NCER Working Paper Series Econometric Issues when Modelling with a Mixture of I(1) and I(0) Variables

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1 NCER Working Paer Series Economeric Issues when Modelling wih a Mixure of I() and I(0) Variables Lance A Fisher Hyeon-seung Huh Adrian R Pagan Working Paer #97 Ocober 203

2 Economeric Issues when Modelling wih a Mixure of I() and I(0) Variables # Lance A. Fisher, Hyeon-seung Huh, Adrian R. Pagan Summary This aer considers srucural models when boh I() and I(0) variables are resen. I is necessary o exend he radiional classificaion of shocks as ermanen and ransiory, and we do his by inroducing a mixed shock. The exra shocks coming from inroducing I(0) variables ino a sysem are hen classified as eiher mixed or ransiory. Condiions are derived uon he naure of he SVAR in he even ha hese exra shocks are ransiory. We hen analyse wha haens when here are mixed shocks, finding ha i changes a number of ideas ha have become esablished from he coinegraion lieraure. The ideas are illusraed using a well-known SVAR where here are mixed shocks. This SVAR is re-formulaed so ha he exra shocks coming from he inroducion of I(0) variables do no affec relaive rices in he long-run and i is found ha his has major imlicaions for wheher here is a rice uzzle. I is also shown how o handle long-run arameric resricions when some shocks are idenified using sign resricions. Key Words: Mixed models, ransiory shocks, mixed shocks, long-run resricions, sign resricions, insrumenal variables JEL Classificaion: C32, C36, C5 Corresonding Auhor Hyeon-seung Huh School of Economics Yonsei Universiy 50 Yonsei-ro, Seodaemun-gu, Seoul, Reublic of Korea, Tel: hshuh@yonsei.ac.kr Dearmen of Economics, Macquarie Universiy. lance.fisher@mq.edu.au School of Economics, Yonsei Universiy. hshuh@yonsei.ac.kr School of Economics, Universiy of Sydney, and Melbourne Insiue of Alied Economic and Social Research, Universiy of Melbourne. aagan@unimelb.edu.au # We would like o hank Helmu Lükeohl for his commens on an earlier version of he aer ha led o he disincions we have made beween shocks in his version. Two anonymous referees also made valuable criicisms ha have grealy imroved he aer.

3 . Inroducion Economeric Issues when Modelling wih a Mixure of I() and I(0) Variables I seems likely ha macroeconomeric modelling will involve a mixure of variables ha are I() and I(0). However mos exbooks and alied work deal wih he case when all series are I(), while reviews such as Juselius (2006) make he assumion ha all series are eiher I() or I(2). So here aears o be no sysemaic examinaion of he esimaion issues raised by a mixure of I() and I(0) variables. When here is no co-inegraion srucural models are generally formulaed in erms of changes in he I() variables. Wih co-inegraion resen some of he changes in I() variables are relaced by error correcion (EC) erms when seing u a srucural VAR (SVAR). When here are only I() variables resen in a sysem, and here is co-inegraion, radiionally shocks have been classified as ermanen and ransiory. Now, when I(0) variables are resen in he sysem, hey will be in levels, and some assumion needs o be made abou he naure of he exra shocks arising from he inroducion of hese variables ino he sysem. They could eiher be urely ransiory or could have ermanen effecs on all or some of he I() variables. Hence we need o make a disincion beween hese cases. When here is a mixure of I() and I(0) variables, secion 2 suggess a classificaion of shocks ino ermanen, ransiory and mixed. If he exra shocks have ermanen effecs hen we encouner a difficuly se ou in secion 2.. The sandard definiion of he number of ermanen shocks involves he rank of he marix showing he long-run imac of shocks uon he I() variables a infiniy - wha we will call he long-run imac marix. As secion 2. shows, when here are ermanen effecs of mixed shocks, his definiion can no longer aly. I is his fac ha gives rise o our erminology of mixed shocks. We kee he erm ermanen shocks for when here are jus I() variables and use he mixed aellaion o accoun for shocks arising from he inroducion of he I(0) variables ha have ermanen effecs. Secion 2.2 hen looks a he case where he inroduced shocks are ransiory. If he exra shocks induced by he resence of I(0) variables are urely ransiory hen we demonsrae ha his requires a aricular ye of model design. Secifically one needs o force he srucural equaions o have changes raher han he levels of I(0) variables, which exends resul found in Pagan and Pesaran (2008). A good deal of emirical work seems o have his siuaion in mind bu does no seem o recognize ha he sysem needs o be designed o ensure ha he shocks are ransiory. 2

