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Modelli autoregressivi e carte di controllo ewma: effetto della correzione della stima dei parametri

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(6)  . ƒ ‘Ž–†‹ ‹‡œ‡–ƒ–‹•–‹ Š‡ . ‘”•‘†‹ƒ—”‡ƒ’‡ ‹ƒŽ‹•–‹ ƒ‹ –ƒ–‹•–‹ ƒ‡‹ˆ‘”ƒ–‹ ƒ .   ‡•‹†‹ƒ—”‡ƒ . ‘†‡ŽŽ‹—–‘”‡‰”‡••‹˜‹‡ ƒ”–‡†‹ ‘–”‘ŽŽ‘ ǣ‡ˆˆ‡––‘†‡ŽŽƒ ‘””‡œ‹‘‡†‡ŽŽƒ•–‹ƒ †‡‹’ƒ”ƒ‡–”‹  .  ‡Žƒ–‘”‡ǣ”‘ˆǤ —‹†‘ƒ•ƒ”‘––‘ ƒ—”‡ƒ†‘ǣƒ‘Ž‘ƒ”  .  ‘ ƒ†‡‹ ‘ʹͲͲͺȀʹͲͲͻ.

(7) .              . .

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(12) ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤ͵ . 

(13) ͳǣ

(14) ǯ

(15) 

(16) 

(17) 

(18) ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͷ ͷǤͷ

(19) †‡–‹ˆ‹ ƒœ‹‘‡†‹—’”‘ ‡••‘•–‘ ƒ•–‹ ‘ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͻ ͷǤ͸ ‡‡•‹†‡‹’”‘ ‡••‹ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͷ͸ ͷǤ͹‘†‡ŽŽ‹ƒ‡†‹ƒ‘„‹Ž‡ȋȌǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͷͺ ͷǤͺ‘†‡ŽŽ‹—–‘”‡‰”‡••‹˜‹ȋȌǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤ͸Ͷ ͷǤͻ‘†‡ŽŽ‹ƒ‡†‹ƒ‘„‹Ž‡ƒ—–‘”‡‰”‡••‹˜‹ȋȌǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤ͸ͺ ͷǤͼ–‹ƒ‡’”‡˜‹•‹‘‹†‡‹‘†‡ŽŽ‹ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤ͸ͽ . 

(20) ʹǣ

(21) 

(22) 

(23) 

(24) 

(25) 

(26) 

(27) ǯǣ 

(28) 

(29) 

(30) 

(31) ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤ͵͵ ͸Ǥͷ‘–”‘ŽŽ‘–ƒ–‹•–‹ ‘†‡Ž”‘ ‡••‘ȋȌǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤ͹͹ ͸Ǥ͸‡ ƒ”–‡†‹ ‘–”‘ŽŽ‘ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤ͹ͻ ͸Ǥ͹ƒ ƒ”–ƒ†‹ ‘–”‘ŽŽ‘ǣ†‡ˆ‹‹œ‹‘‡‡’”‘‰‡––ƒœ‹‘‡ǤǤǤǤǤǤǤǤǤǤǤǤǤͺͶ ͸Ǥͺ’’”‘ ‹‘–”ƒ‹–‡‹”‡•‹†—‹‹’”‡•‡œƒ†‹†ƒ–‹ƒ—–‘ ‘””‡Žƒ–‹ǤǤǤǤǤǤͺͼ . 

(32) ͵ǣ

(33)  .  

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(38) ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͶͺ ͹Ǥͷ”‘’”‹‡–†‡‰Ž‹•–‹ƒ–‘”‹†‹—’”‘ ‡••‘ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͺ; ͹Ǥ͸‡ ‹ Š‡’‡”Žƒ ‘””‡œ‹‘‡ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͻͷ   .  ϭ.

(39) 

(40) Ͷǣ

(41) 

(42) 

(43) 

(44) ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͷ͹ ͺǤͷ ‡Ž–ƒ†‡‹’ƒ”ƒ‡–”‹ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͻͽ ͺǤ͸

(45) –‡”’”‡–ƒœ‹‘‡‡†‹• —••‹‘‡†‡‹”‹•—Ž–ƒ–‹ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͼͶ . 

(46) ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤ͹͵ . 

(47) ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤ͹Ͷ . 

(48) 

(49)  

(50) ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͺ͵    .                  Ϯ.

(51)

(52) –”‘†—œ‹‘‡.   

(53)  — ‘–‡•–‘ †‘˜‡ •‹ ”‹ ‡” ƒ — ‹‰Ž‹‘”ƒ‡–‘ ‘–‹—‘ †‡ŽŽƒ “—ƒŽ‹– ‡‹ ’”‘ ‡••‹ ‹†—•–”‹ƒŽ‹ǡ –”‘˜‡” —ƒ •—ƒ ‰‹—•–‹ˆ‹ ƒœ‹‘‡ Ž̵ƒ’’Ž‹ ƒœ‹‘‡ †‹ ‡–‘†‘Ž‘‰‹‡†‹”‡––‡ƒŽ‘–”‘ŽŽ‘–ƒ–‹•–‹ ‘†‡Ž”‘ ‡••‘ǡ‹“—ƒ–‘”ƒ’’”‡•‡–ƒ‘ — ’”‹‘ ˜ƒŽ‹†‘ •–”—‡–‘ ƒ •‘•–‡‰‘ †‡ŽŽ̵ƒ––‹˜‹– †‡ ‹•‹‘ƒŽ‹ ƒ† ‘‰‹ Ž‹˜‡ŽŽ‘ ‘”‰ƒ‹œœƒ–‹˜‘ ’‡” ‹Ž ”ƒ‰‰‹—‰‹‡–‘ †‡ŽŽƒ “—ƒŽ‹–Ǥ ŽŽƒ „ƒ•‡ †‡ŽŽ‡ –‡ ‹ Š‡ –”ƒ†‹œ‹‘ƒŽ‹ Žǯ‹’‘–‡•‹ ˆ‘†ƒ‡–ƒŽ‡ ° Žǯ‹†‹’‡†‡œƒ †‡ŽŽ‡ ‘••‡”˜ƒœ‹‘‹ǡ ‘ ‹ ‘””‡Žƒœ‹‘‡ǡƒ‹‘Ž–‡•‹–—ƒœ‹‘‹”‡ƒŽ‹ǡ“—‡•–ƒ ‘†‹œ‹‘‡‘°•‘††‹•ˆƒ––ƒǤ

(54) ‡–‘†‹ Žƒ••‹ ‹’‡”‹Ž‘‹–‘”ƒ‰‰‹‘†‡ŽŽƒ“—ƒŽ‹–†‹—’”‘ ‡••‘ǡ ‘‡Ž‡ ƒ”–‡†‹ ‘–”‘ŽŽ‘ ‹ ’”‡•‡œƒ †‹ †ƒ–‹ ‘””‡Žƒ–‹ǡ ˆ‘”‹• ‘‘ ”‹•—Ž–ƒ–‹ ‘ ’”‘’”‹ƒ‡–‡ ƒ ‡––ƒ„‹Ž‹Ǥ‹ ‘•‡‰—‡œƒ‡‰Ž‹—Ž–‹‹ƒ‹ǡ•‘‘•–ƒ–‹ ‘†‘––‹†‹˜‡”•‹•–—†‹ƒŽ ˆ‹‡ †‹ ‡Žƒ„‘”ƒ”‡ ’”‘ ‡†—”‡ ‹ ‰”ƒ†‘ †‹ ƒ†ƒ––ƒ”•‹ ƒ ’”‘ ‡••‹ ƒ—–‘ ‘””‡Žƒ–‹Ǥ ”ƒ “—‡•–‹ƒ’’”‘ ‹ǡ—‘ Š‡•‹°”‹˜‡Žƒ–‘‡••‡”‡—–‹Ž‡ǡ°•–ƒ–‘†‡• ”‹˜‡”‡†‹”‡––ƒ‡–‡ Žƒ •–”—––—”ƒ †‹ ‘””‡Žƒœ‹‘‡ ‘ ƒ†‡‰—ƒ–‹ ‘†‡ŽŽ‹ †‹ •‡”‹‡ –‡’‘”ƒŽ‹ǡ ‘ ‹ “—ƒŽ‹ ”‹—‘˜‡”‡Žƒ’”‡•‡œƒ†‹ƒ—–‘ ‘””‡Žƒœ‹‘‡‡•— ‡••‹˜ƒ‡–‡ƒ’’Ž‹ ƒ”‡Žƒ ƒ”–ƒ †‹ ‘–”‘ŽŽ‘ƒ‹”‡•‹†—‹Ǥǯƒ••—–‘•— —‹•‹„ƒ•ƒ“—‡•–ƒ–‡ ‹ ƒǡ° Š‡Ž‡•–‹‡†‡‹ ’ƒ”ƒ‡–”‹ǡ Š‡ †‡• ”‹˜‘‘ ‹ †ƒ–‹ǡ •‹ƒ‘ ‹Ž ’‹î ˜‹ ‹‡ ’‘••‹„‹Ž‹ ƒŽ ˜‡”‘ ˜ƒŽ‘”‡ †‡Ž ’ƒ”ƒ‡–”‘Ǥ˜˜‹ƒ‡–‡“—‡•–ƒ•‹–—ƒœ‹‘‡’‘”–‡”ƒ†ƒ˜‡”‡†‡‰Ž‹‡””‘”‹†‹•–‹ƒǡ ‡†‹ ‘•‡‰—‡œƒ“—‡•–‹”‹ ƒ†”ƒ‘‡ŽŽ‡ƒ’’Ž‹ ƒœ‹‘‹†‹“—ƒŽ‹–ǡ ‘‡Ž‡ ƒ”–‡†‹ ‘–”‘ŽŽ‘Ǥ —‹†‹ Žǯ‘„‹‡––‹˜‘ †‹ “—‡•–‘ Žƒ˜‘”‘ ° “—‡ŽŽ‘ †‹ ’”‘’‘””‡ •–‹ƒ–‘”‹ ƒŽ–‡”ƒ–‹˜‹ Š‡ ”‹•—Ž–‹‘‡‘ †‹•–‘”–‹†‡‰Ž‹—•—ƒŽ‹ ‡†‹ ‡•ƒ‹ƒ”‡Žǯ‡ˆˆ‡––‘ †‡ŽŽƒ.  ϯ.

(55) ‘””‡œ‹‘‡ ’‡” Žƒ ‘•–”—œ‹‘‡ †‹ ƒ”–‡ †‹ ‘–”‘ŽŽ‘Ǥ ǯ‹†‡ƒ †‹ ˆ‘†‘ “—‡ŽŽƒ †‹ ”‹ ‘””‡”‡ƒŽŽ‡ ƒ”–‡†‹ ‘–”‘ŽŽ‘ȋš’‘‡–‹ƒŽŽ›‡‹‰Š–‡†‘˜‹‰˜‡”ƒ‰‡Ȍ „ƒ•ƒ–‡•—‹”‡•‹†—‹’‡”†ƒ–‹ƒ—–‘ ‘””‡Žƒ–‹ǡ‹’”‡•‡œƒ†‹‹ ‡”–‡œœƒ•—Ž‘†‡ŽŽ‘Ǥ

(56)  ƒŽ–”‹ ƒ„‹–‹ ƒ’’Ž‹ ƒ–‹˜‹ ‡̵ •–ƒ–‘ †‹‘•–”ƒ–‘ Š‡ Žƒ˜‘”ƒ”‡ ‘ •–‹ƒ–‘”‹ ‡‘ †‹•–‘”–‹ †‡‰Ž‹ —•—ƒŽ‹ǡ ‹’‹‡‰ƒ–‹ ’‡” Žƒ •–‹ƒ †‡‹ ’ƒ”ƒ‡–”‹ †‹ ‘†‡ŽŽ‹ ƒ—–‘”‡‰”‡••‹˜‹ȋȌ‡†‹‘†‡ŽŽ‹ƒ—–‘”‡‰”‡••‹˜‹ƒ‡†‹ƒ‘„‹Ž‡ȋȌǡ’‘”–ƒƒ ”‹•—Ž–ƒ–‹•—’‡”‹‘”‹Ǥ‡”–ƒ–‘Ž‘• ‘’‘†‹“—‡•–‘Žƒ˜‘”‘ ‘•‹•–‡‡ŽŽ̵ƒƒŽ‹œœƒ”‡•‡ — ‡˜‡–—ƒŽ‡ ‘””‡œ‹‘‡ †‡‰Ž‹ •–‹ƒ–‘”‹ǡ ’‘••ƒ ’‘”–ƒ”‡ ƒ ‘ Ž—•‹‘‹ ‹‰Ž‹‘”‹ ƒ Š‡‡Ž ƒ•‘†‡ŽŽ‡ ƒ’’Ž‹ ƒœ‹‘‹ƒŽ ‘–”‘ŽŽ‘ †‹“—ƒŽ‹–Ǥ‡Žƒ’‹–‘Ž‘ͳ•ƒ”ƒ‘ –”ƒ––ƒ–‹‘†‡ŽŽ‹’‡”ŽǯƒƒŽ‹•‹†‡ŽŽ‡•‡”‹‡–‡’‘”ƒŽ‹ǡ ‘‹”‡Žƒ–‹˜‹‡–‘†‹†‹•–‹ƒ‡ ’”‡˜‹•‹‘‡Ǥ ‡Žƒ’‹–‘Ž‘ ʹ•ƒ”ƒ‘‹–”‘†‘––‹ ‘ ‡––‹ˆ‘†ƒ‡–ƒŽ‹•—ŽŽƒ–‡‘”‹ƒ †‡Ž‘–”‘ŽŽ‘–ƒ–‹•–‹ ‘†‹—ƒŽ‹–ȋȌ‡–”ƒ“—‡•–‹ǡ‹ƒ‹‡”ƒƒ’’”‘ˆ‘†‹–ƒǡŽ‡ ƒ”–‡ †‹ ‘–”‘ŽŽ‘  ‡ Ž‡ Ž‘”‘ ’”‘’”‹‡–Ǥ

(57)  •‡‰—‹–‘ ‡Ž ƒ’‹–‘Ž‘ ͵ •ƒ” ’”‡•‡–‡—ǯƒ”‰‘‡–ƒœ‹‘‡•—ŽŽ‡’”‘’”‹‡–†‡‰Ž‹•–‹ƒ–‘”‹ƒ—–‘”‡‰”‡••‹˜‹—•—ƒŽ‹ ‡ •ƒ”ƒ‘ ’”‘’‘•–‹ •–‹ƒ–‘”‹ ƒŽ–‡”ƒ–‹˜‹ ’‡” Žƒ ‘””‡œ‹‘‡ †‡ŽŽƒ †‹•–‘”•‹‘‡Ǥ

(58) ˆ‹‡ ‡Ž ƒ’‹–‘Ž‘ Ͷ ˜‡””ƒ‘ ƒƒŽ‹œœƒ–‹ ‡ †‹• —••‹ ‹ ”‹•—Ž–ƒ–‹ †‡ŽŽ‡ •‹—Žƒœ‹‘‹ •˜‘Ž–‡ǡ ‡ŽŽ‡ “—ƒŽ‹ ‰Ž‹ •–‹ƒ–‘”‹ –”ƒ†‹œ‹‘ƒŽ‹ †‡‹ ’ƒ”ƒ‡–”‹ ƒ—–‘”‡‰”‡••‹˜‹ •‘‘ •–ƒ–‹‡••‹ƒ ‘ˆ”‘–‘ ‘“—‡ŽŽ‹’”‘’‘•–‹ƒŽ ƒ’‹–‘Ž‘’”‡ ‡†‡–‡Ǥ          ϰ.

