| Standard goodness-of-fit tests are biased towards acceptance of any hypothesized distribution if the test statistics do not contain explicit corrections for the fact that estimates of model parameters are used rather than unknown true values. Goodness-of-fit tests that use only the most extreme distributional deviation are not as efficient as those that use all the entire distribution.; Whether or not the true distribution has infinite variance, the bias can be avoided by Lagrange Multiplier goodness-of-fit tests proposed herein. If a sample is independent and identically distributed according to a distribution F (with time series data a transformation can be applied to estimate an IID series) then the distribution transform of the data produces a histogram that is approximately uniform over the unit interval. Large deviations from uniformity provide evidence against F. The construction of an alternative hypothesis space surrounding the null hypothesis ensures that deviations in any direction can be detected.; Such tests can be constructed so that they have more power against alternative hypotheses and less size distortion than standard tests. They achieve these improvements by correcting for the presence of unknown model parameters. The test statistic is asymptotically chi-squared. Exact finite sample sizes are calculated employing Monte Carlo simulations; however, for samples with as few as 30 observations, size distortion is quite low.; Unknown model parameters can be estimated by the maximum likelihood principle without asymptotically biasing the test. Furthermore, the test meets the optimality conditions of the Neyman-Pearson lemma against any simple alternative hypothesis in its parameter space. It is an omnibus test with the null hypothesis nested in the space of alternatives.; Tests against many non-standard distributions are conducted including symmetric stable distributions, generalized Student-t distributions, generalized error distributions (GED), and mixtures of Gaussian distributions.; These econometric tests are not restricted to economic or financial studies, but can be applied in any discipline employing econometric or statistical techniques. With these tests, economists and other researchers will have a new tool yielding better results on more data sets, with or without obvious outliers. |
