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I read that if x1, x2 are 2 random variables with different excess kurtosis, their joint distribution cant be elliptical. Is there an intuition or proof of that? It is not very clear to me.

Edit- in light of comment by Thomas Lumley below,

Let x1 and x2 be two random variable such that

  1. $E[x_1]$ = $E[x_2]$ = 0
  2. $E[x_1^2]$ = $E[x_2^2]$ = 1
  3. $E[x_1^3]$ = $E[x_2^3]$ = 0
  4. $E[x_1^4]$ <> $E[x_2^4]$
  5. $E[x_1 *x_2] = \rho $

The statement says that there can't be a set of linear combinations that have equal kurtosis either

i.e there is no POSITIVE DEFINITE Matrix

$W = \begin{bmatrix}a_1 & a_2\\b_1 & b_2\end{bmatrix}$

$ y_1 = a_1 * x_1 + a_2 * x_2$

$ y_2 = b_1 * x_1 + b_2 * x_2$

such that

$ \frac{E[y_1^4]}{[E[y_1^2]]^2} = \frac{E[y_2^4]}{[E[y_2^2]]^2}$

Is there any intuition or proof of it?

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2 Answers 2

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If a distribution is elliptical, it can be rescaled to be spherically symmetric. That means there is a positive definite matrix $A$ with entries $a_{ij}$ such that $a_{11}X_1+a_{12}X_2$ and $a_{21}X_1+a_{22}X_2$ are uncorrelated and have the same distribution. But they can't have the same distribution, because they have different excess kurtosis, so no elliptical symmetry.

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    $\begingroup$ +1. To be explicit, your readers need to understand that univariate rescaling cannot change the excess kurtosis. $\endgroup$
    – whuber
    Jul 14, 2020 at 12:38
  • $\begingroup$ Thanks @ThomasLumley. However it is not very clear why a11X1+a12X2 and a21X1+a22X2 can't have same kurtosis/ distribution $\endgroup$
    – dayum
    Jul 14, 2020 at 18:24
  • $\begingroup$ The spherical rescaling can't be done if the variables are correlated? $\endgroup$
    – develarist
    Aug 3, 2020 at 1:12
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I read that if $X_1, X_2$ are 2 random variables with different excess kurtosis, their joint distribution cant be elliptical. Is there an intuition or proof of that? It is not very clear to me.

In my opinion the best intuition is the follow. Elliptical distributions deal with some stylized facts in finance returns, like: fat tails and tail dependence. However, in multivariate case, there is a restriction about the form of the marginals. Especially about the heaviness of the tails.

The most useful example in my opinion is the t-Student distribution: https://en.wikipedia.org/wiki/Multivariate_t-distribution

Given location vector $\mu$ and scale matrix $\Sigma$ we focus on the tail index $v$ (real positive number). Now, the point is that location and scale are free but the tail index must be common to any marginal.

So, under the hypothesis that $v>4$, the excess kurtosis is $6/(v-4)$ for any marginal. Similar conditions hold for all Elliptical distributions.

Moreover, it seems me that from this example emerge that the widespread use of kurtosis for addrees fat tails problem is not a good practice. Indeed, precisely from fat tails, the existence of fourth moments is questionable and instability of their estimator is related. Focus on something like the tail index is much better.

This article give some insight about Elliptical distribution: https://www.researchgate.net/publication/237633988_Tail_Conditional_Expectations_for_Elliptical_Distributions

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