Class: regARIMA
Convert regression model with ARIMA errors to ARIMAX model
ARIMAX = arima(Mdl)
[ARIMAX,XNew]
= arima(Mdl,Name,Value)
The arima
object function converts a specified regression model with ARIMA errors (regARIMA
model object) to the equivalent ARIMAX model (arima
model object). To create an ARIMAX model directly, see arima
.
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 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.