Generate vector error-correction (VEC) model forecast error variance decomposition (FEVD)
The fevd
function returns the forecast error
decomposition (FEVD) of the variables in a VEC(p – 1)
model attributable to shocks to each response variable in the system. A fully
specified vecm
model object
characterizes the VEC model.
The FEVD provides information about the relative importance of each innovation in
affecting the forecast error variance of all response variables in the system. In contrast,
the impulse response function (IRF) traces the effects of an innovation shock to one variable
on the response of all variables in the system. To estimate the IRF of a VEC model
characterized by a vecm
model object, see irf
.
returns the orthogonalized FEVDs of the response variables that compose the
VEC(p – 1) model Decomposition
= fevd(Mdl
)Mdl
characterized by a fully
specified vecm
model object.
fevd
shocks variables at time 0, and returns the FEVD for
times 1 through 20.
uses additional options specified by one or more name-value pair arguments. For example,
Decomposition
= fevd(Mdl
,Name,Value
)'NumObs',10,'Method',"generalized"
specifies estimating a generalized
FEVD for times 1 through 10.
[
uses any of the input argument combinations in the previous syntaxes and returns lower and
upper 95% confidence bounds for each period and variable in the FEVD.Decomposition
,Lower
,Upper
] = fevd(___)
If you specify series of residuals by using the E
name-value
pair argument, then fevd
estimates the confidence bounds
by bootstrapping the specified residuals.
Otherwise, fevd
estimates confidence bounds by
conducting Monte Carlo simulation.
If Mdl
is a custom vecm
model object
(an object not returned by estimate
or
modified after estimation), fevd
might require a sample size for
the simulation SampleSize
or presample responses
Y0
.
If Method
is "orthogonalized"
, then
fevd
orthogonalizes the innovation shocks by applying the
Cholesky factorization of the model covariance matrix Mdl.Covariance
.
The covariance of the orthogonalized innovation shocks is the identity matrix, and the
FEVD of each variable sums to one, that is, the sum along any row of
Decomposition
is one. Therefore, the orthogonalized FEVD represents
the proportion of forecast error variance attributable to various shocks in the system.
However, the orthogonalized FEVD generally depends on the order of the variables.
If Method
is "generalized"
, then the resulting
FEVD, then the resulting FEVD is invariant to the order of the variables, and is not based
on an orthogonal transformation. Also, the resulting FEVD sums to one for a particular
variable only when Mdl.Covariance
is diagonal[5]. Therefore, the generalized FEVD
represents the contribution to the forecast error variance of equation-wise shocks to the
response variables in the model.
If Mdl.Covariance
is a diagonal matrix, then the resulting generalized and orthogonalized FEVDs are identical. Otherwise, the resulting generalized and orthogonalized FEVDs are identical only when the first variable shocks all variables (in other words, all else being the same, both methods yield the same value of Decomposition(:,1,:)
).
NaN
values in Y0
, X
, and
E
indicate missing data. fevd
removes
missing data from these arguments by list-wise deletion. Each argument, if a row contains at
least one NaN
, then fevd
removes the entire
row.
List-wise deletion reduces the sample size, can create irregular time series, and can
cause E
and X
to be unsynchronized.
The predictor data X
represents a single path of exogenous multivariate time series. If you specify X
and the VAR model Mdl
has a regression component (Mdl.Beta
is not an empty array), fevd
applies the same exogenous data to all paths used for confidence interval estimation.
fevd
conducts a simulation to estimate the confidence bounds
Lower
and Upper
.
If you do not specify residuals E
, then
fevd
conducts a Monte Carlo simulation by following
this procedure:
Simulate NumPaths
response paths of length
SampleSize
from Mdl
.
Fit NumPaths
models that have the same structure as
Mdl
to the simulated response paths. If
Mdl
contains a regression component and you
specify X
, the fevd
fits the NumPaths
models to the simulated response
paths and X
(the same predictor data for all
paths).
Estimate NumPaths
FEVDs from the
NumPaths
estimated models.
For each time point t =
0,…,NumObs
, estimate the confidence intervals
by computing 1 – Confidence
and
Confidence
quantiles (the upper and lower
bounds, respectively).
If you specify residuals E
, then
fevd
conducts a nonparametric bootstrap by following
this procedure:
Resample, with replacement, SampleSize
residuals
from E
. Perform this step NumPaths
times to obtain NumPaths
paths.
Center each path of bootstrapped residuals.
Filter each path of centered, bootstrapped residuals through
Mdl
to obtain NumPaths
bootstrapped response paths of length
SampleSize
.
Complete steps 2 through 4 of the Monte Carlo simulation, but replace the simulated response paths with the bootstrapped response paths.
[1] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
[2] Johansen, S. Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford: Oxford University Press, 1995.
[3] Juselius, K. The Cointegrated VAR Model. Oxford: Oxford University Press, 2006.
[4] Lütkepohl, H. New Introduction to Multiple Time Series Analysis. New York, NY: Springer-Verlag, 2007.
[5] Pesaran, H. H., and Y. Shin. "Generalized Impulse Response Analysis in Linear Multivariate Models." Economic Letters. Vol. 58, 1998, pp. 17–29.