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Properties of Jarque-Bera test

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6 = 1# H HHIL H HHQM H HJHQ H HJHK H HHQQ H HJM H HJMN H HJJO H HJ H HHQO H HHQN

6 = 2# H HHPJ H HHOQ H HJHI H HJKJ H HJLM H HJMN H HJLN H HJML H HJJI H HJMJ H HJHQ 6 = ]# H HHPM H HJHN H HJJP H HJIM H HJKP H HJLM H HJKI H HJKK H HJLO H HJKI H HJMI

6 = 3# H HHIL H HJJK H HJKJ H HJMM H HJII H HJLJ H HJL H HJKP H HJLM H HJJM H HJMN *:EBG7d#FE>A:Bo?#:I=#GF9GE9F>@:=?L# # "FK97#1#(<;@B@GF9#GB@>@GF9#8F9E7?#D:B#>A7#NFBOE7PQ7BF#>7?>#:D#>A7#<E9>@8FB@F>7#=:B<F9@>C# 6 = 1# 6# = 2# #######p# =# 5>6# 5>53# 5>56# 5>556# 5>6# 5>53# 5>556# 5>5556# 6# 1# 2# ]# 3# c# b# e# \# 15# ]>b6]\# b>662c# 6]>\b6\# 1\>1beb# c>b\3# \>2b\c# 6e>15cc# 2b>61\1# 25# 3>3]5\# e>6eec# 6b>e]2b# 2e>3\]]# e>1213# 66>]112# 16>36\3# ]c>5b1# ]5# c>5]be# e>32\1# 6\>5321# ]]>b116# e>3ce2# 61>63b3# 12>16]6# 3b>cbee# 35# c>1]c2# e>\33\# 6e>b3cc# ]\>cec# e>b55]# 61>6b6# 1]>2e13# 36>b31c# c5# c>25\b# \>6\11# 6e>13]\# ]c>b3ec# \>]3]# 61>\3cc# 13>c32e# c5>51\]# b5# c>c16\# \>31]\# 6\>]b12# ]c>265c# \>3121# 62>52c\# 1]>ec1e# ]b>cbeb# e5# c>b3\3# \>\136# 15>6ece# ]b>21b]# \>c]3]# 62>5c36# 1]>e\2]# 3]>\]2# \5# c>bb\# \>cb1# 15>5616# ]5>ec]# \>ebbc# 62>]b36# 1]>6163# ]]>666\# 655# c>\6c1# \>]621# 6e>]35\# ]2>16e1# \>e]6\# 62>6c11# 12>2e61# ]\>]32\# 665# c>\3]# \>2cc# 6e>62\1# ]]>6e23# 65>51]3# 62>2cc3# 12>\11]# 3]>6e]2# 615# c>\513# \>2]c1# 6e>265]# ]6>51b2# \>b252# 62>6113# 12>2]12# ]3>b5c#

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6# 1# 2# ]# 3# c# b# e# \#

6 = ]# 6 = 3#

# e>\c1\# 61>1b\\# 16>\561# ]6>1\e# 66>251b# 6]>31]1# 1]>\b5e# 35>ccb# 25# 65>]c3c# 6]>2\21# 1c>]1e# 3e>3ee6# 61>b3]]# 6c>bbeb# 25>\ce6# 3b>\1]3# ]5# 66>6b]c# 63>6b6e# 1b>ec5c# 3]>e\51# 62>\2\2# 6e>6321# 21>11c6# 3b>6623# 35# 66>\be3# 6c>633# 1\>cbeb# 31>3622# 6]>1bb]# 6e>3b66# 26>1116# c3>13]3# c5# 66>\32b# 6c>6# 1e>\c\# 3\>221b# 6]>c2e# 6\>53ee# 2]>6\16# c]>5e3b# b5# 61>]\# 6c>]666# 1\>1b]3# ce>]5cb# 63>1\b]# 6\>3c\\# 22>5]2\# c\>e65\# e5# 61>362b# 6c>b3bc# 1e>bb23# c3>]\6e# 63>21b3# 6\>b3ec# 22>e6b]# c1>1e36# \5# 61>ce3# 6c>2b52# 1e>\ee# 3c>1621# 63>256e# 6\>2cb6# 22>211c# cb>6b5\# 655# 61>b\6]# 6c>b2eb# 1\>61# 3e>31b2# 63>]221# 6\>]c5c# 21>ceb]# c6>365\# 665# 61>cc2b# 6c>1ec3# 1\>1\]2# c6>5]\3# 63>]6\e# 6\>2\\2# 25>]b21# c]>2b\c# 615# 61>cbe\# 6c>]2]3# 1b>b]e1# ]e>32c3# 63>c21e# 6\>\b6b# 26>32\e# 31>66]\#

