fevdCompute#

Purpose#

Compute forecast error variance decomposition.

Format#

fevd = fevdCompute(irf)#
fevd = fevdCompute(result, n_ahead)
Parameters:

  • irf (struct) – an instance of an irfResult structure from irfCompute().

  • result (struct) – alternatively, a varResult or bvarResult structure (IRF is computed internally).

  • n_ahead (scalar) – number of horizons (required when passing result).

  • quiet (scalar) – Optional keyword, set to 1 to suppress printed output. Default = 0.

Returns:

fevd (struct) –

An instance of a fevdResult structure containing:

fevd.fevd

Array of (n_ahead+1) mxm matrices. fevd.fevd[h+1][i, j] is the fraction of variable i’s forecast error variance at horizon h explained by shock j. Each row sums to 1.0.

fevd.n_ahead

Scalar, number of horizons computed.

fevd.m

Scalar, number of variables.

fevd.var_names

Mx1 string array, variable names.

Examples#

From Pre-Computed IRF#

new;
library timeseries;

data = loadd(getGAUSSHome("pkgs/timeseries/examples/macro.dat"));
result = varFit(data, 4);

irf = irfCompute(result, 20, quiet=1);

struct fevdResult fevd;
fevd = fevdCompute(irf);

Direct from Estimation Result#

new;
library timeseries;

data = loadd(getGAUSSHome("pkgs/timeseries/examples/macro.dat"));
result = varFit(data, 4);

// Skip the explicit IRF step
fevd = fevdCompute(result, 20);

Accessing Decomposition#

new;
library timeseries;

data = loadd(getGAUSSHome("pkgs/timeseries/examples/macro.dat"));
result = varFit(data, 4);
fevd = fevdCompute(result, 20, quiet=1);

// Fraction of GDP variance explained by each shock at h=20
print "GDP variance decomposition at h=20:";
print fevd.var_names';
print fevd.fevd[21, 1, .];

// Verify rows sum to 1
print "Sum:" sumc(fevd.fevd[21, 1, .]');

// Track how FFR's contribution to GDP evolves over horizons
print "FFR contribution to GDP over time:";
for h (0, 20, 1);
    print h;; print "  ";; print fevd.fevd[h+1, 1, 3];
endfor;

Remarks#

The FEVD partitions the h-step-ahead forecast error variance of each variable into contributions from each orthogonal shock. At horizon h, row i of fevd.fevd[h+1, i, .] gives the fraction of variable i’s forecast uncertainty attributable to each shock. Each row sums to 1.0.

At h=0 (impact), the decomposition reflects the contemporaneous Cholesky structure: variable 1’s variance is 100% from its own shock, other variables’ variance includes contributions from earlier-ordered variables.

As h increases, the decomposition typically converges to long-run shares that reflect the relative importance of each shock in driving each variable.

This function accepts either a pre-computed irfResult (if you already computed IRFs) or an estimation result (computes IRFs internally). Both produce identical results.

Model#

The FEVD partitions the h-step forecast error variance of variable \(i\) into contributions from each orthogonal shock \(j\):

\[\text{FEVD}_{i,j}(h) = \frac{\sum_{\ell=0}^{h-1} (\Theta_\ell[i,j])^2}{\sum_{\ell=0}^{h-1} \sum_{k=1}^{m} (\Theta_\ell[i,k])^2}\]

where \(\Theta_\ell\) is the structural IRF at horizon \(\ell\). Each row sums to 1.

At \(h \to \infty\), the FEVD converges to the long-run variance shares.

Algorithm#

  1. Compute Cholesky IRF matrices \(\Theta_0, \ldots, \Theta_{h-1}\) (from irfCompute() or internally).

  2. For each horizon, compute cumulative squared responses and normalize.

Complexity: \(O(h \cdot m^2)\) on top of the IRF computation.

Troubleshooting#

FEVD shares don’t change much across horizons: The model has weak dynamic interactions — shocks are mostly absorbed within the first few periods. This is common in growth-rate data.

One shock dominates everything: Check the variable ordering. With Cholesky identification, the first variable’s shock can absorb variance that should be attributed to other shocks. Try girfCompute() or svarIdentify() for alternative decompositions.

Verification#

FEVD verified against R vars::fevd() at \(10^{-6}\) tolerance on a 2-variable VAR(1), confirming row-sum-to-one property and individual shares at h=1 and h=10.

See gausslib-var/tests/r_benchmark.rs.

References#

  • Lutkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer. Section 2.3.3.

Library#

timeseries

Source#

fevd.src

See also

Functions irfCompute(), hdCompute(), irfPlotData()

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