This article develops the locally uniformly most powerful unbiased Lagrange multiplier test of normality of regression disturbances within the family of power exponential distributions. The small sample power properties of the test are compared in a Monte Carlo study with 6 well-known tests across 12 alternative nonnormal distributions. In addition, the finite sample power properties for nonnormal alternatives within the power exponential family are summarized by estimating response surfaces. The results suggest that the proposed text is computationally convenient and possesses relatively attractive power properties even against alternatives outside the power exponential family. © 1986 Taylor and Francis Group, LLC.