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An Evaluation of Efficiency of Some Estimators for the First Order Autoregressive Models

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A C T A U N I V E R S I T A T I S L 0 D Z I E N 8 I S ___________________ FO LIA OECUNOMICA 4 8 , 1 9 P S ____________________

A n d rz e j To m aszew iczu, M n jld Hamza A l- N a e fiir

AN EVALUATION OF E F F IC IE N C Y OF SOME ESTIMATORS FOR THE F IR S T ORDER AUTOREGRESSIVE MODELS

1. In t r o d u c t io n

A M o n te -C erlo s tu d y we9 done In resp o n se to c a l l s of C h a n d a [ 6 ] . H e n d r y , T r l v l d l [ 9 ] , A i g- n • r [ 2 ] . and D e n t , M i n [ f l ] to compare and e v a ­ lu a t e th e e f f i c i e n c i e s o f the proposed e s t im a t o r » o f unknown pa­ ra m e te rs u s in g :

1) o r d in a r y l e a s t s q u a re s (O L S ), 2 ) m o d ifie d l e a s t s q u a re s (MOO. L S ) ,

3 ) a p p ro x im a te maximum lik o lih o o d (A P R .M L ), 4 ) e x a c t maximum l i k e l i h o o d (EXACT M L ), f o r th e s t a t i o n a r y Markov m odel.

Mont» C a r lo e x p e rim e n ts w ere con d u cted f o r N - 100, 200 and 1000 sa m p le s. The r » » u l t s o re p re s e n te d In 10 T a b le ® and 10 F i ­ g u re s and th e c o rre s p o n d in g comment».

2. E s t i m a t i on P ro c e d u re «n u 11xi • i •ni i n O T m r r ,M"

The a u t o r e g r e s s iv e »cheme uaad In t h i s p a p e r in th e d l f t e , t im e - » t a t lo n a r y , f i r s t o r d e r a u t o r e g r e s s iv e p r o c e s s w it h * e r o mean, i . e .

* D ^ , L e c t u r » r a t th e I n e t i t u t e o f E c o n o m e tric » and s t a t i ­ s t i c s . U n i v e r s i t y o f t$d± .

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( 2 . 1 ) Z ( t ) - ( ^ j Z i t - l ) a ( t ),

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The th r e e r o o t * o f the p o ly n o m ia l g(<f>j) “ 0 ®aY btt ° c » t e by c o n s id e r in g (i ) q (c* > )» 0 0 , ( l l ) g(-oo) « -coi . ( i l l ) g ( 0 ) - - ♦ t^ ) ' T k ♦ K, ♦ K- d y ) g ( i ) --- “ - K ^ T ^ i T ^ “ " Y * [ Z ( t > ' - 2 ( t - l ) ] i /■I-' J • t-2 (T - 1 ) £ Z2 ( t - 1 ) t-3 < 0 , and

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4. S im u la t io n

M o n te - C a rlo e x p e rim e n ts were d e sig n e d to compare end e v a lu ­ a te the e f f i c i e n c i e s o f the e s t im a t o r s p roposed in s e c t io n 2. The e x p e rim e n ts were con d u cted (on 00RA-1304 com puter w ith the use o f 41 MAR 2 own program ), f o r N » 100 , 200 end 1000 sam­ p le s f o r the f o llo w in g v a lu e s o f sam ple s iz e

T - 15, 20, 25, 30, 50 and 100 and a u t o c o r r e la t io n c o e f f i c i e n t v a lu e s <f>^ • 0 . 0 , ¿ 0 . 5 and +0.9. P a ra m e te r c h o ic e s a re mado In on a tte m p t to p r o v id e r e p r e s e n t a ­ t io n th ro u g h o u t s t a t i o n a r i t y r e g io n s and th e v a r ia n c e o f n o ie e I s f ix e d a t u n i t y . Our f in d in g s o re r e p o rte d in 10 T a b le s (4 .1 - • 4 .1 0 ) and 16 F ig u r e s (4 . 1 - 4 . 1 6 ). 5. R e s u lt s and C o n c lu s io n s

