A C T A U N I V E R ' j I T A T I S L C C Z 1 I M ’• I i FO LIA OECONOMIO 48. 1*105
W ie s ła w D ę b s k i* , W ła d y s ła w M ilo *
AN ALGORITHM OF THE ESTIMATION METHODS OF FIN A L f o r m's PARAMETERS
OF SIMULTANEOUS LINEAR ECONOMETRIC MODELS
1. In t r o d u c t io n
We l i m i t ou r c o n s id e r a t io n to thu coso o f s im u lta n e o u s l i - n oar e c o n o m e tric models w it h a maximum lo g e q u a l to 1. The e s t im a t o r s fo rm u la « used in th e a lg o r it h m b elo n g to the c la s s o f n o n i t e r a t i v e e s t im a t o r s o f c o n s tr a in e d l e a s t s q u a re s method. L e t us assume t h a t the e m p i r ic a l s t r u c t u r a l form of the s im u l tan eou s l i n e a r e c o n o m e tric model in g iv e n by
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b ( l )21 h ( l ) 12 22 b ( l ) ‘ *** lm h ( l > *** 2m Bo " •xm h ( o ) h (o ) _b« l m2 ' * * 1 • B1 " mxm h ( D . «1 b ( D •2 *•* b ( l ) mm • re th e m a tr ic e s o f th e e s t im a t o r s o f the p a ra m e te rs s ta n d in g a t th e endogenous v a r i a b l e s In th e p e r io d s t , t- 1 , r e s p e c t i v e l y ,* D r . , L e c t u r e r a t the 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 i v e r s i t y o f Ł ó d ź .
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•** n 'i> 9 ink a re th * m a tr ic e s o f the e s t im a t o r » o f th e p a ra m e te r« s ta n d in g v a r i a b l e s in the p e r io d s t . t X11 X12 . . . xl t • • • Xl n " X21 X22 . . . x2t • • • X2
n Xt “ kxn _ xk i \ г . . . xkt • • • Xkn Xl , l - 1 Xl,2 - 1 . . . xl f t-1 . . . xA X2
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- 1 X2 , 2-1 . . . x~ t-1 . . . x2 Xk . l - 1 Xk,2 -1 Xk . t-1 . . . xka re th s m a tr ic e s o f the o b 'ja r v n tlo f . on tho exogenous v a r iobl«-: In tho p e r io d s t , t -
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, r e s p o c t i v a l y ,e ) p - the number o f e q u a tio n s (t o g e t h e r w ith th® i d e n t i t i e s ) » f ) n - the number o f o b s e r v a t io n s on the endogenous and exogen-
ouo v a r i a b l e s ,
g ) k - the number o f exogenous v a r i a b l e s , h ) V V 1 2 • * * * V 1 1 I c > « • • V 2 i V r. 2 v • • • 2 t C C M > • • • e C X > B > 1 V m2 * • * V rot v • • • n>n
la tho m a tr ix of r e s id u a l s In tho p e r io d t ( i f any e q u a tio n i s »»n I d e n t i t y th en no r e s id u a l w i l l a p p e a r ).
The a lg o r it h m a d d i t i o n a l l y r e q u ir e s as th e in p u t In f o r m a t io n ! a ) tho p a ra m e te r 0 w h ich moons th e number o f r e c u r s iv e re p e t i t i o n s (3 i s u s u a l l y l e s s o r oq u ol to n ~ l ) ,
b) the m a t r ic e s X t _ 2 < •••$
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d e n o te th e m a tr ic e s o f ob- s e r v u t io n s on the exogenous v a r i a b l e s in the p e r io d t -2
,..., t -0 (e a c h of the-se m a tr ic e s has a d im e n sio n kxn and i t i s formed in tho same way at. the m a tr ix X^ o r xt _^ )* The f i n a l form c o r r e sponding to ( i ) i s as f o l l o w s : U ) Y t . o . X , . F j X , . , ♦ F 2x t _ 2 , . . . . f ^ 0 , V w h e re : Dy i s the mxk m a tr ix o f th e d i r e c t (im p a c t ) m u l t i p l i e r s , F j , F2
» •••» F j a ra the mxk m a tr ic e s o f th e i n d i r e c t ( d e la y , la g g e d ) m u l t i p l i e r s ,Wt I s tho mxn m a tr ix o f the r e s i d u a l s o f the f i n a l form .