(59) ƒ’‹–‘Ž‘ͳ ‘†‡ŽŽ‹’‡”ŽǯƒƒŽ‹•‹†‡ŽŽ‡•‡”‹‡–‡’‘”ƒŽ‹    ͳǤͳ

(60) †‡–‹ˆ‹ ƒœ‹‘‡†‹—’”‘ ‡••‘•–‘ ƒ•–‹ ‘

(61) †ƒ–‹ƒ —‹˜‡‰‘‘ƒ’’Ž‹ ƒ–‡Ž‡–‡ ‹ Š‡‹ˆ‡”‡œ‹ƒŽ‹ Š‡ ‘’‘‰‘‘‹Ž„ƒ‰ƒ‰Ž‹‘ †‹ —‘ •–ƒ–‹•–‹ ‘ ’‘••‘‘ ‡••‡”‡ †‹ †—‡ –‹’‹ǣ ”‘••Ǧ•‡ –‹‘ǡ ‡Ž ƒ•‘ ‹ —‹ Ž‡ ‘••‡”˜ƒœ‹‘‹†‹ —‹†‹•’‘‹ƒ‘•‹ƒ‘”‡Žƒ–‹˜‡ƒ†‹†‹˜‹†—‹†‹˜‡”•‹ǡ‘’’—”‡•‡”‹‡ •–‘”‹ Š‡ǡ “—ƒ†‘ ‹Ö Š‡ ƒ„„‹ƒ‘ •‘‘ ‘••‡”˜ƒœ‹‘‹ǡ •— —ƒ ‘ ’‹î ‰”ƒ†‡œœ‡ǡ ’”‘–”ƒ––‡‡Ž–‡’‘ͳǤ‡Ž’”‹‘ ƒ•‘ǡ’‡”‡•‡’‹‘ǡ’‡•ƒ”‡ƒ†—‹•‹‡‡†‹ †ƒ–‹ ‘••‡”˜ƒ–‹ ‘‡ —ƒ †‡ŽŽ‡ ’‘••‹„‹Ž‹ ”‡ƒŽ‹œœƒœ‹‘‹ †‹  ˜ƒ”‹ƒ„‹Ž‹ ƒ•—ƒŽ‹ ‹†‹’‡†‡–‹‡†‹†‡–‹ Š‡‘°—ǯ‹’‘–‡•‹–”‘’’‘‹•‘•–‡‹„‹Ž‡Ǥ‡”‹Ž‡˜‘’‡•‘‡ •–ƒ–—”ƒ†‹‹†‹˜‹†—‹ǡ‘ ǯ°”ƒ‰‹‘‡†‹’‡•ƒ”‡ Š‡ǣ ͳǤ Ž‡ ƒ”ƒ––‡”‹•–‹ Š‡ˆ‹•‹ Š‡†‡ŽŽǯ‹Ǧ‡•‹‘‹†‹˜‹†—‘•‹ƒ‘‹“—ƒŽ Š‡‘†‘ ‘‡••‡ƒ“—‡ŽŽ‡†‡‰Ž‹ƒŽ–”‹‹†‹˜‹†—‹ȋ‹†‹’‡†‡œƒȌǢ ʹǤ Žƒ ”‡Žƒœ‹‘‡ ˆ”ƒ ’‡•‘ ‡ ƒŽ–‡œœƒ Š‡ ˜ƒŽ‡ ’‡” Žǯ‹Ǧ‡•‹‘ ‹†‹˜‹†—‘ •‹ƒ †‹˜‡”•ƒ†ƒ“—‡ŽŽƒ Š‡˜ƒŽ‡’‡”–—––‹‰Ž‹ƒŽ–”‹ȋ‹†‡–‹ ‹–ȌǤ

(62) “—‡•–‹ ƒ•‹ǡ ‹•‡”˜‹ƒ‘†‡Ž ‘ ‡––‘†‹”‡ƒŽ‹œœƒœ‹‘‡†‹—ƒ˜ƒ”‹ƒ„‹Ž‡ ƒ•—ƒŽ‡ ‘‡‡–ƒˆ‘”ƒ†‡ŽŽǯ‹Ǧ‡•‹ƒ‘••‡”˜ƒœ‹‘‡ǡ‡Žǯƒ’’ƒ”ƒ–‘‹ˆ‡”‡œ‹ƒŽ‡ƒ’’”‘’”‹ƒ–‘° —‰—ƒŽ‡ƒ“—‡ŽŽ‘•–ƒ†ƒ”†ǡ‹ —‹Žǯ‹†‹’‡†‡œƒ‡Žǯ‹†‡–‹ ‹– ‹ ‘•‡–‘‘†‹†‹”‡ Š‡ . ˆ ȋ šͳ ǡ…ǡ š  Ȍ = ∏ ˆ ȋ š ‹ Ȍ  ‹ =ͳ.  ͳ. †‹”Žƒ˜‡”‹–ǡ— ƒ•‘‹–‡”‡†‹‘°†ƒ–‘†ƒ‹ ‘•‹††‡––‹†ƒ–‹’ƒ‡Žǡƒ‘ ‡‡‘ —’‹ƒ‘“—‹Ǥ.  ϱ.

(63)  ‹‘° Š‡ Žƒ ˆ—œ‹‘‡ †‹ †‡•‹– †‡Ž ‘•–”‘ ƒ’‹‘‡ ° •‡’Ž‹ ‡‡–‡ Žƒ ’”‘†—––‘”‹ƒ†‡ŽŽ‡ˆ—œ‹‘‹†‹†‡•‹–†‡ŽŽ‡•‹‰‘Ž‡‘••‡”˜ƒœ‹‘‹ȋŽ‡“—ƒŽ‹ˆ—œ‹‘‹ •‘‘ –—––‡ —‰—ƒŽ‹ȌǤ ‘–ƒ–‡ Š‡ “—‡•–‘ –‹’‘ †‹ ”ƒ‰‹‘ƒ‡–‘ ° ’‡”ˆ‡––ƒ‡–‡ ƒ’’”‘’”‹ƒ–‘‡ŽŽƒƒ‰‰‹‘”’ƒ”–‡†‡‹ ƒ•‹‹ —‹‹†ƒ–‹†ƒ‘‹‘••‡”˜ƒ–‹’”‘˜‡‰ƒ‘ †ƒ—‡•’‡”‹‡–‘ ‘–”‘ŽŽƒ–‘ǡ†‡Ž–‹’‘†‹“—‡ŽŽ‹ Š‡—•ƒ‘‹‡†‹ ‹‘‹„‹‘Ž‘‰‹Ǥ

(64) Ž ƒ•‘†‡ŽŽ‡•‡”‹‡–‡’‘”ƒŽ‹ǡ–—––ƒ˜‹ƒǡ’”‡•‡–ƒ—ƒ†‹ˆˆ‡”‡œƒ ‘ ‡––—ƒŽ‡†‹„ƒ•‡ Š‡ ”‹ Š‹‡†‡ —ƒ ‡•–‡•‹‘‡ †‡‹ ‘ ‡––‹ ’”‘„ƒ„‹Ž‹•–‹ ‹ †ƒ —–‹Ž‹œœƒ”‡Ǥ —‡•–ƒ †‹ˆˆ‡”‡œƒ ‘•‹•–‡ ‡Ž ˆƒ––‘ Š‡ ‹Ž –‡’‘ Šƒ —ƒ †‹”‡œ‹‘‡ǡ ‡ “—‹†‹ ‡•‹•–‡ Žƒ •–‘”‹ƒǤ

(65)  — ‘–‡•–‘ †‹ •‡”‹‡ •–‘”‹ Š‡ǡ ‹ˆƒ––‹ǡ Žƒ ƒ–—”ƒŽ‡ –‡†‡œƒ †‹ ‘Ž–‹ ˆ‡‘‡‹ƒ†‡˜‘Ž˜‡”•‹‹‘†‘’‹î‘‡‘”‡‰‘Žƒ”‡’‘”–ƒƒ’‡•ƒ”‡ Š‡‹Ž†ƒ–‘ ”‹Ž‡˜ƒ–‘‹—†ƒ–‘‹•–ƒ–‡–•‹ƒ’‹î•‹‹Ž‡ƒ“—‡ŽŽ‘”‹Ž‡˜ƒ–‘ƒŽŽǯ‹•–ƒ–‡–Ǧͳ’‹—––‘•–‘ Š‡‹‡’‘ Š‡†‹•–ƒ–‹Ǣ•‹’—Ö†‹”‡ǡ‹— ‡”–‘•‡•‘ǡ Š‡Žƒ•‡”‹‡–‡’‘”ƒŽ‡ Š‡ ƒƒŽ‹œœ‹ƒ‘ Šƒ Dz‡‘”‹ƒ †‹ •±dzǤ —‡•–ƒ ƒ”ƒ––‡”‹•–‹ ƒ ° ‰‡‡”ƒŽ‡–‡ ‹†‹ ƒ–ƒ ‘Ž‘‡†‹’‡”•‹•–‡œƒʹǡ‡†‹ˆˆ‡”‡œ‹ƒ‹ ƒ’‹‘‹†‹•‡”‹‡•–‘”‹ Š‡†ƒ“—‡ŽŽ‹ ”‘••Ǧ •‡ –‹‘ ‹ ƒ‹‡”ƒ ‡––ƒǡ ’‡” Š± ‡‹ ’”‹‹ Žǯ‘”†‹‡ †‡‹ †ƒ–‹ Šƒ —ǯ‹’‘”–ƒœƒ ˆ‘†ƒ‡–ƒŽ‡ǡ‡–”‡‡‹•‡ ‘†‹‡••‘°†‡Ž–—––‘‹””‹Ž‡˜ƒ–‡Ǥ‘•–”—‡–‘ Š‡ —–‹Ž‹œœ‹ƒ‘’‡”ˆƒ”ˆ”‘–‡ƒŽŽǯ‡•‹‰‡œƒ†‹–”‘˜ƒ”‡—ƒ‡–ƒˆ‘”ƒ’”‘„ƒ„‹Ž‹•–‹ ƒ’‡” Ž‡•‡”‹‡–‡’‘”ƒŽ‹‘••‡”˜ƒ–‡°‹Ž’”‘ ‡••‘•–‘ ƒ•–‹ ‘Ǥƒ†‡ˆ‹‹œ‹‘‡†‹’”‘ ‡••‘ •–‘ ƒ•–‹ ‘ ‘ ”‹‰‘”‘•ƒǡ ƒ ‹–—‹–‹˜ƒǡ ’—Ö ‡••‡”‡ Žƒ •‡‰—‡–‡ǣ — ’”‘ ‡••‘ •–‘ ƒ•–‹ ‘ ° — ˜‡––‘”‡ ƒŽ‡ƒ–‘”‹‘ †‹ †‹‡•‹‘‡ ‹ˆ‹‹–ƒǤ  ƒ’‹‘‡ †‹  ‘••‡”˜ƒœ‹‘‹ ‘•‡ —–‹˜‡ ‡Ž –‡’‘ ‘ ˜‹‡‡ “—‹†‹ ’‡•ƒ–‘ –ƒ–‘ ‘‡ —ƒ ”‡ƒŽ‹œœƒœ‹‘‡ †‹  ˜ƒ”‹ƒ„‹Ž‹ ƒ•—ƒŽ‹ †‹•–‹–‡ǡ “—ƒ–‘ ’‹—––‘•–‘ ‘‡ ’ƒ”–‡ †‹ —ǯ—‹ ƒ”‡ƒŽ‹œœƒœ‹‘‡†‹—’”‘ ‡••‘•–‘ ƒ•–‹ ‘ǡŽƒ —‹‡‘”‹ƒ°†ƒ–ƒ†ƒŽ‰”ƒ†‘ †‹ ‘‡••‹‘‡ˆ”ƒŽ‡˜ƒ”‹ƒ„‹Ž‹ ƒ•—ƒŽ‹ Š‡Ž‘ ‘’‘‰‘‘Ǥƒ†‡ˆ‹‹œ‹‘‡ƒ’’‡ƒ †ƒ–ƒ ”‡†‡ ‘˜˜‹‡ —ƒ •‡”‹‡ †‹ ’”‘’”‹‡– †‡‹ ’”‘ ‡••‹ •–‘ ƒ•–‹ ‹ ’‹—––‘•–‘.  ʹ. 

(66)  ‡”–‹ ‘–‡•–‹ǡ˜‹‡‡†‡––ƒƒ Š‡‹•–°”‡•‹ȋ‘‹•–‡”°•‹Ȍ’‡”‹†‹ ƒ”‡ ‹” ƒŽƒ•–‡••ƒ ‘•ƒǤ.  ϲ.

(67) ‹’‘”–ƒ–‹ ’‡” ‹Ž •‡‰—‹–‘ǣ †ƒ–‘ — ’”‘ ‡••‘ •–‘ ƒ•–‹ ‘ ‹Ž —‹ –Ǧ‡•‹‘ ‡Ž‡‡–‘ ‹†‹ Š‹ƒ‘ ‘ xt ǡ ͳǤ ° ’‘••‹„‹Ž‡ †‡ˆ‹‹”‡ ‘ ‡––—ƒŽ‡–‡ —ƒ ˆ—œ‹‘‡ †‹ †‡•‹– ’‡” ‹Ž ’”‘ ‡••‘ f (..., xt -1 , xt , xt +1 ,...) Ǣ ʹǤ °’‘••‹„‹Ž‡ƒ”‰‹ƒŽ‹œœƒ”‡–ƒŽ‡ˆ—œ‹‘‡†‹†‡•‹–’‡”‘‰‹•‘––‘‹•‹‡‡ †‡ŽŽ‡•—‡ ‘’‘‡–‹Ǣ†ƒ“—‡•–‘ ‘•‡‰—‡ Š‡•‘‘†‡ˆ‹‹–‡Ž‡ˆ—œ‹‘‹†‹ †‡•‹– ƒ”‰‹ƒŽ‹ ’‡” ‘‰—ƒ †‡ŽŽ‡ xt ǡ ƒ ƒ Š‡ ’‡” ‘‰‹ ‘’’‹ƒ †‹ ‡Ž‡‡–‹ ( xt , xt +1 )  ‡ ‘•¿ ˜‹ƒǢ ‹Ž ˆƒ––‘ ’‘‹ Š‡ Ž‡ ‘••‡”˜ƒœ‹‘‹ ‘ •‹ƒ‘ ‹†‹’‡†‡–‹ ˆ”ƒ Ž‘”‘ ˆƒ •¿ Š‡ Žƒ †‡•‹– †‡Ž ƒ’‹‘‡ ‘ •‹ ’—Ö ’‹î ”ƒ’’”‡•‡–ƒ”‡ ‘‡—ƒ•‡’Ž‹ ‡’”‘†—––‘”‹ƒ†‡ŽŽ‡ƒ”‰‹ƒŽ‹Ǣ ͵Ǥ •‡ Ž‡ ˆ—œ‹‘‹ †‹ †‡•‹– ƒ”‰‹ƒŽ‹ Šƒ‘ ‘‡–‹ǡ ° ’‘••‹„‹Ž‡ †‹”‡ǡ ƒ† ‡•‡’‹‘ǡ Š‡ E ( xt ) = µ t , Var ( xt ) = σ t , Cov ( xt , xt -1 ) = γ t ,t −1 ‡ ‘•¿˜‹ƒǢ 2. ͶǤ ƒŽŽ‘ •–‡••‘ ‘†‘ǡ ° ’‘••‹„‹Ž‡ †‡ˆ‹‹”‡ ˆ—œ‹‘‹ †‹ †‡•‹– ȋ ‘‹ ”‡Žƒ–‹˜‹ ‘‡–‹Ȍ ‘†‹œ‹‘ƒŽ‹Ǥ ‡ ’”‘’”‹‡– ƒ’’‡ƒ †‡• ”‹––‡ ˆƒ‘ ”‹ˆ‡”‹‡–‘ ƒ‹ ’”‘ ‡••‹ •–‘ ƒ•–‹ ‹ ‘‡ •–”—––—”‡ ’”‘„ƒ„‹Ž‹•–‹ Š‡Ǥ —ƒ†‘ ’‡”Ö ˜‘‰Ž‹ƒ‘ —–‹Ž‹œœƒ”‡ “—‡•–‡ •–”—––—”‡ ‘‡„ƒ•‡’‡”’”‘ ‡†—”‡‹ˆ‡”‡œ‹ƒŽ‹ǡ•‹ƒ’”‘‘†—‡’”‘„Ž‡‹ǣ ͳǤ ‡ “—‡ŽŽƒ Š‡ ‘••‡”˜‘ ȋ’‡”ƒŽ–”‘ ‘ ‡ŽŽƒ •—ƒ ‹–‡”‡œœƒȌ ° —ƒ •‘Žƒ ”‡ƒŽ‹œœƒœ‹‘‡ †‡ŽŽ‡ ‘Ž–‡ ’‘••‹„‹Ž‹ǡ Žƒ ’‘••‹„‹Ž‹– Ž‘‰‹ ƒ †‹ ˆƒ”‡ ‹ˆ‡”‡œƒ •—Ž ’”‘ ‡••‘ ‘ ’—Ö ‡••‡”‡ †ƒ–ƒ ’‡” • ‘–ƒ–ƒǢ ‹ˆƒ––‹ǡ ‘ ǯ° ‘†‘ †‹ †‹”‡ “—ƒŽ‹ ƒ”ƒ––‡”‹•–‹ Š‡ †‡ŽŽƒ •‡”‹‡ ‘••‡”˜ƒ–ƒ •‘‘ •’‡ ‹ˆ‹ Š‡ †‹ “—‡ŽŽƒ ”‡ƒŽ‹œœƒœ‹‘‡ǡ ‡ “—ƒŽ‹ ‹˜‡ ‡ •‹ ”‹’”‡•‡–‡”‡„„‡”‘ ƒ Š‡ ‘••‡”˜ƒ†‘‡ ƒŽ–”‡Ǥ ʹǤ ‡ ƒ Š‡ ˆ‘••‡ ’‘••‹„‹Ž‡ —•ƒ”‡ —ƒ •‘Žƒ ”‡ƒŽ‹œœƒœ‹‘‡ ’‡” ˆƒ”‡ ‹ˆ‡”‡œƒ •—ŽŽ‡ ƒ”ƒ––‡”‹•–‹ Š‡ †‡Ž ’”‘ ‡••‘ǡ ° ‡ ‡••ƒ”‹‘ Š‡ ‡••‘ •‹ƒ •–ƒ„‹Ž‡ ‡Ž.  ϳ.

