Econometric Methods for Modelling Systems with a Mixture of I(1) and I(0) Variables # Summary

Transcription

1 Economeric Mehods for Modelling Sysems 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. The srucural shocks associaed wih eiher se of variables could be ermanen or ransiory. We herefore classify he shocks as (P,P0) and (T,T0), where P/T disinguishes ermanen and ransiory, while /0 means hey are aached o eiher I() or I(0) variables. We firs analyse wha haens when here are P0 shocks. This is done using a sequence of examles and shows a variey of oucomes ha differ from sandard resuls in he coinegraion lieraure. Then condiions are derived uon he naure of he SVAR in he even ha T0 (and no P0) shocks are resen. Following his a general mehod ha allows for eiher P0 or T0 shocks is described and relaed o he lieraure ha reas I(0) variables as coinegraing wih hemselves. Finally, we urn o an examinaion of a well known emirical SVAR where here are P0 shocks. This SVAR is re formulaed so ha he exra shock coming from he inroducion of an I(0) variable does no affec relaive rices in he long run i.e. i is T0, 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 in he resence of P0 shocks when some shocks are idenified using sign resricions. Key Words: Mixed models, ransiory shocks, ermanen 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.

2 Economeric Mehods for Modelling Sysems wih a Mixure of I() and I(0) Variables. Inroducion 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 coinegraion srucural models are generally formulaed in erms of changes in he I() variables. Wih coinegraion 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 coinegraion, srucural shocks have been radiionally classified as ermanen and ransiory. We will refer o such shocks as P and T resecively. 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 ransiory or could have ermanen effecs on all or some of he I() variables. We will refer o hese as T0 and P0 shocks resecively. Secion 2. resens some simle examles o show wha he effecs are of having P0 shocks resen in he srucural sysem. I sars wih no P0 shocks, being jus a sandard coinegraed wo I() variable model. Then an I(0) variable is added wih a P0 shock and i is shown ha some of he familiar resuls from coinegraion analysis no longer aly. A hird examle incororaes hree I() variables and a single I(0) variable wih a P0 shock, while a fourh examle reurns o he firs examle and adds on wo I(0) variables wih P0 shocks. Jus as wih he second examle, he hird and fourh examles also show ha radiional resuls from he coinegraion lieraure no longer need aly. Secion 2.2 hen looks a he case where he inroduced shocks from I(0) variables are ransiory i.e. T0. I is shown ha his requires a aricular ye of model design. Secifically one needs o force some of he srucural equaions o have changes raher han he levels of I(0) variables, which exends he resul found in Pagan and Pesaran (2008). A good deal of emirical work seems o have his siuaion in mind bu does no recognize ha he sysem needs o be designed o ensure ha he shocks are ransiory. 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, 2

3 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 real variables in he SVAR bu which 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 coinegraing wih hemselves. Tha sraegy requires he inroducion of seudo coinegraing vecors as well as rue ones. We firs use his device in secion 2.2, when he exra shocks arising from he I(0) variables are assumed o have urely ransiory effecs, i.e. are T0. Then, using a simle examle in secion 2.3, we show ha his mehod rovides he correc comuaion of he ermanen comonen of he I() variable, even when he shock coming from he inroducion of he I(0) variable has ermanen effecs i.e. when i is P0. Secion 2.4 hen inroduces a general mehod for comuing he ermanen comonen of he I() variables when he shocks coming from I(0) variables are eiher ransiory (T0) or ermanen (P0). This mehod allows us o relae he ransiory comonen in he I() series o he error correcion erms and lagged I(0) variables. I also allows us o esablish ha he seudo coinegraion aroach o calculaing he ermanen comonen in he I() variables alies generally. I is no ossible o sudy he many aers ha feaure srucural models wih P0 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. He works wih hree ermanen shocks among he I() variables and one coming from he I(0) variable. This P0 shock is ha semming from he inroducion of an I(0) nominal ineres rae ino he sysem. He regards he ineres rae shock as having no effec uon ouu in he long run, bu i is allowed o affec boh he general 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 3

