PART TWO
CH APTER FO U R
IN F E R E N C E IN LIN EA R M ODELS W ITH GEN ERA TED R E G R E S S O R S : A M ONTE CARLO CO M PA RISO N U S IN G O L S , IV, FIM L
AND BO O TSTR A P ESTIM ATES
4.1. I n t r o d u c t i o n
P ag an (1984) rev iew ed th e tw in issu e s o f e ffic ie n c y and s t a t i s t i c a l in fe re n c e f o r lin e a r m odels w ith g e n e ra te d r e g r e s s o r s , t h a t is, m odels w ith unobserved e x p la n a to ry v a ria b le s , using a num ber of illu s tr a tiv e m odels. It w as d e m o n s tra te d t h a t th e a p p lic a tio n of th e 2 - s te p m ethod to such m odels could lead to in e ffic ie n c y and to in c o r r e c t con clu sio n s being d raw n fro m h y p o th esis te s t s . While th e c o e ffic ie n t e s tim a te s using th e 2 - s te p m ethod a re c o n s is te n t, th e co nv en tio n ally pro g ram m ed s ta n d a r d e r r o r s of th e s e OLS c o e ffic ie n t e s tim a te s a r e o fte n in c o n s is te n t. F o r m ost of th e s e m odels, Pagan (1984) exam ined w hen th e OLS and In stru m e n ta l V a ria b le s (IV) 2 - s te p m etho d s w e re e f f ic ie n t and when s ta n d a r d e r r o r s w e re c o n s is te n t. A su m m ary o f th e p r o p e r tie s of th e 2 - s te p e s tim a to r f o r v a rio u s m odels is p ro vid ed in T ab le 4.1. Since th e s e m odels and v a ria tio n s o f them a r e used e x te n siv ely (and also in c o rre c tly ) in p r a c tic e ( fo r tw o r e c e n t ex am p les, see C ard (1990) and Fam a (1990)), it is e s s e n tia l t h a t t h e i r sm all sam ple p r o p e r tie s be com pared and e v a lu a te d th ro u g h th e use o f Monte C arlo e x p e rim e n ts.
In th is c h a p te r a num ber o f sirriple th e o r e tic a l exam p les is used to in v e s tig a te th e e m p iric a l d iffe re n c e s t h a t can a r is e in th e c a lc u la tio n of th e s ta n d a r d e r r o r s using th e 2 - s te p OLS m ethod, as opposed to th e 2 - s te p IV o r Full In fo rm a tio n Maximum Likelihood
(FIML) m ethods. FIML e s tim a te s a re c o n s is te n t and a sy m p to tic a lly e f f ic ie n t, and th e s ta n d a r d e r r o r s a re a s y m p to tic a lly valid. Even th ou gh th e fo rm u la f o r th e C o rre c t OLS s ta n d a r d e r r o r s is known f o r th e lin e a r m odels exam ined h e re , it w ould also seem v alu ab le to ex am ine th e e m p iric a l u se fu ln e ss of tw o v a ria n ts o f th e p a ra m e tr ic b o o ts tr a p m ethod in p rov id ing a lte rn a tiv e e s tim a te s of s ta n d a r d e r r o r s . The b o o ts tr a p is a very u se fu l c o m p u te r-in te n s iv e m ethod f o r o b ta in in g e s tim a te s o f s ta n d a r d e r r o r s in s itu a tio n s w h ere it is c o m p u ta tio n a lly d if f ic u lt to c a lc u la te th e c o r r e c t s ta n d a r d e r r o r s , o r when th e a n a ly tic a l e x p re ssio n s a r e in tr a c ta b le . The m ethod is s im ila r to a Monte C arlo e x p e rim e n t, w ith a r t i f i c i a l d a ta g e n e ra te d by s e ttin g th e t r u e p a ra m e te rs a t th e ir e s tim a te d v alu es and choosing th e e r r o r d is tr ib u tio n as th e e m p iric a l d is tr ib u tio n o f th e r e s id u a ls . Since th e sam ple d a ta a re random d ra w in g s fro m a p a r t i c u l a r p o p u latio n , ran d om d ra w in g s fro m th e sam ple may also be view ed a s rando m d ra w in g s fro m th e po p u latio n . The p ro b a b ility d is tr ib u tio n o f a p a r tic u la r s t a t i s t i c can th e n be e s tim a te d by th e e m p iric a l fre q u e n c y d is tr ib u tio n o f i ts values fro m a r t i f i c i a l b o o ts tr a p sam p les (of th e sam e size) ta k e n ran d o m ly , and w ith re p la c e m e n t, fro m th e o rig in a l sam ple.
Two v a ria b le a d d itio n d ia g n o stic t e s t s f o r th e g e n e ra te d r e g r e s s o r s tr u c tu r a l e q u a tio n a r e also p re s e n te d f o r each of th e m odels, u sing a lte r n a tiv e m ethods o f e s tim a tio n . T hese t e s t s a r e a t e s t o f s e r ia l c o rr e la tio n (see B reusch and G odfrey (1986)) and th e RESET t e s t f o r m odel m is s p e c ific a tio n (see R am sey (1969, 1974)). Since th e fu n c tio n a l fo rm is assu m ed c o rr e c tly s p e c ifie d and th e e r r o r s assu m ed u n c o rre la te d (bu t n o t n e c e s s a rily h o m o sced astic) in e s tim a tio n and te s tin g o f th e m odels exam ined h e re , it is e s s e n tia l
t h a t th e s e tw o assu m p tio n s be te s te d .
