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:j‘_gh\kdbcK<FheqZgh\BG KL:LBKLBQ?KDB? F?LH>U IJH=GHABJH<:GBY Mq_[gh_ ihkh[b_ Jhklh\-gZ->hgm 2001

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Page 1: KL:LBKLBQ?KDB? F?LH>U IJH=GHABJH

:j`_gh\kdbc�K�<���FheqZgh\�B�G�

KKLL::LLBBKKLLBBQQ??KKDDBB?? FF??LLHH>>UU IIJJHH==GGHHAABBJJHH<<::GGBBYY

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Jhklh\-gZ->hgm 2001

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M>D�>�������������@������� A80 �E� :j`_gh\kdbc�K�<���FheqZgh\�B�G��KlZlbklbq_kdb_�f_lh^u�ijh]ghab�

jh\Zgby��Mq_[gh_�ihkh[b_� �Jhkl��]hk��wdhg��mgb\��–�Jhklh\-g�>���– 2001. –����k��– ISBN 5-7972-0379-0.

<�mq_[ghf�ihkh[bb�baeh`_gu�\�kbkl_fZlbabjh\Zgghf�\b^_��deZkkbnb�dZpby�ijh]ghah\��ZgZeba�\j_f_gguo�jy^h\��f_lh^u�\u^_e_gby�lj_g^Z�b�i_jbh�^bq_kdbo�dhe_[Zgbc��Z^Zilb\gu_��wdki_jlgu_�f_lh^u�ijh]ghabjh\Zgby��Hkh�[h_�\gbfZgb_�m^_e_gh�fh^_eyf� klZpbhgZjguo�b�g_klZpbhgZjguo�\j_f_gguo�jy^h\�b�bo�b^_glbnbdZpbb��Z�lZd`_�\hijhkZf�hp_gdb�Z^_d\Zlghklb�b�lhqghklb�ijh]ghah\��Ih�dZ`^hfm�jZa^_em�ijb\_^_gZ�ebl_jZlmjZ��dhgljhevgu_�\hijhku�b�aZ^Zgby�^ey�kZfhklhyl_evghc�jZ[hlu��

>ey�klm^_glh\�\mah\��h[mqZxsboky�ih�wdhghfbq_kdbf�ki_pbZevghklyf��\�jZfdZo�bamq_gby�dmjkZ�©Wdhghf_ljbdZª��

Arzhenovsky S.V., Molchanov I.N. Statistical methods of forecasting. The manual. Rostov-on-Don: RSEU, 2001. – 74 p. – ISBN 5-7972-0379-0.

In the manual are systematized the following: classification of the forecasts, time-series analysis, methods of trend and periodic fluctuation separation, adaptive, expert methods of forecasting. The special attention is given to models of stationary and non-stationary time-series and their identification, and also an estimation of adequacy and accuracy of the forecasts. On each section the literature, control ques-tions and tasks for independent work are given.

This book will serve as a text for the students training on economic specialties and studying Econometrics.

AZf_qZgby�b�ij_^eh`_gby�ijhkbf�gZijZ\eylv�ih�Z^j_km�� ��������]�Jhklh\-gZ->hgm��me��;hevrZy�KZ^h\Zy������d�������dZn��KFbI.

E−mail: [email protected] [email protected]

Bgl_jg_l: http:// www.srstu.novoch.ru/K_PriclMath.html http:// www.appleclub.donpac.ru

J_p_ga_glu� E�B�Gb\hjh`dbgZ��^hdlhj�wdhghfbq_kdbo�gZmd��ijhn_kkhj��aZ\��dZn_^�

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Ml\_j`^_gh�\�dZq_kl\_�mq_[gh]h�ihkh[by�j_^Zdpbhggh-ba^Zl_evkdbf�kh\_lhf�J=WM

ISBN 5-7972-0379-0. �Jhklh\kdbc�]hkm^Zjkl\_gguc�wdhghfb�

q_kdbc�mgb\_jkbl_l�©JBGOª������ �:j`_gh\kdbc�K�<���FheqZgh\�B�G�������

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]^_� l –�dhebq_kl\h�nZa�\�iheghf�k_ahgghf�pbde_��l ���ijb�f_kyqguo�^Zgguo��l ��ijb�d\ZjlZevguo�^Zgguo�b�l�i���� lts −+τˆ -�hp_gdZ�k_ahgghklb��Hp_gdZ�dhwn�

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Page 29: KL:LBKLBQ?KDB? F?LH>U IJH=GHABJH

29

( )1,11,011,0 ˆˆˆ

ˆ −−−

++= ttlt

tt aa

s

ya βα ,

( ) 1,121,0,02,1 ˆˆˆˆ −− +−= tttt aaaa βα ,

ltt

tt s

a

ys −+= ˆ

ˆˆ 3

,03 βα ,

]^_� α1, α2, α3 –� iZjZf_lju� wdkihg_gpbZevgh]h� k]eZ`b\Zgby� ��α1, α2, α3<1, β1=1-α1, β2=1-α2, β3=1-α3��Ba�wlh]h�ke_^m_l��qlh�\�jZkkfZljb\Z_fhf�f_lh^_�bk�

ihevam_lky�ljb�iZjZf_ljZ�k]eZ`b\Zgby�� Ijh]gha�y\ey_lky�nmgdpb_c�ijhreuo�b�l_dmsbo�^Zgguo��iZjZf_ljh\�α1,

α2, α3��Z�lZd`_�gZqZevguo�agZq_gbc� 0,10,0 ˆ,ˆ aa , 0s .

Mbgl_jk�ij_^eZ]Z_l�hilbfZevgu_�agZq_gby�iZjZf_ljh\�Z^ZilZpbb�gZoh�

^blv�wdki_jbf_glZevgh�k�ihfhsvx�k_ldb�agZq_gbc�α1, α2, α3��i_j_[hj�agZq_�

gbc�iZjZf_ljh\�hl� ��^h� �� k�g_dhlhjuf�rZ]hf��gZijbf_j�� �����ih�fbgbfmfm�

klZg^Zjlgh]h�hldehg_gby�hrb[db�k]eZ`b\Zgby� Wdhghfbq_kdb_�^Zggu_�qZs_�\k_]h�kh^_j`Zl�k_ahgghklv�bf_ggh�\�fmev�

lbiebdZlb\ghc�nhjf_��H^gZdh�L_ce�b�<_c^`�ij_^eh`beb�Z^^blb\gmx�fh^_ev�

k_ahgghklb��dhlhjZy�hlghkbl_evgh�ijhs_�\�\uqbkebl_evghc�j_ZebaZpbb��I_j�

\hgZqZevguc�\j_f_gghc�jy^�\�wlhf�kemqZ_��dZd�ijZ\beh��aZf_gy_lky�eh]Zjbn�

fZfb�agZq_gbc��Fh^_ev�bf__l�\b^�

,

,)(

,11,0,0

,0

ttt

ttt

aaa

usaty

+=++=

τ

]^_� ta ,0 -�\_ebqbgZ�mjh\gy�ijhp_kkZ�ihke_�webfbgbjh\Zgby�k_ahgguo�dhe_[Z�

gbc�� ta ,1 -�Z^^blb\guc�dhwnnbpb_gl�jhklZ�� ts -�Z^^blb\guc�dhwnnbpb_gl�k_�

ahgghklb��ut –�[_euc�rmf�� Ijh]ghagZy�fh^_ev�L_ceZ�b�<_c^`Z�bf__l�\b^�

lttt saaty −+++= ττ τ ˆˆˆ)(ˆ ,1,0 .

Hp_gdZ�dhwnnbpb_glh\�fh^_eb� ( ) ( )1,11,0111,0 ˆˆˆˆ −−− ++−= ttttt aasya βα ,

( ) 1,121,0,02,1 ˆˆˆˆ −− +−= tttt aaaa βα ,

( ) ltttt says −+−= ˆˆˆ 3,03 βα ,

]^_� α1, α2, α3 –� iZjZf_lju� wdkihg_gpbZevgh]h� k]eZ`b\Zgby� ��α1, α2, α3<1, β1=1-α1, β2=1-α2, β3=1-α3���<�^Zgghf�f_lh^_�lZd`_�bkihevam_lky�ljb�iZjZf_ljZ�

k]eZ`b\Zgby��

Ebl_jZlmjZ�d�jZa^_em����>����������������@� Dhgljhevgu_�\hijhku.

