Modify regARIMA Model Properties

Modify Properties Using Dot Notation

If you create a regression model with ARIMA errors using regARIMA, then the software assigns values to all of its properties. To change any of these property values, you do not need to reconstruct the entire model. You can modify property values of an existing model using dot notation. To access the property, type the model name, then the property name, separated by '|.|' (a period).

Specify the regression model with ARIMA(3,1,2) errors

$\begin{array}{rcl} y_{t} & = & c + u_{t} \\ (1 - ϕ_{1} L - ϕ_{2} L^{2} - ϕ_{3} L^{3}) {(1 - L)}^{D} u_{t} & = & (1 + θ_{1} L + θ_{2} L^{2}) ε_{t} . \end{array}$

Mdl = regARIMA(3,1,2);

Use cell array notation to set the autoregressive and moving average parameters to values.

Mdl.AR = {0.2 0.1 0.05};
Mdl.MA = {0.1 -0.05}

Mdl = 
  regARIMA with properties:

     Description: "ARIMA(3,1,2) Error Model (Gaussian Distribution)"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
       Intercept: NaN
            Beta: [1×0]
               P: 4
               D: 1
               Q: 2
              AR: {0.2 0.1 0.05} at lags [1 2 3]
             SAR: {}
              MA: {0.1 -0.05} at lags [1 2]
             SMA: {}
        Variance: NaN

Use dot notation to display the autoregressive coefficients of Mdl in the Command Window.

ARCoeff = Mdl.AR

ARCoeff=1×3 cell array
    {[0.2000]}    {[0.1000]}    {[0.0500]}

ARCoeff is a 1-by-3 cell array. Each, successive cell contains the next autoregressive lags.

You can also add more lag coefficients.

Mdl.MA = {0.1 -0.05 0.01}

Mdl = 
  regARIMA with properties:

     Description: "ARIMA(3,1,3) Error Model (Gaussian Distribution)"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
       Intercept: NaN
            Beta: [1×0]
               P: 4
               D: 1
               Q: 3
              AR: {0.2 0.1 0.05} at lags [1 2 3]
             SAR: {}
              MA: {0.1 -0.05 0.01} at lags [1 2 3]
             SMA: {}
        Variance: NaN

By default, the specification sets the new coefficient to the next, consecutive lag. The addition of the new coefficient increases Q by 1.

You can specify a lag coefficient to a specific lag term by using cell indexing.

Mdl.AR{12} = 0.01

Mdl = 
  regARIMA with properties:

     Description: "ARIMA(12,1,3) Error Model (Gaussian Distribution)"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
       Intercept: NaN
            Beta: [1×0]
               P: 13
               D: 1
               Q: 3
              AR: {0.2 0.1 0.05 0.01} at lags [1 2 3 12]
             SAR: {}
              MA: {0.1 -0.05 0.01} at lags [1 2 3]
             SMA: {}
        Variance: NaN

The autoregressive coefficient 0.01 is located at the 12th lag. Property P increases to 13 with the new specification.

Set the innovation distribution to the t distribution with NaN degrees of freedom.

Distribution = struct('Name','t','DoF',NaN);
Mdl.Distribution = Distribution

Mdl = 
  regARIMA with properties:

     Description: "ARIMA(12,1,3) Error Model (t Distribution)"
      SeriesName: "Y"
    Distribution: Name = "t", DoF = NaN
       Intercept: NaN
            Beta: [1×0]
               P: 13
               D: 1
               Q: 3
              AR: {0.2 0.1 0.05 0.01} at lags [1 2 3 12]
             SAR: {}
              MA: {0.1 -0.05 0.01} at lags [1 2 3]
             SMA: {}
        Variance: NaN

If DoF is NaN, then estimate estimates the degrees of freedom. For other tasks, such as simulating or forecasting a model, you must specify a value for DoF.

To specify a regression coefficient, assign a vector to the property Beta.

