A C T A . U N I V E R S I T A T I S L O D Z I E N S I S POLIA OECONOMICA 5 4 И 9 8 6 __________________
A n d rz e j T om aszew icz*, Abdul M a jid Hamza A l- N a a lr
ON THE ESTIMATION IN SIMPLE LINEAR REGRESSION MODEL WITH AUTOREGRESSIVE MOVING AVERAGES (ЛЕМА) ERROR
1 . I n t r o d u o t l o n
Suppose t h a t a r e s p o n s e y^ f o llo w s t h e m odoil
y t - B0 + + e t , t ■ 1 , 2 , . . . ( D The s im p le l i n e a r e q u a ti o n (1 ) s t a t e s t h a t I n p e r i o d t , t h e ▼alue o f y , t h e r e s p o n s e , i s d e te rm in e d by f o u r f a o t o r s t t h e po-p u l a t i o n o o n s ta n t B0 , t h e po-p o po-p u l a t io n r e g r e s s i o n o o e f f i o i e n t B , , t h e l e v e l o f z , and t h e l e v e l o f • , t h e d i s t u r b a n c e t e r r a . The d i s t u r b a n c e te rm i s assum ed t o h av e c e r t a i n p r o p e r t i e s i n o r d e r t o c a r r y o u t s t a t i s t i c a l e s t i m a t i o n and t e a t s o f s i g n i f i c a n c e , D e p a r tu r e s from t h e s e a s s u m p tio n s b r i n g some o f t h e c h a r a c t e -r i s t i c p -r o b le m s . P o -r i n s t a n c e , u s u a l l y , one assu m es t h a t a l l p a i -r s o f v a lu e s o f e ^ , w h e th e r a d ja c e n t i n tim e o r n o t , a r e n o t c o r -r e l a t e d . A d e p a -r t u -r e f-rom t h i s a s s u m p tio n g i v e s r i s e t o t h e Problem o f a u t o c o r r e l a t i o n . T h is p ro b le m h a s b e e n s t u d i e d b y a num ber o f s t a t i s t i c i a n s , * o r i n s t a n c e , A n d e r s o n ( 1 9 4 2 ), C o c h r a n (1 9 4 9 \ Q u e - a o u i 1 1 e ( 1 9 4 9 ) , D u r b i n ( 1 9 5 0 ), H a n n a n (1957X T h.** 1 1 and N a g a r ( 1 9 6 1 ) , and i n r e c e n t y e a rs , B o z and P i e r -0 • (1 9 7 -0 ) e t u d i e d t h e d i s t r i b u t i o n o f r e s i d u a l s w hich f o llo w a rnized ARMA m o d el, P i e r c e (1971) d e v e lo p e d a m ethod f o r e s t i -m a tin g t h e p a r a -m e te r s by u s in g t h e f i r s t o r d e r te r -m s i n T a y l o r 's
« p a n s i o n and a p p l i e d i t t o a n ARMA o f t h e f i r s t o r d e r and N u
-* L e c t u r e r , I n s t i t u t e o f E c o n o m e tric s and S t a t i s t i c s , U n iv e r-s i t y o f Ł ódź.
r 1 ( 19 7 9 ) proposed, a m ethod f o r f i n d i n g a p p ro x im a te e s tim a te * f o r t h e p a r a m e te r s w hioh la b a s i c a l l y r e l a t e d t o t h e l e a s t s q u a r e s m ethod w ith some m o d i f i c a t i o n s .