Page 30: KL:LBKLBQ?KDB? F?LH>U IJH=GHABJH

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���<�q_f�hlebqb_�Z^Zilb\guo�f_lh^h\�ijh]ghabjh\Zgby�hl�hklZevguo" ���<�dZdbo�kemqZyo�hijZ\^Zggh�ijbf_g_gb_�f_lh^Z�wdkihg_gpbZevgh]h�

k]eZ`b\Zgby? ���Ij_bfms_kl\Z�Z^Zilb\guo�ihebghfbZevguo�fh^_e_c�J�;jZmgZ� ���DZd�ih\ukblv�lhqghklv�ijh]ghabjh\Zgby�k�bkihevah\Zgb_f�Z^Zilb\�

guo�fh^_e_c" ���H[tykgbl_��dZd�\u[bjZ_lky�iZjZf_lj�k]eZ`b\Zgby" ���<�q_f�ij_bfms_kl\Z�fh^_eb�L_ceZ-<_c^`Z"

AZ^Zgby�^ey�kZfhklhyl_evghc�jZ[hlu.

1��Bkoh^gu_�^Zggu_�kh^_j`Zl�jy^�^bgZfbdb��oZjZdl_jbamxsbc�^h[uqm�

]ZaZ�\�JN�ih�f_kypZf�aZ�����-�����]]���fej^��f3:

F_kyp 1 2 3 4 5 6 7 8 9 10 11 12 1996 56,8 53,2 56,3 51,7 46,9 44,3 44 42,2 44,2 52,5 52,6 56,1 1997 57,4 51,5 54,2 48,7 45 39,3 37,9 37,6 40,7 48,6 53,8 56,9

Ihkljhcl_�ih� ^Zgguf� aZ� ����� ]�� Z^Zilb\gmx�ihebghfbZevgmx�fh^_ev�

i_j\h]h� ihjy^dZ�� ijbgy\� 0,0a =51,7, 0,1a = -0,2, α ����� GZc^bl_� klZg^Zjlgh_�kj_^g_d\Z^jZlbq_kdh_�hldehg_gb_�hrb[db�j_ljhki_dlb\gh]h�ijh]ghaZ�

2��Ihkljhcl_�fh^_ev�OhevlZ�ih�^Zgguf�aZ������]���ijbgy\� 0,0a =60, 0,1a = -

0,2, α1=α2 ����� GZc^bl_� klZg^Zjlgh_� hldehg_gb_� hrb[db� j_ljhki_dlb\gh]h�

ijh]ghaZ��KjZ\gbl_�j_amevlZlu�k�ihemq_ggufb�\�aZ^Zq_��� 3��Ihkljhcl_�fh^_ev�Mbgl_jkZ��Bkoh^gu_�^Zggu_�-�\�aZ^Zq_����GZqZev�

gu_�mkeh\by� - 0,0a =60, 0,1a = -0,2, 0s ���IZjZf_lju�fh^_eb�ijbgylv�jZ\gufb�

α1=α2=0,1, α3 �����KjZ\gbl_�j_amevlZlu�k�jZkq_lZfb�\�aZ^ZqZo���b����K^_eZcl_�

\u\h^u� 4��Ihkljhcl_�fh^_ev�L_ceZ-<_c^`Z�ih�^Zgguf�aZ^Zqb� ���<u[_jbl_�gZ�

qZevgu_�mkeh\by�b�agZq_gby�iZjZf_ljh\�fh^_eb��bkoh^y�ba�fbgbfZevgh]h�agZ�

q_gby�klZg^Zjlgh]h�hldehg_gby�hrb[db�j_ljhki_dlb\gh]h�ijh]ghaZ��KjZ\gbl_�

ihemq_gguc�j_amevlZl�k�jZkq_lZfb�\�aZ^Zq_����K^_eZcl_�\u\h^�

���Fh^_eb�klZpbhgZjguo�b�g_klZpbhgZjguo�\j_f_gguo�jy^h\� b�bo�b^_glbnbdZpby

6.1. Fh^_eb�Z\lhj_]j_kkbb�ihjy^dZ�p (AutoRegressive - AR(p) models)

>hklZlhqgh� qZklh� wdhghfbq_kdb_� ihdZaZl_eb�� ij_^klZ\e_ggu_� \� \b^_�

\j_f_ggh]h�jy^Z��bf_xl�keh`gmx�kljmdlmjm��Fh^_ebjh\Zgb_�lZdbo�jy^h\�im�

l_f�ihkljh_gby�fh^_eb�lj_g^Z��k_ahgghklb�b�i_jbh^bq_kdhc�khklZ\eyxs_c�g_�

ijb\h^bl�d�m^h\e_l\hjbl_evguf�j_amevlZlZf��Jy^�hklZldh\�qZklh�bf__l�klZlb�

klbq_kdb_� aZdhghf_jghklb��GZb[he__� jZkijhkljZg_ggufb� fh^_eyfb� klZpbh�

Page 31: KL:LBKLBQ?KDB? F?LH>U IJH=GHABJH

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gZjguo�jy^h\�y\eyxlky�fh^_eb�Z\lhj_]j_kkbb�b�fh^_eb�kdhevays_]h�kj_^g_]h� ;m^_f�jZkkfZljb\Zlv�deZkk�klZpbhgZjguo�\j_f_gguo�jy^h\��AZ^ZqZ�kh�

klhbl�\�ihkljh_gbb�fh^_eb�hklZldh\�\j_f_ggh]h�jy^Z�ut�b�ijh]ghabjh\Zgby�_]h�

agZq_gbc�� :\lhj_]j_kkbhggZy�fh^_ev�ij_^gZagZq_gZ� ^ey� hibkZgby� klZpbhgZjguo�

\j_f_gguo�jy^h\��KlZpbhgZjguc�ijhp_kk�m^h\e_l\hjy_l�mjZ\g_gbx�Z\lhj_]�

j_kkbb�[_kdhg_qgh]h�ihjy^dZ��k�^hklZlhqgh�[ukljh�m[u\Zxsbfb�dhwnnbpb�

_glZfb��<�qZklghklb��ihwlhfm�Z\lhj_]j_kkbhggZy�fh^_ev�^hklZlhqgh�\ukhdh]h�

ihjy^dZ� fh`_l� ohjhrh� Ziijhdkbfbjh\Zlv� ihqlb� ex[hc� klZpbhgZjguc� ijh�

p_kk��<�k\yab�k�wlbf�fh^_ev�Z\lhj_]j_kkbb�qZklh�ijbf_gy_lky�^ey�fh^_ebjh�

\Zgby�hklZldh\�\�lhc�beb�bghc�iZjZf_ljbq_kdhc�fh^_eb��gZijbf_j��j_]j_kkb�

hgghc�fh^_eb�beb�fh^_eb�lj_g^Z� Fh^_ev�Z\lhj_]j_kkbb�ihjy^dZ���AR(1)��fZjdh\kdbc�ijhp_kk�� FZjdh\kdbfb�gZau\Zxlky�ijhp_kku��\�dhlhjuo�khklhygb_�h[t_dlZ�\�dZ�

`^uc�ke_^mxsbc�fhf_gl�\j_f_gb�hij_^_ey_lky�lhevdh�khklhygb_f�\�gZklhy�

sbc�fhf_gl�b�g_�aZ\bkbl�hl�lh]h��dZdbf�iml_f�h[t_dl�^hklb]�wlh]h�khklhygby��

<�l_jfbgZo�dhjj_eypbhggh]h�ZgZebaZ�^ey�\j_f_gguo�jy^h\�fZjdh\kdbc�ijh�

p_kk�fh`gh�hibkZlv�ke_^mxsbf�h[jZahf��kms_kl\m_l�klZlbklbq_kdb�agZqbfZy�

dhjj_eypbhggZy�k\yav�bkoh^gh]h�jy^Z�k�jy^hf��k^\bgmluf�gZ�h^bg�\j_f_gghc�

bgl_j\Ze�b�hlkmlkl\m_l�k�jy^Zfb��k^\bgmlufb�gZ�^\Z��ljb�b�l��^��\j_f_gguo�

bgl_j\ZeZ��<�b^_Zevghf�kemqZ_�wlb�dhwnnbpb_glu�dhjj_eypbb�jZ\gu�gmex� :\lhj_]j_kkbhggZy�fh^_ev�i_j\h]h�ihjy^dZ�hij_^_ey_lky�khhlghr_gb�

_f�� u(t)=µ u(t-1)+ε(t) , (5)

]^_�µ -�qbkeh\hc�dhwnnbpb_gl� |µ|<1, ε(t) –�ihke_^h\Zl_evghklv�kemqZcguo�\_�

ebqbg��h[jZamxsbo�©[_euc�rmfª��E(ε(t))=0, E(ε(t)ε(t+τ))=

≠=0,0

0,2

ττσ ).