*:EBG7d#FE>A:Bo?#:I=#GF9GE9F>@:=?L# # # "FK97#2#"A7#?@M7#:D#>A7#NFBOE7PQ7BF#>7?>#D:B#>A7#<E9>@8FB@F>7#=:B<F9#H@?>B@KE>@:=#D:B#6 j#14#24#]4#34## <#j#546V#5453V#5456#F=H#0#j#154#254#]54#354#c54#b54#e54#\54#6554#6654#615L# *F<;97#?@M7#[0^# -@<7=?@:=# 15# 25# ]5# 35# c5# b5# e5# \5# 655# 665# 615# 6 = 1# 5>5\\e# H JHJO H JHHQ H JHJL H JHJK H JHHL 5>5\\3 H JHHP 5>5\\3 5>5\\1# H JHHM

6 = 2# 5>6566# 5>5\b]# 5>5\e]# 5>652# 5>5\e\ 5>5\bc 5>5\e2 5>656] 5>5\\c 5>5\\c# H JHJM

6 = ]# 5>5\\c# H JHJP H J H JHHN 5>6551 5>5\e3 H JHHK 5>\b\# 5>\\]# H JHJP H JHHQ

6 = 3# 5>5\be# 5>5\e1# 5>5\\]# H JHHM 5>6553 5>5\b\ 5>5\\e 5>5\e2 5>5\\1 H JHJJ H JHHK

.# = #5L53#

6 = 1# 5>5]\6# 5>5]\b# 5>5]\e# H HI H HIJM 5>5]\e 5>5]ee H HIHN 5>5]ee H HIHP# H HIHP

6 = 2# 5>536b# 5>5]\1# 5>5]be# H HIMM H HIHN 5>5]e\ 5>5]\b H HIJ 5>5]ee 5>5]\3# H HIJP

6 = ]# 5>5]\e# H HIHJ 5>5]\3# 5>5]\\ 5>5]\\ 5>5]\6 H HIJ H HIHK 5>5]\1 H HIJ H HIHP

6 = 3# 5>53# H HIOM 5>5]\2# 5>5]\3 H HIHO 5>5]be H HIHM 5>5]e3 5>5]\6 H HIM H HIHN

.# = #5456#

6 = 1# 5>55\b# H HJHJ 5>55\\# H HJ H HJHI 5>55\c 5>55\b 5>56# 5>55\c 5>55\c# 5>55\\#

6 = 2# 5>5653# 5>55\e# 5>55\b# H HJHM H HJHI 5>55e\ 5>55\2 H HJJ H HJHL 5>55\3# 5>55\e#

6 = ]# 5>55\b# H HJHO H HJHJ 5>55\3 H HJHL H HJHL 5>56# 5>55\\ 5>55\] 5>55\2# H HJHK

6 = 3# 5>55\\# 5>55e\# 5>55\b# H HJHL H HJ 5>55\e H HJHN H HJHK 5>55\e H HJJ H HJHK

.# = #5L556 6 = 1# 5>555e# H HHJM 5>555\# 5>556# 5>555\ 5>556# 5>556# 5>555\ H HHJJ 5>555\# 5>556# 6 = 2# 5>555\# 5>556# 5>556# 5>556# 5>556# 5>555\ 5>556# H HHJJ H HHJJ 5>555\# 5>556# 6 = ]# 5>556# 5>555\#H HHJM H HHJM H HHJJ 5>5561 5>555\ 5>556# 5>555\ 5>555\# H HHJM 6 = 3# 5>555\# 5>555\#H HHJJ H HHJJ H HHJJ 5>556# 5>556# 5>556# 5>555e 5>556# H HHJJ *:EBG7d#FE>A:Bo?#:I=#GF9GE9F>@:=?L#

