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Прогнозирование по линейной регрессионной модели с коррелированными возмущениями оцененной с помощью процедуры Кохрейна - Оркатта.

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€�‚ƒ„…†‡�‰�‹� �‚��‚� �„‚�‰�‰‡‰�‹�
‘’“”•”’– —•�•‘™-�’–›�’–•�›�‘•� “’�‘,
”™ ›“–. ΅•“’“�™Ά£¤ �“•Ά›¥�•–›–
¦¥• §¥’Ά•–›¨©�–Ά›
�™��•¤�‘™¤ ΅›”›¥’ ••
tannuola@Gmail.com
Sergey V. Aleksandrovich
the physicist's candidate mathematical
sciences, the associate professor.
Financial University under the Government
of the Russian Federation
tannuola@Gmail.com
«¬­®―­°±¬­²³―±΄ µ­
ΘΙΚΛΜΝΞΟΠΡ� ΙΡ ΟΣΛ ΤΠΡΛΝΚ
¶±―΄·―­· ¬΄®¬΄ΈΈ±­――­·
ΚΛ�ΚΛΞΞΠΙΡ ΥΙΦΛΤ ΧΠΟΣ
Ή­Ί΄¶± Έ »­¬¬΄¶±¬­²³――Ό½
ΟΣΛ ΜΙΚΚΛΤΝΟΛΦ ΦΠΞΟΨΚΩΝΡΜΛΞ
Ή± ²­°ΉΎΏ΄―±ΐΉ±Α
ΛΞΟΠΥΝΟΛΦ ΨΞΠΡ�
­Β΄―΄――­· Έ µ­Ή­ΏΓΔ
ΟΣΛ ΪΚΙΜΛΦΨΚΛ ΙΫ
µ¬­Β΄ΊΎ¬Ό »­Ε¬΄·―³ Ζ
ΜΙΜΣΚΝΡΛ Ζ ΙΚΜΨΟΟ
­¬»³ΗΗ³
άέέήίΰαβγδ εζηθιηκλ μθνμθξξζοηηκλ πορθςσ ξ
τομμθςζμουκηηφπζ υοχπψωθηζλπζϊ οϋθηζυκθό
πκλ ξ ύοποωσώ ύμοϋθρψμφ �ομθιηκ & Sμτκό
κϊ μκξξπομθηκ τκτ ρζηκπζ θξτκλ πορθςσϊ υ
τοομοι θτψωθθ χηκ θηζθ ολξηλθποι ύθμθό
πθηηοι χκυζξζ ο θτψωθνο χηκ θηζλ ολξό
ηλώωθι ύθμθπθηηοιϊ ςκνουφ χηκ θηζι ολξό
ηλθποι ζ ολξηλώωθι ύθμθπθηηφϊ κ κτθ ο
οϋθηοτ τοζϋζθηου τομμθςλϋζζ υοχπψωθό
ηζιϊ ύοςψ κθπφ ηκ υξθ κύκ ύμοϋθρψμφ
0ξορλ ζχ οι πορθςζϊ ύοςψ θηκ ζ πθοροπ
πκθπκζ θξτοι ζηρψτϋζζ ροτκχκηκ οωκλ
ομπψςκ ρςλ υφ ζξςθηζλ ο θ ηονο θχψξςουό
ηονο ύμονηοχκ χηκ θηζι ολξηλθποι ύθμθό
πθηηοι ύο ύμονηοχηοπψ χηκ θηζώ ολξηλώό
ωθι ύθμθπθηηοι ζ θτψωθπψ ζ ςκνουφπ ηκςώό
ρκθπφπ χηκ θηζλπ ολξηλθποι ζ ολξηλώό
ωθι ύθμθπθηηφ ύμζ ςώοπ τοςζ θξυθ ζθμκό
ϋζι υ ύμοϋθρψμθ �ομθιηκ & Sμτκκ
δ U ! " #$%
9 $" "$'(9ϊ $!$" ' $%
@ 9"' ) 9% & B9'$$ϊ 9 "" "=!9 ! " #%9% $% 9'$ *' + $%
"@"$ *( "@" $% 9'$ *'
+ $% ,@$ = *(ϊ " *' + $%
"@"$ " ,@$ = *(ϊ # $!$ + $% 9 ++9$ + 9 $ +
