An Empirical Study on Long-run Neutrality of Money in Japanese Economy

1. 　I nt r oduc t i on

Mone t a r y e c onomi s t s l ong ha ve t hought t ha t gove r nme nt i nj e c t i ons of mone y i nt o a ma c - r oe c onomy ha ve a c e r t a i n ne ut r a l e f f e c t . 　The ma i n i de a i s t ha t c ha nge s i n t he mone y s t oc k e ve nt ua l l y c ha nge nomi na l pr i c e s a nd nomi na l wa ge s , ul t i ma t e l y l e a vi ng i mpor t a nt r e a l v a r i - a bl e s , l i ke r e a l out put , r e a l c ons umpt i on e xpe ndi t ur e s , r e a l wa ge s , a nd r e a l i nt e r e s t r a t e s , una f f e c t e d. 　Si nc e e c onomi c de c i s i on ma ki ng i s ba s e d on r e a l f a c t or s , t he l ong- r un e f f e c t of i nj e c t i ng mone y i nt o t he ma c r o e c onomy i s of t e n de s c r i be d a s ne ut r al — i n t he e nd, r e a l v a r i - a bl e s do not c ha nge a nd s o e c onomi c de c i s i on ma ki ng i s a l s o unc ha nge d. 　Ho w l ong s uc h a pr oc e s s t a ke s , a nd wha t mi ght ha ppe n i n t he me a nt i me , a r e hot l y de ba t e d que s t i ons . 　Al t hough t he r e a r e ma ny c l a s s i c a l hypot he s e s t o t he e f f i c a c y of mone t a r y pol i c y , one hypot he s i s t ha t i s wi de l y a c c e pt e d a mong t he e c onomi s t s a nd pol i c yma ke r s i s t he l ong- r un ne ut r a l i t y of mone y . 　 A f or ma l de f i ni t i on of t he l ong- r un ne ut r a l i t y ( LRN) of mone y i s t ha t a pe r ma ne nt , une xpe c t e d ( e xoge nous ) c ha nge t o t he l e ve l of mone y s uppl y ha s no e f f e c t on t he l e ve l of r e a l out put i n t he l ong r un. 　Unde r LRN, c ha nge s i n t he mone y s uppl y ma y or ma y not ha ve s hor t - r un r e a l e f f e c t s . 　Re l a t e d t o t he LRN of mone y i s t he l ong- r un s upe r ne ut r a l i t y ( LRSN) of mone y ,

i n J a pa ne s e Ec onomy

Md. J a ha nur Ra hma n* a nd Tos hi hi s a Toyoda **

(Received on May 8,2007)

Abs t r ac t

Pr e vi ous t e s t s of t he l ong- r un ne ut r a l i t y of mone y hypot he s i s ha v e ge ne r a l l y r e l i e d on s e a - s ona l l y a dj us t e d da t a a nd ove r l ooke d t he i mpor t a nt i s s ue s of s e a s ona l i t y . 　Thi s pa pe r a na l ys e s t he l ong- r un ne ut r a l i t y of mone y i n J a pa n us i ng qua r t e r l y s e a s ona l l y una dj us t e d da t a , whi c h pe r mi t s a n e xa mi na t i on of t he e f f e c t s of s e a s ona l i t y a nd t he r obus t ne s s of pr e vi ous e mpi r i c a l r e s ul t s . 　Fi s he r a nd Se a t e r ( 1993) me t hodol ogy i s us e d wi t h bot h s e a s ona l l y una dj us t e d a nd a dj us t e d J a pa ne s e r e a l GDP a nd nomi na l mone y s uppl y t o t e s t t he l ong- r un ne ut r a l i t y of mone y hypot he s i s . 　Us i ng t wo me a s ur e s of mone y s t oc k, na me l y M1 a nd M2, i t i s s hown t ha t t he hypot he s i s i s s uppor t e d us i ng M2 a s t he me a s ur e of mone y s uppl y, whi l e i t i s r e j e c t e d us i ng M1.

* Ph. D. St ude nt , Fa c ul t y of Ec onomi c Sc i e nc e s , Hi r os hi ma Shudo Uni v e r s i t y , J a pa n.

** Pr of e s s or , Fa c ul t y of Ec onomi c Sc i e nc e s , Hi r os hi ma Shudo Uni v e r s i t y , J a pa n.

whi c h oc c ur s whe n pe r ma ne nt , e xoge nous c ha nge s i n t he gr owt h r a t e of mone y s uppl y l e a ve t he l e v e l of r e a l out put una f f e c t e d.

The s e l ong- r un ne ut r a l i t y pr e pos i t i ons ha ve be e n s t udi e d e xt e ns i ve l y , bot h t he or e t i c a l l y a nd e mpi r i c a l l y a nd v a r i ous e c onome t r i c pr oc e dur e s a r e a v a i l a bl e f or t e s t i ng t he s e hypot he s e s . 　 Mode r n t he or e t i c a l f ounda t i on wa s pr o vi de d by Fr i e dma n ( 1969a , 1969b) . 　Muc h of t he e xi s t - i ng e mpi r i c a l l i t e r a t ur e ha s be e n mot i v a t e d by t he wor k of Mc Ca l l um ( 1984) who, dr a wi ng on t he r e ma r ks ma de by Sa r ge nt ( 1971) a nd Luc a s ( 1972, 1976) , s ho ws t ha t t he ne ut r a l i t y r e s ul t s obt a i ne d f r om f r e que nc y- doma i n me t hods a r e uni nf or ma t i ve be c a us e of t he pr obl e m of obs e r va - t i ona l e qui v a l e nc e . 　Mc Ca l l um ( 1984) c l a i ms t ha t wi t hout kno wi ng t he t i me s e r i e s pr ope r t i e s of t he mone y s uppl y, bot h f r e que nc y- doma i n a nd r e duc e d- f or m e s t i ma t e s a r e una bl e t o di s c r i mi - na t e e mpi r i c a l l y be t we e n l ong- r un non- ne ut r a l i t i e s a nd t he e f f e c t s a r i s i ng f r om a ut or e gr e s s i ve mone y s uppl y s pe c i f i c a t i ons of t he t ype de ve l ope d by Luc a s ( 1972) . 　He l uc i dl y poi nt s out t ha t a v a l i d t e s t of t he ne ut r a l i t y hypot he s i s c a n onl y be c onduc t e d us i ng c r os s - e qua t i on r e s t r i c t i ons i n a bi v a r i a t e v e c t or a ut or e gr e s s i on ( VAR) .

I n r e s pons e t o t hi s , Fi s he r ( 1988) , Fi s he r a nd Se a t e r ( 1993) , a nd a l s o i n a s e r i e s of pa pe r s by Ki ng a nd Wa t s on ( 1992, 1994, 1997) ha ve a dva nc e d t he domi na nt a ppr oa c he s i n t hi s l i ne of r e s e a r c h. 　The ne ut r a l i t y of mone y hypot he s i s ha s be e n t e s t e d f or nume r ous c ount r i e s us i ng t he i r me t hodol ogi e s . 　For e xa mpl e , Bos c he n a nd Ot r ok ( 1994) , Bos c he n a nd Ot r ok ( 1994) , Ol e ka l ns ( 1996) , Se r l e t i s a nd Kr a us e ( 1996) , Ha ug a nd Luc a s ( 1997) , Coe a nd Na s on ( 2003) , She l l e y a nd Wa l l a c e ( 2006) us e d Fi s he r a nd Se a t e r me t hod, whi l e We be r ( 1994) , J e f f e r s on ( 1997) , Se r l e t i s a nd Kous t a s ( 1998, 2001) e mpl oye d Ki ng a nd Wa t s on me t hods f or t e s t i ng t he LRN a nd LRSN of mone y .

I n s pi t e of t he pr ogr e s s i n r e s e a r c h on l ong- r un ne ut r a l i t y of mone y , onl y a ve r y l i mi t e d numbe r of c ompr e he ns i v e s t udi e s a r e a v a i l a bl e i n J a pa n. 　Ya ma da ( 1997) s ho ws t ha t mone t a r y ne ut r a l i t y hol ds i n t e r ms of r e a l out put , by a ppl yi ng Fi s he r a nd Se a t e r ’ s ( 1993) pr oc e dur e t o t he J a pa ne s e s e a s ona l l y a dj us t e d qua r t e r l y da t a f r om 1957q1 t o 1995q1. 　Us i ng t wo t ype s of da t a s e t s f or c e nt ur y- l ong a nnua l da t a c ove r i ng 119 ye a r s f r om 1885 t hr ough 2003 a s we l l a s pos t - wa r s e a s ona l l y a dj us t e d qua r t e r l y da t a ove r t he pe r i od 1955q2 – 2003q4, Oi e t al . ( 2004) ha ve f ound e vi de nc e s uppor t i ng l ong- r un ne ut r a l i t y , e s pe c i a l l y f or t he c a s e of M2 a s a me a s ur e of mone y s t oc k. 　The y us e d Ki ng a nd Wa t s on’ s ( 1997) pr oc e dur e .

But , a n ove r l ooke d i mpor t a nt i s s ue i n a l l pr e vi ous pa pe r s i n J a pa n i s t ha t of s e a s ona l i t y . 　

