# regARIMA

Create regression model with ARIMA time series errors

## Description

The `regARIMA` function returns a `regARIMA` object specifying the functional form and storing the parameter values of a regression model with ARIMA time series errors for a univariate response process yt.

`$\begin{array}{c}{y}_{t}=c+{X}_{t}\beta +{u}_{t}\\ a\left(L\right)A\left(L\right){\left(1-L\right)}^{D}\left(1-{L}^{s}\right){u}_{t}=b\left(L\right)B\left(L\right){\epsilon }_{t},\end{array}$`

Because they completely specify the model structure, the key components of a `regARIMA` object are the:

• Regression model coefficients c and β

• Polynomial degrees of the ARIMA disturbances ut, for example, the AR polynomial degree p and the degree of integration D

Given only polynomial degrees, the regression model contains only a constant. All parameters, such as the model constant, and error model coefficients and innovation-distribution parameters, are unknown and estimable unless you specify their values. `regARIMA` determines the number of coefficients in the regression model by the number of variables in the supplied predictor data or by other specifications.

To estimate a model containing unknown parameter values, pass the model and data to the `estimate` object function. To work with an estimated or fully specified `regARIMA` object, pass it to an object function.

Alternatively, you can:

## Creation

### Syntax

``Mdl = regARIMA``
``Mdl = regARIMA(p,D,q)``
``Mdl = regARIMA(Name=Value)``

### Description

````Mdl = regARIMA` creates a regression model containing degree 0 ARIMA disturbances. The regression model contains an intercept; the software determines the number of regression coefficients when you fit the model to data by using `estimate`. The innovations are iid Gaussian random variables with a mean of 0 and unknown variance.```

example

````Mdl = regARIMA(p,D,q)` creates a regression model with ARIMA(`p`,`D`,`q`) disturbances. The disturbance model contains nonseasonal AR polynomial lags from 1 through `p`, a degree `D` nonseasonal integration polynomial, and nonseasonal MA polynomial lags from 1 through `q`. The regression model contains an intercept; the software determines the number of regression coefficients when you fit the model to data by using `estimate`. The innovations are iid Gaussian random variables with a mean of 0 and unknown variance.This shorthand syntax provides an easy way to create a model template in which you specify the degrees of the nonseasonal polynomials explicitly. The model template is suited for unrestricted parameter estimation. After you create a model, you can alter property values using dot notation.```

