A C T A U N I V E R S I T A T I S L 0 D Z I E N S I 6 F O L IA OECOf.OMICA 48, 1985
C ze sław a J a c k i e w i c z * , H ol ln u K lo p a c z * ^ , E l ż b i e t a Ż ó łto w s k a '*
THE ESTIMATION OF C ES PRODUCTION FUNCTION PARAMETERS BY THE A X IA L DOUBLE ITERA TIO N METHOD
1 . In t r o d u c t io n
The e a r l i e r i n v e s t i g a t i o n s [ l ] , [
2
] , [7
] on the e s t im a t io n methods f o r CES p r o d u c tio n f u n c t io n in tho form( 1 ) Y - a [S K * ^ + ( l - 5 ) L~^ ] ^ + E, w h e r e t Y - p r o d u c tio n o u t p u t , K - f ix e d a s s e t s , L - em ploym ent, « - p r o d u c tio n s c a le p a ra m e te r , S - d i s t r i b u t i o n p a ra m e te r, v - f u n c t io n hom ogeneity p a ra m e te r, g - s u b s t i t u t i o n p a ra m o te r, E - random term , p ro ve d t h a t the e s t im a t io n o f a fo u r- p a ra m e te r f u n c t io n can be l im it e d to tho e s t im a t io n o f a tw o-p aram eter f u n c t io n , a f t e r h a v in g e s tim a te d tho p a ra m e te r v by K m e n t a method (¡.3 J ) and d e te rm in e d tho e s t im a t e s o f p a ra m e te r a d i r e c t l y from the f i r e t e q u a tio n o f the system o f norm al e q u a t io n s . H ence, tho
* D r . , L e c t u r e r a t th e I n s t i t u t e o f E c o n o m e t r i c s a n d o t a -t i s -t l e s , U n i v e r s i -t y o f Ł ó d ź . _ .
* * S e n io r A s s i s t a n t a t th e I n s t i t u t e of 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 .
la w i J a c k lo t f ic z . (l.-iin a K lupaoz. S i a b i e t a fco lto vak *
above problem hoo b-.-ron c o n iin e d to tho d e te r m in a tio n o f such v a lu e s c f A , b , r f o r w hich the f u n c t io n
n ( 2 ) Q (A , b, r ) x Y ] Yi
i-1
would a t t a i n i t s minimum o t on a p r i o r i g iv e n v a lu e o f c and c t th e v a lu e s o f A d e te rm in e d a c c o r d in g to
¿ Yi [ bKi r ♦
A “ ^ r r r --- T T ^ " '
(Tho form o f ( 3 ) was g e n e ra te d in [
2
] from th e system o f norm al e q u a tio n s b u i l t f o r th e f u n c t io n ( 2 ) ) .To d e te rm in e the v a lu e s o f b and r a so c o l le d d o u b le i t e r a t i o n wethod was su g g ested in [
2
] . The method c o n s is t s in a o te p - b y - s te p s e a r c h in g f o r extrem e v a lu e s o f b and r . Then, s i m i l a r c a l c u l a t i o n s a re made f o r tho o th o r p a ra m e te r. The who l e i t e r a t i o n p r o c e s s , c o n s t r u c t in g two i n t e r r e l a t o d b lo c k s , i s c a r r i e d ou t so lo ng t h a t th e two norm al e q u a tio n s f o r th e fu n c t io n (2
) a re s o lv e d w it h a p re d e te rm in e d a c c u r a c y .. The M o nte-C orlo e x p e rim e n ts co n firm e d t*>e n u m e r ic a l e f f i c i e n cy o f the d ou b le i t e r a t i o n method, i . e . r e g a r d le s s th e c h o i ce o f a s t a r t i n g p o in t the p re d e te rm in e d p a ra m e te rs w ere a lw a y s o b ta in e d . H ow ever, th e i t e r a t i v e p ro c e s s - a lth o u g h c o n v e rg in g w it h the p re d e te rm in e d v a lu e 3 o f p a ra m e te rs - was too alo w ar.d c o s t l y . T h e r e fo r e i n v e s t i g a t i o n s a re c a r r i e d o u t to make i t more n u m e r ic a lly e f f i c i e n t s in c e the s h o rte n in g o f tim e need ed f o r c a l c u l a t i o n s i s n e c e s s a r y a ls o in the ca s e when tho p ro p e r t i e s o f e s t im a t o r s o f model p a ra m e te rs a re s tu d ie d by means o f n u m e r ic a l e x p e rim e n ts .
