Generate vector autoregression (VAR) model forecast error variance decomposition (FEVD)

The `fevd`

function returns the forecast error
decomposition (FEVD) of the variables in a VAR(*p*) model
attributable to shocks to each response variable in the system. A fully specified
`varm`

model object characterizes the VAR model.

To estimate or plot the FEVD of a dynamic linear model characterized by structural,
autoregression, or moving average coefficient matrices, see `armafevd`

.

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 VAR model
characterized by a `varm`

model object, see `irf`

.

`Decomposition = fevd(Mdl)`

`Decomposition = fevd(Mdl,Name,Value)`

`[Decomposition,Lower,Upper] = fevd(___)`

returns the orthogonalized FEVDs of the response variables that compose the
VAR(`Decomposition`

= fevd(`Mdl`

)*p*) model `Mdl`

, characterized by a fully specified
`varm`

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 periods 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 `varm`

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 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 [4]. 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 (that is, 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]
Lütkepohl, H. "Asymptotic Distributions of Impulse Response Functions and Forecast Error Variance Decompositions of Vector Autoregressive Models." *Review of Economics and Statistics*. Vol. 72, 1990, pp. 116–125.

[3]
Lütkepohl, H. *New Introduction to Multiple Time Series Analysis*. New York, NY: Springer-Verlag, 2007.

[4]
Pesaran, H. H., and Y. Shin. "Generalized Impulse Response
Analysis in Linear Multivariate Models." *Economic Letters.* Vol. 58, 1998,
pp. 17–29.