A C T A U N I V E R S I T A T Ï S L O D Z I E N S I S F O L IA OECCMOMICA 40, 1 9 0 5 ... ...
C z e sław a J a c k i e w i c z * H a lin a K l e p a c z * * E l ż b i e t a Ż ó łto w s k a *
THE TWO-STAGE IT E R A T IV E METHOD FOR ESTIM ATING THE CES PRODUCTION FUNCTION
1 . In t r o d u c t io n
C o n s id e r tw o - f a c to r CES p r o d u c tio n f u n c t io n in th e form _ v
( 1 ) f ( K , L ) - a [ s k “ £ ♦ (1 - 5 ) L * ff] ^ ,
K , L d e n o tin g o u t la y s o f c o p i t a l and la b o u r , r e s p e c t i v e l y . A s suming th a t the o u tp u t o f p r o d u c tio n p r o c e s s Y is» a random v a r i a b l e depending on a random term e tho two s im p le s t s t o c h a s t i c m odels a re
( 2 ) Y » f ( K , L ) ♦ E
and
( 3 ) Y - f ( K , L ) e E .
The c h o ic e o f ( 2 ) o r ( 3 ) f o r the model d e s c r ib in g p r o d u c tio n p ro c e s s d e te rm in e s th e methods o f e s t im a t io n t h a t can bo a p p lie d to th e e s t im a t io n o f p r o d u c tio n f u n c t io n p a ra m e te rs .
Out o f w e l l known and commonly a c c e p te d methods o f e s tim a t i o n , th e G a u ss- N ew to n 's end M a r q u e r d t'o methods a re a p p lie d to th e model ( 2 ) and K m en ta 's to th e model ( 3 ) .
The methods a re based on a l i n e a r i z a t i o n o f the CES fun c-* 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 .
**fc1.A. , S e n io r A s s is t a n t 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 Łó d ź .
t io n . I r G ause-N ew ton 'e -nothod the CES f u n c t io n I s expanded (a s o f u n c t io n o f V 3 rio b lo i3 a,6,\>,(3) in T o y lo r s e r i e s around the p o in t (otQ, 8q , v q , up to tlio torme c o n t a in in g f i r c t p a r t i o l
co-r i v o t a s . N e x t, a c c e p tin g a c r i t e r i o n o f m in im iz in g th e sum o f r e s id u a l s q u a re s '1, i t e r a t i v e p ro co d u ro i s a p p lie d to f in d e s t i m ates a , b, c , r o f p a ra m a to rs a , 5 , v,£> r e e p o c t l v e l y . Gauas-New- t o n 'e method i s th u s a s im p le i t e r a t i v e p ro ce d u re w hich does not e n s u re c o n v e rg e n c e , M a r q u a r d t's method 13 a co m b in a tio n o f l i n e a r iz a t io n p r i n c i p l e w it h g r a d ie n t method. The method ia ip lle s the c o n ve rg e n c e o f r e s id u a l s q u a re s o b ta in e d f o r a l i n e a r i z e d form .
The K m en ta 's method i s a o n e - s tu p - p ro c e d u re c o n s is t in g in 8n e x p a n sio n o f the n a t u r a l lo g a r ith m o f the f u n c t io n ( l ) ( t r e a t e d as a f u n c t io n o f £>) in M a c la u r in o e r lo 3 w ith the a c c u r a c y up to th e f i r s t two term s o f t h i s .exp ansion . I n v e s t i g a t i o n s c a r r lo d out in [ 3 ] , [ 4 ] on p r o p e r t ie s o f the CES f u n c t io n p a ra m e te r e s tim a t e s o b ta in e d when th e above method i s a p p lie d , p ro ved tho e x is t e n c e o f the b ia s in p a ra m e te r e s t im a te s f o r 6 m a ll sa m p le s. A p a rt from th a t the e s tim a te s showed a ls o h ig h v a r i a t i o n c o e f f i c i e n t s . Thu w o rs t r e s u l t s o f e s t im a t io n wore o b ta in e d f o r tho p a ra m e te r £ .
¿ ^ p ly in g K m e n ta 'a method e n a b le s , h o w e v e r, g a t t in g a c o r r e c t e s t im a te or th e p n ram o ter v . I t s b io s and v a r i a t i o n a re s m a ll. Such a s i t u a t i o n mokes i t p o s s ib le to use th e e s t im a te of p a ra m e te r v as an a p r i o r i in fo r m a tio n In a n o th e r C ES fu n c t io n p a ra m e te rs e s t im a t io n .
Tho method o f e s t im a t io n o f th e CES p r o d u c tio n f u n c t io n p r e s en te d in t h i s p a p e r makes use o f on a p r i o r i in fo r m a tio n about the v a lu e of p a ra m e te r v . The method b e lo n g s to a group o f le a s t s q u a re s m ethods. More p r e c i s e l y , the p roblem o f s e e k in g the minimum o f a n o n - lin e a r c r i t e r i o n - f u n c t i o n (w h ic h i s n ot a sq u a re fo rm ) c o n s is t s in s o lv in g a n o n - lin e a r system o f norm al e- q u a t io n s .