example

````Mdl = regARIMA(Name=Value)` sets properties and polynomial lags using name-value arguments. For example, ```regARIMA(ARLags=[1 4],AR={0.5 –0.1})``` creates a regression model containing an unknown model intercept and innovations variance, and AR(4) disturbances, where the lag 1 nonseasonal AR coefficient is `–0.5` and the lag 4 nonseasonal AR coefficient is `0.1`.This longhand syntax allows you to create more flexible models. For example, you can create a regression model with seasonal errors by using only longhand syntax. `regARIMA` infers all disturbance model polynomial degrees from the properties that you set. Therefore, property values that correspond to polynomial degrees must be consistent with each other.```

### Input Arguments

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The shorthand syntax provides an easy way for you to create model templates of regression models with nonseasonal ARIMA errors. Model templates are suitable for unrestricted parameter estimation. For example, to create a regression model with ARMA(2,1) errors containing an unknown model intercept and innovations variance, enter:

`Mdl = regARIMA(2,0,1);`
To impose equality constraints on parameter values during estimation, or include seasonal components, set the appropriate property values using dot notation.

Nonseasonal autoregressive polynomial degree for the error model, specified as a nonnegative integer.

Data Types: `double`

Degree of nonseasonal integration (the degree of the nonseasonal differencing polynomial) for the error model, specified as a nonnegative integer. The `D` input argument sets the property D.

Data Types: `double`

Nonseasonal moving average polynomial degree for the error model, specified as a nonnegative integer.

Data Types: `double`

Name-Value Arguments

Specify optional pairs of arguments as `Name1=Value1,...,NameN=ValueN`, where `Name` is the argument name and `Value` is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose `Name` in quotes.

The longhand syntax enables you to create seasonal error models or models in which some or all coefficients are known. During estimation, `estimate` imposes equality constraints on any known parameters.

Example: `regARIMA(ARLags=[1 4],AR={0.5 –0.1})` creates a regression model containing an unknown model intercept and innovations variance, and AR(4) disturbances, where the lag 1 nonseasonal AR coefficient is `–0.5` and the lag 4 nonseasonal AR coefficient is `0.1`, symbolically, $1-0.5{L}^{1}+0.1{L}^{4}$.

Lags associated with the nonseasonal AR polynomial coefficients for the error model ut, specified as a numeric vector of unique positive integers. The maximum lag is p.

`AR{j}` is the coefficient of lag `ARLags(j)`, where `AR` is the value of the property AR.

Example: `ARLags=4` specifies the nonseasonal AR polynomial $1-{\varphi }_{4}{L}^{4}$.

Example: `ARLags=1:4` specifies the nonseasonal AR polynomial $1-{\varphi }_{1}{L}^{1}-{\varphi }_{2}{L}^{2}-{\varphi }_{3}{L}^{3}-{\varphi }_{4}{L}^{4}$.

Example: `ARLags=[1 4]` specifies the nonseasonal AR polynomial $1-{\varphi }_{1}{L}^{1}-{\varphi }_{4}{L}^{4}.$

Data Types: `double`

Lags associated with the nonseasonal MA polynomial coefficients for the error model ut, specified as a numeric vector of unique positive integers. The maximum lag is q.

`MA{j}` is the coefficient of lag `MALags(j)`, where `MA` is the value of the property MA.

Example: `MALags=3` specifies the nonseasonal MA polynomial $1+{\theta }_{3}{L}^{3}$.

Example: `MALags=1:3` specifies the nonseasonal MA polynomial $1+{\theta }_{1}{L}^{1}+{\theta }_{2}{L}^{2}+{\theta }_{3}{L}^{3}.$

Example: `MALags=[1 3]` specifies the nonseasonal MA polynomial $1+{\theta }_{1}{L}^{1}+{\theta }_{3}{L}^{3}$.

Data Types: `double`

Lags associated with the seasonal AR polynomial coefficients for the error model ut, specified as a numeric vector of unique positive integers. The maximum lag is ps.

`SAR{j}` is the coefficient of lag `SARLags(j)`, where `SAR` is the value of the property SAR.

Specify `SARLags` as the periodicity of the observed data, not as multiples of the Seasonality property. This convention does not conform to standard Box and Jenkins  notation, but it is more flexible for incorporating multiplicative seasonality.

Example: `SARLags=[4 8]` specifies the seasonal AR polynomial $1-{\Phi }_{4}{L}^{4}-{\Phi }_{8}{L}^{8}.$

Data Types: `double`

Lags associated with the seasonal MA polynomial coefficients for the error model ut, specified as a numeric vector of unique positive integers. The maximum lag is qs.

`SMA{j}` is the coefficient of lag `SMALags(j)`, where `SMA` is the value of the property SMA.

Specify `SMALags` as the periodicity of the observed data, not as multiples of the Seasonality property. This convention does not conform to standard Box and Jenkins  notation, but it is more flexible for incorporating multiplicative seasonality.

Example: `SMALags=4` specifies the seasonal MA polynomial $1+{\Theta }_{4}{L}^{4}.$

Data Types: `double`

Note

Polynomial degrees are not estimable. If you do not specify a polynomial degree, or `regARIMA` cannot infer it from other specifications, `regARIMA` does not include the polynomial in the model.

## Properties

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You can set writable property values when you create the model object by using name-value argument syntax, or after you create the model object by using dot notation. For example, to create the fully specified regression model with ARMA(2,1) disturbances

`$\begin{array}{l}{y}_{t}=1+3{x}_{1}+5{x}_{2}+{u}_{t}\\ {u}_{t}=0.3{u}_{t-1}-0.15{u}_{t-2}+{\epsilon }_{t}+0.2{\epsilon }_{t-1},\end{array}$`

enter:

```Mdl = regARIMA(Intercept=1,Beta=[3; 5],AR={0.3 -0.15},MA=0.2); Mdl.Variance = 1;```

Note

• `NaN`-valued properties indicate estimable parameters. Numeric properties indicate equality constraints on parameters during model estimation. Coefficient vectors can contain both numeric and `NaN`-valued elements.

• You can specify polynomial coefficients as vectors in any orientation, but `regARIMA` stores them as row vectors.

## Regression Model Properties

Regression model intercept c, specified as a numeric scalar.

Example: `Intercept=1`

Data Types: `double`

Regression component coefficients β associated with predictor variables xt, specified as a numeric vector.

The default indicates one of the following conditions:

• `estimate` infers the size of `Beta` from the number of columns of the specified predictor data `X`. Therefore, if you plan to fit all regression coefficients to data, you do not need to specify `Beta`.

