advanced-algorithmic-trading

Chapter 8
Serial Correlation
In the previous chapter we considered how time series analysis models could be used to eventually
allow us create trading strategies. In this chapter we are going to look at one of the most
important aspects of time series, namely serial correlation (also known as autocorrelation).
Before we dive into the definition of serial correlation we will discuss the broad purpose of
time series modelling and why we are interested in serial correlation.
When we are given one or more financial time series we are primarily interested in forecasting
or simulating data. It is relatively straightforward to identify deterministic trends as well as
seasonal variation and decompose a series into these components. However, once such a time
series has been decomposed we are left with a random component.
Sometimes such a time series can be well modelled by independent random variables. However,
there are many situations, particularly in finance, where consecutive elements of this random
component time series will possess correlation. That is, the behaviour of sequential points in the
remaining series affect each other in a dependent manner. One major example occurs in meanreverting
pairs trading. Mean-reversion shows up as correlation between sequential variables in
time series.
Our task as quantitative modellers is to try and identify the structure of these correlations,
as they will allow us to markedly improve our forecasts and thus the potential profitability
of a strategy. In addition identifying the correlation structure will improve the realism of any
simulated time series based on the model. This is extremely useful for improving the effectiveness
of risk management components of the strategy implementation.
When sequential observations of a time series are correlated in the manner described above
we say that serial correlation (or autocorrelation) exists in the time series.
Now that we have outlined the usefulness of studying serial correlation we need to define it
in a rigourous mathematical manner. Before we can do that we must build on simpler concepts,
including expectation and variance.
8.1 Expectation, Variance and Covariance
Many of these definitions will be familiar if you have a background in statistics or probability,
but they will be outlined here specifically for purposes of consistent notation.
The first definition is that of the expected value or expectation:
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