Historical Decomposition

Definition

Historical decomposition attributes realized values of observed variables to the cumulated contributions of each identified structural shock over a given sample period. Starting from a no-shock baseline forecast, it answers: "of the deviation between actual outcomes and the baseline, how much is explained by each structural force?"

Key Ideas

The baseline ("no-shock scenario") is the model's conditional forecast from a given date assuming all future shocks are zero ( $\varepsilon_{t_0+h} = 0$ for all $h \geq 1$ ).
Adding one shock group at a time produces "ex post forecasts" that progressively close the gap between the baseline and the realized path.
When all identified shock groups are included, the model exactly replicates the realized path by construction.
Interpretation is entirely determined by the identification scheme: in a Bayesian VAR (BVAR), shocks are labeled only to the extent the identification restrictions allow; in a Dynamic Stochastic General Equilibrium (DSGE) model, every shock has an economic name.
BVAR and DSGE decompositions can be compared as a robustness check: agreement on the dominant shock group strengthens a causal narrative.

How It Works

VAR Historical Decomposition

Given a structural vector autoregression (VAR) $y_t = \sum_\ell B_\ell y_{t-\ell} + A\varepsilon_t$ with identified shocks $\varepsilon_t$ , decompose the $h$ -step realization from $t_0$ as:

$y_{t_0+h} = \hat{y}_{t_0+h\mid t_0}^{\text{no-shock}} + \sum_{g}\Delta_g(t_0,h)$

where the contribution of shock group $g$ is:

$\Delta_g(t_0,h) = \sum_{j=1}^h \Theta_j^g\,\varepsilon_{t_0+j}^g$

$\Theta_j^g$ are the structural impulse response function (IRF) matrices restricted to columns in group $g$ , and $\varepsilon_t^g$ are the estimated shock realizations.

BVAR example (Adolfson et al. 2005 — Cholesky identification): "Foreign" shocks are those entering through the equations for foreign gross domestic product (GDP) growth, foreign Consumer Price Index (CPI) inflation, and the foreign interest rate, which are ordered first. "Domestic" shocks are the residual. This minimal two-group decomposition attributed most of the 2003 low-inflation surprise in Sweden to foreign shocks.

DSGE Historical Decomposition

In an estimated DSGE, all structural shocks $\varepsilon_t^g$ are recoverable from the posterior distribution via the Kalman smoother. The decomposition proceeds identically using the model's state-space IRF matrices. Shock groups are fully labeled:

Monetary policy shocks: unexpected deviations from the central bank's reaction function
Technology shocks: unit-root and stationary productivity disturbances
Markup shocks: cost-push and price-setting disturbances
Foreign shocks: shocks to foreign output, inflation, interest rate
Preference shocks: household taste shifters
Fiscal shocks: government expenditure disturbances

DSGE example (Adolfson et al. 2005): For 2003–2004 Swedish low inflation, the DSGE confirms that foreign shocks dominated in 2003. In 2004–2005, technology shocks and foreign shocks jointly account for most of the forecast error. Markup shocks are negligible throughout — ruling out the "increased competition" explanation for low inflation.

Why It Matters

Provides a structural narrative of what caused an economic outcome — essential for central bank communication and policy learning.
Distinguishes between causes that lie outside policymakers' control (foreign shocks, technology) and those within it (monetary policy, markup distortions).
Exposes the limits of a purely statistical BVAR: it can assign shocks to broad groups (foreign vs. domestic) but cannot label sub-components without additional restrictions.
Comparing BVAR and DSGE decompositions builds confidence when they agree and raises questions when they diverge.

Open Questions

Decompositions are sensitive to the identification scheme — Cholesky ordering, sign restrictions, and long-run restrictions can yield very different shock attributions.
In DSGEs, markup shocks often absorb residuals that are not explained by other shocks; findings about their (ir)relevance may be model-specific.
Extension to nonlinear models (Markov-switching VARs, particle-filter DSGEs) is methodologically nontrivial.