stationarity and random walk tests
TRANSCRIPT
8/8/2019 STATIONARITY AND RANDOM WALK TESTS
http://slidepdf.com/reader/full/stationarity-and-random-walk-tests 1/16
Rckc|oc UDFgdcry2 Gdyzxoc` Uxmek Ic|kgx Gaaoeog`ey2 C` O`oxocn Zwlnoe Maag|o`d Zg|uzgexoqg. ZfH
xfguou. Ux|cxfenyhg W`oqg|uoxy. WK.3??6
383
8.5 UXCXOM@C\OX_ C@H \C@HMI TCNK XGUXU
8.5.3 O@X\MHWEXOM@
O` xfg z|gug`x ugexom`. tg gvcio`g xfg zmuuolonoxy ma xfg igc`)|gqg|uom`
z|meguu ma xfou xoig ug|ogu toxfo` xfg a|cigtm|k ma xguxo`d oxu uxcxom`c|oxy Dg`g|cnny.
ic`y gem`migx|oe z|mlngiu ec` lg |cough oa c xoig ug|ogu ou `m`)uxcxom`c|y C
uzw|omwu |gncxom`ufoz cim`d xfg ngqgnu ma xfg qc|oclngu emwnh lg xfg |guwnx ma uwef
`m`)uxcxom`c|oxy Oa xfg ug|ogu c|g `m`)uxcxom`c|y. uxc`hc|h |gd|guuom`u xfcx x|y xm
gvznco` xfg qc|oclng lgfcqomw| tonn lg igc`o`dnguu; xfg uxc`hc|h g||m|u ma xfg
zc|cigxg|u tmwnh lg o`em||gex. c`h xfg qc|oc`eg ma am|gecuxu o`xm xfg awxw|g tmwnh lg
o`ao`oxg; xfcx ou. xfg uyuxgi tonn `mx lg c`efm|gh & ugg. Hoekgy; Bc`ug` c`h Xfm|`xm`
3??82?)3:/
Xfg|gam|g. gem`mioe qc|oclngu uwef cu uxmek z|oegu c`h-m| |gxw|`u. ufmwnh lg
imhoaogh lgam|g lgo`d wugh o` |gd|guuom` c`cnyuou C emiim` imhoaoecxom` ma c xoig
ug|ogu qc|oclng o`qmnqgu ao|ux hoaag|g`eo`d Fmtgqg|. xfg ngqgn ma c qc|oclng c`h ao|ux
hoaag|g`eo`d tonn lg qg|y hoaag|g`x o` xg|iu ma igc` c`h qc|ocxom` Hoaag|g`eo`d
tmwnh lg z|gag||gh. oa xfg ao|ux hoaag|g`egu ma c ugx ma qc|oclngu tg|g uxcxom`c|y. toxf
xfg qc|oclngu xfgiugnqgu lgo`d `m`)uxcxom`c|y Xfg`. m`g ec` gvzgex xfcx xfg
qc|oclngu icy lg emo`xgd|cxgh Xfg emo`xgd|cxom` ma uwef qc|oclngu ou uxcxom`c|y. xfwu
xfgy ec``mx imqg xmm ac| a|mi gcef mxfg| M` xfg em`x|c|y. c aconw|g ma emo`xgd|cxom`
uwddguxu xfcx xfgug qc|oclngu fcqg `m nm`d |w` |gncxom`ufoz @gqg|xfgnguu. uxclng
no`gc| |gncxom`ufozu cim`d qc|oclngu c|g. `mtchcyu. xfg amewu ma emo`xgd|cxom` xguxu
Cu c |guwnx. xfg clug`eg ma emo`xgd|cxom` cim`d qc|oclngu hmgu `mx `geguuc|ony igc`
xfcx xfg|g ou `m uxclng nm`d)|w` |gncxom`ufozu cim`d qc|oclngu Ox umngny uwddguxu xfcx
xfg|g ou `m uxclng nm`d)|w` no`gc| |gncxom`ufoz cim`d xfgi Ceem|ho`dny. xfg
8/8/2019 STATIONARITY AND RANDOM WALK TESTS
http://slidepdf.com/reader/full/stationarity-and-random-walk-tests 2/16
Rckc|oc UDFgdcry2 Gdyzxoc` Uxmek Ic|kgx Gaaoeog`ey2 C` O`oxocn Zwlnoe Maag|o`d Zg|uzgexoqg. ZfH
xfguou. Ux|cxfenyhg W`oqg|uoxy. WK.3??6
38<
uxcxom`c|oxy ma xfg wxonough xoig ug|ogu o` xfou ugexom` c|g o`qguxodcxgh Xguxu ma |c`hmi
tcnku. cx xfg am|icn ngqgn. c|g em`hwexgh lcugh m` xfg Hoekgy c`h Awnng| &3?4?/
igxfmhmnmdy c`h xfg qc|oc`eg)|cxom xgux ma Nm c`h Iceko`ncy &3?66/
8.5.< XGUX MA UXCXOM@C\OX_ LCUGH M@ EM\\GNMD\CI
Cx xfg o`am|icn ngqgn. tgck uxcxom`c|oxy ec` lg xguxgh wuo`d xfg em||gnmd|ci ma
c xoig ug|ogu. tfoef ou c d|czf ma cwxmem||gncxom` cx qc|omwu ncdu Am| c uxcxom`c|y
xoig ug|ogu. xfg em||gnmd|ci xczg|u maa swoekny. tfg|gcu am| `m`)uxcxom`c|y xoig ug|ogu
ox hogu ma d|chwcnny [ugg Dwbc|cxo &3??7/ c`h Uxgtc|x &3??3/Y Xm guxclnouf xfg
em||gnmd|ci ma mw| qc|oclngu. tg uxc|x ly wuo`d xfg Cwxmem||gncxom` Aw`exom`. tfoef
ou a|gswg`xny hgao`gh cu2 - k k : . xfg |cxom ma xfg ucizng emqc|oc`eg xm ucizng
qc|oc`eg Fg`eg. tg fcqg mlxco`gh Aodw|g 8)<. houzncyo`d xfg ucizng em||gnmd|ci ma
xfg ngqgnu c`h xfgo| ao|ux hoaag|g`egu ma xfg gngqg` Gdyzxoc` uxmek o`hoegu Xfg
em||gnmd|ci ou ufmt` wz xm <7 ncdu Xfg gqohg`eg a|mi xfgug em||gnmd|ci aodw|gu
ufmtu xfcx xfg guxoicxgh cwxmem||gncxom`u hog hmt` swoekny am| xfg ngqgnu cu tgnn cu
xfg ao|ux hoaag|g`egu ma xfg qc|oclngu. c`h xfg` czzgc| xm anwexwcxg o` c `m`)uyuxgicxoe
tcy c|mw`h c`h enmug xm rg|m Tg |gcef xfg xg`xcxoqg em`enwuom` xfcx xfg qc|omwu
uxmek o`hoegu emwnh lg tgckny uxcxom`c|yIm|gmqg|. tg bwhdg xfg uxcxouxoecn
uod`oaoec`eg ma c`y ly oxu uxc`hc|h g||m|3 Xclng 8)3: onnwux|cxgu xfcx imux ma xfg
emgaaoeog`xu ma xfg ngqgnu. am| N\3. N\<. N\5. N\7. N\1. N\4. c`h N\3: c|g
o`hoqohwcnny uxcxouxoecnny o`uod`oaoec`x
3Am| gvcizng. mw| hcxc uorg ou ` 9473. oiznyo`d c qc|oc`eg ma 3-473 m| uxc`hc|h g||m| ma 3- 473 9
<48: Xfg`. amnnmto`d xfg z|mzg|xogu ma xfg uxc`hc|h `m|icn houx|olwxom`. xfg ?7' em`aohg`eg o`xg|qcn
am| c`y ufmwnh lg 3?1 &<48:/ 9 ::4< m` goxfg| uohg ma rg|m Xfwu. oa acnnu o`uohg xfg
o`xg|qcn &)::4<. ::4</. tg hm `mx |gbgex xfg fyzmxfguou xfcx xfg x|wg ou rg|m Lwx oa ox nogu mwxuohg
xfou em`aohg`eg o`xg|qcn. xfg` tg ec` |gbgex xfg fyzmxfguou xfcx xfg x|wg ou rg|m
k
k
k
k
8/8/2019 STATIONARITY AND RANDOM WALK TESTS
http://slidepdf.com/reader/full/stationarity-and-random-walk-tests 3/16
Rckc|oc UDFgdcry2 Gdyzxoc` Uxmek Ic|kgx Gaaoeog`ey2 C` O`oxocn Zwlnoe Maag|o`d Zg|uzgexoqg. ZfH
xfguou. Ux|cxfenyhg W`oqg|uoxy. WK.3??6
385
Aodw|g 8)< Xfg Em||gnmd|ci ma xfg Ngqgnu ma xfg Gngqg` Gdyzxoc` Hcony \gxw|` O`hoegu &3??8)3??1/
@mxgu2\39Cd|oewnxw|g. \<9Io`o`d. \59Em`ux|wexom`. \89Ic`wacexw|o`d. \79X|c`uzm|xcxom`. \19X|chg. \49Ao`c`eg.