"$'(9 @ "'9"
$
+ $%
@ 9"' -"
$% ! "ϊ + !' + 99'$ $% @ $ '9 "$ @"9$ + $% "@"$ *(ϊ "@"
$% + 9$ *' + $% ,@$ = *(
" $% 9'$ " " (*" *' +
$% "@"$ " ,@$ = *( #
($" " @ *" = '!( + $$ $% @ 9 + ) 9% & B9'$$
ήΰ ςζηθιηκλ μθνμθξξζοηηκλ
./12345 ! "ϊ 9 $"
"$'(9ϊ @ 9"' + ) 9% & B9'$$ϊ
!$% " + !$%!$9 "'9$ ϊ '9 "$ + 9$
πορθςσϊ τομμθςζμουκηηφθ υοχπψωθηζλϊ ύμοϋθό
ρψμκ �ομθιηκ & Sμτκκϊ πθορ πκθπκζ θό
ξτοι ζηρψτϋζζϊ θχψξςουηοθ ύμονηοχζμουκηζθ
678:;<6> ?7A6:CD> ? E6F>GG> 78:DFH IHD:CD7C 6:J6:KKH7DD7C L7;:IH K >MG7F766:IN8H:C KI<O>CDPA
c M7QL<R:DHC T6:;KG>MIN:G K7V7C HG:6>8H7DDPC T678:KK, D> F>W;7L XG>T: F7G767J7 78:DHM>YGKN T>6>L:G6P IHD:CD7C 6:J6:KKH7DD7C L7;:IH, T:6:L:DDPLH F7G767C NMINYGKN T6:7V6>Q7M>DDP: T:6:L:DDP: HKA7;D7C L7;:IH, H F7XZZH8H:DG >MG7F766:IN8HH T:6M7J7 T76N;F> KI<O>CDPA M7QL<R:DHC [1; 2, K. 136].
[>O>I\DPL ]>J7L XG7C T678:;<6P NMIN:GKN T6HL:D:DH: 7VPOD7J7 L:G7;> D>HL:D\]HA FM>;6>G7M
(^[?) ;IN 78:DFH T>6>L:G67M HKA7;D7C IHD:CD7C 6:J6:KKH7DD7C L7;:IH, T:6:L:DDPLH F7G767C NMINYGKN
D>VIY;>:LP: QD>O:DHN T:6:L:DDPA, H T7I<O:DH: K77GM:GKGM<YRHA 7KG>GF7M (78:D7F KI<O>CDPA M7QL<R:DHC):
e1; e2 ;...e n .
T76N;F>
?
_ F>O:KGM: 78:DFH F7XZZH8H:DG> >MG7F766:IN8HH KI<O>CDPA M7QL<R:DHC T:6M7J7
HKT7I\Q<:GKN :J7 ^[?-78:DF>, T7I<O:DD>N T7 6:J6:KKHH
et = ? et ?1 + ? t .
`>G:L MPT7IDN:GKN >MG76:J6:KKH7DD7: T6:7V6>Q7M>DH: T:6M7J7 T76N;F> T:6:L:DDPA L7;:IH, 78:DHM>:GKN IHD:CD>N 6:J6:KKH7DD>N L7;:I\ T7 T6:7V6>Q7M>DDPL T:6:L:DDPL, D>A7;NGKN 7KG>GFH, H T7 DHL
MD7M\ 78:DHM>:GKN F7XZZH8H:DG >MG7F766:IN8HH KI<O>CDPA M7QL<R:DHC. a7I<O:DD>N 78:DF> F7XZZH8H:DG> >MG7F766:IN8HH K6>MDHM>:GKN K T6:;P;<R:C.
bG:6>8H7DDPC T678:KK Q>F>DOHM>:GKN T6H <KI7MHH K7MT>;:DHN 78:D7F F7XZZH8H:DG> >MG7F766:IN8HH D> T7KI:;D:C H T6:;T7KI:;D:C HG:6>8HNA K Q>;>DD7C G7OD7KG\Y.