4 moneary olicy shock has a ermanen effec on he real oil rice. When his shock is aken o be ransiory he uzzles re aear. Now he mos common case where P0 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. Accordingly, when Peersman (2005) moved o sign resricions o idenify his shocks, he resuling imulse resonses o moneary shocks show long run effecs on real variables. This would be rue of many oher sudies wih I() variables using sign resricions o idenify shocks. Therefore, in secion 4 we look a how one can imose a zero long run resricion on a P0 shock 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 coinegraion 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 srucural shocks i.e. for he second srucural shock and, for he level of he firs variable y j y, he long run effec will be lim j. 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. We will use a se of examles o illusrae some of he issues ha arise when he shocks coming from he I(0) variables are ermanen. In our firs examle here are only wo I() variables and y j here is coinegraion. Then he long run resonse marix C lim j corresonding o he variables will have rows y and columns reresening he srucural shocks. Thus he (,2) If y is I() hen i would ener he SVAR as y. 4

5 y j elemen of C is C2 j lim. Now in his sysem a ermanen ransiory decomosiion for 0 he srucural shocks exiss so he C marix akes he form C 0, where he indicae nonzero elemens. This sysem has one ermanen shock and one ransiory shock. Sandard resuls from he coinegraion lieraure (Lükeohl, 2006,.369) are; (i) he number of indeenden ermanen comonens among he I() series (someimes called sochasic rends) is equal o he rank of C ; (ii) he number of ermanen shocks is also equal o he rank of C ; and (iii) he number of ransiory shocks is equal o he number of coinegraing vecors. All of hese resuls hold in his examle. I is useful o hink of he firs examle above as being he srucure for a marke model in which y is he log of quaniy and y he log of rice. The suly shock has ermanen effecs while he demand shock is ransiory. Now suose ha an I(0) variable is added o he sysem and ha he new shock has ermanen effecs on he second of he I() variables. This involves he addiion of a hird variable o he firs examle. In erms of he marke model i migh be ermed sunsos and hese are assumed o have a ermanen effec on rices. Consequenly, he suly and sunsos shocks have ermanen effecs bu he demand shock is ransiory. In his se u he long run resonse marix for he hree variables will be C Because he Beveridge Nelson decomosiion has y C, i means ha y C and y C C. Hence here are wo ermanen comonens (or, as someimes said, wo sochasic rends), which agrees wih he rank of C. However here is no linear combinaion of y and y ha is I(0), as one canno eliminae he influence of he hird shock. Hence here is no coinegraion beween he wo I() variables. Accordingly, he resence of a P0 shock has removed he coinegraion ha reviously held beween he I() variables. This is because he DGP of changed i.e. in erms of he marke model he ermanen sunsos shocks resul in rices and quaniies no being coinegraed, bu when sunsos are absen here is coinegraion beween y has 2 Clearly we can differeniae beween all he srucural 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 firs and hird shocks. 5

6 0 hose variables. I is also ineresing o noe ha if C 0 and c c ; 2 c c23, hen he wo sochasic rends would be idenical and so here is only one indeenden one i.e. coinegraion would be re esablished. Of course in his case one needs some shor run informaion o differeniae beween he wo ermanen shocks. The nex examle adds a furher I() variable o he second examle and assumes ha here are wo coinegraing relaions among he hree of hem. This will generae a C marix of he form C In his case he number of ermanen comonens is wo (which equals he rank of C ) and he number of ermanen shocks is wo. Examining coinegraion we find ha here is one coinegraing vecor as here will be a linear combinaion of y and y 3 ha is zero. This occurs because each deends only on he firs srucural shock. One canno find a combinaion of y wih eiher ermanen comonen in he oher wo variables ha would be zero. Noice however ha here are wo ransiory shocks and so he sandard relaion beween he number of coinegraing vecors and he number of ransiory shocks is now broken. This is because he DGP of y has changed. Finally, reurn o he firs examle and add wo exra I(0) variables ha have P0 shocks. This resuls in C Insecion shows ha he rank of C is wo and, wriing ou he ermanen comonens for he firs and second variables, shows why hey are indeenden. Bu now here are hree ermanen shocks he firs, hird and fourh. So his illusraes he fac ha he number of indeenden ermanen comonens can be less han he number of ermanen shocks when here are P0 shocks. 6