G e n e r a ll y , we can make the f o llo w in g o b s e r v a t io n » from the* t a b le s and f i g u r e s .

a . W ith r e s p e c t to th e b ia s o f th e e s t im a t e :

( i ) o r d in a r y l e a s t s q u a re s e o tlm o to r c l e a r l y e x h i b it s le a s t a b s o lu te b ia s even f o r sam ples as om oll os 15 (e x c e p t when $ ^ ■ • 0 . 0 ) . /

( i i ) The m o d ifie d l e a s t s q u a re s e s t im a te based on Q u e n o u ll- l e ' s fo rm u la [1 4 ] w i l l im prove th e s i t u a t i o n s i g n i f i c a n t l y (e x ­ c e p t when • 0 .0 and 0 . 5 ) .

( i l l ) The a p p ro x im a te maximum l i k e l i h o o d and e x a c t maximum l i k e l i h o o d e s t im a t o r s e x h i b i t a p p ro x im a te ly the same b ia s e sp e ­ c i a l l y f o r in te r m e d ia t e and la r g e v a lu e s o f T.

( l v ) F o r <(>^ ■ 0 .0 and la r g e v a lu e s o f T ■ 100, the O LS, ap­ p ro x im a te and e x a c t maximum l i k e l i h o o d e s t im a t o r » e x h i b i t th e •ame b ia s .

( v ) The a b s o lu te b ia s f o r th e o r d in a r y l e a B t s q u a r e s , m o d if i­ ed l e a s t s q u a r e s , a p p ro x im a te and maximum l i k e l i h o o d e s t im a t o r «

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d e c re a s e s and In c r e a s e s w it h |'£ j| and d e c re a s e s as T in c r e a s e s ; o r we can say t h a t the e s t im a te has a b ia s w hich in c r e e s o s . l i n e a r l y w it h the p a ra m e te rs Ф^ end | ( c f . T a b le s 4 . 1 , 4 .3 , 4 . 5 , 4 , 7 , 4 .9 and F ig u r e s 4 . 1 , 4 . 2 , 4 . 3 , 4 . 7 , 4 . 8 , 4 . 9 , 4 .1 0 , 4 . 1 1 ) .

b . W ith r e s p e c t to th e mean sq u are e r r o r o f th e e s t im a te s ( i ) The mean sq u a re e r r o r s (M S E ) f o r the OLS e s t im a te s a re c o n s id e r a b ly o & ia lle r than th o se f o r th e m o d ifie d o n e s.

( i i ) On the b a s is o f USE the OLS e s t i n a t o r i s b e s t o n ly f o r <f>x • *ptQt w h ile f o r o t h e r v a lu e s o f the l i k e l i h o o d e s t i -

mote has s m a lle r MSE th an th e l e a s t s q u a re « .

( i l l ) F o r in te r m e d ia t e and la r g e v a lu e * o f T th e HSE f o r the a p p ro x im ate and e x a c t l i k e l i h o o d o s t l n e t e a a r e a p p ro x im a te ly th e в а я в .

( i v ) O r d in a r y , m o d ifie d l e a s t s q u a r e s , a p p ro x im a te end e x a c t maximum l i k e l i h o o d e s t im a t o r s , each has a HSE w h ic h l e s m e lle r f o r la r g e v a lu e s o f Ф^ and T , i . e . HSE 1» d e c r e a s in g а » | I end T a re in c r e a s i n g , ( c f . T a b le s 4 . 2 , 4 . 4 , 4 . 6 , 4 . 8 , 4 .1 0 and F ig u r e s 4 .4 , 4 . 5 , 4 .6 , 4 .1 2 , 4 .1 3 , 4 . H , 4 .1 5 , 4 . 1 6 ) .

c . The e x p e rim e n t i s re - c o n d u cte d f o r N • 100 and N • 200 r e p l i c a t i o n s . We n o t ic e t h a t new in fo r m a tio n wee e l i c i t e d , w it h no changes in r e l a t i v e p e rfo rm a n ce o f th e e s t im a t o r s on th e c r i ­ t e r i a exam ined, and so th e ra n k in g o f e e t lm e to r a n e v e r v e r i e d (c f . T a b le s 4 . 1 , 4 . 2 , 4 , 3 , 4 . 4 , 4 ,5 and 4 , 6 ) .

