vgxma - Convert VARMA model to pure VMA model

Synopsis

SpecMA = vgxma(Spec)

SpecMA = vgxma(Spec,nMA,MAlag,Cutoff)

Description

vgxma converts a VARMA model into a pure vector moving average (VMA) model. This function works only for VARMA models and does not handle exogenous variables (VARMAX models).

Required Input Argument

Spec

A multivariate time series specification structure for an n-dimensional VARMA time series process, as created by vgxset.

Optional Input Arguments

`nMA`	Number of MA lags for the output specification structure. `vgxma` truncates an infinite-order VMA model to `nMA` lags. If specific MA lags are not given by `MAlag`, the lags are `1:nMA`. To use `MAlag`, set `nMA` to `[]` or to the number of specific lags.
`MAlag`	A positive integer vector of specific MA lags for the output specification structure. `MAlag` must be of length `nMA`, unless `nMA` is `[]`.
`Cutoff`	The cutoff for the infinity norm below which trailing lags are removed. The default is `0`, which does not remove any lags and uses the values for `nMA` and `MAlag`.

If neither nMA nor MAlag is specified, vgxma uses the maximum lags of the AR or MA lags of the input Spec.

Note If a large number of lags is needed to form a pure VMA representation (with unit roots close to 1), a large number of initial values is also needed for propagation.

Output Argument

SpecMA

A transformed multivariate time series specification structure that consists of a pure vector moving average (VMA) model with nMA lags. Logical indicators for model parameter estimation ("solve" information) in Spec are not passed on to SpecMA.

Example

Start with a 2-dimensional VARMA(2, 2) specification structure in Spec:

load vgxexample Spec

Convert Spec into a pure VMA(2) model in SpecMA:

SpecMA = vgxma(Spec);

Display the original specification structure in Spec and compare with the new specification structure in SpecMA:

vgxdisp(Spec, SpecMA)

Model 1 - Information
  Model : 2-D VARMA(2,2) with No Additive Constant
          Conditional mean is AR-stable and is MA-invertible
Model 2 - Information
  Model : 2-D VMA(2) with No Additive Constant
          Conditional mean is MA-invertible

       Parameter        Model 1        Model 2
  -------------- -------------- --------------
      AR(1)(1,1)       0.373935               
           (1,2)       0.124043               
           (2,1)       0.375488               
           (2,2)       0.259077               
      AR(2)(1,1)      0.0754758               
           (1,2)     -0.0972418               
           (2,1)      0.0687406               
           (2,2)      0.0155532               
      MA(1)(1,1)       0.205242       0.579177 
           (1,2)      -0.239925      -0.115882 
           (2,1)     -0.0881847       0.287303 
           (2,2)     -0.0617094       0.197368 
      MA(2)(1,1)     -0.0682232       0.259465 
           (1,2)      0.0107276      -0.105364 
           (2,1)      -0.155213       0.205435 
           (2,2)     -0.0040213      0.0191531 
          Q(1,1)           0.08           0.08 
          Q(2,1)           0.01           0.01 
          Q(2,2)           0.03           0.03

Obtain the first 4 MA lags in SpecMA:

SpecMA = vgxma(Spec, 4);

vgxdisp(Spec, SpecMA);

Model 1 - Information

Model : 2-D VARMA(2,2) with No Additive Constant
          Conditional mean is AR-stable and is MA-invertible
Model 2 - Information
  Model : 2-D VMA(4) with No Additive Constant
          Conditional mean is MA-invertible

       Parameter        Model 1        Model 2
  -------------- -------------- --------------
      AR(1)(1,1)       0.373935               
           (1,2)       0.124043               
           (2,1)       0.375488               
           (2,2)       0.259077               
      AR(2)(1,1)      0.0754758               
           (1,2)     -0.0972418               
           (2,1)      0.0687406               
           (2,2)      0.0155532               
      MA(1)(1,1)       0.205242       0.579177 
           (1,2)      -0.239925      -0.115882 
           (2,1)     -0.0881847       0.287303 
           (2,2)     -0.0617094       0.197368 
      MA(2)(1,1)     -0.0682232       0.259465 
           (1,2)      0.0107276      -0.105364 
           (2,1)      -0.155213       0.205435 
           (2,2)     -0.0040213      0.0191531 
      MA(3)(1,1)                      0.138282 
           (1,2)                    -0.0649623 
           (2,1)                      0.194931 
           (2,2)                     -0.039497 
      MA(4)(1,1)                     0.0754946 
           (1,2)                     -0.039006 
           (2,1)                      0.123456 
           (2,2)                    -0.0415703 
          Q(1,1)           0.08           0.08 
          Q(2,1)           0.01           0.01 
          Q(2,2)           0.03           0.03

Obtain just the 99th lag and display the result:

SpecMA = vgxma(Spec, 1, 99);

vgxdisp(SpecMA);

Model - Information
  Model:  2-D VMA(1) with No Additive Constant
          Conditional mean is MA-invertible
          Moving average lags: 99

  MA(99) Moving Average Matrix:
    2.09723e-030  -1.03631e-030 
    3.16333e-030  -8.85453e-031 
  Q Innovations Covariance:
            0.08           0.01 
            0.01           0.03