{\color{red}b = -0.9} \ ; \color{blue} a = \{1.85,1.895,1.9,1.9005\}\]

The IRFs of the AR(2) Process


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More Sophisticated Examples

  • A similar reasoning can be applied to our rather more general model: \[X_{t+1}= A + BX_{t}+ C\varepsilon_{t+1} \tag{3}\]

  • .. where \(B , C\) are \(n\times n\) matrices, while \(X_{t+1}, X_{t}, A, \varepsilon_{t+1}\) are \(n\times 1\) vectors.

  • Consider the following VAR(3) model: \[X_{t+1}=\left[\begin{array}{c} z_{t+1} \\ w_{t+1} \\ v_{t+1} \end{array}\right]\]

  • In this example we take matrices \(A, B\) and \(C\) given by: $$

    \[\begin{gathered} A=\left[\begin{array}{c} 0.0 \\ 0.0 \\ 0.0 \end{array}\right], \quad B=\left[\begin{array}{rrr} 0.97 & 0.10 & -0.05 \\ -0.3 & 0.8 & 0.05 \\ 0.01 & -0.04 & 0.96 \end{array}\right] , \quad C=\left[\begin{array}{lll} {\color{blue}1.0} & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 \end{array}\right] . \end{gathered}\]

    $$

  • The initial state of our system (or its initial conditions) are: \(z_{1}=0, w_{1}=0\) and \(v_{1}=0\), that is: \[X_{1}=[0,0,0]\]

  • The shock only hits the variable \(z_t\) \((\)notice the blue entry in matrix \(C)\), and we assume that the shock occurs in period \(t=3\).

  • What happens to the dynamics of the three endogenous variables? See next figure.

The IRFs of our VAR(3) Process


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AR(1): A Sequence of Shocks


Consider the same AR(1) as in eq. (2). But now impose a sequence of 200 shocks.

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Implications of a Linear Structure


  • In the previous examples, the structure of all our models was linear.
  • This has a crucial implication:
    • The shock’s magnitude did not alter the dynamics produced by the shock itself.
    • Only the structure of the model would lead to different outcomes.
  • This does not usually occur if the structure of the model is non-linear. In this case, the magnitude of the shock may produce different outcomes even if the system’s structure remains the same.
  • We do not have time to cover this particular point.
  • But be careful: if the structure of the model is non-linear, large shocks can not be simulated … in a linearized version of the original system.

3. Macroeconomics: Why are IRFs important ?

Macroeconomics: real experiments are impossible

  • In physics, chemistry, biology, etc., we can perform experiments to study the behavior of a system when it is hit by a shock.
  • In macroeconomics, we cannot perform such experiments.
  • So, we construct abstractions – models – to simulate how the economy is supposed to work.
  • We assume that the economy is on its steady-state and we impose a certain shock to an endogenous variable.
  • The IRFs give us the reaction of the entire economy. So, in a sense, IRFs represent our “experiments”.

IRFs everywhere


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A good example:

  • Boehl, Gregor (2025). The Political Economy of Monetary Financing without Inflation, Working Paper, University of Bonn.
    • An initial steady state
    • A new steady state
    • IRFs as the transition process

3. Readings

  • For this point, there is no compulsory reading. However, any modern textbook on time series will cover this subject.
  • At an introductory level, see sections 11.8 and 11.9 of the textbook: Diebold, F. X. (1998). Elements of forecasting. South-Western College Pub, Cincinnati.
  • At a more advanced level, see, e.g., section 2.3.2 of the textbook: Lütkepohl, H. (2007). New introduction to multiple time series analysis (2nd ed.), Springer, Berlin.