## Create Regression Models with ARMA Errors

### Default Regression Model with ARMA Errors

This example shows how to apply the shorthand `regARIMA(p,D,q)`

syntax to specify the regression model with ARMA errors.

Specify the default regression model with ARMA(3,2) errors:

$$\begin{array}{l}{y}_{t}=c+{X}_{t}\beta +{u}_{t}\\ {u}_{t}={a}_{1}{u}_{t-1}+{a}_{2}{u}_{t-2}+{a}_{3}{u}_{t-3}+{\epsilon}_{t}+{b}_{1}{\epsilon}_{t-1}+{b}_{2}{\epsilon}_{t-2}.\end{array}$$

Mdl = regARIMA(3,0,2)

Mdl = regARIMA with properties: Description: "ARMA(3,2) Error Model (Gaussian Distribution)" Distribution: Name = "Gaussian" Intercept: NaN Beta: [1×0] P: 3 Q: 2 AR: {NaN NaN NaN} at lags [1 2 3] SAR: {} MA: {NaN NaN} at lags [1 2] SMA: {} Variance: NaN

The software sets each parameter to `NaN`

, and the innovation distribution to `Gaussian`

. The AR coefficients are at lags 1 through 3, and the MA coefficients are at lags 1 and 2.

Pass `Mdl`

into `estimate`

with data to estimate the parameters set to `NaN`

. The `regARIMA`

model sets `Beta`

to `[]`

and does not display it. If you pass a matrix of predictors ($${X}_{t}$$) into `estimate`

, then `estimate`

estimates `Beta`

. The `estimate`

function infers the number of regression coefficients in `Beta`

from the number of columns in $${X}_{t}$$.

Tasks such as simulation and forecasting using `simulate`

and `forecast`

do not accept models with at least one `NaN`

for a parameter value. Use dot notation to modify parameter values.

### ARMA Error Model Without an Intercept

This example shows how to specify a regression model with ARMA errors without a regression intercept.

Specify the default regression model with ARMA(3,2) errors:

$$\begin{array}{l}{y}_{t}={X}_{t}\beta +{u}_{t}\\ {u}_{t}={a}_{1}{u}_{t-1}+{a}_{2}{u}_{t-2}+{a}_{3}{u}_{t-3}+{\epsilon}_{t}+{b}_{1}{\epsilon}_{t-1}+{b}_{2}{\epsilon}_{t-2}.\end{array}$$

Mdl = regARIMA('ARLags',1:3,'MALags',1:2,'Intercept',0)

Mdl = regARIMA with properties: Description: "ARMA(3,2) Error Model (Gaussian Distribution)" Distribution: Name = "Gaussian" Intercept: 0 Beta: [1×0] P: 3 Q: 2 AR: {NaN NaN NaN} at lags [1 2 3] SAR: {} MA: {NaN NaN} at lags [1 2] SMA: {} Variance: NaN

The software sets `Intercept`

to 0, but all other parameters in `Mdl`

are `NaN`

values by default.

Since `Intercept`

is not a `NaN`

, it is an equality constraint during estimation. In other words, if you pass `Mdl`

and data into `estimate`

, then `estimate`

sets `Intercept`

to 0 during estimation.

You can modify the properties of `Mdl`

using dot notation.

### ARMA Error Model with Nonconsecutive Lags

This example shows how to specify a regression model with ARMA errors, where the nonzero ARMA terms are at nonconsecutive lags.

Specify the regression model with ARMA(8,4) errors:

$$\begin{array}{l}{y}_{t}=c+{X}_{t}\beta +{u}_{t}\\ {u}_{t}={a}_{1}{u}_{1}+{a}_{4}{u}_{4}+{a}_{8}{u}_{8}+{\epsilon}_{t}+{b}_{1}{\epsilon}_{t-1}+{b}_{4}{\epsilon}_{t-4}.\end{array}$$

Mdl = regARIMA('ARLags',[1,4,8],'MALags',[1,4])

Mdl = regARIMA with properties: Description: "ARMA(8,4) Error Model (Gaussian Distribution)" Distribution: Name = "Gaussian" Intercept: NaN Beta: [1×0] P: 8 Q: 4 AR: {NaN NaN NaN} at lags [1 4 8] SAR: {} MA: {NaN NaN} at lags [1 4] SMA: {} Variance: NaN

The AR coefficients are at lags 1, 4, and 8, and the MA coefficients are at lags 1 and 4. The software sets the interim lags to 0.

Pass `Mdl`

and data into `estimate`

. The software estimates all parameters that have the value `NaN`

. Then `estimate`

holds all interim lag coefficients to 0 during estimation.

### Known Parameter Values for a Regression Model with ARMA Errors

This example shows how to specify values for all parameters of a regression model with ARMA errors.

Specify the regression model with ARMA(3,2) errors:

$$\begin{array}{l}{y}_{t}={X}_{t}\left[\begin{array}{l}2.5\\ -0.6\end{array}\right]+{u}_{t}\\ {u}_{t}=0.7{u}_{t-1}-0.3{u}_{t-2}+0.1{u}_{t-3}+{\epsilon}_{t}+0.5{\epsilon}_{t-1}+0.2{\epsilon}_{t-2},\end{array}$$

where $${\epsilon}_{t}$$ is Gaussian with unit variance.

Mdl = regARIMA('Intercept',0,'Beta',[2.5; -0.6],... 'AR',{0.7, -0.3, 0.1},'MA',{0.5, 0.2},'Variance',1)

Mdl = regARIMA with properties: Description: "Regression with ARMA(3,2) Error Model (Gaussian Distribution)" Distribution: Name = "Gaussian" Intercept: 0 Beta: [2.5 -0.6] P: 3 Q: 2 AR: {0.7 -0.3 0.1} at lags [1 2 3] SAR: {} MA: {0.5 0.2} at lags [1 2] SMA: {} Variance: 1

The parameters in `Mdl`

do not contain `NaN`

values, and therefore there is no need to estimate `Mdl`

using `estimate`

. However, you can simulate or forecast responses from `Mdl`

using `simulate`

or `forecast`

.

