Srucural Inference Wih Long-run Recursive Empirical Models John W. Keaing * Universiy of Kansas Deparmen of Economics 213 Summerfield Hall Lawrence, KS 66045 jkeaing@ukans.edu July 6, 1999 Absrac: This paper invesigaes condiions under which empirical models ha use long-run recursive idenifying assumpions will obain srucural impulse response funcions. I presen a class of srucures defined as long-run parially recursive. If an economic srucure falls ino his class, hen cerain long-run recursive empirical models are able o idenify some of he srucural responses. This sufficien condiion is firs shown in a vecor auoregression. A well-known example from he lieraure is used o illusrae his ype of srucure and some applicaions of he resul. Then he resul is shown in models of coinegraed ime series. Necessary condiions for a long-run recursive model o idenify srucure are addressed in he conclusion. Keywords : long-run muliplier, long-run parially recursive srucure, moving average represenaion, vecor auoregression, coinegraion JEL Classificaions : C51,C32,E00 * I hank Mark Dwyer, Ed Greenberg, Mark Wason, he referees and seminar paricipans a UCLA, he Universiy of Kansas, he Midwes Economerics Group Meeing and he Midwes Macroeconomics Conference for beneficial commens. Any errors would of course be aribuable o me.
1. Inroducion Recursive models have a long hisory in empirical macroeconomic research. 1 More recenly Sims (1980) began using hese models o idenify vecor auoregressions. The coefficiens for his shor-run recursive sysem are obained by he Cholesky decomposiion of he covariance marix for VAR innovaions. A pracical benefi of recursive models is ha, in general, parameers are idenifiable. However, Cooley and LeRoy (1985) and ohers have criicized Sims s approach for being "aheoreical". In response o his criicism, economiss have devised various srucural approaches o VAR modeling. One such approach idenifies economic srucure using long-run resricions derived from he seady-sae properies of heoreical models. This mehod is paricularly aracive o economiss who believe heory describes long-run equilibrium phenomena beer han shor-run dynamics. Following Blanchard and Quah (1989), he empirical work based on long-run resricions has frequenly used recursive idenificaion assumpions. A purely recursive long-run muliplier marix is easily esimaed by means of he Cholesky decomposiion of a marix consruced from he covariance marix for VAR innovaions and he sum of VAR coefficiens. Since he se of recursive sysems is quie limied compared o he se of all possible economic srucures, some migh be concerned ha he criicisms leveled agains shor-run recursive orderings may also apply o long-run recursive orderings. 2 Blanchard and Quah address a number of poenial pifalls wih long-run idenificaion resricions in an appendix. Faus and Leeper (1997) exend Blanchard and Quah s invesigaion ino he usefulness of long-run recursive orderings for economic analysis. 3 Oher sudies invesigae poenial economeric difficulies wih models based on long-run resricions. Such models are ofen be esimaed by means of an insrumenal variables mehod developed by Shapiro and Wason (1988) in which residuals from he srucural equaions serve as he insrumens. Pagan and Roberson (1998) and Sare (1997) invesigae 1
some of he problems ha arise when he residuals are nearly uncorrelaed wih he variables ha require insrumens. This paper is also concerned wih he usefulness of empirical models ha employ long-run idenificaion resricions. Bu, in conras o oher research, I invesigae condiions under which long-run recursive empirical models are able o idenify srucural behavior. The frequen use of long-run recursive models in empirical research is a primary moivaion for his sudy. I presen a class of economic srucures ha permis long-run recursive empirical models o idenify srucure. If he economic srucure is long-run block recursive, he equaions in a leas one of hese blocks can be recursively ordered and he srucural shocks are uncorrelaed, hen cerain long-run recursive empirical models will obain some srucural impulse responses. Economies which saisfy hese hree condiions are called long-run parially recursive srucures. If he chosen long-run ordering is consisen wih his underlying srucure, he empirical model will yield srucural responses for each shock from he paricular block of long-run recursive srucural equaions. The oher shocks from his empirical model will no idenify srucural effecs. The finding ha he block of long-run recursive equaions can occur anywhere wihin a blockrecursive srucure has no been shown before. 4 The paper is composed as follows. Secion 2 describes a popular mehod for consrucing long-run recursive orderings in VARs wih differenced daa and shows he relaionship beween a long-run recursive ordering and a general economic srucure. Secion 3 uses linear projecion argumens o prove ha a long-run parially recursive srucure will permi long-run recursive orderings o idenify some srucural responses in a VAR model wih differenced daa. Secion 4 presens examples of long-run parially recursive srucures, based on he economic heory developed in Amed, Ickes, Wang and Yoo (1993), o illusrae his class of srucures and o clarify some ways ha he resul from Secion 3 may be used. Secion 5 exends he resul o models wih coinegraed daa. Hence, he sufficien condiion applies 2