I n such a p ro c e d u re the sum o f r e s id u a l e q u re s i s a sq u a re
2. E s t im a t io n or CES Fu n c t i o n P a ra m e te rs by Tivo-Stage I t o r a t l v e Method
CES p r o d u c tio n f u n c t io n c o n s t i t u t e s a n o n - lin e a r model f o r the sample {(Y.^, L ^ ) , i ■ 1 , , nj . I t has the form
_ v ( 4 ) Y i " a K~£ ♦ ( l - S ) L“ ^ J £ + , where s Y tm o u t p u t . K - f ix e d a s s e t s , L mt em ploym ent, a - p a ra m e te r o f the s c a le o f p r o d u c tio n , 6 - d i s t r i b u t i o n p a ra m e te r. V m hom ogeneity p a ra m e te r. e m s u b s t i t u t i o n p a ra m e te r, c - random term . T a k in g in t o c o n s id e r a t io n tho n o c e a s it y o f econom ic i n t e r p r e t a t i o n the p a ra m e te rs o f tho above f u n c tio n sh o u ld f u l f i l the f o llo w in g c o n d it io n s » a e ( 0 , + o o ),6 e ( 0 , 1 ) , v e ( 0 , + o o X g c (- l» 0 )u u (0 ,+ a o ). F o r £ * 0 p r o d u c tio n f u n c t io n ( 4 ) becomes the Cobb- -O o ug las t y p e .
The problem o f e s t im a t in g tho p a ra m e te r o f f u n c t io n (a ) u sin g th e l e a s t s q u a re s m ethod, g iv o n a p r i o r i v , re d u c e s to s e e k in g the v a lu e s A , b, r such t h a t the f u n c tio n
n ( 5 ) 3 ( A , b , r ) - Y { y ^ - A [bK^r ♦ ( l - b ) L " r ] r j 2 1>1 a t t a i n s minimum. The e x is t e n c e o f optimum v a lu e s A , b , r o f f u n c t io n ( 5 ) i s , f i r s t o f a l l , c o n d it io n e d by th e e x is t e n c e o f th e f o llo w in g s y stem o f e q u a t io n s ' s o lu t io n »
(
6
)
a » T K -2 £ <Y, - « ¡ 1) 0 1 . 0 , 1-1 a *I V
2Av n ? Z J V1 ■ 1-1 K " r - L " r 1 L 1 0 , 31 - L ( Y 1 - M j o ' x - °-3 r 1-1 whare > g , - b K . r + ( l « b ) L . r , <=1 • g* - -b K " r In Kt - ( l - b ) L " ' In L t .The s o lu t io n o f system ( 6 ) I s e q u iv a le n t to f in d in g o f zero - p o in t s o f the system o f f u n c t io n s - r C7> n ? a • X X - « i » 8 ! ' 1-1 K~r - L~r F 2 • Z ( r i - « t , s i i- 1 ^ n f 3 • E ( r i - " V v i - i where F ^ , F , a re n o n - lln e a r f u n c t io n s o f e s t im a te s A , b, r so t h a t the a n a l y t i c way o f d e t e r m in a t io n o f z e r o - p o ln ts e s tim a t o r s ' v a lu e s i s h i g h ly c o m p lic a te d end p r a c t i c a l l y im p o s s ib le * O n ly the v a lu e o f A i s In t h i s ca s e e a s y to be e s t a b lis h '*
\ •
7 1
ad2 (a s a f u n c t io n o f b and r ) . Making use o f tho f i r s t equ a t io n o f ( 6 ) , wo have ( 0 ) A - A ( b , r ) Y,G
iWj __
Z Gi1-1
So th e &ystam o f e q u a tio n s ( 6 ) assumes the form
( 9 ) n K. - L , £ ( V , - A (b ,r> C 1) O l - * * 1 « ! 9 i n Z (y. - A ( b , r ) G ) o ! - o 1- 1 1 - r
In o r d e r to s o lv e th e above system we o h o ll a p p ly the f o llo w in g I t e r a t i v e method.