4 Thus sudies ha have eiher he growh rae of ouu or he change in he nominal exchange rae in he SVAR, along wih he inflaion rae and he level of ineres raes, need o design he SVAR o ensure ha moneary olicy shocks do no have long-run effecs on ouu or relaive rices such as he real exchange rae. Canova, Gambei and Paa (2007), del Negro and Schorfheide (2004), Smes (997), and he FAVAR SVAR of Bernanke e al. (2005) are examles of aers ha have growh raes in he SVAR bu do no ensure ha moneary olicy has zero long-run effecs on real variables. We don hink ha he researchers working wih hese sysems inended such oucomes bu hey did no formulae SVAR secificaions which ensured ha he shocks were ransiory. Secion 2.3 urns o an examinaion of a device ha has been suggesed as a way of handling mixures of I() and I(0) variables, namely reaing he I(0) variables as co-inegraing wih hemselves. Tha sraegy requires he inroducion of seudo co-inegraing vecors as well as rue ones. Alhough we use his device in secion 2.2 his is done when he exra shocks arising from he I(0) variables are assumed o have urely ransiory effecs. However, a simle examle in secion 2.3 shows ha his mehod would rovide incorrec answers regarding he ermanen comonens of he I() variables if he shocks coming from he inroducion of he I(0) variables have ermanen effecs. Secion 2.4 hen inroduces a general mehod for comuing he ermanen comonen when here are mixed shocks. I is no ossible o sudy he many aers ha feaure srucures wih mixed shocks. Consequenly, in secion 3 we illusrae some of he oucomes by looking a an influenial sudy by Peersman (2005) which had his feaure. Peersman ses u a SVAR involving hree I() variables and one I(0) variable, wih no co-inegraion beween he I() variables. Peersman works wih hree ermanen shocks and one mixed shock. The mixed shock is ha coming from he inroducion of he nominal ineres rae. He imoses ha ineres rae shocks have no effec uon ouu in he long-run, bu hey are allowed o affec boh he rice level and he rice of oil in he long-run. As here is nohing imosed on he model o say ha hese wo rices change by he same amoun in he long run hen he real rice of oil mus be affeced by ineres rae shocks a ha horizon. This is he same mechanism ha makes he real exchange rae resond in he long-run o moneary olicy shocks in he alicaion by Smes (997) menioned earlier. Examining he imlicaions of his resul in Peersman s case we find ha he absence of rice and ouu uzzles in his esimaed model sems from he fac ha a moneary olicy shock has ermanen real oil rice effecs. When his shock is aken o be ransiory he uzzles re-aear. 3

5 Now he mos common case where mixed shocks arise may be when sign raher han arameric resricions are alied o idenify imulse resonses, since hese deermine only he signs of he resonses for a finie number of eriods, and nohing is said abou he long-run oucomes. Consequenly, when SVARs are adoed which include growh raes of ouu, he moneary olicy shocks found from sign resricions will almos always have a long run imac on he level of ouu. Thus, when Peersman (2005) moved o sign resricions o idenify his moneary olicy shocks he resuling imulse resonses show long-run effecs on real variables. This would be rue of many oher sudies wih I() variables using sign resricions o idenify shocks. Consequenly, in secion 4 we look a how one can imose he consrain ha mixed shocks are ransiory wihin a sign resricions framework. Once again we use Peersman s se u o illusrae he aroach. Secion 5 hen concludes. 2. The naure of shocks in srucural models wih I(0) and I() variables 2.. Definiions of shocks wih mixures of variables When all variables are I() and here is co-inegraion beween hem, shocks can be searaed ino wheher hey are ermanen or ransiory. These erms describe he long-run effecs on he variables of he shocks i.e. for a shock ε and, for he level of a variable y, he long-run effec will y+ j be lim ( ) j ε. Secifically, when a shock is alied ha lass only for a single eriod i is called ransiory if i has a zero effec on all he variables a infiniy. A ermanen shock is required o have a non-zero long-run effec on a leas one of he variables. This allows for he ossibiliy ha a ermanen shock may have a zero long-run effec uon some of he I() variables. When one adds I(0) variables o his sysem we will see below ha i is necessary o augmen he classificaion. Consequenly we will classify he exra shocks induced ino he sysem by he resence of he I(0) variables as being eiher mixed or ransiory. To areciae he need for he exended erminology suose we had a sysem wih hree I() y+ j variables and no coinegraion. Then he long-run resonse marix C lim ( ) j ε = will have rows corresonding o he variables y and columns reresening he shocks. Thus he (,3) elemen of C y, + j is C3 ( 3, ) j ε = lim. Now his Cmarix migh ake he form If y is I() hen i would ener he SVAR as y. 4

6 0 C = 0, 0 where he indicae non-zero values. Consequenly, as he rank of his marix is generally hree, radiional heory says ha here are hree ermanen shocks. Now suose ha an I(0) variable is added o he sysem and ha he new shock has ermanen effecs on all hree I() variables. This means ha he long-run resonse marix for he four variables will be C =, since he fourh variable is I(0), and so he long-run resonses of i o all shocks are zero. Now he rank of his marix will sill be a mos hree. So, if one used he radiional definiion of he rank of he long-run marix as being he number of ermanen shocks, one has he difficuly of describing he naure of he fourh shock. I clearly fails he definiion of a ransiory shock. Indeed, i looks much like he original hree ermanen shocks. Bu, if we call i ermanen, hen here would be four such shocks and his is in conflic wih he rank of he marix. Therefore, we need o give i a new descrior, and we will refer o i as a mixed shock, since i arises in he conex of a mixure of I(0) and I() variables. Secifically, he exra shock associaed wih he addiion of an I(0) variable o a sysem of I() variables is mixed if i has a long-run effec on a leas one of he I() variables. If, however, i has a zero long-run effec on all of he I() variables, i is ransiory, and he long-run resonse marix for he four variable case would look like C = The rank of his marix is hree and so he rule delivers he correc number of ermanen shocks. 2 Clearly we can differeniae beween all shocks here jus using he long-run resonses. If however he (,3) elemen in C had been non-zero hen we would need some shor-run resricion o searae he hird and fourh shocks. 5

