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Figure 10.18: Residuals of MA(3) Model Fitted to S&P500 Daily Log Prices
10.6 Autogressive Moving Average (ARMA) Models of order
p, q
We have introduced Autoregressive models and Moving Average models in the two previous
sections. Now it is time to combine them to produce a more sophisticated model.
Ultimately this will lead us to the ARIMA and GARCH models that will allow us to predict
asset returns and forecast volatility. These models will form the basis for trading signals and
risk management techniques.
If you’ve read the previous sections in this chapter you will have seen that we tend to follow
a pattern for our analysis of a time series model. I’ll repeat it briefly here:
• Rationale - Why are we interested in this particular model?
• Definition - A mathematical definition to reduce ambiguity.
• Correlogram - Plotting a sample correlogram to visualise a models behaviour.
• Simulation and Fitting - Fitting the model to simulations, in order to ensure we have
understood the model correctly.
• Real Financial Data - Apply the model to real historical asset prices.
However, before delving into the ARMA model we need to discuss the Bayesian Information
Criterion and the Ljung-Box test, two essential tools for helping us to choose the correct model
and ensuring that any chosen model is a good fit.