Assuming oomo s t a r t i n g v a lu e s b ■ b ^ ° \ r - r the i t e r a t i v e p r o c e s s p ro c e e d s In such a way (w h ere k d e n o te s the g lv o n i t e r a t i o n n u m b er), th a t
1 ° f o r o d e te rm in e d b « b In the s u c c e s s iv e I t e r a t i o n s such r - r ^ k + l) l a found th a t l F » ( b , r^ ) I < W, where W
**8 I s an a p r i o r i im p lie d v a l u e , f o r exam ple W » 1 0 .
2 ° f o r the p r e v i o u s l y d e te rm in e d r ■ r 14+1 ^ in th e s u c c e s s i v e i t e r a t i o n s such b ■ k ^ + 1 ) 18 found t h a t | F g ( r ^b )j
< Wj ,
3 ° the I t e r a t i o n s d e s c r ib e d In 1 ° and 2 ° a r e re p e a te d by tu r n s u n t i l f o r k - I T , ond th u s b ^ * \ r , th e a b s o lu te v a lu e s o f 3 J/ 3 A , d » / 3 b , 9 i / d r become s m a lle r than IV.
The d e te rm in e d v a lu e s r ^^ and b k o re o b ta in e d ao e r e s u l t o f s u c c e s s iv e a p p ro x im a tio n s u s in g the ta n g e n t method I . e . f o r d e te rm in e d v a lu e s o f b (k 1 ) and r ( k “ 1 ) we i n t e r c e p t th e tan
-2 The v a lu e o f A o b ta in e d i n t h i s way c o n s t i t u t e s a good e s t im a te o f p a ra m e te r o* \ i t i s su p p o rted by the r e s u l t s o f many e x p e rim e n ts .
(k - 1 }
gunt of the f u n c t io n F ^ (b , r ) o f the v a r i a b l e r j th e p o in t c f i n t e r s e c t i o n of the tan g e n t and Or a x is i s o new v a lu e o f the v a r i a b l e r in a f i r u t ste p o f th e I n t e r n a l in t e r a c
-(k - 1 )
t io n - so we g o t r^ . Next a new ta n g e n t o f F 3 ie c o n s tr u c ted In n o in t (b , r i 'K’*i b. The a b s c is s a o f the p o in t o f in
-(k - 1 )
t o r a o c t io n w it h Or a x is d e to rrain o e a new v a lu a r^ , and so on. The i n t e r n a l i t e r a t i o n s a re c o n tin u e d u n t i l in a aucces- c-ive s te p IT T such a v a lu e ■ r ^ i s produced that|a#/dr|< < W. (k - 1 ) Thtf s e r i e s o f v a lu e s r^ i s i t e r a t i v e l y c a lc u lp t o d u sin g tho r e c u r s iv e fo rm u la _ (k - 1 ) „ (k - 1 ) i r J - i F 3 ( b ( k ‘ 1 ) , r < y b H2 ( b ( k _ 1 ) , r ^ " 1*) f o r J » l , . . . , IT T , w h e re : H. 3F,
2 " dr'
VA r 2i 1-1 2AGt G ; ) l n ( g ts y
h )r ?7 ,+
V , x f l i * 8 i - ( a t )2 - r L (Y i ' AGi ) G i 1-1 S i g * # - bk” r l n 2K ♦ ( l - b ) L ~ r l n 2L , n 2 (Y . - 2AG.) • G ' A - * * . A - i 1 1____ J L a 2 dr 11Z of
i - 1 1 ( k ) ( k )I n the sane way the v a lu e b i s e s t a b l i s h e d , a t r d e te rm in ed , i . e . bi k ) - b2TB1 } ' i f o n lV I a ^ / a b I < W. f o r b - b ^ ' ^ a n d r * r ( L ' . S u c c e s s iv e v a lu e s b j f o r J » l , . . . . IT B o re os-t c b li'ih o d u s in g os-th e fo rm u la
F ( b ( k " 1) r ( k ) ) . ( I t - l } , ( k - 1 ) 2 1 -1 *
b.
- b
- — T i c m— nrrr
•Mi (bj - I
• r
}
w h e r e t M 1 “ T b ~ “ X ! (YiS l " A i G i" , - r , - r 2AG^S^) 1-1 11■ z
,- r < » » - « i > Gt - L- r 1-1 - 3G s " -5F' * 3A Al * "aE*Such an I t e r a t i v e way o f f in d in g the s o lu t io n o f th e system ( 9 ) I s the base f o r e la b o r a t in g th e e s t im a t io n method f o r p a ra m eters A , b , r , w h ile c r i t e r i o n (5 ) i s assumed. The method la c a l l e d the tw o -stag e i t e r a t i v e method (T S IM ).
To e s t a b l i s h th e p r o p e r t ie s of th e p ro s e n te d CES f u n c tio n e s t im a t io n method a M o nte-C orlo e x p e rim e n t was u se d . F o r t h i3 the sam ple space ft - { ( Y , K. L ) } f u l f i l l i n g r e l a t i o n ( l ) has been c o n s t r u c t e d . The c o n s t r u c t io n c o n s is t e d in f in d in g t h e o r e t i c a l v a l u e s , o f the endogenous v a r i a b l e YT1# 1-1, . . . . n f o r the g iv e n v a l u e s , o f exogenous v a r i a b l e s L ^ ) , 1-1, . . . , n and, of p a ra m e te rs a , 5 ,v,£> ( assumed a p r i o r i ) , the t h e o r e t i c a l v a lu e s of
the endogenous v a r i a b l e i " l # •••* n } a c c o r d in g to the fo rm u la
YT^ - a [Sl<^ + ( l - 5) L“e ]
^ •
3. G e n e ra tin g o f Sam ple Spg^gThe c a lc u la t e d v a lu e s YT^ w ere thon added to th e random term C so t h a t the v a lu e 3 of endogenous v a r i a b l e Y ware e q u a l to
YjL - YT^ ♦ C^, i - 1 , . . . . n.