(7)

ZB:;7B>@7?#:D#>A7#NFBOE7PQ7BF#"7?> e6# "FK97#]#(<;@B@GF9#;:I7B#:D#>7?>?#E=H7B#F9>7B=F>@87#7<;@B@GF9#W7@KE99#H@?>B@KE>@:=#6#j#14#24#]4#3V#p#j#546V# 5453V#5456#F=H#0#j#154#254#]54#354#c54#b54#e54#\54#6554#6654#615# *F<;97#?@M7#[0^# -@<7=?@:=# 15# 25# ]5# 35# c5# b5# e5# \5# 655# 665# 615# 6#j#1# 5>6# 5>53\1 5>5\66# 5>611]# 5>6]c# 5>6bb# 5>151b 5>11\6 5>1321 5>1e1c 5>25bc# 5>22c# 5>53# 5>5]1# 5>5c3c# 5>ee6# 5>6533 5>622# 5>6]e2 5>6cbe 5>6e\e 5>15cc 5>113b# 5>1]b3# 5>56# 5>515b 5>23c# 5>535\# 5>5c5c 5>5bc\ 5>5e3] 5>5\c6 5>65c6 5>66c3 5>61]\# 5>62\# 5>556# 5>565] 5>56e1# 5>51b1# 5>5223 5>5]1e 5>5]1\ 5>531c 5>53bb 5>5c# 5>5c2c# 5>5b61#

6# = #2#

5>6# 5>53]c 5>5e]# 5>66be# 5>6]6# 5>6cb# 5>6\12 5>1151 512\\# 5>1c\c 5>1e\e# 5>263\# 5>53# 5>2b]# 5>53bb# 5>5e2]# 5>6523 5>616\ 5>6]]\ 5>6c12 5>6e51 5>6\b6 5>16]# 5>121e# 5>56# 5>56\# 5>521b# 5>5]3]# 5>53bc 5>5ce1 5>5e1] 5>5\5] 5>652c 5>666c 5>6113# 5>612e# 5>556# 5>55b\ 5>56c2# 5>51]\# 5>51\\ 5>52c\ 5>5]26 5>5]e3 5>532b 5>5c56 5>5c]1# 5>5b6#

6# = #]#

5>6# 5>532\ 5>5\]1# 5>61]\# 5>63\1 5>6\6c 5>11# 5>1356 5>1e66 5>26cb 5>2]\b# 5>2e16# 5>53# 5>52c1 5>5c\\# 5>5\2]# 5>66ee 5>6]52 5>6c2b 5>6eb2 5>16bc 5>1]]1 5>1c\]# 5>1\1# 5>56# 5>6ee# 5>52\e# 5>5336# 5>5b52 5>5eb] 5>5\ee 5>6652 5>61be 5>6]1c 5>63]]# 5>6b]c# 5>556# 5>55\1 5>56\\# 5>51\3# 5>52b3 5>5]b3 5>53c\ 5>5c61 5>5b56 5>5be1 5>5e]e# 5>5\1b#