\69Ug|qoegu. \?9Zwlnoe Uwlue|ozxom`. \3:9Enmugh Uwlue|ozxom`. \339Dg`g|cn o`hgv. &N9 xfg nmd ma xfg o`hgv ngqgn/
8/8/2019 STATIONARITY AND RANDOM WALK TESTS
http://slidepdf.com/reader/full/stationarity-and-random-walk-tests 4/16
Rckc|oc UDFgdcry2 Gdyzxoc` Uxmek Ic|kgx Gaaoeog`ey2 C` O`oxocn Zwlnoe Maag|o`d Zg|uzgexoqg. ZfH
xfguou. Ux|cxfenyhg W`oqg|uoxy. WK.3??6
388
Aodw|g 8)5 Xfg em||gnmd|ci ma xfg Ao|ux Hoaag|g`egu ma xfg Gngqg` Gdyzxoc` Hcony \gxw|`
O`hoegu &3??8) 3??1/
@mxgu2\39Cd|oewnxw|g. \<9Io`o`d. \59Em`ux|wexom`. \89Ic`wacexw|o`d. \79X|c`uzm|xcxom`. \19X|chg. \49Ao`c`eg.
\69Ug|qoegu. \?9Zwlnoe Uwlue|ozxom`. \3:9Enmugh Uwlue|ozxom`. \339Dg`g|cn o`hgv. &H 9 xfg ao|ux hoaag|g`eg/
8/8/2019 STATIONARITY AND RANDOM WALK TESTS
http://slidepdf.com/reader/full/stationarity-and-random-walk-tests 5/16
Rckc|oc UDFgdcry2 Gdyzxoc` Uxmek Ic|kgx Gaaoeog`ey2 C` O`oxocn Zwlnoe Maag|o`d Zg|uzgexoqg. ZfH
xfguou. Ux|cxfenyhg W`oqg|uoxy. WK.3??6
387
Xclng 8)3: Xgux am| Cwxmem||gncxom` Emgaaoeog`x ma xfg Ngqgnu
Cwxmem||gncxom` emgaaoeog`x
Ncdu N\3 N\< N\5 N\8 N\7 N\1 N\4 N\6 N\? N\3: N\33
3 ::< )::: ::3 :37 ::: )::: ::4 :3: :<: ::7 :<1
< ::7 )::: :53 ::6 ::: )::: )::6 :<< :<1 ::5 :<65 ::1 )::: ::5 :3< ::: ::3 )::: :31 :34 ::5 :<8
8 ::6 )::: ::8 ::? ::: )::: )::3 :33 :37 ::3 :34
7 ::6 )::: ::3 ::1 ::: )::: )::1 ::? :37 ::< :37
1 ::8 )::: ::8 ::7 ::: )::: ::8 ::7 ::7 ::< :38
4 ::8 )::: ::5 ::< ::: )::: :3: ::? :34 ::5 :35
6 ::1 )::: ::< ::8 ::: )::: )::3 ::8 ::6 ::8 :35
? ::8 )::: )::: ::3 ::: )::: )::? ::7 ::8 ::< :3:
3: ::: )::: )::: ::5 ::: ::: ::5 ::1 ::8 ::3 ::6
33 ::: )::: )::: ::< ::: )::: ::5 ::8 ::4 ::< :35
3< ::< )::: )::: ::< ::: )::: )::3 ::4 ::7 ::< ::<
35 ::6 )::: )::: )::< ::: ::6 )::3 ::4 ::< ::5 ::138 ::7 )::: )::: ::< ::: )::: ::6 ::4 ::7 ::5 ::6
37 )::< )::: )::3 ::< ::: )::3 )::3 ::3 ::6 ::3 ::6
31 )::5 )::: )::: ::3 ::: )::: ):37 ::8 ::8 ::< ::6
34 ::6 )::: )::: ::< ::5 )::: ::3 ::: ::5 ::< ::1
36 ):33 )::: )::: ::: ::: )::: ::< ::< ::8 ::< ::8
3? :3: )::: )::: )::< ::: )::: ::3 )::3 )::: ::3 ::3
<: )::1 )::: )::: )::3 ::: )::: ::: )::< )::5 ::: )::<
<3 ::5 )::: )::: ::: ::: )::: ::3 ::5 ::< )::3 ::<
<< ::8 )::: )::: )::< )::< )::: )::: ::3 ::< )::3 ::5
<5 )::3 )::: )::: )::5 ::: ::< ::3 ::3 ::3 ::3 ::<
<8 )::3 )::: )::: ::: ::: )::: ::< ::5 )::< ::< ::3<7 )::7 )::: )::: ::: ::: )::: ::< ::< )::3 ::< ::3@mxgu2 @mxgu2\39Cd|oewnxw|g. \<9Io`o`d. \59Em`ux|wexom`. \89Ic`wacexw|o`d. \79X|c`uzm|xcxom`. \19X|chg. \49Ao`c`eg.
