Suppose you have 3 stocks, say X Y Z. You also know that

X is cointegrated to Y using some test (say ADF)


Y is cointegrated to Z.

However, no transitivity, and no threesome cointegration whatsoever (in other words, neither X is directly cointegration to Z, nor is there a 3 symbols cointegration using Johansen).

Is there a way to generalize pair trading to make a dynamic portfolio of X Y Z?

Intuitively I would say yes, by thinking of two pairs, XY and YZ. But I don't see yet a good strategy managing both efficiently.

  • 1
    $\begingroup$ Are you okay with upper and lower bounds on the other correlations? $\endgroup$
    – will
    Commented Jul 23, 2019 at 22:39

1 Answer 1


Assuming we are talking about Pearson correlation, then we may apply the triangle inequality. Let $\rho(X,Y)$ denote the correlation between $X$ and $Y$. Then,

$(1-\rho(X,Z))^{1/2}\le (1-\rho(X,Y))^{1/2} + (1-\rho(Y,Z))^{1/2}$

  • $\begingroup$ great! Actually in my OP I conflated two questions: one is for CORRELATION, which you have answered, another one is for COINTEGRATION. Any clue how to proceed if X Y Z are not Pearson correlated, but are cointegrated (Y cointegrated to X, Z to Y) ? $\endgroup$ Commented Jul 24, 2019 at 14:23
  • $\begingroup$ I edited the question to specify cointegration. However, I still like you answer $\endgroup$ Commented Jul 24, 2019 at 14:26

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