The v a lu e s o f th e random v a r i a b l e s w ere g e n e ra te d from the norm al d i s t r i b u t i o n N ( 0 , d^.The p a ra m e te r d^ woe d e te rm in e d as
cfc m S ( Y T ) y { / R 2 - l ,
where ¿ ( Y T ) d e n o te s the s ta n d a rd d e v ia t io n o f YT arid R2 la the c o e f f i c i e n t d e te rm in in g the p a r t o f v a r i a t i o n o f v a r i a b l e Y be in g e x p la in e d by YT, i . e . i t l a e q u iv a le n t to the t h e o r e t i c a l d e te r m in a tio n c o e f f i c i e n t f o r model ( 1 ) .
D raw ing I P tim e s the r e a l i z a t i o n s o f E we g e t th e sample s p a c e :
A ( j ( , L^) , 1 » 1, • . . , * *■* i ^ ) *
- {(Y < °> ... Y^S ) J : a - 1 ...i p} .
The space ft c o n s t it u t e s the base f o r the d e te r m in a tio n o f the sequence o f e s t im a te s { AB } » { bs } • ( r a } •••• ***) o f p a ra m eters a ,8 ,g g e n e r a t in g t h i s s p a c e ,
A few ty p e s of sam ple sp a ce s w ere in v e s t i g a t e d , th e s o u rc e o f v a r i a t i o n b ein g « 1 ° th e range o f c o r r e l a t i o n o f th e v a r i a b l e « K and L , 2 ° the v a lu e s o f p a ra m e te rs a , 6, g , 3 ° th e v a lu e o f R2 , 4 ° sam ple s iz e ( n ) , 5 ° th e q u a n t i t y I P .
I n the f i r s t ca se th e v a r i a t i o n o f th e sam ple sp ace c o n s is t e d in th e c h o ic e o f two d i f f e r e n t s e t s o f v a lu e s (K , L )^ and ( K , L ) 2, c o r r e l a t i o n c o e f f i c i e n t s betw een K and L b e in g 0 ,0 3 and 0 . 9 7 , r e s p e c t i v e l y . The e a t ( K , L )^ was chosen from random number t a b le s , but the s e t ( K , L )g c o rre s p o n d s ( w it h th e a c o u ra c y to the assumed s c a l e ) to th e r e a l v a lu e s o f f ix e d a s s e t s and employment in the P o l is h economy in th e y e a r s 1958-1977.
The ?wo-Stago Jteracivo .'.ethod ¿or Satim.tinfi the CES ______ 75 - - 0 .5 , - 0 .2 , 0 . 2 , 0 . 5 , 1 .0 f o r ot - 2 .0 , S - 0 . 4 , V - 1 .0 . Such a co n ce p t le duo to th e econom ic Im p o rta n ce of tho p a ra m eter £ as w e ll as to the w o rst e s t im a te s of t h is p a ra m e te r o b ta in e d u sin g o t h e r m ethods. F o r v a lu o e 0 .9 9 , 0 , ‘.'0, 0 .9 5 and 0 .9 0 w ere assumed. I n v e s t i g a t i o n s v<ere made fo r e a n p le a oi 20, 25 and 30 e le m e n ts .
On the b a s is o f soquoncos o f e s t im a te s { Aa } * { bs } * { r s } ^ s " ■ 1 , . . . . I P ) » the p r o p e r t ie s o f p a ra m e te r e s t im a te s a re e s t a b lis h e d f o r e v e r y typ e o f sam ple s p a c e . Moan v a lu o e of the e s tim a te d b ia e , v a r i a t i o n c o e f f i c i e n t , tho v a r ia n c e about the mean and about th e r e a l v a lu e o f p a ra m e te r wero a n a ly s e d .
4 . E f f i c i e n c y o f t he Tw o-Stage I t e r a t i v e Method f o r a D e te rm in i s t i c Mod e l
I n i t i a l l y the p ro c e s s o f th a TSIM c o n ve rg e n c e май a n a ly s e d f o r the cose of d e t e r m i n i s t i c m od els, i . e . when R ® 1 .0 (th e n У я Y T ). The a n a l y s i s was e x p e cte d to g iv e answ ers to the f o l lo w in g q u e s t io n s :
1 . I s the c o n ve rg e n c e p ro c e s s dependent on tho c h o ic e of s t a r t i n g p o i n t s ?