(68) –‡’‘ǡ ‹‘° Š‡‹•—‘‹ ‘‘–ƒ–‹’”‘„ƒ„‹Ž‹•–‹ ‹’‡”ƒ‰ƒ‘‹˜ƒ”‹ƒ–‹ǡ’‡” Ž‘‡‘ƒŽŽǯ‹–‡”‘†‡Ž‹‘‹–‡”˜ƒŽŽ‘†‹‘••‡”˜ƒœ‹‘‡Ǥ —‡•–‡†—‡“—‡•–‹‘‹ ‘†— ‘‘ƒŽŽƒ†‡ˆ‹‹œ‹‘‡†‹†—‡’”‘’”‹‡– Š‡‹’”‘ ‡••‹ •–‘ ƒ•–‹ ‹ ’‘••‘‘ ƒ˜‡”‡ ‘ ‘ ƒ˜‡”‡ǣ •–ƒœ‹‘ƒ”‹‡– ‡† ‡”‰‘†‹ ‹–Ǥ ‹ ’ƒ”Žƒ †‹ ’”‘ ‡••‘ •–‘ ƒ•–‹ ‘ •–ƒœ‹‘ƒ”‹‘ ‹ †—‡ •‡•‹ǣ •–ƒœ‹‘ƒ”‹‡– ˆ‘”–‡ ȋƒ Š‡ †‡––ƒ •–”‡––ƒȌ ‡ •–ƒœ‹‘ƒ”‹‡– †‡„‘Ž‡Ǥ ‡” †‡ˆ‹‹”‡ Žƒ •–ƒœ‹‘ƒ”‹‡– ˆ‘”–‡ǡ ’”‡†‹ƒ‘ ‹ ‡•ƒ‡ — •‘––‘‹•‹‡‡ “—ƒŽ—“—‡ †‡ŽŽ‡ ˜ƒ”‹ƒ„‹Ž‹ ƒ•—ƒŽ‹ Š‡ ‘’‘‰‘‘ ‹Ž ’”‘ ‡••‘Ǣ “—‡•–‡ ‘ †‡˜‘‘ ‡ ‡••ƒ”‹ƒ‡–‡ ‡••‡”‡ ‘•‡ —–‹˜‡ǡ ƒ ’‡” ƒ‹—–ƒ”‡ Žǯ‹–—‹œ‹‘‡ǡ ˆƒ ‹ƒ‘ ˆ‹–ƒ Š‡ Ž‘ •‹ƒ‘Ǥ ‘•‹†‡”‹ƒ‘ ’‡” ‹Ö —ƒ Dzˆ‹‡•–”ƒdz ƒ’‡”–ƒ •—Ž ’”‘ ‡••‘ †‹ ƒ’‹‡œœƒ ǡ ‘••‹ƒ — •‘––‘‹•‹‡‡ †‡Ž –‹’‘ k. Wt = ( xt ,..., xt + k -1 ) Ǥ —‡•–ƒ ° ƒ–—”ƒŽ‡–‡ —ƒ ˜ƒ”‹ƒ„‹Ž‡ ƒ•—ƒŽ‡ ƒ  †‹‡•‹‘‹ǡ. ‘—ƒ•—ƒˆ—œ‹‘‡†‹†‡•‹– Š‡ǡ‹‰‡‡”ƒŽ‡ǡ’—Ö†‹’‡†‡”‡†ƒ–Ǥ‡’‡”Ö ‹Ö k. k. k. ‘ƒ ƒ†‡ǡƒŽŽ‘”ƒŽƒ†‹•–”‹„—œ‹‘‡†‹ Wt °—‰—ƒŽ‡ƒ“—‡ŽŽƒ†‹ Wt +1 ǡ Wt + 2 ‡ ‘•¿˜‹ƒǤ ‹ƒ‘ ‹ ’”‡•‡œƒ †‹ •–ƒœ‹‘ƒ”‹‡– ˆ‘”–‡ “—ƒ†‘ “—‡•–ƒ ‹˜ƒ”‹ƒœƒ ˜ƒŽ‡ ’‡” “—ƒŽ•‹ƒ•‹ Ǥ

(69)  ƒŽ–”‹ –‡”‹‹ǡ “—ƒ†‘ — ’”‘ ‡••‘ ° •–ƒœ‹‘ƒ”‹‘ ‹ •‡•‘ ˆ‘”–‡ Ž‡ ƒ”ƒ––‡”‹•–‹ Š‡†‹•–”‹„—œ‹‘ƒŽ‹†‹–—––‡Ž‡ƒ”‰‹ƒŽ‹”‹ƒ‰‘‘ ‘•–ƒ–‹ƒŽ’ƒ••ƒ”‡ †‡Ž–‡’‘Ǥƒ•–ƒœ‹‘ƒ”‹‡–†‡„‘Ž‡ǡ‹˜‡ ‡ǡ”‹‰—ƒ”†ƒ•‘Ž‘ˆ‹‡•–”‡†‹ƒ’‹‡œœƒʹǣ 2. •‹ Šƒ •–ƒœ‹‘ƒ”‹‡– †‡„‘Ž‡ •‡ –—––‡ Ž‡ ˜ƒ”‹ƒ„‹Ž‹ ƒ•—ƒŽ‹ †‘’’‹‡ Wt = ( xt , xt + k ) ǡ Šƒ‘ ‘‡–‹ ’”‹‹ ‡ •‡ ‘†‹ ‘•–ƒ–‹ ‡Ž –‡’‘͵Ǣ †ƒ “—‡•–‘ †‹• ‡†‡ Š‡ ‡•‹•–‘‘ ƒ Š‡–—––‹‹ ‘‡–‹•‡ ‘†‹‹ ”‘ ‹ƒ–‹ E ( xt ⋅ xt + k ) ‘  “—ƒŽ—“—‡ǡ ‡ ƒ Šǯ‡••‹‘†‹’‡†‘‘†ƒ–ȋƒ Š‡•‡’‘••‘‘†‹’‡†‡”‡†ƒȌǤ†‹•’‡––‘†‡‹ ‘‹ǡ —ƒ †‡ˆ‹‹œ‹‘‡ ‘ ‹’Ž‹ ƒ ŽǯƒŽ–”ƒǢ ƒ† ‡•‡’‹‘ǡ — ’”‘ ‡••‘ ’—Ö ‡••‡”‡ •–ƒœ‹‘ƒ”‹‘‹•‡•‘ˆ‘”–‡ƒ‘’‘••‡†‡”‡‘‡–‹Ǣ˜‹ ‡˜‡”•ƒǡŽƒ ‘•–ƒœƒ‡Ž –‡’‘ †‡‹ ‘‡–‹ ‘ ‹’Ž‹ ƒ Š‡ Ž‡ ˜ƒ”‹‡ ƒ”‰‹ƒŽ‹ ƒ„„‹ƒ‘ Žƒ •–‡••ƒ †‹•–”‹„—œ‹‘‡Ǥ

(70)  — ƒ•‘ǡ –—––ƒ˜‹ƒǡ Ž‡ †—‡ †‡ˆ‹‹œ‹‘‹ ‘‹ ‹†‘‘ǣ “—‡•–‘ ƒ•‘ǡ ’ƒ”–‹ ‘Žƒ”‡–‡ ‹’‘”–ƒ–‡ ’‡” Ž‡ ƒ’’Ž‹ ƒœ‹‘‹ ’”ƒ–‹ Š‡ǡ ° “—‡ŽŽ‘ ‹ —‹ ‹Ž  ͵. 1’‡”“—‡•–‘‘–‹˜‘ Š‡Žƒ•–ƒœ‹‘ƒ”‹‡–†‡„‘Ž‡˜‹‡‡ƒ Š‡†‡ˆ‹‹–ƒ•–ƒœ‹‘ƒ”‹‡–‹ ‘˜ƒ”‹ƒœƒǤ.  ϴ.

(71) ’”‘ ‡••‘°‰ƒ—••‹ƒ‘ǡ‘••‹ƒ“—ƒ†‘Žƒ†‹•–”‹„—œ‹‘‡ ‘‰‹—–ƒ†‹— “—ƒŽ—“—‡ •‘––‘‹•‹‡‡ †‹ ‡Ž‡‡–‹ †‡Ž ’”‘ ‡••‘ ° —ƒ ‘”ƒŽ‡ —Ž–‹˜ƒ”‹ƒ–ƒǤ ‡ — ’”‘ ‡••‘ ° ‰ƒ—••‹ƒ‘ǡ •–ƒ„‹Ž‹”‡ Š‡ ° •–ƒœ‹‘ƒ”‹‘ ‹ •‡•‘ †‡„‘Ž‡ ‡“—‹˜ƒŽ‡ ƒ •–ƒ„‹Ž‹”‡Žƒ•–ƒœ‹‘ƒ”‹‡–‹•‡•‘ˆ‘”–‡Ǥƒ–ƒŽƒ’‡”˜ƒ•‹˜‹–†‡‹’”‘ ‡••‹‰ƒ—••‹ƒ‹ ‡ŽŽ‡ƒ’’Ž‹ ƒœ‹‘‹ƒ‹†ƒ–‹ǡ†ƒ—’—–‘†‹˜‹•–ƒ‘’‡”ƒ–‹˜‘•‹ƒ†‘––ƒ‰‡‡”ƒŽ‡–‡ Žƒ †‡ˆ‹‹œ‹‘‡ †‹ •–ƒœ‹‘ƒ”‹‡– †‡„‘Ž‡ǡ ‡ “—ƒ†‘ •‹ ’ƒ”Žƒ †‹ •–ƒœ‹‘ƒ”‹‡– •‡œƒ ƒ‰‰‡––‹˜‹ǡ°ƒ’’—–‘ƒ“—‡•–ƒ Š‡ ‹•‹”‹ˆ‡”‹• ‡Ǥ ‡” “—ƒ–‘ ”‹‰—ƒ”†ƒ Žǯ‡”‰‘†‹ ‹–ǡ ° —ƒ ‘†‹œ‹‘‡ Š‡ Ž‹‹–ƒ Žƒ ‡‘”‹ƒ †‡Ž ’”‘ ‡••‘ǣ — ’”‘ ‡••‘ ‘ ‡”‰‘†‹ ‘ ° — ’”‘ ‡••‘ Š‡ Šƒ ƒ”ƒ––‡”‹•–‹ Š‡ †‹ ’‡”•‹•–‡œƒ ‘•¿ƒ ‡–—ƒ–‡†ƒˆƒ”•¿ Š‡—•‡‰‡–‘†‡Ž’”‘ ‡••‘ǡ’‡”“—ƒ–‘ Ž—‰‘ǡ•‹ƒ‹•—ˆˆ‹ ‹‡–‡ƒ†‹”‡ƒŽ — Š±•—ŽŽ‡•—‡ ƒ”ƒ––‡”‹•–‹ Š‡†‹•–”‹„—–‹˜‡Ǥ

(72) — ’”‘ ‡••‘ ‡”‰‘†‹ ‘ǡ ƒŽ ‘–”ƒ”‹‘ǡ Žƒ ‡‘”‹ƒ †‡Ž ’”‘ ‡••‘ ° †‡„‘Ž‡ •— Ž—‰Š‹ ‘”‹œœ‘–‹ ‡ ƒŽŽǯƒ—‡–ƒ”‡ †‡ŽŽǯƒ’‹‡œœƒ †‡Ž ƒ’‹‘‡ ƒ—‡–ƒ ‹ ‘†‘ •‹‰‹ˆ‹ ƒ–‹˜‘ƒ Š‡Žǯ‹ˆ‘”ƒœ‹‘‡‹‘•–”‘’‘••‡••‘Ǥ ‡ ‘†‹œ‹‘‹ •‘––‘ Ž‡ “—ƒŽ‹ — ’”‘ ‡••‘ •–‘ ƒ•–‹ ‘ •–ƒœ‹‘ƒ”‹‘ ° ‡”‰‘†‹ ‘ •‘‘ –”‘’’‘ ‘’Ž‡••‡’‡”‡••‡”‡†‡• ”‹––‡„”‡˜‡‡–‡Ǥ‘•–ƒœ‹ƒŽ‡–‡ǡ•‹’—Ö†‹”‡ Š‡—’”‘ ‡••‘°‡”‰‘†‹ ‘•‡‡˜‡–‹Ž‘–ƒ‹ˆ”ƒŽ‘”‘’‘••‘‘‡••‡”‡ ‘•‹†‡”ƒ–‹ ‹†‹’‡†‡–‹Ǣ‘••‡”˜ƒ†‘‹Ž’”‘ ‡••‘’‡”—Žƒ••‘†‹–‡’‘ƒ„„ƒ•–ƒœƒŽ—‰‘ǡ° ’‘••‹„‹Ž‡ ‘••‡”˜ƒ”‡ “—ƒ•‹ –—––‡ Ž‡ •‘––‘•‡“—‡œ‡ Š‡ ‹Ž ’”‘ ‡••‘ ° ‹ ‰”ƒ†‘ †‹ ‰‡‡”ƒ”‡Ǥ

(73) Ž ˆƒ––‘ Š‡ ‡˜‡–‹ Ž‘–ƒ‹ ˆ”ƒ Ž‘”‘ ‡Ž –‡’‘ ’‘••ƒ‘ ‡••‡”‡ ‘•‹†‡”ƒ–‹ ‹†‹’‡†‡–‹ †ƒ — ’—–‘ †‹ ˜‹•–ƒ ’”ƒ–‹ ‘ ° ’‘‹ •’‡••‘ •‹–‡–‹œœƒ–‘ ‡ŽŽƒ •‡‰—‡–‡ ’”‘’”‹‡– †‡‹ ’”‘ ‡••‹ ‡”‰‘†‹ ‹ ȋ Š‡ ƒ ˜‘Ž–‡ ˜‹‡‡ —•ƒ–ƒ ‘‡ †‡ˆ‹‹œ‹‘‡†‹’”‘ ‡••‘‡”‰‘†‹ ‘Ȍǣ  1 n ∑ Cov ( xt , xt -k ) = 0  n →+∞ n k =1 lim. 