The p u rp o s e o f t h i s p a p e r I s t o i n v e s t i g a t e t h e p r o p e r t i e s o f t h e m odel ( 1 ) , e s p e c i a l l y when t h e e r r o r s e t f o llo w a low o r d e r ARMA. T h a t i s b e c a u s e , I n p r a o t i o e , i t I s f r e q u e n t l y t r u e t h a t an a d e q u a te r e p r e s e n t a t i o n o f a o t u a l l y o c c u r r i n g s t a t i o n a r y tim e s e r i e s c a n be o b ta in e d i n m ixed m o d e ls , i n w hioh t h e o r d e r o f a u t o r e g r e s s i v e p r o o e s s p and t h e o r d e r o f m oving a v e ra g e *q a r e n o t g r e a t e r th a n 2 and o f t e n l e s s t h a n 2 (B o x and J e n к i n s
1 9 7 0 ). So ARMA ( 3 ) i s s t u d i e d i n some d e t a i l , a n d some new r e s u l t s a r e o b t a i n e d . A tte m p ts a r e made t o o o n s tr u o t s u i t a b l e ex a m p les! a r t i f i c i a l exam p les and econom ic d a t a e x a m p le s.
2 . The P r o p e r t i e s
I n t h e m odel ( 1 ) su p p o s e t h a t
et - <T1( B ) e ( B ) a t , (2 )
w here a ^ ’ s a r e i n d e p e n d e n tly and n o r m a lly d i s t r i b u t e d random v a r i -a b l e s w ith z e r o me-ans -and v -a r l -a n o e tí 2 , Ф -and в a r e p o ly n o m ia ls s u c h t h a t i Ф (В ) - 1 - ^ B - ig B 2 , 0(B) • 1 - 0,B - O ^ 2 and В i s t h e b a c k s h i f t o p e r a t o r ( l a g o p e r a t o r ) d e f i n e d by B ^ ft -■ f t ^ f o r аду f u n c t i o n f ^ and f o r 3 * 1 , 2 . The f o ll o w i n g a r e eome o f t h e c h a r a c t e r i z a t i o n s f o r t h e p ro p o s e d m odel f o r e ^ . 1 . U sin g e q u a ti o n (2 ) t h e s e c o n d o r d e r a u t o r e g r e s e l v e , t h e se c o n d o r d e r m oving a v e ra g e p r o o e s s ARMA ( 3 ) oan b e w r i t t e n a s t
(1 - Ф,В - Ф ^ 2 ^ - (1 - 0,B - OgB2 ) « ^ , o r
et - V t - 1 + V t - 2 + at - V t - 1 - 62 S - 2 ‘
(3)
2 . M u lt i p ly i n g e q u a ti o n ( 3 ) by е ^ - к and t a k i n g e x p e o t a t l o n s , i t c o u ld b e o b t a i n e d t h a t
тк - V k - 1 + М к - 2 + f a e ( k ) - е 1 * 1) * V a e < k - 2 >»
(4)
"here f k - cov(et , et _k ) eind f ae(k) i s the cross covariance
be-tween a and e at la g d ifferen ce k, defined by E( at et-k^* ^ nce
et-k deP*nds only on shocks which have occurred up to time t - k ,
we obtain
r o 2f 1 ( Ф ,, e p k < 0fae(k ) . j 0 2
k . 0
(5)
О к > 03. I t fo llo w s that
fo * *1
U+ М г + Ö2 - V a e C“1) " 02*ае(“2 ) '
(6)
Í1 * *1 to + V i - е 1 ° 2 ” V a e ( - 1 ) ’ (7)t2
M o - e2°2*
'
(8)
And f o r к > 2 , i k - Ф1Г|С_ 1 + $ 2i k - 2 ' " h i c h doee n o t ln v o lv e th e m oving a v e ra g e p a ra m e te rs » T h e r e f o r e , a f t e r l a g 2 , t h e a u to -c o v a r ia n -c e s and c o n s e q u e n tly t h e a u t o c o i T e la t io n c o e f f i c i e n t s behave a s th o s e f o r t h e AR p r o c e s s . And so we r e a c h t h e same c o n -c l u s i o n a s A n d e r s o n (1976)»4* M ultiplying equation (3) f i r s t by a^_^