2. What 1з an a ve ra g o number o f i t e r a t i o n s and a v e ra g e tim e o f re a c h in g th e assumed r e a l p o i n t ?
3. I n what way the I t e r a t i v e p r o c e s s i s in f lu e n c e d by tho range o f c o r r e l a t i o n o f exogenous v a r i a b l e s К and L ?
N u m e ric a l e x p e rim e n ts c a r r i e d out on OORA 1304 com puter u sin g # CES 4 program p roved the co n v e rg e n c e o f the i t e r a t i v e p ro c e s s . The number of i t e r a t i o n s depends on th e c h o ic e o f a c c u r a c y (w ), a s w e l l as on th e c h o ic e o f s t a r t i n g p o i n t , more p r e c i s e l y on r , and on tho range o f c o r r e l a t i o n o f v a r i a b l e s . The r e s u l t s a llo w us to p u t o u t s e v e r a l i n t e r e s t i n g c o n c lu s io n ^ .
1 ° A l l p a ra m e te r e s t im a te s a re c o n v e rg e n t to the assumed p a ra m e te r v a l u e s . The o n ly e x c e p tio n l e in the ca s e when s t a r t in g Гф and r e a l g a re th e numbers w it h o p p o s ite s ig n s i . e . fo r
^ I t I s o b v io u s t h a t th e s m a lle r the v a lu e W th e l a r g e r the number o f I t e r a t i o n s .
s t a r t i n g r Q e (- 1 , 0 ) and r e a l v a lu e o f p a ra m o te r £ > 0 (a n d i n v e r e e l y ) and in c r e a s e (o r d e c r e a a e ) up to 0 o r r o b ta in e d in s u c c e s s iv e i t e r a t i o n s i s o b s e rv e d . And, in t u r n , w it h r ten d in g to 0 the i t e r a t i v e p ro c e s s i s no lo n g e r c o n v e rg e n t. I n e v e r y i t e r a t i o n the v a lu e s o f r " t r a v e l “ about 0 w hich la im p aled by n u m e r ic a l p r o p e r t ie s o f on e x p r e s s io n o f the f o llo w in g form 4 «
A [b K“ r ♦ (1 - b ) L " r ] r .
In th ltf c a s e when TSIM i s a p p lie d , th e s ig n o f s t a r r i n g v a lu e r sh o u ld bo changed. I f th e s i t u a t i o n I s s t i l l th e same a f t e r the change o f the s ig n then th e h y p o th e s is sh o u ld be a c c e p te d t h a t th e r e l a t i o n between the o u tp u t and p r o d u c tio n f a c t o r s i s of th e Cobb-Douglas t y p e , i . e . a f u n c t io n w it h the e l a s t i c i t y of s u b s t i t u t i o n o' ■ 1 . i t f o llo w s t h a t e x p e c tin g th e e l a s t i c i t y o f s u b s t i t u t i o n g r e a t e r th an [ l ] we sh o u ld assume p o s i t i v e s t a r t « * n9 r Q ; e x p e c tin g tf < 1 we sh o u ld choose n e g a t iv e r .
’* o
2 ° The co n ve rg e n c e o f th e I t e r a t i v e p ro c e s s does n o t depend on a c h o i- e o f s t a r t i n g v a lu e s ( i f o n ly the c o n d it io n on the s ig n o f p a ra m e te r r i s s a t i s f i e d ) . I t was o b s e rv e d , t h a t the number o f i t e r a t i o n s i s s m a lle r when s t a r t i n g v a lu e r i s g r e a t e r than g f o r £ > 0 , and s m a lle r than £ f o r £ < 0 .
. 3 ° The e s t im a t e s o f p a ra m e te r S become s t a b le much so o n e r th an th e e s t im a t e s o f p a ra m e te r £ . So t h a t f o r s t a r t i n g b e- q u a l to 5 thg e s t im a te o f r I s e q u a l to th e r e a l v a lu e o f pa ra m e te r q in th e f i r s t i t e r a t i o n I T a l r e a d y .
4 ° The co n ve rg e n c e in th e prop osed method does riot depend on the range o f c o r r e l a t i o n o f exogenous v a r i a b l e s . I n both c a s e s (¿?K L * “ 0 .0 3 6 , £k l - 0 .9 6 6 ) the a p r i o r i assumed r e a l v a lu e e were o b t a in e d . Bu t th e range o f c o r r e l a t i o n in f lu e n c e s s i g n i f i c a n t l y the number o f I t e r a t i o n s . I t i e o b v io u s t h a t the g r e a t e r the c o r r e l a t i o n c o e f f i c i e n t , the g r e a t e r th e number o f l t e - r a t io n e .
