## Create Multiplicative ARIMA Models

These examples show how to create various multiplicative seasonal
autoregressive integrated moving average (ARIMA) models by using the `arima`

function.

### Seasonal ARIMA Model with No Constant Term

This example shows how to use `arima`

to specify a multiplicative seasonal ARIMA model (for monthly data) with no constant term.

Specify a multiplicative seasonal ARIMA model with no constant term,

$$(1-{\varphi}_{1}L)(1-{\Phi}_{12}{L}^{12})(1-L{)}^{1}(1-{L}^{12}){y}_{t}=(1+{\theta}_{1}L)(1+{\Theta}_{12}{L}^{12}){\epsilon}_{t},$$

where the innovation distribution is Gaussian with constant variance. Here, $$(1-L{)}^{1}$$ is the first degree nonseasonal differencing operator and $$(1-{L}^{12})$$ is the first degree seasonal differencing operator with periodicity 12.

Mdl = arima('Constant',0,'ARLags',1,'SARLags',12,'D',1,... 'Seasonality',12,'MALags',1,'SMALags',12)

Mdl = arima with properties: Description: "ARIMA(1,1,1) Model Seasonally Integrated with Seasonal AR(12) and MA(12) (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 26 D: 1 Q: 13 Constant: 0 AR: {NaN} at lag [1] SAR: {NaN} at lag [12] MA: {NaN} at lag [1] SMA: {NaN} at lag [12] Seasonality: 12 Beta: [1×0] Variance: NaN

The name-value pair argument `ARLags`

specifies the lag corresponding to the nonseasonal AR coefficient, $${\varphi}_{1}$$. `SARLags`

specifies the lag corresponding to the seasonal AR coefficient, here at lag 12. The nonseasonal and seasonal MA coefficients are specified similarly. `D`

specifies the degree of nonseasonal integration. `Seasonality`

specifies the periodicity of the time series, for example `Seasonality`

= 12 indicates monthly data. Since `Seasonality`

is greater than 0, the degree of seasonal integration $${D}_{s}$$ is one.

Whenever you include seasonal AR or MA polynomials (signaled by specifying `SAR`

or `SMA`

) in the model specification, `arima`

incorporates them multiplicatively. `arima`

sets the property `P`

equal to *p* + *D* + $${p}_{s}$$ + *s* (here, 1 + 1 + 12 + 12 = 26). Similarly, `arima`

sets the property `Q`

equal to *q* + $${q}_{s}$$ (here, 1 + 12 = 13).

Display the value of `SAR`

:

Mdl.SAR

`ans=`*1×12 cell array*
{[0]} {[0]} {[0]} {[0]} {[0]} {[0]} {[0]} {[0]} {[0]} {[0]} {[0]} {[NaN]}

The `SAR`

cell array returns 12 elements, as specified by `SARLags`

. `arima`

sets the coefficients at interim lags equal to zero to maintain consistency with MATLAB® cell array indexing. Therefore, the only nonzero coefficient corresponds to lag 12.

All of the other properties of `Mdl`

are `NaN`

-valued, indicating that the corresponding model parameters are estimable, or you can specify their value by using dot notation.

### Seasonal ARIMA Model with Known Parameter Values

This example shows how to specify a multiplicative seasonal ARIMA model (for quarterly data) with known parameter values. You can use such a fully specified model as an input to `simulate`

or `forecast`

.

Specify the multiplicative seasonal ARIMA model

$$(1-.5L)(1+0.7{L}^{4})(1-L{)}^{1}(1-{L}^{4}){y}_{t}=(1+.3L)(1-.2{L}^{4}){\epsilon}_{t},$$

where the innovation distribution is Gaussian with constant variance 0.15. Here, $$(1-L{)}^{1}$$ is the nonseasonal differencing operator and $$(1-{L}^{4})$$ is the first degree seasonal differencing operator with periodicity 4.

