arima
Class: regARIMA
Convert regression model with ARIMA errors to ARIMAX model
Syntax
ARIMAX = arima(Mdl)
[ARIMAX,XNew]
= arima(Mdl,Name,Value)
Description
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.
Input Arguments

Regression model with ARIMA time series errors, as created by 
NameValue Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Namevalue arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.

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

ARIMAX model equivalent to the regression model with ARIMA errors 

Updated predictor data matrix for the regression component of
Each column of 
Examples
Algorithms
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
, thenarima
returnsXNew
in a certain format. Suppose that the nonzero autoregressive lag term degrees ofMdl
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]
andXNew
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
, thenarima
returnsXNew
as an empty matrix without rows and one plus the number of nonzero autoregressive coefficients in the difference equation ofMdl
columns.