Fh^_ev�����gZau\Z_lky�lZd`_�fZjdh\kdbf�ijhp_kkhf�� Bf__f:

E(u(t))≡0. (6.1) >hdZ`_f�lh`^_kl\h�������

ut=µut-1+εt=µ(µut-2+ εt-1)+εt=µ2 ut-2+µεt-1+ εt=…=∑∞

=−

0kkt

kεµ .

Hldm^Z�k�mq_lhf�E(ε(t�� ��ihemqbf�E(u(t))=0. r(u(t)u(t±τ))=µτ. (6.2) Du(t)=σ2/(1-µ2). (6.3) cov(u(t)u(t±τ))=µτDu(t). (6.4)

Ba�������ke_^m_l��qlh�ijb�|µ|�[ebadhf�d�_^bgbp_�^bki_jkby�u(t��[m^_l�gZ�

Page 32: KL:LBKLBQ?KDB? F?LH>U IJH=GHABJH

32

fgh]h�[hevr_�^bki_jkbb�εt��Wlh�agZqbl� �mqblu\Zy� ������µ=r(u(t)u(t±1))=r(1) – bgl_jij_lZpby�iZjZf_ljZ�µ���qlh�\�kemqZ_�kbevghc�dhjj_eypbb�khk_^gbo�agZ�q_gbc�jy^Z�u(t��jy^�keZ[uo�\hafms_gbc�εt�[m^_l�ihjh`^Zlv�jZafZrbklu_�dh�

e_[Zgby�hklZldh\�u(t). Mkeh\b_�klZpbhgZjghklb�jy^Z�����hij_^_ey_lky�lj_[h\Zgb_f�|µ|<1. :\lhdhjj_eypbhggZy�nmgdpby��:DN��r(τ��fZjdh\kdh]h�ijhp_kkZ�hij_^_�

ey_lky�khhlghr_gb_f�������� QZklgZy�Z\lhdhjj_eypbhggZy�nmgdpby�

rqZkl(τ)=r(u(t)u(t+τ)) | u(t+1)=u(t+2)=…=u(t+τ-1)=0 fh`_l�[ulv�\uqbke_gZ�ih�nhjfme_��rqZkl(2)=(r(2)-r2(1))/(1-r2(1)���>ey�\lhjh]h�b�\ur_� ihjy^dh\� �kf�� >�@�� k�� ����� ����� ^he`gh� [ulv� rqZkl(τ)=0 ∀τ ����«� ��Wlh�

m^h[gh�bkihevah\Zlv�^ey�ih^[hjZ�fh^_eb� ����� _keb�\uqbke_ggu_�ih�hp_g_g�

guf� g_\yadZf� u(t)=yt- tf � \u[hjhqgu_� qZklgu_� dhjj_eypbb� klZlbklbq_kdb� g_�

agZqbfh�hlebqZxlky�hl�gmey�ijb�τ ����«��lh�bkihevah\Zgb_�fh^_eb�AR����^ey�hibkZgby�kemqZcguo�hklZldh\�g_�ijhlb\hj_qbl�bkoh^guf�^Zgguf��

B^_glbnbdZpby�fh^_eb��Lj_[m_lky�klZlbklbq_kdb�hp_gblv�iZjZf_lju�

µ�b�σ2�fh^_eb�����ih�bf_xsbfky�agZq_gbyf�bkoh^gh]h�jy^Z�yt��<u^_ey_f�g_�

kemqZcgmx� khklZ\eyxsmx� tf � b� ihemqZ_f� g_\yadb� ttt fyu ˆˆ −= ��GZoh^bf� ^bk�

i_jkbx� g_\yahd� ( )∑=

−=n

tt uu

n 1

2ˆ1γ �� ]^_� ∑

==

n

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nu

1

ˆ1

� �^ey� [hevrbgkl\Z� f_lh^h\�

\u^_e_gby� tf � Z\lhfZlbq_kdb� u ����Bkihevamy�f_lh^�fhf_glh\�b� ������� �������

ihemqbf�nhjfmeu�^ey�hp_gdb�iZjZf_ljh\�fh^_eb�����

( )( )

γµ

ˆ

ˆˆˆ

1

111

1 ∑−

=+− −−

=

n

tttn uuuu

,

( )γµσ ˆˆ1ˆ 22 −= .

Fh^_ev�Z\lhj_]j_kkbb�ihjy^dZ���– AR(2) (ijhp_kk�XeZ). :\lhj_]j_kkbhggZy�fh^_ev�\lhjh]h�ihjy^dZ�hij_^_ey_lky�khhlghr_gb�

_f�� u(t)=µ1u(t-1)+µ2u(t-2)+ε(t), (7)

]^_�ε(t) –�ihke_^h\Zl_evghklv�kemqZcguo�\_ebqbg��h[jZamxsbo�[_euc�rmf� Mfgh`bf�h[_�qZklb�nhjfmeu�����ih�hq_j_^b�gZ�u(t-���b�u(t-���b�\havf_f�

fZl_fZlbq_kdh_�h`b^Zgb_�hl�^\mo�khhlghr_gbc� E(u(t)u(t-1))=µ1E(u(t-1)u(t-1))+µ2E(u(t-2)u(t-1))+E(ε(t)u(t-1)), E(u(t)u(t-2))=µ1E(u(t-1)u(t-2))+µ2E(u(t-2)u(t-2))+E(ε(t)u(t-2)) beb γ(1)-µ1Du(t)-µ2γ(1)=0, γ(2)-µ1γ(1)-µ2Du(t)=0,

Page 33: KL:LBKLBQ?KDB? F?LH>U IJH=GHABJH

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beb�ihke_�^_e_gby�h[hbo�mjZ\g_gbc�gZ�Du(t):

=−−=−−

.0)1()2(

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21

21

µµµµ

rr

rr (8)

JZaj_rZy�kbkl_fm�mjZ\g_gbc�����hlghkbl_evgh�µ1�b�µ2��ihemqbf�

.)1(1

)1()2(

,)1(1

))2(1)(1(

2

2

2

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r

rr

r

rr

−−=

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µ (9)

JZaj_rZy�kbkl_fm�mjZ\g_gbc�����hlghkbl_evgh�r����b�r�����ihemqbf�

.1

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)1(2

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µµµ

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?keb�h[_�qZklb�nhjfmeu�����mfgh`blv�gZ�u(t-τ��b�\aylv�fZl_fZlbq_kdh_�h`b^Zgb_��lh�ihemqbf�j_dmjj_glgmx�nhjfmem�^ey�\uqbke_gby�agZq_gbc�:DN�

�ZgZeh]�������� r(τ)=µ1r(τ-1)+µ2r(τ-2). (11)

Mfgh`bf�h[_�qZklb�nhjfmeu�����gZ�u(t��b�\havf_f�fZl_fZlbq_kdh_�h`b�^Zgb_�

E(u(t)u(t))=µ1E(u(t-1)u(t))+µ2E(u(t-2)u(t))+E(ε(t)u(t)), Du(t)=µ1γ(1)+µ2γ(2)+E[ε(t)(µ1u(t-1)+µ2u(t-2)+ε(t))], Du(t)=µ1γ(1)+µ2γ(2)+σ2, σ2=Du(t)-µ1γ(1)-µ2γ(2)=Du(t)[1-µ1r(1)-µ2r(2)]={ih (10)}=

=Du(t)

−−−

−2

2212

22

21

111

µµµµ

µµ =Du(t) ( )( ) ( )

+−+−2

221

22 1

111

µµµµµ ⇒

σ2=Du(t) ( )[ ].11

1 21

22

2

2 µµµµ −−

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dhwnnbpb_glZf�Z\lhdhjj_eypbb������

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µµµ

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µ

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Page 34: KL:LBKLBQ?KDB? F?LH>U IJH=GHABJH

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rqZkl(τ)=0 ∀τ ��� ���«� ��?keb� \uqbke_ggu_� ih� hp_g_gguf� g_\yadZf� u(t)=yt- tf

