Forecast-Error Variance Decompositions

Why Forecast-Error Variance Decompositions Matter

An impulse response describes how a variable reacts over time to a shock. A forecast-error variance decomposition (FEVD) answers a related but distinct question: at a chosen forecast horizon, what fraction of one variable’s forecast uncertainty is attributed to each shock? It summarizes the contributions of shocks over all periods that enter that forecast error, rather than reporting the response at just one period.

An FEVD is conditional on a model and a shock definition. In particular, when shocks are correlated in the reduced form, dividing forecast uncertainty among them requires a convention or an identified structural representation. The resulting percentages are not properties of the data alone.

Structural Forecast Errors and Their Variances

Consider a stable \(K\)-variable VAR written in its moving-average form, \(y_t=\mu+\sum_{h=0}^{\infty}\Psi_h u_{t-h}\), where \(u_t\) is the reduced-form innovation, \(E(u_t)=0\), and \(E(u_tu_t')=\Sigma\). Stability ensures that the moving-average representation exists. Write the structural representation of the innovations as \(u_t=B\varepsilon_t\), where \(B\) is an impact matrix and the structural shocks satisfy \(E(\varepsilon_t)=0\) and \(E(\varepsilon_t\varepsilon_t')=I\). Thus \(\Sigma=BB'\). The response matrix at horizon \(h\) is \(C_h=\Psi_hB\); its \(ij\) entry is the response of variable \(i\) at horizon \(h\) to a one-unit structural shock \(j\).

For a forecast made at time \(T\), let \(H\) be a positive integer denoting the number of steps ahead. The \(H\)-step forecast error includes the innovations arriving over the next \(H\) periods. Its structural representation is \(y_{T+H}-E_T(y_{T+H})=\sum_{h=0}^{H-1}C_h\varepsilon_{T+H-h}\). Here \(E_T\) denotes the forecast conditional on information available at time \(T\). The sum stops at \(H-1\), because the shock at \(T+1\) has response horizon \(H-1\) at time \(T+H\), while the shock at \(T+H\) has horizon zero.

The structural shocks at distinct dates are uncorrelated, and shocks at the same date have identity covariance. Consequently, the forecast-error variance of variable \(i\) is the sum of the squared responses to every structural shock at every included horizon:

The vector \(e_i\) has a one in position \(i\) and zeros elsewhere. Each squared response is a contribution to variance, not a signed effect on the level of the variable. Positive and negative impulse responses therefore both add nonnegative amounts to the FEVD.

The numerator is shock \(j\)’s accumulated contribution to variable \(i\)’s \(H\)-step forecast-error variance. The denominator is the total forecast-error variance for variable \(i\). Shares are between zero and one, and \(\sum_{j=1}^{K}\theta_{ij}(H)=1\). Multiplying by 100 expresses a share as a percentage.

Why the Formula Adds Squared Responses

The decomposition follows directly from the forecast-error representation. For one variable, write its error as a sum of terms \(a_{hj}\varepsilon_{j,T+H-h}\), where \(a_{hj}=e_i'C_he_j\). Squaring this sum and taking expectations produces squared terms and cross-products. The cross-products have expectation zero: different dates are uncorrelated, and distinct structural shocks at the same date have zero covariance. Since each structural shock has variance one, the expected squared term is \(a_{hj}^2\). Hence the total variance is the sum of squared impulse responses. Grouping those terms by shock gives the numerator for each FEVD share; dividing by their total makes the shares add to one.

This argument relies on the structural shocks being mutually uncorrelated and having the stated unit variances. If a different scaling is used for structural shocks, the response coefficients and their variance contributions must be scaled consistently. The decomposition also inherits the model’s identification assumptions: a different admissible impact matrix \(B\) can yield different shock-specific shares.

Orthogonalized Decompositions

An orthogonalized FEVD uses impulse responses to uncorrelated shocks. When the structural impact matrix \(B\) is identified from the VAR, the formula above applies directly to \(C_h=\Psi_hB\). A common special case is a Cholesky decomposition of the reduced-form innovation covariance matrix: choose a lower-triangular \(P\) such that \(\Sigma=PP'\), and use orthogonalized responses \(\Psi_hP\). This choice makes the contemporaneous shock allocation depend on the ordering of variables. It is an identification convention, not a neutral calculation that removes the need to justify how shocks are defined.

Generalized Decompositions

When reduced-form innovations are correlated, an alternative is the generalized FEVD. It measures the response associated with a one-standard-deviation innovation in component \(j\), while allowing the other components to move according to their conditional covariance with that innovation. It does not first transform the innovations into mutually orthogonal structural shocks. For reduced-form impulse responses \(\Psi_h\), define the unnormalized generalized contribution and its variance-scaled value by

where \(\sigma_{jj}\) is the variance of reduced-form innovation \(j\). The denominator in \(q_{ij}(H)\) is the \(H\)-step forecast-error variance of variable \(i\). Because reduced-form innovations are correlated, the \(q_{ij}(H)\) values generally do not sum to one across shocks. The conventional normalized generalized FEVD is

For each variable \(i\), the normalized generalized shares sum to one. This final normalization produces a useful percentage allocation, but it does not make the underlying correlated innovations orthogonal. Generalized decompositions are invariant to variable ordering, unlike a Cholesky decomposition, but they answer a different attribution question from an orthogonalized structural FEVD.

Reading and Reporting an FEVD

An FEVD should always be reported with the variable, horizon, shock definition, and identification method. For example, “At a two-step horizon, shock 1 accounts for 97.84% of the forecast-error variance of \(y_1\) under the stated orthogonalized VAR” is interpretable. A bare statement that “shock 1 explains 97.84%” omits the outcome and horizon, and may suggest a stronger causal conclusion than the decomposition warrants. FEVD shares describe the allocation of model-based forecast uncertainty; they do not by themselves establish that a shock is an externally manipulated cause.

Comparisons across horizons can show how the model’s attribution changes as additional responses enter the forecast error. Comparisons across specifications or identifications should be made cautiously: changing variable order in a Cholesky decomposition, changing structural restrictions, or changing the model can change the reported shares. Generalized shares avoid ordering dependence but use the correlated-innovation convention and row normalization instead.

Common Mistakes

• Including the wrong horizons. For an \(H\)-step forecast error, sum responses from \(h=0\) through \(H-1\), not through \(H\). The latter includes one extra shock period.
• Adding signed responses. A negative response still contributes positively to variance. Square each response before summing.
• Calling a generalized share an orthogonal shock contribution. Generalized innovations remain correlated. Describe the result as a normalized generalized attribution, rather than as an orthogonalized structural decomposition.
• Assuming generalized values automatically sum to one. The generalized variance-scaled values need not add to one; apply the row normalization before reporting percentage shares.
• Ignoring identification. An FEVD depends on the impact matrix or generalized-shock convention. Percentages are not independent of these modeling choices.