*Time Series Analysis and Control Examples* |

The subroutines TSMULMAR, TSMLOMAR, and
TSPRED analyze multivariate time series.
The periodic AR model, TSPEARS, can also be
estimated by using a vector AR procedure, since
the periodic AR series can be represented as the
covariance-stationary vector autoregressive model.
The stationary vector AR model is estimated
and the order of the model (or each variable) is
automatically determined by the minimum AIC procedure.
The stationary vector AR model is written

Using the **L****D****L**' factorization
method, the error covariance is decomposed as

where **L** is a unit lower triangular
matrix and **D** is a diagonal matrix.
Then the instantaneous response model is defined as

where **C** = **L**^{-1}, **A**_{i}^{*} = **L**^{-1}**A**_{i} for
*i* = 0,1, ... ,*p*, and .
Each equation of the instantaneous response model can
be estimated independently, since its error covariance
matrix has a diagonal covariance matrix **D**.
Maximum likelihood estimates are obtained through the
least squares method when the disturbances are normally
distributed and the presample values are fixed.
The TSMULMAR call estimates the instantaneous response model.
The VAR coefficients are computed using the
relationship between the VAR and instantaneous models.

The general VARMA model can be transformed as an
infinite order MA process under certain conditions.

In the context of the VAR(*p*) model, the coefficient
can be interpreted as the *m*-lagged response
of a unit increase in the disturbances at time *t*.

The lagged response on the one-unit increase in the
orthogonalized disturbances is denoted

where **L**_{j} is the *jth* column of the unit
triangular matrix **L** and **X**_{t} = [**y**_{t-1}, ... ,**y**_{t-p}].
When you estimate the VAR model using the TSMULMAR call,
it is easy to compute this impulse response function.
The MSE of the *m*-step prediction is computed as

Note that .
Then the covariance matrix of is decomposed

where *d*_{ii} is the *ith* diagonal element
of the matrix **D** and *n* is the number of variables.
The MSE matrix can be written

Therefore, the contribution of the *ith*
orthogonalized innovation to the MSE is

The *ith* forecast error variance decomposition
is obtained from diagonal elements of the matrix **V**_{i}.
The nonstationary multivariate series can
be analyzed by the TSMLOMAR subroutine.
The estimation and model identification procedure is
analogous to the univariate nonstationary procedure,
which is explained in the "Nonstationary Time Series" section.

A time series *y*_{t} is periodically correlated
with period *d* if E*y*_{t} = E*y*_{t+d}
and E*y*_{s} *y*_{t} = E*y*_{s+d}*y*_{t+d}.
Let *y*_{t} be autoregressive of period *d*
with AR orders (*p*_{1}, ... ,*p*_{d}), that is,

where is uncorrelated with mean
zero and ,
*p*_{t} = *p*_{t+d}, , and
.Define the new variable such that *x*_{jt} = *y*_{j+d(t-1)}.
The vector series, **x**_{t} = (*x*_{1t}, ... ,*x*_{dt})',
is autoregressive of order *p*, where
*p* = max_{j}int((*p*_{j} - *j*)/*d*) + 1.
The TSPEARS subroutine estimates the periodic
autoregressive model using minimum AIC vector AR modeling.
The TSPRED subroutine computes the one-step or
multistep forecast of the multivariate ARMA model
if the ARMA parameter estimates are provided.
In addition, the subroutine TSPRED produces the (intermediate
and permanent) impulse response function and performs
forecast error variance decomposition for the vector AR model.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.