Eve n mos t of t he pr e vi ous r e s e a r c he s us e d s e a s ona l l y a dj us t e d da t a i n t e s t i ng l ong- r un ne ut r a l i t y

hypot he s e s . 　But i n qua r t e r l y obs e r ve da t a , s e a s ona l i t y ha s be e n a n i s s ue t ha t ha s a t t r a c t e d c on-

s i de r a bl e r e s e a r c h i nt e r e s t i n t he mode l i ng of e c onomi c t i me s e r i e s . 　Re c e nt l y , s ome publ i s he d pa pe r s ha ve a r gue d t ha t mode l i ng, i ns t e a d of r e movi ng, s e a s ona l i t y ma y be be ne f i c i a l f or e c o- nomi c a na l ys i s , f or e xa mpl e , Hyl l e be r g ( 1992) , Le ong e t al . ( 2000) .

Tr a di t i ona l l y , s e a s ona l i t y ha s be e n e xpl i c i t l y r e move d us i ng s e a s ona l a dj us t me nt pr oc e - dur e s , s uc h a s us i ng dummy va r i a bl e s or pr e f i l t e r i ng t he s e r i e s us i ng pe r i od- t o- pe r i od di f f e r - e nc e s or t he ARI MA X- 11 me t hodol ogy . 　The s e me t hodol ogi e s ha ve i mpor t a nt dr a wba c ks : Ghys e l s a nd Pe r r on ( 1993) , a nd Fr a ns e s ( 1996) s ho ws t ha t s e a s ona l a dj us t me nt pr oc e dur e s c a n r e move c yc l i c a l f l uc t ua t i ons i n t he da t a , a s s e a s ona l i t y a nd bus i ne s s c yc l e s a r e t ypi c a l l y c or r e l a t - e d, a nd c a n t he r e by di s t or t t he da t a . 　Mor e ove r , s e a s ona l a dj us t me nt s c a n a f f e c t t he po we r of uni t r oot a nd c oi nt e gr a t i on t e s t s . 　I f s e a s ona l i t y i s de t e r mi ni s t i c , r e movi ng i t wi t h t he a i d of dummy v a r i a bl e s ha s no e f f e c t wha t s oe ve r on uni t r oot t e s t s ( Di c ke y e t al . , 1984) . 　Ho we ve r , whe n s e a s ona l e f f e c t s a r e s t oc ha s t i c , s t a nda r d f i l t e r s c a n gr e a t l y a f f e c t t he po we r of uni t r oot t e s t s . 　Ghys e l s ( 1990) s ho ws t ha t r e movi ng s e a s ona l i t y us i ng t he X- 11 me t hod or t he “ v a r i a - t i on i n s pe r i ods ” i nduc e s e xc e s s pe r s i s t e nc e i n t he s e r i e s a nd c ons e que nt l y r e duc e s t he powe r of uni t r oot t e s t s t o r e j e c t nons t a t i ona r i t y . 　Ol e ka l ns ( 1994) e xt e nds t hi s r e s ul t t o t he c a s e s i n whi c h dummi e s or band- pas s f i l t e r s a r e us e d t o r e move s e a s ona l i t y. 　Abe ys i nghe ( 1994) s hows t ha t r e movi ng s t oc ha s t i c s e a s ona l i t y wi t h dummy va r i a bl e s l e a ds t o t he s pur i ous r e gr e s s i on pr obl e m.

The i s s ue s of l ong- r un ne ut r a l i t y ha ve be e n t he he a r t of de ba t e s ove r t he r e a l e f f e c t s of macr oeconomi c pol i ci es and ar e s t i l l ver y cont r over s i al t opi cs among macr oeconomi c r e s e a r c he r s . 　The e mpi r i c a l t e s t a bi l i t y of t he s e hypot he s e s i s i mpor t a nt f or pol i c y f or mul a t i on a nd de s i gn, s uc h a s t he e f f e c t i ve ne s s i n mone t a r y pol i c y , whi l e s t he de t e r mi na t i on of t he or de r of i nt e gr a t i on, s e a s ona l or nons e a s ona l , i s c r uc i a l f or t he s e e xi s t i ng me t hods . 　So t hi s pa pe r pr e - s e nt s t he e c onome t r i c t r e a t me nt of s e a s ona l i t y f or t he pur pos e of t e s t i ng f or t he l ong- r un ne ut r a l - i t y of mone y . 　Fi s he r a nd Se a t e r ( 1993) a ppr oa c h i s us e d t o i n v e s t i ga t e t he l ong- r un ne ut r a l i t y of mone y i n J a pa n ove r t he e xt e nde d pe r i od t o 2006.

¹⁾

　Cons e que nt l y, bot h qua r t e r l y s e a s ona l l y una dj us t e d a nd a dj us t e d t i me s e r i e s da t a on r e a l GDP , nomi na l mone t a r y v a r i a bl e s M1 a nd M2 a r e us e d i n t hi s pa pe r t o e xa mi ne t he e f f e c t s of s e a s ona l i t y a nd t he r obus t ne s s of pr e vi ous e mpi r i c a l r e s ul t s i n J a pa n.

1) We a l s o us e d Ki ng a nd Wa t s on ( 1997) me t hod. The r e s ul t s ( not s hown) a r e qua l i t a t i ve l y s a me a s

t hos e obt a i ne d f r om Fi s he r a nd Se a t e r ( 1993) me t hod.

2. 　Ove r vi e w of Long- Run Ne ut r al i t y Te s t s

The r e ha s l ong be e n i nt e r e s t i n t e s t i ng l ong- r un pr opos i t i ons a bout t he l i nk be t we e n mone y a nd r e a l or nomi na l va r i a bl e s , whi c h a r e a t t he he a r t of c l a s s i c a l ma c r oe c onomi c s . 　I n t hi s s e c - t i on, t he hi s t or y of s uc h t e s t s i s r e vi e we d a nd t he a ppr oa c h t a ke n i n t hi s pa pe r i s hi ghl i ght e d.

One not a bl e e a r l y s t r a nd of r e s e a r c h on t he s e i s s ue s wa s a t t he Fe de r a l Re s e r ve Ba nk of St . Loui s dur i ng t he 1960s . 　The St . Loui s r e s e a r c he r s be ga n t he e mpi r i c a l s t udy of t he r e l a t i on- s hi p be t we e n nomi na l i nc ome a nd t he mone y s t oc k i n a dyna mi c r e gr e s s i on f r a me wor k,

( 1) whe r e Y

_t

i s l og of nomi na l i nc ome , m

_t

i s l og of mone y s t oc k, x

_t

a r e ot he r v a r i a bl e s t ha t a f f e c t nomi na l i nc ome , e

_t

i s a n e r r or t e r m, a nd i n t he l a g ope r a t or L a nd D = 1 - L i ndi c a t e s a f i r s t di f f e r e nc e .

The St . Loui s r es ear cher s wer e mot i vat ed t o s t udy s uch di s t r i but ed l ag model s by Fr i e dma n’ s ( 1969) a r gume nt t ha t t he r e wa s a l a g i n t he e f f e c t of mone t a r y a c t i ons on t he ma c r o e c onomy . 　I n t he we l l - kno wn wor k of Ande r s on a nd J or da n ( 1968) , t he y s ought t o de t e r mi ne t he na t ur e of t he l a gs i n t he e f f e c t s of mone t a r y pol i c y i n e s t i ma t i ng t he B c oe f f i c i e nt s i n Equa - t i on ( 1) . 　The y f ound t ha t t he r e wa s a l e s s t ha n one - f or - one s hor t - r un e f f e c t of mone y on nomi - na l i nc ome , i . e . , B

₀

< 1. 　To c a l c ul a t e t he e f f e c t of a s us t a i ne d ( l ong- r un) c ha nge i n t he l e ve l of mone y on t he pa t h of nomi na l i nc ome , t he y c a l c ul a t e d dyna mi c mul t i pl i e r s a s f ol l o ws :

La t e r St . Loui s a na l ys i s , Ande r s e n a nd Ka r nos ky ( 1972) , us e d t hi s r e gr e s s i on f r a me wor k t o t e s t LRN a s f ol l o ws . 　The y i ma gi ne d a pe r ma ne nt c ha nge i n t he l e v e l of mone y ha v e no l ong- r un e f f e c t i n t he l e v e l of nomi na l i nc ome i f

The y a l s o i mpl e me nt e d t he c ompa r a bl e t e s t f or LRN us i ng l og of r e a l i nc ome ( y

_t

) a s

The St . Loui s a ppr oa c h wa s c ont r ove r s i a l . 　Not a bl y , Ando a nd Modi gl i a ni ( 1990) c r i t i - c i z e d t he St . Loui s r e gr e s s i on f or not r e c ogni z i ng t ha t nomi na l i nc ome a nd t he mone y s t oc k we r e s i mul t a ne ous l y de t e r mi ne d. 　Ho we v e r , Si ms ( 1972) pr o vi de d s ome s uppor t i n a bi v a r i a t e

Δ Y

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B L B L

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^{C L}

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∂

∂

⁺

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∑

=

Y

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B

t

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s t s

t

i i

Y

s

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0

c ont e xt f or t he St . Loui s r e gr e s s i on, bui l di ng on Gr a nge r ’ s ( 1969) e a r l i e r wor k on t e s t i ng f or c a us a l i t y . 　The or e t i c a l l y , Si ms e s t a bl i s he d t ha t i t wa s onl y l e gi t i ma t e t o r un t he r e gr e s s i on

f or t he pur pos e s of t he St . Loui s r e s e a r c he r s i f t he r e v e r s e r e gr e s s i on

di s pl a ye d D c oe f f i c i e nt s t ha t we r e z e r o. 　Looki ng a t nomi na l i nc ome a nd mone y e mpi r i c a l l y , he f ound e vi de nc e t ha t t he D c oe f f i c i e nt s we r e s t a t i s t i c a l l y i ns i gni f i c a nt .

