A generalized skewness statistic for stationary ergodic martingale differences

We present a class of generalized skewness statistics depending on a parameter β > 0 and containing the usual skewness statistic when β = 3, but providing greater flexibility for modelling and testing skewness when β ≠ 3. The statistics’ suitability for financial applications is illustrated using a large data set from the Australian share market. Data is assumed to be observations on stationary ergodicmartingale differences with possibly leptokurtic marginals, rather than independent identically distributed samples. The statistics can be studentized for use in hypothesis testing. Proof is provided of their asymptotic distributions undermild assumptions. Rates of convergence and power of the tests against skewed alternatives are assessed using simulation.