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ARIMAX Model Specifications

Specify ARIMAX Model Using Name-Value Pairs

This example shows how to specify an ARIMAX model using arima.

Specify the ARIMAX(1,1,0) model that includes three predictors:

$$(1 - 0.1L){(1 - L)^1}{y_t} = x_t^\prime {\left[ {\begin{array}{*{20}{c}}3&{ - 2}&5\end{array}} \right]^\prime } + {\varepsilon _t}.$$

model = arima('AR',0.1,'D',1,'Beta',[3 -2 5])
model = 

    ARIMAX(1,1,0) Model:
    ---------------------
    Distribution: Name = 'Gaussian'
               P: 2
               D: 1
               Q: 0
        Constant: NaN
              AR: {0.1} at Lags [1]
             SAR: {}
              MA: {}
             SMA: {}
            Beta: [3 -2 5]
        Variance: NaN

The output shows that the ARIMAX model, model, has the following qualities:

  • Property P in the output is the sum of the autoregressive lags and the degree of integration, i.e., P = p + D = 2.

  • Beta contains three coefficients corresponding to the effect that the predictors have on the response.

  • The rest of the properties are 0, NaN, or empty cells.

Be aware that if you specify nonzero D or Seasonality, then Econometrics Toolbox™ differences the response series $y_t$ before the predictors enter the model. Therefore, the predictors enter a stationary model with respect to the response series $y_t$ . You should preprocess the predictors $x_t$ by testing for stationarity and differencing if any are unit root nonstationary. If any nonstationary predictor enters the model, then the false negative rate for significance tests of $\beta$ can increase.

Specify ARMAX Model Using Dot Notation

This example shows how to specify a stationary ARMAX model using arima.

Specify the ARMAX(2,1) model

$${y_t} = 6 + 0.2{y_{t - 1}} - 0.3{y_{t - 2}} + 3{x_t} + {\varepsilon _t} + 0.1{\varepsilon _{t - 1}}$$

by including one stationary exogenous covariate in arima.

 model = arima('AR',[0.2 -0.3],'MA',0.1,'Constant',6,'Beta',3)
model = 

    ARIMAX(2,0,1) Model:
    ---------------------
    Distribution: Name = 'Gaussian'
               P: 2
               D: 0
               Q: 1
        Constant: 6
              AR: {0.2 -0.3} at Lags [1 2]
             SAR: {}
              MA: {0.1} at Lags [1]
             SMA: {}
            Beta: [3]
        Variance: NaN

The output shows the model that you created, model, has NaN values or an empty cell ({}) for the Variance, SAR, and SMA properties. You can modify it using dot notation. For example, you can introduce another exogenous, stationary covariate, and specify that the variance of the innovations as 0.1:

$${y_t} = 6 + 0.2{y_{t - 1}} - 0.3{y_{t - 2}} + x_t^\prime \left[ {\begin{array}{*{20}{c}}3\\{ - 2}\end{array}} \right] + {\varepsilon _t} + 0.1{\varepsilon _{t - 1}};\;\;{\varepsilon _t}\sim N(0,0.1).$$

Modify model:

model.Beta=[3 -2];
model.Variance=0.1
model = 

    ARIMAX(2,0,1) Model:
    ---------------------
    Distribution: Name = 'Gaussian'
               P: 2
               D: 0
               Q: 1
        Constant: 6
              AR: {0.2 -0.3} at Lags [1 2]
             SAR: {}
              MA: {0.1} at Lags [1]
             SMA: {}
            Beta: [3 -2]
        Variance: 0.1

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