Structural vector autoregressive (SVAR) models have emerged as a dominant research strategy in empirical macroeconomics. Despite their advantages, just-identified SVAR models suffer from (i) the great number of parameters, (ii) the resulting uncertainty associated with impulse responses, (iii) the existence of alternative observationally-equivalent just-identified models and (iv) the lack of identification of the imposed causal ordering of the variables of the system. In this paper we propose general-to-specific reductions of just-identified SVAR models to overcome these limitations. We show that the computer-automated model selection algorithm embodied in PcGets can be used for an efficient implementation of the underlying methodology. The application of the proposed reduction strategy to a US monetary system demonstrates the feasibility of PcGets for the analysis of large macroeconomic data sets.