### Regression Model with ARMA Errors and t Innovations

This example shows how to set the innovation distribution of a regression model with ARMA errors to a *t* distribution.

Specify the regression model with ARMA(3,2) errors:

$$\begin{array}{l}{y}_{t}={X}_{t}\left[\begin{array}{l}2.5\\ -0.6\end{array}\right]+{u}_{t}\\ {u}_{t}=0.7{u}_{t-1}-0.3{u}_{t-2}+0.1{u}_{t-3}+{\epsilon}_{t}+0.5{\epsilon}_{t-1}+0.2{\epsilon}_{t-2},\end{array}$$

where $${\epsilon}_{t}$$ has a *t* distribution with the default degrees of freedom and unit variance.

Mdl = regARIMA('Intercept',0,'Beta',[2.5; -0.6],... 'AR',{0.7, -0.3, 0.1},'MA',{0.5, 0.2},'Variance',1,... 'Distribution','t')

Mdl = regARIMA with properties: Description: "Regression with ARMA(3,2) Error Model (t Distribution)" Distribution: Name = "t", DoF = NaN Intercept: 0 Beta: [2.5 -0.6] P: 3 Q: 2 AR: {0.7 -0.3 0.1} at lags [1 2 3] SAR: {} MA: {0.5 0.2} at lags [1 2] SMA: {} Variance: 1

The default degrees of freedom is `NaN`

. If you don't know the degrees of freedom, then you can estimate it by passing `Mdl`

and the data to `estimate`

.

Specify a $${t}_{5}$$ distribution.

Mdl.Distribution = struct('Name','t','DoF',5)

Mdl = regARIMA with properties: Description: "Regression with ARMA(3,2) Error Model (t Distribution)" Distribution: Name = "t", DoF = 5 Intercept: 0 Beta: [2.5 -0.6] P: 3 Q: 2 AR: {0.7 -0.3 0.1} at lags [1 2 3] SAR: {} MA: {0.5 0.2} at lags [1 2] SMA: {} Variance: 1

You can simulate or forecast responses from `Mdl`

using `simulate`

or `forecast`

because `Mdl`

is completely specified.

In applications, such as simulation, the software normalizes the random *t* innovations. In other words, `Variance`

overrides the theoretical variance of the *t* random variable (which is `DoF`

/(`DoF`

- 2)), but preserves the kurtosis of the distribution.

### Specify Regression Model with ARMA Errors Using Econometric Modeler App

In the **Econometric Modeler** app, you can specify the predictor variables
in the regression component, and the error model lag structure and innovation
distribution of a regression model with
ARMA(*p*,*q*) errors, by following these
steps. All specified coefficients are unknown but estimable parameters.

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.In the models gallery, in the

**Regression Models**section, click**RegARMA**.The

**RegARMA Model Parameters**dialog box appears.Choose the error model lag structure. To specify a regression model with ARMA(

*p*,*q*) errors that includes all AR lags from 1 through*p*and all MA lags from 1 through*q*, use the**Lag Order**tab. For the flexibility to specify the inclusion of particular lags, use the**Lag Vector**tab. For more details, see Specifying Univariate 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 a regression model with AR(3) errors for the unemployment rate containing all consecutive AR lags from 1 through its order, 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

**Regression Models**section, click**RegARMA**..

In the

**regARMA Model Parameters**dialog box, on the**Lag Order**tab, set**Autoregressive Order**to`3`

.In the

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

To specify a regression model with MA(2) errors for the unemployment rate containing all MA lags from 1 through its order, Gaussian-distributed innovations, 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

**Regression Models**section, click**RegARMA**.In the

**regARMA Model Parameters**dialog box, on the**Lag Order**tab, set**Moving Average Order**to`2`

.In the

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

To specify the regression model with ARMA(8,4) errors for the unemployment rate containing nonconsecutive lags

$$\begin{array}{c}{y}_{t}=c+{\beta}_{1}CO{E}_{t}+{\beta}_{2}CPIAUCS{L}_{t}+{u}_{t}\\ \left(1-{\alpha}_{1}L-{\alpha}_{4}{L}^{4}-{\alpha}_{8}{L}^{8}\right){u}_{t}=\left(1+{b}_{1}L+{b}_{4}{L}^{4}\right){\epsilon}_{t}\end{array},$$

where

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

**Regression Models**section, click**RegARMA**.In the

**regARMA Model Parameters**dialog box, click the**Lag Vector**tab:In the

**Autoregressive Lags**box, type`1 4 8`

.In the

**Moving Average Lags**box, type`1 4`

.

In the

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

To specify a regression model with ARMA(3,2) errors for the unemployment rate containing all consecutive AR and MA lags through their respective orders, 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

**Regression Models**section, click**RegARMA**.In the

**regARMA Model Parameters**dialog box, click the**Lag Order**tab: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 the**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.

## See Also

### Apps

### Objects

### Functions

## Related Examples

- Analyze Time Series Data Using Econometric Modeler
- Specifying Univariate Lag Operator Polynomials Interactively
- Create Regression Models with ARIMA Errors
- Specify the Default Regression Model with ARIMA Errors
- Create Regression Models with AR Errors
- Create Regression Models with MA Errors
- Create Regression Models with ARIMA Errors
- Create Regression Models with SARIMA Errors
- Specify ARIMA Error Model Innovation Distribution