379
defgehigj klm nopelqrlqffmesset klqeulnveonsmm kqlqtqsswx klqeulnveonsswq kqlqtqsswq vnomfyp ep vsnzqsm{ klqeulnvjqtwx kqlqtqsswx o pqgj|m{ m klq}w}j|mq tetqspw olqtqsm, pe kefhq sqfgehigmx mpqln~m{ oevsmgnqp fhesny ovnmtefoyvi tq}j mfxe}swtm m klqeulnveonsswtm kqlqtqsswtm.
€ foyvm f �pmt oevsmgn‚p vsnzmpqhiswq fhesefpm klm klersevmleonsmm vsnzqsm{ vnomfmte{ kqlqtqsse{.
ƒnssny lnuepn kefoy|qsn owoe}j tqpe}et tnpqtnpmzqfge{ ms}jg~mm eu|q{ „eltjhw }hy owzmfhqsmy pezqzsere uqvjfheosere klersevn vnomfmte{ kqlqtqsse{ ke klersevsetj vsnzqsm‚ sqvnomfmte{
kqlqtqsse{ m pqgj|mt m hnreowt snuh‚}nqtwt vsnzqsmyt sqvnomfmte{ m vnomfmte{ kqlqtqsswx, n pngq ep e~qseg knlntqpleo hmsq{se{ lqrlqffmesse{ te}qhm m ge�„„m~mqspn nopegellqhy~mm kqloere kely}gn, klm h‚uet gehmzqfpoq mpqln~m{ o kle~q}jlq …exlq{sn ? †lgnppn.
ƒhy klefpepw lnffteplmt hmsq{sj‚ knlsj‚ te}qhi lqrlqffmm, o gepele{ fhjzn{swq oevtj|qsmy
?t
ke}oqlqsw nopelqrlqffmm kqloere kely}gn:
r}q
yt = ? + ? xt + ? t , ? t = ?? t ?1 +? t ,
xt ? vsnzqsmy sqvnomfmte{ kqlqtqsse{ X ; yt ? vsnzqsmy vnomfmte{ kqlqtqsse{ Y ; ? t ? fhjzn{swq oevtj|qsmy nopelqrlqffmessere jlnosqsmy, t = 1; 2;...; n ? , ? , ? ? knlntqplw te}qhm,
? ? ge�„„m~mqsp nopegellqhy~mm (?1 < ? < 1) .
€ feepoqpfpomm f kle~q}jle{ …exlq{sn ? †lgnppn:
de owuelezswt }nsswt klemvoe}mpfy e~qsgn knlntqpleo
fgeq jlnosqsmq lqrlqffmm:
?
m
?
te}qhm m snxe}mpfy �tkmlmzq-
yt = a + bxt + et ,
r}q
a = ?? ; b = ?? ; ?
e~qsgm knlntqpleo te}qhm;
et = ??t ?
e~qsgm fhjzn{swx oevtj|qsm{ (efpnpgm).
de efpnpgnt lqrlqffmm e~qsmonqpfy te}qhi
et = ? et ?1 + ? t
m snxe}mpfy e~qsgn ge�„„m~mqspn nopelqrlqffmm r1 = ?? .
‡ kete|i‚ e~qsgm
r1 = ?? owkehsyqpfy nopelqrlqffmesseq klqeulnveonsmy kqloere kely}gn
yt?(1) = yt ? r1 yt ?1 ; xt? (1) = xt ? r1 xt ?1 .
dqlowq snuh‚}qsmy oeffpnsnohmon‚pfy ke „eltjhnt
y1?(1) = y1 1 ? r12 ; x1?(1) = x1 1 ? r12 ,
r}q
1 ? r12
? keklnogn dln{fn ? �msfpqsn [2, f. 136; 3].
†~qsmonqpfy te}qhi
m snxe}ypfy e~qsgm
yt?(1) = ?1 + ?1 xt?(1) + ? t(1)
a1 = ??1 ; b1 = ??1 ; et(1) = ??t(1) .
de efpnpgnt lqrlqffmm e~qsmonqpfy te}qhi
et(1) = ?1et(1)?1 + ? t(1)
m snxe}mpfy e~qsgn ge�„„m~mqspn nopelqrlqffmm r2 = ??1 .
dehjzqsseq vsnzqsmq e~qsgm ge�„„m~mqspn nopegellqhy~mm flnosmonqpfy f e~qsge{, kehjzqsse{
sn klq}w}j|qt �pnkq, m kle~q}jln keopelyqpfy, snzmsny f kjsgpn 3).
dle~q}jln vngnszmonqpfy, ger}n lnvsefpi tq}j klq}w}j|q{ m kefhq}j‚|q{ e~qsgntm ? fpnsqp
tqsi‰q h‚uere snkqlq} vn}nssere zmfhn.
