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,

where the innovation distribution is Gaussian with constant variance. Here, is the first degree nonseasonal differencing operator and is the first degree seasonal differencing operator with periodicity 12.

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

model = ARIMA(1,1,1) Model Seasonally Integrated with Seasonal AR(12) and MA(12): -------------------------------------------------------------------------- Distribution: Name = 'Gaussian' P: 26 D: 1 Q: 13 Constant: 0 AR: {NaN} at Lags [1] SAR: {NaN} at Lags [12] MA: {NaN} at Lags [1] SMA: {NaN} at Lags [12] Seasonality: 12 Variance: NaN

The name-value pair argument `ARLags`

specifies the lag corresponding to the nonseasonal AR coefficient,
. `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
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* +
+ *s* (here, 1 + 1 + 12 + 12 = 26). Similarly, `arima`

sets the property `Q`

equal to *q* +
(here, 1 + 12 = 13).

Display the value of `SAR`

:

model.SAR

ans = 1×12 cell array Columns 1 through 11 [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] [0] Column 12 [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 elements in `model`

have value `NaN`

, indicating that these coefficients need to be estimated or otherwise specified by the user.

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

where the innovation distribution is Gaussian with constant variance 0.15. Here, is the nonseasonal differencing operator and is the first degree seasonal differencing operator with periodicity 4.

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

model = ARIMA(1,1,1) Model Seasonally Integrated with Seasonal AR(4) and MA(4): -------------------------------------------------------------------------- Distribution: Name = 'Gaussian' P: 10 D: 1 Q: 5 Constant: 0 AR: {0.5} at Lags [1] SAR: {-0.7} at Lags [4] MA: {0.3} at Lags [1] SMA: {-0.2} at Lags [4] Seasonality: 4 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 of the parameters in the model have a value. Therefore, the model does not contain any `NaN`

values. The functions `simulate`

and `forecast`

*do not* accept input models with `NaN`

values.

`arima`

| `estimate`

| `forecast`

| `simulate`

- Specify Multiplicative ARIMA Model
- Modify Properties of Conditional Mean Model Objects
- Specify Conditional Mean Model Innovation Distribution
- Model Seasonal Lag Effects Using Indicator Variables

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