Mdl = arima('Constant',0,'AR',0.5,'D',1,'MA',0.3,... 'Seasonality',4,'SAR',-0.7,'SARLags',4,... 'SMA',-0.2,'SMALags',4,'Variance',0.15)

Mdl = arima with properties: Description: "ARIMA(1,1,1) Model Seasonally Integrated with Seasonal AR(4) and MA(4) (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 10 D: 1 Q: 5 Constant: 0 AR: {0.5} at lag [1] SAR: {-0.7} at lag [4] MA: {0.3} at lag [1] SMA: {-0.2} at lag [4] Seasonality: 4 Beta: [1×0] Variance: 0.15

The output specifies the nonseasonal and seasonal AR coefficients with opposite signs compared to the lag polynomials. This is consistent with the difference equation form of the model. The output specifies the lags of the seasonal AR and MA coefficients using `SARLags`

and `SMALags`

, respectively. `D`

specifies the degree of nonseasonal integration. `Seasonality`

= 4 specifies quarterly data with one degree of seasonal integration.

All parameter values are specified, that is, no object property is `NaN`

-valued.

### Specify Multiplicative ARIMA Model Using Econometric Modeler App

In the **Econometric
Modeler** app, you can specify the lag structure, presence of a constant,
and innovation distribution of a
SARIMA(*p*,*D*,*q*)×(*p _{s}*,

*D*,

_{s}*q*)

_{s}*model by following these steps. All specified coefficients are unknown but estimable parameters.*

_{s}At the command line, open the

**Econometric Modeler**app.econometricModeler

Alternatively, open the app from the apps gallery (see

**Econometric Modeler**).In the

**Time Series**pane, select the response time series to which the model will be fit.On the

**Econometric Modeler**tab, in the**Models**section, click the arrow to display the models gallery.In the

**ARMA/ARIMA Models**section of the gallery, click**SARIMA**. To create SARIMAX models, see Create ARIMA Models That Include Exogenous Covariates.The

**SARIMA Model Parameters**dialog box appears.Specify the lag structure. Use the

**Lag Order**tab to specify a SARIMA(*p*,*D*,*q*)×(*p*,_{s}*D*,_{s}*q*)_{s}model that includes:_{s}All consecutive lags from 1 through their respective orders, in the nonseasonal polynomials

Lags that are all consecutive multiples of the period (

*s*), in the seasonal polynomialsAn

*s*-degree seasonal integration polynomial

Use the

**Lag Vector**tab for the flexibility to specify particular lags for all polynomials. For more details, see Specifying Univariate Lag Operator Polynomials Interactively. Regardless of the tab you use, you can verify the model form by inspecting the equation in the**Model Equation**section.

For example, consider this SARIMA(2,1,1)×(2,1,1)_{12} model.

$$\left(1-{\varphi}_{1}L-{\varphi}_{2}{L}^{2}\right)\left(1-{\Phi}_{12}{L}^{12}-{\Phi}_{24}{L}^{24}\right)(1-L)(1-{L}^{12}){y}_{t}=c+\left(1+{\theta}_{1}L\right)\left(1+{\Theta}_{12}{L}^{12}\right){\epsilon}_{t},$$

where *ε _{t}* is a series of
IID Gaussian innovations.

The model includes all consecutive AR and MA lags from 1 through their respective
orders. Also, the lags of the SAR and SMA polynomials are consecutive multiples of
the period from 12 through their respective specified order times 12. Therefore, use
the **Lag Order** tab to specify the model.

In the

**Nonseasonal**section:Set

**Degree of Integration**to`1`

.Set

**Autoregressive Order**to`2`

.Set

**Moving Average Order**to`1`

.

In the

**Seasonal**section:Set

**Period**to`12`

.Set

**Autoregressive Order**to`2`

. This input specifies the inclusion of SAR lags 12 and 24 (that is, the first and second multiples of the value of**Period**).Set

**Moving Average Order**to`1`

. This input specifies the inclusion of SMA lag 12 (that is, the first multiple of the value of**Period**).Select the

**Include Seasonal Difference**check box.

Verify that the equation in the

**Model Equation**section matches your model.

To exclude a constant from the model and to specify that the innovations are Gaussian, follow the previous steps, and clear the

**Include Constant Term**check box.To specify

*t*-distributed innovations, follow the previous steps, and click the**Innovation Distribution**button, then select`t`

.

