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Suppose I have two variables and that their dependency structure is Gaussian copula. Suppose that the parameter of the Gaussian copula is 0.8 and the corresponding Kendall's tau is 0.7. How can I interpret the result? How can I describe this dependency from the plot?

Can I said that there are strong correlation at the middle values while these two variables are not correlated at the tails. Also, can I said the relationship is positive which means that if the first variable increases then the second variables increases as well. If yes, why this increase is not apply for extremes. That is, saying the variables are increase together that means they also should increased even at the extremes.

Any help, ideas, please?

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How rigorous do you want the exposition to be? In simple terms, even though we often talk about "dependence" and "correlation" in intuitive terms, they have formal definitions. In particular, Kendall's tau has a formal mathematical definition, and it measures the probability of concordance minus the probability of disconcordance, where concordance between $(X,Y)$ is defined as the probability that $X_1 > X_2$ and $Y_1 > Y_2$ jointly, where $(X_1,Y_1)$, $(X_2,Y_2)$ are independent random variables with the same distribution of $(X,Y)$. As such, even if $(X,Y)$ have a high value for kendall's tau, they can certainly have little concordance at the tails (or any other part of the distribution) as long as the mass of being at those parts are low enough. If I have no association at a certain part of the distribution, but the density is really low, then an overall measure of kendall's tau may say little about the dependence at certain areas.

To answer your question more explicitly, if you know that the copula is a Gaussian copula, then knowing the parameter $.8$ is in some sense all you need to know. A commonly studied feature of copulas is the upper (lower) tail dependence, which precisely measures the 'relationship' of the copula at the tails. It can be shown that for gaussian copulas, the tail dependence is $0$. So in some sense, this is why the correlation in the middle does not apply at the extremes--it has to do with your choice of copula. For a rigorous proof of this, and also an intuitive visualization, check out this post:

Why is Gaussian Copula's Tail Dependence Zero?

Generally, copulas measure arbitrarily complicated and intricate relationships, and there's no reason why correlation in the middle implies anything about correlation at the tails, and measures such as Kendall's tau provide information about the overall 'relationship.' Maybe not the best comparison, but it's sort of similar to how for a random variable, the mean and variance provide information about the variable, but it says little about tail behavior, and so just how tail dependence exits as a concept for copulas, so does concepts like kurtosis for a random variable.

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  • $\begingroup$ Thank you so much for your great help. One point still confuse me and do not know how interpret it. So, can I said that " $X_1$ and $X_2$ has positive gaussian dependency at a short period (at the middle), while they are independence at their extreme. That is, if $X_1$ increase then this independent from $X_2$. In other words, can I said that $X_1$ does not affected by $X_2$ at extreme. My main problem still in how to interpret this dependency in more details. This is the second part of my question. $\endgroup$ – Maryam Jun 14 at 17:58
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    $\begingroup$ @Maryam Yes, you can say such a statement at the limit, but for any finite interval (ie the xth upper percentile of the joint distribution), there will still be a dependence, but at the limit this disappears. It's not clear what it means to "describe the dependence," but typically in your case there is a positive dependence between $X_1,X_2$, but this dependence tapers off at the extreme, and at the limit, there is tail independence. $\endgroup$ – doubled Jun 14 at 20:21
  • $\begingroup$ I really do not know how to thank you for your great help. I just have two variables and would like to see is there a relationship between them. Mainly I would like to find does $X_1$ affect $X_2$. That is, if $X_1$ increases what happen to $X_2$. I fit copula and that the relationship between them is Gaussian. So, I need to say what will happen to $X_2$ if $X_1$ increases. $\endgroup$ – Maryam Jun 15 at 17:00
  • $\begingroup$ @Maryam It seems like you have all the tools you need, what in particular you still stuck on? $\endgroup$ – doubled Jun 15 at 19:04
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    $\begingroup$ @Maryam Not sure what "time" really means here, but if $(X_1,X_2)$ have a corresponding gaussian copula, then you can say that $X_1,X_2$ are related, but that the relationship decreases as you move to the limits, and at the limit, they are unrelated. $\endgroup$ – doubled Jun 18 at 2:33

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