The p op er p r e s e n t s a m o d if ic a t io n o f th e d o u b le i t e r a t i o n m ethod, f u r t h e r c a l l e d th e a x i a l d o u b le i t e r a t i o n method.
2. The A x ia l Oouble I t e r a t i o n Method
I t f o llo w s from the i n v e s t i g a t i o n s [
5
] th a t th e c r i t e r i o n f u n c t io n Q ( A (b , r ) , b . r ) in the form ( 2 ) d e f in e s the p a ra b o lo id - sh ap ed a r e a , and l t e l e v e l l l n e e a re the c u rv e s c lo s e to e l l i p s e s w it h common axes (fro m the e a r l i e r s tu d y [4
] i t fo llo w e d t h a t the " a x i s " o f l e v e l l i n e s symmetry was a p p ro x im a te ly a s t r a i g h t l i n e ) . The p o in t o f i n t e r s e c t i o n o f th e se axes d e t e r m ined, w it h some n u m e ric a l a c c u r a c y , the p o in t in w hich fu n c t io n Q ( A ( b , r ) , b , r ) a t t a in e d i t s minimum. T h is was the ba s i s f o r m o d ify in g the way o f d e te r m in a tio n o f the extrem e v a lu e s o f b and r , i . e . f o r assum ing a n o th e r way o f s o lu t io n o f the system o f e q u a t io n s :( 4 )
F 2( A ( b , r ) , b , r ) - — Q ( A ( b , r ) . b , r ) ■ 0 ,
F 3( A ( b , r ) , b . r ) - Q ( A ( b , r ) . b , r ) - 0 .
T h is way was d i f f e r e n t from the d o u b le i t e r a t i o n method p ro posed and d is c u s s e d in [
2
] . N am ely, a t g iv e n r Q p o in t bQ i s lo o k e d f o r in such a way t h a t the f u n c t io n F 2 d i f f e r s from z e ro l e s s than by a g iv e n a c c u r a c y VVW, and s i m i l a r l y - f o r th e g i v en v a lu e o f r a c o rre s p o n d in g v a lu e o f b^ i * s e a rc h e d f o r .The o b ta in e d p o in t s (b Q, r Q) and (b ^ , r^ ) a re the b a s is f o r d e te rm in in g the s t r a i g h t l i n e c o rre s p o n d in g to th e " a x l e ” o f sym m etry f o r l e v e l l i n e s . T h is a x ie I s d e te rm in e d as a s t r a i g h t l i ne by th e e q u a t io n ! ( 5 ) b - D1 r ♦ 02 , w here s b. - b ° 1 ' r r - " F I ' ° 2 " bo - V o ’
Assum ing t h a t the e t r a l g h t l i n e ( 5 ) d e te rm in e s the a x is o f e l l i p s e s , on t h i s l i n e we s e a rc h th e p o in t ( b ( r ) , r ) in w h ich the f u n c t io n Q h as a c o n d i t i o n a l minimum. T h is c o rre s p o n d s to the d e te r m in a tio n o f a z e ro p la c e o f a r e s p e c t iv e d e r i v a t i v e o f
_ C' I ■ ~ k i bw i c a , • H»i lin » i\!u uz, 2l».bleta Ż o l t o w a ^ .
¡u n ctx o n . ( A ( b ( r ) , r ) , b ( r ) , r ) , I . e . to tho u o .lt.iio n of titO e q u a tio n
( t-) Nr ^ * F j ( A , b , r ) + D1F 2( A , b , r ) • O ,
w heror
A • A ( b , r ) , b •• D^r D g .
P o r a n u m e ric a l r e a l i z a t i o n o f the a x i a l d o u b le I t e r a t i o n ftiwthod the program # CE3 7 has Loon d e v e lo p e d (< h ls program
i i a v a i l a b l e a t the L i b r a r y o f Pro g ram s, I n s t i t u t e of Econome t r i c s end 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 ). The r e n u lt a o f Mon- t * - C a r lo e x p e rim e n ts o b ta in e d u sin g the progiem # C ES 7 a llo w uo to e v a lu a t e p r o p e r t ie s o f the p re s e n te d method, and e s p e c i a l l y to answ er tho f o llo w in g q u e s t io n s :
1. Does tho d e te rm in e d s t r a i g h t l i n e ( 5 ) depend on the as- tu^ed v a lu o s o f s t a r t i n g p o i n t s ? i f s o . In whot w ay?
