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Figure 10.9: Correlogram of First Order Differenced Daily Logarithmic Returns of S&500 Closing
Prices.
However we already knew this because we can see that there is significant serial correlation
in the volatility. For instance looking at the chart of returns displays the highly volatile period
around 2008.
This motivates the next set of models, namely the Moving Average MA(q) and the Autoregressive
Moving Average ARMA(p, q). We will learn about both of these in the next couple of
sections of this chapter. As is repeatedly mentioned these models will ultimately lead us to the
ARIMA and GARCH family of models, both of which will provide a much better fit to the serial
correlation complexity of the S&P500.
This will allows us to improve our forecasts significantly and ultimately produce more profitable
strategies.
10.5 Moving Average (MA) Models of order q
In the previous section we considered the Autoregressive model of order p, also known as the
AR(p) model. We introduced it as an extension of the random walk model in an attempt to
explain additional serial correlation in financial time series.
Ultimately we realised that it was not sufficiently flexible to truly capture all of the autocorrelation
in the closing prices of Amazon Inc. (AMZN) and the S&P500 US Equity Index. The
primary reason for this is that both of these assets are conditionally heteroskedastic, which means
that they are non-stationary and have periods of "varying variance" or volatility clustering, which
is not taken into account by the AR(p) model.
In the next chapter we will consider the Autoregressive Integrated Moving Average (ARIMA)