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 (T0) (covered in secion 2.2) and hen ermanen (P0) (covered in secions 2.3 and 2.4) Shocks associaed wih I(0) Variables are Transiory This secion shows how o rea I(0) variables in srucural models ha conain coinegraing relaionshis among he n I() variables when he exra shocks coming from he I(0) variables are aken o be ransiory. We will ofen refer o hese exra shocks as hose associaed wih he I(0) variables. By his we mean ha he addiion of q I(0) variables o he srucural sysem necessiaes he addiion of q srucural equaions and q shocks so ha he laer will be hose associaed wih he I(0) variables. For simliciy, consider a srucural VAR(2) model of n variables of he form Ax Ax Ax () where A i are n n marices of unknown coefficiens, A 0 is non singular and is an n vecor of srucural shocks wih mean zero and covariance marix D n. We assume ha here are n variables which are I() and q which are I(0) giving n n q, while among he I() variables here are r coinegraing relaions. We refer o he laer as rue i.e. he 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 r coinegraing relaions among he I() variables, here are mn r indeenden I() rocesses driving he n I() variables and he m shocks driving hese are he ermanen shocks in a seudo SVECM. Wihou loss of generaliy, le x x x x 3, where x is he m vecor of I() variables whose srucural shocks are known o have ermanen effecs, x is he r vecor of I() variables whose srucural shocks are known o have 7

8 ransiory effecs, and x 3 is he q vecor of I(0) variables whose srucural shocks will be assumed o have ransiory effecs. Then here are r qransiory shocks in he SVECM. Le 0 2 0, 0 I q where is an n( r q) marix. The marices and 2 are m r and r r, resecively. The firs column of block marices in 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 can be ariioned as 2 2, 3 3 where is an n( r q) marix. The sub marices and q r, resecively. Similarly, he sub marices,, 2 and 2 and are of dimension m r, r r 3 3 are of dimension m q, r q and q q, resecively. The firs column of block marices in 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. The VAR model of () can now be wrien as he seudo SVECM A x x A x. (2) 0 2 The r vecor of rue error correcion erms,, can be wrien as x x. 2 (3) 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 (3), we have x, and x 3. From 8

9 x x, 2 from which i follows ha x ( ) ( x ), (4) 2 rovided he r r marix 2 is non singular. 3 The firs m equaions in (2) are A x A x A x x A x A x A x, (5) where he A marices are ariioned conformably wih srucural shocks wih ermanen effecs. Pagan and Pesaran roved ha x. These equaions conain he 0 in (5), so ha he srucural equaions wih he ermanen shocks do no conain he lagged rue error correcion (EC) erms. Here we show addiionally ha 0 when he srucural shocks associaed wih he I(0) variables are ransiory, i.e. for he SVAR(2) in in x in (5), one obains x, and x 3. Using (4) o eliminae he erms ( A A ( ) ) x A ( ) A x x ( A A ( ) ) x A ( ) A x (6) Defining w x x 3 he SVAR(2) can be exressed as Bw Bw Bw (7) Pariioning (7) ino he form conformable wih he ariion used in (5), he firs m equaions will be B x B B x B x B B x B x B B x I may be necessary o ake care in seing he sysem u o ensure ha is non singular. To ake an 2 examle, suose here are hree I() variables in he sysem wih coinegraing vecor 0. Then, if we selec he firs and second equaions as he wo whose srucural shocks have ermanen effecs, 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 and, in he laer, 2 2 and so are non singular. 9

10 which can be wrien as, Bx B2 B3x3 (8) B x ( B B B ) ( B B B ) x B x B B x Comaring (8) wih (6), we ge ( B B B ), (9) ( B B B ) (0) Now (7) can be wrien in lag oeraor form as BLw ( ), 2 where B( L) B BL B L and L is he lag oeraor. I hen follows ha he moving average reresenaion will be 0 2 w B L C L ( ) ( ), () 2 3 where CL ( ) CCLCLCL. Hence C() B() imlies ha C() B() In. (2) By assumion shocks o he error correcion erms 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 r null marix. Under he resricions, (3) becomes C() B2() 02, from which i follows ha B2() 02, since C () has full rank m. Bu 0 2 B2 () ( B2 B2 B2 ) so ha B2() 02 means 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 0