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-1

*1 OLS MOO. OLS APR. ML. EXACT ML.

1 2 3 4 5 6 - 0 .9 0.1151 0.0034 0 .1 71 2 0.2013 0.012 0 .0 1 4 0 .0 1 3 0 .0 1 3 - 0 .5 0 .0 18 6 0 .0 79 3 0.0530 0.0613 0 .0 1 4 0 .0 1 7 0 .0 1 7 0 .0 14 15 U » U 0.0131 0.0168 0.0122 0.0119 0 .0 1 5 0 .0 1 7 0.0 1 5 0.0 15 0 .0 3 5 7 0 .0 55 4 0.0689 0.0002 0 .0 1 4 0 .0 1 7 0 .0 1 4 0 .0 1 4 0.1229 0.0219 0 .1 78 4 0.1 96 7 0.012 0 .0 1 4 0 .0 1 3 0 .0 1 3 0.0 62 2 0.0 00 6 0.0911 0.0900 •U# v 0 .0 0 9 0.010 0.010 0.010 0.0250 0.0 24 8 0.0414 0.0 41 2 -0*5 0.012 0 .0 1 3 0 . 0 1 1 0.011 30 0.0 0 .0 0 6 6 0.0063 0 .0 06 4 0.0 06 4 0.012 0 .0 1 3 0 .0 1 3 0.012 0.0190 0.0356 0.0 35 6 0.0366 0 . 5 0.012 0 .0 1 3 0 . 0 1 1 0 . 0 1 1 0.0 72 0 0 .0 13 2 0.1 00 5 0.1034 w # w 0 .0 0 8 0.010 0.010 0.010 0 .0 2 3 5 0.0 02 3 0 .0 3 2 3 0.0320 - 0 .9 0 .0 0 6 0 .0 0 6 0 .0 0 7 *. ' 0 .0 0 6 T i b l t 4 .7 R e la t io n s h ip between th e a b s o lu te v a lu e o f the b ia s and v a lu e s o f (1000 r e p l i c a t i o n s )

(16)

T a b le 4 . 7 . (co ntfll.) 1 2 3 4 5 6 - 0 .5 0.0034 0 .0148 0.0084 0.0 08 4 0 .0 0 8 0 .0 09 0 .0 0 8 0 .0 0 8 100 n n 0 .0 02 7 0.0 02 2 0.0027 0.0 02 7 0 .0 09 0.0 0 9 0.0 0 9 0.0 0 9 0 .0 0 6 7 0 .0 1 0 7 0 .0 1 1 7 . 0.0 11 7 U * D 0 .0 0 8 0.0 0 9 0 .0 0 8 0 .0 0 8 Ci Q 0.0199 0.0 01 8 0.0288 0.0285 0 .0 0 6 0 .0 0 6 0 .0 0 6 0 .0 0 6 T a b l e 4. 8 R e la t io n s h ip between th e м е п s q u a re s e r r o r and th e v a lu e s o f ф^ <1000 r e p l i c a t i o n « ) T *1 MSE

OLS MOO. OLS APR. ML. EXACT ML.

15 - 0 .9 0.1542 0.1892 0.1862 0.2 06 8 ¿ • 1Л 0.2071 0.2855 0.1 94 5 0.1 93 4 0 .0 0.2261 ■ 0 .2 8 7 5 0.2 09 9 0 .2 0 9 2 0 .5 0.2070 0 .2 81 6 0.1961 0 .1 95 3 0 .9 0.1539 0 .1 8 7 2 0.1 87 5 0.2019 30 - 0 .9 0 .0 86 6 0 .1 00 5 0.0 98 6 0.0999 - 0 .5 0 .1 3 2 7 0.1579 0.1 29 7 0.1 29 9 0 .0 0 .1 50 7 0.1 73 8 0 .1 45 5 0.1 45 5 0 .5 0.1341 0 .1 64 5 0 .1 30 6 0 .1 30 6 0 .9 0.0967 0 .1 05 0 0.1091 0 .1 13 0 100 i - 0 .9 0.0410 0 .0 44 5 0.0 43 6 0 .0 4 3 7 - 0 .5 0 .0 69 6 0.0760 0.0690 0.0690 0 .0 0 .0 8 2 3 0 .0 8 6 4 0 .0 8 1 5 0 .0 8 1 5 0 .5 0 .0 70 9 0 .0 7 6 6 0 .0 70 4 0.0 70 4 0 .9 0.0381 0.0 40 8 0 .0 4 0 7 0.0408