The i nt e r pr e t a t i on of t he St . Loui s r e gr e s s i ons wa s a l s o c a l l e d i nt o que s t i on by t he a na l ys e s of Sa r ge nt ( 1971) a nd Luc a s ( 1972) . 　St udyi ng a n e c onomy i n whi c h onl y una nt i c i pa t e d mone - t a r y c ha nge s ha d r e a l e f f e c t s a nd whi c h ot he r wi s e di s pl a ye d t he LRN pr ope r t i e s , Luc a s ( 1972) s ho we d t ha t r e s t r i c t i ons on s ums of c oe f f i c i e nt s di d not pr ovi de a wa y of t e s t i ng t he c l a s s i c a l pr opos i t i ons . 　To i l l us t r a t e Luc a s ’ s poi nt , c ons i de r a n e c onomy i n whi c h t he be ha vi or of r e a l a nd nomi na l i nc ome i s gi v e n by

whe r e f i s a pos i t i ve pa r a me t e r a nd 0 < q < 1. 　Tha t i s : una nt i c i pa t e d mone t a r y e xpa ns i ons r a i s e r e a l i nc ome a nd r a i s e nomi na l i nc ome l e s s t ha n one - f or - one . 　I f t he mone y s uppl y i s gi v e n by t he f i r s t - or de r a ut or e gr e s s i on

i t t he n f ol l o ws t ha t t he r a t i ona l e xpe c t a t i ons s ol ut i ons f or r e a l a nd nomi na l out put a r e

( 2a ) ( 2b) Unde r Luc a s ’ s a s s umpt i on t ha t t he mone y s uppl y pr oc e s s i s s t a t i ona r y ( | r | < 1) , t he s um of c oe f - f i c i e nt s i n t he r e a l out put e qua t i on i s i nc ons i s t e nt wi t h ne ut r a l i t y ( f ( 1 - r ) > 0) . 　The s e i mpl i - c a t i ons oc c ur de s pi t e t he f a c t t ha t t he mode l i s one wi t h a s t r ong f or m of ne ut r a l i t y . 　On t he ba s i s of t hi s f i ndi ng, Luc a s a nd Sa r ge nt a r gue d t ha t i f l ong- r un v a r i a t i on i n mone y i s not a pa r t of t he e n vi r onme nt t ha t s ha pe s t he be ha vi or a l r e s pons e s of e c onomi c a ge nt s , t he n a r e duc e d f or m a na l ys i s — s uc h a s dyna mi c r e gr e s s i ons or ve c t or a ut or e gr e s s i ons — c a n ne ve r pr ovi de a ns we r s a bout t he e f f e c t of l ong- r un v a r i a t i ons i n mone y .

Conc e r n a bout c a us a l i t y a nd t he Luc a s c r i t i que c a s t a s ha dow ove r a ppl i e d r e s e a r c h on l ong- r un ( LR) t e s t s f or ne a r l y t wo de c a de s . 　Howe ve r , Fi s he r a nd Se a t e r ( 1993) poi nt e d out

A L Y ( )

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_t

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Y

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m

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,

y

_t

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_t−1

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_{y t,}

Y

_t

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_t

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+ e

_{Y t,}

t ha t t he pe s s i mi s m wa s not ne c e s s a r i l y j us t i f i e d i f e c onomi s t s a r e c onc e r ne d a bout whe t he r LR hypot he s e s he l d i n a pa r t i c ul a r hi s t or y a nd t he hi s t or i c a l da t a c ont a i ne d l ong- r un v a r i a t i on i n mone y . 　Fi s he r a nd Se a t e r ( 1993) e mpl oy a bi v a r i a t e s t r uc t ur a l ve c t or a ut or e gr e s s i on ( SVAR) mode l t o t e s t l ong- r un ne ut r a l i t y a nd s uppe r ne ut r a l i t y unde r t he i de nt i f i c a t i on a s s umpt i on t ha t nomi na l v a r i a bl e i s e xoge nous i n t he l ong- r un. 　The y de mons t r a t e t he r e s t r i c t i ons i mpl i e d by t he s e pr e pos i t i ons a r e c ondi t i ona l on t he or de r of i nt e gr a t i on of t he v a r i a bl e s c ompr i s e d i n t he a na l ys i s . 　Spe c i f i c a l l y, ne ut r a l i t y t e s t s a r e pos s i bl e onl y i f nomi na l a s we l l a s r e a l va r i a bl e s a r e a t l e a s t i nt e gr a t e d of s a me or de r . 　Mor e ove r , s uppe r ne ut r a l i t y t e s t s a r e pos s i bl e onl y i f t he or de r of i nt e gr a t i on of t he nomi na l v a r i a bl e i s e qua l t o one pl us t he or de r of i nt e gr a t i on of t he r e a l va r i a bl e . 　Ba s e d on t he s e t wo i de a s , i nt e gr a t i on a nd i de nt i f i c a t i on, Ki ng a nd Wa t s on ( 1992, 1997) pr o vi de a wa y of t e s t i ng l ong- r un ne ut r a l i t y by me a ns of a s s umi ng di s t i nc t s t r uc - t ur a l a s s umpt i ons a bout t he e c onomy, whi l e pa yi ng s pe c i a l a t t e nt i on t o t he i nt e gr a t i on a nd c oi n- t e gr a t i on pr ope r t i e s of t he da t a . 　The i r me t hodol ogy doe s not a s s ume mone y e r oge ne i t y , a nd pr opos e s t ha t t he i mpa c t of mone y on out put , or out put on mone y be t e s t e d unde r a l t e r na t i ve i de nt i f yi ng r e s t r i c t i ons i n t wo va r i a bl e SVAR mode l s . 　Al t hough a l l t he s e t e s t pr oc e dur e s di f f e r i n t he de t a i l s , e a c h wa s ba s e d on t he c or e i de a t ha t l ong- r un pr opos i t i ons a r e t e s t a bl e i f t he r e i s s ui t a bl e l ong- r un va r i a t i on i n mone y. 　As a n e xa mpl e , s uppos e t ha t t he mone y s t oc k i s a s s ume d t o be a r a ndom wa l k ( r = 1) i n t he Luc a s mode l j us t c ons i de r e d. 　Thi s a s s umpt i on me a ns t ha t a l l c ha nge s i n mone y a r e una nt i c i pa t e d a nd pe r ma ne nt . 　Eva l ua t i ng t he e xpr e s s i ons a bo v e a t r = 1, i t t he n f ol l o ws t ha t t he s um of c oe f f i c i e nt s on t he mone t a r y v a r i a bl e s i n ( 2a ) i s z e r o a nd t he s um i n ( 2b) i s one a s s ugge s t e d by pr i or ne ut r a l i t y t e s t s .

Ne ut r a l i t y t e s t s ba s e d on SVAR a r e mor e c ompl i c a t e d t ha n t he s i mpl e Luc a s e xa mpl e on t hr e e di me ns i ons . 　Fi r s t , t he y a l l o w r e a l out put t o be pot e nt i a l l y a f f e c t e d by r e a l s hoc ks i n t he l ong- r un. 　Se c ond, t he y a l l o w f or mone y gr o wt h t o r e s pond t o i t s o wn l a gs a nd t o l a gs of out - put gr owt h. 　Thi r d, t he y a l l ow f or s hor t - r un i nt e r a c t i ons of r e a l a nd nomi na l va r i a bl e s . 　Al l of t he s e c ons i de r a t i ons a r e r e f l e c t e d i n t he f ol l o wi ng s t r uc t ur a l v e c t or a ut or e gr e s s i on,

( 3)

I n t hi s s t r uct ur al VAR, t her e ar e t wo s t r uct ur al s hocks e

_y,t

and e

_m,t

wi t h mean 0 and whe r e . 　The f or me r r e f e r s t o r e a l pr oduc t i vi t y s hoc ks whi l e t he l a t t e r r e f e r s t o mone t a r y s hoc ks . 　I n a ddi t i on, bot h v a r i a bl e s a r e t r e a t e d e ndoge nous l y , whi c h

π

_{yy t}

π

_{ym t}

α

_{yy i t i}

α ε

i p

ym i t i y t

i p

y m y m

Δ = Δ + Δ

₋

+ Δ +

= −

∑

^,

∑

= ^{, ,}

1 1

π

_{mm t}

π

_{my t}

α

_{my i t i}

α ε

i p

mm i t i m t

i p

m y y m

Δ = Δ + Δ

₋

+ Δ +

= −

∑

^,

∑

= ^{, ,}

1 1

cov( , ) ε ε ′ = Σ

_ε

, ε = ( ε ε

_{y t,}

,

_{m t,}

) ′

mi t i ga t e s t he c a us a l i t y pr obl e ms pr e vi ous l y di s c us s e d i n t he c ont e xt of t he St . Loui s r e gr e s s i on.

