You can express all stationary stochastic processes in the general linear form [1]
$${y}_{t}=\mu +{\epsilon}_{t}+{\displaystyle \sum _{i=1}^{\infty}{\psi}_{i}{\epsilon}_{t-i}.}$$
The innovation process, $${\epsilon}_{t}$$, 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 $${\epsilon}_{t}={\sigma}_{t}{z}_{t}$$. Here, z_{t} is an independent and identically distributed (iid) series with mean 0 and variance 1, and $${\sigma}_{t}^{2}$$ is the variance of the innovation process at time t. Thus, $${\epsilon}_{t}$$ is an uncorrelated series with mean 0 and variance $${\sigma}_{t}^{2}$$.
arima
model objects have two properties for
storing information about the innovation process:
Variance
stores the form of $${\sigma}_{t}^{2}$$
Distribution
stores the parametric
form of the distribution of z_{t}
If $${\sigma}_{t}^{2}={\sigma}_{\epsilon}^{2}$$ for all times t, then $${\epsilon}_{t}$$ is an independent process with constant variance, $${\sigma}_{\epsilon}^{2}$$.
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, $${\sigma}_{t}^{2}$$, 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,
$${z}_{t}=\sqrt{\frac{\nu -2}{\nu}}{T}_{\nu},$$
where $${T}_{\nu}$$ 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
. 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.
arima
| egarch
| garch
| gjr
| struct