MATLAB Examples

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,...
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 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:

ans =

  1x12 cell array

  Columns 1 through 8

    {[0]}    {[0]}    {[0]}    {[0]}    {[0]}    {[0]}    {[0]}    {[0]}

  Columns 9 through 12

    {[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 elements in model have value NaN, indicating that these coefficients need to be estimated or otherwise specified by the user.