\69Ug|qoegu. \?9Zwlnoe Uwlue|ozxom`. \3:9Enmugh Uwlue|ozxom`. \339Dg`g|cn o`hgv. c`h xfg ngxxg| N |gag|u xm xfg nmd ma c
qc|oclng Xfg ?7' em`aohg`eg o`xg|qcn am| ou &)::4<. ::4</
Xfwu. tg emwnh `mx |gbgex xfg fyzmxfguou xfcx xfg x|wg ou rg|m. lwx ox icy lg
|gbgexgh am| N\8. N\6. N\?. c`h N\33 Xfg|gam|g. umig ma xfgug xoig ug|ogu uggiu
xm lgfcqg cu tfoxg `moug. ugg. Xclng 8)3: O` m|hg| xm xgux xfg bmo`x fyzmxfguou xfcx cnn
xfgk cwxmem||gncxom` emgaaoeog`xu c|g uoiwnxc`gmwuny gswcn xm rg|m. tg wug xfg S
uxcxouxoe hgqgnmzgh ly Lmv c`h Zog|eg &3?4:/ . tfoef ou hgao`gh cu
S `k
k
i
<
3
&835/
k
8/8/2019 STATIONARITY AND RANDOM WALK TESTS
http://slidepdf.com/reader/full/stationarity-and-random-walk-tests 6/16
Rckc|oc UDFgdcry2 Gdyzxoc` Uxmek Ic|kgx Gaaoeog`ey2 C` O`oxocn Zwlnoe Maag|o`d Zg|uzgexoqg. ZfH
xfguou. Ux|cxfenyhg W`oqg|uoxy. WK.3??6
381
tfg|g ` ou xfg ucizng uorg c`h i ou xfg ncd ng`dxf Am| mw| hcxc. xfg S uxcxouxoe lcugh
m` <7 ncdu ou onnwux|cxgh o` Xclng 8)33 M`ny. xfg S uxcxouxoe am| xfg ngqgnu ma N\<.
N\7 c`h N\1 hmgu `mx gvegghu xfg e|oxoecn qcnwg a|mi xfg efo)uswc|g xclng cx c`y
|gzm|xgh ngqgn ma uod`oaoec`eg Fmtgqg|. Am| xfg mxfg| qc|oclngu. cnn lgo`d fodfny
uod`oaoec`x; xfg z qcnwgu ma mlxco`o`d uwef efo)uswc|g qcnwgu c|g z|cexoecnny rg|m
Xfg|gam|g. tg icy `mx |gbgex xfg `wnn fyzmxfguou xfcx cnnk c|g cnn rg|m. am| xf|gg xoig
ug|ogu &og N\<. N\7. c`h N\1/ Fmtgqg|. tg emwnh uoizny |gbgex xfg `wnn fyzmxfguou
xfcx cnn c|g cnn rg|m C qc|oc`x ma xfg Lmv)Zog|eg S uxcxouxoe ou xfg Nbw`d)Lmv
&3?46/ uxcxouxoe. tfoef ou hgao`gh cu2
NL `&` </` k
k
<
k 3
i
&
/
i V < &838/
Cnxfmwdf o` nc|dg ucizngu lmxf S c`h NL uxcxouxoe amnnmt xfg efo)uswc|g houx|olwxom`
toxf i ha. xfg NL uxcxouxoe fcu lgg` amw`h xm lg im|g zmtg|awn xfc` S uxcxouxoe o`
uicnn ucizngu Ecnewncxo`d NL uxcxouxoe. tg emwnh `mx |gbgex xfg `wnn fyzmxfguou xfcx
cnn c|g rg|m. am| gcef ma mw| xoig ug|ogu Xfwu. lcugh m` xfg NL uxcxouxoe. xfg
mqg|cnn em`enwuom` ou xfcx xfg uxmek |gxw|`u xoig ug|ogu icy |gz|gug`x uxcxom`c|oxy
Xclng 8)33 Xgux am| Cwxmem||gncxom` Lcugh m` Lmv)Zog|eg c`h Nbw`d)Lmv Uxcxouxoeu
N\3 N\< N\5 N\8 N\7 N\1 N\4 N\6 N\? N\3: N\33
S 7754 ::5: 46:4: 8476: :?1: 711: 738?: 3:74?: 36536 <?13<: <47:3:
NL :57: :::: 45:: :1:: :::3 ::5: :15: <81: 146: 33738: 3<85:
@mxgu2 Xfg bmo`x `wnn fyzmxfguou ou xfcx cnn xfgk cwxmem||gncxom` emgaaoeog`xu c|g uoiwnxc`gmwuny gswcn xm rg|m Xfg mxfg|
ngxxg|u c|g hgao`gh o` Xclng 8)3
@gqg|xfgnguu. xfg wug ma d|czfoecn c`h em||gnmd|ci gqohg`eg ou w`|gnoclng o`
icko`d o`ag|g`eg clmwx xfg uxcxom`c|oxy c`h w`ox |mmxu. c`h tg `mt xw|` xm xfg am|icn
xguxo`d ux|cxgdogu. gvcio`o`d gcef ma xfgug ug|ogu am| xfg z|gug`eg ma w`ox |mmxu
8.5.5 XFG CWDIG@XGH HOEKG_)AWNNG\ IGXFMHMNMD_
k
k
8/8/2019 STATIONARITY AND RANDOM WALK TESTS
http://slidepdf.com/reader/full/stationarity-and-random-walk-tests 7/16
Rckc|oc UDFgdcry2 Gdyzxoc` Uxmek Ic|kgx Gaaoeog`ey2 C` O`oxocn Zwlnoe Maag|o`d Zg|uzgexoqg. ZfH
xfguou. Ux|cxfenyhg W`oqg|uoxy. WK.3??6
384
Cx xfg am|icn ngqgn. xfg ao|ux. uxcxom`c|oxy ma c xoig ug|ogu ou efgekgh ly
giznmyo`d xfg Cwdig`xgh Hoekgy)Awnng| &CHA/ uxcxouxoe. hgqgnmzgh ly Hoekgy c`h
Awnng| &3?4?/ Xfg|gam|g. tg hg`mxg \x cu xfg ufc|g |gxw|` o`hgv. c`h amnnmt xfg
ux|cxgdy mwxno`gh o` Fmnhg` c`h Zg|ic` &3??8/ ma em`ux|wexo`d c `wilg| ma uxcxouxoeu
lcugh wzm` xfg amnnmto`d xf|gg |gd|guuom` gswcxom`u2
\x 9 \x)3 + gx x 9 3. <. &8373/
\x 9 \x)3 + + gx x 9 3. <. &837</
\x 9 \x)3 + + x + gx x 9 3. <. &8375/
O` xfgug gswcxom`u. ox ou cuuwigh xfcx xfg houxw|lc`eg xg|i. gx . ou OOH z|meguu
Hoekgy c`h Awnng| &3?63/ enc|oaogu xfcx xfg noioxo`d houx|olwxom`u c`h e|oxoecn qcnwgu
xfcx xfgy mlxco` w`hg| xfg cuuwizxom` xfcx gx ou c` OOH z|meguu c|g cnum qcnoh tfg` gx
ou cwxm|gd|guuoqg oa cwdig`xgh Hoekgy)Awnng| & CHA / |gd|guuom` ou |w` Xfg|gam|g. O`
cuuwio`d xfg hcxc c|g dg`g|cxgh ceem|ho`d xm &8373/ toxf 9 3 c`h xfcx gx ou c
uxcxom`c|y cwxm|gd|guuoqg ma m|hg| z
gx 9 3gx)3 + <gx)< ++zgx)z + x &831/
tfg|g hgao`gu c` OOH z|meguu. c`h em`uohg| xfg |gzc|cigxg|oucxom` qg|uom` ma
\x 9 \x)3 + + x + gx x 9 3. <.
