# Documentation

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## MA Model Specifications

### Default MA Model

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

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

Specify the default MA(3) model:

`model = arima(0,0,3)`
```model = ARIMA(0,0,3) Model: -------------------- Distribution: Name = 'Gaussian' P: 0 D: 0 Q: 3 Constant: NaN AR: {} SAR: {} MA: {NaN NaN NaN} at Lags [1 2 3] SMA: {} Variance: NaN ```

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

### MA Model with No Constant Term

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

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

where the innovation distribution is Gaussian with constant variance.

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

The `MALags` name-value argument specifies the lags corresponding to nonzero MA 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 MA coefficients and scalar variance.

You can modify the created model variable, or input it (along with data) to `estimate`.

### MA Model with Nonconsecutive Lags

This example shows how to specify an MA(q) model with nonzero coefficients at nonconsecutive lags.

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

where the innovation distribution is Gaussian with constant variance.

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

The output shows the nonzero AR coefficients at lags 1 and 4, as specified. The property `Q` is equal to `4`, the number of presample innovations needed to initialize the MA model. The unconstrained parameters are equal to `NaN`.

Display the value of `MA`:

```model.MA ```
```ans = 1x4 cell array {[NaN]} {[0]} {[0]} {[NaN]} ```

The `MA` 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.

### MA Model with Known Parameter Values

This example shows how to specify an MA(q) model with known parameter values. You can use such a fully specified model as an input to `simulate` or `forecast`.

Specify the MA(4) model

where the innovation distribution is Gaussian with constant variance 0.15.

```model = arima('Constant',0.1,'MA',{0.7,0.2},... 'MALags',[1,4],'Variance',0.15)```
```model = ARIMA(0,0,4) Model: -------------------- Distribution: Name = 'Gaussian' P: 0 D: 0 Q: 4 Constant: 0.1 AR: {} SAR: {} MA: {0.7 0.2} at Lags [1 4] SMA: {} Variance: 0.15 ```

Because all parameter values are specified, the created model object has no `NaN` values. The functions `simulate` and `forecast` don't accept input models with `NaN` values.

### MA Model with a t Innovation Distribution

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

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

where the innovation process follows a Student's t distribution with eight degrees of freedom.

```tdist = struct('Name','t','DoF',8); model = arima('Constant',0,'MALags',1:2,'Distribution',tdist)```
```model = ARIMA(0,0,2) Model: -------------------- Distribution: Name = 't', DoF = 8 P: 0 D: 0 Q: 2 Constant: 0 AR: {} SAR: {} MA: {NaN NaN} at Lags [1 2] SMA: {} Variance: NaN ```

The value of `Distribution` is a `struct` array with field `Name` equal to `'t'` and field `DoF` equal to `8`. When you specify the degrees of freedom, they aren't estimated if you input the model to `estimate`.