Hermite polynomials and Jarque–Bera

(a) Using $E Z^{2m} = (2m-1)!!$ for $Z \sim N(0,1)$, verify $E[He_3(Z)^2] = 6$, $E[He_4(Z)^2] = 24$ and $E[He_3(Z)He_4(Z)] = 0$, where $He_3(z) = z^3 - 3z$ and $He_4(z) = z^4 - 6z^2 + 3$. (b) A year (250 days) of a stock's returns has sample skewness $-0.5$ and excess kurtosis 3. Compute the Jarque–Bera statistic and its $p$-value. (c) Explain why rejecting normality with a single year of data does not tell you the tail index.