Shor t - r un i nt e r a c t i ons of y

_t

a nd m

_t

a r e gove r ne d by t he p c oe f f i c i e nt s , whi l e t he dyna mi c i nt e r a c - t i ons a r e go v e r ne d by t he a c oe f f i c i e nt s . 　The a bo v e s ys t e m c a n be wr i t t e n a s

whe r e L i s l a g ope r a t or , t ha t i s , . 　As s umi ng bot h y a nd m a r e I ( 1) a nd not c oi nt e gr a t - e d, t he l ong- r un r e s pons e s of y

_t

a nd m

_t

t o t he s hoc ks e

_y,t

a nd e

_m,t

, a r e t he s ol ut i ons t o t he a bove s ys t e m, t ha t c a n be a ppr oxi ma t e d a s

whe r e . 　The l ong-

r un e f f e c t of mone t a r y s hoc k e

_m

i s

a nd t he l ong- r un e f f e c t of r e a l s hoc k e

_y

i s

Ba s e d on t he a bo v e s pe c i f i c a t i ons , t he l ong- r un ne ut r a l i t y of mone y wi t h r e s pe c t t o r e a l out put c a n be t e s t e d us i ng t he f ol l o wi ng l ong- r un mul t i pl i e r :

( 4) The e xpr e s s i on i n e qua t i on ( 4) i s a t t he he a r t of Fi s he r a nd Se a t e r ( 1993) a nd Ki ng a nd Wa t s on ( 1997) t e s t s . 　Fi s he r a nd Se a t e r ( 1993) t e r m t he r a t i o a s l ong- r un de r i v a t i ve ( LRD) of y wi t h r e s pe c t t o m , a nd Ki ng a nd Wa t s on ( 1997) c a l l e d t he l ong- r un e l a s t i c i t y of out put ( y ) wi t h r e s pe c t t o pe r ma ne nt e xoge nous c ha nge i n mone y ( m ) .

I f bot h mone y a nd r e a l GDP a r e i nt e gr a t e d of or de r one , I ( 1) , ( or s a me or de r ) t he n t he i mpl i c a t i on of l ong- r un ne ut r a l i t y of mone y i s t ha t

π α π α

π α π

yy yy i

i i p

yy ym i

i i p

my my i

i i

p

mm

L L

L

− − +

− + −

= =

=

∑ ∑

∑

, ,

,

( )

( )

1 1

1

α α

ε

mm i

ε

i i

p

t t

y t

L

m t

y m

,

,

= ,

∑

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

⎡

⎣ ⎢ ⎤

⎦ ⎥ = ⎡

⎣ ⎢

⎢

⎤

⎦ ⎥

1

⎥

Δ Δ L x

ⁱ _t

= x

_{t i−}

lim

^{, ,}

, ,

k

t k y t t k m t

t k y t t k m t

y y

m

m m

→∞

+ +

+ +

∂ ∂ ∂ ∂

∂ ∂ ∂ ∂

⎡

⎣ ⎢

⎢

⎤

⎦ ⎥

⎥ =

ε ε

ε ε ϕ π

_{mm i}^p _{mm i ym i}^p _{ym i}

my i my i

p

yy i yy i

p

− +

+ −

⎡

⎣

⎢

^{= =}

= =

∑ ∑

∑ ∑

α π α

π α π α

, ,

, ,

1 1

1 1

⎢⎢

⎤

⎦

⎥ ⎥

ϕ = π α π α π α π α

− ∑

₌

− ∑

₌

− + ∑

₌

+

1

1 1 1

(

_{yy yy i,}

)(

_,

) (

_,

)(

i p

mm i mm i

p

my i my i

p

ym i yym i

p ,

)

∑

=¹

lim

^,

,

, , k

t k m t

t k m t

ym i ym i

p

yy yy

y

→∞

m

+ +

∂ ∂

=

∂ ∂

⎡

⎣ ⎢

⎢

⎤

⎦ ⎥

⎥ = +

−

∑

ε

ε ϕ π α

π α

1 i ii p

∑

=

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

1

lim

^,

,

, , k

t k y t

t k y t

mm i mm i

p

my my

y m

m

→∞

+ +

∂ ∂

=

∂ ∂

⎡

⎣ ⎢

⎢

⎤

⎦ ⎥

⎥ = +

−

∑

ε

ε ϕ α

π α

1 i ii p

∑

=

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

1

.

γ ε

ε

π α

π α

ym k

t k m t

t k m t

ym i ym i

p

yy i yy i

p

y

= ∂ m ∂

∂ ∂ = +

→∞

−

+ +

=

=

lim

^,

∑

,

, , 1

∑

1

∑

( 5) Long- r un s uppe r ne ut r a l i t y ( LRSN) c a n be t e s t e d whe n mone y i s I ( 2) a nd r e a l out put i s I ( 1) . 　Us i ng t he gr owt h of mone y ( D m

_t

) i n s t e a d of mone y ( m

_t

) i n SVAR mode l i n Equa t i on ( 3) , t he LRD c a n be obt a i ne d a s

( 6) Not e t ha t t he LRD i s not de f i ne d f or t he c a s e i n whi c h . 　The r e f or e , a ne c e s s a r y c ondi t i on f or t e s t i ng LRN i s t ha t t he r e ha ve be e n pe r ma ne nt s hoc ks t o mone y s uppl y . 　Mone y t he r e f or e mus t be a t l e a s t f i r s t or de r i nt e gr a t e d, or I ( 1) , t o a ppl y t he t e s t . 　 Equa t i ons ( 4) a nd ( 6) a l s o s how t ha t i f t he r e ha ve no pe r ma ne nt s hoc ks t o r e a l out put , t he n a nd e qua l t o z e r o r e s pe c t i ve l y , t ha t i s , t he LRD i s e qua l t o z e r o. 　The r e f or e , i f t he out put i s I ( 0) , LRN a nd LRSN c a nnot be r e j e c t e d.

3. 　I de nt i f yi ng Re s t r i c t i ons

By i t s na t ur e , t he l ong- r un mul t i pl i e r g

_ym

i s a s t r uc t ur a l pa r a me t e r , whi c h r e qui r e s i de nt i f y- i ng a s s umpt i ons t o e s t i ma t e i t . 　The s e i de nt i f yi ng a s s umpt i ons a r e mos t e a s i l y di s c us s e d i f we f ol l o w t he a c t ua l pr a c t i c e us e d i n s ome of t he pr i or l i t e r a t ur e . 　To be gi n, s uppos e t ha t we e s t i - ma t e a r e duc e d f or m VAR a s

I n t hi s r e duc e d f or m VAR, u

_y,t

a nd u

_m,t

a r e e r r or t e r ms wi t h me a n 0 a nd , whe r e . 　Compa r i ng t hi s e xpr e s s i on wi t h ( 3) , we not e a s t r uc t ur a l r e l a t i ons hi p be t we e n r e duc e d f or m s hoc ks u

_t

a nd s t r uc t ur a l s hoc ks e

_t

t ha t t a ke s t he f or m

wi t h c o v a r i a nc e ma t r i x

LRD = = ∂ ∂

∂ ∂ = +

→∞

−

+ +

∑

=

γ ε

ε

π α

π α

ym k

t k m t

t k m t

ym i ym i

p

yy yy i

y

lim m

^,

,

, , 1 ii p

∑

=¹

^{= 0}

γ ε

ε

π α

π α

y m k

t k m t

t k m t

ym i y m i

p

yy yy

y

Δ

m

Δ

Δ

Δ

= ∂ Δ ∂

∂ ∂ = +

−

→∞

+ +

∑

=

lim

^,

,

, , 1 i ii p

∑

=¹

lim(

_,

)

k

m

t k m t

→∞

∂

+

∂ ε = 0

lim(

_,

)

k

y

t k m t

→∞

∂

+

∂ ε lim( /

_,

)

k

y

t k m t

→∞

∂

+

∂ ε

_Δ

Δ y

_t

b

_{yy i}

Δ y

_{t i}

b Δ m u

i p

ym i t i y t

i p

=

₋

+ +

= −

∑

^,

∑

= ^{, ,}

1 1

Δ m

_t

b

_{my i}

Δ y

_{t i}

b Δ m u

i p

mm i t i m t

i p

=

₋

+ +

= −

∑

^,

∑

= ^{, ,}

1 1

cov( , ) u u ′ = Σ

_u

u = ( u

_{y t,}

, u

_{m t,}

) ′

u u

y t m t

yy ym

my mm

y t m t ,

,

, ,

⎡

⎣ ⎢

⎢

⎤

⎦ ⎥

⎥ = −

−

⎡

⎣ ⎢

⎢

⎤

⎦ ⎥

⎥

⎡

⎣ ⎢

⎢

⎤

⎦ ⎥

⎥ π π

−

π π

ε ε

1

a nd t he c or r e s pondi ng pa r a me t e r s a r e

The bi v a r i a t e s t r uc t ur a l VAR of or de r p , a s i n e qua t i on ( 3) , ha s 2