11 C () B () C () B () C () B () 0, (4) where 03 is an m q null marix. Using he same reasoning, i follows ha B3() 03, and noing ha B () ( B B B ), (0) imlies ha 0. Thus, when he srucural shocks coming from he I(0) variables are ransiory, he m srucural 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 and Pesaran resul for he SVAR involving x and ha he levels of he EC erms were relaced by heir differences in he srucural equaions for x A Simle Examle for Consrucing he Permanen Comonen of an I() Variable when Shocks are P0 or T0. In order o see he effecs of a P0 shock, i is useful o consider he following simle sysem: y z (5) z, z 2 (6) where on y is I(), z is I(0), he shocks and are whie noise and has a ermanen effec y. Hence, using he Beveridge Nelson decomosiion o find he ermanen comonen of we have y y y E y y E ( z ) y z, j j j j j so ha, from (5) and (6), y y z z z z [( ) z ]. This shows ha y is whie noise as i is he sum of wo whie noise erms. Turning o he case where has ransiory effecs requires he re secificaion of (5) o

12 y. z (7) Then, by he Beveridge Nelson decomosiion, j j j y y E ( z ) And, using he fac ha z z E z j means z 0 ( z is I(0) wih zero mean), as well as being whie noise, we ge j y y ( z z ) y z, so ha y y z. This shows ha has only a ransiory effec on any y. 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 as 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 any srucural equaions 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. Tha would enail having he firs difference of he I(0) variables and no heir levels, on he RHS of he srucural equaions for y. Noice ha if one had used an SVAR wih y and z, his would change he secificaion of he second equaion in (6) 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 firs sysem, (5) and (6), and in he second one, (7) and (6)? In boh sysems here is no coinegraion so ha will be he seudo co inegraing vecor 0. The seudo SVECM for boh sysems has equaions in y and z. For he firs sysem, he seudo SVECM comrises (5) along wih 2

13 z ( ) z, (8) and he corresonding seudo VECM form is y z e (9) z ( ) z e, (20) where he seudo VECM residuals are e and e. 2 In coinegraion analysis, Johansen s formula (995) can be used o obain he change in he ermanen comonen of he series y as y z A() e, (2) where 0, 0, and A () is he (2x2) marix whose elemens are he sum of he coefficiens on y, z and heir lags in each equaion block. In he sysem (9) and (20), 0,, 0 A() and, and A() I2, so ha y e e e, 2 (22) which is he exression we obained earlier by direc mehods for he sysem (5) and (6). Now consider he second sysem. Here he seudo SVECM comrises (7) and (8) which has a seudo VECM form comrising (20) and y ( ) z e, (23) where e and e as before. Now ( ) and, so ha y e e e2, which is wha we obained earlier by direc mehods for he second sysem where he second shock was ransiory. This analysis illusraes ha he seudo coinegraion aroach will work, rovided he SVECM is se u aroriaely, and wha is aroriae will deend on wheher he shocks arising from he 3

14 I(0) variables are assumed o have ermanen or ransiory effecs. When hey are allowed o have ermanen effecs, he srucural equaions for he I() variables mus be secified in he levels of he I(0) variables. When hey are resriced o have ransiory effecs, hese equaions mus be secified o include changes of he I(0) variables. Care migh need o be aken if one uses he seudo coinegraion aroach in conjuncion wih sandard coinegraion sofware since i would be rare for he SVECM o be formulaed as involving boh changes and levels of he I(0) variables i.e. yically he sofware would handle he case when he exra shocks are ermanen bu no when hey are ransiory. We now derive a formula for he comuaion of he ermanen comonen in an I() variable when here is a general VAR sysem i.e. in which shocks can be ransiory or ermanen. This formula also allows us o see he maing from he changes in I() variables and he levels of I(0) variables o he ransiory comonen of an I() variable. I can also be used o allow us o esablish for he general case ha he coinegraion aroach gives he correc esimae of he ermanen comonen. This laer feaure is demonsraed in Aendix General formula for comuaion of ermanen comonens of I() series when he Exra Shocks are eiher P0 or T0. We will consider he following VAR sysem y Ay Gz e (24) z Fz y e, 2 (25) where z has boh any rue EC erms as well as he I(0) variables in i. To raionalize (24) and (25) hink of he case where all variables, y are I() and here is co inegraion. Then we would have y Ay y and his can be wrien as y ec Ay ec, hereby giving an equaion ha has he form ec ( I ) ec A y. So his has he srucure of (25) 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 which we denoe as EK. This will be y y E y, so we need o look a he j j 4

