Variance Decomposition

Definition

The Variance Decomposition Function (VDF), also called Forecast Error Variance Decomposition (FEVD), allocates the $j$ -step-ahead forecast error variance of each variable to the individual structural shocks. It answers: what fraction of variable $v$ 's unpredictability at horizon $j$ is attributable to shock $k$ ?

Key Ideas

Complements Impulse Response Functions (IRFs): IRFs show how variables respond to each shock; variance decompositions show how much each shock contributes to overall variability.
Shares the same identification dependence as IRFs — the decomposition is only meaningful relative to a structural identification of $A_0$ .
At horizon $j = 1$ (one step ahead), the decomposition reflects only the contemporaneous impact matrix $\Theta_0 = A_0^{-1}$ .
As $j \to \infty$ , the decomposition converges to the unconditional variance shares.

How It Works

Moving Average Representation

From the structural vector autoregression (VAR), the variables admit the MA representation:

$x_t = \Theta(L)\,\varepsilon_t = \sum_{i=0}^{\infty} \Theta_i\,\varepsilon_{t-i}$

where $\Theta_i$ is the $n \times n$ matrix of impulse responses at horizon $i$ (see Impulse Response Function) and $\varepsilon_t \sim (0, \Sigma_\varepsilon)$ with $\Sigma_\varepsilon$ diagonal (independent structural shocks).

Forecast Error

The $j$ -step-ahead forecast error for $x_{t+j}$ given information at $t$ is:

$x_{t+j} - \mathbb{E}_t[x_{t+j}] = \sum_{i=0}^{j-1} \Theta_i\,\varepsilon_{t+j-i}$

since all $\varepsilon_{t+j-i}$ for $i < j$ are unknown at time $t$ . The forecast error variance matrix is:

$\text{Var}(x_{t+j} - \mathbb{E}_t[x_{t+j}]) = \sum_{i=0}^{j-1} \Theta_i\,\Sigma_\varepsilon\,\Theta_i'$

Variance Decomposition Formula

Let $\theta_{vs,i}$ denote the $(v,s)$ element of $\Theta_i$ and $\sigma_s^2 = [\Sigma_\varepsilon]_{ss}$ the variance of structural shock $s$ . The $j$ -step forecast error variance of variable $v$ attributable to shock $k$ is:

$\text{VDF}(v,\, k,\, j) = \frac{\displaystyle\sum_{i=0}^{j-1} \theta_{vk,i}^2\,\sigma_k^2}{\displaystyle\sum_{i=0}^{j-1} \sum_{s=1}^{n} \theta_{vs,i}^2\,\sigma_s^2} \times 100$

When $\Sigma_\varepsilon = I_n$ (unit-variance structural shocks, as is conventional after normalization), this simplifies to:

$\text{VDF}(v,\, k,\, j) = \frac{\displaystyle\sum_{i=0}^{j-1} \theta_{vk,i}^2}{\displaystyle\sum_{i=0}^{j-1} \sum_{s=1}^{n} \theta_{vs,i}^2} \times 100$

The decomposition is exhaustive: $\sum_{k=1}^n \text{VDF}(v,k,j) = 100$ for all $v$ and $j$ .

For the Differenced VAR

When the VAR is estimated in first differences (permanent-shock case), the cumulative impulse responses $\Gamma_i = \sum_{\ell=0}^{i} \Theta_\ell$ govern the level responses (see Impulse Response Function). The variance decomposition for the level of $x$ replaces $\Theta_i$ with $\Gamma_i$ in the formula above:

$\text{VDF}_\text{level}(v,\, k,\, j) = \frac{\displaystyle\sum_{i=0}^{j-1} \gamma_{vk,i}^2\,\sigma_k^2}{\displaystyle\sum_{i=0}^{j-1} \sum_{s=1}^{n} \gamma_{vs,i}^2\,\sigma_s^2} \times 100$

Macro vs. Latent Decomposition in Term Structure Models (Ang-Piazzesi 2001)

In an Affine Term Structure Model with observable macro factors $X_t^o$ and latent yield factors $X_t^u$ , the forecast error variance of yield $n$ at horizon $h$ decomposes across all factors:

$\Omega_{jk,h} = \frac{\sum_{i=0}^{h-1} \Psi_{jk,i}^2}{\mathrm{MSE}(\hat{Y}_{t+h|t})} \tag{ATSM-VD}$

where $\Psi_{jk,i}$ is the $(j,k)$ element of the yield-IRF matrix at lag $i$ and $\mathrm{MSE}(\cdot)$ is the mean squared error (MSE) (see Appendix C of Ang-Piazzesi). Key empirical findings (US monthly data, 1952–2000):

Yield maturity	Macro share (unconditional)	Dominant macro factor
1-month	83%	Inflation (48.9% at $h=1$ )
12-month	$\approx 78\%$ (at $h=60$ )	Inflation
60-month	$\approx 40\%$	Latent "level" (near unit-root)

The level latent factor dominates long-yield variance at all horizons — macro variables cannot span the first principal component (PC) of the yield curve.
Inflation consistently outperforms real activity across all maturities and horizons.
Real activity's explanatory share rises with the forecast horizon; inflation's share falls for long yields.

Empirical Pattern (Keating 1992)

In the long-run identified model with variables $(p, y, r, m)$ :

Supply shocks explain 17% of output variance at 1 quarter, rising to $\approx 90\%$ at 48 quarters.
Demand (IS) shocks dominate short-run output movements and long-run interest rate variance.
Money demand shocks have strong effects on money but virtually none on output or prices.

Why It Matters

Variance decompositions provide a quantitative summary of shock importance that complements the directional information in IRFs. They are the standard diagnostic for questions such as: "How much of the business cycle is driven by monetary policy shocks?" or "Does technology or demand account for most output fluctuations?"

Open Questions

Like IRFs, variance decompositions are estimated with substantial uncertainty; confidence intervals (via Monte Carlo or Bayesian methods) are wide, especially at long horizons.
The decomposition adds up to 100% only because shocks are assumed orthogonal; with partially identified models (sign restrictions), shares are set-identified and can overlap.