## ARIMAX Model Specifications

### Create ARIMAX Model Using Name-Value Pairs

This example shows how to specify an ARIMAX model using `arima`

.

Specify the ARIMAX(1,1,0) model that includes three predictors:

$$(1-0.1L)(1-L{)}^{1}{y}_{t}={x}_{t}^{\prime}{\left[\begin{array}{ccc}3& -2& 5\end{array}\right]}^{\prime}+{\epsilon}_{t}.$$

Mdl = arima('AR',0.1,'D',1,'Beta',[3 -2 5])

Mdl = arima with properties: Description: "ARIMAX(1,1,0) Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 2 D: 1 Q: 0 Constant: NaN AR: {0.1} at lag [1] SAR: {} MA: {} SMA: {} Seasonality: 0 Beta: [3 -2 5] Variance: NaN

The output shows that the ARIMAX model `Mdl`

has the following qualities:

Property

`P`

in the output is the sum of the autoregressive lags and the degree of integration, i.e.,`P`

=`p`

+`D`

=`2`

.`Beta`

contains three coefficients corresponding to the effect that the predictors have on the response.`Mdl`

does not store predictor or response data. You specify the required data when you operate on`Mdl`

.The rest of the properties are 0,

`NaN`

, or empty cells.

Be aware that if you specify nonzero `D`

or `Seasonality`

, then Econometrics Toolbox™ differences the response series $${y}_{t}$$ before the predictors enter the model. Therefore, the predictors enter a stationary model with respect to the response series $${y}_{t}$$. You should preprocess the predictors $${x}_{t}$$ by testing for stationarity and differencing if any are unit root nonstationary. If any nonstationary predictor enters the model, then the false negative rate for significance tests of $$\beta $$ can increase.

### Specify ARMAX Model Using Dot Notation

This example shows how to specify a stationary ARMAX model using `arima`

.

Specify the ARMAX(2,1) model

$${y}_{t}=6+0.2{y}_{t-1}-0.3{y}_{t-2}+3{x}_{t}+{\epsilon}_{t}+0.1{\epsilon}_{t-1}$$

by including one stationary exogenous covariate in `arima`

.

Mdl = arima('AR',[0.2 -0.3],'MA',0.1,'Constant',6,'Beta',3)

Mdl = arima with properties: Description: "ARIMAX(2,0,1) Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 2 D: 0 Q: 1 Constant: 6 AR: {0.2 -0.3} at lags [1 2] SAR: {} MA: {0.1} at lag [1] SMA: {} Seasonality: 0 Beta: [3] Variance: NaN

The output shows the model `Mdl`

has `NaN`

values or an empty cell (`{}`

) for the `Variance`

, `SAR`

, and `SMA`

properties. You can modify the model using dot notation at the command line. For example, you can introduce another exogenous, stationary covariate, and specify the variance of the innovations as 0.1

$${y}_{t}=6+0.2{y}_{t-1}-0.3{y}_{t-2}+{x}_{t}^{\prime}\left[\begin{array}{c}3\\ -2\end{array}\right]+{\epsilon}_{t}+0.1{\epsilon}_{t-1};\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}{\epsilon}_{t}\sim N(0,0.1).$$

Modify the model by specifying an innovations variance of 0.1 and regression coefficient values.

Mdl.Beta = [3 -2]; Mdl.Variance = 0.1; Mdl

Mdl = arima with properties: Description: "ARIMAX(2,0,1) Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 2 D: 0 Q: 1 Constant: 6 AR: {0.2 -0.3} at lags [1 2] SAR: {} MA: {0.1} at lag [1] SMA: {} Seasonality: 0 Beta: [3 -2] Variance: 0.1

### Specify ARIMAX or SARIMAX Model Using Econometric Modeler App

In the **Econometric Modeler** app, you can specify the seasonal and nonseasonal lag structure, presence of a constant, innovation distribution, and predictor variables of an ARIMA(*p*,*D*,*q*) or a SARIMA(*p*,*D*,*q*)×(*p _{s}*,

*D*,

_{s}*q*)

_{s}*model by following these steps. All specified coefficients are unknown but estimable parameters.*

_{s}At the command line, open the

**Econometric Modeler**app.econometricModeler

Alternatively, open the app from the apps gallery (see

**Econometric Modeler**).In the

**Time Series**pane, select the response time series to which the model will be fit.On the

**Econometric Modeler**tab, in the**Models**section, click the arrow to display the models gallery. For strictly nonseasonal models, click**ARIMAX**; for seasonal models, click**SARIMAX**. ARIMAX and SARIMAX models must contain at least one predictor variable.The

dialog box appears. This figure shows theModel Parameters`Type`

**SARIMAX Model Parameters**dialog box. All variables in the**Time Series**pane, except the chosen response variable, appear in the**Predictors**section.Specify the lag structure. Use the

