Forecast Error Variance Decomposition¶

Quick Reference

Class: panelbox.var.fevd.FEVDResult Method: PanelVARResult.fevd() Import: from panelbox.var import PanelVAR (FEVD accessed via results) Stata equivalent: irf table fevd R equivalent: vars::fevd()

Overview¶

Forecast Error Variance Decomposition (FEVD) measures the proportion of the forecast error variance of each variable that is attributable to shocks from each variable in the system. While IRFs trace the effect of a single shock over time, FEVD answers: What fraction of the uncertainty in forecasting variable \(i\) at horizon \(h\) is due to shocks in variable \(j\)?

At each horizon \(h\) and for each variable \(i\), the FEVD sums to 100%:

\[ \sum_{j=1}^{K} \omega_{ij}(h) = 1 \]

where \(\omega_{ij}(h)\) is the share of forecast error variance of variable \(i\) at horizon \(h\) explained by shocks to variable \(j\).

For Cholesky-identified FEVD:

\[ \omega_{ij}(h) = \frac{\sum_{s=0}^{h} (\Phi_s^{\text{orth}}[i,j])^2}{\sum_{s=0}^{h} \sum_{k=1}^{K} (\Phi_s^{\text{orth}}[i,k])^2} \]

FEVD provides a complementary perspective to IRFs: IRFs show the direction and magnitude of effects, while FEVD shows their relative importance.

Quick Example¶

from panelbox.var import PanelVARData, PanelVAR

# Estimate model
var_data = PanelVARData(df, endog_vars=["gdp", "inflation", "rate"],
                         entity_col="country", time_col="year", lags=2)
model = PanelVAR(data=var_data)
results = model.fit(cov_type="clustered")

# Compute FEVD
fevd = results.fevd(periods=20, method="cholesky")

# View decomposition for GDP
print(fevd["gdp"])    # Returns (periods+1, K) array

# Summary at selected horizons
print(fevd.summary(horizons=[1, 5, 10, 20]))

# Plot
fevd.plot()

When to Use¶

Relative importance: Which variables drive the most uncertainty in your variable of interest?
Exogeneity assessment: If a variable's forecast error is dominated by own shocks at all horizons, it behaves as (nearly) exogenous
Shock dominance: At what horizon do external shocks start to dominate own shocks?
Model validation: FEVD patterns should be consistent with economic theory and IRF findings

Key Assumptions

Same identification assumptions as IRFs (Cholesky ordering or Generalized)
VAR must be stable for well-defined FEVD
Cholesky FEVD depends on variable ordering (same as Cholesky IRFs)
Generalized FEVD is order-invariant but requires normalization (rows may not sum to 1 before normalization)

Detailed Guide¶

Computing FEVD¶

fevd = results.fevd(
    periods=20,               # Number of horizons
    method="cholesky",        # "cholesky" or "generalized"
    order=None,               # Custom ordering (Cholesky only)
)

Parameters:

Parameter	Type	Default	Description
`periods`	`int`	`20`	Number of horizons to compute
`method`	`str`	`"cholesky"`	`"cholesky"` or `"generalized"`
`order`	`list[str]`	`None`	Variable ordering for Cholesky

Identification Methods¶

Cholesky FEVDGeneralized FEVD

Based on orthogonalized IRFs from Cholesky decomposition. Shares sum exactly to 100% at every horizon.

fevd = results.fevd(periods=20, method="cholesky")

# Custom ordering
fevd = results.fevd(
    periods=20,
    method="cholesky",
    order=["rate", "inflation", "gdp"],
)

Based on Generalized IRFs (Pesaran-Shin). Order-invariant but raw shares may not sum to 100%. PanelBox automatically normalizes the shares so each row sums to 1.

fevd = results.fevd(periods=20, method="generalized")

Accessing Results¶

# Access FEVD for a specific variable
gdp_fevd = fevd["gdp"]           # np.ndarray (periods+1, K)
# gdp_fevd[h, j] = share of GDP variance at horizon h from shock j

# Convert to DataFrame
df_gdp = fevd.to_dataframe(variable="gdp")
df_gdp_horizons = fevd.to_dataframe(variable="gdp", horizons=[1, 5, 10, 20])
df_all = fevd.to_dataframe(horizons=[1, 5, 10, 20])

