Historical Decomposition

Historical Decomposition (HD) decomposes observed variable movements into contributions from individual structural shocks over time. While FEVD answers "which shocks matter in general?", HD answers "which shocks drove this specific episode?" – making it an indispensable tool for narrative analysis of macroeconomic events.

• Frequentist HD: Exact additive decomposition with verification of the identity $y_t = \sum_j \text{HD}_j(t) + \text{initial}(t)$
• Bayesian HD: Posterior distributions over shock contributions with credible intervals
• Accessor functions: contribution(), total_shock_contribution(), and verify_decomposition() for programmatic analysis

using MacroEconometricModels, Random
Random.seed!(42)
fred = load_example(:fred_md)
Y = to_matrix(apply_tcode(fred[:, ["INDPRO", "CPIAUCSL", "FEDFUNDS"]]))
Y = Y[all.(isfinite, eachrow(Y)), :]
Y = Y[end-59:end, :]
model = estimate_var(Y, 4; varnames=["INDPRO", "CPIAUCSL", "FEDFUNDS"])
post = estimate_bvar(Y, 4; n_draws=100, varnames=["INDPRO", "CPIAUCSL", "FEDFUNDS"])

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Quick Start

Recipe 1: Basic historical decomposition

# HD with Cholesky identification
hd = historical_decomposition(model)
report(hd)

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Recipe 2: Verify the decomposition identity

hd = historical_decomposition(model)

# The identity y_t = sum_j HD_j(t) + initial(t) holds to machine precision
verified = verify_decomposition(hd)

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Recipe 3: Extract individual shock contributions

hd = historical_decomposition(model)

# Contribution of monetary shock (shock 3) to output (variable 1)
monetary_to_output = contribution(hd, 1, 3)

# Total shock-driven component for variable 1 (excludes initial conditions)
total = total_shock_contribution(hd, 1)

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Recipe 4: Bayesian historical decomposition

bhd = historical_decomposition(post; method=:cholesky)
report(bhd)

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Recipe 5: HD visualization

hd = historical_decomposition(model)
nothing # hide

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plot_result(hd)

Frequentist HD

Historical decomposition derives from the structural VMA representation of the VAR. The observed data at time $t$ is decomposed into the cumulative effect of all past structural shocks plus a contribution from initial (pre-sample) conditions (Kilian & Lutkepohl 2017, Chapter 4).

\[y_t = \sum_{s=0}^{t-1} \Theta_s \varepsilon_{t-s} + \text{initial conditions}\]

where:

• $y_t$ is the $n \times 1$ vector of observed variables at time $t$
• $\Theta_s = \Phi_s P$ are the $n \times n$ structural MA coefficients at lag $s$
• $\Phi_s$ are the reduced-form MA coefficients from the VMA representation
• $P = L Q$ is the $n \times n$ structural impact matrix (Cholesky factor $L$ times rotation $Q$)
• $\varepsilon_t = Q' L^{-1} u_t$ are the $n \times 1$ structural shocks
• The initial conditions capture the contribution of pre-sample values through the VAR dynamics

Contribution Formula

The contribution of shock $j$ to variable $i$ at time $t$ is:

\[\text{HD}_{ij}(t) = \sum_{s=0}^{t-1} (\Theta_s)_{ij} \, \varepsilon_j(t-s)\]

where:

• $\text{HD}_{ij}(t)$ is the contribution of shock $j$ to variable $i$ at time $t$
• $(\Theta_s)_{ij}$ is the $(i,j)$ element of the structural MA coefficient at lag $s$
• $\varepsilon_j(t-s)$ is the realized structural shock $j$ at time $t-s$

Additive Identity

The decomposition satisfies an exact identity — the sum of all shock contributions plus initial conditions recovers the observed data:

\[y_{i,t} = \sum_{j=1}^{n} \text{HD}_{ij}(t) + \text{initial}_i(t)\]

The verify_decomposition() function checks this identity to machine precision (default tolerance $10^{-10}$). A failure indicates a numerical problem in the MA coefficient computation.

Code Example

# Historical decomposition with Cholesky identification
hd = historical_decomposition(model)
report(hd)

# Verify the additive identity holds
verified = verify_decomposition(hd)

# Extract contribution of monetary policy shock (shock 3) to output (variable 1)
monetary_to_output = contribution(hd, 1, 3)

# Total shock-driven component of INDPRO (excludes initial conditions)
total = total_shock_contribution(hd, 1)

# Tabular output for the last 10 periods
print_table(stdout, hd, 1; periods=(hd.T_eff-9):hd.T_eff)

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The HD reveals which structural shocks drove specific historical episodes. Large positive contributions from the monetary shock to industrial production indicate periods of accommodative policy, while large negative contributions signal contractionary episodes. The initial conditions component is typically large at the beginning of the sample and decays toward zero as the sample grows, reflecting the diminishing influence of pre-sample values.

