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Figure 10.23: Correlogram of the residuals of the best fitting ARMA(p,q) Model, p = 3 and
q = 2
The corelogram does indeed look like a realisation of DWN. Finally, we perform the Ljung-Box
test for 20 lags to confirm this:
> Box.test(resid(final.arma), lag=20, type="Ljung-Box")
Box-Ljung test
data: resid(final.arma)
X-squared = 13.1927, df = 20, p-value = 0.869
Notice that the p-value is greater than 0.05, which states that the residuals are independent
at the 95% level and thus an ARMA(3,2) model provides a good model fit.
Clearly this should be the case since we have simulated the data ourselves. However this is
precisely the procedure we will use when we come to fit ARMA(p,q) models to the S&P500 index
in the following section.
10.6.7 Financial Data
Now that we have outlined the procedure for choosing the optimal time series model for a
simulated series it is rather straightforward to apply it to financial data. For this example we
are going to once again choose the S&P500 US Equity Index.
Let us download the daily closing prices using quantmod and then create the log returns
stream:
> require(quantmod)