2. .What I s the In f lu e n c e o f the c o r r e l a t i o n l e v e l o f v a r i a b le s K and L on tho v a lu e s ta k e n by and 0^7
3. What l a the r e l a t i o n s h i p between the d i r e c t i o n s o f tho s t r a i g h t l i n a ( 5 ) and tho v a lu a 3 o f p a ra m e te r £ d e te rm in in g the s u b s t i t u t i o n e l a s t i c i t y d » in th e f u n c t io n ?
4. What i s the b e h a v io u r o f the e s t im a te s o f f u n c t io n p a ra m eters ( 1 ) in r a l o t i o n to th e number o f o b s e r v a t io n s (n ) and ra n dów term E ?
3, A N u m e ric a l Model
To ans.vor th e above q u e s tio n s the f o llo w in g n u m e r ic a l model o f e x p e rim e n ts has been c o n s tr u c te d «
u) tho sequence o f n v a lu e s L^, l_2 , . . . , L^, , has biien d e f in e d ,
b ) as3u nin g s d i f f e r e n t l e v e l s o f c o r r e l a t i o n between v a r i a b le s K and i~ (m easured by th e v a lu e s o f c o r r e l a t i o n c o e f f i c i e n t s ¿-(K, L ) t h t o Lequencoo of the v a lu e s on v a r i a b l e K, 1 .e. ^ s “ *' K 4. , . 4- ¿i K w ere d e te rm in e d ,p
Cj f o r the a c t of p a l r o j i L ^ *<l 8 ) , 1 - 1 , n
the £i03umod v a lu e s o f p a ra m e te rs a , 5 » £ the v a lu e s o f Y T ^e " - a [ S K & * ( l - © l J ] - 1/« w ere c a l c u l a t e d .
P a ra m e te rs a and 5 , as shown in [ 5 j » can bo d e te rm in e d o t an a r b i t r a r y l e v e l by assum ing measuring u n i t s f o r K, L and V. In the e x p e rim e n ts v a r i o u s l e v e l s w ere assuatod f o r tb a parom etsr £ •
In t h i e way the s s e t s o f p o in t s Dfi 1_^, Ki s ' YTi s ) - 1 • *• . . . . n j on r e s p e c t iv e f u n c t i o n a l s u r f a c e s w ere o b ta in e d .
I n tho e x p e rim e n ts tho f o llo w in g lo v o la o f p a ra m e te r« wore assum ed: cur 2, S t 0 . 4 , Q : - 0 .3 , 0 . 2 , 1. (o (K , L ) i 0 .3 2 4 , 0 .7 2 3 , 0 .9 3 0 , and s t a r t i n g v a l u e s : r Q t 0 . 3 , 0 . 5 , T j j - 0 .5 , - 0 .4 , - 0 .3 , - 0 .2 , - O . i , 0 . 1 , 0 .1 5 , 0 .1 9 , 0 .2 0 0.21, 0 .2 5 , 0 . 3 , 0 . 4 , 0 . 5 , 0 . 6 , 0 . 7 , 0 , 0 , 0 . 9 , 1 .0 , 1 . 2 , 1 .5 . The r e s p o c t lv e p o in t s fcQ * b C r^ ) and b^ ■ b ( r ^ ) w ere d e te rm in e d
in such a way th a t
( 7 ) g ( A ( b , r ) , b, r ) | < 1 0 ” f>.
The s e t a 0 Q viero t r e a t e d no sam p le s. Thoy s e rv o d f o r ch ock in g th a n u m e ric a l e f f i c i e n c y o f tho a x i a l d o u b le i t e r a t i o n mothou ( AOXM) a t v a r io u s c h o ic e o f s t a r t i n g p o in t s r Q and r j . They w ere a ls o a b a s is f o r tho a n a l y s i s o f the a c c u r a c y o f AOIM o- e tim a t e s o f f u n c t io n p a ra m e te rs ( l ) i n th e c a s e o f randon t e r n
o c c u r r in g in the v a r i a b l e Y. m I P
F o r each s e t 0 „ 8 the I P new s e t s o f D l# * " * ° s * 8 ° e w ere d e te rm in a d so th a t
° a m m { ( L i * K i ' Y l a m )* 1 - 1 ... n } *
where Y , „ m i e a r e a l i z a t i o n o f a random v a r i a b l e d e fin e d as
10
( 8 ) Yt 3 m - YTifl ♦ C j® , m - 1 ... IP ,
w it h the d l o t r l b u t i o n N (0 , d j.)(c was g e n e ra te d u sin g the p ro c e d u re NQRGEfJ w it h a lp h a ) . The v a r i a n c e was chosen so os to be O .liv and l. O ij o f v a r i a b i l i t y o f Y . F o r each s o t Da a sequence o f s e t s i ° c ra« m ■ 1, . . . , I p J was o b t a in e d .
