Л О Т А 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 54,1 9 8 6________
Iwona K onarzew ska*, W ładysław H i l o * *
SOUE NOTES ON APPLICABILITY
OP VARIANCE-DECOUPOSITION-PROPORTIONS METHOD 1. I n t r o d u c t i o n L e t u s c o n s i d e r t h e e q u a t i o n o f l i n e a r r e g r e s s i o n o f t h e form y « (30 + p , ! , + . . . + ßkx k + ę , CD where X * [X1§ Xg, . . . , XjJ i s a random v e c t o r o f e x p l a n a t o r y v a r i a b l e s , Y i s an e x p l a i n e d v a r i a b l e and £ i s a d i s t u r b a n c e te r m , tfe assume a b o u t t h e s e random v a r i a b l e s t n e f o l l o w i n g : X i s n o r -m a lly d i s t r i b u t e d w i t h t h e e x p e c t e d v a l u e 1(1) * ц » [ ц ^ , i i j i •••» ] nnd w i t h t h e v a r i a n c e - c o v a r i a n c e m a t r i x » ( X) « % 5 £ i s n o r m a l ly d i s t r i b u t e d w i t h t h e e x p e c t e d v a l u e £(£)= 0 and t h e v a r i a n -o ce v a r
(£ ) ■
6; £
i s i n d e p e n d e n t fromX.
C o n s e q u e n t l y , t h e d ep e n -d e n t v a r i a b l e Y i s n o r m a l l y d i s t r i b u t e d w i t h t h e e x p e c t e d v a l u e t (Y) в + P0 v a r l a n c e v a r(Y ) « + 62 , where Px = “ iP-j» P2» • • • » ßjj] • t , ® a r e t h e o p e r a t o r s o f e x p e c t e d v a l u eand v a r i a n c e - c o v a r i a n c e .
P a r a m e t e r s ß0 , ß1 , . . . , ßk o f t h e model ( 1 ) c an be e s t i m a t e d when we have a m a t r i x o f sam ple o b s e r v a t i o n s on e x p l a n a t o r y V a r i a b l e s and a v e c t o r o f sam ple o b s e r v a t i o n s on e x p l a i n e d v a r i a -b l e . U nder t h e a s s u m p t io n t h a t t h e m a t r i x o f o b s e r v a t i o n s on e x p l a n a t o r y v a r i a b l e s i s a f i x e d , n o t-ra n d o m m a t r i x we c a n o b t a i n t h e f o l l o w i n g model jPAo » (A'U'Ck+.l) , Sf y - * p + S , k 0 < к + 1 , n 0 =* n , (2 ) * S e n i o r A s s i s t a n t , I n s t i t u t e o f E c o n o m e t r i c s and S t a t i s t i c s ,
U niversity
o f Łódź. * * 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 t r i c s and S t a t i s t i c s , U n i v e r —a ity
o f Łódź. Г4Л41<?y - JT y ( « P ,
62
I ) ) , w here j t n x ^k+1) - a s e t o f r e a l n z к + 1 m a t r i c e s , S - a proba-ł b i l i t y sp a o e w ith t h e c o m p le te m easu re * - [1 : * | , 1 - t h e colum n v e o t o r o f u n i t s , X £ Л м к - a m a t r i z o f o b s e r v a t io n s on e x p la n a to r y v a r i a b l e s , p*> [30
Px ] i P € & - v e c t o r o f p a-r a m e t e a-r s , Y, E - a-random n z 1 v e c t o a-r s , k Q - r a n k ( e ) , n Q = ra n k(2 ( Y )).
The m odel (2 ) can be t r e a t e d a s a sam ple r e a l i z a t i o n o f th e m odel ( 1 ) .
