The vector autoregressive moving-average model with exogenous variables is called the VARMAX(,,) model. The form of the model can be written as

where the output variables of interest, , can be influenced by other input variables, , which are determined outside of the system of interest. The variables are referred to as dependent, response, or endogenous variables, and the variables are referred to as independent, input, predictor, regressor, or exogenous variables. The unobserved noise variables, , are a vector white noise process.

The VARMAX(,,) model can be written

where

are matrix polynomials in in the backshift operator, such that , the and are matrices, and the are matrices.

The following assumptions are made:

• , , which is positive-definite, and for .

• For stationarity and invertibility of the VARMAX process, the roots of and are outside the unit circle.

• The exogenous (independent) variables are not correlated with residuals , . The exogenous variables can be stochastic or nonstochastic. When the exogenous variables are stochastic and their future values are unknown, forecasts of these future values are needed to forecast the future values of the endogenous (dependent) variables. On occasion, future values of the exogenous variables can be assumed to be known because they are deterministic variables. The VARMAX procedure assumes that the exogenous variables are nonstochastic if future values are available in the input data set. Otherwise, the exogenous variables are assumed to be stochastic and their future values are forecasted by assuming that they follow the VARMA(,) model, prior to forecasting the endogenous variables, where and are the same as in the VARMAX(,,) model.

State-Space Representation

Another representation of the VARMAX(,,) model is in the form of a state-variable or a state-space model, which consists of a state equation

and an observation equation

where

and

On the other hand, it is assumed that follows a VARMA(,) model

The model can also be expressed as

where and are matrix polynomials in , and the and are matrices. Without loss of generality, the AR and MA orders can be taken to be the same as the VARMAX(,,) model, and and are independent white noise processes.

Under suitable conditions such as stationarity, is represented by an infinite order moving-average process

where .

The optimal minimum mean squared error (minimum MSE) -step-ahead forecast of is

For ,

The VARMAX(,,) model has an absolutely convergent representation as

or

where , , and .

The optimal (minimum MSE) -step-ahead forecast of is

for with . For ,

where .

Define . For with , you obtain

From the preceding relations, a state equation is

and an observation equation is

where

and

Note that the matrix and the input vector are defined only when .