Note: This page has been translated by MathWorks. Click here to see

To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

This example shows how to use the shorthand `arima(p,D,q)`

syntax to specify the default ARMA(*p*, *q*) 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 default, all parameters in the created model object have unknown values, and the innovation distribution is Gaussian with constant variance.

Specify the default ARMA(1,1) model:

model = arima(1,0,1)

model = arima with properties: Description: "ARIMA(1,0,1) Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 1 D: 0 Q: 1 Constant: NaN AR: {NaN} at lag [1] SAR: {} MA: {NaN} at lag [1] SMA: {} Seasonality: 0 Beta: [1×0] Variance: NaN

The output shows that the created model object, `model`

, has `NaN`

values for all model parameters: the constant term, the AR and MA coefficients, and the variance. You can modify the created model object using dot notation, or input it (along with data) to `estimate`

.

This example shows how to specify an ARMA(*p*, *q*) model with constant term equal to zero. Use name-value syntax to specify a model that differs from the default model.

Specify an ARMA(2,1) model with no constant term,

$${y}_{t}={\varphi}_{1}{y}_{t-1}+{\varphi}_{2}{y}_{t-2}+{\epsilon}_{t}+{\theta}_{1}{\epsilon}_{t-1},$$

where the innovation distribution is Gaussian with constant variance.

model = arima('ARLags',1:2,'MALags',1,'Constant',0)

model = arima with properties: Description: "ARIMA(2,0,1) Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 2 D: 0 Q: 1 Constant: 0 AR: {NaN NaN} at lags [1 2] SAR: {} MA: {NaN} at lag [1] SMA: {} Seasonality: 0 Beta: [1×0] Variance: NaN

The `ArLags`

and `MaLags`

name-value pair arguments specify the lags corresponding to nonzero AR and MA coefficients, respectively. The property `Constant`

in the created model object is equal to `0`

, as specified. The model has default values for all other properties, including `NaN`

values as placeholders for the unknown parameters: the AR and MA coefficients, and scalar variance.

You can modify the created model using dot notation, or input it (along with data) to `estimate`

.

This example shows how to specify an ARMA(*p*, *q*) model with known parameter values. You can use such a fully specified model as an input to `simulate`

or `forecast`

.

Specify the ARMA(1,1) model

$${y}_{t}=0.3+0.7\varphi {y}_{t-1}+{\epsilon}_{t}+0.4{\epsilon}_{t-1},$$

where the innovation distribution is Student's *t* with 8 degrees of freedom, and constant variance 0.15.

tdist = struct('Name','t','DoF',8); model = arima('Constant',0.3,'AR',0.7,'MA',0.4,... 'Distribution',tdist,'Variance',0.15)

model = arima with properties: Description: "ARIMA(1,0,1) Model (t Distribution)" Distribution: Name = "t", DoF = 8 P: 1 D: 0 Q: 1 Constant: 0.3 AR: {0.7} at lag [1] SAR: {} MA: {0.4} at lag [1] SMA: {} Seasonality: 0 Beta: [1×0] Variance: 0.15

Because all parameter values are specified, the created model has no `NaN`

values. The functions `simulate`

and `forecast`

don't accept input models with `NaN`

values.

In the **Econometric
Modeler** app, you can specify the lag structure, presence of a constant,
and innovation distribution of an ARMA(*p*,*q*)
model 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

**Data Browser**, select the response time series to which the model will be fit.On the

**Econometric Modeler**tab, in the**Models**section, click**ARMA**.The

**ARMA Model Parameters**dialog box appears.Specify the lag structure. To specify an ARMA(

*p*,*q*) model 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 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.

For example:

To specify an ARMA(2,1) model that includes a constant, includes all AR and MA lags from 1 through their respective orders, and has a Gaussian innovation distribution:

Set

**Autoregressive Order**to`2`

.Set

**Moving Average Order**to`1`

.

To specify an ARMA(2,1) model that includes all AR and MA lags from 1 through their respective orders, has a Gaussian distribution, but does not include a constant:

Set

**Autoregressive Order**to`2`

.Set

**Moving Average Order**to`1`

.Clear the

**Include Constant Term**check box.

To specify an ARMA(2,1) model that includes all AR and MA lags from 1 through their respective orders, includes a constant term, and has

*t*-distributed innovations:Set

**Autoregressive Order**to`2`

.Set

**Moving Average Order**to`1`

.Click the

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

.

The degrees of freedom parameter of the

*t*distribution is an unknown but estimable parameter.To specify an ARMA(8,4) model containing nonconsecutive lags

$${y}_{t}-{\varphi}_{1}{y}_{t-1}-{\varphi}_{4}{y}_{t-4}-{\varphi}_{8}{y}_{t-8}={\epsilon}_{t}+{\theta}_{1}{\epsilon}_{t-1}+{\theta}_{4}{\epsilon}_{t-4},$$

where

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

**Lag Vector**tab.Set

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

.Set

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

.Clear the

**Include Constant Term**check box.

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

- Econometric Modeler App Overview
- Specifying Lag Operator Polynomials Interactively
- Specify Conditional Mean Models
- Modify Properties of Conditional Mean Model Objects
- Specify Conditional Mean Model Innovation Distribution