tfoef ou
x \ 9 x \x)3 + gx &834/
Doqg` xfg gswcxom` am| gx o` &831/ tg |gt|oxg &834/ cu
x \ 9 x \x)3 +3gx)3 + <gx)< ++zgx)z + x
tfoef ec` lg t|oxxg`. uo`eg & \x 9 \x)3 + gx / toxf 93 doqgu gx 9 \x ) \x)3. cu
x \ 9 x \x)3 +3& \x)3 ) \x)</+ <& \x)< ) \x)5/+ ++z& \x)z ) \x)z)3/ + x
8/8/2019 STATIONARITY AND RANDOM WALK TESTS
http://slidepdf.com/reader/full/stationarity-and-random-walk-tests 8/16
Rckc|oc UDFgdcry2 Gdyzxoc` Uxmek Ic|kgx Gaaoeog`ey2 C` O`oxocn Zwlnoe Maag|o`d Zg|uzgexoqg. ZfH
xfguou. Ux|cxfenyhg W`oqg|uoxy. WK.3??6
386
xfcx ou x \ ou |gd|guugh m` \x)3. \x)3. \x)<. \x)<.. \x)z cu tgnn cu c` o`xg|egzx
c`h xoig x|g`h ou |gswo|gh O` uwef c ecug. tg tc`x xm xgux
F :2 9 : cdco`ux F C2 > :.
Aodw|g 88 z|gug`xu c hgeouom` x|gg uwiic|y ma xfg xgux z|meghw|g giznmygh o` xfou
ugexom` Xfg guxoicxgh Cwdig`xgh Hoekgy)Awnng| uxcxouxoeu c|g |gzm|xgh o` Xclng 8)3<
am| xfg ngqgnu ma xfg qc|oclngu Cnn o`hoegu czzgc| xm |gbgex xfg `wnn fyzmxfguou xfcx 9
:. cx xfg 7 zg|eg`x ngqgn Xfcx ou. xfg uxmek |gxw|`u ug|ogu hmgu `mx gvfolox c w`ox |mmx.
tfoef ou c`mxfg| tcy ma ucyo`d xfcx xfg gngqg` o`hoegu ma Gdyzxoc` uxmek |gxw|`u c|g
uxcxom`c|y c`h `mx mlgyo`d xfg xfgm|y ma c |c`hmi tcnk
Fmtgqg|. Now c`h Fg &3??3/ fcqg o`hoecxgh xfcx xfg qc|oc`eg)|cxom xgux ma Nm
c`h IceKo`ncy &3?66/ ou im|g zmtg|awn xfc` xfg Hoekgy)Awnng| m| Lmv)Zog|eg xguxu
w`hg| cnxg|`cxoqg fyzmxfguou o`qmnqo`d C\&3/. C\OIC&3.3.:/ m| C\OIC&3.3.3/ Am|
gvcizng. xfg|g c|g umig oizm|xc`x hgzc|xw|gu a|mi xfg |c`hmi tcnk xfcx xfg Hoekgy)
Awnng| w`ox |mmx xgux ec``mx hgxgex Im|g oizm|xc`xny. tfg` xfg cxx|olwxg ma o`xg|gux ou
w`em||gncxgh`guu ma o`e|gig`xu. xfg qc|oc`eg)|cxom xgux ou im|g czz|geocxg xfc` xfg
w`ox |mmx xgux [Nm c`h IceKo`ncy &3?66/Y Xfg|gam|g. xfg qc|oc`eg)|cxom xgux ou
mwxno`gh c`h giznmygh o` xfg amnnmto`d ugexom`
8/8/2019 STATIONARITY AND RANDOM WALK TESTS
http://slidepdf.com/reader/full/stationarity-and-random-walk-tests 9/16
Rckc|oc UDFgdcry2 Gdyzxoc` Uxmek Ic|kgx Gaaoeog`ey2 C` O`oxocn Zwlnoe Maag|o`d Zg|uzgexoqg. ZfH
xfguou. Ux|cxfenyhg W`oqg|uoxy. WK.3??6
38?
Aodw|g 8)8 C Hgeouom` X|gg ma W`ox \mmx Xguxo`d
Guxoicxg
\ x \ \ gx x x o
o
o s
x
3
3
Wug xm xgux
F:2 & . . / 9 & . :. 3/ cdco`ux FC2& . . / & . :. 3/
5
Ou F: |gbgexgh=
Xgux am| 93. wuo`d x5 toxf xfg
e|oxoecn qcnwgu xckg` a|mi xfg
uxc`hc|h `m|icn xclngu
ou 9 3 =
Hgeouom`2
ou `m`)rg|m c`h ou 3. um xfcx xfg
ug|ogu fcu c w`ox |mmx c`h c no`gc|
Hgeouom`2
Xfg|g ou `m w`ox |mmx
Ou 9 : =
Xfg ug|ogu ou uxcxom`c|y
toxf `m no`gc| x|g`h. lwx
zmuuolny toxf c`
o`xg|egzx
Xfg ug|ogu ou uxcxom`c|y
toxf c no`gc| x|g`h. c`h
zmuuolny toxf c`
o`xg|egzx
Wuo`d x)uxcx. Ou 9 : =
Xfg ug|ogu ou
uxcxom`c|y toxf `m
no`gc| x|g`h. lwx
toxfmwx o`xg|egzx
Xfg ug|ogu ou
uxcxom`c|y toxf c
no`gc| x|g`h. c`h
toxf c `m`)rg|m
Xfg|g ou c w`ox |mmx & 93/ toxf `m
x|g`h & 9 :/. lwx toxf zmuuolng h|oax
Xgux am| 93. wuo`d x5 toxf xfge|oxoecn qcnwgu xckg` a|mi xclng
67< o` Awnng| &3?41/
Wug < xm xgux
F:2 & . . / 9 &:. :. 3/
cdco`ux FC2& . . / &:. :. 3/
Ou F: |gbgexgh=
Xfg ug|ogu ou c |c`hmitcnk toxfmwx h|oax
Xfg ug|ogu ou c |c`hmitoxf h|oax
Guxoicxg \x 9 \x)3 + + gx
Wug xfg xgux am|
F:2 &. / 9 & :. 3/
cdco`ux FC2&. / &:. 3/
3
Xfg ug|ogu ou c
|c`hmi tcnk
toxfmwx h|oax
Xfg ug|ogu ou c
|c`hmi tcnk
toxf h|oax
ygu `m
ygu`m
ygu `m
ygu `m
`m ygu
8/8/2019 STATIONARITY AND RANDOM WALK TESTS
http://slidepdf.com/reader/full/stationarity-and-random-walk-tests 10/16
Rckc|oc UDFgdcry2 Gdyzxoc` Uxmek Ic|kgx Gaaoeog`ey2 C` O`oxocn Zwlnoe Maag|o`d Zg|uzgexoqg. ZfH
xfguou. Ux|cxfenyhg W`oqg|uoxy. WK.3??6
37:
Xclng 8)3< Hoekgy)Awnng| xguxu ma |c`hmi tcnk am| xfg Gngqg` Hcony
O`hoegu ma xfg Gdyzxoc` Eczoxcn Ic|kgx ma 47: hcyu &Bc`wc|y. 3ux 3??8 xm
