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10.1 How Will We Proceed?
In this chapter we are going to outline some new time series concepts that will be needed for
the remaining methods, namely strict stationarity and the Akaike information criterion
(AIC).
Subsequent to these new concepts we will follow the traditional pattern for studying new time
series models:
• Rationale - The first task is to provide a reason why we are interested in a particular
model, as quants. Why are we introducing the time series model? What effects can it
capture? What do we gain (or lose) by adding in extra complexity?
• Definition - We need to provide the full mathematical definition (and associated notation)
of the time series model in order to minimise any ambiguity.
• Second Order Properties - We will discuss (and in some cases derive) the second order
properties of the time series model, which includes its mean, its variance and its autocorrelation
function.
• Correlogram - We will use the second order properties to plot a correlogram of a realisation
of the time series model in order to visualise its behaviour.
• Simulation - We will simulate realisations of the time series model and then fit the model
to these simulations to ensure we have accurate implementations and understand the fitting
process.
• Real Financial Data - We will fit the time series model to real financial data and consider
the correlogram of the residuals in order to see how the model accounts for serial correlation
in the original series.
• Prediction - We will create n-step ahead forecasts of the time series model for particular
realisations in order to ultimately produce trading signals.
Nearly all of the chapters written in this book on time series models will fall into this pattern
and it will allow us to easily compare the differences between each model as we add further
complexity.
We will begin by looking at strict stationarity and the AIC.
10.2 Strictly Stationary
We provided the definition of stationarity in the chapter on serial correlation. However, because
we are going to be entering the realm of many financial series, with various frequencies, we need
to make sure that our eventual models take into account the time-varying volatility of these
series. In particular we need to consider their heteroskedasticity.
We will come across this issue when we try to fit certain models to historical series. Generally,
not all of the serial correlation in the residuals of fitted models can be accounted for without
taking heteroskedasticity into account. This brings us back to stationarity. A series is not
stationary in the variance if it has time-varying volatility, by definition.
This motivates a more rigourous definition of stationarity, namely strict stationarity: