advanced-algorithmic-trading

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8.2 Correlation
Correlation is a dimensionless measure of how two variables vary together, or "co-vary". In
essence, it is the covariance of two random variables normalised by their respective spreads. The
(population) correlation between two variables is often denoted by ρ(x, y):
ρ(x, y) = E[(x − µ x)(y − µ y )] σ(x, y)
= (8.2)
σ x σ y σ x σ y
The denominator product of the two spreads will constrain the correlation to lie within the
interval [−1, 1]:
• A correlation of ρ(x, y) = +1 indicates exact positive linear association
• A correlation of ρ(x, y) = 0 indicates no linear association at all
• A correlation of ρ(x, y) = −1 indicates exact negative linear association
As with the covariance, we can define the sample correlation, Cor(x, y):
Cor(x, y) =
Cov(x,y)
sd(x)sd(y)
(8.3)
Where Cov(x, y) is the sample covariance of x and y, while sd(x) is the sample standard
deviation of x.
8.2.1 Example: Sample Correlation in R
We will use the same x and y vectors of the previous example. The following R code will calculate
the sample correlation:
> cor(x,y)
[1] 0.5796604
The sample correlation is given as 0.5796604 showing a reasonably strong positive linear
association between the two vectors, as expected.
8.3 Stationarity in Time Series
Now that we have outlined the definitions of expectation, variance, standard deviation, covariance
and correlation we are in a position to discuss how they apply to time series data.
Firstly, we will discuss a concept known as stationarity. This is an extremely important
aspect of time series and much of the analysis carried out on financial time series data will
concern stationarity. Once we have discussed stationarity we are in a position to talk about
serial correlation and construct some correlogram plots.
We will begin by trying to apply the above definitions to time series data, starting with the
mean/expectation:
Definition 8.3.1. Mean of a Time Series. The mean of a time series x t , µ(t), is given as the
expectation E(x t ) = µ(t).