Now we c a r r y o u t t h e p r o c e s s o f s t a n d a r d i z a t i o m t h e m odel (1 ) w ith r e s p e c t t o t h e o r e t i o a l means and o o v a r l a n c e s , t h e m odel ( 2 ) w ith r e s p e c t t o sam ple means and c o v a r i a n c e s . We o b t a i n th e f o llo w in g form s o f s t a n d a r d i z e d v e r s i o n s o f c o n s id e r e d r e l a t i o n s ! Y* - (b*X* + . . . + (i*kXk + £ * , (1 a ) where < V , X * ° **ПХ*( 0 * 1 ) ' 1 * 1 * k ’ ** " 1 Í f Y*e “ 1 ■ r ' i ■ - / Ä f Ť P l • a ( x * ) - - a m a t r i z o f s im p le c o r r e l a t i o n c o e f f i c i e n t s b etw e en v a r i a b l e s X^, 1 = 1 , k ; OJTJL0 * ( JL0* * , S , Y* - x * p £ + 2*. k 0 < k , n Q < n , <?у , - 0^Y*Cx* p ; , J j M )j, ( 2 a )
w here X,f - M x(D *;-1 , Y* o - J - MY, И - I & - •} 1 • ! ' ,
D
=[d i á ] i . J
■1. k -
J-x ’ Mx,
D* n d i a g ( d ] { 2 ... d j £ 2) ,d « у - - y ’**y, у - a sample r e a liz a tio n of the random veotor Y,
4
— x * ’x*-
a m a t r ix o f sam ple s im p le c o r r e l a t i o n o o e f f l o l e n t s betw een v a r i a b l e s 1 ^ , 1 ■ 1 , k .Th« s ta n d a r d iz e d v e r s i o n s (1a) o r (2 a ) a r e u s u a l l y t h e b a s e t o s tu d y t h e p ro b lem o f ex tre m e d e p e n d e n c ie s among e x p la n a to r y v a r i a b l e s o a l l e d m u l t l o o l l i n e a r i t y p ro b le m . I n g e n e r a l t h e m ain e f f e c t s o f s t a n d a r d i z a t i o n o f t h e e q u a tio n C1) i s r e p a r a m e t r i z a - t i o n and t h e f a c t t h a t t h e v a r i a n o e - c o v a r i a n c e m a tr ix becomes th e m a tr ix o f s im p le c o r r e l a t i o n c o e f f i c i e n t s . P o r t h e m odel (2 ) t h e e f f e c t o f s t a n d a r d i z a t i o n i s much d e e p e r - w e o b s e rv e a change o f p a ra m e te r v e c t o r , m a tr ix
^ x * 'x *
becom es s im p le sam ple c o r r e l a r t i o n m a tr ix and a d d i t i o n a l l y t h e n o rm al d i s t r i b u t i o n o f t h e v e o to r Y o h an g es t o s i n g u l a r n o rm al o f t h e v e c t o r 1* b e c a u s e o f ldem po- te n c y o f t h e m a t r ix M. 2 . M u l t l o o l l i n e a r i t y - d i a g n o s t i c a l m ea su res S t a n d a r l z a t i o n r e d u c e s a l l m odel v a r i a b l e s t o t h e same s o a l e . f u r t h e r o n , we a n a ly z e p ro b le m s o f I n te r d e p e n d e n c ie s among m odel v a r i a b l e s onthe
b a s i s o f s ta n d a r d iz e d v e r s i o n s (1 a) and ( 2 a ) . I nth e
c a s e o f ( 1 a ) , we c a napeak
a b o u t e t o c n a s t l o a l m u l t l c o l l i - n e a r i t y when ra n k ( $ * ) < k . I n t h e c a s e o f(2 a ),
we ca n sp eak ab o u t n u m e ric a l m u l t i c o l l i n e a r l t y when ra n k( x * ’x * ) < k .