�ng gng lnfftnplmonqpfy uqvjfheoseq klersevmleonsmq, pe klq}kehnrnqpfy, zpe klersevseq vsnzqsmq
xt +1 = x‹
sqvnomfmte{ kqlqtqsse{
X
pezse mvoqfpse m vn klq}qhntm owuelgm fhjzn{swq oevtj|q-
smy m mx e~qsgm (efpnpgm) lnosw sjh‚ [4. f. 184]. �qeuxe}mte sn{pm klersevseq vsnzqsmq
y?t +1 = y?
vnomfm-
te{ kqlqtqsse{ Y .
€ kle~qffq owkehsqsmy kle~q}jlw …exlq{sn ? †lgnppn, eulnvj‚pfy kefhq}eonpqhisefpm sn{}qsswx e~qseg knlntqpleo te}qhm:
{a1 ; b1 ; r1} , {a2 ; b2 ; r2 } ,? {aN ; bN ; rN } ,
r}q
{ai ; bi ; ri } ? feepoqpfpoqsse vsnzqsmy e~qseg knlntqpleo hmsq{se{ lqrlqffmm m ge�„„m~mqspn nopegellqhy~mm fhjzn{swx oevtj|qsm{ sn i ? � mpqln~mm, N ? setql vngh‚zmpqhise{ mpqln~mm.
†uevsnzmt zqlqv
i
y? t(+i )1 = y? (i ) ? klersevseq vsnzqsmq kqlqtqsse{ Y , kehjzqsseq kefhq owkehsqsmy
�pnkeo kle~q}jlw.
380
���‘’�“”��•–’”— ��™�— �—�•�› ”�—�–�””
� ��� ™�΅Ά–—:
yt?(1) = a1 + b1 xt? (1) + et(1) ; yt?(1) = yt ? r1 yt ?1 ; xt? (1) = xt ? r1 xt ?1 .
£�—¤�•–�—�¥’�, ™ ΅Ά—�� ��‘�, Ά�� �¦”¤–— �— “’–Ά—’”—
? (1)
?
yt?+(1)
1 = yt +1 ? r1 yt ; xt +1 = xt +1 ? r1 xt
��¤™�–•�©© “’–Ά—’”©
¬–­ ­–­
xt +1 = x®
yt?+(1)1
”
et(1)
+1
�–•’� ’΅�§, ���΅Ά–— :
y? t?+(1)1 = a1 + b1 xt?+(1)
1 .
¨
xt?+(1)1 ”“ �—�•�« ¤•΅« •��–¦—’”› • ��—�¥—, ���΅Ά–— :
y?t +1 ? r1 yt = a1 + b1 ( xt +1 ? r1 xt ) .
y? t +1 = y? (1) , ―—“΅™��•’�› ��Ά—Ά’�› ���‘’�“ ” ——� •”¤:
”
y? (1) = a1 + b1 ( x° ? r1 xt ) + r1 yt
”�” ¦—
y? (1) = a1 + b1 x± + r1 ( yt ? b1 xt ) .
���‘’�“”��•–’”— ��™�— •����› ”�—�–�””
� ��� ™�΅Ά–—:
? (2)
yt?(2) = a2 + b2 xt? (2 ) + et(2) ; yt?(2) = yt?(1) ? r2 yt??(1)
= xt?(1) ? r2 xt??(1)
1 ; xt
1 .
£�—¤�•–�—�¥’�:
? (1)
? (1)
? (1)
? (1)
yt?+(2)
; xt?+(2)
1 = yt +1 ? r2 yt
1 = xt +1 ? r2 xt
”
y? t?+(12) = a 2 + b2 xt?+(12) .
