E q u a t i o n s ( 2 . 1 3 ) - ( 2 . 1 6 ) may be r e c a s t a s :
( 2 . 2 0 )
Qt = “ l ^ t + xu ß i - ( 1 /y)Ap^
( 2 . 2 1 )
° t = a 2Pt + X2 t ß2 ( 1/y)Ap~
( 2 . 2 2 ) Ap^ = Apt i f APt " 0
= 0 o t h e r w i s e .
( 2 . 2 3 ) Ap~ = 0 i f Ap > 0
= - Apt o t h e r w i s e .
T h i s s y s t e m c o n s i s t s o f f o u r e q u a t i o n s and f o u r endogenous v a r i a b l e s :
Qt , p t , Ap* and Ap . However t h e s y st e m i s c l e a r l y n o n - l i n e a r i n t h e endogenous v a r i a b l e s as e v i d e n c e d by t h e i d e n t i t i e s ( 2 . 2 2 ) and ( 2 . 2 3 ) . The i d e n t i t i e s may be i g n o r e d so t h a t ( 2 . 2 0 ) and ( 2 . 2 1 ) a r e e s t i m a t e d . The f i r s t l e a s t s q u a r e s a p p r o a c h t o e s t i m a t i o n o f ( 2 . 2 0 ) and ( 2 . 2 1 ) i s s u g g e s t e d by P a i r and J a f f e e (1972) and i n v o l v e s e s t i m a t i n g t h e f o l l o w i n g r e d u c e d form e q u a t i o n s : ( 2 . 24 ) Apt = X ir + v Vt S . t . Ap > 0 (ttil) ) &Pt = Xt u^ + v? t Vt s . t . Apt < 0 (tft|<2) ( 2 . 2 5 )
w h e r e c o n s i s t s of e x o g e n o u s v a r i a b l e s and i ncludes X , X and
. T h e n u s i n g Ap^ , Ap^ and p^ ^ , p^ is c o n s t r u c t e d and (2.20)
and (2.21) e s t i m a t e d us i n g these instrum e n t s . N e i t h e r the IV or OLS
e s t i m a t o r o f (2.20) and (2.21) is c o n s i s t e n t as A m c m i y a (1974) has p o i n t e d
out. T h i s r e s u l t s f rom the fact that even if p l i m ^ X!u. = 0 for i = 1,2
1 1 1
1 V 4
this will not n e c e s s a r i l y be true for p l i m — ) X. u. .
1 lZ 11
The r e g r e s s i o n s :
(2.26) +
A p t - V l + V lt t = 1 (1) T
(2.27) Ap' -
V 2
+ v 2t t = 1 (1) Twill y i e l d c o n s i s t e n t e s t i m a t e s w i t h IV and OLS e s t i m a t i o n o f (2.20) and (2.21)
b e i n g identical. H a v i n g o b t a i n e d A p * and Ap^ , p^ is o b t a i n e d u s i n g
the identity: (2.28) p t = P t _ 1 + A p* - A p ” or o f t e n m o r e c o n v e n i e n t l y by e s t i m a t i o n of: (2.29) pt = xt 1'o + vot w h e r e X must i n c l u d e p t -1 If X XT ) o t h e r va r i a b l e s , then u s i n g (2.29): and s i m i l a r l y for (2.30) p = X(X'X) ] X'p = X(X'X) ] X' (p + A p + -Ap )
and if p _ . is i n c l u d e d in X , it is e a s y to show that X(X'X) ^ X' p = p ,
J “ 1 ~ 1.
so that it is c l e a r that (2.28) and (2.30) give the same result for p^ .
*♦ T h e c o n d i t i o n p l i m j X'u is s u f f i c i e n t for c o n s i s t e n c y a l t h o u g h A m e m i y a c o n s i d e r s E ( X ’u) = 0 w h i c h w o u l d require X to be non-
This technique,from now on called the non-linear estimator (abbreviated as NL estimator), forms instruments for the non-linear functions Ap^ and Ap^_ by regressing the functions on the exogenous variables of the system.
An alternative is suggested by Bowden (1970a) and entails estimation of
^ A /X *4"
(2.29) to obtain p^_ and then the use of p^ and p^_ ^ to form Ap^ and
/\—
Ap^ . This technique is called the NAIVE technique.
Also suggested by Bowden (1978a, 1978b) is a conditional expectations technique (CE technique). This also, as a first step, requires estimation
/N
of (2.29) from which it is possible to obtain m (conditionally on X-| t and Pt_j) as Pt - Pt _I where m = E(Ap^_ | X^t ,X2t ,Pt_ i) • After some manipulation it is possible to replace Ap^ by the instrument:
A a /\ 2 -v O ( 1 .A , N 91
m tN ( 0 , m t ,0") - -j== expj- y (mt/o) j
o r : (2.31)
where
m [1-N(m 0,O2) ] - cxpj-y(m /o)2
t t /2tF l 2 t
N(x,u,cr) .2
2tt*ö
expj - \ (y-u/o)2 J^dy .
The estimated o" is obtained as the estimated variance of Ap^_ from (2.29) The instrument for Ap^_ is then:
(2.32) n\ N(nV ° ’S } + ox,,|-i(mt/5)2
By using a slightly revised notation it is possible to examine the properties of these alternative estimators. Some of the results stem from those in Kelejian (1971), Edgerton (1972) and Bowden (1978a). Write the i^1 structural equation of a simultaneous model as :
(2.33) = 8i (Yi ’Xi)0i + xi^i + ui where
(yi 1 yiT) (Y,
il Y ) 'iTJ (xil xiT) '
and = (u^...u^.)' and are TX1 , TXH^ , TxlC and Txl respectively. Also is a functional operator for the i ^ 1 equation and g^(Y^,X^) is TxG. . Then 0i is G ^ l and 3- is K ^ l . Also define X = (X.:Xt) as the full set of exogenous variables in the system. Assume that
X ’u i v
---- ---x N(0,^) and 1 im — X'X = Q which is finite and non-singular.5
/F
T-x» 1
XX
Equation (2.33) may be written even more compactly as:
(2.34) y . = Z .6. + u.
i l i l
where Zi = [g^ (Y^
,
X^) : X F and 6^ = [0|:ßM .65 By --- > itismeant convergence in distribution.
6 The correspondence between this model and that in (2.20) - (2.23) is easy to show. X will include p_^ and if h is a function such that
h(x) = x if x > 0 = 0 otherwise
then g1 (p,X1)01 = o^p - (l/y)h(p-p_1)
(p:h(p-p_1)) 1 1 / Y
and g2 (p,X2)02 = a 2p + (1/y) (p-p^ - h(p-p_1)
(P: (P-P.p - h(p-p_1))
U /yJ
The simultaneous estimation approach first chooses Z. , the instrument for , and then at stage two obtains the OLS estimator:
(2.35) 6 p 3 = (Z!Z.) 1Z.yi
or the IV estimator:
''TV " -l"
(2.36) 6|V = (Z!Z.) Z.y. .
Mechanically none of the three techniques in question differ at stage two; it is at stage one that they differ, and this may result in different
properties in relation to consistency, efficiency and asymptotic distribution. These properties are now considered, as well as the effects of some possible modifications to the specification of the underlying model.