• The model does not include regression coefficients.

Example: `Beta=[0.5 NaN 3]` specifies three regression coefficients. During estimation, `estimate` fixes β1 to 5 and β3 to 3, and it fits β2 to the data associated with the second predictor variable.

Data Types: `double`

## Error Model Properties

Compound AR polynomial degree of the error model, specified as a nonnegative integer.

`P` does not necessarily conform to standard Box and Jenkins notation  because `P` captures the degrees of the nonseasonal and seasonal AR polynomials (properties `AR` and `SAR`, respectively), nonseasonal integration (property `D`), and seasonality (property `Seasonality`). Explicitly, `P` = p + D + ps + s. `P` conforms to Box and Jenkins notation for models without integration or a seasonal AR component (`D` = `0` and `SAR` = `{}`).

`P` specifies the number of lagged observations required to initialize the AR components of the model.

Data Types: `double`

Compound MA polynomial degree of the error model, specified as a nonnegative integer.

`Q` does not necessarily conform to standard Box and Jenkins notation  because `Q` captures the degrees of the nonseasonal and seasonal MA polynomials (properties `MA` and `SMA`, respectively). Explicitly, `Q` = q + qs. `Q` conforms to Box and Jenkins notation for models without a seasonal MA component (`SMA` = `{}`).

`Q` specifies the number of lagged innovations required to initialize the MA components of the model.

Data Types: `double`

Nonseasonal AR polynomial coefficients ϕ for the error model ut, specified as a cell vector. Cells contain numeric scalars or `NaN` values. A fully specified nonseasonal AR polynomial must be stable.

Coefficient signs correspond to the model expressed in difference-equation notation. For example, for the nonseasonal AR polynomial $\varphi \left(L\right)=1-0.5L+0.1{L}^{2},$ specify `AR={0.5 –0.1}`.

If you do not set the `ARLags` name-value argument, `AR{j}` is the coefficient of lag `j`, `j` = 1,…,p, where p = `numel(AR)`.

Otherwise, if `ARLags` = `arlags`, with p = `max(arlags)`, the following conditions apply:

• The lengths of `AR` and `arlags` must be equal.

• `AR{j}` is the coefficient of lag `arlags(j)`, for each `j`.

• `regARIMA` stores `AR` as a length p cell vector. All cells that do not correspond to lags in `arlags` contain `0`.

The default value of `AR` depends on other specifications:

• If you use the shorthand syntax to specify `p` > 0, `AR` is a length `p` cell vector, where each cell contains a `NaN` value.

• If you specify `ARLags`, `AR` is a length p cell vector. `AR{j}` = `NaN` for each lag `arlags(j)`. All other cells contain `0`.

• Otherwise, `AR` is an empty cell vector `{}`, meaning the model does not contain a nonseasonal AR polynomial.

The coefficients in `AR` correspond to coefficients in an underlying `LagOp` lag operator polynomial, and they are subject to a near-zero tolerance exclusion test. If a coefficient is `1e–12` or below, `regARIMA` excludes that coefficient and its corresponding lag in `ARLags` from the model.

Example: `AR={0.8}` sets the only AR lag coefficient associated with lag `ARLags(1)` to `0.8`.

Example: `regARIMA(AR={0.2 0 0.1})` sets the error model, in difference-equation form, to ${u}_{t}=0.2{u}_{t-1}+0.1{u}_{t-3}+{\epsilon }_{t}$.

Example: `regARIMA(AR={NaN –0.1},ARLags=[4 8])` sets the AR lag polynomial to $1-{\varphi }_{4}{L}^{4}+0.1{L}^{8}$, where ϕ4 is unknown and estimable.

Data Types: `cell`

Nonseasonal MA polynomial coefficients θ for the error model ut, specified as a cell vector. Cells contain numeric scalars or `NaN` values. A fully specified nonseasonal MA polynomial must be invertible.

If you do not set the `MALags` name-value pair argument, `MA{j}` is the coefficient of lag `j`, `j` = 1,…,q, where q = `numel(MA)`.

Otherwise, if `MALags` = `malags`, with q = `max(MALags)`, the following conditions apply:

• The lengths of `MA` and `malags` must be equal.

• `MA{j}` is the coefficient of lag `malags(j)`, for each `j`.

• `regARIMA` stores `MA` as a length q cell vector. All cells that do not correspond to lags in `malags` contain `0`.