���–¦–© ��—�―�–“�•–’’�— “’–Ά—’”© �—�— —’’�« Ά—�—“ ”™«�¤’�— ’–―�§¤–— �— “’–Ά—’”©, ���΅Ά–— :
yt? ( 2) = yt? (1) ? r2 yt??(1)1 = yt ? r1 yt ?1 ? r2 ( yt ?1 ? r1 yt ? 2 ) = yt ? ( r1 + r2 ) yt ?1 + r1r2 yt ? 2 ,
xt? ( 2) = xt? (1) ? r2 xt??(1)1 = xt ? r1 xt ?1 ? r2 ( xt ?1 ? r1 xt ? 2 ) = xt ? ( r1 + r2 ) xt ?1 + r1r2 xt ? 2
.
²�© ’–―�§¤—’”© ™ ’� —�� t + 1 ���΅Ά—’’�— •��–¦—’”© ��”’” –§� •”¤:
yt?+(12) = yt?+(1)1 ? r2 yt? (1) = y?t(+2)1 ? r1 yt ? r2 ( yt ? r1 yt ?1 ) = y? t(2)
+1 ? ( r1 + r2 ) yt + r1r2 yt ?1 ,
xt?+(12) = xt?+(1)1 ? r2 xt? (1) = xt +1 ? r1 xt ? r2 ( xt ? r1 xt ?1 ) = xt +1 ? ( r1 + r2 ) xt + r1r2 xt ?1 .
��¤™�–•�©© “’–Ά—’”©
yt?+(2)
1
”
xt?+(2)
1
• ΅�–•’—’”—
y? t? ( 2) = a 2 + b2 xt? ( 2) , ���΅Ά–— :
y? t(+2)1 ? ( r1 + r2 ) yt + r1 r2 yt ?1 = a2 + b2 ( xt +1 ? (r1 + r2 ) xt + r1r2 xt ?1 ) .
£ ΅Ά—�� ••—¤—’’�« �―�“’–Ά—’”›
y?
”�” ¦—
( 2)
y? t(+2)1 = y? ( 2)
”
xt +1 = x³ , ���΅Ά–— ―—“΅™��•’�› ��Ά—Ά’�› ���‘’�“:
= a2 + b2 x΄ ? b2 ( r1 + r2 ) xt + b2 r1r2 xt ?1 + ( r1 + r2 ) yt ? r1 r2 yt ?1 ,
y? ( 2) = a2 + b2 xµ + ( r1 + r2 )( yt ? b2 xt ) ? r1r2 ( yt ?1 ? b2 xt ?1 )
���‘’�“”��•–’”— ��™�— ��—�¥—› ”�—�–�””
� ��� ™�΅Ά–—:
yt? (3) = a2 + b2 xt? (3) + et(3) ; yt? (3) = yt? ( 2) ? r3 yt??(12) ; xt? (3) = xt? (2) ? r3 xt??(12)
”
.
? (3)
y? t?+(3)
1 = a 2 + b2 xt +1 .
���–¦–© ��—�―�–“�•–’’�— “’–Ά—’”© �—�— —’’�« Ά—�—“ ”™«�¤’�— ’–―�§¤–— �— “’–Ά—’”©, ���΅Ά–— :
? ( 2)
? ( 2)
yt?+(3)
= y? t +1 ? ( r1 + r2 + r3 ) yt + ( r1r2 + r1 r2 + r1r2 ) yt ?1 ? r1r2 r3 yt ? 2 ,
1 = yt +1 ? r3 y t
? ( 2)
? (2)
xt?+(3)
= xt +1 ? (r1 + r2 + r3 ) xt + ( r1 r2 + r1r2 + r1r2 ) xt ?1 ? r1r2 r3 xt ? 2 .
1 = xt +1 ? r3 xt
�����’©© –’–��‘”Ά’�— ��—�―�–“�•–’”©, ���΅Ά–— ―—“΅™��•’�› ��Ά—Ά’�› ���‘’�“:
y? (3) = a3 + b3 x¶ ? b3 (r1 + r2 + r3 ) xt + b3 (r1r2 + r1r3 + r2 r3 ) ? b3 r1r2 r3 xt ?2 +
+ (r1 + r2 + r3 ) yt ? (r1r2 + r1r3 + r2 r3 ) yt ?1 + r1r2 r3 yt ? 2
”�” ¦—:
y? (3) = a3 + b3 x· + ( r1 + r2 + r3 )( yt ? b3 xt ) ? ( r1r2 + r1r3 + r2 r3 )( yt ?1 ? b3 xt ?1 ) + r1r2 r3 ( yt ? 2 ? b3 xt ? 2 ) .
