A Comparative Analysis of Normality Tests for Time Series Data

The assumption of normality is fundamental to many statistical methods, and violation of this assumption may result in biased interpretations and inferences. This paper evaluates the performance of seven formal normality tests for time series data including Shapiro-Wilk (SW), Kolmogorov-Smirnov (KS), Jarque-Bera (JB), Lilliefors (LF), Cramer-von Mises (CvM), Vasicek-Song (VS) and Anderson-Darling (AD) in various settings. A simulation study with symmetric and asymmetric distributions was carried out to find the detection rates, considering the most important time series characteristics, including trend, seasonality, and presence of outliers. The tests were evaluated based on their ability to correctly classify normal and non-normal data, and their performance was summarized using accuracy metrics. The results indicate that Cramer-von Mises, Lilliefors, Anderson-Darling and Vasicek-Song tests provide superior performance as compared to the other tests in various situations. Shapiro-Wilk test showed good performance for data with trend and seasonality and Kolmogorov-Smirnov test was most effective in detecting non-normality. Anderson-Darling, Lilliefors and Cramer-von Mises test demonstrated good performance with the presence of outliers. These findings suggest that one should select normality tests based on the specifics of the data. We suggest supplementing formal tests with graphical procedures to have a more detailed and reliable test of distributional assumptions.