vgxar - Convert VARMA model to pure VAR model

Synopsis

SpecAR = vgxar(Spec)

SpecAR = vgxar(Spec,nAR,ARlag,Cutoff)

Description

vgxar converts a VARMA model into a pure vector autoregressive (VAR) 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

`nAR`	Number of AR lags for the output specification structure. `vgxar` truncates an infinite-order VAR model to `nAR` lags. If specific AR lags are not given by `ARlag`, the lags are `1:nAR`. To use `ARlag`, set `nAR` to `[]` or to the number of specific lags.
`ARlag`	A positive integer vector of specific AR lags for the output specification structure. `ARlag` must be of length `nAR`, unless `nAR` 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 `nAR` and `ARlag`.

If neither nAR nor ARlag is specified, vgxar 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 VAR representation (with unit roots close to 1), a large number of initial values is also needed for propagation.

Output Argument

SpecAR

A transformed multivariate time series specification structure that consists of a pure vector autoregressive (VAR) model with nAR lags. Logical indicators for model parameter estimation ("solve" information) in Spec are not passed on to SpecAR.

Example

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

load vgxexample Spec

Convert Spec into a pure VAR(2) model in SpecAR:

SpecAR = vgxar(Spec);

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

vgxdisp(Spec, SpecAR)

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 VAR(2) with No Additive Constant
          Conditional mean is AR-stable

       Parameter        Model 1        Model 2
  -------------- -------------- --------------
      AR(1)(1,1)       0.373935       0.579177 
           (1,2)       0.124043      -0.115882 
           (2,1)       0.375488       0.287303 
           (2,2)       0.259077       0.197368 
      AR(2)(1,1)      0.0754758     -0.0426874 
           (1,2)     -0.0972418      -0.015377 
           (2,1)      0.0687406     -0.0176683 
           (2,2)      0.0155532      0.0134923 
      MA(1)(1,1)       0.205242                
           (1,2)      -0.239925                
           (2,1)     -0.0881847                
           (2,2)     -0.0617094                
      MA(2)(1,1)     -0.0682232                
           (1,2)      0.0107276                
           (2,1)      -0.155213                
           (2,2)     -0.0040213                
          Q(1,1)           0.08           0.08 
          Q(2,1)           0.01           0.01 
          Q(2,2)           0.03           0.03

Instead of just the default number of AR lags (which is two), obtain the first four AR lags in SpecAR:

SpecAR = vgxar(Spec, 4);

vgxdisp(Spec, SpecAR);

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 VAR(4) with No Additive Constant
          Conditional mean is AR-stable

       Parameter        Model 1        Model 2
  -------------- -------------- --------------
      AR(1)(1,1)       0.373935       0.579177 
           (1,2)       0.124043      -0.115882 
           (2,1)       0.375488       0.287303 
           (2,2)       0.259077       0.197368 
      AR(2)(1,1)      0.0754758     -0.0426874 
           (1,2)     -0.0972418      -0.015377 
           (2,1)      0.0687406     -0.0176683 
           (2,2)      0.0155532      0.0134923 
      AR(3)(1,1)                     0.0409534 
           (1,2)                   -0.00362997 
           (2,1)                     0.0861962 
           (2,2)                    -0.0177161 
      AR(4)(1,1)                    0.00955252 
           (1,2)                   -0.00469931 
           (2,1)                     0.0022339 
           (2,2)                   -0.00374581 
      MA(1)(1,1)       0.205242                
           (1,2)      -0.239925                
           (2,1)     -0.0881847                
           (2,2)     -0.0617094                
      MA(2)(1,1)     -0.0682232                
           (1,2)      0.0107276                
           (2,1)      -0.155213                
           (2,2)     -0.0040213                
          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:

SpecAR = vgxar(Spec, 1, 99);

vgxdisp(SpecAR);

Model - Information
  Model:  2-D VAR(1) with No Additive Constant
          Conditional mean is AR-stable
          Autoregression lags: 99

  AR(99) Autoregression Matrix:
    8.06035e-045  -2.39247e-045 
    1.44771e-044  -4.29698e-045 
  Q Innovations Covariance:
            0.08           0.01 
            0.01           0.03