4. L o c a t i on o f the L e v e l A x is
On th e l o v e l a x is th e ro a re p o in t s w ith c o o r d in a t e s ( r , b ( r o) ) . ' They were d e te rm in e d assuming r Q and d e f in in g b ( r ) In such a way t h a t r e l a t i o n ( 7 ) h o ld s . T a b le 1 p r e s e n t s c o o rd in a te© o f th e p o in t s on the l e v e l axes d e te rm in e d f o r t h r e e v a r io u s f u n c t io n a l a re a s d i f f e r e n t i a t e d by the c o r r e l a t i o n d eg ree o f K and L . The T a b le a ls o p r e s e n t s the v a lu e s o f r e s id u a l s o ( r )Q o b ta in e d as a d i f f e r e n c e between th e v a lu o s o f b ( r ) and bCr ) .
* 0 0
The v a lu e s o f b ( r Q) were c a lc u la t e d from the LSM r e g r e s s io n l i n e »
( 9 ) b ( r ) - o j r ♦
0*
d e te rm in e d f o r the o b ta in e d s e t s o f p o in t s w ith c o o r d in a t e s ( r , b ( r 0 » . T h ree r e g r e s s io n l i n e s f o r th r e e s u b s e ts y 1 »K1 »L i 1 - 1 , n c o rre s p o n d in g to t h r e e l e v e l s £ ( K , L ) - 0 .3 2 4 , 0 .7 2 7 , 0 .9 3 5 w ere d e te rm in e d . These a r e : ( 1 0 ) b ■ 0 .1 1 9 B 2 9 6 r + 0.375885 f o r £> (K,L) - 0 .3 2 4 , (1 1 ) b - 0.1372636r V 0.3726333 f o r g ( K , L ) - 0 .7 2 7 , ( 1 2 ) b • 0.1547703r ♦ 0.3697843 f o r g ( K , L ) - 0 .9 3 6 .From the r e s u l t s p re s e n te d In T a b le 1 i t f o llo w s t h a t in each ca se b eing c o n s id e re d th e v a lu e s o f b ( r Q ) in c r e a s e w it h the i n c r e a s e in r . A t a f ix e d l e v e l o f r the v a lu e s o f b ( r ) in
-u O O .
c r e a s e w it h th e In c r e a s e in th e c o r r e l a t i o n d eg ree o f v a r i a b l e s K , L when r Q > ^ , and t h e y d e c re a s e when r Q< ^ . H ence, the d e te rm in e d r e g r e s s io n l i n e s (1 0 ) - (1 2 ) a r e c h a r a c t e r iz e d by the In c r e a s e o f In t e r c e p t v a lu e o f 0 * and the d e c re a e e o f random term D* f o r h ig h e r v a lu e s o f £>(K, L ) .