7 We now invesigae he imlicaions of including boh I(0) and I() variables ogeher in sysems when he shocks associaed wih he former are, firsly, ransiory (covered in secion 2.2) and hen mixed (covered in secions 2.3 and 2.4) Shocks associaed wih I(0) variables are ransiory This secion shows how o rea I(0) variables in srucural models ha conain coinegraing relaionshis among he I() variables when he exra shocks coming from he I(0) variables are aken o be ransiory. For simliciy, consider a srucural VAR(2) model of nvariables of he form A x = A x + A x + ε () where A i are n n marices of unknown coefficiens, A 0 is non-singular and ε is an n vecor of srucural shocks wih mean zero and covariance marix D n. We assume ha here are n q variables which are I() and qwhich are I(0), while among he I() variables here are ( < n q) coinegraing relaions. We refer o he laer as he rue or acual coinegraing relaions as disinc from he q seudo coinegraing relaions coming from he reamen of each of he I(0) variables as coinegraing wih iself. This is robably he sandard way of handling I(0) variables in SVECMs ha is currenly in he lieraure. We will refer o hese srucures as seudo-svecms. Because here are coinegraing relaions among he I() variables, here are m = n q srucural shocks wih ermanen effecs in a seudo-svecm. Wihou loss of generaliy, le x x, = x2 x 3 where x is he m vecor of I() variables whose srucural shocks are known o have ermanen effecs, x 2 is he vecor of I() variables whose srucural shocks are known o have ransiory effecs, and x 3 is he q vecor of I(0) variables whose srucural shocks are ransiory by assumion. Le β 0 % β = β2 0, 0 I q 6

8 where % β is an n ( + q) marix, and noe ha here are + qransiory shocks in he SVECM model. The marices β and β 2 are m and, resecively. The firs column of block marices in β % are he coefficiens in he rue coinegraing relaions among he I() variables, while he second column gives he seudo coinegraing relaions. The laer involve a coefficien of one on a given saionary variable, and a coefficien of zero on all he remaining variables, and so are reresened by he ideniy marix. Analogously he loadings vecor α% can be ariioned as α δ % α = α2 δ 2, α3 δ 3 where α% is an n ( + q) marix. The sub-marices α, and q, resecively. Similarly, he sub-marices α and α are of dimension m, 2 δ, 3 δ 2 and δ are of dimension m q, q and q q, resecively. The firs column of block marices in α% shows he loadings on he rue coinegraing relaions for each grou of srucural equaions while he second shows he loadings on he I(0) variables. 3 The VAR model of () can now be wrien as he seudo- SVECM A x = % α % β x + A x + ε. (2) 0 2 The vecor of rue error correcion erms, ξ, can be wrien as ξ = β x + β x. (3) 2 2 Following he develomen in Pagan and Pesaran, we roceed o exress he SVECM model of (2) as a srucural vecor auoregressive (SVAR) model of order wo in he variables x, ξ and x 3. From (3), we have ξ = β x + β x, 2 2 from which i follows ha x = ( β ) ( ξ β x ), (4) 2 2 7

9 rovided he marix β 2 is non-singular. 3 The firs m equaions in (2) are A x + A x + A x = α ξ δ x + A x + A x + A x + ε, (5) where he A marices are ariioned conformably wih x. These equaions conain he srucural shocks wih ermanen effecs. Pagan and Pesaran roved ha α = 0 in (5), so ha he srucural equaions wih he ermanen shocks do no conain he lagged rue error correcion erms. Here we show addiionally ha δ = 0when he srucural shocks associaed wih he I(0) variables are ransiory (have zero long-run effec on all he I() variables), i.e. for he SVAR(2) in x, ξ and x 3. Using (4) o eliminae he erms in x2 in (5), one obains ( A A ( β ) β ) x + A ( β ) ξ + A x = α ξ δ x + ( A A ( β ) β ) x + A ( β ) ξ + A x + ε (6) Defining w ( x x ) = ξ he SVAR(2) can be exressed as 3 B w = B w + B w + ε (7) Pariioning (7) ino he form conformable wih he ariion used in (5), he firs m equaions will be B x + B ξ + B x = B x + B ξ + B x + B x + B ξ + B x + ε which can be wrien as, B x + B2 ξ + B3 x 3 (8) = B x + ( B + B B ) ξ + ( B + B B ) x + B x B ξ B x + ε Comaring (8) wih (6), we ge 3 I may be necessary o ake care in seing he sysem u o ensure ha β is non-singular. To ake an 2 β = 0. examle, suose here are hree I() variables in he sysem wih coinegraing vecor ( ) Then, if we selec he firs and second equaions as he wo whose srucural shocks have ermanen effecs, β = 0. So we would need o choose eiher he firs and hird or he second and hird as he wo variables 2 whose associaed equaions have ermanen shocks. In he former case β = and, in he laer, β = 2 2 and so are non-singular. 8

10 α = ( B + B B ), (9) δ = ( B + B B ). (0) Now (7) can be wrien in lag oeraor form as B( L) w = ε, 2 where B( L) = B B L B L and L is he lag oeraor. I hen follows ha he moving average reresenaion will be 0 2 w = B L ε = C L ε () ( ) ( ) 2 3 where C( L) = C + C L + C L + C L +K. Hence C() = B() imlies ha C() B() = In. (2) By assumion shocks o he error correcion erms ξ are ransiory, so i mus be he case ha C () 0 2 =, where C () is ariioned analogously o he ariioned marices in (8). When shocks o he I(0) variables are ransiory, i is he case ha C () 0 3 =. These boh lace resricions on he B marices. To deermine wha hey are mulily he firs row of C() wih he second column of B () o obain he equaion C () B () + C () B () + C () B () = 0, (3) where 02 is an m null marix. Under he resricions, (3) becomes C () B 2() = 02, from which i follows ha B 2() = 02, since C () has full rank m. Bu 0 2 B2 () = ( B2 B2 B2 ) so ha B 2() = 02 means α = 0 from (9). This is he Pagan and Pesaran resul. 2 Similarly, mulilying he firs row of C() wih he hird column of B () gives C () B () + C () B () + C () B () = 0, (4) where 03 is an m q null marix. Using he same reasoning, i follows ha B 3() = 03, and noing B () = ( B B B ), his means δ = 0 by (0). Thus, when he srucural shocks coming from he I(0) variables are ransiory, he msrucural equaions wih he ermanen shocks do no conain he levels of he I(0) variables, only heir differences. This case is an exension of he Pagan 9