(17)

R e la t io n s h ip b e t w e n th e v a lu o s o f th e a b s o lu te b io s and the v a lu e s o f T (10C0 r e p l i c a t i o n s )

1 B IA S t

*1 T OLS KOO. OLS APR. ML. ¡¿XivCT ML.

15 0.1151 0.0034 0.1712 0.2013 20 0.0 99 6 0.0156 0.1417 0.1488 —П Q 25 0 .0 7 7 9 0.0 1 0 3 0,1121 0 .1 1 5 2 • v * ? 30 0.0 6 2 2 0 .0 0 0 6 0.0911 0.0 90 0 50 0 .0 4 2 3 0.0 01 5 0.0 5 9 8 0.0 5 8 8 100 0 .0 2 3 5 0 .0 0 2 3 0 .0 3 2 3 0.0320 15 0.0 1 8 6 0 .0 7 9 3 0.0 53 0 0 .0 6 1 3 20 0.0 26 0 0 .0 4 8 4 0.0 51 0 0 .0 5 3 2 25 0 .0 1 8 3 0 .0 4 3 3 0 .0 3 0 4 0.0 3 8 5 - 0 .5 30 0 .0 2 5 0 0.0 24 8 0.0 41 4 0.0 41 2 50 0 .0 1 4 6 0 . 0 ie 4 0.0 2 4 5 0.0 2 4 4 100 0 .0 0 3 4 0 .0 1 4 8 0.0 0 8 4 0 .0 0 8 4 15 0.0131 0 .0 1 6 8 0.0 1 2 2 0.0119 20 0 .0 1 3 6 0.0 15 3 0.0 12 9 0.0 13 0 25 0 .0 0 0 3 0.0 04 4 0 .0 0 0 3 0 .0 0 0 3 0 .0 30 0 .0 0 6 6 0.0 0 6 3 0.0 0 6 4 0.0 0 6 4 50 0 .0 0 1 2 0 .0 0 2 5 0.0011 0.0 01 1 100 0 .0 0 2 7 0 .0 0 2 2 0 .0 0 2 7 0.0 0 2 7 15 0 .0 3 5 7 0 .0 5 5 4 0.0689 0.0 80 2 20 0 .0 1 5 2 0 .0 5 7 7 0 .0 4 0 8 0.0 45 2 25 0.0 31 8 0.0 26 2 0 .0 5 1 3 0.0 51 2 0 . 5 30 0 .0 1 9 0 0 .0 3 5 6 0.0 3 5 6 0.0 3 6 6 50 0.0 0 2 0 0 .0 3 2 9 0.0121 0.0121 100 0 .0 0 6 7 0 .0 1 0 7 0 .0 1 1 7 0.0 1 1 7 15 0.1 2 2 9 0.0219 0 .1 7 8 4 0.1 9 6 7 20 0.0 8 9 8 0.0 0 3 0 0-1324 0.1 4 2 4 25 0 .0 8 4 4 0 .0 1 7 5 0 .1 1 8 3 0.1 19 8 0 . 9 30 0 .0 7 2 0 0.0 13 2 0 .1 0 0 5 0.1 03 4 50 0 .0 4 3 5 0 .0 0 2 6 0.0 61 0 0.0 60 0 100 0 .0 1 9 9 0.0 0 1 8 0.0 2 8 8 0 .0 2 8 5

(18)

T a b l e 4 .1 0

R e la t io n s h ip between th e moon s q u a re s e r r o r end the v a lu e s o f T (1000 r e p l i c a t i o n » )

*x

T MSE

OLS MOD. OLS APR. ML. EXACT ML.