and
secondly by
at»2
taking expectations we find
ía e ( - 1) - (^1 - Ô,) O2
(9 )
fcnd
гГае(“2) * f * i * $161 + $2 ~ e2 ) o 2 ‘
(10)
S u b stitu tin g th ese in equations (6) and (7 ) lead s to
Q - Ф 2 ) [1 e f e 22 - 262Ф2 ] - + e ^ j 2
Ф« 0 . + 0 2 Ф« “ Op0-i o
*i ■ - r ^ V t0 -
------
-łg г 1 ° 2
t i »
and фр + ф ? - ф? > т - * ~ * 2 в2 * в 1$1 “ е 2ф2 * Ф1е 2 ~ ^ I e i e 2 2 ®2 " 1 - Ф2 ^ О - (1 - ф2 ) ° * (1 3 ) 3» L e a s t S q u a r e s E s t im a t o r s То o b t a i n t h e l e a s t s q u a r e s e s t l m a t o r a o f t h e o o e f f l o l e n t s , we f i r s t r e w r i t e e q u a ti o n (1 ) a s fo llo w * 9 (B ) yt ■ S ł i ( B0 + B1 * t ) + * t* (1 4 ) Then t h e p ro b lem I s o a r r l e d o u t by m in im is in g t - ( » « - » o - v t ) ] 2 - ч » We f i n d t h a t ЪБ~ " ” 2 ^ { й й } ( yt ~ B0 " B1 * t ) ' ] * “ 2 Z L { e f B ) } ( yt “ Bo “ Bi * t ) * t < ÖV ÔB and H ; ■ 2 I L ( yt “ B0 " B1 * t ) 2 » e - 1 , 2. (1 6 ) F o llo w in g t h e a p p r o z l o a t l o n e p ro p o s e d by H u r l (1 9 7 9 ) we o b t a i n s- в г - ф .в 141 - Ф-в1* 2 + г е . в 14-1 + г е - в 1* 2 , r - 1, 2 ( e ( B ) ] z 1 ^ and Ш И р В ^ Be _ 2 ф Be+1 _ 2 ф_Вв+2 + Зв.В 8* 1 + ЗврВ8*2 . (1 7 ) { e (B )} 3 1 r 1 . An I n i t i a l e s t i m a t o r f o r ? ■(_ 8*0»** o b ta in e d ЬУ *he l * a e t s q u a re s m ethod f o r t h e m odel y^ • Bq +• B^x^ + e ^ . 2 . D e f in in g Bt - ( y t - b Q - b 1xt ) 2 , t h e norm al e q u a t i o n s ( 1 6 ), ta k e t h e form nbQA(B) ♦ b 1 J ^ A Í B ) ^ - ^ A ( B ) y t , (1 8 ) b 0 ^ A ( B ) » t + b 1 2 ^ / A(® )x t " *1 ľ ‘ t - r - 1 + Ф2 ^ ^ - г - 2 ~ 2Ô1 F ’ t - r - 1 “ 202 ^ Zt - r - 2 " ■ V 1 z t ( 2®) ę x- r ' 2® 1 ^ * t - . . 1 ♦ *>2 £ Zt - s - 2 “ » 1 1 “ 3e2 ^ at a 2 -e e - 1 . 2 . ^21) 3 . S o lv in g e q u a tio n s (2 0 ) and (2 1 ) s n i n i t i a l e s t i m a t o r o f St* C^1$2®1®2)^ 1,8 o b t a i n e d * nam ely Д О ) ОС ( i / 0 ) i 2 í 0 ) é ( 0 ) 9 ( 0 ) ) T . 4 . A f t e r o b t a i n in g t h e i n i t i a l e s ti m a t e o f л , t h e s e t s o f e q u a tio n s i n (1 8 ) and ( 1 9 ) a r e em ployed t o o b t a i n a f i r s t a p p ro x i-mate e s t i m a t o r b ^ ^ and s o t h e f i r s t a p p ro x im a te e s t i m a t o r of cC *e th e n o b t a i n e d . And so on i t e r a t i v e l y t h e s e s t e p s a r e r e p e a te d u a t ü we h av e | ь ( ш) - b ( m- 1) | < 5 , and |«<n ) - сс(ш- 1 ) 1 < 52 « o r
A n d rtej Tomaszawlcz, Abdul Majid Hamza A l-H a s lr
4 . The D i s t r i b u t i o n o f t h e E s tim a to r s
D e fin e a random v a r i a b l e ut euch t h a t ф(.В)и^ - a t , where 0 (B ) . 1 - ф,В - íg B 2 and a t i e H ID (0, o ) 2 .