²

´ ( p + 1) unkno wns i n t he c oe f f i c i e nt s a nd 3 ( = 2 ´ ( 2 + 1) / 2) unkno wns i n t he c ov a r i a nc e ma t r i x of t he r e s i dua l . 　Me a n- whi l e , t he bi va r i a t e r e duc e d- f or m VAR of or de r p pr ovi de s e s t i ma t e s of 2

²

´ p pa r a me t e r va l ue s f or t he c oe f f i c i e nt s a nd t hr e e va l ue s f or t he c ova r i a nc e ma t r i x of t he r e duc e d f or m e r r or s . 　 Ac c or di ngl y , 2

²

= 4 i de nt i f yi ng r e s t r i c t i ons mus t be pl a c e d t o i de nt i f y t he s t r uc t ur a l s hoc ks e

_y,t

a nd e

_m,t

. 　St a nda r di z i ng p

_yy

a nd p

_mm

t o uni t y a nd us i ng t he a s s umpt i on t ha t t he s t r uc t ur a l s hoc ks a r e mut ua l l y unc or r e l a t e d, i . e . , , one a ddi t i ona l i de nt i f yi ng r e s t r i c t i on i s r e qui r e d. 　For t hi s r e a s on, t o c omput e t he g

_ym

, t he di f f e r e nt r e s e a r c he r s i mpos e d di f f e r e nt i de n- t i f yi ng a s s umpt i ons .

Fi s her and Seat er ( 1993) as s umed t hat mone y i s l ong- r un e xogenous by i mpos i ng . 　Tha t i s , t he r e a l s hoc k e

_y,t

i s a s s ume d not t o a f f e c t t he va r i a t i on i n mone y i n t he l ong- r un. 　For ma l l y, gi ve n t hi s a s s umpt i on, t he s e c ond e qua t i on of ( 3) gove r ni ng t he l ong- r un r e s pons e of mone y i mpl i e s t ha t : t he l ong- r un r e s pons e of mone y doe s not de pe nd on t he l ong- r un r e s pons e of out put . 　Us i ng t hi s a s s umpt i on, Fi s he r a nd Se a t e r de mons t r a t e t ha t g

_ym

c a n be c ons i s t e nt l y e s t i ma t e d a s t he s l ope c oe f f i c i e nt i n t he f ol l o wi ng OLS r e gr e s s i on:

( 7) Tha t i s , LRD c a n be a ppr oxi ma t e d by . 　St a nda r d pr a c t i c e i s t o e s t i ma t e b

_k

us i ng OLS f or e a c h va l ue of k t a ki ng va l ue s of one t hr ough a pr e de t e r mi ne d uppe r l i mi t . 　Wi t h T obs e r va - t i ons , t he 95- pe r c e nt c onf i de nc e i nt e r v a l s t he n a r e c ons t r uc t e d f or t he b

_k

’ s f r om a t - di s t r i b ut i on wi t h T / k de gr e e s of f r e e dom us i ng s t a nda r d e r r or s c or r e c t e d f or s e r i a l c or r e l a t i on by t he Ne we y- We s t pr oc e dur e . 　Long- r un ne ut r a l i t y of mone y i s r e j e c t e d i f z e r o l i e s out s i de t he c onf i de nc e i nt e r v a l a s k be c ome l a r ge .

For t he LRSN of mone y, whi c h i s t e s t a bl e i f m i s I ( 2) a nd y i s I ( 1) , t he e s t i ma t or of g

_ym

i s t he s l ope c oe f f i c i e nt i n

( 8)

var( ) cov( , )

cov( , ) var( )

, , ,

, , ,

u u u

u u u

y t y t m t

y t m t m t

yy

⎡

⎣ ⎢

⎢

⎤

⎦ ⎥

⎥

= π −−

−

⎡

⎣ ⎢

⎢

⎤

⎦ ⎥

⎥ π

−

π π

ε ε ε

ε ε

ym

my mm

y t y t m t

y t m t

1

var( ) cov( , )

cov( ,

, , ,

, ,

)) var( ε

_,

)

π π

π π

m t

yy ym

my mm

⎡

⎣ ⎢

⎢

⎤

⎦ ⎥

⎥

−

−

⎡

⎣ ⎢

⎢

⎤

⎦ ⎥

⎥

′

⁻¹

b b

b b

yy i ym i

my i mm i

yy ym

my mm

yy i

, ,

, ,

⎡

,

⎣ ⎢

⎢

⎤

⎦ ⎥

⎥ = −

−

⎡

⎣ ⎢

⎢

⎤

⎦ ⎥

⎥ π π

−

π π

α

1

α α

α α

ym i

my i mm i

i p

,

, ,

; , , ,

⎡

⎣ ⎢

⎢

⎤

⎦ ⎥

⎥ = 1 2 …

cov( ε

_{y t,}

, ε

_{m t,}

) = 0

( π

_my

+ ∑

_i^p₌¹

α

_{my i,}

) = ⁰

( y

_t

− y

_{t k− −}1

) = α

_k

+ β

_k

( m

_t

− m

_{t k− −}1

) + ε

_kt

lim

k k

→∞

β

( y

_t

− y

_{t k}− −1

) = α

_k

+ β

_k

( Δ m

_t

− Δ m

_{t k}− −1

) + ε

_kt

Ki ng a nd Wa t s on ( 1997) e mpha s i z e t he r e l a t i ons hi p be t we e n t e s t r e s ul t s a nd i de nt i f yi ng i s s ue s . 　The y a na l yz e l ong- r un ne ut r a l i t y pr opos i t i ons a c r os s a r a nge of pos s i bl e i de nt i f i c a t i ons of t he i r bi va r i a t e s t r uc t ur a l VAR s ys t e m, i n a n e f f or t t o unde r s t a nd t he r obus t ne s s of va r i ous c onc l us i ons t o di f f e r i ng a s s umpt i ons . 　The y us e d one of t he f ol l o wi ng i de nt i f yi ng r e s t r i c t i ons i n t he i r LR t e s t s .

( i ) t he i mpa c t e l a s t i c i t y of y wi t h r e s pe c t t o m i s kno wn ( i . e . , p

_ym

i s kno wn) , ( i i ) t he i mpa c t e l a s t i c i t y of m wi t h r e s pe c t t o y i s kno wn ( i . e . , p

_my

i s kno wn) , ( i i i ) t he l ong- r un e l a s t i c i t y of y wi t h r e s pe c t t o m i s kno wn ( i . e . , g

_ym

i s kno wn) , ( i v) t he l ong- r un e l a s t i c i t y of m wi t h r e s pe c t t o y i s kno wn ( i . e . , g

_my

i s kno wn) .

²⁾

Fi r s t , t he s hor t - t e r m r e s t r i c t i on s pe c i f i e s a c ont e mpor a ne ous r e l a t i ons hi p be t we e n e ndoge nous va r i a bl e s a nd s hoc ks by i mpos i ng r e s t r i c t i ons on t he s hor t - t e r m e l a s t i c i t y, s uc h a s s uppos e p

_ym

= 0 or p

_my

= 0. 　The f or me r r e s t r i c t i on i ndi c a t e s s hor t - r un ne ut r a l i t y whe r e by out put doe s not r e a c t c ont e mpor a ne ous l y t o t he s hoc k t o t he mone y s t oc k. 　I n c ont r a s t , t he l a t t e r r e s t r i c t i on i ndi - c a t e s t he s i t ua t i on whe r e by t he mone y s t oc k doe s not c ont e mpor a ne ous l y a c c ommoda t e c ha nge s i n out put , a nd out put be c ome s t he pr e de t e r mi ne d v a r i a bl e . 　Se c ond, t he l ong- t e r m r e s t r i c t i on s pe c i f i e s a l ong- t e r m r e l a t i ons hi p be t we e n e ndoge nous va r i a bl e s a nd s hoc ks by i mpos i ng r e s t r i c - t i ons on t he l ong- t e r m e l a s t i c i t y , f or e xa mpl e , c ons i de r g

_my

= 1. 　Thi s a s s umpt i on i s c ons i s t e nt wi t h l ong- r un pr i c e s t a bi l i t y unde r t he a s s umpt i on of s t a bl e v ol a t i l i t y . 　Addi t i ona l l y , l ong- r un ne ut r a l i t y ( g

_ym

= 0) i s a ppl i c a bl e a s a n i de nt i f yi ng r e s t r i c t i on. 　Va r yi ng t he v a l ue s of p

_ym

, p

_my

a nd g

_my

, t he y f ound t he e vi de nc e of s uppor t i ng l ong- r un ne ut r a l i t y of mone y f or r e a l out put i n t he pos t wa r U. S. da t a .

