Coskewness

In probability theory and statistics, coskewness is a measure of how much two random variables change together. Coskewness is the third standardized cross central moment, related to skewness as covariance is related to variance. In 1976, Krauss and Litzenberger used it to examine risk in stock market investments. The application to risk was extended by Harvey and Siddique in 2000.

If two random variables exhibit positive coskewness they will tend to undergo extreme positive deviations at the same time. Similarly, if two random variables exhibit negative coskewness they will tend to undergo extreme negative deviations at the same time.

Definition

For two random variables X and Y there are two non-trivial coskewness statistics:

$S(X,X,Y)={\frac {\operatorname {E} \left[(X-\operatorname {E} [X])^{2}(Y-\operatorname {E} [Y])\right]}{\sigma _{X}^{2}\sigma _{Y}}}$

and

$S(X,Y,Y)={\frac {\operatorname {E} \left[(X-\operatorname {E} [X])(Y-\operatorname {E} [Y])^{2}\right]}{\sigma _{X}\sigma _{Y}^{2}}}$

where E[X] is the expected value of X, also known as the mean of X, and $\sigma _{X}$ is the standard deviation of X.

Properties

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

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

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

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

where S_X is the skewness of X and $\sigma _{X}$ 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 are completely symmetric in isolation (S_X = 0 and S_Y = 0 and likewise for other odd moments).

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.

Example

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

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

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:

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

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

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

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

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

Using the skewness sum formula above, we have

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

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

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

See also

• Moment (mathematics)
• Cokurtosis

Further reading

• Harvey, Campbell R.; Akhtar Siddique (2000). "Conditional Skewness in Asset Pricing Tests" (PDF). The Journal of Finance. 55 (3): 1263–1295. 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.