We have to m e n tio n t h a t i f we a r e d e a l i n g w ith s t o c h a s t i o a l m u l t i o o l l i - a * a r i t y i n t h e (1 a ) th e n i n i t s sam p le r e a l i z a t i o n m odel (2 a ) t h e r e w i l l a p p e a r n u m e r ic a l a u l t i o o l l i n e a r i t y w ith p r o b a b i l i t y o n e. On t h e o t h e r h a n d , i f one i s d e a l i n g w ith n u m e ric a l r a u l t i - o o l l l n e a r i t y i n t h e c a s e o f (2 a ) t h e r e i s no c e r t a i n t y w h eth er i t l a c a u se d by s t a t i s t i c a l d ep en d en cy among v a r i a b l e s X^, i » 1 , к o r w h e th e r i t i s a p r o p e r t y o f t h e i n d i v i d u a l sam p le m a tr ix o f 8t a t l e t i c a l d a t a ( i n t h e s e n s e t h a t e x p a n d in g t h e m a tr ix * by a new row o f o b s e r v a t io n s on e x p la n a to r y v a r i a b l e s we ca n g e t r i d ° * f m u l t i c o l l i n e a r l t y p r o b le m ). The p ro b lem o f n o n - f u l l ra n k o f th e x ^ ’x * m a tr ix i s r e l a t e d t o e x a c t l i n e a r depen d en cy b etw een oolumns o f t h e m a t r ixx * .
The p a ra m e te rs o f t h i s l l n e á r r e l a t i o n - a h lp a r e t h e e le m e n ts o f t h e e ig e n v e c t o r o fx ’x
m a t r ix co n n e c -te d w ith t h e e ig e n v a lu e e q u a l t o z e r o . I t c a n be e a s i l y shown by U sing t h e m a t r ixx * ’ x*
s p e c t r a l d e c o m p o s itio n .L e t Л - d ia g (X1# . . . , xk ) b e t h e d ia g o n a l m a tr ix w ith t h e e ig e n v a lu e s o f ж *’ж* on t h e m ain d ia g o n a l and V ■ [ v ^ , . . . , V, 1 С be t h e m a t r ix o f n o r m a lis e d e ig e n v e o to r e o o rre e p o n d -К J in g t o X j, . . . » Xjj* I t h o ld e
A a V’K*’**V
t3 )
and i i• ( ж‘ т Л
( £
r n ) i T 7 1 ‘-The e i n g u l a r i t y o f ж * ’ж* m a t r ix means t h a t a t l e a a t one o f e ig e n v a lu e e J - T 7~k, e a y X- f i s e q u a l t o * e r o . T h e r e f o r e
n / k v 2
*e - 0 V r s . ) • 0 . (4 )
w hich means t h a t t h e e le m e n te v r a r ■ i , к » r e t h e p a r a m e te r s o f l i n e a r r e l a t i o n s h i p b e tw e en colum ns o f ж*.
The e x a c t m u l t i o o l l i n e a r i t y Is a r a r e phenomenon i n e o o n o m e tri- c a l m o d e ls . We a r e d e a l i n g m o s tly w ith n e a r - m u l t i c o l l i n e a r i t y p r o b le m s . We c a n a p p ly v a r i o u s m e a su re s o f t h e s t r e n g t h o f n e a r - m u l t i o o l l i n e a r i t y . T h e se m e a su re s oan b e d i v id e d i n t o two g ro ups» n u m e r io a l and e t a t i e t i c a l . H u m e rio a l m ea e u re s a r e b a s e d on c o n d i-t i o n num ber o f х * ’ж * and ж* m a i-t r i x , Ae a o o n d i t i o n num ber we u s e ( s e e B e l s l e y , K u h , W e l s o h 1 9 8 0 )
* « * ) . ( 5 )
w h e r e Amax( * * ’ * * ) and Х ^ С ж ^ 'ж * ) a r e r e s p e c t i v e l y t h e m axim al and m in im al e ig e n v a lu e o f
ж*’ж*.