The default value of `MA` depends on other specifications:

• If you use the shorthand syntax to specify `q` > 0, `MA` is a length `q` cell vector, where each cell contains a `NaN` value.

• If you specify `MALags`, `MA` is a length q cell vector. `MA{j}` = `NaN` for each lag `malags(j)`. All other cells contain `0`.

• Otherwise, `MA` is an empty cell vector `{}`, meaning the error model does not contain a nonseasonal MA polynomial.

The coefficients in `MA` correspond to coefficients in an underlying `LagOp` lag operator polynomial, and they are subject to a near-zero tolerance exclusion test. If a coefficient is `1e–12` or below, `regARIMA` excludes that coefficient and its corresponding lag in `MALags` from the model.

Example: `MA=0.8` sets the only MA lag coefficient associated with lag `MALags(1)` to `0.8`.

Example: `regARIMA(MA={0.2 0.1})` sets the error model to ${u}_{t}={\epsilon }_{t}+0.2{\epsilon }_{t-1}+0.1{\epsilon }_{t-2}.$

Example: `regARIMA(MA={NaN –0.1},MALags=[4 8])` sets the MA lag polynomial to $1+{\theta }_{4}{L}^{4}-0.1{L}^{8}$, where θ4 is unknown and estimable.

Data Types: `cell`

Seasonal AR polynomial coefficients Φ for the error model ut, specified as a cell vector. Cells contain numeric scalars or `NaN` values. A fully specified seasonal AR polynomial must be stable.

Coefficient signs correspond to the model expressed in difference-equation notation. For example, for the seasonal AR polynomial $\Phi \left(L\right)=1-0.5{L}^{4}+0.1{L}^{8},$ specify `SAR={0.5 –0.1}`.

If you do not set the `SARLags` name-value argument, `SAR{j}` is the coefficient of lag `j`, `j` = 1,…,ps, where ps = `numel(SAR)`.

Otherwise, if `SARLags` = `sarlags`, with ps = `max(sarlags)`, the following conditions apply:

• The lengths of `SAR` and `sarlags` must be equal.

• `SAR{j}` is the coefficient of lag `sarlags(j)`, for each `j`.

• `regARIMA` stores `SAR` as a length ps cell vector. All cells that do not correspond to lags in `sarlags` contain `0`.

The default value of `SAR` depends on the value of `SARLags`:

• If you specify `SARLags`, `SAR` is a length ps cell vector. `SAR{j}` = `NaN` for each lag `SARLags(j)`. All other cells contain `0`.

• Otherwise, `SAR` is an empty cell vector `{}`, meaning the error model does not contain a seasonal AR polynomial.

The coefficients in `SAR` correspond to coefficients in an underlying `LagOp` lag operator polynomial, and they are subject to a near-zero tolerance exclusion test. If a coefficient is `1e–12` or below, `regARIMA` excludes that coefficient and its corresponding lag in `SARLags` from the model.

Example: `SAR=0.8` sets the only SAR lag coefficient associated with lag `SARLags(1)` to `0.8`.

Example: `regARIMA(SAR={0.2 0.1},Seasonality=4)` sets the error model to $\left(1-0.2{L}^{1}-0.1{L}^{2}\right)\left(1-{L}^{4}\right){u}_{t}={\epsilon }_{t}$.

Example: `regARIMA(SAR={NaN –0.1},SARLags=[4 8],Seasonality=4)` sets the SAR lag polynomial to $\left(1-{\Theta }_{4}{L}^{4}-0.1{L}^{8}\right)\left(1-{L}^{4}\right)$, where Φ4 is unknown and estimable.

Data Types: `cell`

Seasonal MA polynomial coefficients for the error model, specified as a cell vector. Cells contain numeric scalars or `NaN` values. A fully specified seasonal MA polynomial must be invertible.

If you do not set the `SMALags` name-value argument, `SMA{j}` is the coefficient of lag `j`, `j` = 1,…,qs, where qs = `numel(SMA)`.

Otherwise, if `SMALags` = `smalags`, with qs = `max(smalags)`, the following conditions apply:

• The lengths of `SMA` and `SMALags` must be equal.

• `SMA{j}` is the coefficient of lag `smalags(j)`, for each `j`.

• `regARIMA` stores `SMA` as a length qs cell vector. All cells that do not correspond to lags in `smalags` contain `0`.

The default value of `SMA` depends on other specifications:

• If you specify `SMALags`, `MA` is a length q cell vector. `MA{j}` = `NaN` for each lag `MALags(j)`. All other cells contain `0`.

