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In Econometrics Toolbox™, the general form of the innovation
process is
A conditional variance model
specifies the parametric form of the conditional variance process.
The innovation distribution corresponds to the distribution of the
independent and identically distributed (iid) process *z _{t}*.

For the distribution of *z _{t}*,
you can choose a standardized Gaussian or standardized Student's

where *T _{ν}* follows
a Student's

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 object
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`. `garch`, `egarch`,
and `gjr` create the required data structure.

To illustrate, consider specifying a GARCH(1,1) model:

Mdl = garch(1,1)

Mdl = GARCH(1,1) Conditional Variance Model: -------------------------------------- Distribution: Name = 'Gaussian' P: 1 Q: 1 Constant: NaN GARCH: {NaN} at Lags [1] ARCH: {NaN} at Lags [1]

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 a GJR(2,1) model with an iid Student's *t* innovation
distribution, with unknown degrees of freedom:

GJRMdl = gjr('GARCHLags',1:2,'ARCHLags',1,'LeverageLags',1,... 'Distribution','t')

GJRMdl = GJR(2,1) Conditional Variance Model: -------------------------------------- Distribution: Name = 't', DoF = NaN P: 2 Q: 1 Constant: NaN GARCH: {NaN NaN} at Lags [1 2] ARCH: {NaN} at Lags [1] Leverage: {NaN} at Lags [1]

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 a GARCH(1,1) model with an iid Student's *t* distribution
with eight degrees of freedom:

GARCHMdl = garch('GARCHLags',1,'ARCHLags',1,... 'Distribution',struct('Name','t','DoF',8))

GARCHMdl = GARCH(1,1) Conditional Variance Model: -------------------------------------- Distribution: Name = 't', DoF = 8 P: 1 Q: 1 Constant: NaN GARCH: {NaN} at Lags [1] ARCH: {NaN} at Lags [1]

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, `model.Distribution.DoF =
8` is not a valid assignment. However, you can get the individual
fields.

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

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

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

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

You can get the individual `Distribution` fields:

tDoF = Mdl.Distribution.DoF

tDoF = 8

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

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

Mdl = GARCH(1,1) Conditional Variance Model: -------------------------------------- Distribution: Name = 'Gaussian' P: 1 Q: 1 Constant: NaN GARCH: {NaN} at Lags [1] ARCH: {NaN} at Lags [1]

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

- Specify GARCH Models Using garch
- Specify EGARCH Models Using egarch
- Specify GJR Models Using gjr
- Modify Properties of Conditional Variance Model Objects

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