A Test of Normality. 1 n S 2 3. n 1. Now introduce two new statistics. The sample skewness is defined as:

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1 A Test of Normality Textbook Referece: Chapter. (eighth editio, pages 59 ; seveth editio, pages 6 6). The calculatio of p values for hypothesis testig typically is based o the assumptio that the populatio distributio is ormal. Therefore, a test of the ormality assumptio may be useful to ispect. A variety of tests of ormality have bee developed by various statisticias. Oe of these tests will be described here. To start, the calculatio of descriptive statistics is reviewed. A data set has the umeric observatios: x, x,..., x. Familiar descriptive statistics are the sample mea: x x i i ad the sample variace: s (x i i x) Now itroduce two ew statistics. The sample skewess is defied as: (xi x) i S ( ~ ) where ~ (x i i x) Skewess gives a measure of how symmetric the observatios are about the mea. For a ormal distributio the skewess is 0. A distributio skewed to the right has positive skewess ad a distributio skewed to the left has egative skewess. The sample kurtosis is defied as: (xi x) i K ( ~ ) Kurtosis gives a measure of the thickess i the tails of a probability desity fuctio. For a ormal distributio the kurtosis is. Excess kurtosis is defied as: EK K It follows that, for a ormal distributio, the excess kurtosis is 0. Eco 5 Normality Test Eco 5 Normality Test

2 A fat tailed or thick tailed distributio has a value for kurtosis that exceeds. That is, excess kurtosis is positive. This is called leptokurtosis. The graph below compares the shape of the probability desity fuctio for the stadard ormal distributio (mea 0 ad variace ) ad a fat tailed distributio, also with mea 0 ad variace. The above calculatio formula for skewess ad kurtosis are cosidered suitable for large samples. Formula that icorporate small sample adjustmets are available. The adjusted calculatio formula for skewess is: g (xi x) i ( )( ) (s ) The adjusted calculatio formula for excess kurtosis is: g ( ) xi x ( )( )( ) s i ( ) ( )( ) Microsoft Excel fuctios are: SKEW reports skewess usig the formula g KURT reports excess kurtosis usig the formula g. Note: the fat tailed distributio draw above is the logistic distributio with probability desity fuctio: b exp( x / b) exp( x / b) with b This distributio has mea 0, variace, coefficiet of skewess equal to 0, ad coefficiet of kurtosis equal to.. Eco 5 Normality Test Eco 5 Normality Test

3 The Jarque Bera test for ormality is ow preseted. Cosider testig the ull hypothesis: H 0 : ormal distributio, skewess is zero ad excess kurtosis is zero; agaist the alterative hypothesis: H : o ormal distributio. The Jarque Bera test statistic is: S (EK) JB 6 The critical values ca be foud from the Appedix Table for the chi square distributio as: Sigificace Level Critical Value The presetatio of this test of ormality is valid for large samples. For small samples the decisio rule ca be viewed as approximate. It turs out that this test statistic ca be compared with a (chi square) distributio with degrees of freedom. The ull hypothesis of ormality is rejected if the calculated test statistic exceeds a critical value from the () distributio. 5 Eco 5 Normality Test 6 Eco 5 Normality Test

4 Example: A stock market data set has daily percetage returs observed for the year 997 for two compaies Barrick Gold ad Bak of New York. The sample has observatios for = 5 tradig days. For each compay, a exercise is to test for ormality of the daily returs. Various statistics are give i the table below. Both the small sample ad large sample versios of the skewess ad excess kurtosis statistics are preseted to give emphasis to the methodology. Barrick Gold Small sample statistics Bak of NY Skewess g Excess Kurtosis g.8 0. For Barrick Gold, the Jarque Bera test statistic of 8.7 exceeds the critical values for ay reasoable sigificace level to lead to the coclusio that the daily returs do ot follow a ormal distributio. Sice the excess kurtosis statistic is greater tha zero, the appearace is that the daily returs follow a distributio that features leptokurtosis. Researchers have suggested that the leptokurtosis arises from a patter of volatility i fiacial markets where periods of high volatility are followed by periods of relative stability. A p value for the test statistic is calculated as a chi square distributio probability ad, with Microsoft Excel, is computed with the fuctio: CHISQ.DIST.RT(test_statistic, ) degrees of freedom Large sample statistics Skewess S Excess Kurtosis EK. 0.8 Jarque-Bera test for ormality calculated with the large sample statistics JB test statistic 8.7. p-value < Eco 5 Normality Test 8 Eco 5 Normality Test

5 For the Bak of New York, the calculatio of a p value for the Jarque Bera test statistic is illustrated i the graph below. A statistical result is that the (chi square) distributio with two degrees of freedom is a expoetial distributio. It is clear that the calculated p value is greater tha ay usual sigificace level (such as = 0.0, 0.05 or 0.0) to suggest that there is o evidece to reject the ull hypothesis of a ormal distributio for the daily returs of the Bak of New York. 9 Eco 5 Normality Test

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