11 and Pesaran resul for he SVAR involving x and ξha he levels of he EC erms were relaced by heir differences in he srucural equaions for x Shocks associaed wih I(0) variables are mixed I is useful o look a a simle examle in order o areciae he fac ha one needs o aroach mixed shocks wih care. To his end consider he sysem y = δ z + ε (5) z = γ + ε (6), z 2 where y is I(), z is I(0) and he shocks ε and ε 2 are whie noise. Hence, using he Beveridge- Nelson decomosiion o find he ermanen comonen of y we have γδ y y E y y E z y z = + + j = + ( δ + j + ε + j ) = + j= j= γ, so ha y = y + δγ δγ z δ z ε z γ = + + γ This shows ha, unlike he radiional case reaed in co-inegraion heory (wih shocks being ermanen and ransiory) where y is whie noise, now he change in y is generally serially correlaed, owing o he imac of he mixed shocks on he ermanen comonen. Only if he second shock is ransiory ( δ = 0) will he change in y be whie noise i.e. ε. Now, seing δ = 0would eliminae z from he firs srucural equaion (5). I is however ossible o allow z o aear in his equaion, and also o make he second shock ε 2 ransiory, by secifying he equaion as y = δ z + ε. Then = + δ + j + ε + j = + δ = δ j= y y E ( z ) y ( z z ) y z, using he facs ha = + + j j= z z E z, z = 0 ( z is I(0) wih zero mean) and ε is whie noise. Therefore, 0

12 y y z = δ = δ δ = 0, ε ε ε ha is ε 2 has only ransiory effecs. Of course, his is he resul ha we esablished more generally in he revious sub-secion viz. ha he I(0) variables have o be enered in differences if exra shocks in he sysem coming from he inroducion of he I(0) variables are o have ransiory effecs. I would generally be he case ha he SVAR secified by emirical researchers would involve y and z, and so i is clear ha such a SVAR would no incororae a srucural equaion ha had z on he RHS. One needs o modify exising SVAR rograms o ge ha effec i.e. o make he exra shocks from he I(0) variables have ransiory effecs. Noice ha if one had used an SVAR wih y and z, his would change he secificaion of he second equaion o an AR() in z. However his is a very differen secificaion. In an earlier secion we noed ha ofen i had been suggesed ha I(0) variables can be handled by using he idea of seudo co-inegraing vecors. Would his aroach give a correc esimae of he ermanen comonen of y in he simle sysem above? In his sysem here is no coinegraion so ha β will be he seudo co-inegraing vecor β = ( 0 ). So he firs issue is wha he seudo-svecm is like. In his aroach z is reaed as if i is I(), in which case one seudo- SVECM migh be y = δ z + ε (7) z = ( γ ) z + ε (8) 2 bu, as we noed above, in his formulaion he shocks ε 2 mus be ransiory and canno be mixed. An alernaive seudo-svecm hen is y = δ z + ε z = ( γ ) z + ε 2 which has he seudo-vecm form y = δγ z + e (9)

13 z = ( γ ) z + e (20) 2 δγ leading o α =. γ In he sandard aroach o exracing ermanen comonens, and here is a single ermanen shock, he laer becomes a mulile of α e, where e are he seudo-vecm residuals and α α = 0. For he seudo VECM in (9) and (20), e = δε 2 + ε and ε. 2 = e2 Consequenly, using α ( δγ / ( γ ) ) = and we see ha he imlied ermanen shock would be roorional o δε + ε [ δγ / ( γ )] ε, and so i would deend on ε. However, because he sandard formulaion from co-inegraion also imlies ha is roorional o α, using he y e seudo EC aroach would redic ha y is whie noise, whereas we have already seen in his case ha i is serially correlaed. Hence he aroach gives an incorrec esimae of he ermanen comonen General formula for comuaion of ermanen comonens of I() series when here is a mixure of I() and I(0) series We will consider he following VAR sysem y = A y + Gz + e (2) z = Fz + Φ y + e (22) 2 where z has boh he EC erms and he I(0) variables in i. To raionalize (2) and (22) hink of he case where all variables y are I() and here is co-inegraion. Then we would have y = A y + αβ ' y and his can be wrien as β ' y = ec = β ' A y + β ' αec, hereby giving an equaion ha has he form ec = ( I + β ' α) ec + β ' A y. So his has he srucure of (22) wih ec being included in z. Alhough we are working wih a firs order sysem, higher order sysems can be handled simly by reducing hem o a firs order form in he sandard way. Now he ermanen comonen of y is second erm. This will be y y E y so we need o look a he = + j= + j. (23) E y = E ( A y + Gz + e ) + j + j + j + j j= j= 2