15 0.1542 0 .1 89 2 0.1862 0.2068 20 0.1239 0.1412 0 .1 47 7 0 .1 55 5 . 25 0.1043 0.1174 0.1211 0.1250 - 0 .9 30 0 .0 9 6 6 0.1 00 5 0.0986 0.0999 50 0.0 62 2 0.0 67 5 0.0693 0 .0 69 6 100 0.0410 0.0 44 5 0.0436 0.0437 15 0.2071 0.2 05 5 0.1 94 5 0.1934 20 0.1698 0.2 19 4 0.1634 0.1631 25 0 .1466 0.1829 0.1418 0.1420 - 0 .5 30 0 .1327 0.1579 0.1 29 7 0.1299 50 0.1018 0.1151 0 .1 00 3 0.1004 100 0.0696 0.0760 0.0690 0,0690 15 0.2261 0.2875 0.2099 0.2092

20

0.1878 0.2302 0.178Q 0,1779

25

0.1658 0.1 92 3 0.1588 0.1 58 8

0.0

30 0 .1 50 7 0.1738 0.1455 0 .1 45 5

50

0.1160 0.1282 0 .1 13 6 0.1136

100

0.0023 0 .0 86 4 0 .0 8 1 5 0.0 81 5 15 0.2070 0.2 81 6 0.1961 0 .1953 20 0.1693 ' 0.2214 0.1614 0.1613 25 0.1531 0 .1 86 8 0.1488 0 .1489 0 .5 30 0.1341 0.1645 0.1 30 6 0.1306 50 0.0984 0.1151 0.0 96 2 0.0962 100 0.0709 0.0 76 6 0.0 70 4 0 .0 70 4

15

0.3 539 0 .1 87 2 0.1 87 5 0.2019 20 0 .1 18 6 0 .1415 0 .1 39 8 0 .1 5 0 3 • ? S 0 .1 06 0 0 .1 16 7 0 .1 24 6 0.1 27 6 0 .9 30 0.0967. 0.1 05 0 0.1091 0.1130 50 0.0626 0 .0 67 2 0 .0 6 9 5 0 .0 69 7 100 0.0381 0 .0408 0 .0 4 0 7 0.0408

(19)
(20)

I BIAS I

f

0.2000 OUS MOO. OLS APR. ML. EXACT ML. 0.1500 0.1000 F i g . 4 . 2 . R e la t io n s h ip between th e a b e o lu ta v a lu a o f th « b ia s and v a lu e s o f (T -m 3 0 )

(21)

(BIAS I

1

0.2000 ■ — 01.S --- -- MOD. OLS •... APR. ML. --- EXACT ML. 0.1500 0.1000 0.0500

(22)

M SE

I

0.3000

(23)

1

о.зооо

0.2500

i 0.2000

(24)

M|E

o. 3000

F i g . 4 . 6 . R e le t lo n e h lp betw een th e mean aq ueroe e r r o r

and the v a lu e s o f f»1 vT - l o o ) 0.2500 0.2000 0.1500 - OLS - MOD. OLS • • APB. ML. - EXACT ML.

(25)
(26)
(27)
(28)

I BIAS i

• . ■ . _ • «. *1 •- ,* * ■ ’ * * ' , i

* F l a . 4 .1 0 . R e la tio n s h ip between tha v a lu e u of tha a b e o lu te b ia s and the v a lu e s of T (0 ^ « 0 , 5 )

(29)

F l 9. 4 .1 1 . R e la t lo n e h ip between t h e 'v a l u e s o f the absolute» b ia s and th e v a lu e s o f T ( p ^ * 0 . 9 ;

(30)

м$е

F ig . 4 .1 2 , t te lo t t o n c h lp between the moan s q u a re s e r r o r end the v & lu e s o f ' T ( 0 ^ * - 0 . 9 )

(31)

MSE 0.3000 -Ч \ \ --- OLS х ---MOD. OIS V ... APR. ML. °- 2500 \ ---EXACT ML. \ \ \ V К N e \ 0.2000 - \ 4n

4

• * \ V v \ N •• \ 4 v ‘\V \ \'

