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

syntax to specify the default AR(
) model,

By default, all parameters in the created model object have unknown values, and the innovation distribution is Gaussian with constant variance.

Specify the default AR(2) model:

model = arima(2,0,0)

model = ARIMA(2,0,0) Model: -------------------- Distribution: Name = 'Gaussian' P: 2 D: 0 Q: 0 Constant: NaN AR: {NaN NaN} at Lags [1 2] SAR: {} MA: {} SMA: {} Variance: NaN

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

, has `NaN`

values for all model parameters: the constant term, the AR 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 AR(*p*) model with constant term equal to zero. Use name-value syntax to specify a model that differs from the default model.

Specify an AR(2) model with no constant term,

where the innovation distribution is Gaussian with constant variance.

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

model = ARIMA(2,0,0) Model: -------------------- Distribution: Name = 'Gaussian' P: 2 D: 0 Q: 0 Constant: 0 AR: {NaN NaN} at Lags [1 2] SAR: {} MA: {} SMA: {} Variance: NaN

The `ARLags`

name-value argument specifies the lags corresponding to nonzero AR coefficients. The property `Constant`

in the created model object is equal to `0`

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

values as placeholders for the unknown parameters: the AR coefficients and scalar 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 AR(*p*) model with nonzero coefficients at nonconsecutive lags.

Specify an AR(4) model with nonzero AR coefficients at lags 1 and 4 (and no constant term),

where the innovation distribution is Gaussian with constant variance.

model = arima('ARLags',[1,4],'Constant',0)

model = ARIMA(4,0,0) Model: -------------------- Distribution: Name = 'Gaussian' P: 4 D: 0 Q: 0 Constant: 0 AR: {NaN NaN} at Lags [1 4] SAR: {} MA: {} SMA: {} Variance: NaN

The output shows the nonzero AR coefficients at lags 1 and 4, as specified. The property `P`

is equal to `4`

, the number of presample observations needed to initialize the AR model. The unconstrained parameters are equal to `NaN`

.

Display the value of `AR`

:

model.AR

ans = 1×4 cell array [NaN] [0] [0] [NaN]

The `AR`

cell array returns four elements. The first and last elements (corresponding to lags 1 and 4) have value `NaN`

, indicating these coefficients are nonzero and need to be estimated or otherwise specified by the user. `arima`

sets the coefficients at interim lags equal to zero to maintain consistency with MATLAB® cell array indexing.

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

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(1,0,1) Model: -------------------- Distribution: Name = 't', DoF = 8 P: 1 D: 0 Q: 1 Constant: 0.3 AR: {0.7} at Lags [1] SAR: {} MA: {0.4} at Lags [1] SMA: {} 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.

This example shows how to specify an AR(
) model with a Student's *t* innovation distribution.

Specify an AR(2) model with no constant term,

where the innovations follow a Student's *t* distribution with unknown degrees of freedom.

model = arima('Constant',0,'ARLags',1:2,'Distribution','t')

model = ARIMA(2,0,0) Model: -------------------- Distribution: Name = 't', DoF = NaN P: 2 D: 0 Q: 0 Constant: 0 AR: {NaN NaN} at Lags [1 2] SAR: {} MA: {} SMA: {} Variance: NaN

The value of `Distribution`

is a `struct`

array with field `Name`

equal to `'t'`

and field `DoF`

equal to `NaN`

. The `NaN`

value indicates the degrees of freedom are unknown, and need to be estimated using `estimate`

or otherwise specified by the user.

`arima`

| `estimate`

| `forecast`

| `simulate`

| `struct`

- Specify Conditional Mean Models Using arima
- Modify Properties of Conditional Mean Model Objects
- Specify Conditional Mean Model Innovation Distribution

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