%% 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.

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

% 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:

Mdl.SAR

% 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.
% Copyright 2015 The MathWorks, Inc.