On th e b a s is o f th e r e e u l t s o b ta in e d f o r th e d e t e r m i n i s t i c model I t sho uld be s t a t e d t h a t th e two s ta g e i t e r a t i v e method
p r o v id e s good r e s u l t s and I s e f f i c i e n t . H ow ever, f o r s t o c h a s t i c modela the b ia s and e f f e c t i v e n e s s o f p a ra m e te r e s t im a te s sh o u ld be in v e s t ig a t e d . The f o llo w in g p a r t o f the p a p e r I s devotod to t h i s p rob lem .
•*
5. P r o p e r t ie s o f TSIM f o r a S t o c h a s t ic Model
The sam ple ep acee and M o n te - C a rlo e x p e rim e n ts wore a p p lie d to e s t a b l i s h th e p r o p e r t ie s of TSIM e e t im a to s . P a r t i c u l a r a t t e n t io n was p a id to b a s ic c h a r a c t e r i s t i c s o f tho o b ta in e d se q u en ces o f p a ra m e te r e s t im a t e s { Ag }» { b 8 }« { r 8 }• E s p e c i a l l y tho f o llo w in g v a lu e e hove been a n a ly e e d t mean v a lu e s from th e I P r e p e t i t i o n s , v a r ia n c e s and s ta n d a rd d e v ia t io n s from th e sample ( c a l c u l a t e d in r e l a t i o n to the m ean), v a r i a t i o n c o e f f i c i e n t s f o r th e means, th e v a lu e s o f b ia s f o r the means, v a r ia n c e s and s ta n d a rd d e v ia t io n s c a lc u l a t e d in r e l a t i o n to the p a ra m e te rs . The same c h a r a c t e r i s t i c s havo been d e te rm in e d f o r th e sum o f s q u a re s o f r e s i d u a l s ( Q ) , e s t im a te s o f d e te r m in a tio n c o e f f i c ie n t R2 from th e sam ples and th e v a lu e F d e te rm in e d as a sum o f s q u a re s o f p a r t i a l d e r i v a t e s of tho c r i t e r i o n f u n c t io n $ .
Tha o b ta in e d r e s u l t s , b ein g r a t h e r p r e li m in a r y , a re b e t t e r than e x p e cte d in v ie w o f a s m a ll number o f u n d e rta k e n e x p e r i m ents. The p a ra m e te r e s t im a t e s were supposed to be b ia s e d f o r two r e a s o n s : the b a s ic sam ple c o v e re d
20
- e lem o n te end was a s m a ll o n e , and , s e c o n d ly , in a common w id e ly - a c c e p te d o p in io n , the i t e r a t i v e methods a re b ia s e d . I n th e e x p e rim e n ts c a r r ie d outfor
R2 - 0 .9 9 0 th e s t a t i s t i c a l l y s i g n i f i c a n t p a ra m e te r b ia s has n o t been o b se rve d os mean p a ra m e te r e s t im a te s d id not d i f f e r from the r e a l v a lu e s o f th e s e p a ra m e te rs more than by one s ta n d a rd de v i a t i o n f o r the mean. The dependence o f mean p a ra m e te r e s t im a t e s on th e number o f r e p e t i t i o n s I P i s p re s e n te d in F ig s 1-3.The mean v a lu e o f th e sum o f r e s i d u a l s q u a re s Q and the mean R2 showed a v e r y s i m i l a r b e h a v io u r . The c h a r a c t e r i s t i c ob t a in e d in I P r e p e t i t i o n s w ere i d e n t i c a l f o r d i f f e r e n t s t a r t i n g v a l u e s . The e x p e rim e n ta l r e s u l t s f o r R2 ■ 0 .9 9 , a ■ 2.0 , S ■ - 0 . 4 , £> » 0 .2 o b ta in e d in s u c c e s s iv e r e p e t i t i o n s I P ■ 50, ICO, 200 a re shown in T a b le 1. I t i s w o r th w h ile to n o te t h a t the
a
■rl
T a b l e 1
Mean sample p oram nter e s tim a te s