**Lag Order**tab to specify a model that includes:All consecutive lags from 1 through their respective orders, in the seasonal polynomials

Lags that are all consecutive multiples of the period (

*s*), in the seasonal polynomialsAn

*s*-degree seasonal integration polynomial

For the flexibility to specify the inclusion of particular lags, use the

**Lag Vector**tab. For more details, see Specifying Lag Operator Polynomials Interactively. Regardless of the tab you use, you can verify the model form by inspecting the equation in the**Model Equation**section.In the

**Predictors**section, choose at least one predictor variable by selecting the**Include?**check box for the time series.

For example, suppose you are working with the `Data_USEconModel.mat`

data set and its variables are listed in the **Time Series** pane.

To specify an ARIMAX(3,1,2) model for the unemployment rate containing a constant, all consecutive AR and MA lags from 1 through their respective orders, Gaussian-distributed innovations, and the predictor variables

**COE**,**CPIAUCSL**,**FEDFUNDS**, and**GDP**:In the

**Time Series**pane, select the`UNRATE`

time series.On the

**Econometric Modeler**tab, in the**Models**section, click the arrow to display the models gallery.In the models gallery, in the

**ARMA/ARIMA Models**section, click**ARIMAX**.In the

**ARIMAX Model Parameters**dialog box in the**Nonseasonal**section of the**Lag Order**tab, set**Degree of Integration**to`1`

.Set

**Autoregressive Order**to`3`

.Set

**Moving Average Order**to`2`

.In the

**Predictors**section, select the**Include?**check box for the**COE**,**CPIAUCSL**,**FEDFUNDS**, and**GDP**time series.

To specify an ARIMAX(3,1,2) model for the unemployment rate containing all AR and MA lags from 1 through their respective orders, Gaussian-distributed innovations, no constant, and the predictor variables

**COE**and**CPIAUCSL**:In the

**Time Series**pane, select the`UNRATE`

time series.On the

**Econometric Modeler**tab, in the**Models**section, click the arrow to display the models gallery.In the models gallery, in the

**ARMA/ARIMA Models**section, click**ARIMAX**.In the

**ARIMAX Model Parameters**dialog box, in the**Nonseasonal**section of the**Lag Order**tab, set**Degree of Integration**to`1`

.Set

**Autoregressive Order**to`3`

.Set

**Moving Average Order**to`2`

.Clear the

**Include Constant Term**check box.In the

**Predictors**section, select the**Include?**check box for the**COE**and**CPIAUCSL**time series.

To specify an ARMA(8,1,4) model for the unemployment rate containing nonconsecutive lags

$$\left(1-{\varphi}_{1}L-{\varphi}_{4}{L}^{4}-{\varphi}_{8}{L}^{8}\right)\left(1-L\right){y}_{t}=\left(1+{\theta}_{1}L+{\theta}_{4}{L}^{4}\right){\epsilon}_{t}+{\beta}_{1}CO{E}_{t}+{\beta}_{2}CPIAUCS{L}_{t},$$

where

*ε*is a series of IID Gaussian innovations:_{t}In the

**Time Series**pane, select the`UNRATE`

time series.On the

**Econometric Modeler**tab, in the**Models**section, click the arrow to display the models gallery.In the models gallery, in the

**ARMA/ARIMA Models**section, click**ARIMAX**.In the

**ARIMAX Model Parameters**dialog box, click the**Lag Vector**tab.Set

**Degree of Integration**to`1`

.Set

**Autoregressive Lags**to`1 4 8`

.Set

**Moving Average Lags**to`1 4`

.Clear the

**Include Constant Term**check box.In the

**Predictors**section, select the**Include?**check box for the**COE**and**CPIAUCSL**time series.

To specify an ARIMA(3,1,2) model for the unemployment rate containing all consecutive AR and MA lags through their respective orders, a constant term, the predictor variables

**COE**and**CPIAUCSL**, and*t*-distributed innovations:In the

**Time Series**pane, select the`UNRATE`

time series.**Econometric Modeler**tab, in the**Models**section, click the arrow to display the models gallery.In the models gallery, in the

**ARMA/ARIMA Models**section, click**ARIMAX**.In the

**ARIMAX Model Parameters**dialog box, in the**Nonseasonal**section of the**Lag Order**tab, set**Degree of Integration**to`1`

.Set

**Autoregressive Order**to`3`

.Set

**Moving Average Order**to`2`

.Click the

**Innovation Distribution**button, then select`t`

.In the

**Predictors**section, select the**Include?**check box for**COE**and**CPIAUCSL**time series.

The degrees of freedom parameter of the

*t*distribution is an unknown but estimable parameter.

After you specify a model, click **Estimate** to estimate all unknown parameters in the model.