ARIMAX =
arima
(Mdl)
[ARIMAX,XNew]
= arima(Mdl,Name,Value)
converts
the univariate regression model with ARIMA time series errors ARIMAX
= arima
(Mdl
)Mdl
to
a model of type arima
including
a regression component (ARIMAX).
[
returns
an updated regression matrix of predictor data using additional options
specified by one or more ARIMAX
,XNew
]
= arima(Mdl
,Name,Value
)Name,Value
pair arguments.

Regression model with ARIMA time series errors, as created by 
Specify optional commaseparated pairs of Name,Value
arguments.
Name
is the argument
name and Value
is the corresponding
value. Name
must appear
inside single quotes (' '
).
You can specify several name and value pair
arguments in any order as Name1,Value1,...,NameN,ValueN
.

Predictor data for the regression component of The last row of Each column of 

ARIMAX model equivalent to the regression model with ARIMA errors 

Updated predictor data matrix for the regression component of
Each column of 
Let X denote the matrix of concatenated predictor
data vectors (or design matrix) and β denote
the regression component for the regression model with ARIMA errors, Mdl
.
If you specify X
, then arima
returns XNew
in
a certain format. Suppose that the nonzero autoregressive lag term
degrees of Mdl
are 0 < a_{1} < a_{2} <
...< P, which is the largest lag term degree.
The software obtains these lag term degrees by expanding and reducing
the product of the seasonal and nonseasonal autoregressive lag polynomials,
and the seasonal and nonseasonal integration lag polynomials
$$\varphi (L){(1L)}^{D}\Phi (L)(1{L}^{s}).$$
The first column of XNew
is Xβ.
The second column of XNew
is a
sequence of a_{1} NaN
s,
and then the product $${X}_{{a}_{1}}\beta ,$$ where $${X}_{{a}_{1}}\beta ={L}^{{a}_{1}}X\beta .$$
The jth column of XNew
is
a sequence of a_{j} NaN
s,
and then the product $${X}_{{a}_{j}}\beta ,$$ where $${X}_{{a}_{j}}\beta ={L}^{{a}_{j}}X\beta .$$
The last column of XNew
is a sequence
of a_{p} NaN
s,
and then the product $${X}_{p}\beta ,$$ where $${X}_{p}\beta ={L}^{p}X\beta .$$
Suppose that Mdl
is a regression
model with ARIMA(3,1,0) errors, and ϕ_{1} =
0.2 and ϕ_{3} = 0.05.
Then the product of the autoregressive and integration lag polynomials
is
$$(10.2L0.05{L}^{3})(1L)=11.2L+0.02{L}^{2}0.05{L}^{3}+0.05{L}^{4}.$$
This implies that ARIMAX.Beta
is [1
1.2 0.02 0.05 0.05]
and XNew
is
$$\left[\begin{array}{ccccc}{x}_{1}\beta & NaN& NaN& NaN& NaN\\ {x}_{2}\beta & {x}_{1}\beta & NaN& NaN& NaN\\ {x}_{3}\beta & {x}_{2}\beta & {x}_{1}\beta & NaN& NaN\\ {x}_{4}\beta & {x}_{3}\beta & {x}_{2}\beta & {x}_{1}\beta & NaN\\ {x}_{5}\beta & {x}_{4}\beta & {x}_{3}\beta & {x}_{2}\beta & {x}_{1}\beta \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {x}_{T}\beta & {x}_{T1}\beta & {x}_{T2}\beta & {x}_{T3}\beta & {x}_{T4}\beta \end{array}\right],$$
where x_{j} is the jth row of X.
If you do not specify X
, then arima
returns XNew
as
an empty matrix without rows and one plus the number of nonzero autoregressive
coefficients in the difference equation of Mdl
columns.