The Directed Acyclic Graph (DAG) is a commonly used tool to encode the causal relationship between random variables.  Estimation of the DAG structure is often a challenging problem as the computational complexity scales exponentially in the graph size when the total ordering of the DAG is unknown.  To reduce the computational cost, and also with the aim of improving the estimation accuracy via the bias-variance trade-off, we propose a two-step approach for
estimating the DAG, when data are generated from a linear structural equation model.  In the first step, we infer the moral graph of the DAG via estimation of the inverse covariance matrix, which reduces the space that one would search for the DAG.  In the second step, we apply
an existing method for estimating the DAG to the reduced space. Preliminary numerical results indicate that the proposed method compares favorably with existing methods in terms of both
computational cost and estimation accuracy.