Normal but Skewed?

We propose a multivariate normality test against skew normal distributions using higher-order log-likelihood derivatives which is asymptotically equivalent to the likelihood ratio but only requires estimation under the null. Numerically, it is the supremum of the univariate skewness coefficient test over all linear combinations of the variables. We can simulate its exact finite sample distribution for any multivariate dimension and sample size. Our Monte Carlo exercises confirm its power advantages over alternative approaches. Finally, we apply it to the joint distribution of US city sizes in two consecutive censuses finding that non-normality is very clearly seen in their growth rates.