This paper develops the uniform asymptotic theory for local projection (LP) regression when the true lag order of the model is unknown and potentially infinite. The theory allows for varying degrees of persistence in the data, growing response horizons, and general conditionally heteroskedastic martingale-difference shocks. Based on the theory, we make two main contributions. First, we show that LPs can achieve semiparametric efficiency at a given horizon under classical assumptions on the data, provided that the controlled lag order diverges. Thus the commonly perceived efficiency loss of LPs can become asymptotically negligible with many controls. Second, we propose LP-based inference procedures for (level and cumulated) impulse responses that possess robustness properties not shared by existing methods. Inference methods using two distinct standard errors are considered. The uniform validity for the first method depends on a zero fourth-order cumulant condition on shocks, while that of the second holds more generally for conditionally heteroskedastic martingale-difference shocks. We propose a bootstrap procedure that improves finite-sample performance and extend the standard error construction to structural responses.