re-T a b l e 1 V a lu e s o f c o o r d in a t e s b ( r ) f o r s e le c t e d v a lu e s o f r c *• £ ( K , L ) - 0 .3 2 4 / p (K ,L ) - 0 .7 27 ç ( K , L ) - С>.936 . 0 b ( r Q) e ( r 0 ) b ( r o ) n ( r o ) cr О • ( r o ) 1.0 0 0.4950 -0.00C71 0.5 09 3 -0.00060 0.5 25 2 ♦0.00065 0 .9 0 0.4 83 4 -0.00033 0.4961 -0.00007 0.5098 ♦0.00072 0 .8 0 0.4 71 7 -0.00005 0.4 82 6 ♦0.00016 0.4 94 3 ♦0.00070 0 .7 0 0.4599 +0.00013 0.4691 ♦0.00038 0.4 78 7 ♦0.00058 0 .6 0 0.4481 +0.00032 0.4554 ♦0.00041 0.4630 ♦0.00035 0 .5 0 0.4361 ♦0.00030 0.4 41 6 ♦0.00033 0 .4 47 2 ♦0.00003 0 .4 0 0.4241 ♦0.00028 0.4278 ♦0.00026 0 .4 31 4 -0.00029 0 .3 0 0.4121 ♦0.00027 0.4139 ♦0.00009 0.4 15 7 -0.00052 0 .2 5 0.4060 ♦0.00018 0.4069 ♦0.00005 0.4078 -0.00068 0 .2 1 0.4012 ♦0.00015 0.4014 -0.00006 0.4 01 6 -0.00069 0 .2 0 0.4000 ♦0.00015 0.4000 -0.00009 0.4000 -0.00074 0 .1 9 0.3908 ♦0.00015 0.3986 -0.00011 0.3904 -0.00079 0 .1 5 0.3 94 0 ♦0.00014 0.3931 -0.00012 0.3922 -0.00080 0 .1 0 0.3879 ♦0.00003 0.3861 -0.00026 0.3044 -0.0008G -0 .1 0 0.3637 -0.00020 0.3585 -0.00041 0.3536 -0.00071 -0 .2 0 0.3517 -0.00022 0.3448 -0.00038 0.3385 -0.00033 - 0 .3 0 0 .3 39 7 -0.00024 0.3 31 3 -0.00015 0.3 23 6 ♦0.00025 -0 .4 0 0 .3 2 7 7 -0.00025 0.3 17 6 ♦0.00007 0.3089 ♦0.00102 •*0.50 0.3159 -0.00007 0.3 04 6 ♦0.00060 0.2945 ♦0.00210 N o t e i I n the column e ( r 0) t h e v a lu e s o f r e s i d u a l s I . e . e ( r 0 ) - b ( r 0) - b ( r 0 ) a re g iv e n .
s l d u a l s s ( r o ) . N am ely, th e g ro u p in g o f r e s i d u a l s o f the same s ig n I s an e v id e n c e t h a t th e r e l a a u t o c o r r e l a t i o n In the r e s id u a l s s e r i e s . F r o « a fo rm a l p o in t o f v ie w t h i s a u t o c o r r e l a t i o n can p r o v id e e v id e n c e t h a t , f i r s t , t h e r e a re s y s t e m a t ic e r r o r s in n u m e r ic a l c a l c u l a t l o n a , and s e c o n d ly , th e re can be an e rro n e o u s h y p o th e s is t h a t th e symmetry a x is o f l e v e l l i n e s i s a s t r a i g h t l i n e . Changing a b s o lu te v a lu e s o f r e s i d u a l s back up th e second v a r i a n t r a t h e r , a lth o u g h th e y do n ot e x c lu d e th e f i r s t one. I t i s w o r th w h ile to n o te t h a t w it h th e in c r e a s e o f c o r r e l a t i o n d e g re e
of variables tho distribution of positivo and nogative rooiduals
changos and their absoluto valuó incre&303.
T a b lo o 2-4 p ro e o n t the v a lu e s o f c o o f f i c i a n t s 0^ and D2 of • t r o ig h t l i n o s d e to rm in a d in ( 6 ) . They a ro o b ta in e d in such a way t h a t a l l o f them have ono common p o in t r Q ■ 0 .5 and b ( r ) and the p o in t ( r ^ , b ( r ^ ) ch an gos. I n most o f the in v e s t ig a t e d
T a b l e 2 C o e f f i c i e n t s and 02 o f s t r a i g h t l i n e ( 5 ) d e te rm in e d f o r ( r 0 ,b o )- ( 0 . 5 , 0 .4 3 6 1 ) a t g ( K , L ) * 0 .3 24 r
i ri
° 1 ° 2 d * ( l , ( 0 . 2 , 0 . 4 ) ) 1.0 0 0.117630 0.377326 0.000846 ( P ) 0 .9 0 0.118138 0.377072 0.000695 ( P ) 0 .8 0 0.118747 0.376727 0.000473 ( P ) 0 .7 0 0.119035 0.376624 0.000428 ( P ) 0 .6 0 0.119415 0.376434 0.000314 ( P ) 0 .4 0 0.120039 0.376121 0.000129 ( P ) 0 .3 0 0.120282 0.376000 0.000056 ( P ) 0 .2 5 0.120381 0.375951 0.000027 ( P ) 0 .2 1 0.120455 0.375916 0.000005 ( P ) 0 .2 0 0.120471 0.375906 0.000000 0 .1 9 0.120486 0.375898 0.000005 ( n ) 6 .1 5 0.120547 0.375868 0.000023 ( n ) 0 .1 0 0.120608 * 0.375038 0.000041 ( n ) - 0 .1 0 0.120716 0.375783 0.000073 ( n ) -0 .2 0 0.120687 0.375790 0.000064 ( n ) * 0 .3 0 0.120599 0.375841 0.000039 ( n ) -0 .4 0 0.120453 0.375915 0.000007 ( p ) -0 .5 0 0.120248 0.376018 0.000008 ( p )* Sym bol d 0 , ( 0 . 2 , 0 . 4 ) ) d e n o te s the d is t a n c e betneun the r e a l p o in t ( 0 . 2 . 0 . 4 ) and the s t r a i g h t l i n e i l г b - D, r ♦ 0, sym- b o ls ( p ) and ( n ) in th e f o u r t h column d en o te t h a t th e p o in t ( 0 . 2 , 0 . 4 ) i s b elow th e s t r a i g h t l i n e 1 and above the s t r a i g h t l i n e 1, r e s p e c t i v e l y .