\u[hjhqgu_�qZklgu_�dhjj_eypbb�klZlbklbq_kdb�g_agZqbfh�hlebqZxlky�hl�gm�

ey�ijb�τ ������«��lh�bkihevah\Zgb_�fh^_eb�AR����^ey�hibkZgby�kemqZcguo�hk�lZldh\�g_�ijhlb\hj_qbl�bkoh^guf�^Zgguf�

B^_glbnbdZpby�fh^_eb��Lj_[m_lky�klZlbklbq_kdb�hp_gblv�iZjZf_lju�

µ1, µ2�b�σ2�fh^_eb�����ih�bf_xsbfky�agZq_gbyf�bkoh^gh]h�jy^Z�yt��<u^_ey_f�

g_kemqZcgmx�khklZ\eyxsmx� tf �b�ihemqZ_f�g_\yadb� ttt fyu ˆˆ −= ��GZoh^bf�^bk�

i_jkbx� g_\yahd� ( )∑=

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\u^_e_gby� tf � Z\lhfZlbq_kdb� u ��� b� Z\lhdhjj_eypbb� )1(r , )2(r :

( )( ) 2,1,ˆˆˆ)(ˆ1

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kn

tkttkn γ ��Ih^klZ\b\� \� ���� )1(r �b� )2(r ��ihemqbf�

hp_gdb�iZjZf_ljh\� 1µ �b� 2µ ��bkihevamy�dhlhju_�ih������[m^_f�bf_lv�hp_gdm�σ2.

Fh^_eb�Z\lhj_]j_kkbb�j�ihjy^dZ�– AR(p��ijb�p≥���kf���gZijbf_j��>�@��k��834-837):

u(t)=µ1u(t-1)+µ2u(t-2)+…+ε(t). (13) Ijbf_j��=jZnbd�i_j\hc� jZaghklb� jy^Z�� ohjhrh� hibku\Zxs_cky�fh^_�

evx�AR�����ij_^klZ\e_gghc�gZ�jbk�����]jZnbd�\u[hjhqghc�Z\lhdhjj_eypbhgghc�nmgdpbb��:DN��i_j\hc�jZaghklb�wlh]h�jy^Z�ij_^klZ\e_g�gZ�jbk�����x

Jbk��� Jbk���

6.2. Fh^_eb�kdhevays_]h�kj_^g_]h�ihjy^dZ�q

(Moving Average - MA(q) models)

Ijb�jZkkfhlj_gbb�wdkihg_gpbZevgh]h�k]eZ`b\Zgby�j_qv�reZ�h�lhf��qlh�

gZ�ihdZaZl_ev�\�l_dmsbc�fhf_gl�\j_f_gb�hdZau\Z_l�\ha^_ckl\b_�agZq_gb_�ih�

dZaZl_ey�\�ij_^u^msb_�fhf_glu��Ohly�\ha^_ckl\b_�hl^Ze_gguo�we_f_glh\�g_�

agZqbl_evgh��gh�\�kmff_�hgh�fh`_l�hdZau\Zlv�kms_kl\_ggh_�\ebygb_�gZ�fh�

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35

^_ev��Mq_klv�wlh�\ha^_ckl\b_�\hafh`gh�\�fh^_eb�kdhevays_]h�kj_^g_]h��Fh�

^_ebjh\Zgb_�\ha^_ckl\by�\k_o�ij_^r_kl\mxsbo�we_f_glh\�jy^Z�gZ�ihdZaZl_ev�

\�l_dmsbc�fhf_gl�hkgh\Zgh�gZ�ij_^ihkued_�h�lhf��qlh�\�hrb[dZo�fh^_eb�aZ�

g_kdhevdh�ij_^r_kl\mxsbo�i_jbh^h\�khkj_^hlhq_gZ�bgnhjfZpby�h�\k_c�ij_�

^uklhjbb�jy^Z� Fh^_evx�kdhevays_]h�kj_^g_]h�ihjy^dZ�q�gZau\Z_lky�ijhp_kk�

u(t)=ε(t)-θ1ε(t-1)-θ2ε(t-2)-…-θqε(t-q). (14) <�qZklghklb��fh^_eb�ihjy^dZ���b���khhl\_lkl\_ggh�bf_xl�\b^�

u(t)=ε(t)-θε(t-1), (15) u(t)=ε(t)-θ1ε(t-1)-θ2ε(t-2). (16)

I_j_oh^�hl�nhjfu������d�nhjf_������hkms_kl\ey_lky�k�ihfhsvx�ihke_�

^h\Zl_evghc�ih^klZgh\db�\�ijZ\mx�qZklv�nhjfmeu������\f_klh�u(t-1), u(t-2), … bo�\ujZ`_gbc��\uqbke_gguo�ih�nhjfme_� �����^ey�fhf_glh\�\j_f_gb� t-1, t-2, «��Wlh�hagZqZ_l�^\hckl\_gghklv�\�ij_^klZ\e_gbb�ZgZebabjm_fh]h�\j_f_ggh]h�

jy^Z�–�^\_� wd\b\Ze_glgu_�nhjfu�ebg_cgh]h�ijhp_kkZ� -�b�h[jZlbfhklv�AR�b�MA�fh^_e_c�

<� dZq_kl\_� ijbf_jZ� jZkkfhljbf�fh^_ev� kdhevays_]h� kj_^g_]h� i_j\h]h�

ihjy^dZ�–�F:�����>ZggZy�fh^_ev�hibku\Z_lky�khhlghr_gb_f�������Fh`gh�ih�

dZaZlv��qlh�klZpbhgZjghklv�u(t��h[_ki_qb\Z_lky�ijb�ex[hf�agZq_gbb�iZjZf_ljZ��Fh^_ev�h[jZlbfZ��ij_^klZ\bfZ�\�\b^_�fh^_eb�Z\lhj_]j_kkbb�[_kdhg_qgh]h�

ihjy^dZ��ijb�mkeh\bb�|θ|<1. :\lhdhjj_eypbhggZy�nmgdpby�

=+−

=.2,0

,1,1)( 2

τ

τθθ

τr

QZklgZy� dhjj_eypbhggZy�nmgdpby� ijhp_kkZ�F:����� hij_^_eyxsZy� kl_�

i_gv� l_kghlu�dhjj_eypbhgghc� k\yab�f_`^m�u(t��b�u(t±τ��ijb�nbdkbjh\Zgguo�agZq_gbyo�\k_o�ijhf_`mlhqguo�we_f_glh\�wlh]h�jy^Z��aZ^Z_lky�\ujZ`_gb_f�

)1(2

2

1

1)( +−

−−= ττ

θθθτ

qZklgr .

B^_glbnbdZpby�fh^_eb F:�����Lj_[m_lky� klZlbklbq_kdb� hp_gblv� iZ�

jZf_lju�θ�b�σ2�fh^_eb������ih�bf_xsbfky�agZq_gbyf�bkoh^gh]h�jy^Z�yt��<u�

^_ey_f�g_kemqZcgmx�khklZ\eyxsmx� tf �b�ihemqZ_f�g_\yadb� ttt fyu ˆˆ −= ��GZoh�

^bf� hp_gdm� Z\lhdhjj_eypbb� )1(r : ( )( ) ( )∑∑=

=+− −−−=

n

ttn

n

tttn uuuuuur

1

211

111

1 ˆˆˆ)1(ˆ .

Ih^klZ\eyy� )1(r � \� \ujZ`_gb_� ^ey� Z\lhdhjj_eypbhgghc� nmgdpbb�� bf__f��

θ2+(1/ )1(r )θ�� ���d\Z^jZlgh_�mjZ\g_gb_�^ey�θ��Ba�^\mo�j_r_gbc�ijb\_^_ggh]h�

d\Z^jZlgh]h�mjZ\g_gby��θ1θ2 ���h^gh�[m^_l�f_gvr_�_^bgbpu�–�_]h�b�\u[bjZ_f�

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36

\�dZq_kl\_�bkdhfhc�hp_gdb�iZjZf_ljZ�\�fh^_eb�F:(1).

Hp_gdZ�σ2 ihemqZ_lky�ba�nhjfmeu��

( )2

1

21

2

ˆ1

ˆˆ

θσ

+

−=

∑=

n

ttn uu

.