14 M Now le us consider L = y + j and define j= M K = y + j= j. Then i is clear ha L = K + y ym. Thus, as M, E ( L ) = E ( K ) + y. Consequenly, when M, we can wrie (23) above as E K = ( A E K + A y ) + GE z (24) + j j= ( ) ( ) + j j= E K = I A A y + I A GE z (25) This makes sense since, if G = 0, hen he shocks e 2 have no ermanen effecs. Now from (22) (26) E z = E ( F z + Φ y + e ) = FE z + ΦE y + e. + j 2+ j 2+ j 2+ j 2+ j 2+ j 2 j= j= j= j= j= j= Using he same mehodology as above we le Q z + P = Q + z z, enabling us o exress (26) as + M M M = j= j, P z j= 2 + j =, so ha E Q = FE Q + Fz + E Φ y + e 2+ j 2 j= = FE Q + Fz + Φ E L + Φ y + e 2 E Q = I F Fz + Φ E L + Φ y + e ( ) ( 2 ) = I F Fz + Φ E K + Φ y + Φ y + e ( ) ( 2 ) Now relacing z j= + by Q j in (25) when M, we ge E K = ( I A ) A y + ( I A ) GE Q, whereuon using he exression for EQ yields E K = ( I A ) A y + ( I A ) G( I F ) ( Fz + Φ E K + Φ y + Φ y + e ). 2 Defining R I A G I F = ( ) ( ) we have 3

15 E K = ( I A ) A y + R( Fz + Φ E K + Φ y + Φ y + e ) 2 and so E K = ( I RΦ) [( I A ) A + RΦ] y + ( I RΦ ) Rz This gives us y = y + E K. Le us look a he simle examles in he beginning of secion 2.3. The firs model has he form y = δ z + ε = δγ z + ε + δε = δγ z + e 2 z = γ z + ε = γ z + e 2 2 Then G = δγ, F = γ, A = 0, Φ = 0, R = G( I F) so ha δγ EK = Rz = z γ as we found earlier. Thus he second shock has a ermanen effec. Looking a he second model where y = δ z + ε = δ ( γ ) z + ε + δε = δ ( γ ) z + e 2 z = γ z + ε = γ z + e 2 2 hen G = δ ( γ ), F = γ, A = 0, Φ = 0, R = G( I F) and E K = Rz = δ z y y EK = + = δ δ = 0, e e e which shows he second shock only has a ransiory effec, as found reviously. 3. An illusraion of he reamen of mixed shocks in SVARs In an influenial aer, Peersman (2005) esimaed an SVAR model of four variables o invesigae he role layed by he underlying srucural shocks in he early millennium slowdown exerienced in he Unied Saes and Euroe. The VAR consised of he oil rice ( o ), ouu ( y ), consumer rices 4

16 ( ) (all in log levels) and he shor-erm nominal ineres rae ( s ). The oil rice, ouu and consumer rices are reaed as I() variables and he shor-erm ineres rae as an I(0) variable. There was no evidence for a coinegraing relaion among he I() variables. In view of hese roeries of he daa, Peersman followed common racice and secified an SVAR in he firs difference of he I() variables and in he level of he saionary variable. To exacly idenify he SVAR, Peersman imosed wo long-run and four conemoraneous resricions. Under hese resricions, he srucural shock o oil rices was inerreed as an oil rice shock, o ouu as a suly shock, o consumer rices as a demand shock and o he ineres rae as a moneary olicy shock. The wo long-run resricions are ha he demand and moneary olicy shocks have a zero long-run effec on ouu, and hese disinguish hose shocks from he oil rice and suly shocks. In order o disinguish he moneary olicy shock from he demand shock, Peersman imosed he resricion ha he moneary olicy shock has a zero conemoraneous effec on ouu. Finally, he assumed ha he change in oil rices does no deend on he conemoraneous change in ouu, consumer rices and he ineres rae. These serve o differeniae he suly shock from he oil rice shock and also imly ha suly, demand and moneary olicy shocks have a zero conemoraneous effec on oil rices. Here he moneary olicy shock is, in our erminology, a mixed shock, as i arises from he inroducion of he I(0) ineres rae variable and i is ermied o have a long-run effec on some of he I() variables, secifically, consumer and oil rices. The SVAR was secified wih hree lags and each equaion included a consan and a ime rend. Peersman esimaed he SVAR by maximum likelihood mehods for he samle 980Q 2002Q2. Figure (a) of his aer (2005,.89) shows he imulse resonses of he variables o he srucural shocks ou o a 28 quarer horizon ( he differences beween 28 and 200 quarer horizons are small, so we will use he 28 quarer resuls as showing he long-run resonses). An insecion of his figure reveals several feaures. Firs, in resonse o a moneary olicy shock which raises he shor-erm ineres rae, boh consumer rices and oil rices fall over all horizons i.e. here are no rice uzzles. While ouu increases by only a small amoun iniially, i hen falls over he nex four quarers, afer which i sars o gradually recover o is level rior o he moneary olicy shock. Third, he moneary olicy shock has a long-run effec on relaive rices since oil rices fall roorionaely more han consumer rices (2% comared wih 0.3%) a he 28 quarer horizon. Fourh, he demand shock has a long-run effec on relaive rices. Oil rices increase by 3% in resonse o a osiive demand shock a he 28 quarer horizon while consumer rices increase by only 0.3%. While here are no ouu and rice uzzles in he resuls, he moneary olicy and 5

