Choose ARMA Lags Using BIC

This example shows how to use the Bayesian information criterion (BIC) to select the degrees p and q of an ARMA model. Estimate several models with different p and q values. For each estimated model, output the loglikelihood objective function value. Input the loglikelihood value to aicbic to calculate the BIC measure of fit (which penalizes for complexity).

Simulate ARMA time series

Simulate an ARMA(2,1) time series with 100 observations.

Mdl0 = arima('Constant',0.2,'AR',{0.75,-0.4},...
               'MA',0.7,'Variance',0.1);
rng('default')
Y = simulate(Mdl0,100);

figure
plot(Y)
xlim([0,100])
title('Simulated ARMA(2,1) Series')

Plot Sample ACF and PACF

Plot the sample autocorrelation function (ACF) and partial autocorrelation function (PACF) for the simulated data.

figure
subplot(2,1,1)
autocorr(Y)
subplot(2,1,2)
parcorr(Y)

Both the sample ACF and PACF decay relatively slowly. This is consistent with an ARMA model. The ARMA lags cannot be selected solely by looking at the ACF and PACF, but it seems no more than four AR or MA terms are needed.

Fit ARMA(p,q) Models to Data

To identify the best lags, fit several models with different lag choices. Here, fit all combinations of p = 1,...,4 and q = 1,...,4 (a total of 16 models). Store the loglikelihood objective function and number of coefficients for each fitted model.

LOGL = zeros(4,4); % Initialize
PQ = zeros(4,4);
for p = 1:4
    for q = 1:4
        Mdl = arima(p,0,q);
        [EstMdl,~,logL] = estimate(Mdl,Y,'Display','off');
        LOGL(p,q) = logL;
        PQ(p,q) = p + q;
     end
end

Calculate BIC

Calculate the BIC for each fitted model. The number of parameters in a model is p + q + 1 (for the AR and MA coefficients, and constant term). The number of observations in the data set is 100.

LOGL = reshape(LOGL,16,1);
PQ = reshape(PQ,16,1);
[~,bic] = aicbic(LOGL,PQ+1,100);
reshape(bic,4,4)

ans = 4×4

  108.6241  105.9489  109.4164  113.8443
   99.1639  101.5886  105.5203  109.4348
  102.9094  106.0305  107.6489   99.6794
  107.4045  100.7072  102.5746  102.0209

In the output BIC matrix, the rows correspond to the AR degree (p) and the columns correspond to the MA degree (q). The smallest value is best.

The smallest BIC value is 99.1639 in the (2,1) position. This corresponds to an ARMA(2,1) model, matching the model that generated the data.