# Excess kurtosis is defined as 3 k ek it follows that

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Excess kurtosis is defined as: 3 K EK - = It follows that, for a normal distribution, the excess kurtosis is 0 .

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Econ 325 – Normality Test 3 A fat-tailed or thick-tailed distribution has a value for kurtosis that exceeds 3 . That is, excess kurtosis is positive. This is called leptokurtosis . The graph below compares the shape of the probability density function for the normal distribution and a fat-tailed distribution. 0 normal distribution kurtosis = 3 fat-tailed distribution kurtosis > 3 Econ 325 – Normality Test 4 The above calculation formula for skewness and kurtosis are considered suitable for ‘large samples’. Formula that incorporate ‘small sample’ adjustments are available. The adjusted calculation formula for skewness is: 2 3 2 n 1 i 3 i 1 ) s ( ) x x ( ) 2 n ( ) 1 n ( n g = - - - = The adjusted calculation formula for excess kurtosis is: ) 3 n ( ) 2 n ( ) 1 n ( 3 s x x ) 3 n ( ) 2 n ( ) 1 n ( ) 1 n ( n g 2 n 1 i 4 i 2 - - - - - - - - + = = With Microsoft Excel the function SKEW reports skewness and the function KURT reports excess kurtosis using the formula 1 g and 2 g .
Econ 325 – Normality Test 5 The Jarque-Bera test for normality is now presented. Consider testing the null hypothesis: : H 0 normal distribution, skewness is zero and excess kurtosis is zero; against the alternative hypothesis: : H 1 non-normal distribution. The Jarque-Bera test statistic is: + = 24 ) EK ( 6 S n JB 2 2 It turns out that this test statistic can be compared with a 2 χ ( chi-square) distribution with 2 degrees of freedom.

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