The statistical literature offers many tests of normality, yet I find it surprising that there aren't vastly more.
Perhaps we face a situation here that is analogous to the famous policy of "Don't Ask: Don't Tell." Many investigators seem not to want to know if their normality assumptions might be wrong.
Conceivably they are fearful that lack of normality will leave them without appropriate tools, but this fear is usually ill-founded. Yes, the data may be non-normal. Nevertheless, there are truths to be discovered, and there are tools that aid in that discovery.
Anyway, that's my story and I am sticking to it.
S. Lawford, "Finite-sample quantiles of the Jarque-Bera test." How well does the chi-square limit really work in a finite sample JB test? That is the question that motivates this paper. Along the way one gets a nice review of JB theory and the issues in finite sample approximation.
Yi-Ting Chena and Chung-Ming Kuanb, "A Generalized Jarque-Bera Test of Conditional Normality" This paper and the next take up the issue of conditional normality testing. This issue is more sophisticated that most of our course, but the question is critical if one wants to address the adequacy of models such as those called "Stochastic Volatility Models." One charm of the SV models is that they can produce unconditional values that have the skewness and long-tailed properties that one expects in equity returns. Still, these models are conditionally normal, so we would like to see if the data we have is also conditionally normal.
G. Fiorentini, E. Sentana,and G. Calzolari "On the validity of the Jarque-Bera Normality Test in Conditionally Heteroskedastic Dynamic Regression Models" This is yet more technical. Still, it adds a useful dimension to the Chena and Kuanb discussion.
Jarque, C.M. and A.K. Bera (1980), “Efficient Tests for Normality, Homoskedasticity and Serial Independence of Regression Residuals,” Economic Letters, 6, 255-59
Jarque, C.M. and A.K. Bera (1987), “A Test for Normality of Observations and Regression Residuals,” International Statistical Reviews, 55, 163-72
Normality testing is a subfield of Goodness-of-Fit Testing. The later is a gargantuan subfied of statistics. Moreover, the field is muddy, riddled with brackish streams, dotted by peat bogs, and bordered by quicksand. Few who enter have ever returned.