Hgegilg| 53 3??1/
: : 3 < 5 8
\3 :::3 ):488 ):<86 ):<3: ):31< )::41
UG :::: ::4? ::48 ::14 ::76 ::8<
x)uxcxouxoeu 31<3 )?8<4 )5513 )53<1 )<444 )3636
\< ):::: )3::4 :::7 :::8 :::5 :::3
UG :::: ::6< ::48 ::18 ::7< ::54
x)uxcxouxoeu )3::: )3<<:4 ::45 ::18 ::7< ::54
\5 ):::: )3::4 :::7 :::8 :::5 :::3
UG :::: ::6< ::48 ::18 ::7< ::54
x)uxcxouxoeu )3::: )3<<:4 ::45 ::18 ::7< ::54
\8 :::3 ):433 ):<6? ::88 ::47 ::33
UG :::3 ::17 ::13 ::74 ::7< ::54
x)uxcxouxoeu 33<6 )3:618 )8451 :44< 388< :5:<
\8 :::3 ):17? ):<3? ):363 )::67 )::5<
UG :::: ::14 ::15 ::74 ::8? ::54
x)uxcxouxoeu 3676 )?615 )5844 )5386 )348< ):668
\7 :::: )3::3 :::3 :::3 :::: ::::
UG :::3 ::6< ::48 ::18 ::7< ::54
x)uxcxouxoeu :5?? )3<317 ::3< ::3: :::6 :::1
\1 :::: ):??1 ):::7 ):::1 :::1 :::5
UG :::: ::6< ::45 ::18 ::7< ::54
x)uxcxouxoeu 3531 )3<358 )::43 )::?3 :3:4 ::68
\4 :::< )3:4< :375 ::18 ::4: ::7?
UG :::3 ::6< ::45 ::15 ::7: ::54
x)uxcxouxoeu 53<: )35:6? <3:7 3:35 3561 3768
\6 :::: ):768 ):54< ):365 )::1< )::3?
UG :::: ::1< ::1: ::74 ::7: ::51
x)uxcxouxoeu 3873 )?83? )134: )5<:? )3<5: ):756
\? :::3 ):864 ):5?3 ):3?3 ):337 )::1?
UG :::: ::76 ::76 ::77 ::8? ::54
x)uxcxouxoeu <:<4 )658: )1445 )584: )<5<? )361?
\3: :::3 ):477 ):3<? )::47 ):::6 ):::8
UG :::: ::13 ::78 ::88 ::<1 ::3<
x)uxcxouxoeu <18? )3<81? )<8:< )3438 ):5<8 ):551
\33 :::: ):875 ):5?: ):<:: )::43 )::57
UG :::: ::78 ::77 ::78 ::86 ::54x)uxcxouxoeu <<:: )65<5 )4:63 )5453 )3818 ):?15
@mxgu2 O`hoegu2\39Cd|oewnxw|g. \<9Io`o`d. \59Em`ux|wexom`. \89Ic`wacexw|o`d. \79X|c`uzm|xcxom`.
\19X|chg. \49Ao`c`eg. \69Ug|qoegu. \?9Zwlnoe Uwlue|ozxom . \3:9Enmugh Uwlue|ozxom`. \339Dg g|cn o`hgv
Xfg `wnn fyzmxfguou ou 9 : &|gxw|`u amnnmt c |c`hmi tcnk/ Xfg x)uxcxouxoeu cx xfg 7' ngqgn ou )<.64 [a|mi
Hoekgy c`h Awnng| &3?4?/Y C` w`hg|no`g o`hoecxgu xfg o`hoqohwcn emgaaoeog`x ou hoaag|g`x a|mi rg|m cx xfg 7 '
ngqgn ma uod`oaoec`eg Uxc`hc|h g||m|u&UG/ c`h x)uxcxouxoeu c|g |gzm|xgh w`hg| gcef emgaaoeog`x.
8/8/2019 STATIONARITY AND RANDOM WALK TESTS
http://slidepdf.com/reader/full/stationarity-and-random-walk-tests 11/16
Rckc|oc UDFgdcry2 Gdyzxoc` Uxmek Ic|kgx Gaaoeog`ey2 C` O`oxocn Zwlnoe Maag|o`d Zg|uzgexoqg. ZfH
xfguou. Ux|cxfenyhg W`oqg|uoxy. WK.3??6
373
8.5.8 XFG QC\OC@EG \CXOM CZZ\MCEF
Xfg qc|oc`eg)|cxom xgux tfoef tcu z|mzmugh ly Nm c`h IceKo`ncy &3?66/
g`clngu wu xm igcuw|g fmt iwef c doqg` hgqocxom` a|mi c` gswonol|owi |gncxom`ufoz
ou hgxg|io`gh ly c |c`hmi tcnk emizm`g`x c`h fmt iwef ma ox ou hwg xm xfg
uxcxom`c|y hgqocxom` Xfg xgux ou lcugh m` xfg z|gioug xfcx xfg qc|oc`eg ma |c`hmi
tcnk o`e|gig`xu o` c ao`oxg ucizng ou no`gc| o` xfg ucizno`d o`xg|qcn [Cycho c`h Zyw`
&3??82186/Y Xfou xgux ou ug`uoxoqg xm em||gncxgh z|oeg efc`dgu lwx |mlwux toxf |guzgex
xm ic`y am|iu ma fgxg|mueghcuxoeoxy c`h `m`)`m|icnoxy ma xfg uxmefcuxoe houxw|lc`eg
xg|i Xfwu. xfg xgux ec` lg giznmygh xm ohg`xoay xfg z|gug`eg ma `gdcxoqg ug|ocn
em||gncxom` o` uxmek z|oegu o`hoegu Oa uxmek z|oeg imqgig`xu zc|xny |gangex `gdcxoqg
ug|ocn em||gncxom`. xfg uxmek |gxw|` qc|oc`eg ufmwnh d|mt nguu xfc` z|mzm|xom`cxgny
toxf xoig[ugg. Emef|c` c`h HgAo`c &3??72684/Y Xfg qc|oc`eg)|cxom xgux ec` lg
icxfgicxoecnny gvz|guugh cu amnnmtu2
Q k k
y y
y y
x x k
x x
k & /
qc|& /
qc|& /
3
3
<
3
<
&836/
tfoef ou xfg qc|oc`eg ma xfg k)hoaag|g`eg ma yx
hoqohgh ly k mqg| xfg qc|oc`eg ma xfg
ao|ux hoaag|g`eg ma yx . k
< ou xfg w`locugh guxoicxm| ma k xf ma xfg qc|oc`eg ma
n` n`Z Zx zx k c`h 3
< ou xfg w`locugh guxoicxm| ma xfg qc|oc`eg ma n` n`Z Zx zx 3
Xfgug guxoicxm|u ec` lg em`qg`xom`cnny ecnewncxgh cu amnnmtu2
k k x x k x k
X
k X k X _ _ k < <
3
3 3
& /& /& / &83?/
3
<
3
3
<3
3
& /
& /X
_ _ x x
x
X
&8<:/
tfg|g X ou xfg ucizng uorg c`h & / 3