1 . F o llo w in g W o l d and M a n n (1943 ).Сф 1 <$2 ) T I s asymp-t o asymp-t i c a l l y norm al w iasymp-th теапСФ^ $ 2 ^ u d v a r i a n c e - c o v a r i a n c e m a asymp-tr ix
2 f “ 1 . Where Г ° and Гг - covC x^, u ^ ) - Б r
-» 0 , 1 *
2. Г
oan b e o b ta in e d a l t e r n a t i v e l y ae f o lio w eUt "
1- Ф,в - Ф^2 14 ' (1 - V i l i - TgB) V
w here T1 + T2 » ф., and T.jT2 ■ - Ф2 , o r ___ l _ I T2 h \ “ t ■ T2 - T1 у - T ^ " 1 - ŤýB J * f ъ - i r h : L ( Tľ ' - * ľ % - y 1 г г > З -о T h e r e f o r e Г г - Е ч ^ ^ + г “ * ■ (тг Л , ) 8 Ł ( т ^ ’ • ■ r w - n * в * » - л » г - г*• ■ 5 - V
j i ^ ( * Г 1 - * łł , ) ( * l w ’ - *?t r ł1 )
<«>
2 “ ГГ j - 0 s u b s t i t u t i n g r - 0 , 1 , and a f t e r few e te p e we o b t a i ni
o r o r _____________ г ' о * * 2 ) [ ( i - Фг >2 . * * ](1 - Ф,
)cr‘
r„ - 7---- . \ r , . 8 . ' ż -!>, (24) and1 ' (1 ♦ Ф2 ) [ 0 - Ф2f - Ф ? ] ‘
3 . S i m i l a r l y d e f i n e t h e random v a r i a b l e a u ch t h a t (1 в ^ В -- в ^ 2 ) ^ -- a t , w here é -- (Ô , ©2 )T i s a l s o a s y m p to tic n o rm al
1 2 *1
w ith mean 0 and T a r la n o e - o o v a r ia n c e m a t r ix 2 O Q , w here Q ■ * ( f i ? Qq ) eu ch t h a t Qe " Evt vt+ s* a “ 0 ,1 * U 8ine t h ® 8ame Way a s f o r Г , we o b t a l n t Tt • * г Ь т £ ( * * * ’ - » ł * ' ) * . - ! . (26) • h e r e ♦ Я2 ■ 0^ and A ^ Д2 ■ - 0 2 and а . - , . " г ? £ ( * * * ’ - » D M * * 1 - • f ’ *1) . с г 7 ) С 2 “ Л1 ) > 0 s u b s t i t u t i n g в ■ 0 , 1 w* o b t a i n - - ( г е ) and e 1 q2____________ (2 9 ) I;> - % )< ? ( 1 * вг) I[ 0 - 9г ) 2 - e ?J 4 1 " (1 ♦ * 2 ) [ 0 - ° 2) 2 - • ? ] ' 4 . D e f in e a m a t r i x w j s u c h t h a t wk - с о у (и * . v t + k ) - В u t v t+ k How, f o r к - 0
And a f t e r few s t e p s we o b t a i n .2 ______________..., ( 1 * * г вг ) - (®1 + * 1®гХ* 1 * ® i* a ) w here к - 1, W1 e B * t Tt+ 1 " (T 2 - f y) ( * 2 " Xl ) j ^ ^ +1 " T^+1) ( X^ 2 “
- x j + 2 ) ,
(31)
o r__________(
6, * * & ) ? ________
<32)
1 " 0 - М г ) * - («1 * * 1 « г Х * 1 ł ® 1*г) ' B u t, w here к • - 1 , we h av e *_1 - Eut v t - 1 * (T 2 - 1 \ \ x 2 - x1 ) £ ( Tl +1 ” * ! * % í - x 0 * J e Q o r(Ф. + е.Ф -) e 2
