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):

  • Assume that for :

  • This implies that at . But what happens next, if there are no more shocks?
  • The IRF of provides the answer.
  • The dynamics of will depend crucially on the value of . Six examples:

The IRFs of the AR(1) Process


Another example


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

  • Assume that for :

  • This implies that at

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

  • The dynamics of will depend on the values of and . For simplicity consider:

The IRFs of the AR(2) Process


More Sophisticated Examples

  • A similar reasoning can be applied to our rather more general model:

  • .. where are matrices, while are vectors.

  • Consider the following VAR(3) model:

  • In this example we take matrices and given by:

  • The initial state of our system (or its initial conditions) are: and , that is:

  • The shock only hits the variable notice the blue entry in matrix , and we assume that the shock occurs in period .

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

The IRFs of our VAR(3) Process


AR(1): A Sequence of Shocks


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

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


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.