# The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

# Variance Decomposition

• Helmut Lütkepohl
Reference work entry
DOI: https://doi.org/10.1057/978-1-349-95189-5_2274

## Abstract

Variance decomposition is a classical statistical method in multivariate analysis for uncovering simplifying structures in a large set of variables (for example, Anderson 2003). For example, factor analysis or principal components are tools that are in widespread use. Factor analytic methods have, for instance, been used extensively in economic forecasting (see for example, Forni et al. 2000; Stock and Watson 2002). In macroeconomic analysis the term ‘variance decomposition’ or, more precisely, ‘forecast error variance decomposition’ is used more narrowly for a specific tool for interpreting the relations between variables described by vector autoregressive (VAR) models. These models were advocated by Sims (1980) and used since then by many economists and econometricians as alternatives to classical simultaneous equations models. Sims criticized the way the latter models were specified, and questioned in particular the exogeneity assumptions common in simultaneous equations modelling.

## Keywords

Bayesian methods Bootstrap Choleski decompositions Cointegrated variables Cointegration Factor analysis Forecasting Least squares Maximum likelihood Multivariate analysis Principal components Simultaneous equations models Structural vector autoregressions Variance decomposition Vector autoregressions

## JEL Classifications

C32

Variance decomposition is a classical statistical method in multivariate analysis for uncovering simplifying structures in a large set of variables (for example, Anderson 2003). For example, factor analysis or principal components are tools that are in widespread use. Factor analytic methods have, for instance, been used extensively in economic forecasting (see for example, Forni et al. 2000; Stock and Watson 2002). In macroeconomic analysis the term ‘variance decomposition’ or, more precisely, ‘forecast error variance decomposition’ is used more narrowly for a specific tool for interpreting the relations between variables described by vector autoregressive (VAR) models. These models were advocated by Sims (1980) and used since then by many economists and econometricians as alternatives to classical simultaneous equations models. Sims criticized the way the latter models were specified, and questioned in particular the exogeneity assumptions common in simultaneous equations modelling.

VAR models have the form
$${y}_t={A}_1{y}_{t-1}+\cdots +{A}_p{y}_{t-p}+{u}_t,$$
(1)
where yt = (y1t, …, yKt) (the prime denotes the transpose) is a vector of K observed variables of interest, the Ai’s are (K × K) parameter matrices, p is the lag order and ut is a zero mean error process which is assumed to be white noise, that is, E(ut) = 0, the covariance matrix, $$E\left({u}_t{u}_t^{\prime}\right)={\Sigma}_u$$, is time invariant and the ut’s are serially uncorrelated or independent. Here deterministic terms such as constants, seasonal dummies or polynomial trends are neglected because they are of no interest in the following. In the VAR model (1) all the variables are a priori endogenous. It is usually difficult to disentangle the relations between the variables directly from the coefficient matrices. Therefore it is useful to have special tools which help with the interpretation of VAR models. Forecast error variance decompositions are such tools. They are presented in the following.
An h steps ahead forecast or briefly h-step forecast at origin t can be obtained from (1) recursively for h = 1, 2, …, as
$${y}_{t+h\mid t}={A}_1{y}_{t+h-1\mid t}+\cdots +{A}_p{y}_{t+h-p\mid t}.$$
(2)
Here yt + j|t = yt + j for j ≤ 0. The forecast error turns out to be
$${\displaystyle \begin{array}{ll}{y}_{t+h}-{y}_{t+h\mid t}=& {u}_{t+h}\hfill \\ {}& +\sum \limits_{i=1}^{h-1}{\Phi}_i{u}_{t+h-i}\sim \left(0,{\Sigma}_h={\Sigma}_u+\sum \limits_{i=1}^{h-1}{\Phi}_i{\Sigma}_u{\Phi}_i^{\prime}\right),\hfill \end{array}}$$
that is, the forecast errors have mean zero and covariance matrices Σh. Here the Φi’s are the coefficient matrices of the power series expansion $${\left({I}_K-{A}_1z-\cdots -{A}_p{z}^p\right)}^{-1}={I}_K+{\sum}_{i=1}^{\infty }{\Phi}_i{z}^i.$$ Note that the inverse exists in a neighbourhood of z = 0 even if the VAR process contains integrated and cointegrated variables. (For an introductory exposition of forecasting VARs, see Lütkepohl 2005.)
If the residual vector ut can be decomposed in instantaneously uncorrelated innovations with economically meaningful interpretation, say, ut = t with εt ~ (0, IK), then Σu = BB and the forecast error variance can be written as $${\Sigma}_h={\sum}_{i=0}^{h-1}{\Theta}_i{\Theta}_i^{\prime }$$, where Θ0 = B and Θi = ΦiB; i = 1, 2, …. Denoting the (n,m)th element of Θj by θnm, j, the forecast error variance of the kth element of the forecast error vector is seen to be
$${\sigma}_k^2(h)=\sum \limits_{j=0}^{h-1}\left({\theta}_{k1,j}^2+\cdots +{\theta}_{kK,j}^2\right)=\sum \limits_{j=1}^K\left({\theta}_{kj,0}^2+\cdots +{\theta}_{kj,h-1}^2\right).$$
The term $$\left({\theta}_{kj,0}^2+\cdots +{\theta}_{kj,h-1}^2\right)$$ may be interpreted as the contribution of the jth innovation to the h-step forecast error variance of variable k. Dividing the term by $${\sigma}_k^2(h)$$ gives the percentage contribution of innovation j to the h-step forecast error variance of variable k. This quantity is denoted by ωkj, h in the following. The ωkj, h, j = 1, … , K, decompose the h-step ahead forecast error variance of variable k in the contributions of the εt innovations. They were proposed by Sims (1980) and are often reported and interpreted for various forecast horizons.

