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On Wikipedia one can read:

In probability theory and statistics, partial correlation measures the degree of association between two random variables, with the effect of a set of controlling random variables removed.

Does anyone know of the existence of something like the partial skewness (kurtosis), i.e. a measure of the degree of skewness (kurtosis) of a random variable, with the effect of a series of controlling random variables removed?

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  • $\begingroup$ Well, you could look at the skewness (kurtosis) of the residuals of a regression model controlling for those variables. $\endgroup$
    – kjetil b halvorsen ♦
    Apr 29, 2019 at 11:13
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    $\begingroup$ Well, I actually did that. I used the formulas for $G_1$ and $G_2$ as given on Wikipedia under the articles entitled Skewness and Kurtosis. However, with residual scores, the number of degrees of freedom is different than with raw scores. Therefore, I wonder if I should replace $n$ with $n-k$, where $n$ is equal to the number of observations and $k$ is equal to the number of covariates. $\endgroup$
    – Ad van der Ven
    Apr 29, 2019 at 13:26
  • $\begingroup$ Yes, that could be a good idea, that you should investigate. $\endgroup$
    – kjetil b halvorsen ♦
    Apr 29, 2019 at 13:28
  • $\begingroup$ I've worked it out and you'll hear from me tomorrow. $\endgroup$
    – Ad van der Ven
    Apr 29, 2019 at 14:19
  • $\begingroup$ Now the partial correlation between say two variables $X$ and $Y$ with partialling out the effect of two other independent variables say $U$ and $V$ is equal to the correlation calculated on the residual scores $E_X$ and $E_Y$ associated with $X$ and $Y$. No account is taken of the number of degrees of freedom of the variables $E_X$ and $E_Y$. If we follow the same approach, i.e. not taking into account the reduction in degrees of freedom, then it is sufficient if we simply apply the formulas for $G_1$ and $G_2$ to the residual scores. The question remains whether that would be justified. $\endgroup$
    – Ad van der Ven
    Apr 30, 2019 at 9:06

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I suppose you're interested in the skewness of the residuals $\epsilon$ in the population regression $Y = \alpha + \beta X + \epsilon$, rather than the skewness of the residuals $E$ in the sample regression $Y = a + bX + E$. The third moment of $\epsilon$ can be estimated with equation (6) of Kakwani (1965)(https://www.jstor.org/stable/pdf/1909801.pdf). I didn't check the proof of this article. Denote this estimate with $K_3$. Next, the population variance of $\epsilon$ can be estimated with the mean square error ($MSE$) of the sample regression. A method of moments estimate for the skewness would then be $K_3 / MSE^{3/2}$. The $MSE$ would indeed use $df_{error}$ rather than $n-1$, but the correction in $K_3$ is more elaborate.

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  • $\begingroup$ Do I understand it correctly that according to Kakwani (1965) (jstor.org/stable/pdf/1909801.pdf) an unbiased estimator for the third central moment is $SSE/(n-k)$, where $k$ is equal to the number of covariates and $SSE$ is the sum of squared errors? $\endgroup$
    – Ad van der Ven
    May 1, 2019 at 9:01
  • $\begingroup$ No. You would use inner part of the left hand side of equation (6) of Kakwani. If the sample size is $N$, and the residual values in the sample are $E_1, ..., E_N$, then the suggested estimate for the expectation of ${\epsilon}^3$ would be $K_3 = {1\over{N-k}} \sum_iE_i^3 $, where $k = 3A - 3B + C$, as defined in the article. $\endgroup$
    – Jules Ellis
    May 1, 2019 at 21:57

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