In
econometrics
Econometrics is the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A Dictionary of ...
and other applications of multivariate
time series analysis
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
, a variance decomposition or forecast error variance decomposition (FEVD) is used to aid in the interpretation of a
vector autoregression
Vector autoregression (VAR) is a statistical model used to capture the relationship between multiple quantities as they change over time. VAR is a type of stochastic process model. VAR models generalize the single-variable (univariate) autoregres ...
(VAR) model once it has been fitted.
[Lütkepohl, H. (2007) ''New Introduction to Multiple Time Series Analysis'', Springer. p. 63.] The
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
decomposition indicates the amount of information each variable contributes to the other variables in the autoregression. It determines how much of the forecast error variance of each of the variables can be explained by exogenous shocks to the other variables.
Calculating the forecast error variance
For the VAR (p) of form
:
.
This can be changed to a VAR(1) structure by writing it in companion form (see general matrix notation of a VAR(p))
:
where
::
,
,
and
where
,
and
are
dimensional column vectors,
is
by
dimensional matrix and
,
and
are
dimensional column vectors.
The mean squared error of the h-step forecast of variable
is
:
and where
:*
is the j
th column of
and the subscript
refers to that element of the matrix
:*
where
is a lower triangular matrix obtained by a
Cholesky decomposition
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for effici ...
of
such that
, where
is the covariance matrix of the errors
:*
where
so that
is a
by
dimensional matrix.
The amount of forecast error variance of variable
accounted for by exogenous shocks to variable
is given by
:
See also
*
Analysis of variance
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statisticia ...
Notes
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Multivariate time series