Fh^_ev�kdhevays_]h�kj_^g_]h�\lhjh]h�ihjy^dZ�–�F:����hlebqZ_lky�[he__�

keh`guf�ihkljh_gb_f �kf��>�@��k�����-845). <Z`gh_�ijZdlbq_kdh_�agZq_gb_�bf_xl�ijhp_kku��i_j\Zy��beb�[he__�\u�

khdZy��jZaghklv�dhlhjuo�klZpbhgZjgZ�b�y\ey_lky�ijhp_kkhf�F:(q���Ih^h[gu_�ijhp_kku�mkljh_gu�dZd�kemqZcgu_�dhe_[Zgby�k�g_ihklhygguf�kj_^gbf�mjh\�

g_f��beb� �^ey�\lhjhc�jZaghklb��g_ihklhygguf�m]ehf�gZdehgZ��>ey�ijh]ghab�

jh\Zgby� ih^h[guo� ijhp_kkh\� qZklh� bkihevamxl� f_lh^� wdkihg_gpbZevgh]h�

k]eZ`b\Zgby�

6.3. Fh^_eb�Z\lhj_]j_kkbb-kdhevays_]h�kj_^g_]h (AutoRegressive - Moving Average - ARMA(j, q) models)

GZ�ijZdlbd_�^ey�wdhghfbqghc�iZjZf_ljbaZpbb�ZgZebabjm_fh]h�ijhp_kkZ�

bgh]^Z�[u\Z_l�g_h[oh^bfh�\dexqblv�\�fh^_ev�dZd�qe_gu��hibku\Zxsb_�Z\lh�

j_]j_kkbx��lZd�b�qe_gu��fh^_ebjmxsb_�hklZlhd�\�\b^_�kdhevays_]h�kj_^g_]h��

LZdhc�ebg_cguc�ijhp_kk�bf__l�\b^� u(t)=µ1u(t-1)+…+µju(t-j)+ε(t)-θ1ε(t-1)-…-θqε(t-q) (17)

b�gZau\Z_lky�ijhp_kkhf�Z\lhj_]j_kkbb�-�kdhevays_]h�kj_^g_]h�ihjy^dZ��p, q) – ARMA(p, q).

JZkkfhljbf� \� dZq_kl\_� ijbf_jZ� fh^_ev ARMA(1, 1)�� <� khhl\_lkl\bb� k�fh^_evx�������ijhp_kk�ARMA(1, ���hibku\Z_lky�nhjfmehc�

u(t)=µu(t-1)+ε(t)-θε(t-1) beb u(t)-µu(t-1)=ε(t)-θε(t-1). Ijhp_kk� ARMA(1, ��� klZpbhgZj_g�� _keb� dhj_gv� oZjZdl_jbklbq_kdh]h�

mjZ\g_gby�AR����fh^_eb��-µz ��ih�fh^mex�[hevr_�_^bgbpu��Lh�_klv�^he`gh�

[ulv� |µ|���� H[jZlbfhklv� ijhp_kkZ� ARMA(1, ��� h[_ki_qb\Z_lky� lj_[h\Zgb_f��qlh[u�dhj_gv�oZjZdl_jbklbq_kdh]h�mjZ\g_gby�FA����fh^_eb��-θz ��ih�fh^mex�[ue�[hevr_�_^bgbpu��Lh�_klv�^he`gh�[ulv�|θ|<1.

:DN��

( )( )

≥+−

=−+

−−=

− .2),1()1(

,1,21

1

)(1

2

τµτµ

τµθθ

θµµθτ

τ rr

r

:\lhdhjj_eypbhggZy�nmgdpby�wdkihg_gpbZevgh�m[u\Z_l�hl�gZqZevgh]h�

agZq_gby�r�����ijbq_f�wlh�m[u\Zgb_�fhghlhggh��_keb�µ�iheh`bl_evgh��b�dhe_�[Zl_evgh��agZdhi_j_f_ggh���_keb�µ�hljbpZl_evgh��

Ba�ihke_^g_]h�jZ\_gkl\Z�b�mkeh\bc�klZpbhgZjghklb�b�h[jZlbfhklb�ke_�

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37

^m_l��qlh�r����b�r����^he`gu�m^h\e_l\hjylv�mkeh\byf�

>−><+>

<

.0)1(),1)1(2)(1()2(

,0)1(),1)1(2)(1()2(

,)1()2(

rrrr

rrrr

rr

Wlb�mkeh\by�[u\Zxl�ihe_agufb�ijb�ijh\_jd_�]bihl_au��ih�\u[hjhqguf�

agZq_gbyf�dhwnnbpb_glh\�Z\lhdhjj_eypbb��h�lhf��qlh�ZgZebabjm_fuc�ijhp_kk�

fh`_l�[ulv�hibkZg�ARMA(1, ���fh^_evx� B^_glbnbdZpby�fh^_eb ARMA(1, ����Lj_[m_lky�klZlbklbq_kdb�hp_gblv�

iZjZf_lju�µ, θ�b�σ2�fh^_eb�ih�bf_xsbfky�agZq_gbyf�bkoh^gh]h�jy^Z�yt. WlZi����Bf__f�ihke_�mfgh`_gby�h[_bo�qZkl_c�fh^_eb�gZ�u(t-2):

u(t)u(t-2)-µu(t-1)u(t-2)=ε(t)u(t-2)-θ1ε(t-1)u(t-2) ⇒ E(u(t)u(t-2))-µE(u(t-1)u(t-2))=E(ε(t)u(t-2))-θE(ε(t-1)u(t-2)),

beb�ihke_�^_e_gby�gZ�Du(t):

r(2)-µr(1)=0 ⇒ .)1(ˆ)2(ˆ

ˆr

r=µ

WlZi����<uibku\Z_f�khhlghr_gby� u(t)- µ u(t-1)=ε(t)-θε(t-1),

u(t+1)-µ u(t)=ε(t+1)-θε(t).

<ha\h^y� \� d\Z^jZl� i_j\h_� jZ\_gkl\h�� i_j_fgh`Zy� ihqe_ggh� i_j\h_� kh�

\lhjuf��i_j_oh^y�d�fZl_fZlbq_kdbf�h`b^Zgbyf�ihemq_gguo�\ujZ`_gbc�� E(u2(t))-2µ E(u(t)u(t-1))+µ 2E(u2(t-1))=?(ε2(t))-2θ?(ε(t)ε(t-1))+θ2

?(ε2(t-1)), E(u(t)u(t+1))-µ E(u(t-1)u(t+1))-µ E(u2(t))+ µ 2E(u(t-1)u(t))=

=E(ε(t)ε(t+1))+θ2?(ε(t-1)ε(t)).

AZf_gyy�dh\ZjbZpbb�γ�bo�\u[hjhqgufb�agZq_gbyfb��ihemqbf�kbkl_fm�ba�^\mo�mjZ\g_gbc�hlghkbl_evgh�θ�b�σ2:

( ) ( )( ) ( )

−=+−+

+=−+

.)2(ˆˆˆˆ1)1(ˆ

,1)1(ˆˆ2ˆ1ˆ22

222

θσγγµµγ

θσγµµγ

Ih^_eb\� i_j\h_� mjZ\g_gb_� kbkl_fu� gZ� \lhjh_�� ihemqbf� d\Z^jZlgh_�

mjZ\g_gb_�hlghkbl_evgh�θ:

A=-(1+θ2)/θ��]^_�:= ( )( ) ( ).)2(ˆˆˆˆ1)1(ˆ

)1(ˆˆ2ˆ1ˆ2

2

γγµµγγµµγ+−+

−+

Ba�^\mo�dhjg_c�mjZ\g_gby�\u[bjZ_f�lhl��dhlhjuc�m^h\e_l\hjy_l�mkeh�

\bx�h[jZlbfhklb�|θ|����Hp_gdm�σ2 hij_^_ey_f�ba�ex[h]h�mjZ\g_gby�kbkl_fu�

6.4. Fh^_ev�Z\lhj_]j_kkbb�-�ijhbgl_]jbjh\Zggh]h�kdhevays_]h�kj_^g_]h� (AutoRegressive Integrated Moving Average - ARIMA (j, q, k) models)

Fh^_ev�\i_j\u_�[ueZ�ij_^eh`_gZ�>`�;hdkhf�b�=�>`_gdbgkhf�b�ihwlh�

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38

fm�ba\_klgZ�dZd�fh^_ev�;hdkZ->`_gdbgkZ��Wlh�h^gZ�ba�gZb[he__�ihimeyjguo�fh^_e_c�^ey�ihkljh_gby�djZldhkjhqguo�ijh]ghah\�

;m^_f�jZkkfZljb\Zlv�g_klZpbhgZjgu_��h^ghjh^gu_�\j_f_ggu_�jy^u��Lh�

_klv�jy^u��^ey�dhlhjuo�kemqZcguc�hklZlhd�u(t���ihemqZxsbcky�ihke_�\uqblZ�

gby�ba�jy^Z�y(t��_]h�g_kemqZcghc�khklZ\eyxs_c� f(t���ij_^klZ\ey_l�klZpbhgZj�guc�\j_f_gghc�jy^�

Fh^_ev�;hdkZ->`_gdbgkZ�ij_^gZagZq_gZ�^ey�hibkZgby�g_klZpbhgZjguo�\j_f_gguo�jy^h\�kh�ke_^mxsbfb�k\hckl\Zfb�

Z��\�jZfdZo�Z^^blb\ghc�fh^_eb�y(t��\dexqZ_l� f(t���bf_xsmx�\b^�Ze]_[�

jZbq_kdh]h�ihebghfZ�hl�t�kl_i_gb�k-���ijbq_f�dhwnnbpb_glu�ihebghfZ�fh]ml�[ulv�dZd�klhoZklbq_kdb_��lZd�b�g_klhoZklbq_kdb_��

[��jy^�yk(t), t=1,2,…, n-k��ihemqb\rbcky�ba�y(t��ihke_�ijbf_g_gby�d�g_fm�f_lh^Z�ihke_^h\Zl_evguo�jZaghkl_c��fh`_l�[ulv�hibkZg�fh^_evx�ARMA(j, q).

Ke_^h\Zl_evgh��fh^_ev�;hdkZ->`_gdbgkZ�bf__l�\b^�

yk(t)=µ1yk(t-1)+…+µjyk(t-j)+ε(t)-θ1ε(t-1)-…-θqε(t-q), (18) ]^_ yk(t)=∆ky(t)=y(t)-Ck

1y(t-1)+Ck2y(t-2)-…+(-1)ky(t-k), t=k+1, k+2, …, n. A^_kv ∆k

– k-y ihke_^h\Zl_evgZy jZaghklv ZgZebabjm_fh]h ijhp_kkZ y(t) (∆=y(t)-y(t-1), ∆2=∆y(t)-∆y(t-1) b l.i.).

<\_^_f�hi_jZlhju�k^\b]Z�\h�\j_f_gb��F+yt=yt+1, F−yt=yt-1��Ijbq_f�F+F−=1, Fk

−yt=yt-k, Fk+yt=yt+k, ∆=1−F−� �� Lh]^Z� hi_jZlhj� Z\lhj_]j_kkbb� ihjy^dZ� p AR(p)

bf__l�\b^�

pA (F−, µ)=1-µ1F−-µ2F2

−- … -µpFp

− ,

Z�hi_jZlhj�kdhevays_]h�kj_^g_]h�ihjy^dZ�q MA(q):

qB (F−, θ)=1-θ1F−-θ2F2

−- … -θqFq

− .

Fh^_ev�ARIMA(j, q, k��[m^_l�k�mq_lhf�nhjfmeu������b�\\_^_gguo�hi_jZ�lhjh\�bf_lv�\b^�

pA (F−, µ)∆ky(t)= qB (F−, θ)ε(t). (18Z)

H[hagZqbf�S –�hi_jZlhj�kmffbjh\Zgby�– Sy(t)=y(t)+y(t-1)+y(t-���«���Lh�

]^Z�bf__f� Sy(t)=(1+F−+F2

−+ …)y(t)=(1−F−)-1y(t) beb Sy(t)=∆-1y(t)

b S2y(t)=S(Sy(t)), S2y(t)=(∆2)-1y(t) … Sky(t)=S(Sk-1y(t)), Sky(t)=(∆k)-1y(t). Lh�_klv�hi_jZlhj�6k�y\ey_lky�h[jZlguf�^ey�∆k��Ijbf_gyy�hi_jZlhj�6k�d�

h[_bf�qZklyf����Z���ihemqbf�

pA (F−, µ)Sk{ ∆ky(t)}=Sk{ qB (F−, θ)ε(t)} ⇒ pA (F−, µ)y(t)=Sk{ qB (F−, θ)ε(t)}.

<�ihke_^g_c�nhjfme_�ke_\Z�ij_^klZ\e_g�ijhp_kk�Z\lhj_]j_kkbb��kijZ\Z�-�ijh�kmffbjh\Zggh]h��©ijhbgl_]jbjh\Zggh]hª��kdhevays_]h�kj_^g_]h��

GZ�ijZdlbd_�ijbf_gyxlky�fh^_eb�ARIMA(j, q, k���\�dhlhjuo�j, q, k�g_�ij_\urZxl����GZijbf_j, ARIMA(1, 1, 1):

Page 39: KL:LBKLBQ?KDB? F?LH>U IJH=GHABJH

39

1A (F−, µ)∆y(t)= 1B (F−, θ)ε(t) ⇒ (1-µF−)(yt-yt-1)=(1-θF−)εt ⇒

yt-yt-1-µyt-1+µyt-2=εt-θεt-1 ⇒ y(t)=(1+µ)y(t-1)-µy(t-2)+ε(t)-θε(t-1).

QZklguf�kemqZ_f�fh^_eb�ARIMA�y\ey_lky�fh^_ev�Z\lhj_]j_kkbb�:R(p), ^ey� dhlhjhc� q=k ���>jm]hc� qZklguc� kemqZc� -�fh^_ev� kdhevays_]h� kj_^g_]h�

MA(q��� ^ey� dhlhjhc� p=k ��� <Z`gu_� ki_pbZevgu_� deZkku� fh^_e_c� -� fh^_eb�ARIMA(0, q, k���b�fh^_eb�ARMA(p, q)= ARIMA(j, q, 0).

Fh^_ev�:R����ijb�iheh`bl_evghf�dhwnnbpb_gl_�Z\lhdhjj_eypbb�ij_^�klZ\ey_l�kh[hc�dhe_[Zl_evguc�ijhp_kk�k�ij_h[eZ^Zgb_f�^ebgguo�\heg��?keb�

dhwnnbpb_gl�dhjj_eypbb�hljbpZl_e_g��ijhp_kk�y\ey_lky�kbevgh�hkpbeebjmx�

sbf��Fh^_ev�ARIMA���� ��� ���hibku\Z_l� kemqZcguc�ijhp_kk� k�g_ihklhygguf�mjh\g_f�� _]h� gZbemqrbc� ijh]gha� fh`_l� [ulv� ihkljh_g� k� ihfhsvx� f_lh^Z�

wdkihg_gpbZevgh]h� k]eZ`b\Zgby�ihjy^dZ� ���Ijb� wlhf�hilbfZevgh_� agZq_gb_�

k]eZ`b\Zxs_c�ihklhygghc�k\yaZgh�k�dhwnnbpb_glhf�kdhevays_]h�kj_^g_]h�b�

^bki_jkb_c� hklZldh\�� :gZeh]bqgh_� ml\_j`^_gb_� kijZ\_^eb\h� ^ey� fh^_eb�

ARIMA�����������hibku\Zxs_c�kemqZcguc�ijhp_kk�k�i_j_f_gguf�mjh\g_f�b�m]�

ehf�gZdehgZ��>ey�_]h�ijh]ghabjh\Zgby�g_h[oh^bfh�bkihevah\Zlv�f_lh^�wdkih�

g_gpbZevgh]h�k]eZ`b\Zgby�i_j\h]h�ihjy^dZ� B^_glbnbdZpby�ARIMA �fh^_e_c��KljmdlmjZ�fh^_eb�ARIMA�hibku\Z�

_lky�lj_fy�iZjZf_ljZfb��j, q, k���Djhf_�lh]h��jZagu_�ih�nhjf_�fh^_eb�fh]ml�[ulv�^h\hevgh�[ebadb�^jm]�^jm]m��Ihwlhfm�\_kvfZ�\Z`gh�ih�\hafh`ghklb�ijZ�

\bevgh�hij_^_eblv�kljmdlmjm�fh^_eb��JZkkfhljbf�wlZiu�b^_glbnbdZpbb� 1. Ih^[bjZ_lky� ihjy^hd�fh^_eb� k��>ey� wlh]h� bkihevam_lky� eb[h�f_lh^�

ihke_^h\Zl_evguo�jZaghkl_c��eb[h�ZgZeba�Z\lhdhjj_eypbhgguo�nmgdpbc�ijh�

p_kkh\�∆y(t), ∆2y(t), … -�ihdZ�g_�^hklb]g_f�[ukljh]h�aZlmoZgby��klZpbhgZjgh�klb�� Z\lhdhjj_eypbhgghc�nmgdpbb�^ey�g_dhlhjh]h�k��>`�;hdk�b�=�>`_gdbgk�ij_^eZ]Zxl�\aylv�aZ�\bamZevguc�djbl_jbc�klZpbhgZjghklb�[ukljh_�m[u\Zgb_�

