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Choices for the Variance Model Choices for the Innovation Distribution |

You can express all stationary stochastic processes in the general linear form [1]

The innovation process, , is an uncorrelated—but not necessarily independent—mean zero process with a known distribution.

In Econometrics Toolbox™, the general form for the innovation
process is
. Here, *z _{t}* is
an independent and identically distributed (iid) series with mean
0 and variance 1, and
is the variance
of the innovation process at time

`arima` model objects have two properties for
storing information about the innovation process:

`Variance`stores the form of`Distribution`stores the parametric form of the distribution of*z*_{t}

If for all times

*t*, then is an independent process with constant variance, .The default value for

`Variance`is`NaN`, meaning constant variance with unknown value. You can alternatively assign`Variance`any positive scalar value, or estimate it using`estimate`.A time series can exhibit

*volatility clustering*, meaning a tendency for large changes to follow large changes, and small changes to follow small changes. You can model this behavior with a conditional variance model—a dynamic model describing the evolution of the process variance, , conditional on past innovations and variances.Set

`Variance`equal to one of the three conditional variance model objects available in Econometrics Toolbox (`garch`,`egarch`, or`gjr`). This creates a composite conditional mean and variance model variable.

The available distributions for *z _{t}* are:

Standardized Gaussian

Standardized Student's

*t*with*ν*> 2 degrees of freedom,where follows a Student's

*t*distribution with*ν*> 2 degrees of freedom.

The *t* distribution is useful for modeling
time series with more extreme values than expected under a Gaussian
distribution. Series with larger values than expected under normality
are said to have *excess kurtosis*.

The property `Distribution` in a model stores the distribution name (and degrees of freedom for the *t* distribution). The data type of `Distribution` is a `struct` array. For a Gaussian innovation distribution, the data structure has only one field: `Name`. For a Student's *t* distribution, the data structure must have two fields:

`Name`, with value`'t'``DoF`, with a scalar value larger than two (`NaN`is the default value)

If the innovation distribution is Gaussian, you do not need to assign a value to `Distribution`. `arima` creates the required data structure.

To illustrate, consider specifying an MA(2) model with an iid Gaussian innovation process:

Mdl = arima(0,0,2)

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

The model output shows that `Distribution` is a `struct` array with one field, `Name`, with the value `'Gaussian'`.

When specifying a Student's *t* innovation distribution, you can specify the distribution with either unknown or known degrees of freedom. If the degrees of freedom are unknown, you can simply assign `Distribution` the value `'t'`. By default, the property `Distribution` has a data structure with field `Name` equal to `'t'`, and field `DoF` equal to `NaN`. When you input the model to `estimate`, the degrees of freedom are estimated along with any other unknown model parameters.

For example, specify an MA(2) model with an iid Student's *t* innovation distribution, with unknown degrees of freedom:

Mdl = arima('MALags',1:2,'Distribution','t')

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

The output shows that `Distribution` is a data structure with two fields. Field `Name` has the value `'t'`, and field `DoF` has the value `NaN`.

If the degrees of freedom are known, and you want to set an equality constraint, assign a `struct` array to `Distribution` with fields `Name` and `DoF`. In this case, if the model is input to `estimate`, the degrees of freedom won't be estimated (the equality constraint is upheld).

Specify an MA(2) model with an iid Student's *t* innovation process with eight degrees of freedom:

Mdl = arima('MALags',1:2,'Distribution',struct('Name','t','DoF',8))

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

The output shows the specified innovation distribution.

After a model exists in the Workspace, you can modify its `Distribution` property using dot notation. You cannot modify the fields of the `Distribution` data structure directly. For example, `Mdl.Distribution.DoF = 8` is not a valid assignment. However, you can get the individual fields.

Start with an MA(2) model:

Mdl = arima(0,0,2);

To change the distribution of the innovation process in an existing model to a Student's *t* distribution with unknown degrees of freedom, type:

```
Mdl.Distribution = 't'
```

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

To change the distribution to a *t* distribution with known degrees of freedom, use a data structure:

Mdl.Distribution = struct('Name','t','DoF',8)

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

You can get the individual `Distribution` fields:

DistributionDoF = Mdl.Distribution.DoF

DistributionDoF = 8

To change the innovation distribution from a Student's *t* back to a Gaussian distribution, type:

```
Mdl.Distribution = 'Gaussian'
```

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

The `Name` field is updated to `'Gaussian'`, and there is no longer a `DoF` field.

[1] Wold, H. *A Study in the Analysis
of Stationary Time Series*. Uppsala, Sweden: Almqvist and
Wiksell, 1938.

- Specify Conditional Mean Models Using arima
- Modify Properties of Conditional Mean Model Objects
- Specify Conditional Mean and Variance Models

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