17 demand shocks have a long-run effec on relaive rices. 4 Because i is sandard in mos economic models for demand and moneary olicy shocks o have only ransiory effecs on relaive rices and ouu so ha in he long-run relaive rices and ouu are unaffeced by hese shocks, we would exec ha he SVAR should also be designed o have such roeries. We now urn o how his is o be done. 3.. Design of he SVAR To arrive a a SVAR wih he long-run roeries jus menioned, we begin by relacing he rice of oil wih he relaive rice of oil, defined as ς = o. This is also an I() variable, as Peersman found no co-inegraion beween he I() variables. The resuling SVAR is: 5 ς = a ς + a y + a y + a + a + a s + a s + ε (27) y = a ς + a ς + a y + a + a + a s + a s + ε (28) = a ς + a ς + a y + a y + a + a s + a s + ε (29) s = a ς + a ς + a y + a y + a + a + a s + ε (30) The four long-run resricions we imose are ha demand and moneary olicy shocks have a zero long-run effec on relaive rices and ouu. Wih resec o relaive rices, he resricions are, resecively, a + a = , a + a = (3) , and, wih resec o ouu, hey would be a + a = 0, a + a = 0. (32) We esimaed Peersman s SVAR by IV and relicaed his resuls. We will laer find some rice and ouu uzzles in various SVARs we esimae. There are of course suggesions ha hese uzzles may no be so e.g. i has been argued ha a rise in ineres raes could increase he rice level owing o increased working caial coss. However, mosly such resuls are regarded as abnormal, and so classified as uzzles. We will jus follow he convenional aroach here and classify rises in ouu and rices in resonse o moneary olicy shocks as uzzles. 5 For ease of exosiion, our develomen assumes an SVAR of order one which does no include deerminisic erms. I can be easily generalised o he SVAR we acually esimae which, following Peersman, has hree lags and a consan and ime rend in each equaion. 6

18 These enable demand and moneary olicy shocks o be differeniaed from relaive oil rice and suly shocks. 6 We require wo conemoraneous resricions, one o searae demand from moneary olicy shocks and he oher o searae relaive oil rice from suly shocks. They are, resecively, ha he demand and suly shock have a zero conemoraneous effec on he relaive rice of oil. These are he equivalen of wo of Peersman s shor-run resricions, hough now wih resec o he relaive oil rice. These resricions can be imosed aramerically on (27)-(30). Le he (4 4) marix of conemoraneous ineracions among he variables be denoed by A 0, where he elemens along he rincial diagonal are uniy, so he firs srucural equaion is for he change in he relaive oil rice, he second for he change in ouu, he hird for he change in consumer rices and he fourh for he ineres rae. The relaionshi beween he srucural shocks and he reduced form errors ( e ) is given by ε = A0 e. Le he elemen in he ih row and he jh column of A 0 be ij denoed as a 0. Then he resricion ha he demand shock has a zero conemoraneous effec on relaive oil rices is exressed as 3 a 0 = 0. Because e = A ε he reduced form errors are linear 0 combinaions of he srucural shocks, so ha he resricion 3 a 0 = 0 means ha he demand shock does no aear in he reduced form (VAR) error for relaive oil rices. Consequenly he residuals from he VAR equaion for relaive oil rices ( e ) can be used as an insrumen in he esimaion of he consumer rice equaion. Similarly, he resricion ha he suly shock has a zero conemoraneous effec on relaive oil rices is 2 a 0 = 0, showing ha he VAR relaive oil rice residuals can also be used as an insrumen in he esimaion of he ouu equaion. The wo conemoraneous resricions ogeher wih he four long-run resricions shown in (3) and (32) roduce he correc number of resricions o idenify he SVAR arameers Esimaion Imosing he wo long-run resricions in (32) on (28), he equaion for he change in ouu becomes y = a ς + a ς + a y + a + a s + ε (33) To imlemen hese resricions, we noe, for examle, ha 0 0 a s + ( a + a ) s a s + a s can be exressed as

19 We esimae his equaion using, as insrumens, e ˆ,, s, as well as and. The nex equaion o esimae is he equaion for he relaive rice of oil ha is obained by imosing he resricions in (3) on (27). The resuling equaion is ς y ς = a ς + a y + a y + a + a s + ε (34) I is esimaed using, as insrumens, he residuals ˆ ε 2 from (33), along wih,, y are ˆ, s ς and. The nex equaion esimaed is (29), he equaion for consumer rices. Here he insrumens ˆ, e, as well as,, and s. Finally, he las equaion esimaed is (30), ε ε ˆ 2 ς y he ineres rae equaion. For his, we use he esimaed residuals ˆ ε, ˆ ε 2 and ˆ ε 3 as well as, y, and s, as he insrumens. ς 3.3 Resuls Figure shows he imulse resonses of he U.S. variables o he srucural shocks ou o a horizon of 28 quarers. 7 Noe ha he resonse of he oil rice iself o a shock is simly he sum of he relaive oil rice and consumer rice resonse o ha shock. The idenifying resricions are aaren in he resonses: he demand and moneary olicy shocks have a zero long-run effec on relaive oil rices and ouu, and he suly and demand shocks have a zero conemoraneous effec on relaive oil rices. 8 Wih his secificaion, however, here are rice and ouu uzzles. In resonse o a moneary olicy shock which raises he ineres rae, consumer rices seadily rise and by 28 quarers have increased by 0.3%. The oil rice iniially falls by abou 3%, so here is no relaive oil rice uzzle, and by 28 quarers i has increased by he same roorionae amoun as consumer rices, leaving long-run relaive oil rices unchanged. Ouu iniially rises by around 0.3% following he moneary olicy shock so here is an ouu uzzle. We esimaed several SVARs under oher combinaions of wo zero conemoraneous resricions while mainaining he four long-run resricions. In all he SVAR s, a leas one uzzle was 7 The resonses a 28 quarers are sufficien o show he long-run as hey are indisinguishable from hose a much longer horizons (we generaed resonses ou o 200 quarers and saw no discernible differences). The imulse resonses are shown ogeher wih heir one sandard error bands based on 000 boosraed draws. In he boosra, he forecas values and re-samled residuals from he reduced-form VAR model esimaed wih acual daa were used o consruc arificial ime series for each variable. 8 As a check on our resuls, we also esimaed he model using he shor and long rocedure in RATS Version 8.2. The RATS numerical rocedure confirmed he resuls from IV esimaion and he numerical differences beween he wo ses of imulse resonses were sligh. 8

