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AR Model with No Constant Term AR Model with Nonconsecutive Lags |

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|s 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 = [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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