(74) Ž‹‡ƒ†‹’”‹ ‹’‹‘‡ ‘•‡‰—‡ Š‡ǡ•‡—’”‘ ‡••‘°‡”‰‘†‹ ‘ǡ°’‘••‹„‹Ž‡—•ƒ”‡ Ž‡ ‹ˆ‘”ƒœ‹‘‹ ‘–‡—–‡ ‡Ž •—‘ •˜‘Ž‰‹‡–‘ ‡Ž –‡’‘ ’‡” ‹ˆ‡”‹”‡ Ž‡ ƒ”ƒ––‡”‹•–‹ Š‡Ǥ•‹•–‡—–‡‘”‡ƒȋ–‡‘”‡ƒ‡”‰‘†‹ ‘†‹Ž—–•›Ȍ Š‡†‹ ‡ Š‡ǡ•‡  ϵ.

(75) —’”‘ ‡••‘°‡”‰‘†‹ ‘ǡŽǯ‘••‡”˜ƒœ‹‘‡†‹—ƒ•—ƒ”‡ƒŽ‹œœƒœ‹‘‡ƒ„„ƒ•–ƒœƒŽ—‰ƒ ° ‡“—‹˜ƒŽ‡–‡ǡ ƒ‹ ˆ‹‹ ‹ˆ‡”‡œ‹ƒŽ‹ǡ ƒŽŽǯ‘••‡”˜ƒœ‹‘‡ †‹ — ‰”ƒ —‡”‘ †‹ ”‡ƒŽ‹œœƒœ‹‘‹Ǥ‡ǡƒ†‡•‡’‹‘ǡ—’”‘ ‡••‘‡”‰‘†‹ ‘ xt Šƒ˜ƒŽ‘”‡ƒ––‡•‘ρǡƒŽŽ‘”ƒ Žƒ•—ƒ‡†‹ƒƒ”‹–‡–‹ ƒ‡Ž–‡’‘°—‘•–‹ƒ–‘”‡ ‘•‹•–‡–‡†‹ρ‡“—‹†‹’—Ö ‡••‡”‡•–‹ƒ–‘‹‘†‘ ‘•‹•–‡–‡ ‘‡•‡†‹•’‘‡••‹‘†‹‘Ž–‡”‡ƒŽ‹œœƒœ‹‘‹ †‡Ž’”‘ ‡••‘ƒœ‹ Š±†‹—ƒ•‘ŽƒǤ

(76) Ž‹‡ƒ‰‡‡”ƒŽ‡ǡ•‹’—Ö†‹”‡ Š‡Žǯ‹ˆ‡”‡œƒ° ’‘••‹„‹Ž‡ •‘Ž‘ •‡ ‹Ž ’”‘ ‡••‘ •–‘ ƒ•–‹ ‘ Š‡ •‹ •–ƒ •–—†‹ƒ†‘ ° •–ƒœ‹‘ƒ”‹‘ ‡† ‡”‰‘†‹ ‘Ǥ ƒ †‡––‘ ’‡” ƒŽ–”‘ Š‡ǡ •‡ ‡•‹•–‘‘ †‡‹ ‡–‘†‹ ’‡” •‘––‘’‘””‡ ƒ –‡•– Žǯ‹’‘–‡•‹ †‹ ‘ •–ƒœ‹‘ƒ”‹‡–ǡ Žǯ‹’‘–‡•‹ †‹ ‡”‰‘†‹ ‹– ‘ ° ˜‡”‹ˆ‹ ƒ„‹Ž‡ •‡ •‹ †‹•’‘‡†‹—ƒ•‘Žƒ”‡ƒŽ‹œœƒœ‹‘‡†‡Ž’”‘ ‡••‘ǡˆ‘••‡ƒ Š‡†‹ƒ’‹‡œœƒ‹ˆ‹‹–ƒǤ ‡Ž ƒ•‘†‹’”‘ ‡••‹•–‘ ƒ•–‹ ‹•–ƒœ‹‘ƒ”‹ǡƒ˜”‡‘†—“—‡ Š‡‘‰‹‡Ž‡‡–‘†‡Ž ’”‘ ‡••‘ xt  ƒ˜” — ˜ƒŽ‘”‡ ƒ––‡•‘ ˆ‹‹–‘ ‡ ‘•–ƒ–‡ ρ ‡ —ƒ ˜ƒ”‹ƒœƒ ˆ‹‹–ƒ ‡ ‘•–ƒ–‡ɐʹǤ

(77) ‘Ž–”‡ǡ”‹•—Ž–ƒ‘†‡ˆ‹‹–‡–—––‡Ž‡ ‘˜ƒ”‹ƒœ‡ˆ”ƒ‡Ž‡‡–‹†‹˜‡”•‹†‡Ž ’”‘ ‡••‘ǡ Š‡•ƒ”ƒ‘’ƒ”‹ƒ γ k = E[( xt - µ )( xt -k - µ )] .  ‡ Š‡ •‘‘ ‘–‡ ‘‡ ƒ—–‘ ‘˜ƒ”‹ƒœ‡Ǥ ‹ ”‹ ‘”†‹ Š‡ Žƒ •–ƒœ‹‘ƒ”‹‡– ‰ƒ”ƒ–‹• ‡ Š‡ “—‡•–‡ “—ƒ–‹– ‘ •‘‘ ˆ—œ‹‘‹ †‹ –Ǣ ‡••‡ •‘‘ –—––ƒ˜‹ƒ ˆ—œ‹‘‹ †‹ ǡ ‡† ƒœ‹•‹’ƒ”Žƒ†‹ˆ—œ‹‘‡†‹ƒ—–‘ ‘˜ƒ”‹ƒœƒǡ‹–‡†‡†‘—ƒˆ—œ‹‘‡†‹–ƒŽ‡’‡” —‹ γ ( k ) = γ k Ǥƒ †ƒ •± Š‡ Žǯƒ—–‘ ‘˜ƒ”‹ƒœƒ †‹ ‘”†‹‡ Ͳ ‘ ° Š‡ Žƒ ˜ƒ”‹ƒœƒ ‡ Š‡Žƒ†‡ˆ‹‹œ‹‘‡°–ƒŽ‡’‡” —‹ γ k = γ - k ǡ°˜‡”‹–‹‡”ƒǤŽŽ‘•–‡••‘‘†‘°’‘••‹„‹Ž‡ †‡ˆ‹‹”‡Ž‡ƒ—–‘ ‘””‡Žƒœ‹‘‹ǡ Š‡•‘‘†ƒ–‡†ƒ  ρ0 =. γk γ = k2  γ0 σ.  —‡•–‡“—ƒ–‹–ǡ•‡†‹˜‡”•‡†ƒͲǡ ‘•–‹–—‹• ‘‘Žƒ‡‘”‹ƒ†‡Ž’”‘ ‡••‘ǡ‡•‘‘ ƒ’’—–‘Žǯ‡Ž‡‡–‘ Š‡”‡†‡‹’”‘ ‡••‹•–‘ ƒ•–‹ ‹Ž‘•–”—‡–‘–‡‘”‹ ‘ƒ†ƒ––‘ƒ.  ϭϬ.

(78) ”ƒ’’”‡•‡–ƒ”‡ •‡”‹‡ –‡’‘”ƒŽ‹ ƒ”ƒ––‡”‹œœƒ–‡ †ƒ ’‡”•‹•–‡œƒǤ

(79) ˆƒ––‹ǡ •‡ γ 1 ≠ 0 ǡ ƒŽŽ‘”ƒ•‹Šƒ Š‡ f ( x t xt -1 ) ≠ f ( x t ) .  ‡†‹ ‘•‡‰—‡œƒ E ( x t xt -1 ) ≠ E ( x t ) Ǥ.  ‘–”‡‘ ‡•–‡†‡”‡ Žǯ‹•‹‡‡ †‹ ˜ƒ”‹ƒ„‹Ž‹ ƒ•—ƒŽ‹ •— —‹ ‡ˆˆ‡––—‹ƒ‘ ‹Ž ‘†‹œ‹‘ƒ‡–‘ƒ Š‡ƒ x t -2 ǡ x t -3 ‡ ‡–‡”ƒǤ—‡•–‘‹•‹‡‡†‹˜ƒ”‹ƒ„‹Ž‹ ƒ•—ƒŽ‹ ’”‡†‡ƒ˜‘Ž–‡‹Ž‘‡†‹•‡–‹ˆ‘”ƒ–‹˜‘ƒŽ–‡’‘–Ǧͳǡ‡˜‹‡‡‹†‹ ƒ–‘ ‘ I t -1 Ǥ                  .  ϭϭ.

(80) ͳǤʹ ‡‡•‹†‡‹’”‘ ‡••‹.

(81) ’”‘ ‡••‹ ‘•–‹–—‹• ‘‘Žƒˆƒ‹‰Ž‹ƒ†‹’”‘ ‡••‹•–‘ ƒ•–‹ ‹†‹‰”ƒŽ—‰ƒ’‹î —–‹Ž‹œœƒ–‹ ‹ •–ƒ–‹•–‹ ƒ ‡ ‘ •‘Ž‘Ǥ —‡•–ƒ • ‡Ž–ƒ Šƒ ”ƒ‰‹‘‹ –‡‘”‹ Š‡ ‡ ”ƒ‰‹‘‹ ’”ƒ–‹ Š‡ǡ Š‡•ƒ”ƒ‘‹ŽŽ—•–”ƒ–‡‡Ž•‡‰—‹–‘Ǥ”‹ƒ†‹ƒƒŽ‹œœƒ”‡Ž‡ ƒ”ƒ––‡”‹•–‹ Š‡ ’”‹ ‹’ƒŽ‹†‹–ƒŽ‹’”‘ ‡••‹ǡ–—––ƒ˜‹ƒǡ•‘‘‡ ‡••ƒ”‹‡ƒŽ —‡†‡ˆ‹‹œ‹‘‹†‹„ƒ•‡ǡ Š‡ ˆ‘”ƒ‘Žǯ‘‰‰‡––‘†‡ŽŽ‡’”‘••‹‡•‡œ‹‘‹Ǥƒ–‘‹’”‘ ‡••‹•–‘ ƒ•–‹ ‹ Š‡Ž‡•‡”‹‡ –‡’‘”ƒŽ‹•‘‘ǡ‹„—‘ƒ•‘•–ƒœƒǡ•‡“—‡œ‡†‹—‡”‹Ǥƒ’‹–‡”‘Ž–‘•’‡••‘†‹ †‘˜‡” ƒ‹’‘Žƒ”‡ –ƒŽ‹ •‡“—‡œ‡ǡ ‡ Ž‘ •‹ ˆƒ” ’‡” ‡œœ‘ †‹ ƒ’’‘•‹–‹ ‘’‡”ƒ–‘”‹Ǥ ǯ‘’‡”ƒ–‘”‡ ”‹–ƒ”†‘ ȋ†‡––‘ ƒ Š‡ ‘’‡”ƒ–‘”‡ ƒ ™ƒ”†Ȍ ˜‹‡‡ ‰‡‡”ƒŽ‡–‡ ‹†‹ ƒ–‘ ‘ŽƒŽ‡––‡”ƒ‡ŽŽƒŽ‡––‡”ƒ–—”ƒ•–ƒ–‹•–‹ ƒǢ°—‘’‡”ƒ–‘”‡ Š‡•‹ƒ’’Ž‹ ƒ ƒ•‡“—‡œ‡†‹—‡”‹ǡ‡–”ƒ•ˆ‘”ƒ—ƒ•‡“—‡œƒ x t ‹—ƒŽ–”ƒ•‡“—‡œƒ Š‡Šƒ Žƒ ƒ”ƒ––‡”‹•–‹ ƒ†‹ƒ˜‡”‡‰Ž‹•–‡••‹˜ƒŽ‘”‹†‹ x t ǡƒ•ˆƒŽ•ƒ–‹†‹—’‡”‹‘†‘Ǥ

(82) ˆƒ––‹‹ ˆ‘”—Ž‡ǡ Bx t = x t -1 Ǥ.  ǯƒ’’Ž‹ ƒœ‹‘‡”‹’‡–—–ƒ˜‘Ž–‡†‹˜‹‡‡‹†‹ ƒ–ƒ ‘Žƒ• ”‹––—”ƒǡ‡“—‹†‹•‹ n. Šƒ B x t = xt -n Ǥ‡” ‘˜‡œ‹‘‡•‹’‘‡Ͳ αͳǤǯ‘’‡”ƒ–‘”‡°—‘’‡”ƒ–‘”‡†‹ –‹’‘Ž‹‡ƒ”‡‡Ž•‡•‘ Š‡ǡ•‡ƒ‡„•‘‘ ‘•–ƒ–‹ǡ•‹Šƒ  B (ax t + b ) = aBx t + b = ax t -1 + b Ǥ.  ƒ ƒ”ƒ––‡”‹•–‹ ƒ ’‹î ‹–‡”‡••ƒ–‡ †‹ “—‡•–‘ ‘’‡”ƒ–‘”‡ ° Š‡ Ž‡ •—‡ ’”‘’”‹‡– ƒ’’‡ƒ ‡— ‹ƒ–‡ ’‡”‡––‘‘ǡ ‹ ‘Ž–‡ ‹” ‘•–ƒœ‡ǡ †‹ ƒ‹’‘Žƒ”Ž‘ ƒŽ‰‡„”‹ ƒ‡–‡ ‘‡•‡ˆ‘••‡——‡”‘Ǥ—‡•–‘ƒ˜˜‹‡‡•‘’”ƒ––—––‘“—ƒ†‘•‹ ‘•‹†‡”ƒ‘’‘Ž‹‘‹‡ŽŽǯ‘’‡”ƒ–‘”‡Ǥ ǯƒŽ–”ƒ †‡ˆ‹‹œ‹‘‡ †ƒ ‘ –”ƒŽƒ• ‹ƒ”‡ǡ ° “—ƒ†‘ — ’”‘ ‡••‘ •–‘ ƒ•–‹ ‘ ° †‡ˆ‹‹–‘™Š‹–‡‘‹•‡ȋDz”—‘”‡„‹ƒ ‘dzȌǤ—‡•–‘–‹’‘†‹’”‘ ‡••‘°‹Ž’‹î•‡’Ž‹ ‡ Š‡•‹’—Ö‹ƒ‰‹ƒ”‡ǣ‹ˆƒ––‹ǡ°—’”‘ ‡••‘ Š‡’‘••‹‡†‡‘‡–‹ƒŽ‡‘ˆ‹‘.  ϭϮ.