# Summary table
print(fevd.summary(horizons=[1, 5, 10, 20]))

FEVDResult Attributes:

Attribute	Type	Description
`decomposition`	`np.ndarray`	Shape `(periods+1, K, K)`. `decomposition[h, i, j]` = share of var \(i\) variance at \(h\) from shock \(j\)
`var_names`	`list[str]`	Variable names
`periods`	`int`	Number of horizons
`method`	`str`	Identification method
`ordering`	`list[str]` or `None`	Variable ordering used

Visualization¶

# Stacked area chart (default) -- shows evolution of shares over horizons
fevd.plot(kind="area", backend="matplotlib")

# Stacked bar chart at selected horizons
fevd.plot(kind="bar", horizons=[1, 5, 10, 20])

# Plot specific variables only
fevd.plot(variables=["gdp", "inflation"])

# Interactive Plotly chart
fevd.plot(backend="plotly", theme="professional")

Interpreting FEVD¶

Typical patterns:

Horizon	Pattern	Interpretation
\(h = 1\)	Own shock dominates (~80-100%)	At short horizons, variables are mostly driven by their own shocks
\(h = 5\)	Cross-variable shares increase	External shocks start accumulating effect
\(h = 20\)	Shares stabilize	Long-run decomposition reveals structural importance

Reading the decomposition:

Variable: gdp
Horizon    Shock gdp    Shock inflation    Shock rate
h=1           95.2%            3.1%           1.7%
h=5           72.4%           18.3%           9.3%
h=10          58.1%           25.6%          16.3%
h=20          52.3%           28.4%          19.3%

This tells us: - At \(h=1\), 95% of GDP forecast uncertainty comes from GDP's own shocks - By \(h=20\), inflation shocks explain 28% and interest rate shocks explain 19% of GDP forecast uncertainty - Inflation is a more important driver of GDP uncertainty than the interest rate

Consistency Check

FEVD results should be consistent with IRFs. If the IRF shows that inflation has a large effect on GDP, then inflation shocks should explain a meaningful share of GDP's FEVD. If they don't match, investigate the identification assumptions.

Comparing with IRFs¶

# Compute both IRF and FEVD with same identification
irf = results.irf(periods=20, method="cholesky", order=["rate", "inflation", "gdp"])
fevd = results.fevd(periods=20, method="cholesky", order=["rate", "inflation", "gdp"])

# IRF shows direction and magnitude
print("IRF: GDP response to inflation shock")
print(irf["gdp", "inflation"])

# FEVD shows relative importance
print("\nFEVD: GDP variance decomposition")
print(fevd.to_dataframe(variable="gdp", horizons=[1, 5, 10, 20]))

Configuration Options¶

Method Comparison¶

Feature	Cholesky	Generalized
Ordering-dependent	Yes	No
Shares sum to 100%	Exactly	After normalization
Consistent with IRF method	Yes (use same method)	Yes
Best for	Theory-guided analysis	Exploratory / robustness

Complete Example¶

import pandas as pd
from panelbox.var import PanelVARData, PanelVAR

# Load data
df = pd.read_csv("macro_panel.csv")

# Estimate Panel VAR
var_data = PanelVARData(
    data=df,
    endog_vars=["gdp_growth", "inflation", "interest_rate"],
    entity_col="country",
    time_col="year",
    lags=2,
)
model = PanelVAR(data=var_data)
results = model.fit(cov_type="clustered")

# FEVD with economic ordering: interest rate -> inflation -> GDP
fevd = results.fevd(
    periods=20,
    method="cholesky",
    order=["interest_rate", "inflation", "gdp_growth"],
)

# Detailed summary
print(fevd.summary())

# Export to DataFrame for further analysis
df_fevd = fevd.to_dataframe(horizons=[1, 5, 10, 20])
print(df_fevd)

# Visual analysis
fevd.plot(kind="area", backend="matplotlib", theme="academic")

Tutorials¶

Tutorial	Description	Link
FEVD Analysis	Variance decomposition workflow

References¶

Luetkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer-Verlag, Chapter 2.
Pesaran, H. H., & Shin, Y. (1998). Generalized impulse response analysis in linear multivariate models. Economics Letters, 58(1), 17-29.
Sims, C. A. (1980). Macroeconomics and reality. Econometrica, 48(1), 1-48.