20 aaren in he resonses. When here was a consumer rice uzzle, here was no oil rice uzzle and vice-versa, and i was only in secificaions which resriced he conemoraneous resonse of ouu o he moneary olicy shock o zero ha he ouu uzzle disaeared. 9 I aears ha, once demand and moneary olicy shocks are resriced o have only ransiory effecs on relaive rices, uzzles emerge. Once hese shocks are allowed o have ermanen effecs on relaive rices, he uzzles disaear. Our exerience in oher alicaions is ha his is a common henomenon and i should force emirical researchers o jusify why hey allow nominal shocks o have long-run effecs on real variables and relaive rices. 4. Sign Resricions wih Mixed Shocks In addiion o he arameric aroach, Peersman chose o use he sign resricions mehodology, develoed by Faus (998), Uhlig (2005) and Canova and De Nicoló (2002), o idenify he srucural shocks. The mehod sars by obaining an iniial se of shocks ha are uncorrelaed. Peersman followed radiional racice and obained hese from a recursive model. While his resrics he conemoraneous imacs of he iniial shocks, i leaves he long-run imacs unresriced. In our alicaion, we secify he iniial model o reserve he wo long-run resricions ha he hird and fourh shocks have a zero long-run effec on ouu and hen make he model recursive. In his way, he iniial shocks are orhogonal and have he roery ha he hird and fourh shocks do no have a long-run imac on ouu. To describe our iniial model, we will refer o (27)-(30) for he variable numbers. However, now he relaive rice of oil has o be relaced by he rice of oil, as we are re-considering he resuls from Peersman s original model wih shocks now being idenified using sign resricions. 0 The firs equaion we se u o generae shocks ha are o be he basis of he sign resricions aroach is for ouu i.e. (33). This has imosed on i he wo long-run zero resricions. Bu we need a furher resricion, and ha involves assuming oil rices are ordered afer ouu, so ha 0 a 2 = 0. The nex equaion is for he change in consumer rices (29), and here we assume ha oil rices and ineres raes are ordered afer he general rice level, hereby generaing he resricions 0 a 34 = 0. The oil rice equaion (27) uses he resricion 0 a 3 = 0 and 0 a 4 = 0, ha is ineres raes are ordered afer he oil rice, Finally no resricions are laced on he ineres rae equaion (30). The model jus described is hen esimaed by IV. In esimaion of he consumer rice equaion, ˆ ε 2 is used as an 9 This aern emerged in all secificaions including ones ha lef unresriced he conemoraneous effec of all he shocks on he relaive rice of oil. 0 Again, in he acual alicaion, we follow Peersman and esimae a SVAR wih hree lags and a consan and ime rend in each equaion. 9

21 insrumen; in esimaion of he oil rice equaion, ˆ ε 2 and ˆ ε 3 are used as insrumens; and in he ˆ ineres rae equaion, ˆ ε 2, ε 3 and ε are used as insrumens. ˆ In sign resricions, he iniial shocks from he model jus described are normalized o have uni variance so hey become ˆ ε ˆ ˆ i, = ( εi, / σ i ), i =, 2,3, 4 and arei. i. d(0, I 4). We focus on he grou ˆ ε = ( ˆ ε ˆ ε ) as hese are resriced o have a zero long-run effec on ouu. The nex se is R, 3 4 o linearly combine hese shocks o form a new se of shocks ˆ η = Q ˆ ε, where he (2 2) marix Q is he Givens marix R, R, cosθk sinθk sinθk, θk (0, π ), cosθ k wih he roery ha Q Q = QQ = I2. The Q marix deends on a draw of θ k and, in sign resricions, he number of draws is large. 2 Noe ha he new shocks are uncorrelaed wih each oher. Now le he (4 2) marix C R, j denoe he resonses a horizon j of he variables o a one uni innovaion in each of he shocks in ε ˆR,. Then, for a given draw of he Givens marix, he resonses o a one uni innovaion in each of he new shocks, η, is C, Q '. Noe ha he long-run resonse of ouu o ˆR η, is zero since boh elemens of he second row of CR, are zero. Sign resricions are now used o disinguish beween he wo shocks in η. 3 The resricions we use are aken from Peersman. A osiive moneary olicy shock raises he ineres rae and has a non-osiive effec on oil rices, ouu, and consumer rices. In conras, if all he resonses are non-negaive, i is reaed as a osiive demand shock. 4 We found ha 0.578% of he draws saisfied he sign resricions for demand and moneary olicy shocks. This success rae is a lile lower han wha Peersman reored ( in 30 or 0.769%). ˆR, ˆR, R j Searaing he shocks ino aroriae grous and alying sign resricions o each grou is he aroach aken by Fry and Pagan (20) for co-inegraed sysems in which here are boh ermanen and ransiory shocks. As we are making finer disincions among he shocks, i is naural o ado a similar aroach here, so ha he new shocks will reain he feaures of he iniial shocks. 2 In our alicaion, θk = k( π / 500, 000), k = 0,, 2, K, 500, They could also be used o searae he shocks in he grou ˆ η = ( ˆ η ˆ η ) bu ha is no our focus. 20 U, 2 4 In line wih Peersman, he ime eriod over which he sign resricions are binding is for four quarers on he resonses of ouu and consumer rices and only on he insananeous resonse of oil rices and he ineres rae.

