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Let's consider the following correlation matrix:

$$\begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix}$$

which describes the correlation between 3 random variables $X,Y,Z$. It says that $X$ is strongly correlated to $Y$, $X$ is strongly correlated to $Z$, but $Y$ is not correlated to $Z$.

Though possible, it seems a bit odd. So I would like to understand what the observations of these variables can look like.

Can somebody provide an illustration in terms of three time series of $X, Y, Z$ realizations?

Richard Hardy
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mic
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    Correlation matrices are always positive semidefinite. Your matrix $M$ is not. For instance, take $v=(\sqrt{0.5},-0.5,-0.5)$ (the third eigenvector of $M$), then $v^tMv<0$. Thus, your matrix *cannot* be a correlation matrix. (Good question, nevertheless. +1.) – Stephan Kolassa Nov 12 '15 at 12:00
  • Thanks I just realized. For less extreme case, one can exhibit $Y \sim \mathcal{N}(0,1), Z \sim \mathcal{N}(0,1), X = Y - Z$. – mic Nov 12 '15 at 12:06
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    See this for applicable bounds to rework your example: http://stats.stackexchange.com/questions/72790/bound-for-the-correlation-of-three-random-variables – Christoph Hanck Nov 12 '15 at 15:36

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