I n d e s c r ib in g th e a lg o r it h m we w i l l use two k in d s o f e s t i m ators f o r the p a ra m e te rs o f th e f i n a l form [ i , 2, 3 ] . Tho f i r s t one u ses th e in fo r m a t io n about tho exogenous v a r i a b l e s in the r e c e n t ar.d p a s t p e r io d s and I t does not U6a th e in fo r m a tio n
about the m utu al r e l a t i o n e between th e d i r e c t and I n d i r e c t m u l t i p l i e r s and the p a ra m e te rs o f th e s t r u c t u r a l form of an e c o n o m e tric m odel. The o t h e r method J u s t ta k e s In t o a c cou n t th e above m entioned r e l a t i o n s h i p s . B elo w we s h a l l g iv e • d e s c r i p t i o n o f th e s u c c e s s iv e s te p e o f th e a lg o r it h m w hich In c lu d e s both o f th e s e e s t im a t o r s . The In p u t d s t a t m , n , k , 3 , t « 1 ( I - the s e t o f i n t e g e r s ) , B » B . jC . « Rnx*1 (R - th e s e t o f mxm r e e l m a t r i c e s ) , v ° v e f j" * n W i e K * Co * Cl * Do ' ° l ' F o * R • xt , x t-1 e Rk x n , X 6 p k O + D x n ^ ¿ g e RBIX|c(
3
#1 ) fS i : G iv e n the m a tr ic e s Y t *x t »x t _ i * •••* X t- 3 form the ma t r i x t-1 » • • • • t- 3 where X i s th e k(3+ -l)*n m a tr ix whose b lo c k e a re m a tr ic e s o f o b s e r v a t io n s on th e exogenous v a r i a b l e s in th e p e r io d e t , t - 1 , . . . , t- 3 . S2 : Compute ( X X ' ) ^ X Y ' i f d e t ( X X ') * 0 , i . e . i f n > k ( 3 * l ) 8 ' ( X X ') X Y ' i f d e t ( X X ') - 0 , i . e . i f n < k ( 3 * l )
(t h e sym bol d e n o te s th e M o ore-Penrose g e n e r a liz e d in v e r s e m a tr ix f o r th e c a l c u l a t i o n o f w h ich one o f th e known a lg o r ith m e sh o u ld be u se d , and Y i ■ Y ^ ) .
The m a tr ix 6 c o n s t i t u t e « th e f i r s t group o f e e t lm a t o r s f o r the p a ra m e te rs o f th e f i n a l form o f a l i n e a r e c o n o m e tric m odel. I t h as a d im e n sio n m x k(3+ l) and I t s b lo c k s a r e m e t r lc e s o f d i- r e c t and i n d i r e c t m u l t l p l i e r e .
S 3 : G iv e n Bq , Bj , Gq , Gj^ compute (u n d e r th e c o n d it io n t h a t d e t B # q)
b ) D * -B ” i G (t h e m a tr ix o f d i r e c t m u l t i p l i e r s ) , o o c c ) D j -3 4
1
F o r p ■ 1 , 2 , 3, . . . . 0 c a l c u l a t e . F ■ C P " ^ ( C « b + D „ ) ( t h e m a trlc o o o f i n d i r e c t m u l t i p l i e r ; » ) , p 1 l o i S 5 : F o rm o ' c ; o 1 * l • i F 1C1 • • • •^ Ci
• • » i e '» S 6 t Compute (x x' ) " 1x y'- ( .x^ T 1x y'c i(c1c ' ) *c1+ (x xT 120c'k' (c1c' ) + i f d e t (x x ')» ‘ 0 and d e t i C j C ^ ^ O o r d e t ( C j C ' ) - 0 i . e . n > k ( 0 + l ) ( X X V X Y '- ( X X ') * X Y 'C1(C 1CJ ) * C 1 ♦ ( X X y j i X ' K ' i ^ C ' ) 4^ I f d e t ( x x ') - 0 and d u t i C j C 'l - O o r det ( C j C ' ) - 0 l . c n < k ( 3 * l ) o r In th e s im p la way 5 ' can be e x p re s s e d as K '( C ') * * 1 , i f n > k ( 3 * l ) and d e t 4 O e - e ' e ' i e j C ' ) ^ + ( x x ' J ^ x x ' K ' i C j C ' ) ^ , i f n < k ( o * i ) . e ' « .The m a tr ix 8 has tho same d im e n s io n and s t r u c t u r e as the ma t r i x 8 and i t c o n s t i t u t e s th e second group o f e s t im a t o r s f o r tho p a ra m e te rs of th e f i n a l form o f an e c o n o m e tric m odel.