22 In boh cases however hese low reenion raes sugges ha he daa does no suor he sign resricions. Based on he successful draws, figure 2 reors he median (50 h fracile) resonses o uni shocks. Demand and moneary olicy shocks have a zero long-run effec on ouu by design bu hey clearly have long-run effecs on he relaive rice of oil. In our signs aroach, care needs o be exercised in formulaing he iniial recursive model. Suose we had decided o order he oil rice before ouu. Then his would mean ha he iniial hird and fourh shocks have a zero conemoraneous effec on oil rices. Now hese wo shocks have he requisie zero long-run effecs so we linearly combine hem ogeher o form new shocks. Bu his mus mean ha any new shocks have a zero conemoraneous effec on oil rices. I does no seem reasonable o consrain he demand and moneary shocks o always have such effecs. Consequenly, his led us o ado he ordering described where oil rices came afer ouu and he general rice level. 5. Conclusion The inclusion of I(0) variables in srucural economeric models inroduces addiional shocks which do no fi nealy ino he radiional classificaion of shocks in I() sysems as ermanen or ransiory. We augmen his classificaion by describing shocks associaed wih he I(0) variables as eiher mixed (having a non-zero long-run effec on a leas one I() variable) or ransiory (having a zero long-run effec on all I() variables). We showed ha several well-known resuls follow hrough o seings which include I(0) variables rovided he shocks associaed wih hem are ransiory. The Pagan and Pesaran (2008) resul abou he naure of srucural equaions wih ermanen shocks can be exended o his case, and familiar resuls associaed wih he Beveridge-Nelson decomosiion and from coinegraion analysis aly. However, if he shocks are mixed, familiar resuls may no longer aly. We show ha i is no longer he case ha he change in he ermanen comonen of an I() series defined by he Beveridge- Nelson decomosiion would be whie noise. We also show ha a sandard aroach o handling boh I() and I(0) variables coming from he co-inegraion would no longer give he correc answer for he ermanen comonen when shocks are mixed. Consequenly, we derive a general formula for comuing he ermanen comonen when shocks are mixed. We hen urned o some alicaions, using as he vehicle Peersman s influenial SVAR which feaures a mixed shock. The laer arises from he resence of an I(0) ineres rae variable and is mixed because i is allowed o have a long-run effec on oil and consumer rices, boh of which are I(). In Peersman s SVAR, here are no rice or ouu uzzles bu here is moneary non-neuraliy, 2

23 since he mixed shock affecs relaive rices in he long-run. When he mixed shock is made ransiory ouu and rice uzzles emerge. We conclude ha he absence of rice and ouu uzzles in Peersman s VAR comes abou because he allows he mixed shock o have a long-run effec on relaive rices; ha is, he absence of uzzles comes a he cos of moneary neuraliy. Finally, we show how o aly sign resricions o he SVAR for which he wo long-run zero resricions of Peersman are mainained. 22

24 References Bernanke B, Boivin J, Eliasz P Measuring he effecs of moneary olicy: A facor-augmened vecor auoregressive (FAVAR) aroach. Quarerly Journal of Economics 20: Canova F, De Nicoló G Moneary disurbances maer for business flucuaions in he G-7. Journal of Moneary Economics 49: Canova F, Gambei L, Paa E The srucural dynamics of ouu growh and inflaion: Some inernaional evidence. Economic Journal 7: C67-C9 del Negro M, Schorfheide F Priors from general equilibrium models for VARs. Inernaional Economic Review 45: Faus J The robusness of idenified VAR conclusions abou money. Carnegie-Rocheser Conference Series in Public Policy 49: Fry R, Pagan A. 20. Sign resricions in srucural vecor auoregressions: A criical review. Journal of Economic Lieraure 49: Juselius K The Coinegraed VAR Model: Mehodology and Alicaions. Advanced Texs in Economerics. Oxford Universiy Press: Oxford. Pagan A, Pesaran M Economeric analysis of srucural sysems wih ermanen and ransiory shocks. Journal of Economic Dynamics and Conrol 32: Peersman G Wha caused he early millenium slowdown? Evidence based on auoregressions. Journal of Alied Economerics 20: RATS (Regression Analysis of Time Series), Version Esima: Evanson, Illinois. Shairo M, Wason M Sources of business cycle flucuaions. In NBER Macroeconomics Annual 3: -48. Smes F Measuring moneary olicy shocks in France, Germany and Ialy: The role of he exchange rae. Swiss Journal of Economics and Saisics 33: Uhlig H Wha are he effecs of moneary olicy on ouu? Resuls from an agnosic idenificaion rocedure. Journal of Moneary Economics 52:

25 Figure. Imulses resonses from relaive rice model Figure 2. Imulse resonses based on signs 24