4. 　Te s t s f or Se as onal i t y

Se a s ona l i t y c a n be de t e r mi ni s t i c a nd/ or s t oc ha s t i c . 　For qua r t e r l y da t a , de t e r mi ni s t i c s e a - s ona l i t y a s s ume s t ha t t he da t a ge ne r a t i ng pr oc e s s f or t he v a r i a bl e y

_t

i s

whe r e s

_st

( = 1 i n s e a s on s , 0 e l s e whe r e , f or s = 1, 2, 3, 4) i s a s e a s ona l dummy va r i a bl e . 　I nc l ud- i ng s e a s ona l dummy va r i a bl e s i n a r e gr e s s i on mode l i s a ppr opr i a t e f or va r i a bl e s wi t h de t e r mi ni s - t i c s e a s ona l i t y . 　The a bs e nc e of t he s e dummy v a r i a bl e s wi l l l e a d t o t he s t a nda r d pr obl e m of

y

_t

= γ

_{1 1}

s

_t

+ γ

_{2 2}

s

_t

+ γ

_{3 3}

s

_t

+ γ

_{4 4}

s

_t

+ ε

_t

2) de not e s t he l ong- r un e l a s t i c i t y of mone y ( m ) wi t h r e s pe c t t o pe r ma ne nt e xoge - nous c ha nge i n out put ( y ) .

γ π α

π α

my

my i

p my i

mm i

p mm i

= + ∑

− ∑

=

= 1 1

, ,

bi a s a s s oc i a t e d wi t h t he e xc l us i on of r e l e va nt e xpl a na t or y va r i a bl e s . 　St oc ha s t i c s e a s ona l i t y e xt e nds t he uni t r oot hypot he s i s t o s e a s ona l t i me s e r i e s . 　An i nt e gr a t e d s e a s ona l pr oc e s s i s a pr oc e s s t ha t c ont a i ns uni t r oot s a t t he s e a s ona l f r e que nc i e s , a nd a ppr opr i a t e di f f e r e nc i ng f i l t e r s a r e r e qui r e d f or s e a s ona l l y i nt e gr a t e d pr oc e s s e s .

One me t hod t o t e s t f or s e a s ona l i t y i s due t o Mi r on ( 1994) , a nd i nvol ve s t he us e of a n a ux- i l i a r y r e gr e s s i on of t he f or m

( 9) whe r e e

_t

i s a s s ume d t o be a s t a t i ona r y a nd i n v e r t i bl e ARMA pr oc e s s . 　I f t he l oga r i t hm of y

_t

i s t a ke n a s t he de pe nde nt v a r i a bl e , t he e qua t i on i n v ol ve s t he r e gr e s s i on of t he gr o wt h r a t e of t he va r i a bl e on a s e t of s e a s ona l dummy va r i a bl e s . 　The e xpl a na t or y powe r of t he r e gr e s s or s woul d i ndi c a t e t he e xt e nt of t he s e a s ona l i t y i n t he s e r i e s .

I f s e a s ona l c ompone nt s a r e s t oc ha s t i c , t he va r i a bl e c a n ha ve a uni t r oot not onl y i n i t s l ong- r un be ha vi or , b ut a l s o a t i t s s e a s ona l f r e que nc i e s . 　To t e s t t he uni t r oot pr ope r t i e s of v a r i a bl e s wi t h a s ubs t a nt i a l s e a s ona l pa t t e r n Hyl l e be r g e t al . ( 1990) ha ve pr opos e d a ge ne r a l i z a t i on t o t he Di c ke y- Ful l e r a ppr oa c h, de not e d a s HEGY- t e s t a nd a ppl i e d i n Engl e e t al . ( 1993) . 　The pr oc e - dur e i s ba s e d on t he a s s umpt i on t ha t t he i ndi vi dua l t i me s e r i e s i s ge ne r a t e d by a f i ni t e a ut or e - gr e s s i v e pr oc e s s :

wi t h e

_t

a s a z e r o me a n whi t e noi s e . 　Whe r e D

₄

be i ng t he di f f e r e nc i ng f i l t e r de f i ne d a s D

₄

= ( 1 - L

⁴

) wi t h L be i ng t he us ua l l a g ope r a t or . 　I f t he pr oc e s s y

_t

i s i nt e gr a t e d a nd s e a s ona l l y i nt e gr a t e d t he n t he di f f e r e nc i ng ope r a t or ( 1 - L

⁴

) = ( 1 - L ) ( 1 + L ) ( 1 + L

²

) = ( 1 - L ) ( 1 + L ) ( 1 - i L ) ( 1 + i L )

= 0 ha s t wo r e a l r oot s 1 a nd - 1 a nd t wo c ompl e x r oot s i a nd - i . 　The r e a l r oot 1 i s us ua l z e r o f r e que nc y ( non- s e a s ona l ) r oot whi c h c a n be r e move d by a ppl yi ng t he f i l t e r ( 1 - L ) , whi l e t he r e a l r oot - 1 c or r e s ponds t o uni t r oot a t ½ c yc l e pe r qua r t e r ( s e mi - a nnua l uni t r oot ) t ha t i s e l i mi - na t e d by t he f i l t e r ( 1 + L ) . 　The c ompl e x r oot s i a nd - i c or r e s pond t o uni t r oot s a t ¼ c yc l e ( a nnua l uni t r oot s ) t ha t c a n be f i l t e r e d by me a ns of ( 1 + L

²

) .

To a na l yz e t he uni t r oot s a t a l l s e a s ona l f r e que nc i e s a nd z e r o f r e que nc y c ons i de r i ng t he de t e r mi ni s t i c s e a s ona l i t y i n y

_t

, t he pr oc e s s i s r e a r r a nge d a s

( 10) whe r e t i s de t e r mi ni s t i c t i me t r e nd, s

_1t

, s

_2t

, a nd s

_3t

a r e s e a s ona l dummy va r i a bl e s s hown a s a bo v e , a nd

( 1 − L y )

_t

= γ

_{1 1}

s

_t

+ γ

_{2 2}

s

_t

+ γ

_{3 3}

s

_t

+ γ

_{4 4}

s

_t

+ ε

_t

Δ

4

1

4

y

_t

= − ( L y )

_t

= ε

_t

Δ

4

y

_t

= + + α δ t γ

1 1

s

_t

+ γ

2 2

s

_t

+ γ

3 3

s

_t

+ π

1 1

y

_,t−1

+ π

2 2

y

_,t−1

+ π

3 3

y

_,t−2

+ π

4

y

_33,t−1

+ ε

_t

y

_1,t

= + ( 1 L )( 1 + L y

²

)

_t

y

_2,t

= − − ( 1 L )( 1 + L y

²

)

_t

whe r e t he s i gns i n de f i ni ng y

_2,t

a nd y

_3,t

a r e us e d j us t f or c on ve ni e nc e . 　Gi ve n a qua r t e r l y i nt e - gr a t e d s e r i e s , y

_1,t

he nc e j us t c ont a i ns t he us ua l l ong- r un t r e nd, whi l e y

_2,t

a nd y

_3,t

c ons i s t of t he nons t a t i ona r y bi - a nnua l a nd a nnua l c yc l e s , r e s pe c t i v e l y .

I n or de r t o de t e r mi ne t he di f f e r e nc i ng f i l t e r s t ha t a r e ne c e s s a r y f or t he s t a t i ona r i t y of t he va r i a bl e , Hyl l e be r g e t a l . ( 1990) s ugge s t t he f ol l owi ng pr oc e dur e . 　Es t i ma t e e qua t i on ( 4) by OLS a nd t e s t t he f ol l o wi ng hypot he s e s :

( i ) H

₀

: p

₁

= 0, H

₁

: p

₁

< 0;

( i i ) H

₀

: p

₂

= 0, H

₁

: p

₂

< 0;

( i i i ) H

₀

: p

₃

= p

₄

= 0, H

₁

: p

₃

¹ 0; or / a nd p

₄

¹ 0.

St a nda r d t - t e s t s a r e us e d f or t he f i r s t t wo hypot he s e s ( i ) a nd ( i i ) , whi l e a n F - t e s t i s us e d f or t he t hi r d hypot he s i s ( i i i ) . 　I f t he hypot he s i s ( i ) i s not r e j e c t e d, t he r e i s a non- s e a s ona l or z e r o f r e que nc y uni t r oot i n t he s e r i e s , whi c h r e qui r e s t he f i l t e r ( 1 - L ) f or s t a t i ona r i t y . 　For t he hypot he s i s ( i i ) , t he nul l hypot he s i s i s c ons i s t e nt wi t h a s e mi - a nnua l uni t r oot , i mpl yi ng t ha t a ny s hoc ks t o t he v a r i a bl e wi l l l e a d t o pe r ma ne nt c ha nge s i n t he s e a s ona l pa t t e r n of t he v a r i a bl e a t t he s e mi - a nnua l l e v e l . 　Thi s r e qui r e s t he f i l t e r ( 1 + L ) f or s t a t i ona r i t y . 　A non- r e j e c t i on of t he t hi r d nul l hypot he s i s ( i i i ) i mpl i e s t ha t t he s e r i e s ha s a t l e a s t one of t he t wo uni t r oot s i n t he a nnua l f r e que nc y , r e qui r i ng t he f i l t e r ( 1 + L

²

) f or s t a t i ona r i t y . 　Wi t h a nnua l uni t r oot s , a s hoc k t o t he v a r i a bl e wi l l c ha nge pe r ma ne nt l y t he s e a s ona l pa t t e r n of t he v a r i a bl e a t t he a nnua l l e v e l . 　 The e mpi r i c a l r e j e c t i on of t he t hr e e nul l hypot he s e s i mpl i e s t ha t t he s e r i e s ha s no non- s e a s ona l , s e mi - a nnua l or a nnua l uni t r oot s , r e s pe c t i ve l y, f or qua r t e r l y da t a . 　The a s ympt ot i c di s t r i but i ons of t he HEGY s t a t i s t i c s a r e non- s t a nda r d, a nd a r e f unc t i ona l s of Wi e ne r pr oc e s s e s . 　Hyl l e be r g e t al . ( 1990) c omput e t he a ppr opr i a t e c r i t i c a l va l ue s f or t he s e t e s t s by Mont e Ca r l o s i mul a t i ons .