We rem ind t h a t th e m a tr ix 6 o f th e p a ra m e te rs o f tho f i n a l form , f o r w h ich e s t im a t o r s a re th e m a tr ic e s & and 8 , has th e f o llo w in g s t r u c t u r e
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end i t s b lo c k s o re in t e r p r e t e d a s the d i r e c t (D q) and i n d i r e c t m u l t i p l i e r s ( F j ... F ^ ) . I t sh o u ld be a l s o n o ted t h a t In p r a c
c o l c u l o t l o n o f
6
o r 9 bocauee o f the r e l a t i o n n < k ( 3 + l ) (w h ic h f r e q u e n t ly o c c u r3 in th e e c o n o m e tric m o d els) end i n the s p e c i a l form o f i d e n t i t i o e in c lu d e d In t o th e m odel.The above d e s c r ib e d a lg o r it h m i s a v a i l a b l e a t th e I n e 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 i v e r s i t y o f Ł ó d ź .
BIBLIO G RAPH Y
[ 1 ] M i 1 o W. (1 9 7 8 ), E e ty m a c ja p aram etrów p o s t a c i końcow ej modelu e ko n o m e tryczn e g o , R e s e a rc h under th e c o n t r a c t R. I I I . 9 .
[ 2 ] M i l o W. (1 9 0 0 ), E o t im a t io n o f the F l n o l Form Param e t e r s o f th e L in e a r E c o n o m e tric M od el. P a r t I , P u b l i c a t i o n s E- co n o m d triq u e s, V o l. X I I I , F a s c , t/1 9 6 0 , p . 69-97.
[ 3 ] D ę b s k i W. , M i l o W ., Some Remarks on the E s t i m ation Methods o f F i n a l F o rm 's P a ra m e te rs of M u lt ie q u a t lo - n a l L in e a r E c o n o m e tric Modole (t o be p u b l is h e d ) .
W ie s ła w D ę b s k i, W ła d y s ła w M ilo
ALGORYTM ESTYM ACJI PARAMETRÓW POSTACI KofcoWED W LINIOWYCH WSPÓŁZALEŻNYCH MODELACH EKONOMETRYCZNYCH
W o s t a t n i c h l a t a c h p o w s ta ło k i l k a p r a c na tem at metod e e ty - n s e j i param etrów p o s t a c i końcow ej lin io w y c h m o d eli ekonom etrycz- nych [ l , 2, 3 J , jed n a k ż e żadna z n ic h n ie z a w ie r a a lg o ry tm u po z w a la ją c e g o na o b l i c z e n i e w a r t o ś c i e stym a to ró w p aram etró w t e j po s t a c i . Głównym celem tego a r t y k u łu j e s t p r e z e n t a c ja ta k ie g o w ła d n ie a lg o ry tm u . P r a c a s k ła d a s i ę z dwóch c z ę ś c i . P ie rw s z a J e s t wstępem i z a w ie r a in fo r m a c je w e jś c io w e do a lg o ry tm u eatym u jg ceg o p a ra m e try p o s t a c i k o ń co w e j. Druga n a to m ia s t p r z e d s t a w ia o p la a l gorytm u.