(83) ƒŽ •‡ ‘†‘ ‘”†‹‡Ǣ ‡••‹ •‘‘ ‘•–ƒ–‹ ‡Ž –‡’‘ ȋ“—‹†‹ ‹Ž ’”‘ ‡••‘ ° •–ƒœ‹‘ƒ”‹‘Ȍǡƒ‘†ƒ‘ƒŽ’”‘ ‡••‘ƒŽ —ƒ‡‘”‹ƒ†‹•±Ǥƒ•–‡••ƒ ‘•ƒ•‹ ’—Ö †‹”‡ ‹ ‘†‘ ’‹î ˆ‘”ƒŽ‹œœƒ–‘ ‘‡ •‡‰—‡ǣ — ’”‘ ‡••‘ ™Š‹–‡ ‘‹•‡ǡ ‹Ž —‹ ‡Ž‡‡–‘–Ǧ‡•‹‘‹†‹ Š‡”‡‘ ‘ εt ǡ’”‡•‡–ƒ“—‡•–‡ ƒ”ƒ––‡”‹•–‹ Š‡ǣ  E (ε t ) = µ E (ε t2 )=Var (ε t )=σ 2. γk = 0. . per k > 0.   ™Š‹–‡ ‘‹•‡ ° “—‹†‹ǡ ‹ •‘•–ƒœƒǡ — ’”‘ ‡••‘ ‘’‘•–‘ †‹ — —‡”‘ ‹ˆ‹‹–‘ †‹ ˜ƒ”‹ƒ„‹Ž‹ ƒ•—ƒŽ‹ ƒ ‡†‹ƒ œ‡”‘ ‡ ˜ƒ”‹ƒœƒ ‘•–ƒ–‡Ǣ “—‡•–‡ ˜ƒ”‹ƒ„‹Ž‹ ƒ•—ƒŽ‹ǡ ‹‘Ž–”‡ǡ •‘‘ –—––‡ ‹ ‘””‡Žƒ–‡ Žǯ—ƒ ƒŽŽǯƒŽ–”ƒǤ  ”‹‰‘”‡ǡ “—‡•–‘ ‘ •‹‰‹ˆ‹ ƒ Š‡ ‡••‡ •‹ƒ‘ ‹†‹’‡†‡–‹Ǥ ‡ ’‡”Ö •‹ ’ƒ”Žƒ †‹ ™Š‹–‡ ‘‹•‡ ‰ƒ—••‹ƒ‘ǡ ‘••‹ƒ †‹ — ™Š‹–‡ ‘‹•‡ ‹ —‹ Žƒ †‹•–”‹„—œ‹‘‡ ‘‰‹—–ƒ †‹ –—––‡ Ž‡ ‘’’‹‡ ȋ ε t ǡ ε t +k Ȍ•‹ƒ—ƒ‘”ƒŽ‡„‹˜ƒ”‹ƒ–ƒǡƒŽŽ‘”ƒ•¿Ǥ’”‘ ‡••‘™Š‹–‡‘‹•‡ǡ“—‹†‹ǡ° —’”‘ ‡••‘•–‘ ƒ•–‹ ‘ Š‡‘‡•‹„‹• ‡’‡”•‹•–‡œƒǤ

(84) “—ƒ–‘–ƒŽ‡ǡ•‹’‘–”‡„„‡ ’‡•ƒ”‡ Š‡ •‹ƒ ‹ƒ†‡‰—ƒ–‘ ƒ ”ƒ‰‰‹—‰‡”‡ Ž‘ • ‘’‘ †‹ –”‘˜ƒ”‡ —ƒ •–”—––—”ƒ ’”‘„ƒ„‹Ž‹•–‹ ƒ Š‡’‘••ƒ•‡”˜‹”‡†ƒ‡–ƒˆ‘”ƒ’‡” ƒ’‹‘‹†‹•‡”‹‡–‡’‘”ƒŽ‹ Š‡ǡ ‹˜‡ ‡ǡŽƒ’‡”•‹•–‡œƒ ‡ŽǯŠƒ‘Ǥ

(85) Ž’ƒ••‘‹ ƒ˜ƒ–‹†‡ ‹•‹˜‘•–ƒ ‡Ž ‘•‹†‡”ƒ”‡ ‘•ƒ•— ‡†‡ƒ’’Ž‹ ƒ†‘—’‘Ž‹‘‹‘‡ŽŽǯ‘’‡”ƒ–‘”‡”‹–ƒ”†‘ƒ†—™Š‹–‡‘‹•‡Ǥ         .  ϭϯ.

(86) ͳǤ͵‘†‡ŽŽ‹ƒ‡†‹ƒ‘„‹Ž‡ȋȌ.  ’”‘ ‡••‘ ǡ ‘ ’”‘ ‡••‘ ƒ ‡†‹ƒ ‘„‹Ž‡ ȋ •–ƒ ƒ’’—–‘ ’‡” ‘˜‹‰ ˜‡”ƒ‰‡Ȍǡ°—ƒ•‡“—‡œƒ†‹˜ƒ”‹ƒ„‹Ž‹ ƒ•—ƒŽ‹†‡ˆ‹‹–ƒ–”ƒ‹–‡Žƒˆ‘”ƒ  q yt = ∑ θi ⋅ε t -i = C (B )ε t  i =0.  †‘˜‡ ȋȌ ° — ’‘Ž‹‘‹‘ †‹ ‘”†‹‡ “ ‡ŽŽǯ‘’‡”ƒ–‘”‡ ”‹–ƒ”†‘ ‡ εt  ° — ™Š‹–‡ ‘‹•‡Ǥ ‡‡”ƒŽ‡–‡ǡ‡•‡œƒ’‡”†‹–ƒ†‹‰‡‡”ƒŽ‹–ǡ•‹’‘‡ C (0)=θ0 =1 Ǥ‡ȋȌ° —’‘Ž‹‘‹‘†‹‰”ƒ†‘“ǡ•‹†‹ ‡ƒ Š‡ Š‡ yt °—’”‘ ‡••‘ȋ“ȌǤ•ƒ‹ƒ†‘‹ •—‘‹‘‡–‹ǡ•‹˜‡†‡ Š‡“—‡ŽŽ‘†‹‘”†‹‡’”‹‘•‹• ”‹˜‡ǣ  q  q  E (yt ) = E  ∑ θi ⋅ε t -i  = ∑ θi ⋅ E ε t -i  = 0  i =0  i = 0.   “—‹†‹ — ’”‘ ‡••‘  Šƒ ‡†‹ƒ ͲǤ  ’”‹ƒ ˜‹•–ƒǡ •‹ ’‘–”‡„„‡ ’‡•ƒ”‡ Š‡ “—‡•–ƒ ƒ”ƒ––‡”‹•–‹ ƒŽ‹‹–‹ˆ‘”–‡‡–‡Žǯƒ’’Ž‹ ƒ„‹Ž‹–†‹’”‘ ‡••‹ƒ•‹–—ƒœ‹‘‹ ”‡ƒŽ‹ǡ˜‹•–‘ Š‡ǡ‹‰‡‡”‡ǡ‘°†‡––‘ Š‡Ž‡•‡”‹‡–‡’‘”ƒŽ‹‘••‡”˜ƒ–‡‘• ‹ŽŽ‹‘ ‹–‘”‘ ƒŽ˜ƒŽ‘”‡ͲǤ—––ƒ˜‹ƒǡŽƒŽ‹‹–ƒœ‹‘‡°’‹îƒ’’ƒ”‡–‡ Š‡”‡ƒŽ‡ǡ˜‹•–‘ Š‡ ’‡”‘‰‹’”‘ ‡••‘ xt ’‡” —‹ E ( xt )= µt •‹’—Ö•‡’”‡†‡ˆ‹‹”‡——‘˜‘’”‘ ‡••‘ yt = xt - µt ƒ ‡†‹ƒ —ŽŽƒǤ ‡ yt  ° •–ƒœ‹‘ƒ”‹‘ ‹ ‘˜ƒ”‹ƒœƒǡ ƒŽŽ‘”ƒ „ƒ•–ƒ •–—†‹ƒ”‡ yt ‡’‘‹ƒ‰‰‹—‰‡”‡—‘˜ƒ‡–‡Žƒ‡†‹ƒ’‡”ƒ˜‡”‡ xt Ǥ‡”“—ƒ–‘”‹‰—ƒ”†ƒŽƒ. ˜ƒ”‹ƒœƒǡ‹Žˆƒ––‘ Š‡‹Ž‘‡–‘’”‹‘•‹ƒ—ŽŽ‘ ‹ ‘•‡–‡†‹• ”‹˜‡”Žƒ ‘‡‹Ž ‘‡–‘•‡ ‘†‘ǡ‘••‹ƒ  2  q    Var (yt )= E (yt )=E  ∑ θi ⋅εt -i         i =0 2.   ϭϰ.

(87)  ˜‹Ž—’’ƒ†‘‹Ž“—ƒ†”ƒ–‘ǡ’‘••‹ƒ‘• ‘’‘””‡Žƒ•‘ƒ‹†—‡’ƒ”–‹ǣ  2. q q  q   ∑ θi ⋅ε t -i  = ∑ θi2ε t2−i + ∑ ∑ θi θ j ε t -i ε t - j    i = 0 j ≠i  i =0  i =0.  ƒŽŽƒ’”‘’”‹‡–†‡Ž™Š‹–‡‘‹•‡ǡ Š‡‹Ž˜ƒŽ‘”‡ƒ––‡•‘†‡ŽŽƒ•‡ ‘†ƒ•‘ƒ–‘”‹ƒ° —ŽŽ‘ǡ ‘•‹ Š± q q  q  q E (yt 2 )=E  ∑ θi2ε t2−i  = ∑ θi2E (ε t2−i )= ∑ θi2σ 2 =σ 2 ⋅ ∑ θi2  i =0 i =0 i =0  i = 0.  q. 2 Š‡Šƒ˜ƒŽ‘”‡ˆ‹‹–‘•‡ ∑ θi < +∞ ǡ ‘•ƒ•‡’”‡˜‡”ƒ•‡“°ˆ‹‹–‘Ǥ

(88) ˆ‹‡ǡ ‘— i =0. ”ƒ‰‹‘ƒ‡–‘ †‡Ž –—––‘ ƒƒŽ‘‰‘ ’‡”˜‡‹ƒ‘ ƒŽ ƒŽ ‘Ž‘ †‡ŽŽ‡ ƒ—–‘ ‘˜ƒ”‹ƒœ‡ǣ Žǯƒ—–‘ ‘˜ƒ”‹ƒœƒ†‹‘”†‹‡°†ƒ–ƒ†ƒ   q  q E (yt yt +k )=E  ∑ θi εt -i  ∑ θ j εt - j +k   i =0  j =0 .  q  = ∑ θ i    i =0.  q   ∑ θ j E (εt -i εt - j +k )     j =0 .  ˆ”—––ƒ†‘ ƒ ‘”ƒ —ƒ ˜‘Ž–ƒ Ž‡ ’”‘’”‹‡– †‡Ž ™Š‹–‡ ‘‹•‡ǡ •‹ Šƒ Š‡ E (εt -i εt - j +k )=σ 2  •‡ Œ ε ‹ή ‡ Ͳ ‹ –—––‹ ‰Ž‹ ƒŽ–”‹ ƒ•‹ǡ ‘•‹ Š± Žǯ‡•’”‡••‹‘‡. ’”‡ ‡†‡–‡•‹”‹†— ‡ƒǣ 2. q. γ k = E(yt yt+k ) = σ ∑ θi θi+k  i=0.  †‘˜‡•‹‹–‡†‡ Š‡ θi =0 ’‡”‹ι“Ǥ‹‘–‹ Š‡ǣ •. ǯ‡•’”‡••‹‘‡ ’‡” Žƒ ˜ƒ”‹ƒœƒ ° — ƒ•‘ ’ƒ”–‹ ‘Žƒ”‡ †‡ŽŽƒ ˆ‘”—Žƒ ’”‡ ‡†‡–‡ǡ’‘‡†‘αͲǢ.  ϭϱ.

(89) •. ’‡”ε“ǡŽ‡ƒ—–‘ ‘˜ƒ”‹ƒœ‡•‘‘—ŽŽ‡Ǥ.  ’”‘ ‡••‘ ȋ“Ȍǡ “—‹†‹ǡ ° — ’”‘ ‡••‘ ‘––‡—–‘ ‘‡ ‘„‹ƒœ‹‘‡ †‹ †‹˜‡”•‹ ‡Ž‡‡–‹ †‹ —‘ •–‡••‘ ™Š‹–‡ ‘‹•‡ Š‡ ’”‡•‡–ƒ †‡ŽŽ‡ ƒ”ƒ––‡”‹•–‹ Š‡ †‹ ’‡”•‹•–‡œƒ –ƒ–‘ ’‹î ’”‘— ‹ƒ–‡ “—ƒ–‘ ’‹î ƒŽ–‘ ° ‹Ž •—‘ ‘”†‹‡Ǥ —‡•–ǯ—Ž–‹‘ ’—Ö ƒ Š‡ ‡••‡”‡ ‹ˆ‹‹–‘Ǣ ‹ “—‡•–‘ ƒ•‘ǡ –—––ƒ˜‹ƒǡ Žǯ‡•‹•–‡œƒ †‡‹ ‘‡–‹ •‡ ‘†‹ ȋ‡ “—‹†‹ Žƒ •–ƒœ‹‘ƒ”‹‡–Ȍ ° ‰ƒ”ƒ–‹–ƒ •‘Ž‘ ‡Ž ƒ•‘ ‹ —‹ •‹ q. 2 ˜‡”‹ˆ‹ ƒ ∑ θi < +∞ Ǥ i =0. ƒ–‘’‡”ƒ˜‡”‡—ǯ‹†‡ƒ’‹î ‘ ”‡–ƒǡ’”‡†‹ƒ‘—’”‘ ‡••‘ȋͳȌ†‹‡•‡’‹‘ ‡ ˆƒ ‹ƒ‘‡ — ‰”ƒˆ‹ ‘ǣ Žǯƒ†ƒ‡–‘ †‹ yt  ’‡” †‹˜‡”•‹ ˜ƒŽ‘”‹ †‹ “ ° ”ƒ’’”‡•‡–ƒ–‘‡ŽŽƒˆ‹‰—”ƒͳǤͳǤƒ–—”ƒŽ‡–‡ǡ“—ƒ†‘ θ =0 ‹Ž’”‘ ‡••‘°—™Š‹–‡ ‘‹•‡Ǥ‘‡•‹˜‡†‡ǡƒŽ ”‡• ‡”‡†‹ θ Ž‡ ƒ”ƒ––‡”‹•–‹ Š‡†‹’‡”•‹•–‡œƒ†‹˜‡‰‘‘ ’‹î ˜‹•‹„‹Ž‹ ȋŽƒ •‡”‹‡ –‡’‘”ƒŽ‡ •‹ Dz•—••ƒdzȌ ‡ Žƒ •—ƒ ˜ƒ”‹ƒœƒ ȋ‹•—”ƒ–ƒ ƒ’’”‘••‹ƒ–‹˜ƒ‡–‡†ƒŽŽǯ‘”†‹‡†‹‰”ƒ†‡œœƒ†‡ŽŽ‡‘”†‹ƒ–‡Ȍƒ—‡–ƒǤ  ‹‰—”ƒͳǤͳǣ’”‘ ‡••‘ȋͳȌǦ θ =0 ȋ™Š‹–‡‘‹•‡Ȍ.  .  ϭϲ.

(90) ‡ƒ˜‡••‹‘•‹—Žƒ–‘—’”‘ ‡••‘†‹‘”†‹‡•—’‡”‹‘”‡ǡŽƒ ‘•ƒ•ƒ”‡„„‡•–ƒ–ƒ ƒ ‘” ’‹î ‡˜‹†‡–‡Ǥ ‘•‹†‡”ƒ†‘ ’‹î ƒ ˆ‘†‘ — ’”‘ ‡••‘ ȋͳȌǡ •‹ ’‘••‘‘ ˆƒ”‡ ‘•‹†‡”ƒœ‹‘‹‹–‡”‡••ƒ–‹Ǥ†‡•‡’‹‘ǡŽǯƒ—–‘ ‘””‡Žƒœ‹‘‡†‹‘”†‹‡ͳ†‹— ’”‘ ‡••‘ȋͳȌ°†ƒ–ƒ†ƒŽŽƒˆ‘”—Žƒ  θ Ǥ 1+θ 2. ρ1 =. —‡•–ƒ”‡Žƒœ‹‘‡°”ƒ’’”‡•‡–ƒ–ƒ‰”ƒˆ‹ ƒ‡–‡‡ŽŽƒˆ‹‰—”ƒͳǤʹǢ•‹’—֐‘–ƒ”‡ Š‡ ‹Ž˜ƒŽ‘”‡ƒ••‹‘ Š‡”ƒ‰‰‹—‰‡ ρ1 °ͲǤͷǡ‹ ‘””‹•’‘†‡œƒ†‹ θ =1 Ǥ†‹• ‘”•‘ ƒƒŽ‘‰‘ǡ ‘‹•‡‰‹ ƒ„‹ƒ–‹ǡ˜ƒŽ‡’‡”‹Ž’—–‘†‹‹‹‘Ǥ

