In Econometrics Toolbox™, the general form of the innovation process is $${\epsilon}_{t}={\sigma}_{t}{z}_{t}.$$ 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 t distribution with ν > 2 degrees of freedom. Note that if z_{t} follows a standardized t distribution, then
$${z}_{t}=\sqrt{\frac{\nu -2}{\nu}}{T}_{\nu},$$
where T_{ν} 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.
Tip It is good practice to assess the distributional properties of model residuals to determine if a Gaussian innovation distribution (the default distribution) is appropriate for your data. |
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
. 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, GARCHMdl.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 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.