5. 　Dat a and De s c r i pt i on

We i n ve s t i ga t e t he l ong- r un ne ut r a l i t y of mone y hypot he s e s us i ng pos t wa r da t a f or J a pa n. 　

Bot h s e a s ona l l y una dj us t e d a nd a dj us t e d qua r t e r l y t i me s e r i e s ove r t he pe r i od 1955: q2 - 2006: q1 a r e us e d i n t hi s s t udy . 　Qua r t e r l y r e a l GDP , a nd nomi na l mone y s uppl i e s M1 a nd M2 + CD, a r e us e d a s t he r e a l out put a nd nomi na l mone y s uppl y s e r i e s . 　We us e d Ba nk of J a pa n ( BOJ ) mont hl y mone t a r y s t a t i s t i c s f or M1 a nd M2 + CD. 　Mont hl y s e r i e s we r e us e d t o f or m t he qua r t e r l y da t a . 　Tha t i s , t he obs e r v a t i on f or t he f ol l o wi ng mont h i s us e d a s a pr oxy f or t he e nd- of - qua r t e r mone y. 　For e xa mpl e , our M1 f i gur e f or t he f i r s t qua r t e r of 1990 i s t he va l ue of

y

_3,t

= − − ( 1 L )( 1 + L y )

_t

M1 on Apr i l , 1990. 　Whe r e a s , qua r t e r l y r e a l GDP s e r i e s wa s t a ke n f r om t he Sys t e m of Na t i ona l Ac c ount s ( SNA) . 　The f ol l owi ng s ymbol i c not a t i ons a r e us e d t hr ough out t hi s a r t i c l e .

y

_t

= l og( r e a l GDP) m 1

_t

= l og( M1) m 2

_t

= l og( M2 + CD)

Al t hough m 2 i s t he ma j or i ndi c a t or of boa r d mone t a r y a ggr e ga t e , c ons i de r a t i on of t wo me a s ur e s of mone y s uppl y , na me l y m 1 a nd m 2, s e r ve s a s a s e ns i t i vi t y a na l ys i s of t he pot e nt i a l e f f e c t s of mone y on r e a l out put . 　I t i s s ho wn i n Bul l a r d ( 1994) a nd Ol e ka l ns ( 1996) t ha t t he out c ome of t he t e s t of LRN i s s e ns i t i v e t o me a s ur e t he mone y i n v ol v e d.

6. 　Ti me Se r i e s Pr ope r t i e s of Dat a

I de nt i f i c a t i on of t he or de r s of i nt e gr a t i on of mone y a nd r e a l gdp i s a n i mpor t a nt i s s ue be f or e t e s t i ng l ong- r un ne ut r a l i t y us i ng t he me t hodol ogy di s c us s e d i n a bove . 　Si nc e we us e d s e a s ona l l y una dj us t e d da t a , i t i s a l s o i mpor t a nt t o c he c k we t he r t he s e r i e s a r e s e a s ona l l y i nt e gr a t e d. 　I n t hi s s e c t i on we e mpl oye d HEGY t e s t t o i n ve s t i ga t e t he s e a s ona l i nt e gr a t i on, whi l e DF- GLS a nd KPSS t e s t s we r e us e d t o i de nt i f y t he or de r of nons e a s ona l i nt e gr a t i ons i n t he da t a us e d i n t hi s pa pe r .

Fi gur e 1 c ont a i ns t he pl ot s of t he r e a l GDP y , nomi na l mone y s uppl y m 1 a nd m 2 i n l e v e l . 　 Al l t hr e e va r i a bl e s a r e t r e ndi ng upwa r d. 　I t c a n a l s o be obs e r ve d i n Fi gur e 1 t ha t r e a l GDP e xhi bi t s s ubs t a nt i a l s e a s ona l f l uc t ua t i ons . 　To gue s s whe t he r t he s e a s ona l i t y i s de t e r mi ni s t i c or

Fi gur e 1. Logar i t hm of quar t e r l y unadj us t e d r e al GDP ( y ) and nomi nal mone y

s uppl i e s ( m 1 & m 2)

s t oc ha s t i c , we us e d t he a uxi l i a r y r e gr e s s i on i n Equa t i on ( 9) pr opos e d by Mi r on ( 1994) . 　Re gr e s - s i on ( 9) i s pe r f or me d f or t he r e a l GDP a nd t he t wo mone t a r y v a r i a bl e s m 1 a nd m 2, f or t he f ul l s a mpl e 1955: 2 t o 2006: 1, a s we l l a s t wo s ubs e t s a mpl e s , e a c h wi t h a n a ppr oxi ma t e l y e qua l num- be r of obs e r v a t i ons . 　Cons i de r a t i on of s pl i t s a mpl e s a l l o ws a n e xa mi na t i on of whe t he r t he s e a - s ona l f l uc t ua t i ons ha ve c ha nge d ove r t i me , na me l y , whe t he r t he r e i s s t oc ha s t i c s e a s ona l i t y . 　 Re gr e s s i on r e s ul t s f or t he e s t i ma t e s of t he pa r a me t e r s i n t he a uxi l i a r y r e gr e s s i on, a nd t he i r R

²

va l ue s , a r e pr e s e nt e d i n Ta bl e 1. 　The hi gh R

²

va l ue s s ugge s t t ha t r e a l GDP e xhi bi t s s ubs t a nt i a l s e a s ona l f l uc t ua t i ons . 　As e xpe c t e d, t he e xpl a na t or y po we r s of t he s e a s ona l dummy v a r i a bl e s f or t he t wo mone t a r y v a r i a bl e s m 1 a nd m 2 a r e c ompa r a t i v e l y l o w.

Es t i ma t e s of t he g

_s

( s = 1, 2, 3, 4) c oe f f i c i e nt s c a n be us e d t o obs e r ve t he pa t t e r n of s e a s ona l i t y . 　For t he f ul l a nd s ubs a mpl e s , t he e s t i ma t e d c oe f f i c i e nt s of t he s e a s ona l dummy va r i a bl e s a r e ve r y s i mi l a r f or r e a l GDP a nd nomi na l mone t a r y va r i a bl e m 2. 　Thi s s ugge s t s t ha t t he pa t t e r n of s e a s ona l i t y ha s r e ma i ne d r e l a t i v e l y c ons t a nt o v e r t he e nt i r e s a mpl e f or t he s e t wo s e r i e s , a nd s ugge s t s t ha t s e a s ona l uni t r oot s a r e l i ke l y t o be a bs e nt . 　But t he s i gn a nd s i gni f i - c a nc e l e ve l of f i r s t t wo dummy c oe f f i c i e nt s of nomi na l mone t a r y va r i a bl e m 1 a r e not c ons i s t e nt wi t h f ul l a nd s ubs a mpl e s . 　So t he mone t a r y va r i a bl e m 1 ma y be s us pe c t e d t o ha ve s e a s ona l uni t r oot .

Tabl e 1. 　Se as onal i t y i n r e al GDP y and nomi nal mone y s uppl i e s m 1 and m 2 g

₄

g

₃

g

₂

g

₁

R

²

Sa mpl e Va r i a bl e

0. 1724 ( 22. 657) 0. 0723

( 10. 976) - 0. 0502

( - 5. 193) - 0. 0390

( - 3. 680) 0. 88

1955: 2 - 2006: 1 y

0. 2113 ( 22. 146) 0. 0772

( 9. 398) - 0. 0625

( - 3. 998) - 0. 0587

( - 3. 492) 0. 95

1955: 2 - 1980: 3

0. 1092 ( 7. 711) 0. 0579

( 4. 527) - 0. 0240

( - 1. 585) - 0. 0555

( - 3. 434) 0. 90

1980: 4 - 2006: 1

0. 0826 ( 21. 185) 0. 0414

( 8. 943) 0. 0549

( 17. 555) 0. 0257

( 3. 990) 0. 79

1955: 2 - 2006: 1 m 2

0. 0790 ( 14. 872) 0. 0283

( 4. 760) 0. 0477

( 13. 334) - 0. 0091

( - 0. 957) 0. 84

1955: 2 - 1980: 3

0. 0743 ( 11. 498) 0. 0428

( 5. 673) 0. 0562

( 8. 956) 0. 0412

( 4. 674) 0. 75

1980: 4 - 2006: 1

0. 1247 ( 19. 104) 0. 0154

( 2. 233) 0. 0339

( 5. 192) 0. 0272

( 2. 516) 0. 68

1955: 2 - 2006: 1 m 1

0. 1503 ( 18. 500) 0. 0151

( 1. 686) 0. 0406

( 4. 749) - 0. 02574

( - 1. 473) 0. 83

1955: 2 - 1980: 3

0. 0514 ( 3. 238) - 0. 0364

( - 2. 333) - 0. 0103

( - 0. 667) 0. 0100

( 0. 603) 0. 64

1980: 4 - 2006: 1

Not e : Fi gur e s i n pa r e nt he s e s s ho w t - v a l ue s

Whe n s e a s ona l l y una dj us t e d da t a a r e us e d, t he uni t r oot t e s t s c a nnot de t e c t s e a s ona l uni t r oot s or s t oc ha s t i c s e a s ona l i t y . 　The r e f or e , whe n t he out c ome s of uni t r oot t e s t s s ugge s t t ha t a s e r i e s s houl d be f i r s t - di f f e r e nc e d, t he s e r i e s ma y not ne c e s s a r i l y be s t a t i ona r y due t o t he pos s i - bl e pr e s e nc e of s e a s ona l uni t r oot s . 　As s uc h, t e s t i ng f or s e a s ona l uni t r oot s i n qua r t e r l y obs e r v e d da t a i s of pa r a mount i mpor t a nc e .

