Specify Conditional Mean Model Innovation Distribution

You can express all stationary stochastic processes in the general linear form [2]

`${y}_{t}=\mu +{\epsilon }_{t}+\sum _{i=1}^{\infty }{\psi }_{i}{\epsilon }_{t-i}.$`

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, zt 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 zt

Choices for the Variance Model

• 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.

Choices for the Innovation Distribution

The available distributions for zt 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.

Specify the Innovation Distribution

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 with properties: Description: "ARIMA(0,0,2) Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 0 D: 0 Q: 2 Constant: NaN AR: {} SAR: {} MA: {NaN NaN} at lags [1 2] SMA: {} Seasonality: 0 Beta: [1×0] 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 with properties: Description: "ARIMA(0,0,2) Model (t Distribution)" Distribution: Name = "t", DoF = NaN P: 0 D: 0 Q: 2 Constant: NaN AR: {} SAR: {} MA: {NaN NaN} at lags [1 2] SMA: {} Seasonality: 0 Beta: [1×0] 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 with properties: Description: "ARIMA(0,0,2) Model (t Distribution)" Distribution: Name = "t", DoF = 8 P: 0 D: 0 Q: 2 Constant: NaN AR: {} SAR: {} MA: {NaN NaN} at lags [1 2] SMA: {} Seasonality: 0 Beta: [1×0] Variance: NaN ```

The output shows the specified innovation distribution.

Modify the 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.

`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 with properties: Description: "ARIMA(0,0,2) Model (t Distribution)" Distribution: Name = "t", DoF = NaN P: 0 D: 0 Q: 2 Constant: NaN AR: {} SAR: {} MA: {NaN NaN} at lags [1 2] SMA: {} Seasonality: 0 Beta: [1×0] 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 with properties: Description: "ARIMA(0,0,2) Model (t Distribution)" Distribution: Name = "t", DoF = 8 P: 0 D: 0 Q: 2 Constant: NaN AR: {} SAR: {} MA: {NaN NaN} at lags [1 2] SMA: {} Seasonality: 0 Beta: [1×0] 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 with properties: Description: "ARIMA(0,0,2) Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 0 D: 0 Q: 2 Constant: NaN AR: {} SAR: {} MA: {NaN NaN} at lags [1 2] SMA: {} Seasonality: 0 Beta: [1×0] Variance: NaN ```

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

References

[1] Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.

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