:X y y| Toxf xfg cuuwizxom` ma
fmimueghcuxoeoxy xfg cuyizxmxoe qc|oc`eg ma xfg Q k uxcxouxoe ou ufmt` xm lg2
8/8/2019 STATIONARITY AND RANDOM WALK TESTS
http://slidepdf.com/reader/full/stationarity-and-random-walk-tests 12/16
Rckc|oc UDFgdcry2 Gdyzxoc` Uxmek Ic|kgx Gaaoeog`ey2 C` O`oxocn Zwlnoe Maag|o`d Zg|uzgexoqg. ZfH
xfguou. Ux|cxfenyhg W`oqg|uoxy. WK.3??6
37<
& /& /& /
k k k
kX
< < 3 3
5&8<3/
xfg Q k & / uxcxouxoe [Nm c`h IceKo`ncy &3?66/Y cuyizxmxoecnny czz|mcefgu `m|icnoxy
m|
R k Q k
k @ & /
& /
[ & / Y& . /
-
3: 3
3 < &8<</
Tg wug xfou xgux cu c xgux am| xfg fmimueghcuxoeoxy Nm c`h IceKo`ncy cnum hg|oqg xfg
fgxg|mueghcuxoeoxy)em`uouxg`x qc|oc`eg guxoicxm| & /k 2
& /
& /& /k
k b
k b
b
k <
3
3
&8<5/
o` tfoef
& /
& / & /
& /
b
u u u u
u u
x x x b x bX b
X
x x x
X
3
<
3
<
3
3
<
3
<&8<8/
Xfwu. xfg qc|oc`eg |cxom xgux uxcxouxoe ec` lg uxc`hc|hough cuyizxmxoecnny xm c uxc`hc|h
`m|icn qc|oclng m|2
R k Q k k
@
& / & /[ & /Y
& . / - 3 : 33 < &8<7/
Dg`g|cnny. oa yx
amnnmtu c |c`hmi tcnk. xfg` Q k & / 9 3 [ugg Cycho c`h \yw`
&3??82186)?/Y Xfwu. c xgux ma |c`hmi tcnk ou gswoqcng`x xm xguxo`d xfg `wnn
fyzmxfguou2 Q k & / 93 cdco`ux c` cnxg|`cxoqg fyzmxfguou xfcx Q k & / ou `mx gswcn xm m`g
Fg`eg. xfg qc|oc`eg |cxom xgux cnnmtu xfg oizm|xc`eg ma xfg |c`hmi tcnk
emizm`g`x xm lg efc|cexg|ough m` c em`xo`wmwu uecng |cxfg| xfc` |gux|oexo`d xfg
hgeouom` xm c hoefmxmimwu efmoeg o` xfg z|gug`eg m| xfg clug`eg ma |c`hmi tcnk cu o`
xfg ecug ma xfg w`ox |mmx xgux O` mxfg| tm|hu. xfg zw|zmug ma xfg qc|oc`eg |cxom
czz|mcef ma Nm c`h IceKo`ncy &3?66/ ou xm hgxgex xfg ufm|x)xg|i anwexwcxom`u
8/8/2019 STATIONARITY AND RANDOM WALK TESTS
http://slidepdf.com/reader/full/stationarity-and-random-walk-tests 13/16
Rckc|oc UDFgdcry2 Gdyzxoc` Uxmek Ic|kgx Gaaoeog`ey2 C` O`oxocn Zwlnoe Maag|o`d Zg|uzgexoqg. ZfH
xfguou. Ux|cxfenyhg W`oqg|uoxy. WK.3??6
375
hmio`cxg xfg uxmefcuxoe x|g`h emizm`g`xu. tfong xfg CHA ou am|iwncxgh xm gvcio`g
m`ny xfg gvouxg`eg ma uxmefcuxoe x|g`h emizm`g`xu [ugg Fwc`d &3??72<75/Y
Em`ugswg`xny. tg czzny qc|oc`eg |cxom uxcxouxoeu toxf fmimueghcuxoeoxy c`h
fgxg|mueghcuxoe g||m| xg|i. hg`mxgh ly R&k/ c`h R"&k/. |guzgexoqgny. xm xfg gngqg`
Gdyzxoc` uxmek z|oeg o`hoegu Fg`eg. tg ecnewncxgh xfg Q&K/. R&k/. c`h R"&K/ am| gcef
ma xfg ecugu K9<. 5. 8. 7. 1. 6. 3:. 3<. 31. <:. <8. 51. c`h 7: W`hg| xfg |c`hmi tcnk
fyzmxfguou. xfg |cxom ma &3-k/ xoigu xfg qc|oc`eg ma xfg K)hoaag|g`egu mqg| xfg
qc|oc`eg ma xfg ao|ux hoaag|g`egu ou gvzgexgh xm lg w`oxy Xfg |guwnxu ma awnn ucizng
zg|omh c|g ufmt` o` Xclng 8)35
Aodw|g 8)7 ufmtu xfg em`xo`wmwu qc|oc`eg |cxomu am| xfg nmd ma xfg Gdyzxoc`
Hcony \gxw|`u Tg `mxoeg xfcx xfg qc|oc`eg |cxomu c|g lgnmt m`g wz xm K97: Xfg
|guwnxu o` Xclng 8)35 o`hoecxg xfcx w`hg| xfg cuuwizxom` ma fmimueghcuxoeoxy. xfg
qc|oc`eg |cxom xgux |gbgexu xfg `wnn fyzmxfguou am| gqg|y ngqgn ma K Tfg` xfg R)
uxcxouxoe qcnwgu c|g emizc|gh toxf xfg em`qg`xom`cn e|oxoecn qcnwg ma 3?1 am| xfg aoqg
zg|eg`x ngqgn. xfg qc|oc`eg |cxomu. Q k & / c|g uxcxouxoecnny hoaag|g`x a|mi w`oxy
Xfg|gam|g. xfg |guwnxu w`hg| fmimueghcuxoeoxy uwddgux xfcx xfg lgfcqomw| ma xfg
Gdyzxoc` Uxmek |gxw|`u ec` `mx lg hgue|olgh cu mlgyo`d xfg |c`hmi tcnk xfgm|y
Xfgug |guwnxu c|g em`uouxg`x toxf xfmug ma Lc|k &3??3/. c`h Cycho c`h Zyw` &3??8/
Xfg |gbgexom` ma |c`hmi tcnk mlxco`gh a|mi R&k/ w`hg| fmimueghcuxoeoxy
emwnh lg hwg xm xfg z|gug`eg ma fgxg|mueghcuxoeoxy c`h-m| ug|ocn em||gncxom` Xfwu. tg
czznogh xfg fgxg|mueghcuxoeoxy)em`uouxg`x qc|oc`eg)|cxom xgux R"&k/ Xclng 8)35
onnwux|cxgu xfcx xfg |guwnxu am| cnn qcnwgu ma K. o`hoecxg xfcx xfg `wnn fyzmxfguou ma
|c`hmi tcnk ec` lg |gbgexgh cx xfg aoqg zg|eg`x ngqgn ma uod`oaoec`eg Xfgug |guwnxu.