c a s e s th e r e a l p o in t ( ¿ o » i ) l i e s below th e s e s t r a i g h t l i n e s w h ich back up th e s ta te m e n t t h a t th e d e t e r a in e d o s tim a to a o f p a ra m e te r g can be o v e r e s tim a te d to a g r e a t e x t e n t . H ow ever, when r^ ie
choaan n e a r the a c t u a l v n lu o o f p a ra m e te r 5 the e t r n ig h t l i n e a boing d ete rm in ad n re q u it s n ear to th a p o in t ( ç , 5 ) .
T n b 1 о 3 C o e f f i c i e n t s О} d e to rm in ed f o r ( r Q, b0) end 0-> o f o t r a l g h t li n o ( 5 ) « ( 0 . 5 ‘, 0 .Л 4 1 6 ) J t ^ .( K ,U ) • 0 ,7 2 7 r l ° 1 D2 d ( 1, (.0. 2, 0 . 4 » 1.00 0.135495 0.373047 0.000937 ( p ) 0 .9 0 0.130142 0.373524 0.000746 ( P ) 0.00 0.135724 0.373233 0.000572 ( Р ) 0 .7 0 0.137238 0.372976 0.000420 ( P ) о . во 0.137676 0.372757 0.000209 ( P ) 0 .4 0 0.138320 0.372432 0.000096 ( P ) o .s n 0.138500 0.372350 O.OOOOiiO ( P ) 0 .2 5 0.130599 0.372295 0.000015 ( P ) 0 .2 1 0.138641 0.372274 0.000002 (P> 0 .2 0 0.130650 0.3/2270 0.000000 ( n ) 0 .1 9 0.138658 0.3722C6 0.000002 0 .1 5 0.130670 0.372256 0.000000 ( n ) 0 .1 0 0.130683 0.372254 0.000009 ( n ) -0 .1 0 Q . 138403 0.372353 0.000049 ( Р ) - 0 .2 0 0.130240 0.372471 0.000119 ( P ) -0 .3 0 0.137922 0.372634 0.000216 ( P ) -0 .4 0 0.137507 0.372842 0.000340 <P> - 0 .5 0 0.137002 0.373094 0.000490 (P) .... N 0 t e i C f . T a b le C o e f f i c i e n t s t 2. T ), and D , o f O t r a ig h t l i n e ( 5 ) D0 ) - ( 0 . 5 , 0 .4 4 7 2 ) a t ¿ » ( K . U a b l e A >■ • 0 .9 3 6 d e te rm in e d f o r ( r 0#l r l ° 1 ° 2 d ( l.C O .2 , 0 . 4 ) ) 1 2 3 4 1.00 0.155913 0.369247 0.000424 ( p ) 0 .9 0 0.156469 0.360969 0.000260 ( p ) 0 .8 0 0.156923 0.368742 0.000125 ( p )
T a b le 4 (c o n t d . ) 1 2 3 4 0 .7 0 0.157271 0.368568 0.000022 ( p ) 0.0/0 0.15751 6 0.368445 0.000051 (n ) 0 .4 0 0.157655 0.360376 0.000092 ( n ) 0 .3 0 0.157559 0.368424 0.000063 ( n ) 0 .2 5 0.157465 0.368471 0.000036 ( n ) 0 .2 1 0.157370 0.368518 0.000008 ( n ) 0 .2 0 0.157333 0.368533 0.000000 0 .1 9 0.157316 0.368545 0.000008 ( p ) 0 .1 5 0.157195 0.368606 0.000044 ( p ) 0 .1 0 0.157015 0.368696 0.000098 ( p ) -0 .1 0 0.156008 0.369199 0.000396 ( p ) - 0 .2 0 0.155334 0.369536 0.000596 ( p ) -0 .3 0 0.154551 0.369928 0.000785 ( p ) -0 .4 0 0.153660 0.370374 0.001093 ( p ) -0 .5 0 0.152665 0.370871 0.001388 ( p ) N o t e i C f . T a b le 2. 5. R e s u lt s o f th e H o n te - C a rlo E x p e rim e n ts
Tho M o nto-C ario e x p e rim e n ta con cern ed a choeen sam ple
Ki# L ) . 1 » 1 , . . . . n j f o r w hich the c o r r e l a t i o n c o e f f i c i e n t ^ (k, L ) was about 0 .7 2 3 . Tho sam ple c o n t a in e d : n - 20, 30, 40 e le m e n ts , r e s p e c t i v e l y . I n each sam ple th e re w ere random ly d i s t r ib u t e d the v a lu e s o f v a r i a b l e Y assum ing Y^ d e fin e d by (8 ) in s t e a d o f YT¿ ,