(91) ‘Ž–”‡ǡ•ƒ’’‹ƒ‘†ƒŽŽ‡ ‘•‹†‡”ƒœ‹‘‹ ˆƒ––‡ ‹ ’”‡ ‡†‡œƒ Š‡ –—––‡ Ž‡ ƒ—–‘ ‘””‡Žƒœ‹‘‹ †‹ ‘”†‹‡ ƒ‰‰‹‘”‡†‹ͳ•‘‘—ŽŽ‡Ǥ‘‹ƒ‘ ‹‘”ƒ—’”‘„Ž‡ƒ‹ˆ‡”‡œ‹ƒŽ‡ǣ•‡˜‘Ž‡••‹‘ ”ƒ’’”‡•‡–ƒ”‡ —ƒ ‡”–ƒ •‡”‹‡ –‡’‘”ƒŽ‡ ‘‡ ”‡ƒŽ‹œœƒœ‹‘‡ †‹ — ’”‘ ‡••‘ ȋͳȌǡ ‘‡ ’‘–”‡‘ —–‹Ž‹œœƒ”‡ Ž‡ •–ƒ–‹•–‹ Š‡ ƒŽ ‘Žƒ„‹Ž‹ •—ŽŽƒ •‡”‹‡ ’‡” ”‹ ƒ˜ƒ”‡ †‡ŽŽ‡ •–‹‡ †‡‹ ’ƒ”ƒ‡–”‹ †‡Ž ’”‘ ‡••‘ ȋ‘˜˜‡”‘ǡ ‹Ž ’ƒ”ƒ‡–”‘ θ Ȍǫ ƒ–—”ƒŽ‡–‡ǡ “—‡•–‘ ’”‘ ‡†‹‡–‘ •ƒ”‡„„‡ •‘•–‡‹„‹Ž‡ •‘Ž‘ ‡Ž ƒ•‘ ‹ —‹ Žƒ ‘•–”ƒ •‡”‹‡ –‡’‘”ƒŽ‡ ƒ˜‡••‡ — ‘””‡Ž‘‰”ƒƒ ȋ‹Ž ‰”ƒˆ‹ ‘ †‡ŽŽƒ •–‹ƒ †‡ŽŽƒ ˆ—œ‹‘‡ †‹ ƒ—–‘ ‘””‡Žƒœ‹‘‡Ȍ ‘ ˜ƒŽ‘”‹ ‘†‡”ƒ–‹ ’‡” Žǯƒ—–‘ ‘””‡Žƒœ‹‘‡ †‹ ’”‹‘‘”†‹‡‡–”ƒ• —”ƒ„‹Ž‹’‡”Ž‡ƒŽ–”‡Ǥ‡ ‘•¿ˆ‘••‡ǡ•‹’‘–”‡„„‡ƒ Š‡’‡•ƒ”‡ǣ •‡ ‹Ž ’”‘ ‡••‘ Š‡ Šƒ ‰‡‡”ƒ–‘ ‹ †ƒ–‹ ° ‡ˆˆ‡––‹˜ƒ‡–‡ — ȋͳȌǡ ƒŽŽ‘”ƒ ° •–ƒœ‹‘ƒ”‹‘ ‡† ‡”‰‘†‹ ‘ǡ ’‡” —‹ Žǯƒ—–‘ ‘””‡Žƒœ‹‘‡ ƒ’‹‘ƒ”‹ƒ ‘˜‡”‰‡ ‹ ’”‘„ƒ„‹Ž‹–ƒ“—‡ŽŽƒ–‡‘”‹ ƒǤ

(92) ˆ‘”—Ž‡ǣ ρˆ1. p  →. θ 1+θ 2. .  ’‘‹ Š± “—‡•–ƒ ° —ƒ ˆ—œ‹‘‡ ‘–‹—ƒ †‹ θ ǡ ’‘••‘ ‹˜‡”–‹”Žƒ ‡ –”‘˜ƒ”‡ —‘ •–‹ƒ–‘”‡ ‘•‹•–‡–‡ †‹ θ  ‘Ž ‡–‘†‘ †‡‹ ‘‡–‹ǡ ‘••‹ƒ –”‘˜ƒ”‡ “—‡Ž ˜ƒŽ‘”‡ Š‡•‘††‹•ˆƒŽǯ‡“—ƒœ‹‘‡ .  ϭϳ.

(93) ‹‰—”ƒͳǤʹǣȋͳȌǦ—–‘ ‘””‡Žƒœ‹‘‡†‹’”‹‘‘”†‹‡‹ˆ—œ‹‘‡†‹ θ .   θˆ  1+θˆ2. ρˆ1 =. ‡•‹˜‡†‡ˆƒ ‹Ž‡–‡ Š‡Žƒ•‘Ž—œ‹‘‡°ǣ  θˆ =. (. ). 1 1- 1-4 ρˆ12 Ǥ 2 ρˆ1. 

(94) ‡ˆˆ‡––‹†‹˜ƒŽ‘”‹ ‡‡•ƒ”‡„„‡”‘†—‡ǡƒ•‹°• ‡Ž–‘’‡”Žƒ•‘Ž—œ‹‘‡ ‘’”‡•ƒ ‡ŽŽǯ‹–‡”˜ƒŽŽ‘ ȏǦͳǡͳȐ ’‡” Š± ’”‘’”‹‘ ‹ “—‡•–‘ •’ƒœ‹‘ ”‹–”‘˜‹ƒ‘ –—––‹ ‹ ˜ƒŽ‘”‹ ’‘••‹„‹Ž‹ǡ ‘‡„‡˜‹•‹„‹Ž‡‡Ž‰”ƒˆ‹ ‘’”‡ ‡†‡–‡Ǥ ”ƒǡ‘‹•ƒ’’‹ƒ‘ Š‡ Ž‡ ‘•‡ ‘•–ƒ‘ •‡’”‡ ‘•¿Ǥ1˜‡”‘ Š‡ —’”‘ ‡••‘  †‹ ‘”†‹‡ •—’‡”‹‘”‡ Šƒ ƒ—–‘ ‘˜ƒ”‹ƒœ‡ ’‹î ƒ”–‹ ‘Žƒ–‡ǡ ‡ “—‹†‹ •‹ ’—Ö ‹ƒ‰‹ƒ”‡ Š‡ Žƒ •–‡••ƒ •–”ƒ–‡‰‹ƒ ’‘–”‡„„‡ ‡••‡”‡ ’‡” ‘””‹„‹Ž‡ǡ ƒŽ‡‘ ‹ –‡‘”‹ƒǡƒ ‘†‹œ‹‘‡†‹•’‡ ‹ˆ‹ ƒ”‡—‘”†‹‡†‡Ž’‘Ž‹‘‹‘ȋȌƒ„„ƒ•–ƒœƒƒŽ–‘Ǥ ƒ ‡†‘ — ’ƒ••‘ ’‹î ‹ Žǡ ‹ •‹ ’‘–”‡„„‡ Š‹‡†‡”‡ •‡ Žƒ •—’’‘•‹œ‹‘‡ ˜ƒŽ‡ ’‡”.  ϭϴ.

(95) “—ƒŽ—“—‡ •–”—––—”ƒ †‹ ƒ—–‘ ‘˜ƒ”‹ƒœ‡Ǥ ƒ ”‹•’‘•–ƒ ° ‡Ž –‡‘”‡ƒ †‹ ‘Ž†ǡ Š‡ ‡••‡œ‹ƒŽ‡–‡ †‹ ‡ǣ “—ƒŽ—“—‡ ’”‘ ‡••‘ •–‘ ƒ•–‹ ‘ǡ ’—” Š± •–ƒœ‹‘ƒ”‹‘ǡ ’‘••‹‡†‡ —ƒ •–”—––—”ƒ †‹ ƒ—–‘ ‘˜ƒ”‹ƒœ‡ Š‡ ° ”‡’Ž‹ ƒ„‹Ž‡ ‘ —ƒ ’”‘ ‡••‘ ƒ ‡†‹ƒ‘„‹Ž‡†‹‘”†‹‡‹ˆ‹‹–‘Ǥ—‡•–‘”‹•—Ž–ƒ–‘°†‹‹’‘”–ƒœƒ‡‘”‡ǣ‡••‘ ‹ †‹ ‡ǡ ‹ •‘•–ƒœƒǡ Š‡ “—ƒŽ—“—‡ •‹ƒ Žƒ ˆ‘”ƒ Dz˜‡”ƒdz †‹ — ’”‘ ‡••‘ •–‘ ƒ•–‹ ‘ •–ƒœ‹‘ƒ”‹‘ǡ’‘••‹ƒ‘•‡’”‡”ƒ’’”‡•‡–ƒ”Ž‘ ‘‡—’”‘ ‡••‘ȋƒŽŽ‹‹–‡†‹ ‘”†‹‡ ‹ˆ‹‹–‘ȌǤ 1 ’‡” “—‡•–‘ Š‡ǡ –”ƒ––ƒ†‘ ‹ ’”‘ ‡••‹ ǡ •‹ •–ƒ †‹ ˆƒ––‘ •–—†‹ƒ†‘–—––‹‹’”‘ ‡••‹•–ƒœ‹‘ƒ”‹’‘••‹„‹Ž‹ǡ’‡”Ž‘‡‘’‡”“—ƒ–‘”‹‰—ƒ”†ƒŽ‡ Ž‘”‘ ƒ”ƒ––‡”‹•–‹ Š‡†‹‡†‹ƒ‡†‹ ‘˜ƒ”‹ƒœƒǤ                .  ϭϵ.

(96) ͳǤͶ‘†‡ŽŽ‹—–‘”‡‰”‡••‹˜‹ȋȌ. ǯƒŽ–”ƒ‹’‘”–ƒ–‡ Žƒ••‡†‹’”‘ ‡••‹°†ƒ–ƒ†ƒ‹’”‘ ‡••‹—–‘”‡‰”‡••‹˜‹Ǥ—‡•–‹ ’”‘ ‡••‹ˆ‘”‹• ‘‘ǡ‹— ‡”–‘•‡•‘ǡ—ƒ”ƒ’’”‡•‡–ƒœ‹‘‡’‹î‹–—‹–‹˜ƒ†‹—ƒ •‡”‹‡’‡”•‹•–‡–‡†‹“—‡ŽŽƒ†‡‹’”‘ ‡••‹ǡ’‘‹ Š±Žǯ‹†‡ƒ° Š‡‹ŽŽ‹˜‡ŽŽ‘†‡ŽŽƒ•‡”‹‡ –‡’‘”ƒŽ‡ ƒŽ –‡’‘ – •‹ƒ —ƒ ˆ—œ‹‘‡ Ž‹‡ƒ”‡ †‡‹ ’”‘’”‹ ˜ƒŽ‘”‹ ’ƒ••ƒ–‹ǡ ’‹î — ™Š‹–‡‘‹•‡Ǥ

(97) Ž‘‡†‡”‹˜ƒƒ’’—–‘†ƒŽˆƒ––‘ Š‡—‘†‡ŽŽ‘•‘‹‰Ž‹ƒ‘Ž–‘ ƒ†— ‘†‡ŽŽ‘†‹ ”‡‰”‡••‹‘‡‹ —‹Ž‡˜ƒ”‹ƒ„‹Ž‹ ‡•’Ž‹ ƒ–‹˜‡ •‘‘‹ ˜ƒŽ‘”‹ ’ƒ••ƒ–‹ †‡ŽŽƒ˜ƒ”‹ƒ„‹Ž‡†‹’‡†‡–‡Ǥ  y = ϕ1y1 + ... + ϕ p y p + ε t  t.  ‘ ° ‘œ‹‘•‘ ‘–ƒ”‡ Š‡ǡ ‹ “—‡•–‘ ‘–‡•–‘ǡ ‹Ž ™Š‹–‡ ‘‹•‡ ’—Ö ‡••‡”‡ ‹–‡”’”‡–ƒ–‘‹‘†‘ƒƒŽ‘‰‘ƒŽ†‹•–—”„‘†‹—‘†‡ŽŽ‘†‹”‡‰”‡••‹‘‡ǡ ‹‘° ‘‡ Žƒ†‹ˆˆ‡”‡œƒˆ”ƒ›–‡Žƒ•—ƒ‡†‹ƒ ‘†‹œ‹‘ƒŽ‡Ǣ‹“—‡•–‘ ƒ•‘ǡŽ‡˜ƒ”‹ƒ„‹Ž‹ ƒ•—ƒŽ‹ Š‡ ‘•–‹–—‹• ‘‘Žǯ‹•‹‡‡†‹ ‘†‹œ‹‘ƒ‡–‘•‘‘•‡’Ž‹ ‡‡–‡‹Ž’ƒ••ƒ–‘†‹ ›–Ǥ

(98)  ’”‘ ‡••‹  •‘‘ ‹ — ‡”–‘ •‡•‘ •’‡ —Žƒ”‹ ƒ‹ ’”‘ ‡••‹  ’‡” Š±ǡ •‡ — ’”‘ ‡••‘  ° — ’”‘ ‡••‘ †‡ˆ‹‹–‘ †ƒŽŽǯƒ’’Ž‹ ƒœ‹‘‡ †‹ — ’‘Ž‹‘‹‘ ‡ŽŽǯ‘’‡”ƒ–‘”‡ƒ†—™Š‹–‡‘‹•‡ǡ—’”‘ ‡••‘°†‡ˆ‹‹–‘ ‘‡—’”‘ ‡••‘ Žǯƒ’’Ž‹ ƒœ‹‘‡ƒŽ“—ƒŽ‡†‹—’‘Ž‹‘‹‘‡ŽŽǯ‘’‡”ƒ–‘”‡’”‘†— ‡—™Š‹–‡‘‹•‡Ǥ

(99) •‹„‘Ž‹ A(B )yt = ε t .  †‘˜‡ȋȌ°‹Ž•‘Ž‹–‘’‘Ž‹‘‹‘‹„†‹‰”ƒ†‘’ ‘ȋͲȌαͳǤ ‡” ˆƒ‹Ž‹ƒ”‹œœƒ”‡ ‘ “—‡•–‘ –‹’‘ †‹ ’”‘ ‡••‹ǡ ‹‹œ‹ƒ‘ ‘Ž ‘•‹†‡”ƒ”‡ ‹Ž ƒ•‘ ’‹î•‡’Ž‹ ‡ǣ“—‡ŽŽ‘‹ —‹’αͳ‡‹Ž’”‘ ‡••‘’—Ö‡••‡”‡• ”‹––‘  yt = ϕyt-1 + εt.  →. (1 − Bϕ ) yt = εt . .  ϮϬ.

(100)

(101)  ‘‡–‹ †‹ — ’”‘ ‡••‘ ȋͳȌ ’‘••‘‘ ‡••‡”‡ ”‹ ƒ˜ƒ–‹ ‹ †‹˜‡”•‹ ‘†‹ǣ —‘ ’‹—––‘•–‘ ‹–—‹–‹˜‘ ° “—‡ŽŽ‘ †‹ •—’’‘””‡ Žƒ •–ƒœ‹‘ƒ”‹‡– †‡Ž ’”‘ ‡••‘ǡ ‡ ’‘‹ †‡”‹˜ƒ”‡ Ž‡ ‘•‡‰—‡œ‡ †ƒ “—‡•–ƒ ‹’‘–‡•‹Ǥ —’’‘‹ƒ‘ “—‹†‹ Š‡ ‹Ž ’”‘ ‡••‘ ƒ„„‹ƒ‡†‹ƒ ‘•–ƒ–‡ɊǤ—‡•–ƒ‹’‘–‡•‹‹’Ž‹ ƒ Š‡ǣ  µ = E(yt ) = ϕ E (yt-1 ) + E( ε t ) = ϕµ .  ǯ‡•’”‡••‹‘‡ ’”‡ ‡†‡–‡ ’—Ö ‡••‡”‡ ˜‡”ƒ ‹ †—‡ ƒ•‹ǣ ‘ Ɋ α Ͳǡ ‡Ž “—ƒŽ ƒ•‘ ° ˜‡”ƒ’‡”“—ƒŽ•‹ƒ•‹˜ƒŽ‘”‡†‹ɔǡ‘’’—”‡‡Ž ƒ•‘ɔαͳǡ‡ƒŽŽ‘”ƒŽǯ‡•’”‡••‹‘‡°˜‡”ƒ ’‡” “—ƒŽ•‹ƒ•‹ ˜ƒŽ‘”‡ †‹ Ɋǡ ‡ Žƒ ‡†‹ƒ †‡Ž ’”‘ ‡••‘ ° ‹†‡–‡”‹ƒ–ƒǤ