6# = #3

5>6# 5>53ce 5>655b# 5>6]bc# 5>6ec# 5>11b# 5>1ce\ 5>2661 5>2c6b 5>]5]e 5>]3cb# 5>]\e1# 5>53# 5>5]52 5>5b]]# 5>666b# 5>6]6] 5>6b6e 5>15\e 5>1]2e 5>1bce 5>2653 5>2c53# 5>2\33# 5>56# 5>515\ 5>52\e# 5>5ce6# 5>5ecc 5>65cb 5>61]\ 5>6]b] 5>6b52 5>6\5c 5>115]b# 5>1]2b# 5>556# 5>565c 5>511e# 5>52cb# 5>5362 5>53\1 5>5c\1 5>5e6# 5>5\]2 5>65b2 5>66bb# 5>62c1# *:EBG7d#FE>A:Bo?#:I=#GF9GE9F>@:=?L# # "FK97#3#(<;@B@GF9#;:I7B#:D#>7?>?#E=H7B#F9>7B=F>@87#7<;@B@GF9#nF<<F#H@?>B@KE>@:=#6#j#14#24#]4#3V#pj546V# 5453V#5456#F=H#0#j154#254#]54#354#c54#b54#e54#\54#6554#6654#615# *F<;97#?@M7#[0^# -@<7=?@:=# 15# 25# ]5# 35# c5# b5# e5# \5# 655# 665# 615# 6#j#1#

5>6# 5>6\]2# 5>22b]# 5>]c6b# 5>3c2] 5>c3e] 5>b2c# 5>e5e\ 5>e3b] 5>e\2c 5>\2# 5>\]b\# 5>53# 5>636b# 5>1bc# 5>2\61# 5>]eee 5>3e6c 5>cc6c 5>b2b] 5>b\b# 5>e]2\ 5>eeb6# 5>\622# 5>56# 5>5\2\# 5>6e\c# 5>1e3c# 5>2c\1 5>]36e 5>32]b 5>c5e\ 5>cb3e 5>b1e2 5>bbe3# 5>e6c\# 5>556# 5>3e2# 5>61]6# 5>155e# 5>1cc\ 5>22\# 5>]5b2 5>]b16 5>32ee 5>3e\e 5>c]\1# 5>cee2#

6# = #2#

5>6# 5>6bb3# 5>26e# 5>]363# 5>3c3] 5>cbb1 5>b]cc 5>e61b 5>e3]c 5>e\33 5>\1bc# 5>\36# 5>53# 5>62e]# 5>1c# 5>2e5e# 5>]e\\ 5>c53b 5>cbb2 5>b333 5>e56e 5>e32c 5>e\2\# 5>\6\]# 5>56# 5>5e33# 5>6e5c# 5>1b2b# 5>2b]1 5>]b\c 5>331\ 5>c231 5>c\61 5>b33b 5>e532# 5>e2\6# 5>556# 5>5]be# 5>66b]# 5>6e\# 5>1c]2 5>23e2 5>]6cc 5>35]6 5>3c5b 5>c1be 5>cbc2# 5>b2]3#

6# = #]#

5>6# 5>6\2b# 5>2bb2# 5>3226# 5>cc]6 5>bb]\ 5>e2c\ 5>e\ce 5>\2]\ 5>\336 5>\b]# 5>\e]2# 5>53# 5>6362# 5>26c1# 5>]c2b# 5>3\1c 5>b5\3 5>be23 5>e36e 5>\561 5>\26] 5>\3e# 5>\b5c# 5>56# 5>5\2]# 5>111\# 5>2]21# 5>]c\e 5>3eeb 5>cc\3 5>bc]c 5>e6e1 5>ecb# 5>\5e]# 5>\26e# 5>556# 5>5333# 5>6]b\# 5>1]c2# 5>2356 5>]cc6 5>32\b 5>c2\e 5>b5ec 5>bb]1 5>e13e# 5>ec1#

6# = #3

5>6# 5>11]1# 5>]]]c# 5>c2e\# 5>bcc3 5>ec11 5>\1b2 5>\c13 5>\b\# 5>\\5c 5>\\32# 5>\\b\# 5>53# 5>6b\e# 5>2bcc# 5>3ccb# 5>b5]c 5>e6]e 5>e\2] 5>\2\e 5>\c]e 5>\e26 5>\\53# 5>\\3\# 5>56# 5>662e# 5>1b]2# 5>]]be# 5>3e22 5>b653 5>e61\ 5>ee61 5>\11] 5>\33] 5>\b23# 5>\e2c# 5>556# 5>5cbe# 5>6eec# 5>2236# 5>]3\e 5>3\1] 5>c\\6 5>be3\ 5>e356 5>\56] 5>\26e# 5>\3c1#

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