This article extends the variance ratio test of Lo and MacKinlay (1988) to tests of skewness and kurtosis ratios. The proposed tests are based on generalized methods of moments. In particular, overlapping observations are used and their dependencies (under the IID assumption) are explicitly modelled so that more information can be used in order to make the tests more powerful with better size properties. The proposed tests are particularly relevant to the risk management industry where risk models are estimated using daily data, although multi-period forecasts of tail risks are required for the determination of risk capital. Applications of the tests find significant higher-order nonlinear dependencies in global major equity markets. Failure to correctly model such nonlinear relationships is likely to have a negative impact on the accuracy of forecasts of multi-period tail risks.