���‘’�“”��•–’”— ��™�— N-› ”�—�–�””
Έ–―�§¤–© «–�–­�—� ���΅Ά—’’�« “–•”™” �™�—› ���‘’�“’�« “’–Ά—’”›
Ί ” ’–―�§¤–— �« “’–Ά—’”› �—�— —’’�«
“’–Ά—’”© Ή–­���– x
Ή�� ΅�– ¤�© ���‘’�“–
y?
(N)
, •�Ά”™�—’’�‘� ��™�—
N
y? (1) ; y? ( 2) ; y? (3)
X ” Y , �¦’� ��—¤����¦”�¥, Ά�� �―»–©
��–��• ����—¤΅��, ” ——� •”¤:
381
�� ���‘’�“’�‘�
N
N
i =1
i1 :i2 =1
( i1 <i2 )
y? ( N ) = aN + bN x½ + (Ό ri )( yt ? bN xt ) ? ( Ό ri1 ri2 )(yt ?1 ? bN xt ?1 ) + ... +
N
+(?1)k (
Ό
i1 :i2 ;...;ik =1
( i1 < i2 <...< ik )
ri1 ri2 ...rik )(yt ? k +1 ? bN xt ?k +1 ) + ... + (?1) N r1r2 ...rN ( yt ? N +1 ? bN xt ? N +1 )
(?)
ΎΏΐ ΑΒΓΔ:
N
N
y? ( N ) = y?t?+(1N ) = y?t +1 ? (Ε ri ) yt + ( Ε ri1 ri2 ) yt ?1 + ... + (?1) k (
i =1
i1 :i2 =1
( i1 < i2 )
N
N
xt?+(1N ) = xt +1 ? (Ζ ri ) xt + ( Ζ ri1 ri2 )xt ?1 + ... + ( ?1) k (
i =1
i1 :i2 =1
( i1 <i2 )
N
N
Ε
i1 :i2 ;...;ik =1
( i1 < i2 <...< ik )
N
Ζ
i1 :i2 ;...;ik =1
( i1 < i2 <...< ik )
N
yt?( N ) = yt ? (Η ri ) yt ?1 + ( Η ri1 ri2 ) yt ? 2 + ... + (?1)k (
ri1 ri2 ...rik )xt ? k +1 + ... + ( ?1) N r1r2 ...rN xt ? N +1
N
Η
i =1
i1 :i2 =1
( i1 < i2 )
i1 :i2 ;...;ik =1
( i1 < i2 < ...< ik )
N
N
N
xt?( N ) = xt ? (Θ ri ) xt ?1 + ( Θ ri1 ri2 )xt ? 2 + ... + ( ?1) k (
i =1
i1 :i2 =1
( i1 < i2 )
(?)
ΙΓΚΛΜΝΔ ΞΓΏΔΟΠΟ
ri1 ri2 ...rik ) yt ? k +1 + ... + (?1) N r1r2 ...rN yt ? N +1
Θ
i1 :i2 ;...;ik =1
( i1 < i2 <...< ik )
ri1 ri2 ...rik ) yt ? k + ... + (?1) N r1r2 ...rN yt ? N
ri1 ri2 ...rik )xt ? k + ... + (?1) N r1r2 ...rN xt ? N
ΔΝΒΓΡΓΔ ΔΛΒΝΔΛΒΐ�ΝΣΚΓΤ ΐΥΡΟΚΦΐΐ. ΧΛΚ ΨΩΠΓ ΪΓΚΛΫΛΥΓ άΩέΝ, ΡΠή
N = 1; 2;3 ΑΒΛ ΞΓΏΔΟΠΛ άΝΏΥΛ. ΎΏΝΡΪΓΠΓΜΐΔ, �ΒΓ ΓΥΛ άΝΏΥΛ ΡΠή N ΐ, ΐΣίΓΡή ΐΫ ΑΒΓΰΓ, ΡΓΚΛΜΝΔ
N +1.