• Otherwise, `SMA` is an empty cell vector `{}`, meaning the error model does not contain a seasonal MA polynomial.

The coefficients in `SMA` correspond to coefficients in an underlying `LagOp` lag operator polynomial, and they are subject to a near-zero tolerance exclusion test. If a coefficient is `1e–12` or below, `regARIMA` excludes that coefficient and its corresponding lag in `SMALags` from the model.

Example: `SMA=0.8` sets the only SMA lag coefficient associated with lag `SMALags(1)` to `0.8`.

Example: `regARIMA(SMA{0.2 0.1},Seasonality=4)` specifies the error model $\left(1-{L}^{4}\right){u}_{t}=\left(1+0.2L+0.1{L}^{2}\right){\epsilon }_{t}.$

Example: `regARIMA(SMALags=[1 4],SMA={0.2 0.1},Seasonality = 4)` specifies the error model $\left(1-{L}^{4}\right){u}_{t}=\left(1+0.2L+0.1{L}^{4}\right){\epsilon }_{t}.$

Data Types: `cell`

Degree of nonseasonal integration, or the degree of the nonseasonal differencing polynomial, for the error model specified as a nonnegative integer.

If you use shorthand syntax to create `Mdl`, the input `d` sets `D`.

Example: `D=1`

Example: `regARIMA(0,1,2)` sets `D` to `1`.

Data Types: `double`

Degree of the seasonal differencing polynomial s for the error model, specified as a nonnegative integer.

Example: `Seasonality=12` specifies monthly periodicity.

Data Types: `double`

Variance σ2 of the model innovations process εt, specified as a positive scalar.

`NaN` specifies an unknown and estimable variance, which `estimate` fits to data.

Example: `Variance=1`

Data Types: `double`

## Other Properties

Model description, specified as a string scalar or character vector. `regARIMA` stores the value as a string scalar. The default value describes the parametric form of the model, for example, ```"Regression with ARMA(2,1) Error Model (Gaussian Distribution)"```.

Example: `"Model 1"`

Data Types: `string` | `char`

Conditional probability distribution of the innovation process εt, specified as a string or structure array. `regARIMA` stores the value as a structure array.

DistributionStringStructure Array
Gaussian`"Gaussian"``struct('Name',"Gaussian")`
Student’s t`"t"``struct('Name',"t",'DoF',DoF)`

The `'DoF'` field specifies the t distribution degrees of freedom parameter.

• `DoF` > 2 or `DoF` = `NaN`.

• `DoF` is estimable.

• If you specify `"t"`, `DoF` is `NaN` by default. You can change its value by using dot notation after you create the model. For example, `Mdl.Distribution.DoF = 3`.

• If you supply a structure array to specify the Student's t distribution, then you must specify both the `'Name'` and the `'DoF'` fields.

Example: `Distribution=struct('Name',"t",'DoF',10)`

Since R2023b

Response series name, specified as a string scalar or character vector. `regARIMA` stores the value as a string scalar.

Example: `"StockReturn"`

Data Types: `string` | `char`

## Object Functions

 `estimate` Fit univariate regression model with ARIMA errors to data `infer` Infer residuals of univariate regression model with ARIMA time series errors `summarize` Display estimation results of regression model with ARIMA errors `simulate` Monte Carlo simulation of univariate regression model with ARIMA time series errors `filter` Filter disturbances through regression model with ARIMA errors `impulse` Generate regression model with ARIMA errors impulse response function `forecast` Forecast responses of univariate regression model with ARIMA time series errors `arima` Convert regression model with ARIMA errors to ARIMAX model

## Examples

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Specify the following regression model with ARIMA(2,1,3) errors:

`$\begin{array}{c}{y}_{t}={u}_{t}\\ \left(1-{\varphi }_{1}L-{\varphi }_{2}{L}^{2}\right)\left(1-L\right){u}_{t}=\left(1+{\theta }_{1}L+{\theta }_{2}{L}^{2}+{\theta }_{3}{L}^{3}\right){\epsilon }_{t}.\end{array}$`

