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A note on testing symmetry of the error distribution in linear regression models

In the classical linear regression model, the problem of testing for symmetry of the error distribution is considered. The test statistic is a functional of the difference between the two empirical distribution functions of the estimated residuals and their counterparts with opposite signs. The weak convergence of the difference process to a Gaussian process is established. The covariance structure of this process depends heavily on the density of the error distribution, and for this reason, the performance of a symmetric wild bootstrap procedure is discussed in asymptotic theory and by means of a simulation study.