Thus , t he t hr e e s e r i e s c ons i de r e d i n t hi s pa pe r a r e t e s t e d f or pos s i bl e s e a s ona l uni t r oot s us i ng t he HEGY t e s t us i ng Equa t i on ( 10) . 　The r e s ul t s of HEGY t e s t unde r t hr e e hypot he s e s a r e pr e s e nt e d i n Ta bl e 2. 　I t i s f ound t ha t f or a l l t hr e e va r i a bl e s hypot he s e s ( i i ) a nd ( i i i ) a r e r e j e c t e d a t hi ghl y s i gni f i c a nc e l e ve l , whi l e hypot he s i s ( i ) i s not r e j e c t e d. 　The s e r e s ul t s i mpl y t ha t a l l t hr e e va r i a bl e s a r e not s e a s ona l l y i nt e gr a t e d, na me l y, t he s e a s ona l l y una dj us t e d qua r t e r l y s e r i e s do not c ont a i n s e a s ona l uni t r oot s a t t he s e mi - a nnua l or t he a nnua l f r e que nc i e s a nd t he v a r i a bl e s do, ho we ve r , ha ve uni t r oot s a t t he z e r o or non- s e a s ona l f r e que nc i e s . 　The s e r e s ul t s a r e c ons i s t e nt wi t h t he r e s ul t s of pr e vi ous t e s t s of LRN.

Con ve nt i ona l uni t r oot t e s t s a r e a l s o us e d t o c onf i r m t he I ( 1) na t ur e of t he t hr e e s e r i e s . 　 Two uni t r oot t e s t s a r e pe r f or me d: ( i ) t he a s ympt ot i c a l l y mos t po we r f ul DF- GLS t e s t of El l i ot t

Tabl e 2. 　HEGY t e s t s f or s e as onal i nt e gr at i on

H

₀

: p

₃

= p

₄

= 0 H

₀

: p

₂

= 0

H

₀

: p

₁

= 0 Va r i a bl e

19. 513***

- 2. 654**

- 1. 5039 y

124. 00***

- 7. 2433***

0. 3298 m 2

98. 4274***

- 6. 3419***

- 1. 2096 m 1

Not e : The c r i t i c a l v a l ue s a r e t a ke n f r om HEGY f or 200 obs e r v a t i ons . :

***, ** a nd * i ndi c a t e t he s i gni f i c a nt l e v e l a t 1%, 5% a nd 10%

r e s pe c t i v e l y

Tabl e 3. 　Uni t Root Te s t s f or Le ve l s and Fi r s t Di f f e r e nc e s Se r i e s Adj us t e d Una dj us t e d

v a r i a bl e

KPSS DF- GLS

l a g KPSS

DF- GLS l a g

0. 4248***

- 0. 3252 4

0. 4288***

- 0. 5093 4

y

0. 4515***

- 0. 6521 5

0. 4535***

- 1. 2084 8

m 2

0. 4114***

- 0. 2376 2

0. 4185***

- 0. 6932 8

m 1

0. 0975 - 4. 1079***

2 0. 0515 - 41. 7683***

2 D y

0. 0925 - 3. 3731**

1 0. 0681 - 3. 3507**

6 D m 2

0. 1876**

- 6. 4649***

1 0. 1282*

- 2. 9668**

7 D m 1

Not e : Cons t a nt a nd t r e nd a r e i nc l ude d i n t he mode l s . : ***, ** a nd * i ndi c a t e t he s i gni f i -

c a nt l e v e l a t 1%, 5% a nd 10%, r e s pe c t i v e l y .

e t al . ( 1996) a nd ( i i ) t he Kwi a t ko ws ki e t al . ( 1992) LM t e s t ( KPSS) . 　The nul l hypot he s i s of t he f or me r t e s t i s t ha t a v a r i a bl e ha s a uni t r oot whi l e t ha t of t he l a t e r t e s t i s t ha t a v a r i a bl e i s s t a t i ona r y. 　A c ommon s t r a t e gy i s t o pr e s e nt r e s ul t s of bot h DF- GLS a nd KPSS t e s t s , a nd s how t ha t t he r e s ul t s a r e c ons i s t e nt ( e . g. , t ha t t he f or me r r e j e c t t he nul l whi l e t he l a t e r f a i l t o do s o a nd vi c e - ve r s a ) . 　The l a g l e ngt h i s s e l e c t e d by t he Aka i ke I nf or ma t i on Cr i t e r i a ( AI C) . 　The r e s ul t s a r e s ho wn i n Ta bl e 3. 　We pe r f or m t he s a me t e s t on t he f i r s t di f f e r e nc e d v a r i a bl e s a nd f ound t ha t a l l v a r i a bl e s a r e I ( 1) . 　Tha t i s , bot h t e s t s i ndi c a t e t ha t t he v a r i a bl e s a r e i nt e gr a t e d a t t he z e r o f r e que nc y .

As r eal GDP i s not s eas onal l y i nt e gr at ed, t he s eas onal var i at i ons ar e l i kel y t o be de t e r mi ni s t i c . 　I n or de r t o t e s t t he LRN hypot he s i s us i ng r e a l GDP a s t he out put va r i a bl e , t hr e e s e a s ona l dummy va r i a bl e s a nd a t r e nd t e r m a r e i nc l ude d i n Equa t i on ( 7) t o c a pt ur e t he s e a s ona l v a r i a t i ons . 　Equa t i on ( 7) i s modi f i e d a s f ol l o ws :

( 11) She l l e y a nd Wa l l a c e ( 2006) s ho wn t ha t t he i nc l us i on of t r e nd a nd s e a s ona l dummy i n t he FS e qua t i on doe s not bi a s t he e s t i ma t e s of t he b

_k

c oe f f i c i e nt s .

The Fi s he r a nd Se a t e r ( 1993) me t hodol ogy r e qui r e s mone y be e xoge nous . 　The va l i di t y of t hi s a s s umpt i on i s a s s e s s e d by t e s t i ng i f r e a l GDP gr o wt h Gr a nge r - c a us e s mone y gr o wt h. 　We us e d t he VAR mode l f or t e s t i ng t he Gr a nge r - c a us a l i t y whe r e t he nul l hypot he s i s i s t ha t r e a l GDP gr o wt h doe s not Gr a nge r - c a us e s mone y gr o wt h. 　The l a g l e ngt h wa s de t e r mi ne d us i ng t he AI C c r i t e r i a , wi t h a ma xi mum of 15 l a gs c ons i de r e d. 　The c r i t e r i a s e l e c t 12 a nd 10 l a gs f or m 1 a nd m 2 r e s pe c t i ve l y i n c a s e of s e a s ona l l y una dj us t e d s e r i e s . 　For m 1 a nd m 2 l a gs 3 a nd 9 r e s pe c t i v e l y a r e us e d whe n t he s e r i e s a r e s e a s ona l l y a dj us t e d. 　Ta bl e 4 c ont a i ns t he r e s ul t s f or bot h s e a s ona l l y una dj us t e d a nd a dj us t e d da t a .

( y

_t

− y

_{t k}− −1

) = + + α δ t γ

1 1

s

_t

+ γ

2 2

s

_t

+ γ

3 3

s

_t

+ β

_k

( m

_t

− m

_{t k}− −1

) + ε

_t

As t a bl e s hows , t he nul l hypot he s i s of r e a l GDP doe s not Gr a nge r - c a us e s mone y c a nnot be r e j e c t e d f or e a c h of mone y a nd r e a l GDP s e r i e s . 　So we c onc l ude t ha t f or bot h c a s e s , s e a s on- a l l y una dj us t e d a nd a dj us t e d, mone y i s e xoge nous .

Tabl e 4. 　Gr ange r - c aus al i t y t e s t s f or e r oge ne i t y of mone y c

²

- v a l ue ( a dj us t e d) c

²

- v a l ue ( una dj us t e d)

Hypot he s i s

0. 420 ( 3) [ 0. 9360]

17. 051 ( 12) [ 0. 1477]

y doe s not c a us e s m 1

7. 138 ( 9) [ 0. 6228]

12. 072 ( 10) [ 0. 2803]

y doe s not c a us e s m 2

Not e : ( ) a nd [ ] c ont a i n l a gs a nd p- v a l ue s r e s pe c t i v e l y