# arima

Convert regression model with ARIMA errors to ARIMAX model

## Description

## Examples

## Input Arguments

## Output Arguments

## 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`

,`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$$\varphi (L){(1-L)}^{D}\Phi (L)(1-{L}^{s}).$$

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 $${X}_{{a}_{1}}\beta ={L}^{{a}_{1}}X\beta .$$Column

*j*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$$(1-0.2L-0.05{L}^{3})(1-L)=1-1.2L+0.02{L}^{2}-0.05{L}^{3}+0.05{L}^{4}.$$

This implies that

`ARIMAXMdl.Beta`

is`[1 -1.2 0.02 -0.05 0.05]`

and`XNew`

is$$\left[\begin{array}{ccccc}{x}_{1}\beta & NaN& NaN& NaN& NaN\\ {x}_{2}\beta & {x}_{1}\beta & NaN& NaN& NaN\\ {x}_{3}\beta & {x}_{2}\beta & {x}_{1}\beta & NaN& NaN\\ {x}_{4}\beta & {x}_{3}\beta & {x}_{2}\beta & {x}_{1}\beta & NaN\\ {x}_{5}\beta & {x}_{4}\beta & {x}_{3}\beta & {x}_{2}\beta & {x}_{1}\beta \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {x}_{T}\beta & {x}_{T-1}\beta & {x}_{T-2}\beta & {x}_{T-3}\beta & {x}_{T-4}\beta \end{array}\right],$$

where

*x*is row_{j}*j*of*X*.If you do not specify

`X`

,`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.

## Version History

**Introduced in R2013b**