A Test of Normality. Textbook Reference: Chapter 14.2 (eighth edition, pages 591 3; seventh edition, pages 624 6).

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1 A Test of Normalty Textbook Referece: Chapter 4. (eghth edto, pages 59 ; seveth edto, pages 64 6). The calculato of p-values for hypothess testg typcally s based o the assumpto that the populato dstrbuto s ormal. Therefore, a test of the ormalty assumpto may be useful to spect. A varety of tests of ormalty have bee developed by varous statstcas. Oe of these tests wll be descrbed here. To start, the calculato of descrptve statstcs s revewed. A data set has the umerc observatos: x, x,..., Famlar descrptve statstcs are the sample mea: x. x = x ad the sample varace: s = x) Eco 5 Normalty Test

2 Now troduce two ew statstcs. The sample skewess s defed as: S = ( σ ~ x) ) where σ~ = x) Skewess gves a measure of how symmetrc the observatos are about the mea. For a ormal dstrbuto the skewess s 0. A dstrbuto skewed to the rght has postve skewess ad a dstrbuto skewed to the left has egatve skewess. The sample kurtoss s defed as: K = ( σ ~ x) ) 4 Kurtoss gves a measure of the thckess the tals of a probablty desty fucto. For a ormal dstrbuto the kurtoss s. Excess kurtoss s defed as: EK = K It follows that, for a ormal dstrbuto, the excess kurtoss s 0. Eco 5 Normalty Test

3 A fat-taled or thck-taled dstrbuto has a value for kurtoss that exceeds. That s, excess kurtoss s postve. Ths s called leptokurtoss. The graph below compares the shape of the probablty desty fucto for the stadard ormal dstrbuto (mea 0 ad varace ) ad a fat-taled dstrbuto, also wth mea 0 ad varace. Note: the fat-taled dstrbuto draw above s the logstc dstrbuto wth probablty desty fucto: b exp( x / b) [ + exp( x / b) ] wth b = π Ths dstrbuto has mea 0, varace, coeffcet of skewess equal to 0, ad coeffcet of kurtoss equal to 4.. Eco 5 Normalty Test

4 The above calculato formula for skewess ad kurtoss are cosdered sutable for large samples. Formula that corporate small sample adjustmets are avalable. The adjusted calculato formula for skewess s: g = ( )( ) (s x) ) The adjusted calculato formula for excess kurtoss s: g ( + ) x = ( )( )( ) 4 x s ( ) ( )( ) Mcrosoft Excel fuctos are: SKEW reports skewess usg the formula g KURT reports excess kurtoss usg the formula g. 4 Eco 5 Normalty Test

5 The Jarque-Bera test for ormalty s ow preseted. Cosder testg the ull hypothess: H 0 : ormal dstrbuto, skewess s zero ad excess kurtoss s zero; agast the alteratve hypothess: H : o-ormal dstrbuto. The Jarque-Bera test statstc s: JB = S 6 + (EK) 4 It turs out that ths test statstc ca be compared wth a χ (ch-square) dstrbuto wth degrees of freedom. The ull hypothess of ormalty s rejected f the calculated test statstc exceeds a crtcal value from the χ () dstrbuto. 5 Eco 5 Normalty Test

6 The crtcal values ca be foud from the Appedx Table for the ch-square dstrbuto as: Sgfcace Level α Crtcal Value The presetato of ths test of ormalty s vald for large samples. For small samples the decso rule ca be vewed as approxmate. 6 Eco 5 Normalty Test

7 Example: A stock market data set has daly percetage returs observed for the year 997 for two compaes Barrck Gold ad Bak of New York. The sample has observatos for = 5 tradg days. For each compay, a exercse s to test for ormalty of the daly returs. Varous statstcs are gve the table below. Both the small sample ad large sample versos of the skewess ad excess kurtoss statstcs are preseted to gve emphass to the methodology. Barrck Gold Bak of NY Small sample statstcs Skewess g Excess Kurtoss g Large sample statstcs Skewess S Excess Kurtoss EK. 0.8 Jarque-Bera test for ormalty calculated wth the large sample statstcs JB test statstc 8.7. p-value < Eco 5 Normalty Test

8 For Barrck Gold, the Jarque-Bera test statstc of 8.7 exceeds the crtcal values for ay reasoable sgfcace level to lead to the cocluso that the daly returs do ot follow a ormal dstrbuto. Sce the excess kurtoss statstc s greater tha zero, the appearace s that the daly returs follow a dstrbuto that features leptokurtoss. Researchers have suggested that the leptokurtoss arses from a patter of volatlty facal markets where perods of hgh volatlty are followed by perods of relatve stablty. A p-value for the test statstc s calculated as a ch-square dstrbuto probablty ad, wth Mcrosoft Excel, s computed wth the fucto: CHISQ.DIST.RT(test_statstc, ) degrees of freedom 8 Eco 5 Normalty Test

9 For the Bak of New York, the calculato of a p-value for the Jarque- Bera test statstc s llustrated the graph below. A statstcal result s that the χ (ch-square) dstrbuto wth two degrees of freedom s a expoetal dstrbuto. It s clear that the calculated p-value s greater tha ay usual sgfcace level (such as α = 0.0, 0.05 or 0.0) to suggest that there s o evdece to reject the ull hypothess of a ormal dstrbuto for the daly returs of the Bak of New York. 9 Eco 5 Normalty Test