Among s t a t i s t i c a l m e a su re s we d i s t i n g u i s h « ^ - s im p le c o r r e l a t i o n c o e f f i c i e n t s r ^ ( t h e e le m e n ts o f п ж* ’ж* m a t r i x ) , - p a r t 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 e R ^ В . - I - --- ( 6 ) 11w here r ^ i a Ci» J ) e le m e n t o f
■
—
(ж*’ж*)
,
- sam p le m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t s R^ b etw e en 1 ^ i ■ 1 , к and o t h e r e x p la n a to r y v a r i a b l e s , - v a r i a n c e i n f l a t i o n f a o t o r s (V IP ) r a ' i • r ł l ■ ■ ( 7 ) - m easu re b a s e d on a d d i t i o n a l c o n t r i b u t i o n s o f v a r i a b l e s ( s e e T h e i 1 1971) к w here R i s m u l t i p l e o o r r e l a t l o n c o e f f i c i e n t b e tw e en Y and X « ■ 1 * 1 » • • • » Xfci and R , i s m u l t i p l e o o r r e l a t l o n o o e f f l o i e n t b etw een Y andThe o o e f f l o l e n t s r ^ j and R ^ g iv e ua v e r y s m a ll amount o f in f o r m a t io n i n t h e o a s e when more t h a n two o f e x p la n a to r y v a r i a -b l e s (colum ns o f ж*) p a r t i c i p a t e i n t h e r e l a t i o n s h i p . A d d i t i o n a l l y "h e n a t l e a s t one e ig e n v a lu e o f ж*’ ж* te n d s t o z e r o a l l R ^ n r e e f f e c t e d i n t h e s e n s e t h a t V i f j lim R ,., « + 1 , w hioh c a n X — 0 X J . s e a s i l y shown a p p ly in g s p e o t r a l d e c o m p o s itio n ( s e e K o n a -r z e w s k a , M i l o 1 9 8 2 ). G r e a t e r amount o f in f o r m a t io n oan be o b t a i n e d from t h e a n a l y s i s o f R^ and VTP^ 1 • 1, к ab o u t t h e s t r e n g t h o f d e p e n d e n c ie s and v a r i a b l e s ln c lu d e d .W e h av e t o n o t i c e t h a t R я max R4 i s a bounded m easu re i n t h e ra n g e ( 0 , 1) b u t
' i 1
t h e l i m i t f o r e a c h VIP^ when A—^"*‘ 0 and v ^ a ^ 0 i s + o o .T h e m easu-r e 5 d e v e lo p e d by T h e 1 1 (1971) i s bounded i n t h e r a n g e ( - k + + 1 ,0 ) and i s z e ro i n t h e c a s e o f o r t h o g o n a l i t y o f ж*’ ж* and *k + 1 i n t h e c a s e o f e x tre m e o u l t i c o l l l n e a r l t y . B o th two g ro u p s o u l t i c o l l l n e a r l t y m e a s u re s a r e u s e f u l I n d e te r m in in g t h e num ber ®nd s t r e n g t h o f r e l a t i o n s h i p s among e x p la n a to r y v a r ia b le s .H o w e v e r , t h e a n a l y s i s o f e ig e n v a lu e s and e ig e n v e o to r s o f Jť*' x * m a t r ix e lv e s u s t h e p o s s i b i l i t y t o go d e e p e r i n t o t h e n a t u r e and c o n - | s e q u e n c e s o f t h e o b s e rv e d d e p e n d e n c ie s .
3. V a r i u n c e - D e c o m p o a i t i o n - P r o p o r t i o n a Method
The m ethod a llo w s u e t o f i n d
• t h e num ber o f r e l a t i o n s h i p s among colum ns o f я * ,
w hich v a r i a b l e s (colum ns o f **) a r e in v o lv e d i n an i n d i v i -d u a l r e l a t i o n s h i p ,
- w hioh o f t h e m odel p a r a m e te r s c a n b e e s ti m a t e d by l e a s t s q u a r e s m ethod w ith o u t g r e a t i m p r e c i s i o n .
