This study combines block-recursive restrictions with non-Gaussian and mean independent shocks to derive identifying and overidentifying higher-order moment conditions for structural vector autoregressions. We show that overidentifying higher-order moments can contain additional information and increase the efficiency of the estimation. In particular, we prove that in the non-Gaussian recursive SVAR higher-order moment conditions are relevant and therefore, the frequently applied estimator based on the Cholesky decomposition is inefficient. Even though incorporating information in valid higher-order moments is asymptotically efficient, including many redundant and potentially even invalid moment conditions renders standard SVAR GMM estimators unreliable in finite samples. We apply a LASSO-type GMM estimator to select the relevant and valid higher-order moment conditions, increasing finite sample precision. A Monte Carlo experiment and an application to quarterly U.S. data illustrate the improved performance of the proposed estimator.