Test for independence between variables [closed]

Question

I have the weekly prices of two stocks over a period of 20 years. How could I test the independency between those two stocks?

Answer 1

Testing for independence of continuous variables is a non-trivial problem that is still a subject of active resarch. An old test was developed by Hoeffding in 1948:

W. Hoeffding: "A non-parametric test of independence." The annals of mathematical statistics 19,4, pp. 546-557, 1948

Two random variables X and Y are called independent, if their joint distribution function factorizes: $$P(X<x, Y<y) = F_{12}(x,y) = F_1(x)\cdot F_2(y) = P(X<x, y \in\mathbb{R})\cdot P(Y<y, x\in\mathbb{R})$$ It is thus natural to consider the difference between these two sides, i.e. $\Big(F_{12}(x,y) - F_1(x) F_2(y)\Big)^2$. Hoeffding found an estimator D for the functional $$\Delta = \int dx\int dy\, \Big(F_{12}(x,y) - F_1(x) F_2(y)\Big)^2 f_1(x)\,f_2(y)$$ where $f_1$ and $f_2$ are the probability densities of X and Y, respectively. He found the approximate distribution of the estimator D under the null hypothesis that X and Y are independent.

The test is thus used like any other test: if the computed p-value is less than a threshold (say 0.05), the observation is "unlikely" under the null hypothesis, and your variables are presumably not independent. Unfortunately a high value for p is not necessarily a sign for independence, but you can at least feel somewhat safer ;-)

Example:

> x <- rnorm(20)
> y <- rnorm(20)
> hoeffd(x, y)
...
P
  x      y 
x        0.9933    <--- high value: presumably independent
y 0.9933   
> hoeffd(x, x+y)
...
P
  x y
x   0              <--- low value: most likely not independent
y 0

Edit: Oh sorry, just saw that your data do not stem from two constant distributions, but are time series. In that case, other approaches are more appropriate, like computing the intra-class correlation ICC3 as a measure how similar the curve shapes are.