`Mdl = regARIMA(2,1,3)`
```Mdl = regARIMA with properties: Description: "ARIMA(2,1,3) Error Model (Gaussian Distribution)" SeriesName: "Y" Distribution: Name = "Gaussian" Intercept: NaN Beta: [1×0] P: 3 D: 1 Q: 3 AR: {NaN NaN} at lags [1 2] SAR: {} MA: {NaN NaN NaN} at lags [1 2 3] SMA: {} Variance: NaN ```

The output displays the values of the properties `P`, `D`, and `Q` of `Mdl`. The corresponding autoregressive and moving average coefficients (contained in `AR` and `MA`) are cell arrays containing the correct number of `NaN` values. Because `P` = `p` + `D` = 3, you need three presample observations to initialize the model for estimation.

Define the regression model with ARIMA errors:

`$\begin{array}{l}\begin{array}{c}{y}_{t}=2+{X}_{t}\left[\begin{array}{c}1.5\\ 0.2\end{array}\right]+{u}_{t}\\ \left(1-0.2L-0.3{L}^{2}\right){u}_{t}=\left(1+0.1L\right){\epsilon }_{t},\end{array}\end{array}$`

where ${\epsilon }_{t}$ is Gaussian with variance 0.5.

```Mdl = regARIMA(Intercept=2,AR={0.2 0.3},MA={0.1}, ... Variance=0.5,Beta=[1.5 0.2])```
```Mdl = regARIMA with properties: Description: "Regression with ARMA(2,1) Error Model (Gaussian Distribution)" SeriesName: "Y" Distribution: Name = "Gaussian" Intercept: 2 Beta: [1.5 0.2] P: 2 Q: 1 AR: {0.2 0.3} at lags [1 2] SAR: {} MA: {0.1} at lag  SMA: {} Variance: 0.5 ```

`Mdl` is fully specified to, for example, simulate a series of responses given the predictor data matrix, ${X}_{t}$.

Modify the model to estimate the regression coefficient, the AR terms, and the variance of the innovations.

```Mdl.Beta = [NaN NaN]; Mdl.AR = {NaN NaN}; Mdl.Variance = NaN;```

Change the innovations distribution to a $t$ distribution with 15 degrees of freedom.

`Mdl.Distribution = struct("Name","t","DoF",15)`
```Mdl = regARIMA with properties: Description: "Regression with ARMA(2,1) Error Model (t Distribution)" SeriesName: "Y" Distribution: Name = "t", DoF = 15 Intercept: 2 Beta: [NaN NaN] P: 2 Q: 1 AR: {NaN NaN} at lags [1 2] SAR: {} MA: {0.1} at lag  SMA: {} Variance: NaN ```

Specify the following model:

`$\begin{array}{l}\begin{array}{c}{y}_{t}=1+6{X}_{t}+{u}_{t}\\ \left(1-0.2L\right)\left(1-L\right)\left(1-0.5{L}^{4}-0.2{L}^{8}\right)\left(1-{L}^{4}\right){u}_{t}=\left(1+0.1L\right)\left(1+0.05{L}^{4}+0.01{L}^{8}\right){\epsilon }_{t},\end{array}\end{array}$`

where ${\epsilon }_{t}$ is Gaussian with variance 1.

```Mdl = regARIMA(Intercept=1,Beta=6,AR=0.2,MA=0.1,D=1, ... SAR={0.5,0.2},SARLags=[4, 8],SMA={0.05,0.01},SMALags=[4 8], ... Seasonality=4,Variance=1)```
```Mdl = regARIMA with properties: Description: "Regression with ARIMA(1,1,1) Error Model Seasonally Integrated with Seasonal AR(8) and MA(8) (Gaussian Distribution)" SeriesName: "Y" Distribution: Name = "Gaussian" Intercept: 1 Beta:  P: 14 D: 1 Q: 9 AR: {0.2} at lag  SAR: {0.5 0.2} at lags [4 8] MA: {0.1} at lag  SMA: {0.05 0.01} at lags [4 8] Seasonality: 4 Variance: 1 ```

If you do not specify `SARLags` or `SMALags`, then the coefficients in `SAR` and `SMA` correspond to lags 1 and 2 by default.

```Mdl = regARIMA(Intercept=1,Beta=6,AR=0.2,MA=0.1,D=1, ... SAR={0.5,0.2},SARLags=[4, 8], ... Seasonality=4,Variance=1)```
```Mdl = regARIMA with properties: Description: "Regression with ARIMA(1,1,1) Error Model Seasonally Integrated with Seasonal AR(8) (Gaussian Distribution)" SeriesName: "Y" Distribution: Name = "Gaussian" Intercept: 1 Beta:  P: 14 D: 1 Q: 1 AR: {0.2} at lag  SAR: {0.5 0.2} at lags [4 8] MA: {0.1} at lag  SMA: {} Seasonality: 4 Variance: 1 ```