(102)  “—‡•–‘ •‡ ‘†‘ ƒ•‘•‹†‹ ‡ Š‡‹Ž’”‘ ‡••‘’”‡•‡–ƒ—ƒ”ƒ†‹ ‡—‹–ƒ”‹ƒǡ’‡” Š±‹Ž˜ƒŽ‘”‡ †‹ œ ’‡” —‹ ȋœȌαͲ ° ƒ’’—–‘ ͳǤ  ƒŽ–”‘ ‘†‘ †‹ †‡”‹˜ƒ”‡ ȋ›–Ȍ ° “—‡ŽŽ‘ †‹ ”ƒ’’”‡•‡–ƒ”‡ ›– ‘‡ — ’”‘ ‡••‘ ƒ ‡†‹ƒ ‘„‹Ž‡Ǥ ‡” ˆƒ”Ž‘ǡ —–‹Ž‹œœ‹ƒ‘ ‹ ”‹•—Ž–ƒ–‹ ”‹’‘”–ƒ–‹ ’”‡ ‡†‡–‡‡–‡ •—ŽŽƒ ƒ‹’‘Žƒœ‹‘‡ †‡‹ ’‘Ž‹‘‹Ǥ ‡ ‹ Ž‹‹–‹ƒ‘ ƒ‹ ƒ•‹ ‹ —‹ ȁɔȁ δ ͳ ȋ ‘†‹œ‹‘‡ Š‡ Š‹ƒ”ƒ‡–‡ ‡• Ž—†‡ Žƒ ”ƒ†‹ ‡ —‹–ƒ”‹ƒȌǡ•‹ƒ˜” Š‡  A(B) -1 = (1- Bϕ ) = 1+ Bϕ + B 2 ϕ 2 + ...= C(B) -1. .  ‡“—‹†‹Žƒ”ƒ’’”‡•‡–ƒœ‹‘‡†‹›–•ƒ”  yt = (1+ Bϕ + B 2 ϕ 2 + ...) ⋅ ε t = C(B) ⋅ ε t .  ‹‘° — ’”‘ ‡••‘  ‘ Ʌ‹αɔ‹ǡ Š‡ Šƒ ‡†‹ƒ œ‡”‘Ǥ ‡” “—ƒ–‘ ”‹‰—ƒ”†ƒ ‹ ‘‡–‹•‡ ‘†‹ǡ’”‘ ‡†‹ƒ‘ ‘‡’”‹ƒǢ•—’’‘‹ƒ‘ Š‡‹Ž™Š‹–‡‘‹•‡ƒ„„‹ƒ ˜ƒ”‹ƒœƒ’ƒ”‹ƒɐʹǤ‡‹†‹ Š‹ƒ‘ ‘Žƒ˜ƒ”‹ƒœƒ†‹›–ǡ‡•—’’‘‹ƒ‘ Š‡‡••ƒ ‡•‹•–ƒ‡•‹ƒ ‘•–ƒ–‡‡Ž–‡’‘ǡƒ˜”‡‘ Š‡ .  Ϯϭ.

(103) 2 V = E(yt2 ) = E (ϕ yt-1 + εt )  = ϕ 2V + σ 2 + 2ϕ E(yt-1εt )   .  ǯ—Ž–‹‘ ‡Ž‡‡–‘†‡ŽŽƒ•‘ƒ ° Ͳǡ “—‹†‹”‹•—Ž–ƒ—ƒ ‘„‹ƒœ‹‘‡Ž‹‡ƒ”‡ †‹ ƒ—–‘ ‘˜ƒ”‹ƒœ‡†‹—™Š‹–‡‘‹•‡ȋ–—––‡—ŽŽ‡’‡”†‡ˆ‹‹œ‹‘‡ȌǤ ‡‡†‡†— ‡ Š‡ V=. σ2 1− ϕ2. . 

(104)  ’”‹‘ Ž—‘‰‘ǡ Žǯ‡•’”‡••‹‘‡ ‹ †‹ ‡ Š‡ •‘Ž‘ •‡ ȁɔȁ δ ͳ Šƒ •‡•‘ ’ƒ”Žƒ”‡ †‹ ˜ƒ”‹ƒœƒ •–ƒ„‹Ž‡ ‡Ž –‡’‘ǡ ƒŽ–”‹‡–‹ ‘ •ƒ”‡„„‡ ’‹î ˜ƒŽ‹†ƒ Žǯ—Ž–‹ƒ —‰—ƒ‰Ž‹ƒœƒǤ —‡•–ƒ ‘†‹œ‹‘‡ ‡• Ž—†‡ †ƒŽŽƒ ƒ–‡‰‘”‹ƒ †‡‹ ’”‘ ‡••‹ ȋͳȌ •–ƒœ‹‘ƒ”‹‘ •‘Ž‘“—‡ŽŽ‹ƒ ”ƒ†‹ ‡—‹–ƒ”‹ƒǡƒƒ Š‡“—‡ŽŽ‹ƒ”ƒ†‹ ‡ ‘•‹††‡––ƒ ‡•’Ž‘•‹˜ƒ ‹‘° ‘ȁɔȁεͳǤƒ•‡ ‘†ƒ ‘•‹†‡”ƒœ‹‘‡ƒ• ‡†ƒŽ ‘ˆ”‘–‘†‹ǡ Š‡°Žƒ˜ƒ”‹ƒœƒ ‘ ‘†‹œ‹‘ƒŽ‡†‹›–ǡ ‘ɐʹǡ Š‡°Žƒ˜ƒ”‹ƒœƒ†‹. › ℑ– Ǧͳ Ǥ° –. •‡’”‡ƒ‰‰‹‘”‡†‹ɐʹǡ‡Žƒ†‹ˆˆ‡”‡œƒ°–ƒ–‘ƒ‰‰‹‘”‡“—ƒ–‘’‹îɔ°˜‹ ‹‘ƒ ͳǣ –ƒ–‘ ’‹î ’‡”•‹•–‡–‡ ° ‹Ž ’”‘ ‡••‘ǡ –ƒ–‘ ’‹î Žƒ •—ƒ ˜ƒ”‹ƒœƒ ‘†‹œ‹‘ƒŽ‡ ƒŽ ’”‘’”‹‘’ƒ••ƒ–‘•ƒ”‹‘”‡†‡ŽŽƒ•—ƒ˜ƒ”‹ƒœƒ‘ ‘†‹œ‹‘ƒŽ‡ǤƒŽ‡ƒ†‹”‡ Š‡ Žƒ ‘‘• ‡œƒ †‡Ž ˜ƒŽ‘”‡ †‹ ›–Ǧͷ ”‹†— ‡ Žǯ‹ ‡”–‡œœƒ •—Ž ˜ƒŽ‘”‡ †‹ ›– “—ƒ–‘ ’‹î ’‡”•‹•–‡–‡ ° Žƒ •‡”‹‡Ǥ ‹ƒ‰‘‘ †ƒ ˜‡”‹ˆ‹ ƒ”‡ Ž‡ ƒ—–‘ ‘˜ƒ”‹ƒœ‡ǣ Žǯƒ—–‘ ‘˜ƒ”‹ƒœƒ†‹‘”†‹‡Ͳ°ǡ Š‡ ‘‘• ‹ƒ‘‰‹ǢŽǯƒ—–‘ ‘˜ƒ”‹ƒœƒ†‹‘”†‹‡ ͳ°†ƒ–ƒ†ƒ  γ k = E(yt yt-k ) = E[( ϕ yt-1 + εt )yt-k ] = ϕγ k-1 .  ‡•‹†‡†— ‡ Š‡ γ k = ϕk. σ2 1−ϕ 2. . .  ϮϮ.

(105) ‡ƒ—–‘ ‘””‡Žƒœ‹‘‹ƒ••—‘‘‹“—‡•–‘ ƒ•‘—ƒˆ‘”ƒ‘Ž–‘•‡’Ž‹ ‡ǣ  ρk = ϕ k .   Š‡‹“—‡•–‘ ƒ•‘ °’‘••‹„‹Ž‡†ƒ”‡—ǯ‹–‡”’”‡–ƒœ‹‘‡‹–—‹–‹˜ƒ†‡Ž”‹•—Ž–ƒ–‘ǣ Ž‡ ƒ—–‘ ‘””‡Žƒœ‹‘‹ǡ Š‡ •‘‘ — ‹†‹ ‡ †‡ŽŽƒ ‡‘”‹ƒ †‡Ž ’”‘ ‡••‘ǡ •‘‘ –ƒ–‘ ’‹î‰”ƒ†‹ȋ‹˜ƒŽ‘”‡ƒ••‘Ž—–‘Ȍǡ–ƒ–‘’‹î‰”ƒ†‡ȋ•‡’”‡‹˜ƒŽ‘”‡ƒ••‘Ž—–‘Ȍ°ɔǡ ‘ˆ‡”ƒ†‘ Žǯ‹–‡”’”‡–ƒœ‹‘‡ †‹ ɔ ‘‡ ’ƒ”ƒ‡–”‘ †‹ ’‡”•‹•–‡œƒǤ

(106)  ’‹îǡ •‡„„‡‡ ‹Ž Ž‹‹–‡ ’‡”  Š‡ ˜ƒ ƒŽŽǯ‹ˆ‹‹–‘ ’‘•‹–‹˜‘ ”‹•—Ž–ƒ —‰—ƒŽ‡ ƒ œ‡”‘ǡ ɀ ° •‡’”‡†‹˜‡”•‘†ƒͲǤŽ–”‹ˆƒ––‹•‘‘ Š‡—’”‘ ‡••‘ȋ’Ȍǣ ͳǤ Šƒ‡‘”‹ƒ‹ˆ‹‹–ƒǡƒŽ‡ƒ—–‘ ‘””‡Žƒœ‹‘‹†‡ ”‡• ‘‘ƒŽ ”‡• ‡”‡†‹‹ ’”‘‰”‡••‹‘‡‰‡‘‡–”‹ ƒǢ ʹǤ ‡Ž ƒ•‘†‹Dz‹–‡” ‡––ƒdz†‹˜‡”•ƒ†ƒͲǡŠƒ˜ƒŽ‘”‡ƒ––‡•‘ɊȀȋͳȌǡ†‘˜‡ȋͳȌ° ƒ’’—–‘‹Ž’‘Ž‹‘‹‘ȋœȌ˜ƒŽ—–ƒ–‘‹œαͳƒœ‹ Š±‹œα ‘‡ƒŽ•‘Ž‹–‘Ǥ ǯ—‹ ‘ ƒ•’‡––‘ Š‡ ˜ƒŽ‡ Žƒ ’‡ƒ †‹ •‘––‘Ž‹‡ƒ”‡ †‡Ž ƒ•‘ ‹ —‹ Žǯ‘”†‹‡ †‡Ž ’”‘ ‡••‘ƒ—–‘”‡‰”‡••‹˜‘’•‹ƒƒ‰‰‹‘”‡†‹ͳ° Š‡’”‘ ‡••‹ȋ’Ȍ’‘••‘‘ƒ˜‡”‡ ƒ†ƒ‡–‹ ‹ Ž‹ ‹Ǥ

(107) “—‡•–‘ ƒ•‘ǡ‹Ž’”‘ ‡••‘ƒ••—‡—ƒ†ƒ‡–‘ ‹ Ž‹ ‘‹ —‹ Žǯƒ’‹‡œœƒ†‡ŽŽ‡‘• ‹ŽŽƒœ‹‘‹˜ƒ”‹ƒƒ––‘”‘ƒ†—˜ƒŽ‘”‡‡†‹‘Ǥ‘˜”‡„„‡‡••‡”‡ ‡˜‹†‡–‡ Š‡ ‹ ’”‘ ‡••‹ †‹ “—‡•–‘ –‹’‘ •‘‘ ‹ ƒ†‹†ƒ–‹ ƒ–—”ƒŽ‹ ƒ ‘†‡ŽŽƒ”‡ ˆ‡‘‡‹‡ ‘‘‹ ‹ ƒ”ƒ––‡”‹œœƒ–‹†ƒˆƒ•‹ ‹ Ž‹ Š‡Ǥ        .  Ϯϯ.

(108) ͳǤͷ‘†‡ŽŽ‹—–‘”‡‰”‡••‹˜‹‡†‹ƒ‘„‹Ž‡ȋȌ. ƒ Žƒ••‡†‡‹’”‘ ‡••‹ ‘’”‡†‡•‹ƒ‹’”‘ ‡••‹ Š‡‹’”‘ ‡••‹ ‘‡ ƒ•‘’ƒ”–‹ ‘Žƒ”‡Ǥ’”‘ ‡••‘ȋ’ǡ“Ȍ°‹ˆƒ––‹†‡ˆ‹‹–‘†ƒ  A(B)yt = C(B)εt. .  †‘˜‡’°Žǯ‘”†‹‡†‡Ž’‘Ž‹‘‹‘ȋȌ‡“°Žǯ‘”†‹‡†‡Ž’‘Ž‹‘‹‘ȋȌǤ–”ƒ„‹ •‘‘ —‡”‹ ˆ‹‹–‹Ǥ

(109)  ’”‘ ‡••‹  ‘  •‘‘ “—‹†‹ ƒ•‹ ’ƒ”–‹ ‘Žƒ”‹ ȋ“αͲ ‡ ’αͲ ”‹•’‡––‹˜ƒ‡–‡ȌǤ ‡ ‹Ž ’‘Ž‹‘‹‘ ȋȌ Šƒ –—––‡ Ž‡ •—‡ ”ƒ†‹ ‹ ƒ‰‰‹‘”‹ †‹ ͳ ‹ ‘†—Ž‘ǡƒŽŽ‘”ƒ›–’—Öƒ Š‡‡••‡”‡”ƒ’’”‡•‡–ƒ–‘‹ˆ‘”ƒǡ‘˜˜‡”‘ǣ  yt = A(B)-1C(B)εt = D(B) ⋅ εt .  †‘˜‡ȋȌ°—’‘Ž‹‘‹‘†‹‘”†‹‡‹ˆ‹‹–‘•‡’εͲǤƒŽ‡ ‘†‹œ‹‘‡•—ȋȌ° ‡ ‡••ƒ”‹ƒ‡•—ˆˆ‹ ‹‡–‡ƒˆˆ‹ Š±‹Ž’”‘ ‡••‘•‹ƒ•–ƒœ‹‘ƒ”‹‘ǤŽŽ‘•–‡••‘‘†‘ǡ•‡ ‹Ž ’‘Ž‹‘‹‘ ȋȌ ° ‹˜‡”–‹„‹Ž‡ǡ ƒŽŽ‘”ƒ ›– ƒ‡––‡ —ƒ ”ƒ’’”‡•‡–ƒœ‹‘‡ ƒ—–‘”‡‰”‡••‹˜ƒȋ†‹‘”†‹‡‹ˆ‹‹–‘•‡“εͲȌ  A(B)yt C(B)-1 = A(B) ⋅ yt = εt. . 

(110) “—‡•–‘ ƒ•‘ǡ•‹†‹ ‡ ƒ Š‡ Š‡‹Ž’”‘ ‡••‘°‹˜‡”–‹„‹Ž‡Ǥ ‡ ƒ”ƒ––‡”‹•–‹ Š‡†‡‹ ‘‡–‹ †‹ — ’”‘ ‡••‘ ȋ’ǡ“Ȍ ’‘••‘‘ ‡••‡”‡ ”‹ ƒ˜ƒ–‡ ‹ ‘†‘ ‘ ‡––—ƒŽ‡–‡ •‡’Ž‹ ‡ †ƒŽŽƒ •—ƒ ”ƒ’’”‡•‡–ƒœ‹‘‡ ‹ ‡†‹ƒ ‘„‹Ž‡Ǥ ǯ—‹ ƒ ƒ”ƒ––‡”‹•–‹ ƒ Š‡‹’ƒ”‡†‡‰ƒ†‹‡œ‹‘‡° Š‡•‡ƒ‰‰‹—‰‹ƒ‘—ǯ‹–‡” ‡––ƒǡ •‹†‹‘•–”ƒ Š‡Žƒ‡†‹ƒ†‡Ž’”‘ ‡••‘°ƒ ‘”ƒɊȀȋͳȌǤ Š‡ •‡•‘ Šƒ •–—†‹ƒ”‡ ’”‘ ‡••‹ ǫ

(111)  Ž‹‡ƒ –‡‘”‹ ƒǡ ‡••—ƒǡ ˜‹•–‘ Š‡ ‹Ž –‡‘”‡ƒ†‹”ƒ’’”‡•‡–ƒœ‹‘‡†‹‘Ž† ‹†‹ ‡ Š‡“—ƒŽ—“—‡’”‘ ‡••‘•–ƒœ‹‘ƒ”‹‘ ’—Ö‡••‡”‡”ƒ’’”‡•‡–ƒ–‘ ‘‡—’”‘ ‡••‘Ǥƒ—’—–‘†‹˜‹•–ƒ’”ƒ–‹ ‘ ǯ°.  Ϯϰ.