Mdl.Beta = [1; 3; -5]

Mdl = 
  regARIMA with properties:

     Description: "Regression with ARIMA(12,1,3) Error Model (t Distribution)"
      SeriesName: "Y"
    Distribution: Name = "t", DoF = NaN
       Intercept: NaN
            Beta: [1 3 -5]
               P: 13
               D: 1
               Q: 3
              AR: {0.2 0.1 0.05 0.01} at lags [1 2 3 12]
             SAR: {}
              MA: {0.1 -0.05 0.01} at lags [1 2 3]
             SMA: {}
        Variance: NaN

If you pass Mdl into estimate with the response data and three predictor series, then the software fixes the non-|NaN| parameters at their values, and estimate Intercept, Variance, and DoF. For example, if you want to simulate data from this model, then you must specify Variance and DoF.

Nonmodifiable Properties

Not all properties of a regARIMA model are modifiable. To change them directly, you must redefine the model using regARIMA. Nonmodifiable properties include:

P, which is the compound autoregressive polynomial degree. The software determines P from p, d, p_s, and s. For details on notation, see Regression Model with ARIMA Time Series Errors.
Q, which is the compound moving average degree. The software determines Q from q and q_s
DoF, which is the degrees of freedom for models having a t-distributed innovation process

Though they are not explicitly properties, you cannot reassign or print the lag structure using ARLags, MALags, SARLags, or SMALags. Pass these and the lag structure into regARIMA as name-value pair arguments when you specify the model.

For example, specify a regression model with ARIMA(4,1) errors using regARIMA, where the autoregressive coefficients occur at lags 1 and 4.

Mdl = regARIMA('ARLags',[1 4],'MALags',1)

Mdl = 
  regARIMA with properties:

     Description: "ARMA(4,1) Error Model (Gaussian Distribution)"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
       Intercept: NaN
            Beta: [1×0]
               P: 4
               Q: 1
              AR: {NaN NaN} at lags [1 4]
             SAR: {}
              MA: {NaN} at lag [1]
             SMA: {}
        Variance: NaN

You can produce the same results by specifying a regression model with ARMA(1,1) errors, then adding an autoregressive coefficient at the fourth lag.

Mdl = regARIMA(1,0,1);
Mdl.AR{4} = NaN

Mdl = 
  regARIMA with properties:

     Description: "ARMA(4,1) Error Model (Gaussian Distribution)"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
       Intercept: NaN
            Beta: [1×0]
               P: 4
               Q: 1
              AR: {NaN NaN} at lags [1 4]
             SAR: {}
              MA: {NaN} at lag [1]
             SMA: {}
        Variance: NaN

To change the value of DoF, you must define a new structure for the distribution, and use dot notation to pass it into the model. For example, specify a regression model with AR(1) errors having t-distributed innovations.

Mdl = regARIMA('AR',0.5,'Distribution','t')

Mdl = 
  regARIMA with properties:

     Description: "ARMA(1,0) Error Model (t Distribution)"
      SeriesName: "Y"
    Distribution: Name = "t", DoF = NaN
       Intercept: NaN
            Beta: [1×0]
               P: 1
               Q: 0
              AR: {0.5} at lag [1]
             SAR: {}
              MA: {}
             SMA: {}
        Variance: NaN

The value of DoF is NaN by default.

Specify that the t distribution has 10 degrees of freedom.

Distribution = struct('Name','t','DoF',10);
Mdl.Distribution = Distribution

Mdl = 
  regARIMA with properties:

     Description: "ARMA(1,0) Error Model (t Distribution)"
      SeriesName: "Y"
    Distribution: Name = "t", DoF = 10
       Intercept: NaN
            Beta: [1×0]
               P: 1
               Q: 0
              AR: {0.5} at lag [1]
             SAR: {}
              MA: {}
             SMA: {}
        Variance: NaN

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Regression Models with Time Series Errors