\

0.1500 • V 4 ^ \

•'« Ч

\

■•9Sv \

* ^ 4

'S h N 0 .1 0 0 0

-\

4

V

;

4 ^

' * V

Л 0.0500 • • Щ 15 20 25 30 5C 100

(32)

0.0 500 -—i— 15 —t— 20 —t— 25 T -30 - V 50

" N — >T

to

■ 4 .1 4 . R e la t io n s h ip between th e mepn s q u a re s e r r o r and th e v a lu e s of T » 0 . 0 ) MSE 0.3000-0.2500 — OLS -- HOD. OLS .« APfi. ML. — EXACT ML. 0.2000 0 1500 0.1000 ■

(33)

• * , . ’

F i g . 4 .1 5 . R e la t io n s h ip between th e »een s q u a re s e r r o r ond th e v a lu e s o f t ( 0 4 * 0 . 5 }

(34)
(35)

R e fe re n c je

[ 1 ] A b d e 1-R « z e k

(

1 9 7 2

),

The D i s t r i b u t i o n o f l.oant Squ­ a r e s E s t im a t o r s in F i r o t O rd er S t o c h a s t ic D if f e r e n c e Equtt- t i o n s , The U n i v e r s i t y of K a n s a s , USA.

[ 2 ] A i g n e r D. 0. (1 9 7 1 ). A Conpendium on E s t im a t io n of the A u to re g ro s a lv e - M o v in g A ve rag e M od el, I n t o r n a t . ucon. R e v . 12, p . 34B-371,

[ 3 ] A n d * r s o n T . W. (1 9 7 1 ), S t a t i s t i c a l A n a ly s is o f Time S e r i e s , 0. W ile y and S o n s, Now Y o rk .

[ 4 ] A n d o r e o n T. W. (1 9 5 9 ), On A s y m p to tic D i s t r i b u ­ t io n s o f E s tim a t e s o f P a ra m e te rs o f S t o c h a s t ic D if f e r e n c e li­ q u a t io n s , Ann, M ath. S t c t i s . 30, p. 676-687.

[ 5 ] B o x G. E . P . , 0 o n k i n o G. M. (1 9 7 0 ), Time S e ­ r i e s A n a l y s i s , F o r e c a s t in g and C o n t r o l, Holdon Day I n c . , l>an F r a n c is c o .

[ 6 ] C h a n d a K. C. (1 9 6 1 ), C o m p a ra tive E f f i c i e n c i e s of Methods o f E s t im a t in g P a ra m e te rs in L in e a r A u t o r e g r e s s i v e

3chem es, " B io m e t r ik u " 48, 3 and 4 , p . 427-432.

[ 7 ] C o p a t 0. B . (1 9 C G ), Monte C a r lo R e s u lt s f o r E s t i ­ m ation in S t a b le Markov Time S e c i e s , ORSS, A , 129, 1 .

[ 8 ] D B n t W . , M i n A. 3. (1 9 7 8 ), A Monte C a r lo S tu d y A u t o r e g r e s s iv e I n t e g r a t e d Moving A ve rag e P r o c e s s u s , Econo­ m e t r ic s " 7 , p . 23-55.

[ 9 ] H e n d r y O. F . , T r i v i d i P . K. (1 9 7 1 ), M ax i­ mum L ik u lih o o d EetiTH O tion o f D if f e r e n c e E q u a tio n s w it h Mov­ in g A vo rage E r r o r : a S im u la t io n S tu d y , R e v . o f Econora. S tu d , p . 170.

[ 1 0 ] 0 o o b 0 . L. (1 9 4 4 ), The E le m e n ta r y G a u s s ia n P r o c e s s e s , Ann. M ath. S t a t i s t . , 15 , p . 229-282.

[ 1 1 ] M a 1 i n v a u d E . ( l 9 6 l ) . The E s t im a t io n o f D i s t r i ­ buted Lo g s. A Comment, "E c o n o m e tr ic a " 29 , p . 430-433.

[ 1 2 ] M a n n H. 8. , W e l d A. (1 9 4 3 ), On th e S t a t i s t i c a l T re a tm e n t o f L in e a r S t o c h a s t ic D if f e r e n c e E q u a t io n s , Econo­ m e t r ic s " 11, p . 173-220.

