STATIONARITY AND RANDOM WALK TESTS

8/8/2019 STATIONARITY AND RANDOM WALK TESTS

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\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. \<9Ioò`d. \59Emùx|wexom`. \89Ic`wacexw|o`d. \79X|cùzm|xcxom`.

\19X|chg. \49Ao`cèg. \69Ug|qoegu. \?9Zwlnoe Uwlue|ozxom . \3:9Enmugh Uwlue|ozxom`. \339Dg g|cn o`hgv

Xfg `wnn fyzmxfguou ou 9 : &|gxw|ù 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 uodòaoecèg Uxc`hc|h g||m|u&UG/ c`h x)uxcxouxoeu c|g |gzm|xgh w`hg| gcef emgaaoeog`x.





373

8.5.8 XFG QC\OC@EG \CXOM CZZ\MCEF

Xfg qc|ocèg)|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 |gncxomùfoz

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|ocèg ma |c`hmi

tcnk oè|gig`xu o` c aoòxg ucizng ou no`gc| o` xfg ucizno`d o`xg|qcn [Cycho c`h Zyw`

&3??82186/Y Xfou xgux ou ugùoxoqg 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|lcèg

xg|i Xfwu. xfg xgux ec` lg giznmygh xm ohg`xoay xfg z|gugèg 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|ocèg ufmwnh d|mt nguu xfc` z|mzm|xom`cxgny

toxf xoig[ugg. Emef|c` c`h HgAo`c &3??72684/Y Xfg qc|ocèg)|cxom xgux ec` lg

icxfgicxoecnny gvz|guugh cu amnnmtu2

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ao|ux hoaag|gèg ma yx . k

< ou xfg w`locugh guxoicxm| ma k xf ma xfg qc|ocèg ma

n` n`Z Zx zx k c`h 3

< ou xfg w`locugh guxoicxm| ma xfg qc|ocèg 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

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k X k X _ _ k < <

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tfg|g X ou xfg ucizng uorg c`h & / 3

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fmimueghcuxoeoxy xfg cuyizxmxoe qc|ocèg ma xfg Q k uxcxouxoe ou ufmt` xm lg2





37<

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k k k

kX

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xfg Q k & / uxcxouxoe [Nm c`h IceKo`ncy &3?66/Y cuyizxmxoecnny czz|mcefgu `m|icnoxy

m|

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Tg wug xfou xgux cu c xgux am| xfg fmimueghcuxoeoxy Nm c`h IceKo`ncy cnum hg|oqg xfg

fgxg|mueghcuxoeoxy)emùouxg`x qc|ocèg guxoicxm| & /k 2

& /

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`m|icn qc|oclng m|2

R k Q k k

@

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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 cdcoùx c` cnxg|`cxoqg fyzmxfguou xfcx Q k & / ou `mx gswcn xm m`g

Fgèg. xfg qc|ocèg |cxom xgux cnnmtu xfg oizm|xcèg 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|gugèg m| xfg clugèg ma |c`hmi tcnk cu o`

xfg ecug ma xfg wòx |mmx xgux O` mxfg| tm|hu. xfg zw|zmug ma xfg qc|ocèg |cxom

czz|mcef ma Nm c`h IceKo`ncy &3?66/ ou xm hgxgex xfg ufm|x)xg|i anwexwcxomù





375

hmio`cxg xfg uxmefcuxoe x|g`h emizm`g`xu. tfong xfg CHA ou am|iwncxgh xm gvcio`g

m`ny xfg gvouxgèg ma uxmefcuxoe x|g`h emizm`g`xu [ugg Fwc`d &3??72<75/Y

Emùgswg`xny. tg czzny qc|ocèg |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 Fgèg. 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|ocèg ma xfg K)hoaag|gègu mqg| xfg

qc|ocèg ma xfg ao|ux hoaag|gègu ou gvzgexgh xm lg wòxy Xfg |guwnxu ma awnn ucizng

zg|omh c|g ufmt` o` Xclng 8)35

Aodw|g 8)7 ufmtu xfg em`xo`wmwu qc|ocèg |cxomu am| xfg nmd ma xfg Gdyzxoc`

Hcony \gxw|ù Tg `mxoeg xfcx xfg qc|ocèg |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|ocèg |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|ocèg |cxomu. Q k & / c|g uxcxouxoecnny hoaag|g`x a|mi wòxy

Xfg|gam|g. xfg |guwnxu w`hg| fmimueghcuxoeoxy uwddgux xfcx xfg lgfcqomw| ma xfg

Gdyzxoc` Uxmek |gxw|ù ec` `mx lg hgue|olgh cu mlgyo`d xfg |c`hmi tcnk xfgm|y

Xfgug |guwnxu c|g emùouxg`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|gugèg ma fgxg|mueghcuxoeoxy c`h-m| ug|ocn em||gncxom` Xfwu. tg

czznogh xfg fgxg|mueghcuxoeoxy)emùouxg`x qc|ocèg)|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 uodòaoecèg Xfgug |guwnxu.

fg|g. emào|i xfg |guwnxu ma xfg fmimueghcuxoeoxy xgux





378

Xfg |gbgexom` ma xfg |c`hmi tcnk fyzmxfguou ou fgxg|mueghcuxoeoxy |mlwux c`h

uwddguxu xfcx xfg qc|ocèg |cxomu c|g hoaag|g`x a|mi wòxy Uoèg. xfg qc|ocèg |cxomu

c|g hgenoò`d cu K oè|gcugu. xfgy gvfolox `gdcxoqg cwxmem||gncxom` tfoef o`hoecxgu

xfg z|gugèg ma c igc`)|gqg|uom` z|meguu Xfg ao`ho`d ma igc`)|gqg|uom` cd|ggu toxf

Acic c`h A|gèf &3?66/. c`h A|gèf 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 nmcù 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. Umnòk &3?48/ o`

fou o`qguxodcxom` ma xfg ic|kgx ux|wexw|g ma uxmek z|oegu am| ugqg|cn Gw|mzgc` ic|kgxu

emènwhgu 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|ocèg)|cxom xguxu fcqg |gbgexgh xfg

|c`hmi tcnk o` acqmw| ma c igc`)|gqg|uom` z|meguu Fmtgqg|. xfg gvouxgèg ma

cwxmem||gncxom` o` xfg Gdyzxoc` uxmek z|oegu o`hoegu hmgu `mx `geguuc|ony oizny

o`gaaoeogèy [ugg. Nwecu &3?46/. Ngqoxef &3?4?/. Ngg gx cn &3??1/Y Guug`xocnny.





377

uzw|omwu cwxmem||gncxom` icy cnum lg hwg xm oà|gswg`x m| `m`)uyèf|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 gaaoeogèy ouuwg

giznmyo`d xfg |geg`xny hgqgnmzgh emo`xgd|cxom` z|meghw|g

Xclng 8)35 Guxoicxgu ma Qc|ocèg \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 uodòaoecèg

"Xfg wilg|u o` xfg ao|ux emnwi` |gag| xm xfg ugexm|u m|hg|gh cu o` Xclng 8)3





371

Aodw|g 8)7 Em`xo`wmwu Q&k/ Znmx am| xfg Gngqg` Hcony O`hoegu \gxw|ù ma xfg Gdyzxoc` Eczoxcn

Ic|kgx ma 47: hcyu &Bc`wc|y. 3ux 3??8 xm Hgegilg| 53 3??1/

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STATIONARITY AND RANDOM WALK TESTS

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