Th u s, th r e e s e t s o f sam ples in th e form
{ { ( Y t , 1 ■ 1 , . r t | n j ffl ■ 1 , . . . I P ^ t
,/ere o b ta in e d (a c c o r d in g to th e number n ). They w ere used to de te rm in e the r e a l i z a t i o n o f sample e s t im a t o r s o f e x p e cte d v a lu e s or p a ra m e te r e s t im a te s in model ( l ) and t h e i r s e le c t e d c h a r a c t e r i s t i c s such as v a r ia n c e e s t im a t e s , v a r i a b i l i t y c o e f f i c i e n t s , b ia s v a lu e s , RSME ( v a r ia n c e s o f e v a lu a t io n s oround th e a o t u a l v a lu e s o f p a r a m e te r s ).
The above m easures wore d e te rm in e d f o r I P » 5 , 10, 15, 20, 25, . . . , 95, 100 in o rd e r to o b se rv o t h e i r chongos when the numbar o f sam ples In c r e a s e d and then to fo rm u la te c o rre s p o n d in g h yp o th e ses on tho p r o p e r t ie s o f th e e a tim o t o r s o f model p a ra m e te rs ( l ) o b ta in e d u sin g the a x i a l d ou b le i t e r a t i o n method.
I t f o llo w s from the ex p e rim e n to c a r r ie d o u t by the a u th o rs t h a t w it h the In c r e a s e in tho number e f I P tho f l u c t u a t io n s o f mean e s t im a te s c o rre s p o n d in g to A , b , r d e c re a s e but the e s t i m ates of p a ra m e te r a a re not b ia s e d n u m e r ic a lly , -w h llo th e e s tim a te s o f p a ra m e te r 5 a re g e n e r a l ly o v e r e s t im a t e d . Mean v a lu e s o f b io s a re lo w e r than th e s ta n d a rd d o v i o t io n . Tho e s t im a te s of p o ra m ete r g a re c h a r a c t e r iz e d by h ig h e r v a r i a b i l i t y . The v a lu e o f r i s e s tim a te d to be tho w o r s t. Tho in c r e a s e In th e somple s iz e ( n ) a f f e c t s p o s i t i v e l y tho r e s u l t s o f e s t im a t io n , i . e . w it h th e in c r e a s e o f n tho v a lu e s o f b ia s and r e s p e c t iv e v a r ia n c e s de c r e a s e . BI0LIOGRAPHY [ 1 ] 3 a c k i o w i c z C . , K l e p a c z H . , Ż ó ł t o w s k a £ . (1 9 7 7 ), E s ty m a c ja n ie li n io w y c h p o s t a c i f u n k c j i p r o d u k c ji p r z y w arunkach pobocznych (E s t y m a c ja f u n k c j i p ro d u k c j i typ u C E S ) . P o r t I I , Ł ó d ź , G ra n t R . I I I . 9 . 5 . 4 . [ 2 ] 3 n c k i e w i c z C. , K l o p a c z H . , Ż ó ł t o w s k a E . (1 9 7 0 ), E s ty m a c ja f u n k c j i p r o d u k c ji typ u CES p r z y w y k o rz y s ta n iu i n f o r m a c j i o p r i o r i u zysk anych z różn ych metod e s t y m a c ji teg o typ u f u n k c j i . Metoda p o d w ó jn e j i t e r a c j i . P a r t I , Ł ó d ź , G ra n t R . I I I . 9 . 5 . 4 .
