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Notice that there are some significant peaks, especially at higher lags. This is indicative of a
poor fit. Let us perform a Ljung-Box test to see if we have statistical evidence for this:
> Box.test(resid(spfinal.arma), lag=20, type="Ljung-Box")
Box-Ljung test
data: resid(spfinal.arma)
X-squared = 37.1912, df = 20, p-value = 0.0111
As we suspected the p-value is less than 0.05 and thus we cannot say that the residuals are
a realisation of discrete white noise. Hence there is additional autocorrelation in the residuals
that is not explained by the fitted ARMA(3,3) model.
10.7 Next Steps
As we have discussed all along in this part of the book we have seen evidence of conditional
heteroskedasticity (volatility clustering) in the S&P500 series, especially in the periods around
2007-2008. When we use a GARCH model in the next chapter we will see how to eliminate these
autocorrelations.
In practice, ARMA models are never generally good fits for log equities returns. We need
to take into account the conditional heteroskedasticity and use a combination of ARIMA and
GARCH. The next chapter will consider ARIMA and show how the "Integrated" component
differs from the ARMA model we have been considering in this chapter.