We introduce SpinSVAR, a novel method for estimating a structural vector autoregression (SVAR) from time-series data under sparse input assumption. Unlike prior approaches using Gaussian noise, we model the input as independent Laplacian variables, enforcing sparsity and yielding a maximum likelihood estimator (MLE) based on least absolute error regression. We provide theoretical consistency guarantees for the MLE under mild assumptions. SpinSVAR is efficient: it can leverage GPU acceleration to scale to thousands of nodes. On synthetic data with Laplacian or Bernoulli-uniform inputs, SpinSVAR outperforms state-of-the-art methods in accuracy and runtime. When applied to S&P 500 data, it clusters stocks by sectors and identifies significant structural shocks linked to major price movements, demonstrating the viability of our sparse input assumption.