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Generate ARMA model forecast error variance decomposition

The `armafevd`

function returns the forecast error variance
decomposition (FEVD) of the variables in a univariate or vector (multivariate)
autoregressive, moving average (ARMA or VARMA) model specified by arrays of coefficients or
lag operator polynomials.

The FEVD provides information about the relative importance of each innovation in
affecting the forecast error variance of all 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 IRFs of univariate or multivariate ARMA
models, see `armairf`

.

`armafevd(ar0,ma0)`

`armafevd(ar0,ma0,Name,Value)`

`Y = armafevd(___)`

`armafevd(ax,___)`

`[Y,h] = armafevd(___)`

`armafevd(`

plots, in separate figures, the FEVD of the `ar0`

,`ma0`

)`numVars`

time series
variables that compose an ARMA(*p*,*q*) model, with
autoregressive (AR) and moving average (MA) coefficients `ar0`

and
`ma0`

, respectively. Each figure corresponds to a variable and contains
`numVars`

line plots. The line plots are the FEVDs of that variable,
over the forecast horizon, resulting from a one-standard-deviation innovation shock
applied to all variables in the system at time 0.

The `armafevd`

function:

Accepts vectors or cell vectors of matrices in difference-equation notation

Accepts

`LagOp`

lag operator polynomials corresponding to the AR and MA polynomials in lag operator notationAccommodates time series models that are univariate or multivariate, stationary or integrated, structural or in reduced form, and invertible or noninvertible

Assumes that the model constant

*c*is 0

`armafevd(`

plots the `ar0`

,`ma0`

,`Name,Value`

)`numVars`

FEVDs with additional options specified by one or
more name-value pair arguments. For example,
`'NumObs',10,'Method','generalized'`

specifies a 10-period forecast
horizon and the estimation of the generalized FEVD.

`armafevd(`

plots to the axes specified in `ax`

,___)`ax`

instead of
the axes in new figures. The option `ax`

can precede any of the input argument
combinations in the previous syntaxes.

To accommodate structural ARMA(

*p*,*q*) models, supply`LagOp`

lag operator polynomials for the input arguments`ar0`

and`ma0`

. To specify a structural coefficient when you call`LagOp`

, set the corresponding lag to 0 by using the`'Lags'`

name-value pair argument.For orthogonalized multivariate FEVDs, arrange the variables according to

*Wold causal ordering*[3]:The first variable (corresponding to the first row and column of both

`ar0`

and`ma0`

) is most likely to have an immediate impact (*t*= 0) on all other variables.The second variable (corresponding to the second row and column of both

`ar0`

and`ma0`

) is most likely to have an immediate impact on the remaining variables, but not the first variable.In general, variable

*j*(corresponding to row*j*and column*j*of both`ar0`

and`ma0`

) is the most likely to have an immediate impact on the last`numVars`

–*j*variables, but not the previous*j*– 1 variables.

`armafevd`

plots FEVDs only when it returns no output arguments or`h`

.If

`Method`

is`'orthogonalized'`

, then`armafevd`

orthogonalizes the innovation shocks by applying the Cholesky factorization of the innovations covariance matrix`InnovCov`

. 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`Y`

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.

The resulting FEVD is not based on an orthogonal transformation.

The resulting FEVD of a variable sums to one only when

`InnovCov`

is diagonal [4].

Therefore, the generalized FEVD represents the contribution to the forecast error variance of equation-wise shocks to the variables in the system.

If

`InnovCov`

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`Y(:,1,:)`

).

[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.