w m---—J---1 -Z.l--- (3 3 ) " 1 0 - ф 20г ) 2 - C9 1 + 5 10г ) ( ф 1 + в 1фг ) * c o m p a rin g (32) w ith (33) I t a p p e a r s t h a t w1 í w ^ ^ 5 . We o b t a i n t h a t i - Ъ - ( b o b , ) *l a n o r m a lly d i s t r i b u t e d ( b i v a r l a t e ) w ith mean Б and v a r i a n o e -c o v a r l a n -c e m a t r ix ( 6 2/ n ) B“ 1 , w here
1 <©>
—
к -< *2 >
\ V - 0 1 - 02 У «
* . ( } § ) * I s a s y m p to tic norm al ( 4 - d im e n s io n a l) w ith mean oj, and . v a r i a n c e - c о v a r i a n c e m a t r ix ~ ( ^ | and c o n s e q u e n t ly ( t« 6 2) T
\ w T Q /
i s a s y m p to tio n o rm al ( 7 -d im e n .) w ith mean (]J л 62) T and v a r i a n o e - - o o v a r ia n c e m a t r ix : (3 4 ) R e f e re n c e s A n d e r s o n R . L . , 1942, D i s t r i b u t i o n o f t h 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 s , Ann. U a th . S t a t i s t . , v o l . 1 3 , p . 1-13« A n d e r s o n R . L . , 1954, The P ro b lem o f A u t o c o r r e l a t i o n R e g r e s s io n A n a l y s i s , JASA, v o l . 4 9 , p . 1 1 3 -1 2 9 . B o x 0 . E . P . and J e n k i n s G. M ., 1970, T im e - S e rie s A n a ly s is P o r e c a s t i n g and C o n t r o l , Holden-£>ay, London.
B o x G. E . P . and P i e r c e D. A ., 1970, D i s t r i b u t i o n ° f R e s i d u a l s A u t o c o r r e l a t i o n i n ARMA Time S e r i e s M odel, JASA, * ° 1 . 6 5 , p , 1 5 0 9 -1 5 2 6 . B u t t e r I . P . , K a v • s h R. A. a nd P l a t t R. B ., ^974, M ethods and T e c h n iq u e s o f B u s in e s s P o r e c a s t i n g , P r e n t l c e - " H a ii, I n c . , Hew J e r s e y . C o o h r a n D. a nd O r c u t t G. H ., 1949, A p p l i c a t io n ° f L e a s t S q u a r e s R e g r e s s io n to R e l a t i o n s h i p s C o u n tin g A u t o c o r r e la -t i o n E r r o r T e rm s, JASA, v o l . 4 4 , p . 3 2 -6 1 . B u r b i n J . and W a t s o n G. S . , 1950, 1951, T e s t i n g f ° r S e r i a l C o r r e l a t i o n I n L e a s t S q u a re s R e g r e s s i o n , P t s I and I I , Bi o m e t r i c a , p . 4 0 9 -4 2 8 and 1 5 9 -1 7 7 .
