Coskewness

There are two different measures for the degree of coskewness in data.

Skewness is a special case of the coskewness when the three random variables are identical:

${\displaystyle S(X,X,X)={\frac {\operatorname {E} \left[(X-\operatorname {E} [X])^{3}\right]}{\sigma _{X}^{3}}}={\operatorname {skewness} [X]},}$

For two random variables, X and Y, the skewness of the sum, X + Y, is

${\displaystyle S_{X+Y}={1 \over \sigma _{X+Y}^{3}}{\left[\sigma _{X}^{3}S_{X}+3\sigma _{X}^{2}\sigma _{Y}S(X,X,Y)+3\sigma _{X}\sigma _{Y}^{2}S(X,Y,Y)+\sigma _{Y}^{3}S_{Y}\right]},}$

where S_X is the skewness of X and ${\displaystyle \sigma _{X}}$ is the standard deviation of X. It follows that the sum of two random variables can be skewed (S_X+Y ≠ 0) even if both random variables have zero skew in isolation (S_X = 0 and S_Y = 0).

The coskewness between variables X and Y does not depend on the scale on which the variables are expressed. If we are analyzing the relationship between X and Y, the coskewness between X and Y will be the same as the coskewness between a + bX and c + dY, where a, b, c, and d are constants.

The standardized rank coskewness RS(X, Y, Z) satisfies the following properties:[4]

(1) −1 ≤ RS(X, Y, Z) ≤ 1.

(2) The upper bound of 1 is obtained by the copula given in (3.3) in Bernard, Chen, Rüschendorf and Vanduffel (2023). The lower bound of −1 is obtained by the copula (3.5) in the same paper.

(3) It is invariant under strictly increasing transformations, i.e., when fi, i = 1, 2, 3, are arbitrary strictly increasing functions, RS(X, Y, Z) = RS(f₁ (X), f₂ (Y), f₃ (Z)).

(4) RS(X, Y, Z) = 0 if X, Y and Z are independent.

Let X be standard normally distributed and Y be the distribution obtained by setting X=Y whenever X<0 and drawing Y independently from a standard half-normal distribution whenever X>0. In other words, X and Y are both standard normally distributed with the property that they are completely correlated for negative values and uncorrelated apart from sign for positive values. The joint probability density function is

${\displaystyle f_{X,Y}(x,y)={\frac {e^{-x^{2}/2}}{\sqrt {2\pi }}}\left(H(-x)\delta (x-y)+2H(x)H(y){\frac {e^{-y^{2}/2}}{\sqrt {2\pi }}}\right)}$

where H(x) is the Heaviside step function and δ(x) is the Dirac delta function. The third moments are easily calculated by integrating with respect to this density:

${\displaystyle S(X,X,Y)=S(X,Y,Y)=-{\frac {1}{\sqrt {2\pi }}}\approx -0.399}$

Note that although X and Y are individually standard normally distributed, the distribution of the sum X+Y is significantly skewed. From integration with respect to density, we find that the covariance of X and Y is

${\displaystyle \operatorname {cov} (X,Y)={\frac {1}{2}}+{\frac {1}{\pi }}}$

from which it follows that the standard deviation of their sum is

${\displaystyle \sigma _{X+Y}={\sqrt {3+{\frac {2}{\pi }}}}}$

Using the skewness sum formula above, we have

${\displaystyle S_{X+Y}=-{\frac {3{\sqrt {2}}\pi }{(2+3\pi )^{3/2}}}\approx -0.345}$

This can also be computed directly from the probability density function of the sum:

${\displaystyle f_{X+Y}(u)={\frac {e^{-u^{2}/8}}{2{\sqrt {2\pi }}}}H(-u)+{\frac {e^{-u^{2}/4}}{\sqrt {\pi }}}\operatorname {erf} \left({\frac {u}{2}}\right)H(u)}$

Bernard, Chen, Rüschendorf and Vanduffel (2023) found risk bounds on coskewness for some popular marginal distributions as shown in the following table.[4]

Marginal distributions	Minimum coskewness	Maximum coskewness
N(${\displaystyle \mu _{i}}$, ${\displaystyle \sigma _{i}^{2}}$)	${\displaystyle -{\frac {2{\sqrt {2\pi }}}{\pi }}}$	${\displaystyle {\frac {2{\sqrt {2\pi }}}{\pi }}}$
Student(${\displaystyle \nu }$), ${\displaystyle \nu >3}$	${\displaystyle -{\frac {4(\nu -2){\sqrt {(\nu -2}})\pi \Gamma ({\frac {\nu +1}{2}})}{(3-4\nu +\nu ^{2})\pi \Gamma ({\frac {\nu }{2}})}}}$	${\displaystyle {\frac {4(\nu -2){\sqrt {(\nu -2}})\pi \Gamma ({\frac {\nu +1}{2}})}{(3-4\nu +\nu ^{2})\pi \Gamma ({\frac {\nu }{2}})}}}$
Laplace(${\displaystyle \mu _{i}}$, ${\displaystyle b_{i}}$)	${\displaystyle -{\frac {3{\sqrt {2}}}{2}}}$	${\displaystyle {\frac {3{\sqrt {2}}}{2}}}$
U(${\displaystyle a_{i}}$, ${\displaystyle b_{i}}$)	${\displaystyle -{\frac {3{\sqrt {3}}}{4}}}$	${\displaystyle {\frac {3{\sqrt {3}}}{4}}}$

where ${\displaystyle \Gamma (x)}$ is the gamma function.

• Moment (mathematics)
• Cokurtosis

• Harvey, Campbell R.; Akhtar Siddique (2000). "Conditional Skewness in Asset Pricing Tests" (PDF). The Journal of Finance. 55 (3): 1263–1295. CiteSeerX 10.1.1.46.5155. doi:10.1111/0022-1082.00247.
• Kraus, Alan; Robert H. Litzenberger (1976). "Skewness Preference and the Valuation of Risk Assets". The Journal of Finance. 31 (4): 1085–1100. doi:10.1111/j.1540-6261.1976.tb01961.x.