15 . (26) EK E y E ( Ay Gz e ) j j j j j j Aendix shows ha E K ( I R) [( I A) A R] y ( I R ) Rz (27) where R I A G I F ( ) ( ). (28) Using (27) for EK we can hen comue y from y y E K. (29) The ransiory comonen in y is EK and he formula (27) shows how i is relaed o he error correcion erms and he lagged I(0) variables i.e. for he case G 0. When G 0, he formula shows ha he ransiory comonen is ( I A) Ay, and so is no deenden on z. Furher, we show in aendix 2 ha his formula gives he change in he ermanen comonen of he I() variables as y ( I R) ( I A) [ e G( I F) e ]. (30) Alicaion of (30) o he wo simle sysems in secion 2.3 gives idenical resuls o hose esablished reviously. This is demonsraed in aendix. 3. An illusraion of he reamen of P0 shocks in Peersman s (2005) SVAR 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 ( ) (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. 5

16 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. Hence he moneary olicy shock is, in our erminology, a P0 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. 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 longrun 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 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 4 We esimaed Peersman s SVAR by IV and relicaed his resuls. Noe ha IV esimaion of SVARs was inroduced by Shairo and Wason (988). Peersman s daa was obained from he daa archive of he Journal of Alied Economerics. 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. 6

17 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 o. This is also an I() variable, as Peersman found no co inegraion beween he I() variables. The resuling SVAR is: 5 a a y a y a a a s a s (3) y a a a y a a a s a s (32) a a a y a y a a s a s (33) s a a a y a y a a a s (34) 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 (35) , and, wih resec o ouu, hey would be a a 0, a a 0. (36) 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 shocks 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. 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 0 To imlemen hese resricions, we noe, for examle, ha a s a s can be exressed as 0 0 a s ( a a ) s

18 These resricions can be imosed aramerically on (3) (34). 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 Ae 0. 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 a0 0. Because e A he reduced form errors are linear 0 combinaions of he srucural shocks, so ha he resricion 3 a0 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 a0 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 (35) and (36) roduce he correc number of resricions o idenify he SVAR arameers Esimaion Imosing he wo long run resricions in (36) on (32), he equaion for he change in ouu becomes y a a a y a a s (37) 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 (35) on (3). The resuling equaion is y a a y a y a a s (38) I is esimaed using, as insrumens, he residuals ˆ 2 from (37), along wih,, y are ˆ, s and. The nex equaion esimaed is (33), he equaion for consumer rices. Here he insrumens ˆ, e, as well as,, and s. Finally, he las equaion esimaed is (34), ˆ y 8

19 he ineres rae equaion. For his, we use he esimaed residuals ˆ, ˆ 2 and ˆ 3 as well as, y, and s, as he insrumens. 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 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. 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. 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. 9

20 4. Sign Resricions wih P0 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. Noe ha in our erminology, he hird and fourh shocks are P and P0 shocks, resecively, as hey can have a long run effec on oil and consumer rices. To describe our iniial model, we will refer o (3) (34) 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. (37). 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 a2 0. The nex equaion is for he change in consumer rices (33), and here we assume ha oil rices and ineres raes are ordered afer he general rice level, hereby generaing he resricions 0 a34 0. The oil rice equaion (3) uses he resricion 0 a3 0 and 0 a4 0, ha is ineres raes are ordered afer he oil rice. Finally no resricions are laced on he ineres rae equaion (34). The model jus described is hen esimaed by IV. In esimaion of he consumer rice equaion, ˆ 2 is used as an insrumen; in esimaion of he oil rice equaion, ˆ 2 and ˆ 3 are used as insrumens; and in he ˆ ineres rae equaion, ˆ 2, 3 and 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 ˆ ˆ i, ( i, / i), i, 2, 3, 4 and areiid.. (0, I 4). We focus on he grou 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. 20