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ijb\h^bl�d�jhklm�^bki_jkbb�hrb[hd�b�d�aZf_lghfm�jhklm�^bki_jkbb�ijh]ghaZ�� 2. GZoh^bf�yk(t)=∆ky(t��b�b^_glbnbpbjm_f�ARMA(j, q��fh^_ev��kf��i������� Ijbf_j��>ey�hij_^_e_gby�ihjy^dh\�Z\lhj_]j_kkbb�b�kdhevays_]h�kj_^�

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lhf�ba\_klguo�iZjZf_ljh\�µ1, …, µp, θ1, …, θq�iml_f�ijbf_g_gby�hi_jZpbb�mk�eh\gh]h� mkj_^g_gby� d� h[_bf�qZklyf�jZ\_gkl\Z� ����� k�mq_lhf� ke_^mxsbo�jZ�\_gkl\�

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Jbk������Ijhp_kk�\u[hjZ�ARIMA�fh^_eb

E(yt-i |y1, y2, …, yt)=yt-i ∀ i=0, 1, 2, …, t-1; E(yt+i |y1, y2, …, yt)= )(ˆ tyi ∀ i=1, 2, …;

E(εt+i |y1, y2, …, yt)=0 ∀ i=1, 2, …; E(εt-i |y1, y2, …, yt)=yt-i- )1(ˆ1 −− ity ∀ i=0, 1, 2, … . (20)

Ijbf_j��Fh^_ev�ARIMA(1, 0, ����K�mq_lhf����Z��fh^_ev�bf__l�\b^� y(t)=(1+µ)y(t-1)-µy(t-2)+ε(t),

ARMA(1,0) ?

ARMA(2,1) ?

ARMA(2,0) ?

ARMA(1,1) ?

ARMA(3,2) ?

ARMA(3,0) ?

ARMA(2,2) ?

ARMA(1,2) ?

ARMA(0,2) ?

ARMA(n,n-1) ?

ARMA(n,0) ?

ARMA(n-1, n-1) ?

ARMA(n-2, n-1) ?

ARMA(0, n-1) ?

ARMA(0,1) ?

Page 44: KL:LBKLBQ?KDB? F?LH>U IJH=GHABJH

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]^_�agZq_gb_�µ�ba\_klgh�ihke_�b^_glbnbdZpbb�fh^_eb��Ba�mjZ\g_gby������ih�emqZ_f�

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[m^msb_�agZq_gby�εt�jZ\gufb�gmex� )(ˆ1 ty =(1+µ)yt-µyt-1,

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)(ˆ tyτ =(1+µ) )(ˆ 1 ty −τ -µ )(ˆ 2 ty −τ , τ≥3. x

Ijbf_j��JZkkfhljbf�bkihevah\Zgb_�fh^_eb�ARIMA��������^ey�ijh]ghab�jh\Zgby��Bkoh^gZy�fh^_ev�bf__l�\b^�

y(t)=(1+µ)y(t-1)-µy(t-2)+ε(t)-θε(t-1), ]^_�agZq_gby�µ, θ�ba\_klgu�ihke_�b^_glbnbdZpbb�fh^_eb��Ba�mjZ\g_gby������b�nhjfme������ihemqZ_f��iheZ]Zy�[m^msb_�agZq_gby�εt�jZ\gufb�gmex�

)(ˆ1 ty =(1+µ)yt-µyt-1-θε(t),

)(ˆ tyτ =(1+µ) )(ˆ 1 ty −τ -µ )(ˆ 2 ty −τ , τ≥2.

AgZq_gb_�ε(t��fh`gh�\uqbkeblv�ih�nhjfme_�ε(t)=y(t)- )1(ˆ1 −ty ��ihkljhb\�j_ljhki_d�

lb\guc�ijh]gha� )1(ˆ1 −ty ��<�mijhs_gghf�\ZjbZgl_�fh`gh�iheh`blv�ε(t)=0. x

6.6. Fh^_eb ARCH (AutoRegressive Conditional Heteroscedasticity) b GARCH (General ARCH)

>ey� g_dhlhjuo� \j_f_gguo� jy^h\�� oZjZdl_jbamxsbo� fZdjhwdhghfbq_�

kdb_�ijhp_kku��[ueZ�\uy\e_gZ�aZdhghf_jghklv�\�ih\_^_gbb�kemqZcguo�hklZl�

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kl_jZfb�� G_kfhljy� gZ� wlh�� ]bihl_aZ� h� ]hfhkd_^Zklbqghklb� ^bki_jkbb� ^ey�

[hevrbo� \j_f_gguo� bgl_j\Zeh\� g_� ijhlb\hj_qbeZ� ^Zgguf��>ey� h[tykg_gby�

wlh]h�nZdlZ�[ueb�fh^bnbpbjh\Zgu�fh^_eb�ARMA(j, q). DeZkk� fh^_e_c� \j_f_gguo� jy^h\�� \� dhlhjuo� mqblu\Zxlky� baf_g_gby�

^bki_jkbb�b�dh\ZjbZpbc��gZau\Z_lky�fh^_eyfb�\heZlbevghklb. Imklv�fu�ohlbf�hp_gblv�ARMA�fh^_ev�u(t)=µu(t-1)+ε(t)-θε(t-���b�kijh]�

ghabjh\Zlv� u(t����� Mkeh\guc� ijh]gha� kmlv�� E(u(t+1)|u(t))=µu(t��� >bki_jkby�hrb[db� ijh]ghaZ�� E[(u(t+1)-µu(t))2]=E[(ε(t)-θε(t-1))2]=σ2(1+θ2

��� ?keb� kljhblv�

[_amkeh\guc� ijh]gha�� mkj_^gyy� u(t�� ih� \k_f� ^Zgguf�� lh� ihemqbf� u(t+1)=0. >bki_jkby�hrb[db�[_amkeh\gh]h�ijh]ghaZ��E[(u2(t+1)]=σ2(1+θ2-2µθ)/(1-µ2

���LZd�

dZd�����-µ2��!����lh�[_amkeh\guc�ijh]gha�bf__l�[hevrmx�^bki_jkbx��q_f�mk�

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gby�jy^Z��ij_^ihqlbl_evg__�� Dexq_\hc� hkh[_gghklvx�fh^_eb�ARCH� y\ey_lky� jZa]jZgbq_gb_� mkeh\�

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b�h[sb_�^bgZfbq_kdb_�khhlghr_gby�^ey�p_g�Zdlb\h\��Djhf_�lh]h��k�lhqdb�aj_�

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lb\ghklb�ba-aZ�g_�mq_lZ�]_l_jhkd_^Zklbqghklb�fh`_l�[ulv�kdhev�m]h^gh�[hev�rhc�� b� ijb� khklZ\e_gbb� khpbZevgh-wdhghfbq_kdbo� ijh]ghah\�� dZd� ba\_klgh��fh`gh�bkihevah\Zlv�gZfgh]h�[he__�lhqgmx�hp_gdm�g_hij_^_e_gghklb�hrb[db�

ijh]ghaZ�� _keb� ihemqZlv� __� dZd� mkeh\gmx� ih� l_dms_fm� bgnhjfZpbhgghfm�

fgh`_kl\m� Ke_^my�aZ�J�Wg]ehf� ������]����[m^_f�jZkkfZljb\Zlv�hklZldb�u(t��dZd�mk�

eh\gh�]_l_jhkd_^Zklbqgu_�b�k\yaZggu_�Z\lhj_]j_kkb_c�i_j\h]h�ihjy^dZ� [ ] ( )

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σ

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beb��qlh�lh�`_�� u(t)=ε(t)[µ0+µ1u

2(t-1)], (21a) ]^_�ε(t) –�ghjfZevguc�[_euc�rmf��µ0>0, 0<µ1<1.

