ARIMA Model Specifications

Default ARIMA Model

This example shows how to use the shorthand arima(p,D,q) syntax to specify the default ARIMA(p, D, q) model,

$${\Delta ^D}{y_t} = c + {\phi _1}{\Delta ^D}{y_{t - 1}} +  \ldots  + {\phi _p}{\Delta ^D}{y_{t - p}} + {\varepsilon _t} + {\theta _1}{\varepsilon _{t - 1}} +  \ldots  + {\theta _q}{\varepsilon _{t - q}},$$

where ${\Delta ^D}{y_t}$ is a $D^{th}$ differenced time series. You can write this model in condensed form using lag operator notation:

$$\phi (L){(1 - L)^D}{y_t} = c + \theta (L){\varepsilon _t}.$$

By default, all parameters in the created model object have unknown values, and the innovation distribution is Gaussian with constant variance.

Specify the default ARIMA(1,1,1) model:

model = arima(1,1,1)
model = 

    ARIMA(1,1,1) Model:
    --------------------
    Distribution: Name = 'Gaussian'
               P: 2
               D: 1
               Q: 1
        Constant: NaN
              AR: {NaN} at Lags [1]
             SAR: {}
              MA: {NaN} at Lags [1]
             SMA: {}
        Variance: NaN

The output shows that the created model object, model, has NaN values for all model parameters: the constant term, the AR and MA coefficients, and the variance. You can modify the created model using dot notation, or input it (along with data) to estimate.

The property P has value 2 (p + D). This is the number of presample observations needed to initialize the AR model.

ARIMA Model with Known Parameter Values

This example shows how to specify an ARIMA(p, D, q) model with known parameter values. You can use such a fully specified model as an input to simulate or forecast.

Specify the ARIMA(2,1,1) model

$$\Delta {y_t} = 0.4 + 0.8\Delta {y_{t - 1}} - 0.3\Delta {y_{t - 2}} + {\varepsilon _t} + 0.5{\varepsilon _{t - 1}},$$

where the innovation distribution is Student's t with 10 degrees of freedom, and constant variance 0.15.

tdist = struct('Name','t','DoF',10);
model = arima('Constant',0.4,'AR',{0.8,-0.3},'MA',0.5,...
						'D',1,'Distribution',tdist,'Variance',0.15)
model = 

    ARIMA(2,1,1) Model:
    --------------------
    Distribution: Name = 't', DoF = 10
               P: 3
               D: 1
               Q: 1
        Constant: 0.4
              AR: {0.8 -0.3} at Lags [1 2]
             SAR: {}
              MA: {0.5} at Lags [1]
             SMA: {}
        Variance: 0.15

The name-value pair argument D specifies the degree of nonseasonal integration (D).

Because all parameter values are specified, the created model object has no NaN values. The functions simulate and forecast don't accept input models with NaN values.

See Also

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