For such an interpretation to make sense it is important to have economically meaningful innovations. In other words, a suitable transformation matrix B for the reduced form residuals has to be found. Clearly, B has to satisfy Σu = BB. These relations do not uniquely determine B, however. Thus, restrictions from subject matter theory are needed to obtain a unique B matrix and, hence, unique innovations εt. A number of different possible sets of restrictions and approaches for specifying restrictions have been proposed in the literature in the framework of structural VAR models. A popular example is the choice of a lower-triangular matrix B obtained by a Choleski decomposition of Σu (for example, Sims 1980). Such a choice amounts to setting up a system in recursive form where shocks in the first variable have potentially instantaneous effects also on all the other variables, shocks to the second variable can also affect the third to last variable instantaneously, and so on. In recursive systems it may be possible to associate the innovations with variables, that is, the jth component of εtis primarily viewed as a shock to the jth observed variable. Generally, the innovations can also be associated with unobserved variables, factors or forces and they may be named accordingly. For example, Blanchard and Quah (1989) consider a bivariate model for output and the unemployment rate, and they investigate effects of supply and demand shocks. Generally, if economically meaningful innovations can be found, forecast error variance decompositions provide information about the relative importance of different shocks for the variables described by the VAR model.

Estimation of reduced form and structural form parameters of VAR processes is usually done by least squares, maximum likelihood or Bayesian methods. Estimates of the forecast error variance components, ωkj, h, are then obtained from the VAR parameter estimates. Suppose the VAR coefficients are contained in a vector α, then ωkj, h is a function of α, ωkj, h = ωkj, h(a). Denoting the estimator of α by $$\widehat{\alpha}$$, ωkj, h may be estimated as $${\widehat{\omega}}_{kj,h}={\sigma}_{kj,h}\left(\widehat{\alpha}\right)$$. If $$\widehat{\alpha}$$ is asymptotically normal, that is, $$\sqrt{T}\left(\widehat{\alpha}-\alpha \right)\underrightarrow{d}\mathcal{N}\left(0,{\Sigma}_{\widehat{\alpha}}\right)$$, then, under general conditions, $${\widehat{\omega}}_{kj,h}$$ is also asymptotically normally distributed, $$\sqrt{T}\left({\widehat{\omega}}_{kj,h}-{\omega}_{kj,h}\right)\underrightarrow{d}\mathcal{N}\left(0,{\sigma}_{kj,h}^2=\frac{\partial {\omega}_{kj,h}}{\partial {\alpha}^{\prime }}{\Sigma}_{\widehat{\alpha}}\frac{\partial {\omega}_{kj,h}}{\partial \alpha}\right),$$ provided the variance of the asymptotic distribution is non-zero. Here ∂ωkj, h/∂a denotes the vector of first-order partial derivatives of ωkj, h with respect to the elements of α (see Lütkepohl 1990, for the specific form of the partial derivatives). Unfortunately, $${\sigma}_{kj,h}^2$$ is zero even for cases of particular interest, for example, if ωkj, h = 0 and, hence, the jth innovation does not contribute to the h-step forecast error variance of variable k (see Lütkepohl 2005, Sect. 3.7.1, for a more detailed discussion). The problem can also not easily be solved by using bootstrap techniques (cf. Benkwitz et al. 2000). Thus, standard statistical techniques such as setting up confidence intervals are problematic for the forecast error variance components. They can at best give rough indications of sampling uncertainty. The estimated ωkj, h’s are perhaps best viewed as descriptive statistics.

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