For another example, consider this
SARIMA(12,1,1)×(2,1,1)_{12} model.

$$\left(1-{\varphi}_{1}L-{\varphi}_{12}{L}^{12}\right)\left(1-{\Phi}_{24}{L}^{24}\right)(1-L)(1-{L}^{12}){y}_{t}=c+\left(1+{\theta}_{1}L\right)\left(1+{\Theta}_{12}{L}^{12}\right){\epsilon}_{t}.$$

The model does not include consecutive AR lags, and the lags of
the SAR polynomial are not consecutive multiples of the period. Therefore, use the
**Lag Vector** tab to specify this model:

In the

**SARIMA Model Parameters**dialog box, click the**Lag Vector**tab.In the

**Nonseasonal**section:Set

**Degree of Integration**to`1`

.Set

**Autoregressive Lags**to`1 12`

.Set

**Moving Average Lags**to`1`

.

In the

**Seasonal**section:Set

**Seasonality**to`12`

. The app includes a 12-degree seasonal integration polynomial.Set

**Autoregressive Lags**to`24`

. This input specifies the inclusion of SAR lag 24. The input is independent of the value in the**Seasonality**box.Set

**Moving Average Lags**to`12`

. This input specifies the inclusion of SMA lag 12. The input is independent of the value in the**Seasonality**box.

Verify that the equation in the

**Model Equation**section matches your model.

After you specify a model, click **Estimate** to estimate all
unknown parameters in the model.

### What Are Multiplicative ARIMA Models?

Many time series collected periodically (e.g., quarterly or monthly) exhibit a seasonal trend, meaning there is a relationship between observations made during the same period in successive years. In addition to this seasonal relationship, there can also be a relationship between observations made during successive periods. The multiplicative ARIMA model is an extension of the ARIMA model that addresses seasonality and potential seasonal unit roots [1].

In lag operator polynomial notation, $${L}^{i}{y}_{t}={y}_{t-i}$$. For a series with periodicity *s*, the
multiplicative
ARIMA(*p*,*D*,*q*)×(*p _{s}*,

*D*,

_{s}*q*)

_{s}*is given by*

_{s}$$\varphi (L)\Phi (L){(1-L)}^{D}{(1-{L}^{s})}^{{D}_{s}}{y}_{t}=c+\theta (L)\Theta (L){\epsilon}_{t}.$$ | (1) |

Here, the stable, degree *p* AR operator polynomial $$\varphi (L)=(1-{\varphi}_{1}L-\dots -{\varphi}_{p}{L}^{p})$$, and $$\Phi (L)$$ is a stable, degree *p _{s}* AR
operator of the same form. Similarly, the invertible, degree

*q*MA operator polynomial $${\theta}_{q}(L)=(1+{\theta}_{1}L+\dots +{\theta}_{q}{L}^{q})$$, and $$\Theta (L)$$ is an invertible, degree

*q*MA operator of the same form.

_{s}When you specify a multiplicative ARIMA model using `arima`

,

Set the nonseasonal and seasonal AR coefficients with the opposite signs from their respective AR operator polynomials. That is, specify the coefficients as they would appear on the right side of Equation 1.

Set the lags associated with the seasonal polynomials in the periodicity of the observed data (e.g., 4, 8,... for quarterly data, or 12, 24,... for monthly data), and not as multiples of the seasonality (e.g., 1, 2,...). This convention does not conform to standard Box and Jenkins notation, but is a more flexible approach for incorporating multiplicative seasonality.

The nonseasonal differencing operator, $${(1-L)}^{D}$$ accounts for nonstationarity in observations made in successive
periods. The seasonal differencing operator, $${(1-{L}^{s})}^{{D}_{s}}$$, accounts for nonstationarity in observations made in the same period
in successive years. Econometrics Toolbox™ supports only the degrees of seasonal integration
*D _{s}* = 0 or 1. When you
specify

*s*≥ 0, Econometrics Toolbox sets

*D*= 1.

_{s}*D*= 0 otherwise.

_{s}## References

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

## See Also

### Apps

### Objects

### Functions

## Related Topics

- Specifying Univariate Lag Operator Polynomials Interactively
- Analyze Time Series Data Using Econometric Modeler
- Creating Univariate Conditional Mean Models
- Create Multiplicative Seasonal ARIMA Model for Time Series Data
- Modify Properties of Conditional Mean Model Objects
- Specify Conditional Mean Model Innovation Distribution
- Model Seasonal Lag Effects Using Indicator Variables
- Nonseasonal and Seasonal Differencing