The m ethod was f i r s t l y d e s o r lb e d and a n a ly z e d by B e 1 s - l e y , K u h , W e l s c h (1 9 8 0 ). The m ain I d e a o f t h i s m ethod, c a l l e d v a r la n c e - d e o o m p o s itlo n p r o p o r t i o n s m eth o d , i s a s f o l l o w s . L e t u s r e w r i t e t h e e s ti m a t e d sam p le v a r l a n o e o f t h e e s t i m a t o r b j . . h e r e ( « * ’ ■ [ r 11 . . . r 1* ] , I n t h e form к „2 . V i ) • л 11 ■ L ■ 1-1 1 ( 9 ) w here 6^ i s an e s ti m a t e d v a r l a n o e o f d i s t u r b a n c e te r m . We c o n s t r u c t a m a t r ix ЗГ a s t h e m a t r i x w ith t h e e le m e n ts w h e re i Vi l /X l (10) L & x r r»1 к I t c a n be n o t i c e d t h a t V i . 1 , P o r m a t r io e s x* 1-1
w ith m u tu a lly o r th o g o n a l oolum ns i t h o ld s ЗГ - 1 ^ . P o r a n exam ple we c a n c o n s i d e r t h e m a t r ix 7Г o f t h e form
4 v a r ( b * ) v a r ( b * ) ▼ a r( b j)
b 1 0 0
?2
0 0 .0 1 0 .1where ^ ł ■ 1, 3 are condition Indexes defined as foliow a:
W « * ’ « * ) *?г " ' “ “ ( x * 1* * ) r " 1 * k * C11> The m axim al c o n d i t i o n in d e x i s , by d e f i n i t i o n , e q u a l t o th e m a t r ix к * c o n d i t i o n num ber ЯХх*). I f 7 3 l e " g r e a t " (w hat means t h a t i s g r e a t e r t h a n 15) we can s a y t h a t among t h e e x p la n a to r y v a r i a b l e s e x i s t s one d ep en d en -c y . ( B e l s l e y , K u h , W e l s o h 1980) s t a t e t h a t a s t h e « u l t i p l e c o r r e l a t i o n c o e f f i c i e n t s w hioh c h a r a c t e r i s e t h e depend-ency i n c r e a s e a lo n g t h e p r o g r e s e io n < . 9 , . 9 , .9 9 , .9 9 9 , .9 9 9 9 , . . . t h e c o n d i t i o n in d e x e e I n o r e a s e a lo n g t h e p r o g r e e s io n 3 , Ю , 30, 10 0 , 300, . . . T h is d ep e n d en c y l e b etw e en Xg 810(1 *3 1111(1 e f f e c t s ▼ arian ces o f e s t i m a t o r s bg and b ^ .G e n e r a l ly s t e p s o f t h e v a r i a n c e - • d e e o m p o s itio n p r o p o r t i o n s m ethod c a n b e sum m arized a s f o ll o w s :1 . S t a n d a r d i s a t i o n o f m a t r ix
к .
2 . C om puting o o n d l t l o n num ber and c o n d i t i o n in d e x e s J f t * * ) , J?r
* ■ Г, k.