(112) ‹Ž’”‘„Ž‡ƒ Š‡Žƒ”ƒ’’”‡•‡–ƒœ‹‘‡†‹‘Ž†°ǡ‹‰‡‡”ƒŽ‡ǡ‹ˆ‹‹–ƒǤ—‡•–‘‘° —’”‘„Ž‡ƒƒŽ‹˜‡ŽŽ‘–‡‘”‹ ‘ǡƒŽ‘†‹˜‡–ƒ‡ŽŽƒ’”ƒ–‹ ƒǣŽƒ•‡”‹‡–‡’‘”ƒŽ‡ Š‡ ‘••‡”˜‹ƒ‘˜‹‡‡‹ˆƒ––‹’‡•ƒ–ƒ ‘‡”‡ƒŽ‹œœƒœ‹‘‡†‹—’”‘ ‡••‘•–‘ ƒ•–‹ ‘ǡ‹ —‹ ’ƒ”ƒ‡–”‹ •‘‘ ‹ ‘‡ˆˆ‹ ‹‡–‹ †‡‹ ’‘Ž‹‘‹ ‡ŽŽǯ‘’‡”ƒ–‘”‡  Š‡ ‡ †‡–‡”‹ƒ‘ Ž‡ ƒ”ƒ––‡”‹•–‹ Š‡ †‹ ’‡”•‹•–‡œƒ ȋ’‹î Žƒ ˜ƒ”‹ƒœƒ †‡Ž ™Š‹–‡ ‘‹•‡ȌǤ ‡ •‹ ‘•‹†‡”ƒ —ƒ •‡”‹‡ ‘••‡”˜ƒ–ƒ ‘‡ —ƒ ”‡ƒŽ‹œœƒœ‹‘‡ †‹ — “—ƒŽ Š‡ ’”‘ ‡••‘ •–ƒœ‹‘ƒ”‹‘ǡ —–‹Ž‹œœƒ”‡ — ’”‘ ‡••‘  ’‡” ”‹ƒ••—‡”‡ Ž‡ ƒ”ƒ––‡”‹•–‹ Š‡ †‹ ‡†‹ƒ ‡ ‘˜ƒ”‹ƒœƒ ‘’‘”–ƒ “—‹†‹ ‹Ž ’”‘„Ž‡ƒ ‹ˆ‡”‡œ‹ƒŽ‡ †‹•–‹ƒ”‡——‡”‘’‘–‡œ‹ƒŽ‡–‡‹ˆ‹‹–‘†‹’ƒ”ƒ‡–”‹Ǥ

(113) ˆƒ––‹ǡ•‡’‡•‹ƒ‘ Š‡›–•‹ƒ”ƒ’’”‡•‡–ƒ„‹Ž‡‹ˆ‘”ƒ ‘‡  yt = F(B)εt .  ‹‡–‡ ‹ ƒ••‹ —”ƒ Š‡ ‹Ž ’‘Ž‹‘‹‘ ȋȌ ‘ •‹ƒ †‹ ‘”†‹‡ ‹ˆ‹‹–‘Ǥ ‹ ’—Ö ’‡”Ö ’‡•ƒ”‡ †‹ —•ƒ”‡ —ǯƒ’’”‘••‹ƒœ‹‘‡ †‹ ȋȌǢ ‹ ’ƒ”–‹ ‘Žƒ”‡ǡ ’—Ö †ƒ”•‹ Š‡ •‹ ”‹‡• ƒ‘ƒ–”‘˜ƒ”‡†—‡’‘Ž‹‘‹†‹‘”†‹‡ˆ‹‹–‘ȋ‡’‘••‹„‹Ž‡–‡„ƒ••‘ȌȋȌ‡ ȋȌ–ƒŽ‹’‡” —‹ F(z) ≃. C(z) A(z) .  ‡ Žǯ—‰—ƒ‰Ž‹ƒœƒ ˆ‘••‡ ‡•ƒ––ƒǡ •‹ ’‘–”‡„„‡ ƒŽŽ‘”ƒ • ”‹˜‡”‡ A(B)yt = D(B) ε t ‡ •‡ “—‡•–ǯ—Ž–‹ƒ˜ƒŽ‡•‘Ž‘‹‘†‘ƒ’’”‘••‹ƒ–‘ǡƒŽŽ‘”ƒ•‹ƒ˜”ǣ  A(B)yt = D(B) ε t  *. †‘˜‡ *. εt =. A (B ) ⋅ F (B )ε t  D (B ). .  Ϯϱ.

(114)

(115) Ž’”‘ ‡••‘ ε t ‘ °ǡƒ”‹‰‘”‡ǡ—™Š‹–‡‘‹•‡ǡƒ•‡Ž‡•—‡ƒ—–‘ ‘˜ƒ”‹ƒœ‡‘ *. •‘‘–”‘’’‘‰”ƒ†‹ǡ’—Ö‡••‡”‡ ‘•‹†‡”ƒ–‘–ƒŽ‡ƒ–—––‹‹ˆ‹‹’”ƒ–‹ ‹Ǥ‹’‘–”‡„„‡ †‹”‡ Š‡ ‘•‹†‡”ƒ”‡ ε t —™Š‹–‡‘‹•‡ ‘•–‹–—‹• ‡—ƒ‡–ƒˆ‘”ƒ†‡‹†ƒ–‹ Š‡‘ *. °‘Ž–‘’‹îˆ—‘”˜‹ƒ–‡†‹“—‡ŽŽƒ„ƒ•ƒ–ƒ•—ŽŽƒ”ƒ’’”‡•‡–ƒœ‹‘‡†‹‘Ž†‡ Š‡Šƒ‹Ž ˜ƒ–ƒ‰‰‹‘ †‹ „ƒ•ƒ”•‹ •— — —‡”‘ ˆ‹‹–‘ †‹ ’ƒ”ƒ‡–”‹Ǥ

(116)  ’”ƒ–‹ ƒǡ — ‘†‡ŽŽ‘ ˜‹‡‡ ‘•–”—‹–‘ˆƒ ‡†‘—ǯ‹’‘–‡•‹ƒ’”‹‘”‹•—‹‰”ƒ†‹†‡‹†—‡’‘Ž‹‘‹ȋȌ ‡ ȋȌ ‡ ’‘‹ǡ —ƒ ˜‘Ž–ƒ •–‹ƒ–‹ ‹ ‘‡ˆˆ‹ ‹‡–‹ †‡‹ ’‘Ž‹‘‹ǡ ‡•ƒ‹ƒ†‘ Ž‡ ƒ—–‘ ‘””‡Žƒœ‹‘‹ ƒ’‹‘ƒ”‹‡†‡ŽŽƒ•‡”‹‡Ǥ‡“—‡•–‡‘•‘‘–”‘’’‘‰”ƒ†‹ǡ‘ ‹ •‘‘ ’”‘„Ž‡‹ †‹ •‘”–ƒ ƒ ‘•‹†‡”ƒ”‡ ε t  ‘‡ — ™Š‹–‡ ‘‹•‡Ǥ ǯ‡•‹‰‡œƒ †‹ *. –‡‡” „ƒ••‘ ‹Ž —‡”‘ †‡‹ ’ƒ”ƒ‡–”‹ †‡‹ ’‘Ž‹‘‹ ‘†— ‡ǡ ‹ ‡”–‹ ƒ•‹ǡ ƒ Žƒ˜‘”ƒ”‡ ‘†‡‹‘†‡ŽŽ‹‘–‹ ‘‡‘Ž–‹’Ž‹ ƒ–‹˜‹ǡ Š‡•‹—•ƒ‘•‘’”ƒ––—––‘ ’‡” •‡”‹‡ ƒ”ƒ––‡”‹œœƒ–‡ †ƒ ’‡”•‹•–‡œƒ •–ƒ‰‹‘ƒŽ‡ǡ ‡ Š‡ “—‹†‹ •‘‘ ƒ Š‡ ‘‘• ‹—–‹ ‘‡•–ƒ‰‹‘ƒŽ‹ǡ‘Ǥ                  Ϯϲ.

(117) ͳǤ͸–‹ƒ‡’”‡˜‹•‹‘‹†‡‹‘†‡ŽŽ‹. ‹‘ƒ†‘”ƒƒ„„‹ƒ‘ˆƒ––‘ˆ‹–ƒ Š‡‹Ž’”‘ ‡••‘•–‘ ƒ•–‹ ‘ Š‡•‘˜”ƒ’’‘‹ƒ‘ƒ‹ †ƒ–‹’‡”‹–‡”’”‡–ƒ”Ž‹ˆ‘••‡‰‘˜‡”ƒ–‘†ƒ’ƒ”ƒ‡–”‹‘–‹Ǥ‡“—‡•–‹—Ž–‹‹‘–‹‘ •‘‘ ȋ‡ ‹ ’”ƒ–‹ ƒ ‘ Ž‘ •‘‘ ƒ‹Ȍǡ •‹ ’‘••‘‘ —–‹Ž‹œœƒ”‡ †‡ŽŽ‡ Ž‘”‘ •–‹‡Ǥ ƒ –‡ ‹ ƒ †‹ „ƒ•‡ ’‡” Žƒ •–‹ƒ †‡‹ ’ƒ”ƒ‡–”‹ †‹ — ’”‘ ‡••‘  ° Žƒ ƒ••‹ƒ ˜‡”‘•‹‹‰Ž‹ƒœƒǤ ‹ •‘Ž‹–‘ •‹ ƒ••—‡ Š‡ ‹Ž ’”‘ ‡••‘ •‹ƒ ‘”ƒŽ‡ǡ ‘•‹ Š± Žƒ ˆ‘”ƒ†‡ŽŽƒˆ—œ‹‘‡†‹†‡•‹–†‡ŽŽ‡‘••‡”˜ƒœ‹‘‹°‘–ƒ‡–”ƒ––ƒ„‹Ž‡Ǥ—Ö‡••‡”‡ —–‹Ž‡”‹ Š‹ƒƒ”‡„”‡˜‡‡–‡ ‘•ƒ•‹‹–‡†‡’‡”ˆ—œ‹‘‡†‹˜‡”‘•‹‹‰Ž‹ƒœƒǤƒ ˜‡”‘•‹‹‰Ž‹ƒœƒ ° Žƒ ˆ—œ‹‘‡ †‹ †‡•‹– †‡Ž ƒ’‹‘‡ǡ ƒŽ ‘Žƒ–ƒ ‡Ž ’—–‘ ‘””‹•’‘†‡–‡ ƒŽ ƒ’‹‘‡ ‘••‡”˜ƒ–‘Ǥ ••ƒ †‹’‡†‡” †ƒ — ˜‡––‘”‡ › †‹ ’ƒ”ƒ‡–”‹ ‹ ‘‰‹–‹ǡ Š‡ ‡ †‡–‡”‹ƒ‘ Žƒ ˆ‘”ƒǤ ƒ••‹‹œœƒ†‘ “—‡•–ƒ ˆ—œ‹‘‡•‹‘––‹‡‡Žƒ•–‹ƒ†‹ƒ••‹ƒ˜‡”‘•‹‹‰Ž‹ƒœƒǤ—ƒ†‘‘••‡”˜‹ƒ‘—ƒ ”‡ƒŽ‹œœƒœ‹‘‡ †‹ — ’”‘ ‡••‘ •–‘ ƒ•–‹ ‘ǡ Žƒ ˆ—œ‹‘‡ †‹ ˜‡”‘•‹‹‰Ž‹ƒœƒ ‘ ° ƒŽ–”‘ Š‡Žƒˆ—œ‹‘‡†‹†‡•‹– ‘‰‹—–ƒ†‡ŽŽƒ’ƒ”–‡†‹’”‘ ‡••‘‘••‡”˜ƒ–ƒǡ‘••‹ƒ Žƒˆ—œ‹‘‡†‹†‡•‹–ƒ”‰‹ƒŽ‡†‡Ž˜‡––‘”‡ƒŽ‡ƒ–‘”‹‘ȋšͳǡǤǤǤǡšȌǡ ƒŽ ‘Žƒ–ƒ ‡‹ ˜ƒŽ‘”‹ ‘••‡”˜ƒ–‹Ǥ ‡ •—’’‘‹ƒ‘ Š‡ ‹Ž ’”‘ ‡••‘ •‹ƒ ‰ƒ—••‹ƒ‘ǡ Žƒ ˆ—œ‹‘‡ †‹ ˜‡”‘•‹‹‰Ž‹ƒœƒ‘° Š‡Žƒˆ—œ‹‘‡†‹†‡•‹–†‹—ƒ‘”ƒŽ‡—Ž–‹˜ƒ”‹ƒ–ƒǣ   1  L(ψ) = f(x; ψ) =  T  ∑  2π . -. 1 2. {. }. 1 exp - (x-k)' ∑-1 (x-k)  2.  †‘˜‡ š°‹Ž˜‡––‘”‡ȋšͳǡǤǤǤǡšȌ†‡ŽŽ‡ ‘••‡”˜ƒœ‹‘‹Ǣ ‡ ∑•‘‘ ‹•—‘‹‘‡–‹ ’”‹‹‡•‡ ‘†‹ǡ Š‡†‹’‡†‘‘†ƒ ψ Ǥ†‡•‡’‹‘ǡŽǯ‡Ž‡‡–‘‹Œ†‡ŽŽƒƒ–”‹ ‡ ∑ ‘ ° Š‡ Žǯƒ—–‘ ‘˜ƒ”‹ƒœƒ †‹ ‘”†‹‡ ȁ‹ Ǧ Œȁ Žƒ “—ƒŽ‡ǡ ‘‡ •ƒ’’‹ƒ‘ǡ ° —ƒ ˆ—œ‹‘‡ †‡‹ ’ƒ”ƒ‡–”‹ †‡Ž ’”‘ ‡••‘ Ǥ 1 ’‘••‹„‹Ž‡ †‹‘•–”ƒ”‡ Š‡ ‰Ž‹ •–‹ƒ–‘”‹ †‹ ƒ••‹ƒ ˜‡”‘•‹‹‰Ž‹ƒœƒ †‹ ’”‘ ‡••‹  ‰ƒ—••‹ƒ‹ •‘‘ ‘•‹•–‡–‹ǡƒ•‹–‘–‹ ƒ‡–‡‘”ƒŽ‹‡† ƒ•‹–‘–‹ ƒ‡–‡‡ˆˆ‹ ‹‡–‹Ǥ

(118) ‘Ž–”‡ǡ•‘––‘ ‘†‹œ‹‘‹ ’‹—––‘•–‘ „Žƒ†‡ǡ Ž‡ ’”‘’”‹‡– †‹ ‘•‹•–‡œƒ ‡ ‘”ƒŽ‹– ƒ•‹–‘–‹ ƒ ˜‡‰‘‘ ‘•‡”˜ƒ–‡ ƒ Š‡ “—ƒ†‘ Žƒ ˜‡”ƒ †‹•–”‹„—œ‹‘‡ †‡Ž ’”‘ ‡••‘ ‘ •‹ƒ.  Ϯϳ.

(119) ‘”ƒŽ‡ ȋ•‹ ’ƒ”Žƒ ‹ “—‡•–‘ ƒ•‘ †‹ •–‹‡ †‹ “—ƒ•‹ Ǧ ƒ••‹ƒ ˜‡”‘•‹‹‰Ž‹ƒœƒȌǤ

(120) ‘Ž–”‡ ˜ƒ †‡––‘ Š‡ǡ †‹ •‘Ž‹–‘ǡ ‘ •‹ Žƒ˜‘”ƒ •—ŽŽƒ ˆ—œ‹‘‡ ȋɗȌǡ ƒ •—Ž •—‘ Ž‘‰ƒ”‹–‘ƒ“—‡•–‘°‹””‹Ž‡˜ƒ–‡Ǥ

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