Helper Functions

HistoricalDecomposition Return Values

Keyword Arguments

Bayesian HD

Bayesian HD computes the historical decomposition for each posterior draw from a Bayesian VAR, producing posterior distributions over shock contributions (Kilian & Lutkepohl 2017, Chapter 12). Non-stationary posterior draws are discarded to ensure economically meaningful decompositions.

# Bayesian HD with 68% credible intervals
bhd = historical_decomposition(post; method=:cholesky,
                               quantiles=[0.16, 0.5, 0.84])
report(bhd)

# Access posterior mean contribution of shock 3 to variable 1
monetary_contrib = contribution(bhd, 1, 3; stat=:mean)

# Verify the mean decomposition identity (approximate, due to averaging)
verified = verify_decomposition(bhd)

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The Bayesian HD iterates over all stationary posterior draws, computing the structural MA coefficients and shock contributions for each. The point_estimate array contains the posterior mean (or median) contributions, while the quantiles array stores the full posterior distribution. Wide credible bands around a shock's contribution indicate that the data do not clearly attribute a specific episode to that shock.

BayesianHistoricalDecomposition Return Values

Keyword Arguments

Complete Example

This example combines IRF, FEVD, and HD for a complete structural analysis of a three-variable monetary policy VAR. The workflow moves from shock identification to dynamic responses, variance contributions, and finally episode-level attribution.

# Step 1: Impulse responses — how do variables respond to shocks?
irfs = irf(model, 20; method=:cholesky, ci_type=:bootstrap, reps=50)
report(irfs)

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# Step 2: Variance decomposition — which shocks matter most?
decomp = fevd(model, 20)
report(decomp)
print_table(stdout, decomp, 1; horizons=[1, 4, 8, 20])

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# Step 3: Historical decomposition — which shocks drove specific episodes?
hd = historical_decomposition(model)
report(hd)

# Verify the additive identity
verified = verify_decomposition(hd)

# Monetary policy contribution to output over time
monetary_to_output = contribution(hd, 1, 3)
nothing # hide

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plot_result(irfs)
plot_result(decomp)
plot_result(hd)

The IRFs reveal the dynamic transmission mechanism: a contractionary monetary shock (positive FFR innovation) reduces industrial production with a lag of several quarters. The FEVD shows that monetary shocks explain a modest but non-trivial share of output forecast error variance at business-cycle horizons. The HD identifies specific historical episodes where monetary shocks made large positive or negative contributions to output movements, providing a narrative interpretation of the estimated structural model.

Common Pitfalls

1. HD depends on the full identification scheme. The historical decomposition inherits all identification assumptions from the structural VAR. With Cholesky identification, reordering the variables changes every shock contribution. With sign restrictions, the HD reports a representative draw from the identified set — re-run with multiple seeds to assess robustness.

2. Initial conditions dominate early in the sample. The first $p$ to $2p$ periods of the HD are driven primarily by initial conditions rather than structural shocks, because the VMA representation has not yet accumulated enough lags. Focus interpretation on the middle and end of the sample.

3. Bayesian HD discards non-stationary draws. If many posterior draws are non-stationary (common with diffuse priors or short samples), the effective number of draws for the Bayesian HD is reduced. A warning is issued when more than half of the draws are discarded. Tighten the Minnesota prior or increase n_draws to compensate.

4. The horizon argument caps MA coefficient computation. For frequentist HD, the horizon argument controls how many structural MA coefficients $\Theta_s$ are computed. Setting horizon less than T_eff truncates the decomposition and the additive identity holds only approximately. The default uses the full effective sample size.

5. String indexing requires matching variable names. The contribution(hd, "INDPRO", "FEDFUNDS") syntax requires that the variable names stored in the model match the strings exactly. Use hd.variables and hd.shock_names to inspect the available names.

References

• Kilian, Lutz, and Helmut Lutkepohl. 2017. Structural Vector Autoregressive Analysis. Cambridge: Cambridge University Press. DOI: 10.1017/9781108164818
• Lutkepohl, Helmut. 2005. New Introduction to Multiple Time Series Analysis. Berlin: Springer. ISBN 978-3-540-40172-8.