Regress the US gross domestic product (GDP) onto the US consumer price index (CPI) using a regression model with ARMA(1,1) errors, and summarize the results.

Load the US Macroeconomic data set and preprocess the data.

load Data_USEconModel;
logGDP = log(DataTable.GDP);
dlogGDP = diff(logGDP);
dCPI = diff(DataTable.CPIAUCSL);

Fit the model to the data.

Mdl = regARIMA('ARLags',1,'MALags',1);
EstMdl = estimate(Mdl,dlogGDP,'X',dCPI,'Display','off');

Display the estimates.

summarize(EstMdl)

 
   ARMA(1,1) Error Model (Gaussian Distribution)
 
    Effective Sample Size: 248
    Number of Estimated Parameters: 5
    LogLikelihood: 798.406
    AIC: -1586.81
    BIC: -1569.24
 
                   Value       StandardError    TStatistic      PValue  
                 __________    _____________    __________    __________

    Intercept      0.014776      0.0014627        10.102      5.4238e-24
    AR{1}           0.60527        0.08929        6.7787      1.2124e-11
    MA{1}          -0.16165        0.10956       -1.4755         0.14009
    Beta(1)        0.002044     0.00070616        2.8946       0.0037969
    Variance     9.3578e-05     6.0314e-06        15.515      2.7338e-54

 

Estimate several models by passing the data to estimate. Vary the autoregressive and moving average degrees p and q, respectively. Estimation results contain the AIC, which you can extract and then compare among the models.

Simulate response and predictor data for the regression model with ARMA errors:

where $${\epsilon }_t$$ is Gaussian with mean 0 and variance 1.

Mdl0 = regARIMA('Intercept',2,'Beta',[-2; 1.5],...
    'AR',{0.75, -0.5},'MA',0.7,'Variance',1);

rng(2);            % For reproducibility
X = randn(1000,2); % Predictors
y = simulate(Mdl0,1000,'X',X);

To determine the number of AR and MA lags, create and estimate multiple regression models with ARMA(p, q) errors. Vary p = 1,..,3 and q = 1,...,3 among the models. Suppress all estimation displays. Extract the AIC from the estimation results structure. The field AIC stores the AIC.

pMax = 3;
qMax = 3;
AIC = zeros(pMax,qMax); % Preallocation

for p = 1:pMax
    for q = 1:qMax
        Mdl = regARIMA(p,0,q);
        EstMdl = estimate(Mdl,y,'X',X,'Display','off');
        results = summarize(EstMdl);
        AIC(p,q) = results.AIC;
    end
end

Compare the AIC values among the models.

minAIC = min(min(AIC))

minAIC = 2.9280e+03

[bestP,bestQ] = find(AIC == minAIC)

bestP = 2

bestQ = 1

The best fitting model is the regression model with AR(2,1) errors because its corresponding AIC is the lowest. This model also has the structure of the model used to simulate the data.