αΛ ( N + 1) -Δ ΑΒΛΪΝ ΪΏΓΦΝΡΟΏΩ ΣΪΏΛάΝΡΠΐάΩ άΩΏΛΜΝΥΐή:
y? t?+(1N +1) = a N +1 + bN +1 xt?+(1N +1) ; xt?+(1N +1) = xt?+(1N ) ? rN +1 xt?( N ) ; yt?+(1N +1) = yt?+(1N ) ? rN +1 yt?( N ) ;
ΝΝ ΡΠή
yt?+(1N ) ? rN +1 yt?( N ) = a N +1 + bN +1 ( xt?+(1N ) ? rN +1 xt?+(1N ) );
ΎΓΡΣΒΛάΠήή ά ΪΓΣΠΝΡΥββ ΞΓΏΔΟΠΟ άΩΏΛΜΝΥΐή ΡΠή
ΞΓΏΔΟΠΟ
y?t?+(1N ) , xt?+(1N ) , y?t?( N ) , xt?( N ) ΚΓΒΓΏΩΝ,
ΣΓΣΒΛάΠήή
(?) , ήάΠήβΒΣή άΝΏΥΩΔΐ ΪΓ ΪΏΝΡΪΓΠΓΜΝΥΐβ ΐΥΡΟΚΦΐΐ, ΪΓΠΟ�ΛΝΔ:
N
N
y?t +1 ? (γ ri ) yt + ( γ ri1 ri2 ) yt ?1 + ... + (?1)k (
i =1
i1 :i2 =1
( i1 <i2 )
N
N
γ
i1 :i2 ;...;ik =1
( i1 < i2 <...< ik )
ri1 ri2 ...rik ) yt ? k +1 + ... + (?1) N r1r2 ...rN yt ? N +1 ?
N
? rN +1 ( yt ? (γ ri ) yt ?1 + ( γ ri1 ri2 ) yt ?2 + ... + (?1) k (
i =1
i1 :i2 =1
( i1 < i2 )
N
N
γ
i1 :i2 ;...;ik =1
( i1 < i2 <...< ik )
N
= aN +1 + bN +1 ( xt +1 ? (γ ri ) xt + ( γ ri1 ri2 )xt ?1 + ... + (?1) k (
i =1
i1 :i2 =1
( i1 < i2 )
N
N
? rN +1 ( xt ? (γ ri ) xt ?1 + ( γ ri1 ri2 )xt ?2 + ... + (?1) k (
i =1
i1 :i2 =1
( i1 < i2 )
N
γ
i1 :i2 ;...;ik =1
( i1 < i2 <...< ik )
N
γ
ri1 ri2 ...rik ) yt ? k + ... + (?1) N r1r2 ...rN yt ? N ) =
i1 :i2 ;...;ik =1
( i1 < i2 <...< ik )
ri1 ri2 ...rik )xt ? k +1 + ... + (?1) N r1r2 ...rN xt ? N +1 ?
ri1 ri2 ...rik )xt ? k + ... + (?1) N r1r2 ...rN xt ? N ))
ΎΓΣΠΝ ΪΝΏΝΰΏΟΪΪΐΏΓάΚΐ ΣΠΛΰΛΝΔΩί, ΪΓΠΟ�ΛΝΔ:
N
N
N
i =1
i1 :i2 =1
( i1 < i2 )
i =1
y? t +1 ? (δ ri ) yt ? rN +1 yt + ( δ ri1 ri2 ) yt ?1 + rN +1 (δ ri ) yt ?1 + ... + ( ?1) N +1 r1r2 ...rN rN +1 yt ? N ?
N
N
N
i =1
i1 :i2 =1
( i1 < i2 )
i =1
= aN +1 + bN +1 ( xt +1 ? (δ ri ) xt ? rN +1 xt + ( δ ri1 ri2 )xt ?1 + rN +1 (δ ri ) xt ?1 + ... + (?1) N +1 r1r2 ...rN rN +1 xt ? N )
εΛΒΝΔ, ΓΨζΝΡΐΥήή ΣΟΔΔΩ, ΪΓΠΟ�ΛΝΔ:
382
N +1
N +1
i =1
i1 :i2 =1
( i1 < i2 )
y? t +1 ? (η ri ) yt + ( η ri1 ri2 ) yt ?1 + ... + (?1) N +1 r1r2 ...rN rN +1 yt ? N ?