21 ˆ ( ˆ ˆ ) 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 is he Givens marix Q ˆ R, R,, where he (2 2) marix Q cosk sink sink, k [0, ], cos k wih he roery ha QQ QQ I2. The Q marix deends on a draw of 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, jdenoe he resonses a horizon j of he variables o a one uni innovaion in each of he shocks in ˆR,. Then, for a given draw of he Givens marix, he resonses o a one uni innovaion in each of he new shocks, ˆR,, is C, Q. Noe ha he long run resonse of ouu o ˆR, is zero since boh elemens of he second row of C, are zero. Sign resricions are now used o disinguish beween he wo shocks in. 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%). In boh cases however hese low reenion raes migh 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 ˆR, bu hey clearly have long run effecs on he relaive rice of oil. R j R Searaing he shocks ino aroriae grous and alying sign resricions o each grou is he aroach aken by Fry and Pagan (20) for coinegraed 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, k k ( / 500, 000), k 0,, 2,, 500, They could also be used o searae he shocks in he grou ˆ ( ˆ ˆ ) bu ha is no our focus. U, 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. 2

22 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 can be eiher ermanen i.e. have a non zero long run effec on a leas one I() variable, or ransiory i.e. have a zero long run effec on all I() variables. We denoe he former as P0 shocks and he laer as T0 shocks. I is common racice for researchers o secify SVARs in he firs difference of he I() variables and in he levels of he I(0) variables. In his case we show ha shocks associaed wih he I(0) variables can have ermanen effecs on he I() variables i.e. are P0. I was also demonsraed how o se he SVAR u so ha shocks are ransiory i.e. are made T0, and he mehod can be seen as a generalizaion of he Pagan and Pesaran (2008) resul. I involves secifying he srucural equaions for he I() variables so ha he firs difference of he I(0) variables and no heir lagged levels aear in hese equaions. We hen derived a general exression for finding he ermanen comonen in an I() variable, from which we can see how he I(0) (and error correcion) variables in he P0 shock case ma ino he variable s ransiory comonen. This formula was also used o esablish ha a mehod for calculaing he ermanen comonen of an I() variable by reaing he I(0) variable as co inegraing wih iself would work, rovided ha one was careful in choosing he seudo SVECM sysem. We hen urned o some alicaions, using as he vehicle Peersman s (2005) influenial SVAR which feaures a P0 shock. The laer arises from he resence of an I(0) ineres rae variable and is P0 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, since he P0 shock affecs relaive rices in he long run. When he moneary shock is made ransiory i.e. T0, 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 P0 shock o have a long run 22

23 effec on relaive rices; ha is, he absence of uzzles comes a he cos of moneary nonneuraliy. Finally, we show how o aly sign resricions o he SVAR for which he wo long run zero resricions of Peersman are mainained. 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: Johansen, S Likelihood based Inference in Coinegraed Vecor Auoregressive Models. Oxford Universiy Press: Oxford. Juselius K The Coinegraed VAR Model: Mehodology and Alicaions. Advanced Texs in Economerics. Oxford Universiy Press: Oxford. Lükeohl H New Inroducion o Mulile Time Series Analysis. Sringer. 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:

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

25 Aendix : Derivaion of he Permanen Comonen in he General Case As shown in he ex, he following VAR sysem is considered y Ay Gz e (A) z Fz y e, 2 (A2) where z has boh any rue EC erms as well as he I(0) variables in i. Now he ermanen comonen of y is y y E y, so we need o look a he second erm. This will be j j. (A3) E y E ( Ay Gz e ) j j j j j j M Now le us consider L y j and define K j M y j j. Then i is clear ha L K y y M. Thus, as M, E ( L) E( K) y. Consequenly, when M, we can wrie (A3) above as EK ( AEK Ay) GE z j j ( ) ( ) j j E K I A Ay I A GE z (A4) This makes sense since, if G 0, hen he shocks e have no ermanen effecs. Now from (A2) (A5) E z E ( F z y e ) FE z E y e. j 2j 2j j 2j 2j j j j j j j M M Using he same mehodology as above we le Q z j, P 2 P Q z z, enabling us o exress (A5) as M j z j j, so ha EQ= FEQ Fz E y e 2 j j FE Q Fz E L y e 25

