advanced-algorithmic-trading

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• Use our knowledge of time series to fit an appropriate model that reduces the serial correlation
in the residuals
• Refine the fit until no correlation is present and then subsequently make use of statistical
goodness-of-fit tests to assess the model fit
• Use the model and its second-order properties to make forecasts about future values
• Iterate through this process until the forecast accuracy is optimised
• Utilise such forecasts to create trading strategies
This is our basic process. The complexity will arise when we consider more advanced models
that account for additional serial correlation in our time series.
In this chapter we are going to consider two of the most basic time series models, namely
White Noise and Random Walks. These models will form the basis of more advanced models
later so it is essential we understand them well.
However, before we introduce either of these models, we are going to discuss some more
abstract concepts that will help us unify our approach to time series models. In particular, we
are going to define the Backward Shift Operator and the Difference Operator.
9.2 Backward Shift and Difference Operators
The Backward Shift Operator (BSO) and the Difference Operator will allow us to write
many different time series models in a particularly succinct way that more easily allows us to
draw comparisons between them.
Since we will be using the notation of each so frequently, it makes sense to define them now.
Definition 9.2.1. Backward Shift Operator. The backward shift operator or lag operator, B,
takes a time series element as an argument and returns the element one time unit previously:
Bx t = x t−1 .
Repeated application of the operator allows us to step back n times: B n x t = x t−n .
We will use the BSO to define many of our time series models going forward.
In addition, when we come to study time series models that are non-stationary (that is, their
mean and variance can alter with time), we can use a differencing procedure in order to take a
non-stationary series and produce a stationary series from it.
Definition 9.2.2. Difference Operator. The difference operator, ∇, takes a time series element as
an argument and returns the difference between the element and that of one time unit previously:
∇x t = x t − x t−1 , or ∇x t = (1 − B)x t .
As with the BSO, we can repeatedly apply the difference operator: ∇ n = (1 − B) n .
Now that we’ve discussed these abstract operators, let us consider some concrete time series
models.