3. C h o ice o f l i m i t v a lu e s f * a n d T ľ T f .i . i)** 1 5 , 0 . 5 ) . 4 . C om puting 7Г m a t r i x .
5 . E xam ining e le m e n ts o f rows o f c o r r e s p o n d in g t o цт > * f t h e r e e x i s t a t l e a s t two o f them f o r w h ich IT
E x i s te n c e o f one r e l a t i o n s h i p c a u s e s no p ro b le m s f o r d ia g n o s -i s . P ro b lem s a r -i s e when two o r m ore r e l a t i o n s h i p s c o e x i s t . The * l r s t k in d o f p ro b le m s i s w ith d o m in a tin g d e p e n d e n c ie s (o n e o f " g r e a t " c o n d i t i o n In d e x e s i s much h i g h e r t h a n o t h e r s " g r e a t " con-d i t i o n I n con-d e x e s ) . The se co n con-d k in con-d o f p ro b le m s I s w ith c o m p e tin g d e p e n d e n c ie s (tw o o r more " g r e a t " c o n d i t i o n In d e x e s w ith s i m i l a r c o n d it io n i n d e x e s ) . To s o lv e t h e a r i s i n g p ro b le m s one h a s t o b u ild a u x i l i a r y r e g r e s s i o n s t o c h e ck w hioh v a r i a b l e s a r e in v o lv e d In w hich r e l a t i o n s h i p .
I n o u r o p in io n v a r l a n c e - d e c o m p o s it lo n p r o p o r t i o n s m ethod ^ s p i t e o f t h e above m e n tio n e d p ro b le m s i s a suprem e one i n com-p a r i s o n w ith t h e m ethods b a s e d on t h e a n a l y s i s o f m u l t i p l e c o r -r e l a t i o n c o e f f i c i e n t s b e c a u s e i t g i v e s u s d e e p e -r I n s i g h t i n t o t h e n a tu r e o f d e p e n d e n c ie s and p o s s i b i l i t y o f q u a n t i f i c a t i o n o f n e a r — • o u l t i c o l l i n e a r i t y c o n s e q u e n c e s on e s t i m a t i o n p r e o i s i o n . We s h o u ld h o t f o r g e t a b o u t t h e se c o n d f a c t o r o f sam p le e s t i m a t o r 's v a r i a n c e
- 6 2 . A lth o u g h t h e e s tim a te d v a r i a n c e oan be e x te n d e d by t h e com-p o n e n t c o n n e c te d w ith h ig h o o n d l ti o n i n d e x , a t t h e same tim e i t may be s h ru n k e n to w a rd s z e ro by n e a r - z e r o v s l u e o f ž2 ( i n t h e c a s e o f v e r y h ig h v a lu e o f R2 ) .
Now, we w i l l show two p r a o t i o a l exam ples o f v a ria n o e -d e o o m - p o s i t i o n p r o p o r t i o n s m ethod t o f o llo w in g e q u a t i o n s »
I PSQPPt - f 3 (PSQFPt - 1 , WUQPt , PYPRt , 0 7 5 ) , t - 196 3 -1 9 7 7 , I I ХОШЦ - f 2 (KQMWt , NUQUt , T ) , t - 196 1 -1 9 7 9 , w h e re i FSQFP - means m o n th ly wage I n f u e l and pow er i n d u s t r i e s , WUQF - p r o d u c t i v i t y o f work i n f u e l and pow er i n d u s t r i e s , PYPR - I n d e x o f l i v i n g o o s t s o f mean em ployee's f a m i ly , XQifH I n o f p r o -d u o t io n i n m e t a lu r g io , o h e m io a l an-d m in e r a l i n d u s t r i e s ( m .o .m .) , KQMW - I n o f p r o d u o tlv e o a p l t a l s to o k i n m .o .m . I n d u s t r i e s , WUQM - num ber o f em ployees i n m .o.m . i n d u s t r i e s , I n , T - tim e t r e n d , U75 - dummy v a r i a b l e - U75 - 1 I n 1975, U75 ■ 0 i n o t h e r y e a r s o f sam p le p e r i o d . We o b ta in e d t h e f o ll o w i n g r e s u l t s t I . ' 1 .OOOO 0.9201 1.0000 * # ,x * -o .9 e4 3 O .94O4 1.0000 0.3029 0.3098 0.2984 1.000C 7 b 2 »1 *4 1.0000 0 .0 0 3 2 . O .Ö II4 0.0024 0 .0 1 8 6 5.8290 0.0893 0.8399 0.0210 0.0 02 0 14.8568 0.9067 O .I 465 0.9759 О.ОО36 1.8734 0 .0 0 0 8 0.0022 0 .0 0 0 7 0.9758