26 EQ IF Fz EL y e ( ) ( ) ( I F) ( Fz EK y y e) Now, relacing z by Q in (A4), when M, we ge j j E K ( I A) Ay ( I A) GEQ, whereuon using he exression for EQ gives E K ( I A) Ay ( I A) G( I F) ( Fz E K y y e ). Defining R ( I A) G( I F) (A6) we have E K ( I A) Ay R( Fz E K y y e ) and so E K ( I R) [( I A) A R] y ( I R ) Rz (A7) We will aly hese resuls o he wo simle sysems of secion 2.3 in he ex. The firs sysem had he form y z z z e z z z e. 2 Then G F A R G I F,, 0, 0, ( ) so ha EK Rz z and y e e, 2 as we found in he ex. Thus he second srucural shock has a ermanen effec. 26

27 Looking a he second sysem, we have y z ( ) z ( ) z e z z z e. 2 Then G ( ), F, A 0, 0, R G( I F) so ha EK= Rz z and y e e which shows he second srucural shock only has a ransiory effec, as we said would occur because of he resence of z in he srucural equaion (7) in he ex. Aendix 2: Equivalence of he General Formula in Aendix wih he Case of Treaing he I(0) Variable as Coinegraing wih Iself Here we derive he exression for he change in ermanen comonen of he I() variables from he general formula (A7) of aendix and show ha he exression one would use from coinegraion analysis is equivalen o i. Recall ha, y A y Gz e z Fz y e y y E K E K ( I R) [( I A) A R] y ( I R ) Rz where R ( I A) G( I F) Now y y EK and y ( I R) [( I A) A R ] y ( I R) Rz 2 2 y y y ( AI) y Gze so ha y Ay Gz e ( I R) [( I A) A R][( A I) y Gz e ] ( I R) R z 27

28 A y Gz e ( I R) [( I A ) A R][( A I) y Gz e ] ( I R) R[( F I) z ye ] Firs, collec erms in y A y ( I R) [( I A) A R]( A I) y ( I R) Ry A y ( I R) [( I A) A( A I)] y ( I R) R( A I) y ( I R) Ry A y ( I R) [( I A) A( A I)] y ( I R) RAy A y ( I R) [( I A) A( A I) RA] y ( I R) [( I R) A ( I A) A( A I) RA] y ( I R) [ A ( I A) A( A I)] y ( I R) [ A ( I A) A( I A )] y ( I R) ( I A) [( I A) A A( I A )] y ( I R) ( I A) [ A A A A ] y (0) 0 y 2 2 Second, collec erms in z ( I R) [( I A) A R] Gz Gz ( I R) R[( F I) z ] { G( I R) [( I A) A R] G( I R) R( F I)} z ( ) I R [( I R) G( I A) AG RGR( F I)] z ( ) I R [ G( I A) AG R( F I)] z ( I R) [ G( I A) AGR( I F)] z ( I R) [ G( I A) AG( I A) G( I F) ( I F)] z ( I R) [ G( I A) AG( I A) G] z ( I R) ( I A) [( I A) G AGG] z ( I R) ( I A) [ G AG AGG] z (0) 0 z Therefore, we are lef wih y e ( I R) [( I A) A R] e ( I R) Re { I ( I R) [( I A) A R]} e ( I R) Re ( I R) [( I R) ( I A) A R] e ( I R) Re ( I R) [ I ( I A) A] e ( I R) Re ( I R) ( I A) [( I A) A] e ( I R ) Re ( I R) ( I A) e ( I R) Re 28

29 ( I R) ( I A) e ( I R) ( I A) G( I F) e ( I R) ( I A) [ e G( I F) e ] which we reor as (30) in he ex. In he ECM aroach we would have y [ A() ] e z I and, noing ha, he ermanen comonen of y will be [ A() ] e. Because 0 e e G( I F) e we need o rove ha [ A() ] ( I R ) ( I A ) in order o esablish equivalence beween he wo mehods. Now ( I R) ( I A) Now 0 I, [( I A)( I R)] [( I A)( I ( I A) G( I F) )] [( I A) G( I F) )] I 0, I A 0 A() I, I G( I F) so ha 0 A I G( I F) [ () ] I A I I 0 [ I A G( I F) ] I G( I F) I A which comlees he roof. 29