Multiplicative ARIMA Model Specifications

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,

$$(1 - {\phi _1}L)(1 - {\Phi _{12}}{L^{12}}){(1 - L)^1}(1 - {L^{12}}){y_t} = (1 + {\theta _1}L)(1 + {\Theta _{12}}{L^{12}}){\varepsilon _t},$$

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.

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, $\phi_1$. 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 if 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:

model.SAR
ans = 

  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.

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

$$(1 - .5L)(1 + 0.7{L^4}){(1 - L)^1}(1 - {L^4}){y_t} = (1 + .3L)(1 - .2{L^4}){\varepsilon _t},$$

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.

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.

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