We oan o b s e rv e one n o t v e r y s t r o n g dependenoy b e tw e e n P S Q P P .^ and PYPRę.Only v a r ( b p and v a r ( b ^ ) a r e e f f e c t e d by t h i s r e l a t i o n -s h i p . I I . ' 1.0000 0.9281 1.0000 0 .9 9 8 2 0 .9 4 8 8 1 .0 0 0 i ł , x ł
-4
b í
b 2b 3
1.0000
0.0002
0.0050
0.0001
5.8576
0.0042
0.3546
0.0014
66.5111
0.9957
0.6404
0.9985
S i m i l a r l y , we oan o b s e rv e one s t r o n g d ep en d en cy b etw e en KQMW^ and T - MUQM v a r i a b l e i s a l s o i n t e r r e l a t e d w ith t h e s e two b u t i n • b i t w eak er way. A l l e s t i m a t e s o f p a r a m e te r s a r e e f f e c t e d by n e a r - o u l t i o o l l l n e a r l t у . R e f e re n c e s B e l e l e y D. A ., K u h E . ,W e l s c h R .E ., 1990, Re-g r e s s i o n D i a Re-g n o s t i c s , W ile y . K o n a r z e w s k a I . , M i l o W., 1982 .D ia g n o s ty k a w s p ó ł- l l n l o w o ś c i m iędzy zmiennym i o b ja ś n ia ją o y m i m odelu e k o n o a e try c z n e - 8o - om ów ienie m eto d , t y p e s c r i p t , R. I I I . 9 . 4 .3 »
T h e i 1 H ., 1971, P r i n c i p l e s o f E c o n o m e tric s , W ile y .
Iw ona K o n arzew ska, W ładysław M ilo
KILKA UWAG HA TEMAT MOŻLIWOŚCI STOSOWANIA METODY UDZIAŁÓW W ZDEKOMPONOWANEJ WARIANCJI
W a r t y k u l e rozw ażono m etodę d ia g n o z y związków m iędzy zm ienny-mi o b ja ś n ia ją c y m i lin io w e g o m odelu r e g r e s j i . P o dstaw ą t e j m etody j e s t a n a l i z a num eryczna m a c ie rz y o b s e rw a o ji n a ty c h zm ien n y ch . M etoda j e s t sk o n s tru o w a n a p r z y z a s to s o w a n iu r e g r e s j i w edług w art o ś c i o s o b liw y c h . O b lic z a n e s ą p r o p o r c j e u d z i a ł u każd eg o s k ł a d n i -c a t e j w a r i a n -c j i w -c a ł e j s u m ie . M etoda ta ,n a z y w a n a m etodą u d z i a - Ww w zdekomponowanej w a ria n c ji,w p ro w a d z o n a p r z e z B e 1 s 1 e y . К ц h , W e l s c h ( 1980) p o zw ala obok m o ż liw o ś c i w y k ry c ia i l o a c i związków t a k ż e n a s p e c y f i k a c j ę zm iennych zw ią za n y c h r e l a c j a m i . Ds l ę k i tem u m ożliw a j e s t d ia g n o z a , k t ó r e w s p ó łc z y n n ik i w m odelu ®ogą być oszacow ane w z g lę d n ie p r e c y z y j n i e .
A u to rz y p r z e d s t a w i l i w y n ik i z a s to s o w a n ia t e j m etody n a p r z y -k ła d a c h zbudowanych n a b a z i e ban-ku dan y ch m odelu W-3 g o s p o d a r-k i n arodow ej P o l s k i o r a z w ła s n ą o p i n i ę n a te m a t m o ż liw o ś c i w ykorzy— ■ ta n ia t e j m eto d y .