An e x te n siv e ra n g e of Monte C arlo sim u latio n e x p e rim e n ts is co nd u cted as follow s: a u n ifo rm d is tr ib u tio n and a tr e n d - s ta tio n a r y f i r s t - o r d e r a u to re g re s s iv e p ro c e ss f o r th e e x p e c ta tio n s e q u atio n ; th e e f f e c ts of using th e sam e and d if f e rin g n um bers o f o b s e rv a tio n s f o r th e e x p e c ta tio n s and s tr u c tu r a l e q u a tio n s on th e co m p u ta tio n o f th e c o r r e c t co v a rian c e m a trix ; th e e f f e c ts o f im posing an in c o rre c t co v a rian c e r e s t r i c ti o n b etw een th e e r r o r s of th e e x p e c ta tio n s and s tr u c tu r a l eq u a tio n s; th e e f f e c ts on th e s ta n d a r d e r r o r s of s tr u c tu r a l in s ta b ility , h e te ro s c e d a s tic ity and m is s p e c ific a tio n of th e e x p e c ta tio n s e q u atio n ; and th e e f f e c ts o f d if f e r e n t m ethods of e s tim a tio n on th e fu n c tio n a l m is s p e c ific a tio n and s e r ia l c o rr e la tio n d ia g n o stic te s t s .
4.2 . S i m u l a t i o n E x p e r i m e n t s
T his c h a p te r c o n sid e rs a num ber of sp e c ia l c a se s o f th e fo llo w in g tw o -e q u a tio n model:
yt * * 5 z + 5 z + r (z - l t 2 t - i i t + y2 (z t-1- Z t-1 ) + X 'ß + t Gt * z = z + 7} = W , a + 7i. t t t t t
The f o u r c a se s c o n sid e red a re : Model 1 : 5 = y = y = 0, ß = 0 2 1 2 Model 2: 5 = y = y = 0 2 1 2 Model 3: 5 = y = 0, ß = 0 2 2 Model 4: ß = 0.
T hese f o u r m odels c o rre sp o n d to Models 1, 2, 4 and 5, re sp e c tiv e ly , in P ag an (1984). It is in itia lly assum ed t h a t th e d is tu rb a n c e s o f th e tw o e q u a tio n s a r e d ra w n fro m in d ep en d en t n orm al d is tr ib u tio n s , each
w ith z e ro m ean and c o n s ta n t v a ria n c e , i.e. e ~ NID(0, cr ) and 77 ~
t t
2
NID(0, cr ) f o r t = 1, 2, T. Since in c re a s in g th e v a ria n c e o f th e d istu rb a n c e te r m s re d u c e s th e e x p la n a to ry pow er of th e m odel, tw o
2 2
d if f e r e n t v alu es o f <r a r e used, nam ely <r = 1, 100, to exam ine th e e f f e c ts o f v ary in g th e e x p la n a to ry pow er. The assu m p tio n of independence of th e tw o d is tu rb a n c e s is n e c e s s a ry f o r id e n tific a tio n of a ll o f th e p a ra m e te rs in Models 3 and 4, b u t independence is not n e c e s s a ry f o r Models 1 and 2. F or each of th e m odels, sim u latio n r e s u lts a r e o b tain e d b o th w ith and w ith o u t th e a ssu m p tio n of independence. In th e e x p e c ta tio n s eq u a tio n , th e m a tr ix of e x p la n a to ry v a ria b le s c o n s is ts o f an in te r c e p t and tw o v a ria b le s , W and W , so t h a t W' = (1, W , W ). The W v a ria b le s a re g e n e ra te d by tw o p ro c e s s e s to e s ta b lis h a d e g re e of ro b u stn e s s a c ro s s a lte rn a tiv e m ethods. F ir s t, W and W a r e assum ed to be u n c o rre la te d and a re
it 2t
d ra w n fro m u n ifo rm d is trib u tio n s , i.e. W ~ U niform (10) and W
it 2t
U n iform (5). Second, i t is assum ed t h a t W and W a r e c o rr e la te d
it 2t
and fo llo w a t r e n d - s t a ti o n a r y f i r s t - o r d e r a u to re g re s s iv e p ro c e ss. In b o th c a se s , th e v e c to r o f p a ra m e te rs f o r th e e x p e c ta tio n s e q u a tio n is s e t a t a ' = (1.0, 2 .0 , 2.5).
T h re e w ays in w hich th e d a ta a r e allow ed to v a ry f o r each model a re :
(i) th e g e n e ra tio n o f W and W , i.e. a u n ifo rm d is tr ib u tio n o r a tre n d ;
2
(ii) th e value o f th e e r r o r v a ria n c e , i.e. o* = 1 o r 100;
(iii) th e dependence o r independence o f th e d is tu rb a n c e te rm s . The e ig h t p o ssib le a lte rn a tiv e e x p e rim e n ts a r e given a s E x p e rim en ts 1-8 in T ab le 4.2.
The Monte C arlo r e s u lts a r e b ased on one th o u sa n d re p lic a tio n s . 2