D u r b i n J . , 1 9 6 0 ,E s tim a tio n of P a ra m e te rs i n Time S e r i e s R e g r e s s io n M o d els, JE SS, v o l . 2 2 , p . 139-153*
D u r b i n J . , 1 9 7 0 ,T e s ti n g f o r S e r i a l C o r r e l a t i o n I n L e a s t S q u a re s R e g r e s s io n when Some o f t h e R e g r e s s o r s i r e L agged Depend» e n t V a r i a b l e s , " E o o n o m e trio a " , v o l . 3 8 , p . 4 1 0 -4 2 1 . O r e n a n d e r U ., 1954, On t h e E s t im a t i o n o f R e g r e s s io n C o e f f i c i e n t s i n t h e C ase o f A u t o o o r r e l a t e d D i s tu r b a n c e , A nn. Math. S t a t i s t . , v o l . 2 5 , p . 2 5 2 -2 7 2 . G u r l a n d J . , 1954, An Exam ple o f A u to C o r r e la te d D is -tu r b a n c e s i n L i n e a r R e g r e s s i o n , " E o o n o m e trio a '', v o l . 2 2 , p . 2 1 8 - - 2 2 7 . H a n n a n E . J . , 1 9 57, T e s t i n g f o r S e r i a l C o r r e l a t i o n i n L e a s t S q u a re s R e g r e s s io n , " B lo m e tr lc a " , v o l . 4 4 , Р» 5 7 -6 6 . N u r i W. A ., 1979, E s t im a t i o n I n R e g r e s s io n M odels w ith E r r o r s F o llo w in g an ARMA S c h e m e ,I n te r n a t i o n a l Time S e r i e s M e etin g a t N o ttin g h a m U n i v e r s i t y .
P i e r o e D. A ., 1971, D i s t r i b u t i o n o f R e s id u a ls A u to c o r-r e l a t i o n I n t h e R e g r-r e s s io n Model w ith ARMA E r r o r , JRSS, B . 33, p p . 140- 146. Q u e n o u i l l e M. H ., 1949, A p p ro x im ate T e s t s o f C or-r e l a t i o n i n Time S e or-r i e s , JRSS, B . 1 1 , p . 6 8 . T h e i l H. and H a g a r A. L . ,1 9 6 1 .T e s t i n g In d e p e n d e n c e o f R e g r e s s io n D i s t u r b a n c e s , JASA, p . 7 9 3 -8 0 6 . W a t s o n G. S. a nd H a n n a n B. J . , 1955, S e r i a l C o r-r e l a t i o n i n R e g r-r e s s io n A n a l y s i s , " B io m e tr-r io a " , v o l . 4 3 , p . 436 -4 4 8 . W i s e J . , 1 9 5 6 , R e g r e s s io n A n a ly s is o f R e l a t i o n s h i p s B etw een A u t o o o r r e l a t e d Time S e r i e s , JRSS, B . 1 8 , p . 2 4 0 -2 5 6 . W o l d H. a nd M a n n H. B . , 1943, On t h e S t a t i s t i c a l T re a tm e n t o f L i n e a r S t o c h a s t i o D i f f e r e n c e E q u a t io n s , " E c o n o m e tri- c a " , v o l . 1 1 , p . 1 7 3 -2 2 0 .
A ndrzej T om aszew icz, Abdul M ajid Hamza A l- N a s lr 0 ESTYMACJI PARAMETRÓW LINIOWYCH MODELI . ZE SKŁADNIKAMI LOSOWYMI TYPU ARMA NISKICH RZgDOW
W b a d a n ia c h e m p iry cz n y c h l e t n i e j ą zw ykle p o d staw y do z a ło ż e -n i a , że bad an y s z e r e g czasow y generow any j e s t p r z e z "m ie sz a n y " p r o c e s s to c h a s ty c z n y b ę d ą c y sumą p r o o e s u a u t o r e g r e s y jn e g o i p r o -c e s u á r e d n io h ru-chom y-ch ARMA ( B o x , J e n k i n s 1970)} w n i -n i e j s z e j p r a c y z a n a liz o w a n o n i e k t ó r e w ła s n o ś c i m o d e li t y p u ARMA n i s k i c h rzęd ó w , s z c z e g ó ln i e p r o c e s u ARMA ( 2 , 2 ) .
