Impulse Response Function (IRF)

To isolate the effect of a single structural shock, IRFs require orthogonal innovations. The standard approach uses Cholesky decomposition on the variance-covariance matrix, imposing a recursive causal ordering. This forces a lower-triangular structure where the first variable is strictly exogenous, and subsequent variables respond contemporaneously only to those ordered before them. The economic interpretation of the IRF is highly sensitive to this ordering assumption.

An IRF is fundamentally a dynamic multiplier. It traces the marginal effect of a one-unit innovation at time t on the entire future path of the system.

• Impact Multiplier: The immediate effect at horizon h=0.
• Interim Multipliers: The cumulative effect up to horizon h.
• Long-Run Multiplier: The total accumulated effect as h approaches infinity, representing the permanent shift in equilibrium.

Point estimates of IRFs are meaningless without quantifying uncertainty. Confidence bands are constructed using:

• Analytical (Delta Method): Assumes asymptotic normality, often leading to narrow, overconfident bands.
• Bootstrap (Hall's Percentile): Resamples residuals to generate an empirical distribution, providing more accurate coverage in finite samples.
• Bayesian (Credible Sets): Uses Monte Carlo Markov Chains to sample from the posterior distribution of the coefficients, naturally incorporating parameter uncertainty.

Proposed by Koop, Pesaran, and Potter (1996), GIRFs solve the ordering-dependence problem of Cholesky decompositions. Instead of imposing a structural causal chain, GIRFs integrate out the effects of other shocks using the historically observed distribution of errors. This makes the response invariant to variable ordering, but it abandons the attempt to identify structural shocks, instead measuring a generalized historical correlation dynamic.

A robust alternative to iterating a VAR forward. Local projections estimate the response at horizon h directly using a sequence of single-equation regressions. This method is more robust to model misspecification because it does not force the dynamics to compound through the VAR's lag structure. It is particularly advantageous for estimating non-linear state-dependent IRFs where the response depends on the current phase of the business cycle.

The shape of the IRF reveals the stability of the system.

• Monotonic Decay: The shock dissipates smoothly without oscillation.
• Hump-Shaped: The effect builds over several periods before decaying, indicating a propagation mechanism (e.g., investment accelerator).
• Oscillatory Convergence: Cyclical overshooting before returning to equilibrium.
• Permanent Divergence: The IRF does not return to zero, indicating a unit root or non-stationary process where shocks have a permanent effect on the level of the variable.

The foundational model for generating IRFs. A VAR models each variable as a linear function of its own past values and the past values of all other variables in the system. Key properties:

• Captures linear interdependencies among multiple time series
• Requires stationary data for valid inference
• Order selection via AIC or BIC is critical

Without a properly specified VAR, an IRF has no structural meaning.

A variant that isolates shocks using a Cholesky decomposition of the residual covariance matrix. This imposes a recursive causal ordering, meaning the first variable affects all others contemporaneously, the second affects all but the first, and so on.

• Critical caveat: Results are highly sensitive to variable ordering
• Used when theory dictates a clear causal hierarchy
• Contrast with Generalized IRFs, which are order-invariant

Developed by Pesaran and Shin (1998), this method constructs order-invariant responses by integrating out the effects of other shocks using the observed historical distribution of errors.

• Does not require orthogonalization
• Produces a unique, unambiguous response profile
• Preferred when no clear causal ordering exists among variables
• Commonly used in financial connectedness studies and spillover analysis

Imposes theory-driven restrictions on contemporaneous relationships to identify structural shocks with economic meaning. Unlike recursive Cholesky ordering, SVAR uses restrictions derived from economic theory, institutional knowledge, or long-run neutrality assumptions.

• Enables identification of truly exogenous shocks
• Common in monetary policy analysis (e.g., identifying a Fed funds rate shock)
• Over-identification tests can validate the imposed structure

The natural companion to IRFs. While an IRF traces the dynamic path of a shock, FEVD quantifies the proportion of the forecast error variance of each variable attributable to shocks in every other variable.

• Answers: 'How much of variable X's volatility comes from variable Y?'
• Used extensively in spillover index construction (Diebold & Yilmaz)
• Together with IRFs, provides a complete picture of dynamic interconnectedness

A robust alternative to VAR-derived IRFs proposed by Jordà (2005). Instead of iterating a VAR forward, local projections estimate the response at each horizon h using a separate direct regression.

• More robust to model misspecification
• Easily accommodates nonlinearities and state-dependence
• Does not impose the dynamic restrictions inherent in VARs
• Increasingly preferred in applied macroeconomics and finance for its flexibility