Changes in residual volatility in vector autoregressive (VAR) models can be used for identifying structural shocks in a structural VAR analysis. Testable conditions are given for full identification for the case where the volatility changes can be modelled by a multivariate GARCH process. Formal statistical tests are presented for identification and their small sample properties are investigated via a Monte Carlo study. The tests are applied to investigate the validity of identification conditions in two studies. First, we test an identifying condition employed in a study of the impact of financial market uncertainty on real activity. Second, we illustrate our tests in the context of an investigation of the effects of U.S. monetary policy on exchange rates. In the first application the identification conditions are confirmed, and in the second application they are partly not supported by the data.