N +1
N +1
i =1
i1 :i2 =1
( i1 < i2 )
= aN +1 + bN +1 ( xt +1 ? (η ri ) xt + ( η ri1 ri2 )xt ?1 + ... + ( ?1) N +1 r1r2 ...rN rN +1 xt ? N )
θικλμλν
y? t +1 = y? ( N +1)
ξ οπξριςλν, πρσ
xt +1 = xτ , συσφπλρχψωφσ ϊσψοπλχϋ:
N +1
N +1
i =1
i1 :i2 =1
( i1 < i2 )
y? ( N +1) = aN +1 + bN +1 xύ + (ό ri )( yt ? bN +1 xt ) ? ( ό ri1 ri2 )( yt ?1 ? bN +1 xt ?1 ) +
+... + (?1) N +1 r1r2 ...rN rN +1 ( yt ? N ? bN +1 xt ? N )
ώψχ�σςλρχψωφσ, ξ σ�ν ξ ρσσ, πρσ σκϋοψλ
�ψν
(?) ϊκλςχ�ψξςλ �ψν N , ϊσυλλφλ χχ ϊκλςχ�ψξςσρω
N + 1 , πρσ ξ �συλιςλχρ σκϋοψο (?) .
6λυξϋ σκλσϋ, σκϋοψλ �ψν ςιπξψχφξν ρσπχπφσσ χοψσςφσσ ϊκσφσλ
ϋχφφσ
Y
N
λςξξϋσ ϊχκχ-
ϊσ ϊκσφσφσϋο φλπχφξ x φχλςξξϋσ ϊχκχϋχφφσ
ϋιϋ φλπχφξνϋ φχλςξξϋσ ξ λςξξϋσ ϊχκχϋχφφι
ρςχ
y? ( N )
X ξ ρχυοχϋο ξ ψλσςιϋ φλψ�λχ xt ; xt ?1 ; xt ? 2 ;...; yt ; yt ?1 ; yt ?2 ;... ϊκξ ψσϋ υσψξπχ-
ξρχκλξ ς ϊκσχ�οκχ !σ κχφλ ? κυλρρλ ξϋχχρ ςξ�:
N
N
i =1
i1:i2 =1
( i1 <i2 )
y? ( N ) = aN + bN x
+ ( ri )( yt ? bN xt ) ? ( ri1 ri2 )( yt ?1 ? bN xt ?1 ) + ... +
+ (?1) k (
N
i1:i2 ;...;ik =1
( i1 <i2 <...<ik )
ri1 ri2 ...rik )( yt ?k +1 ? bN xt ?k +1 ) + ... + ( ?1) N r1r2 ...rN ( yt ? N +1 ? bN xt ? N +1 )
:
Literature:
1. D. Cochrane, G.H. Orcutt. Application of Least
Square Regression to Relationships Containing
Auto-Correlated Error Terms, Journal of the
American Statistical Association. 44 (245).
. 32?61. 1949.
1. D. Cochrane, G.H. Orcutt. Application of Least
Square Regression to Relationships Containing
Auto-Correlated Error Terms, Journal of the
American Statistical Association. 44 (245).
. 32?61. 1949.
2. .. . φσςι +υσφσϋχρκξπχυσσ
ϋσ�χψξκσςλφξν. 3. : !σϋ!φξλ, 2007. 432 .
2. L.O. Babeshko. Econometric modeling. M. :
KomKniga, 2007. 432 p.
3. S.J. Prais, C.B. Winsten. Trend Estimators and
Serial Correlation. Cowles Comission Discussion
Paper < 383. . 1?27. 1954.
3. S.J. Prais, C.B. Winsten. Trend Estimators and
Serial Correlation. Cowles Comission Discussion
Paper < 383. . 1?27. 1954.
4. . . "#$%&, '.(. ()*,, -.-. '.&/01.
2υσφσϋχρκξυλ. 4λπλψωφι υοκ. 3. : 5χψσ, 2007.
504 .
4. J.R. Magnus, P.K. Katishev, A.A. Peresetsky.
Econometrics. Initial course. M. : Delo, 2007. 504 p.
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