# 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.

ARIMAX = arima(Mdl) converts the univariate regression model with ARIMA time series errors Mdl to a model of type arima including a regression component (ARIMAX).

[ARIMAX,XNew] = arima(Mdl,Name,Value) returns an updated regression matrix of predictor data using additional options specified by one or more Name,Value pair arguments.

## Input Arguments

 Mdl Regression model with ARIMA time series errors, as created by regARIMA or estimate.

### Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value 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.

 X Predictor data for the regression component of Mdl, specified as the comma-separated pair consisting of 'X' and a matrix. The last row of X contains the latest observation of each series. Each column of X is a separate time series.

## Output Arguments

 ARIMAX ARIMAX model equivalent to the regression model with ARIMA errors Mdl, returned as a model of type arima. XNew Updated predictor data matrix for the regression component of ARIMAX, returned as a matrix. XNew has the same number of rows as X. The last row of XNew contains the latest observation of each series. Each column of XNew is a separate time series. The number of columns of XNew is one plus the number of nonzero autoregressive coefficients in the difference equation of Mdl.

## Examples

expand all

Convert a regression model with ARMA(4,1) errors to an ARIMAX model using the arima converter.

Specify the regression model with ARMA(4,1) errors:

$\begin{array}{l}{y}_{t}=1+0.5{X}_{t}+{u}_{t}\\ {u}_{t}=0.8{u}_{t-1}-0.4{u}_{t-4}+{\epsilon }_{t}+0.3{\epsilon }_{t-1},\end{array}$

where ${\epsilon }_{t}$ is Gaussian with mean 0 and variance 1.

Mdl = regARIMA('AR',{0.8, -0.4},'MA',0.3,...
'ARLags',[1 4],'Intercept',1,'Beta',0.5,...
'Variance',1)
Mdl =
regARIMA with properties:

Description: "Regression with ARMA(4,1) Error Model (Gaussian Distribution)"
Distribution: Name = "Gaussian"
Intercept: 1
Beta: [0.5]
P: 4
Q: 1
AR: {0.8 -0.4} at lags [1 4]
SAR: {}
MA: {0.3} at lag [1]
SMA: {}
Variance: 1

You can verify that the lags of the autoregressive terms are 1 and 4 in the AR row.

Generate random predictor data.

rng(1); % For reproducibility
T = 20;
X = randn(T,1);

Convert Mdl to an ARIMAX model.

[ARIMAX,XNew] = arima(Mdl,'X',X);
ARIMAX
ARIMAX =
arima with properties:

Description: "ARIMAX(4,0,1) Model (Gaussian Distribution)"
Distribution: Name = "Gaussian"
P: 4
D: 0
Q: 1
Constant: 0.6
AR: {0.8 -0.4} at lags [1 4]
SAR: {}
MA: {0.3} at lag [1]
SMA: {}
Seasonality: 0
Beta: [1 -0.8 0.4]
Variance: 1

The new arima model, ARIMAX, is

${y}_{t}=0.6+{Z}_{t}\Gamma +0.8{y}_{t-1}-0.4{y}_{t-4}+{\epsilon }_{t}+0.3{\epsilon }_{t-1},$

where

${Z}_{t}\Gamma =\left[\begin{array}{ccc}0.5{x}_{1}& NaN& NaN\\ 0.5{x}_{2}& 0.5{x}_{1}& NaN\\ 0.5{x}_{3}& 0.5{x}_{2}& NaN\\ 0.5{x}_{4}& 0.5{x}_{3}& NaN\\ 0.5{x}_{5}& 0.5{x}_{4}& 0.5{x}_{1}\\ ⋮& ⋮& ⋮\\ {0.5}_{T}& 0.5{x}_{T-1}& 0.5{x}_{T-4}\end{array}\right]\left[\begin{array}{c}1\\ -0.8\\ 0.4\end{array}\right]$

and ${x}_{j}$ is row j of X. Since the product of the autoregressive and integration polynomials is $\varphi \left(L\right)=\left(1-0.8L+0.4{L}^{4}\right),$ ARIMAX.Beta is simply [1 -0.8 0.4]. Note that the software carries over the autoregressive and moving average coefficients from Mdl to ARIMAX. Also, Mdl.Intercept = 1 and ARIMAX.Constant = (1 - 0.8 + 0.4)(1) = 0.6, i.e., the regARIMA model intercept and arima model constant are generally unequal.