[ o j O u o z c z o k C. (1 9 7 7 ), A n a liz a w ła s n o ś c i estym atorów p aram etrów f u n k c j i p r o d u k c ji typ u CES otrzym an ych metodę Kme- n t y , Ł ó d i , G ra n t R . I I I . 9 . 5 . 4 ,
[
4
] « l e p a c z H . , Ż ó ł t o w s k a E . (1 9 7 9 ), W łasn o ś c i e stym ato ró w u zysk an ych metodę p o d w ó jn e j I t e r a c j i a w sp ó ł z a le ż n o ś ć zm iennych o b j a ś n ia j ą c y c h . P a r t I , Ł ó d ź , G ra n t R . U I . 9 . 5 . 4 .[ 5 ] K l e p a c z H . , Ż ó ł t o w s k a E . (1 9 8 0 ), W łasno ś c i e stym a to ró w u zysk anych metodę p o d w ó jn ej i t e r a c j i a w
spół-CG CsorUwa Ju c k ltw lc z , H alina Kiepacz, B U b io ta Żółtowska
z a le ż n o ś ć zm iennych o b j a ś n ia j ą c y c h . P a r t I I , Ł ó d ź , G ra n t R. I I I . 9 , 5 , 4 .
[ 6 ] K l e p a c z H . , Ż ó ł t o w s k a E . (1 9 0 1 ), W e r y f ik a c j a o s io w e j motody e s t y m a c ji param otrów f u n k c j i C E S, Ł ó d ź , C ro n t П . I l i . 9 . 5 . Л.
[ 7 ] Ż ó ł t o w а к a E . , ( l9 7 G ) , E s ty m a c ja n i e l i n i o w e j p o s ta c i f u n k c j i p r o d u k c ji p r z y w arunkach pobocznych (E s t y m a c ja f u n k c j i p r o d u k c ji typ u C J S ), P a r t I , Łó d ź , G ra n t R . I I I . 9 . 5 . 4 . [ 0 ] 2 ó ł t o w а к o E . (1 9 8 0 ), Udoeikonulonlo metody p o d w ó jn ej
i t e r a c j i . U s p ra w n ie n ia alg orytm ó w o b l ic z e n i o w y c h . 1 oelow o me toda p od w ó jnej i t e r o c j l , Ł ó d ź , G ra n t R. I I I . 9 . 5 . 4 .
C ze sław a J a c k i e w i c z , H a lin a K le p a c z , E l ż b i e t a Ż ó łto w s k a
ESTYMACJA PARAMETRÓW FU N K C JI PROOUKCJI TYPU CES OSIOWĄ METODĄ PODWÓJNEJ IT E R A C JI
W a r t y k u le omówiono motodg e s t y m a c ji param etrów f u n k c j i p ro d u k c j i typ u CES n&zwang oaiowg metod? p o d w ó jn e j I t e r a c j i o ra z w y n ik i eksperym entu M o n te -C a rlo przeprow adzonego d la t e j motody. E k sp e rym e n ty te m ia ły na c e lu zbodonle num erycznych w ła s n o ś c i o- trzym an ych ocen param etrów f u n k c j i C ES, Przoprow adzono j e d la w yb ran ych p rób d w u d z ie s to - , t r z y d z ie s t o - , c z te rd z le s to o ie m e n to - w ych, k tó r y c h w s p ó łc z y n n ik k o r e l a c j i m iędzy zm iennym i o b ja ś n i a j ą c y » ! w y n o si 0 ,7 2 3 . 2 badań ty c h w y n ik a , że w raz ze w z ro s tem i l o ś c i i t e r a c j i m a le ją w ahan ia ocen ś r e d n ic h o d p o w ie d n ich rs - g l i z ^ c j l e stym ato ró w p a ra m e tró w : s k a l i p r o d u k c ji ( a ) , p o d z ia łu ( f f ) , s u b s t y t u c j i ( # ) . Oceny p a ra m e tru a ni© w ykazu ję numerycz- nogo o b c i^ ż s r jla , n a to m ia s t o ce n y p a ra m e tru S oą z r e g u ły p r z e szacow ane. S ro d n le w i e l k o ś c i o b c ię ź e ń sg je d n a k m n ie js z e n iż jsd r.o o d c h y le n ie stan d ard ow e z p r ó b y . W ięk sz y z m ie n n o ś c ią cha r a k t e r y z u je s ię ocen y p a ra m e tru q .