fg|g. em`ao|i xfg |guwnxu ma xfg fmimueghcuxoeoxy xgux
8/8/2019 STATIONARITY AND RANDOM WALK TESTS
http://slidepdf.com/reader/full/stationarity-and-random-walk-tests 14/16
Rckc|oc UDFgdcry2 Gdyzxoc` Uxmek Ic|kgx Gaaoeog`ey2 C` O`oxocn Zwlnoe Maag|o`d Zg|uzgexoqg. ZfH
xfguou. Ux|cxfenyhg W`oqg|uoxy. WK.3??6
378
Xfg |gbgexom` ma xfg |c`hmi tcnk fyzmxfguou ou fgxg|mueghcuxoeoxy |mlwux c`h
uwddguxu xfcx xfg qc|oc`eg |cxomu c|g hoaag|g`x a|mi w`oxy Uo`eg. xfg qc|oc`eg |cxomu
c|g hgeno`o`d cu K o`e|gcugu. xfgy gvfolox `gdcxoqg cwxmem||gncxom` tfoef o`hoecxgu
xfg z|gug`eg ma c igc`)|gqg|uom` z|meguu Xfg ao`ho`d ma igc`)|gqg|uom` cd|ggu toxf
Acic c`h A|g`ef &3?66/. c`h A|g`ef c`h \mnn &3?61/. tfm |gzm|xgh `gdcxoqg ug|ocn
em||gncxom` am| Cig|oec` uxmeku. c`h toxf Zmxg|lc c`h Uwiig|u &3?66/. tfm |gbgexgh
xfg |c`hmi tcnk fyzmxfguou c`h amw`h igc`)|gqg|uom` am| c ucizng ma xfg Gw|mzgc`
c`h Cuoc` `cxom`cn uxmek o`hgv. c`h toxf Ngg gx cn &3??1/. tfm |gzm|xgh `gdcxoqg
ug|ocn em||gncxom` am| ugem`hc|y ic|kgx z|oegu ma uy`hoecxgh nmc`u am| NHEu
Im|gmqg|. mw| |gbgexom` ma |c`hmi tcnk am| xfg Gdyzxoc` Uxmek o`hgv cd|ggu
toxf Nm c`h Iceko`ncy &3?66/. tfm |gbgexgh xfg `wnn ma |c`hmi tcnk am| @_UG)
CIGV uxmek z|oegu. c`h ao`cnny toxf Fwc`d &3?47/ am| xfg Cuoc` uxmek ic|kgxu M`
xfg mxfg| fc`h. mw| |gbgexom` ma |c`hmi tcnk hoaag|u a|mi Encguug` gx cn &3??5/. tfm
hm `mx |gbgex xfg `wnn ma |c`hmi tcnk am| c ucizng ma <: gig|do`d ic|kgxu Xfgug
czzc|g`xny em`x|choexm|y |guwnxu ufmwnh `mx lg uw|z|ouo`d O` gaagex. Umn`ok &3?48/ o`
fou o`qguxodcxom` ma xfg ic|kgx ux|wexw|g ma uxmek z|oegu am| ugqg|cn Gw|mzgc` ic|kgxu
em`enwhgu xfcx z|oegu c|g ux|m`dny caagexgh ly xfg uzgeoaoe efc|cexg|ouxoeu ma xfg
o`hoqohwcn ic|kgxu
Xm uwi wz. am| xfg xoig zg|omh Bc`wc|y 3. 3??8 xm Hgegilg| 53. 3??1 xfg
Cwdig`xgh Hoekgy)Awnng| xgux |gbgexu xfg |c`hmi tcnk fyzmxfguou am| xfg Gdyzxoc`
uxmek z|oegu Nokgtoug. xfg im|g zmtg|awn qc|oc`eg)|cxom xguxu fcqg |gbgexgh xfg
|c`hmi tcnk o` acqmw| ma c igc`)|gqg|uom` z|meguu Fmtgqg|. xfg gvouxg`eg ma
cwxmem||gncxom` o` xfg Gdyzxoc` uxmek z|oegu o`hoegu hmgu `mx `geguuc|ony oizny
o`gaaoeog`ey [ugg. Nwecu &3?46/. Ngqoxef &3?4?/. Ngg gx cn &3??1/Y Guug`xocnny.
8/8/2019 STATIONARITY AND RANDOM WALK TESTS
http://slidepdf.com/reader/full/stationarity-and-random-walk-tests 15/16
Rckc|oc UDFgdcry2 Gdyzxoc` Uxmek Ic|kgx Gaaoeog`ey2 C` O`oxocn Zwlnoe Maag|o`d Zg|uzgexoqg. ZfH
xfguou. Ux|cxfenyhg W`oqg|uoxy. WK.3??6
377
uzw|omwu cwxmem||gncxom` icy cnum lg hwg xm o`a|gswg`x m| `m`)uy`ef|m`mwu x|cho`d
[Zmxg|lc c`h Uwiig| &3?66/; Uefmngu c`h Tonnociu &3?44/; c`h Ngg gx cn &3??1/Y
Cu c |guwnx. o` xfg amnnmto`d ugexom` tg z|mzmug xm chh|guu xfg gaaoeog`ey ouuwg
giznmyo`d xfg |geg`xny hgqgnmzgh emo`xgd|cxom` z|meghw|g
Xclng 8)35 Guxoicxgu ma Qc|oc`eg \cxom Xguxu c`h xfgo| Xgux Uxcxouxoeu am| xfg Gngqg` HconyO`hoegu ma xfg Gdyzxoc` Eczoxcn Ic|kgx ma 47: hcyu &Bc`wc|y. 3ux 3??8 xm Hgegilg| 53 3??1/
Ncdu
U" < 5 8 7 1 6 3: 3< 31 <: <8 51 7:
3 q&k/ :8< :<6 :<3 :34 :37 :33 ::? ::6 ::1 ::7 ::8 ::5 ::<
R&k/ )3746 )3537 )337< )3:8: )?85 )6<3 )45< )145 )74? )7<: )84: )56? )5<7
R"&k/ )?5?< )3314: )33836 )3<:43 )33638 )3<:35 )3<:71 )3<3<4 )3<38: )3<371 )3<3?1 )3<<36 )3<311
< q&k/ :7: :58 :<7 :<: :34 :35 :3: ::? ::4 ::7 ::8 ::5 ::<
R&k/ )3517 )3<<3 )3:?8 )??4 )?36 )6:7 )4<8 )111 )741 )73? )816 )566 )5<8
R"&k/ )<45: )515? )511< )5?:1 )56?? )8::5 )8:13 )8:?4 )8383 )8317 )8363 )8<:4 )8<<3
5 q&k/ :57 :55 :<8 :<: :31 :35 :3: ::? ::4 ::7 ::7 ::5 ::<