Fh^_ev� ������ ���a��gZau\Z_lky� Z\lhj_]j_kkbhgghc�mkeh\gh� ]_l_jhkd_^Z�

klbqghc�fh^_evx�i_j\h]h�ihjy^dZ�– ARCH (1). ARCH fh^_ev�ijhba\hevgh]h�q�ihjy^dZ�bf__l�\b^�

u(t)=ε(t)[µ0+µ1u2(t-1)+µ2u

2(t-2)+…+µqu2(t-q)]. (21[)

GZ�iZjZf_lju�µ0, µ1��«�gZdeZ^u\Zxlky�h]jZgbq_gby��h[_ki_qb\Zxsb_�

[_amkeh\gmx�]hfhkd_^Zklbqghklv�hklZldh\�u(t). Fh^_ev����

a��fh`gh�aZibkZlv�\�\b^_��u2(t)= 2

tσ +(u2(t)- 2tσ )=µ0+µ1u

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E(y2T+τ|yT��� dhlhjZy� ^ey� ARCH (1) kmlv�� ( ) ( ) 2

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1102 ...1 TTT yyy ττ

τ µµµµ ++++= −+E .

Bgh]^Z�^hklZlhqgh�qZklh�yT+τ|yT∼N{0, E(y2T+τ|yT)}.

Ijbf_j��Y\e_gb_�deZkl_jbaZpbb�\heZlbevghklb�kjZam�aZf_lgh�gZ�]jZnb�

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ablguo�nhg^h\uo�bg^_dkh\�6WDQGDUG����^ey�����-����]]��b�6WDQGDUG�DQG�3RRUV����� ^ey� ����-����]]��>hoh^ghklb� \ujZ`_gu� \� ijhp_glZo� b� jZkkqblZgu� dZd�l_fiu�ijbjhklZ�\�g_ij_ju\ghf�\j_f_gb��:gZeba�]jZnbdZ�b�ex[u_�jZamfgu_�

klZlbklbq_kdb_�djbl_jbb�ihdZau\Zxl��qlh�^hoh^ghklb�g_�y\eyxlky�[_euf�rm�

fhf�ih�\j_f_gb��<b^gh��gZijbf_j��qlh�\heZlbevghklv�[ueZ�\ur_�\�����-_�]]���q_f�\�����-_�]]���qlh�ih^l\_j`^Z_lky�j_amevlZlZfb�hp_gb\Zgby��x

Jbk�����

;Zah\Zy�fh^_ev������fh`_l�[ulv�jZkijhkljZg_gZ�\�jZaebqguo�gZijZ\e_�

gbyo��GZb[he__� \Z`guf� y\ey_lky� h[h[s_ggZy�ARCH��gZau\Z_fZy�GARCH�� \�dhlhjmx�\dexq_gu�keZ]Z_fu_��khhl\_lkl\mxsb_�kdhevaysbf�kj_^gbf�

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µµµσθσθσθψσ

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τ, τ<t���<�ijhkl_cr_f�\ZjbZgl_�GARCH (1, 1): u(t)=ε(t)[µ0+µ1u(t-1)+ 2

1−tθσ ]. (22a)

:\lhjhf� fh^_eb� kqblZ_lky� L�;hee_jke_\�� ohly� hgZ� g_aZ\bkbfh� [ueZ�

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knhjfmebjh\ZgZ�K�Lwcehjhf��Fh^_ev����a��hq_gv�ohjhrh�aZj_dhf_g^h\ZeZ�k_�

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ttt uXy += β , t = 1, ..., T.

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ij_^klZ\eyxsZy�kh[hc�GARCH-ijhp_kk��<�fZljbqghc�aZibkb�y = X ββ + ε. x >�G_evkhg�ij_^eh`be�wdkihg_gpbZevgmx�GARCH�fh^_ev��?GARCH):

u(t)=ε(t)exp[µ0+ 21−tθσ +µ1

1

1

t

tu

σ+µ2

1

1

t

tu

σ].

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ghklb�jugdZ��q_f�gZ�\ae_lu�� G_dhlhju_�^jm]b_�\b^u�fh^_e_c ARCH: :[khexlgu_�hklZldb�� 2

tσ =µ0+µ1|u(t-1)|. G_ebg_cgu_��NARCH). 2

tσ =µ0+µ1|u(t-1)|γ. Ihjh]h\u_��TARCH). 2

tσ =µ0+µ12

1−tu +µ22

1−tu dt-1+ 21−tθσ ��]^_�dt ���_keb�u(t)<0

b���\�ijhlb\ghf�kemqZ_� <� hlebqb_� hl� jZkkfhlj_gguo� \ur_�fh^_e_c�ARCH�� dhlhju_� y\eyxlky�

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tσ �aZ\bkbl�g_�hl�ijhreuo�gZ[ex^_gbc��Z�hl�g_dhlhjuo�g_gZ[ex^Z_fuo�

dhfihg_gl�� GZb[he__�rbjhdh�ba\_klgZ�fh^_ev�klhoZklbq_kdhc�\heZlbevghklb��SV –

VWRFKDVWLF�YRODWLOLW\��K�LwcehjZ� u(t)=ε(t)exp[ht/2], ht+1=µ0+µ1ht+ηt,

]^_�Kt –�eZl_glgZy�dhfihg_glZ��dhlhjmx�fh`gh�bgl_jij_lbjh\Zlv�dZd�kemqZc�guc�b�g_mklhcqb\uc�ihlhd�gh\hc�bgnhjfZpbb��dhlhjuc�hq_gv�ljm^gh�fh^_eb�

jh\Zlv�g_ihkj_^kl\_ggh��gh�fh`gh�hp_gblv�ih�^Zgguf�gZ[ex^_gbc��ηt –�]Zmk�kh\kdbc�[_euc�rmf��Fh^_ev�LwcehjZ�gZau\Z_lky�eh]ghjfZevghc�SV�fh^_evx��Hp_gdb�fh^_eb�ihemqZxl�h[h[s_gguf�f_lh^hf�gZbf_gvrbo�d\Z^jZlh\��Ohly�

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klZpbhgZjguc�\j_f_gghc�jy^�–�kemqZcgh_�[em`^Zgb_�kh�k^\b]hf��\�wlhf�kem�qZ_�kqblZxl��qlh�\j_f_gghc�jy^�y(t��bf__l�_^bgbqguc�dhj_gv (unit root).

<uql_f�y(t-���ba�h[_bo�qZkl_c�fh^_eb��∆y(t)=γy(t-1)+ε(t���]^_�γ=µ-���>bdb�b�Nmee_j�jZkkfhlj_eb�ljb�j_]j_kkbb�

∆y(t)=γy(t-1)+ε(t), ∆y(t)=µ0+γy(t-1)+ε(t), ∆y(t)=µ0+γy(t-1)+µ2t+ε(t).

<lhjZy�j_]j_kkby�kh^_j`bl�ihklhygguc�we_f_gl�µ0��Z�lj_lvy��djhf_�wlh�

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gbc��IhemqZxl�hp_gdm�γ��klZg^Zjlgmx�hrb[dm�b�khhl\_lkl\mxs__�agZq_gb_�t – klZlbklbdb��KjZ\gb\Zy�agZq_gb_� t-klZlbklbdb�k�lZ[ebqguf��hij_^_eyxl��ijb�gylv�beb�hldehgblv�H0��Djblbq_kdh_�agZq_gb_�t-klZlbklbdb�bf__l�g_klZg^Zjl�gh_�jZkij_^_e_gb_�b�aZ\bkbl�hl�nhjfu�j_]j_kkbb�b�h[t_fZ�\u[hjdb�>��@��

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∆y(t)=γy(t-1)+∑=

+−∆p

iiti y

21µ +ε(t),

∆y(t)=µ0+γy(t-1)+∑=

+−∆p

iiti y

21µ +ε(t),

1�Fh^_eb�l_hjbb�\j_f_gguo�jy^h\�\�nbgZgkZo�b�wdhghf_ljbd_��H[haj_gb_�ijbdeZ^�

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Page 49: KL:LBKLBQ?KDB? F?LH>U IJH=GHABJH

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∆y(t)=µ0+γy(t-1)+µ2t+ ∑=

+−∆p

iiti y

21µ +ε(t).

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tion, McGraw-Hill, 1997. – 531 p. 27. Greene, W.H. Econometric analysis, Prentice Hall, 4th Edition, 2000. – 1004 p. 28. Hamilton, J.D. Time-Series Analysis, Princeton University Press, 1994. – 820 p. 29. Verbeek, M. A Guide to Modern Econometrics, Wiley, 2000. – 400 p.

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