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Figure 10.22: Correlogram of an ARMA(2,2) Model, with α 1 = 0.5, α 2 = −0.25, β 1 = 0.5 and
β 2 = −0.3
[1] -0.296816 -0.161184
> 0.319 + c(-1.96, 1.96)*0.0792
[1] 0.163768 0.474232
> -0.552 + c(-1.96, 1.96)*0.0771
[1] -0.703116 -0.400884
Notice that the confidence intervals for the coefficients for the moving average component
(β 1 and β 2 ) do not actually contain the original parameter value. This outlines the danger of
attempting to fit models to data, even when we know the true parameter values.
However for trading purposes we only need to have a predictive power that reasonably exceeds
chance, producing sufficient trading revenue above transaction costs in order to be profitable in
the long run.
Now that we have seen some examples of simulated ARMA models we need a mechanism for
choosing the values of p and q when fitting to the models to real financial data.
10.6.6 Choosing the Best ARMA(p,q) Model
In order to determine which order p, q of the ARMA model is appropriate for a series, we need
to use the AIC (or BIC) across a subset of values for p, q, and then apply the Ljung-Box test to
determine if a good fit has been achieved, for particular values of p, q.
To show this method we are going to firstly simulate a particular ARMA(p,q) process. We
will then loop over all pairwise values of p ∈ {0, 1, 2, 3, 4} and q ∈ {0, 1, 2, 3, 4} and calculate
the AIC. We will select the model with the lowest AIC and then run a Ljung-Box test on the