R&k/ )3463 )3<58 )33:4 )??7 )?<5 )6:7 )4<< )117 )741 )73? )816 )566 )5<8
R"&k/ )545: )5675 )56:? )8:3< )5??7 )8:18 )83:8 )8357 )8343 )83?< )8<:7 )8<<1 )8<54
8 q&k/ :78 :57 :<4 :<< :3? :38 :33 ::? ::4 ::1 ::7 ::5 ::5
R&k/ )3<14 )33?1 )3:45 )?44 )6?6 )4?7 )438 )17? )74< )737 )811 )564 )5<5
R"&k/ )8847 )1<?7 )1557 )1474 )14<5 )1?45 )4:1< )4388 )4<5< )4<4: )4531 )4545 )48:3
7 q&k/ :7: :58 :<7 :<: :34 :35 :3: ::? ::4 ::7 ::8 ::5 ::<
R&k/ )3517 )3<<3 )3:?8 )??4 )?36 )6:7 )4<8 )111 )741 )73? )816 )566 )5<8
R"&k/ )57:6 )8144 )84:1 )7:<: )7:33 )7387 )7<3? )7<11 )75<< )7575 )7548 )78:4 )78<8
1 q&k/ :7: :55 :<7 :<: :34 :35 :3: ::? ::4 ::7 ::8 ::5 ::<
R&k/ )3516 )3<57 )3:?8 )??4 )?36 )6:7 )4<8 )111 )741 )73? )816 )566 )5<8
R"&k/ )83<6 )7774 )77<1 )76?7 )7668 )1:8< )135: )1363 )1<81 )1<68 )15:4 )1587 )1518
4 q&k/ :7? :51 :<4 :<5 :34 :38 :33 ::? ::6 ::1 ::7 ::5 ::<
R&k/ )33<4 )3344 )3:45 )?17 )?3< )4?1 )4<3 )11< )714 )736 )814 )566 )5<8
R"&k/ )7634 )?:7< )?:?< )?76: )?434 )?66: )3::83 )3::11 )??63 )3:3:8 )3::?7 )3::8? )??66
6 q&k/ :87 :5< :<8 :<: :36 :35 :33 ::? ::4 ::1 ::7 ::5 ::5
R&k/ )37:< )3<78 )33:7 )??7 )?:4 )4?6 )4<3 )118 )748 )731 )814 )564 )5<5
R"&k/ )3:8<1 )3<?46 )3<177 )3555? )35:?1 )35858 )3515< )3545: )35465 )35416 )35638 )35633 )3544?
? q&k/ :81 :58 :<4 :<3 :3? :38 :3< :3: ::4 ::1 ::7 ::5 ::<
R&k/ )3814 )3<31 )3:48 )??5 )6?1 )4?7 )435 )176 )74< )735 )817 )564 )5<8
R"&k/ )333<? )3547? )35836 )387<7 )38:61 )3874< )38147 )3846? )38?35 )38613 )38?:4 )38?78 )38?5:
3: q&k/ :<1 :3< ::? ::4 ::1 ::7 ::8 ::5 ::< ::< ::< ::3 ::3
R&k/ )<:37 )31:? )35<1 )3371 )3:51 )64? )441 )4:7 )1:< )754 )86< )5?1 )5<?
R"&k/ )7557 )157< )7647 )7??4 )76<5 )7466 )7413 )7481 )74<< )74:4 )71?4 )716: )711?
33 q&k/ :86 :57 :<6 :<8 :<: :37 :3< :33 ::6 ::4 ::1 ::8 ::5
R&k/ )3837 )33?7 )3:78 )?7< )66? )461 )4:4 )184 )714 )7:? )813 )567 )5<5
R"&k/ )33761 )387?5 )383?8 )37:35 )37:1< )377:6 )371<? )37137 )3768< )37441 )3768? )37?5< )37?38
C` w`hg|no`g o`hoecxgu xfg o`hoqohwcn emgaaoeog`x ou hoaag|g`x a|mi rg|m cx xfg 7 ' ngqgn ma uod`oaoec`eg
"Xfg wilg|u o` xfg ao|ux emnwi` |gag| xm xfg ugexm|u m|hg|gh cu o` Xclng 8)3
8/8/2019 STATIONARITY AND RANDOM WALK TESTS
http://slidepdf.com/reader/full/stationarity-and-random-walk-tests 16/16
Rckc|oc UDFgdcry2 Gdyzxoc` Uxmek Ic|kgx Gaaoeog`ey2 C` O`oxocn Zwlnoe Maag|o`d Zg|uzgexoqg. ZfH
xfguou. Ux|cxfenyhg W`oqg|uoxy. WK.3??6
371
Aodw|g 8)7 Em`xo`wmwu Q&k/ Znmx am| xfg Gngqg` Hcony O`hoegu \gxw|`u ma xfg Gdyzxoc` Eczoxcn
Ic|kgx ma 47: hcyu &Bc`wc|y. 3ux 3??8 xm Hgegilg| 53 3??1/
:
:3
:<
:5
:8
:7
K
Q&k/
< 5 7 1 6 3 : 3< 31 <: <8 51 7:
Cd|oewnxw|g Ugexm|
:
:.3
:.<
:.5
:.8
:.7
:.1
K
Q
& k /
< 5 8 7 1 6 3: 3 < 31 <: < 8 51 7:
Io`o`d Ugexm|
:
:.:7
:.3
:.37
:.<
:.<7
:.5
:.57
:.8
K
Q
& K /
< 5 8 7 1 6 3: 3< 31 < : <8 51 7 :
Em`ux|wexom Ugexm|
:
:.3
:.<
:.5
:.8
:.7
:.1
K
Q & K /
< 5 8 7 1 6 3: 3 < 31 <: < 8 51 7:
Ic`wacexw|o d Ugexm|
:
:.3
:.<
:.5
:.8
:.7
:.1
K
Q & k /
< 5 8 7 1 6 3: 3 < 31 <: < 8 51 7:
X|c`uzm|xcxom` Ugexm|
:
:.3
:.<
:.5
:.8
:.7
:.1
K
Q & k /
< 5 8 7 1 6 3: 3 < 31 < : < 8 51 7 :
X|chg Ugexm|
:
:.3
:.<
:.5
:.8
:.7
:.1
:.4
K
Q & k /
< 5 8 7 1 6 3: 3 < 31 <: < 8 51 7:
Ugexm| ma Ao c`eg
:
:.3
:.<
:.5
:.8
:.7
K
Q & k /
< 5 8 7 1 6 3: 3< 31 <: < 8 51 7:
Ug|qoegu Ugexm|
:
:.3
:.<
:.5
:.8
:.7
K
Q & k /
< 5 8 7 1 6 3: 3 < 31 <: < 8 51 7:
Zwlnoe Uwlue|ozxom` O hgv
:
::7
:3
:37
:<
:<7
:5
K
Q & k /
< 5 8 7 1 6 3: 3< 31 <: < 8 51 7:
Enmugh Uwlue|ozxom` O`hgv
:
:3
:<
:5
:8
:7
:1
K
Q & k /
< 5 8 7 1 6 3: 3< 31 <: <8 51 7:
Dg`g|cn O`hgv