For testing normality we investigate the power of several tests, first of all, the well known test of Jarque and Bera (1980) and furthermore the tests of Kuiper (1960) and Shapiro and Wilk (1965) as well as tests of Kolmogorov-Smirnov and Cramér-von Mises type. The tests on normality are based, first, on independent random variables (model I) and, second, on the residuals in the classical linear regression (model II). We investigate the exact critical values of the Jarque-Bera test and the Kolmogorov-Smirnov and Cramér-von Mises tests, in the latter case for the original and standardized observations where the unknown parameters µ and σ have to be estimated. The power comparison is carried out via Monte Carlo simulation assuming the model of contaminated normal distributions with varying parameters µ and σ and different proportions of contamination. It turns out that for the JarqueBera test the approximation of critical values by the chi-square distribution does not work very well. The test is superior in power to its competitors for symmetric distributions with medium up to long tails and for slightly skewed distributions with long tails. The power of the Jarque-Bera test is poor for distributions with short tails, especially if the shape is bimodal, sometimes the test is even biased. In this case a modification of the Cramér-von Mises test or the Shapiro-Wilk test may be recommended.

The case of two variables is considered, where the sample consists of two heterogeneous groups. The behaviour of the pooled sample correlation coefficient is studied. The heterogeneity of the two groups may be interpreted as a hidden qualitative variable. It is shown that, even if the correlation is the same within both groups, the pooled correlation coefficient may be severely biased owing to heterogeneity of other group-specific parameters. In the case of uncorrelatedness, nonsensical correlation may arise from pooled estimation. These and further results are obtained and can be quantified or forecast from an asymptotic formula for the pooled sample correlation coefficient, which is well reproduced in finite sample computer experiments and illustrated with empirical examples.

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