and t h e i r c h a r a c t e r i s t i c s f o r R 2 * 0 .9 9 0 , n * 20 C h a r a c t e r i s t i c s The numbor o f r e p e t i t i o n s ( I P ) f o r of a soquence e s t im a te s 50 100 200-1 2 3 4 5 a. 1.99092 2.00060 1.99977 s2(a ) 0.00082 0.00092 0.00076 s(a ) 0.02858 0.03029 0.02767 K ) s(a ) /a ' 0.01429 0.01513 0.01383 A - a 0.00108 -0.00060 0.00023 s2 (a) 0.00082 0.00002 0.00077 S ( A ) 0.02060 0.03029 0.02767 h r . 0.40330 0.40144 0.40102 S 2 ( b ) 0.00024 0.00027 0.00028 S ( b ) 0.01549 0.01645 0.01665t K )
s(b)/b
0.03040 0.04097 0.04143 b - « -0.00330 -0.00142 -0.00102 S 2 ( b ) 0.00025 0.00027 0.00028s(b)
0.01584 0.01651 0.01674 r I P 0.18041 0.20424 0.19790s
2
(r)
0.00654 0.00855 0.00749 S ( r ) 0.08090 0.09244 0.08656{ % }
s ( r ) / r
0.44843 0.45262 0.43738 F - e 0.01959 -0.00424 0.00210Czesława (JftckioWica, Halina
Klepacz,
E l ż b i e t aŻółtowaks
T o b ie 1 (c o n t d .) 1 2 3 4 5 32 ( r ) O.OOG93 0.00856 0.00750 G ( r ) 0 .0 0 324 0.09254 0.08658 R * , 0.99150 0.99175 0.99172 s2 (r2 ) 4 .7 • 1 0 "6 5 .7 • 1 0 "6 6 .7 • 1 0 "6 s(r2) 0.00217 0.00230 0.00259 { « . } s(r2 ) /r 0.00219 0.00240 0,00261 R2 - R2O
0.00150 0.00175 0.00172 s 2 Cr2 ) Ro 6 .9 • 1 0 "6 8 .7 • 1 0 "6 9 .6 • 1 0 "6 S ( R 2 ) Ro 0.00264 0.00296 0.00311 « I P 361.67 352.22 351.41 s2(q ) 9047.37 10 803.84 11 993.04 s(q) 95 .12 103.94 109.51 W S (Q )/ Q I p 0 ,2 6 0 .2 9 0 .3 1QjP - Q
- 2 .1 9 , -1 1.6 4 -1 2.4 6 s q«Q)
9052.17 10 939.33 12 148.20 SqCQ> 9 5 .14 104.59 110.22F1P
1.5 8•
10‘ 9 1 .8 7•
1 0 "9 1 .8 2•
10**9 s2 (f) 5.21•
Ю " 1? 5 .3 6•
1 0 "18 5 .8 2•
1 0 "18 S t P ) 2.2 8•
1 0 "9 2.3 1•
1 0 "9 2.41.
1 0 "9 S ( F ) / F I p 1 .4 4 1 .2 3 1 .3 2 F I P '0
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10“ 9 1 .8 7 • 10~9 1 .8 2 • 1 0 "9T a b le 1 (c o n td .) 1 2 3 4 5 S JC F ) S o ( F ) 7 .7 3 • 1 0 "9 2.7 8 • 10~9 8 .8 8 • 1 0 "18 2 .9 8 • 1 0 '9 9 .1 3 • 10“ 18 3.0 2 • 1 0 "9
number o f I t e r a t i o n s In s u c c e s s iv e sam pling# woe I d e n t i c a l to t h a t in th e d e t e r m i n i s t i c севе (7 t i l l 1 0 ). I n th e p ro c e s e of s a m p lin g t h e r e w ere g e n e ra te d euch two sam p les (w h ere th e gene r a t o r s t a r t e d from 0.41053835 and 0.65214471) t h a t the number o f I t e r a t i o n s was stopped w it h I T ■ 30 and th e r e s u l t s , e sp e c i a l l y f o r r , s e r i o u s l y d i f f e r e d from the r e a l o n e s . The v a lu e s o f Q and th e I n d i c a t o r F w ere la r g e f o r th e two c o s e s . The two sam p les w ere not tak on in t o a c c o u n t in e s t a b l i s h i n g the moan e s t im a te s ond t h e i r c h a r a c t e r i s t i c s .
C o n c lu s io n s c o n c e rn in g th e c h a r a c t e r i s t i c s o f p aro m o ter e s t i mates o b ta in e d on th e b e o is o f n u m e r ic a l e x p e rim e n te a lw a y s de pend on the зеоро o f th e e x p e rim e n ts , i . e . on th e number o f r e p e t i t i o n s f o r о b a s ic som ple. I t i s о p r i o r i assumed th a t the number I P sh o u ld be l n d l f l n l t e l y l a r g e . P r a c t i c a l l y tho f u i f l i mant o f t h i s c o n d it io n i s alm o st im p o s s ib le because o f tim e- and la b o u r- c o n s u m p tlo n . T h e r e f o r e , a l i m i t a t i o n on th e number of r e p e t i t i o n s i s n e c e s s a r y .
