vgxinfer - Infer innovations of multivariate time series process

Synopsis

W = vgxinfer(Spec,Y)

[W,LLF] = vgxinfer(Spec,Y,X,Y0,W0)

Description

vgxinfer infers innovations of a multivariate time series process from observations of that process.

The functions vgxinfer and vgxproc are complementary. For example, given a specification structure Spec for a stable and invertible process and an innovations process W1,

Y = vgxproc(Spec,W1,X,Y0,W0);
W2 = vgxinfer(Spec,Y,X,Y0,W0);

produces an innovations process W2 identical to W1. Differences might appear, however, if the process in Spec is either unstable or noninvertible.

Input Arguments

`Spec`	A multivariate time series specification structure for an n-dimensional time series process, as created by `vgxset`.
`Y`	nP paths of an n-dimensional time series process with T observations for each path, collected in a T-by-n-by-nP array. Times are ordered by row from oldest to most recent.
`X`	Exogenous inputs. A collection of nPX paths of regression design matrices associated with T observations of an n-dimensional time series process. Each design matrix linearly relates nX exogenous inputs to each of n time series at each observation time. `X` is a T-by-nPX matrix of cell arrays with n-by-nX design matrices in each cell. If `Y` has multiple paths, `X` must contain either a single path or no fewer than the same number of paths as in `Y` (extra paths are ignored).
`Y0`	Presample time series process. nPY0 presample paths of an n-dimensional time series process with TY0 observations for each path, collected in a TY0-by-n-by-nPY0 array. If `Y0` is either empty or if TY0 is less than the maximum `AR` lag in `Spec`, presample values are padded with zeros. If TY0 is greater than the maximum `AR` lag, the most recent observations from the last rows of each path of `Y0` are used. If `Y0` has multiple paths, `Y0` must contain either a single path or no fewer than the same number of paths as in `Y`. Extra paths are ignored.
`W0`	Presample innovations process. nPW0 presample paths of an n-dimensional innovations process with TW0 observations for each path, collected in a TW0-by-n-by-nPW0 array. If `W0` is either empty or if TW0 is less than the maximum `MA` lag in `Spec`, presample values are padded with zeros. If TW0 is greater than the maximum `MA` lag, the most recent observations from the last rows of each path of `W0` are used. If `W0` has multiple paths, `W0` must contain either a single path or no fewer than the same number of paths as in `W`. Extra paths are ignored.

Output Arguments

`W`	Innovations process inferred from time series process. nP paths of an n-dimensional innovations process with T observations for each path, collected in a T-by-n-by-nP array. The number of paths in `W` is equal to the number of paths in `Y`. Times are ordered by row from oldest to most recent.
`LLF`	Total loglikelihood function for T observations of an n-dimensional time series process. If `W` has nP paths, `LLF` is a 1-by-nP vector containing the total loglikelihood function for each path.

Example

Start with a 2-dimensional VARMA(2, 2) specification structure in Spec with time series data, innovations process, and presample data:

load vgxexample Spec Y W Y0 W0

Infer the innovations process from the time series data in Y with the function vgxinfer and compare the result with the original innovations process in W:

W1 = vgxinfer(Spec, Y, [], Y0, W0);
 
norm(W1 - W)

ans =

  2.2123e-016