This study proposes a method to identify factor-augmented vector autoregression models without imposing uncorrelatedness or any timing restrictions among observed and unobserved factors in the vector autoregression system. To this end, we utilize changes in unconditional shock variances following Rigobon (2003). The proposed method can incorporate both observed and unobserved factors in the structural vector autoregression system and allows the contemporaneous matrix to be fully unrestricted. We derive the asymptotic distribution of the impulse response estimator and consider a bootstrap inference method. We also provide two diagnostic tools: a test for the identification condition and a class of overidentifying restrictions tests. A Monte Carlo experiment shows that the asymptotic and bootstrap methods yield a satisfactory coverage rate when the shock of an observed factor is analyzed, although the bootstrap method is more accurate. We apply the proposed method to an empirical example for the effects of U.S. monetary policies on asset prices. A contractionary monetary policy shock induces positive and hump-shaped interest rate responses along the maturity dimension and negative but insignificant stock price responses.