[1 3 ] N e l s o n C. R. ( l 9 7 3 ) , The F i r s t O rd er Moving A ve rag e P r o c e s s , “ E c o n o m e tric a “ 2 , p . 121-141.

(36)

Q u e n o u i l l o M. H. (1 9 4 9 ), A p p rox im ate T e s ts of C o r r e la t i o n In Tlmo S e r i e s , 3R3S, B, 11, p , 68.

O r c u t t G. H . , W 1 n o k u r H. S . (1 9 6 9 ), F i r s t O rd er A u to r e g r e s s io n * I n f e r e n c e , E s t im a t io n and P r e d ic ­ t i o n , “ E c o n o m e tric o " 37, I , p . 1-14.

3 n w e T. (1 9 7 0 ), The E x a c t Moments o f the L e a s t S q u a re s E s t im a t o r f o r the A u t o r e g r e s s iv e M od el, " E c o n o m o t r lc s “ 8, p . 159-172.

S h e n t o n L. R . , D o h n e o n IV. L . (1 9 6 5 ), Mo- o o p ts o f a S e r i a l C o r r e la t i o n C o e f f i c i e n t , ORSS, B . 27, 2 , p . 308-320.

T h r o n b e r H. (1 9 6 7 ), F i n i t e Sam ple H onte C a r lo S t u ­ d i e s : an A u t o r e g r e s s iv e I l l u s t r a t i o n , 3ASA, p . 601-818. W h i t e 0. S . (1 9 6 1 ), A s y m p to tic E x p a n sio n s f o r th e Mean and th e V a r ia n c e o f th e S e r i a l C o r r e l a t i o n C o e f f i c i e n t , "B io - m etriko*' 48, p . 85-94.

A n d rz e j Tom aszew lcz, M a jid Hemza A l- N a s s lr OCENA EFEKTYWNOŚCI PEWNYCH ESTYMATORÓW DLA M OOELJ.AUTOREGRESai PIERWSZEGO RZęOU

• A r t y k u ł p r z e d s ta w ia p o ró w n an ie e fe k t y w n o ś c i n a s tę p u ją c y c h me­ tod e s t y m a c ji d la p aram etró w m o d e li a u t o r e g r e s j i p ie rw s z e g o r z ę ­

d u :

-1 ) zwykła- metoda n a jm n ie js z y c h kw adratów (z m n k ),

2) zm odyfikowana metodą n a jm n ie js z y c h kw adratów (mod mnk), 3 ) p r z y b liż o n a metoda n a jw ię k s z e j w ia r y g o d n o ś c i,

4 ) d ok ład n e metoda n a jw ię k s z e j w ia r y g o d n o ś c i.

R e z u lt a t y eksperym entów M o n t e iC a r lo , p rz e d s ta w io n e w 10 te-* b e la c h i na 16 w y k re s a c h , w s k a z u je , t e

o ) o b c ią ż e n ie rozw ażanych e sty m a to ró w i e s t podobne w p rz y p a d ­ ku m ałych w a r t o ś c i w sp ó łcz yn n ik ó w a u t o k o r e l a c j i 0 ^ : w przyp ad ku j 0 , | 0 , 5 zm odyfikow ana n etod a n a jm n ie js z y c h kw adratów Quenou-

i l l e ' a j e s t le p s z a ;

b) b łę d ś re d n io k w a d ra to w y J a s t z w y k le m n ie js z y d la zank n i * d la pod mnk. E s ty m a to ry zmnk 1 nod mnk maję je d n a k ż e m n ie js z y b łę d ś re d n io k w a d ra to w y n iż e s ty m a to r y p r z y b liż o n e j m etody n a j ­ w ię k s z e j w ia r y g o d n o ś c i i d o k ła d n o j m etody n a jw ię k s z e j w ia ryg o d n o ­ ś c i , k t ó r e j e fe k ty w n o ś ć J e s t podoboa. [ 1 4 ] [ 1 5 ]

[1C]

[1 7 ] [1 8 ] ¿1 9 ]

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