I n t e r e s t i n g r e s u l t s f o llo w from the a n a l y s i s of e m p i r ic a l d i s t r i b u t i o n s o f p a ra m e te r e s t im a te s ( f o r I P * 2 0 0 ). The h lg h o c t s t a b i l i t y showed the e s t im a te o f p a ra m e te r oi : tho v a r i a t i o n c o e f f i c i e n t f o r i t was e q u a l to 1.58Й ( i t was m easured a s a sha re o f a s ta n d a rd d o v i a t lo n In the m ean). The lo w e s t s t a b i l i t y was r e v e a le d by the e s t im a te o f p a ra m e te r £> ; the v a r i a t i o n co e f f i c i e n t - v e r y h ig h - was e q u a l to c i r c a 4 5 .1 % . The v a r i a t i o n c o e f f i c i e n t f o r th e e s t im a te o f p a ra m e te r 5 was e q u a l to 4 .0 8 % . The e m p i r i c a l d i s t r i b u t i o n s of th e e s t im a t e s o f p a r a m e te rs a , 5, g a re shown In F i g s 4 -6. These d i s t r i b u t i o n s show le f t - h a n d e ld o asym m etry f o r A and r i g h t hand s id e ssym m otry f o r b and r .
The above p re s e n te d r e e u l t s c o n s t i t u t e o n ly a p a r t o f tho se w h ich we o b t a in e d . These w hich e re n ot p r e s e n te d h e r e , have v e r y s i m i l a r p r o p e r t i e s . The p roposed tw o-«ta I t e r a t i v e method
50 40 io 20 to w №0 / \
*992 Î008
1040 ' 2072
7" F i g . 4 . Diagram o f e m p i r ic a l d i s t r i b u t i o n « o f e s t im a te A f o r I P - 200 ( R 2 - 0 .9 9 0 , n - 20 ) F i g . 5. Diagram o f e m p i r ic a l d i s t r i b u t i o n o f e s t im a te b f o r I P - 200 ( R 2 - 0 .9 9 0 , n - 2 0 )can be t h e r e f o r e a p p lie d to e s t im a te th e p a ra m e te rs o f CES p r o d u c tio n f u n c t io n , assum ing t h a t a l l c o n d it io n s f o r th e ra n dom term a r e f u l f i l l e d . The a n a l y s i s o f p r o p e r t i e s o f t h i s method in tho ca s e o f w eaker assu m p tio n s I s c a r r i e d o u t a t th e 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 *.6d£.
The p a p e r i s based on th e i n v e s t i g a t i o n s c a r r i e d o u t under tho c o n t r a c t R . I I I . 9 .5 .
The Two-Stage Iteratlvo Method for SaLlmatlng the CES 85 F i g . 6 . D iagram o f e m p i r ic a l d i s t r i b u t i o n o f e s t im a te r f o r I P - 200 ( R 2 - 0 .9 9 0 , n - 20 ) BIBLIO G RAPH Y [ 1 ] O a c k i e w l c z C z . , K l e p a c z H . , Ż ó ł t o w s k a E . (1 9 7 6 ), 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 in f o r m a c j i o p r i o r i u zysk an ych z różn ych metod e s t y m a c ji togo typ u f u n k c j i , The p roblem worked ou t under th e c o n t r a c t R . I I I . 9 , Ł ó d ź .
[ 2 ] O o c k i o w i c z C z , , K l e p a c z H . , Ż ó ł t o fl e k a E . (1 9 7 6 ), 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 a p r i o r i . Metoda p o d w ó jn ej i- t e r a c j i , The p rob lem worked out undor th e c o n t r a c t R . I I I . 9 , Łó d ź .
[ 3 ] J u s z c z a k G . , (1 9 7 6 ), A n a liz a w ła s n o ś c i e stym ato ró w param etrów f u n k c j i p r o d u k c ji typ u CES otrzym an ych metodę Km enty, The p rob lem worked o u t under th e c o n t r a c t R . I I I . 9 , Ł ó d ź .
[ 4 ] K u s l a k G . , Ż ó ł t o w s k a Ę. (1 9 7 6 ), 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 (f u n k c je p r o d u k c ji t y pu C E S ) , The problem worked o u t under th e c o n t r a c t R . I I I . 9 , Łó d ź .
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 ) , The problem worked o u t under the c o n t r a c t R . I I I . 9 , Łód ź.
Czooîo'./a J a c k i e w i c z , H a lin o K le p a c z , E l ż b i e t a Ż ó łto w s k a
DVAJ3T0PNI0V/A ESTYMACYJNA METOOA ESTYM ACJI FU N K C JI PRODUKCJI TYPU CES
.V a r t y k u le p rz e d s ta w io n o p ro p o z y c ję metody e s t y m a c ji f u n k c j i p r o d u k c ji CES z a d d y ty w n lc wprowadzonym s k ła d n ik ie m losowym. Me
toda to j e s t o p a r ta na k la s y c z n e j m etod zie n a jm n ie js z y c h kw adra tów. O trzym any u k ła d n ie li n io w y c h równań norm aln ych ro z w lę z u je s i t « sposób l t e r a c y j n y d w ustopniow o. IV a r t y k u l e p rz e d s ta w io n o rów nie?, w y n ik i eksp erym en tu M o n te - C a rlo uzyskano tę metodę.