Convert a regression model with seasonal ARIMA errors to an ARIMAX model using the arima converter.

Specify the regression model with $ARIMA\left(2,1,1\right)×{\left(1,1,0\right)}_{2}$ errors:

$\begin{array}{c}{y}_{t}={X}_{t}\left[\begin{array}{c}-2\\ 1\end{array}\right]+{u}_{t}\\ \left(1-0.3L+0.15{L}^{2}\right)\left(1-L\right)\left(1-0.2{L}^{2}\right)\left(1-{L}^{2}\right){u}_{t}=\left(1+0.1L\right){\epsilon }_{t},\end{array}$

where ${\epsilon }_{t}$ is Gaussian with mean 0 and variance 1.

Mdl = regARIMA('AR',{0.3, -0.15},'MA',0.1,...
'ARLags',[1 2],'SAR',0.2,'SARLags',2,...
'Intercept',0,'Beta',[-2; 1],'Variance',1,'D',1,...
'Seasonality',2)
Mdl =
regARIMA with properties:

Description: "Regression with ARIMA(2,1,1) Error Model Seasonally Integrated with Seasonal AR(2) (Gaussian Distribution)"
Distribution: Name = "Gaussian"
Intercept: 0
Beta: [-2 1]
P: 7
D: 1
Q: 1
AR: {0.3 -0.15} at lags [1 2]
SAR: {0.2} at lag [2]
MA: {0.1} at lag [1]
SMA: {}
Seasonality: 2
Variance: 1

Generate predictor data.

rng(1); % For reproducibility
T = 20;
X = randn(T,2);

Convert Mdl to an ARIMAX model.

[ARIMAX,XNew] = arima(Mdl,'X',X);
ARIMAX
ARIMAX =
arima with properties:

Description: "ARIMAX(2,1,1) Model Seasonally Integrated with Seasonal AR(2) (Gaussian Distribution)"
Distribution: Name = "Gaussian"
P: 7
D: 1
Q: 1
Constant: 0
AR: {0.3 -0.15} at lags [1 2]
SAR: {0.2} at lag [2]
MA: {0.1} at lag [1]
SMA: {}
Seasonality: 2
Beta: [1 -1.3 -0.75 1.41 -0.34 -0.08 0.09 -0.03]
Variance: 1

Mdl.Beta has length 2, but ARIMAX.Beta has length 8. This is because the product of the autoregressive and integration polynomials, $\varphi \left(L\right)\left(1-L\right)\Phi \left(L\right)\left(1-{L}^{s}\right)$, is

$1-1.3L-0.75{L}^{2}+1.41{L}^{3}-0.34{L}^{4}-0.08{L}^{5}+0.09{L}^{6}-0.03{L}^{7}.$

You can see that when you add seasonality, seasonal lag terms, and integration to a model, the size of XNew can grow quite large. A conversion such as this might not be ideal for analyses involving small sample sizes.

## 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, then arima returns XNew in a certain format. Suppose that the nonzero autoregressive lag term degrees of Mdl are 0 < a1 < a2 < ...< 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

$\varphi \left(L\right){\left(1-L\right)}^{D}\Phi \left(L\right)\left(1-{L}^{s}\right).$

• The first column of XNew is .

• The second column of XNew is a sequence of a1 NaNs, 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 aj NaNs, 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 ap NaNs, 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 ϕ1 = 0.2 and ϕ3 = 0.05. Then the product of the autoregressive and integration lag polynomials is

$\left(1-0.2L-0.05{L}^{3}\right)\left(1-L\right)=1-1.2L+0.02{L}^{2}-0.05{L}^{3}+0.05{L}^{4}.$

This implies that ARIMAX.Beta is [1 -1.2 0.02 -0.05 0.05] and XNew is

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

where xj 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.