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Hi DirtyQuant,
I am curious about the calculation of the skewness. The skewed t distribution by Hansen has zero mean and unit variance. Therefore the skewness calculated with (m_3 - 3*\mu*m_2 + 2*\mu^3) / \sigma^3 should simplify just to m_3. In your code you calculate the skewness by (m_3 - 3*a*m_2 + 2*a^3) / b^3, with a and b as defined by Hansen. I tried to verify the calculations by drawing sample rvs and calculating the skewness of these and your calculations seemed to be better. Now Iām wondering why the mean a and variance b of the not normalized distribution are taken into account for calculations.
Maybe I miss out on something important, can you help me here?
Thanks in advance, kind regards Simon
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Hi Simon,
Glad someone is paying attention!
Have a look at page 5 (1703 of the document) for the skewness and kurtosis formulas, which is what I used. Indeed you are correct that \mu is 0 and \sigma is 0 for the distribution.
a and b are the mean and variance of Y, the original data, whilst we are using Z which is the (GARCH) residual data which has \mu is 0 and \sigma is 1, where:
y_t = r_t - \mu, \\
y_t = \sigma_t z_t, \\
and hence
z_t = y_y / \sigma_t
Have a look at Appendix B and Appendix C which also go through it.
Hope that clears that up. Essentially we are talking about 2 different datasets. The original one, and the residual one.
1-s2.0-S0165188902000799-main.pdf (616.4 KB)