o some of he mos popular mulivariae ime series models. Secion 6 concludes he paper and briefly discusses necessary condiions for a long-run recursive model o idenify srucure. 2. Implemening Long-Run Recursive Orderings This secion examines recursive orderings under general srucural assumpions, and shows ha if he economy's srucure is no recursive in he long run, a long-run recursive ordering will no, in general, be able o exrac srucural impulse response funcions from he reduced form. Presening his resul is a useful saring poin for deermining condiions ha permi long-run recursive orderings o produce srucural responses. The srucural moving average represenaion (MAR) is a convenien ool for sudying economic sysems ha are usually wrien in auoregressive form. This represenaion wries each endogenous variable as a funcion of curren and pas srucural shocks. If y is an n-vecor of difference-saionary ime series and g is he vecor of n srucural shocks, he srucural MAR is: y = θ( L) ε (1) where θ( L) = θ L and for j=0,1,2,...,4 is an n n marix of parameers. For he presenaion of j= 0 j j θ j idenificaion resuls, deerminisic elemens can be omied wihou loss of generaliy. An economerician would like o uncover equaion (1) wih impulse responses from a VAR model. The mulivariae Beveridge and Nelson (1981) decomposiion for y is 5 y = y+ θ( 1) ε + θ ( L) ε j j= 0 * (2) * θ( L) θ( 1) * i where θ ( L) = = θi L wih θ * i = θ and is he iniial condiion for y. L k y 1 i= 0 3 k= i+ 1
If )y is a saionary vecor process, hen he las erm in equaion (2) represens a saionary mulivariae moving average process. Consequenly, his erm has no effec on he level of y asympoically. The second erm in (2) sums he vecor of srucural shocks, indicaing ha each shock may have a permanen effec on y. The magniude and direcion of hese permanen effecs is given by he marix of long-run srucural mulipliers 2(1), which is he marix sum of he srucural parameers in 2(L). The VAR represenaion derived from equaion (1) can be wrien as b( L) y = v (3) where v an n-vecor of VAR innovaions and b( L) = I b L b L... b L 1 2 2 l l wih b j he n n marix of VAR coefficiens on variables lagged j periods and I he n n ideniy marix. This VAR represenaion is derived from he underlying srucural MAR by pre-muliplying (1) by he inverse 6 of 2(L) and hen pre-muliplying he resuling expression by 2(0). Consequenly, he VAR's coefficiens are a funcion of he parameers in he srucural MAR, 7 b( L) = θ( 0) θ( L) 1 (4) and he VAR innovaions are given by = θ( 0) ε. (5) v In mos examples from he lieraure, srucural shocks are assumed o be uncorrelaed whie noise processes. Thus equaion (5) shows ha each innovaion is serially uncorrelaed because i is a linear combinaion of whie noise srucural shocks, and ha hese linear combinaions are based on he conemporaneous srucural parameers 2(0). The diagonal covariance marix for g is convenienly normalized o be an ideniy. If E v is he covariance marix for VAR innovaions, hen (5) implies ha Σ v = θ( 0) θ( 0). (6) Economiss who favor conemporaneous idenificaion resricions make use of his relaionship. For example, Sims (1980) used Cholesky decomposiions of E v for idenificaion purposes. A Cholesky 4
decomposiion is obained by finding he unique 8 lower riangular marix C ha solves Σ v = CC. (7) The firs generaion of srucural VAR models applied conemporaneous resricions on 2(0) derived from economic srucures. Clearly if 2(0) is lower riangular, hen C=2(0) because of he uniqueness of he recursive facorizaion. Mehods oher han he Cholesky decomposiion may be required, however, if one needs o idenify parameers from an economic srucure ha is no recursive. 9 In conras, long-run idenificaion resricions are based on he marix of long-run mulipliers. A relaionship beween srucural long-run mulipliers, conemporaneous srucural parameers and he sum of VAR coefficiens, given by b(1), is obained by leing L=1 in equaion (4): b( 1) = θ( 0) θ( 1) 1. (8) One can solve (8) for 2(0), inser his resul ino (6) and simplify o obain 1 1 b( 1) Σ v[ b( 1) ] = θ( 1) θ( 1). (9) This equaion maps he long-run srucural parameers ino parameers from he reduced form. A long-run recursive ordering is obained by finding he represenaion y = R( L) u, (10) such ha R(1), he marix of long-run mulipliers for u, is riangular. The assumpion ha he u shocks are conemporaneously uncorrelaed yields a diagonal covariance marix which for convenience is normalized o he ideniy marix. The MAR in (10) is mapped ino he VAR represenaion following he same seps used wih he srucural sysem, yielding: b( L) = R ( 0) R ( L) 1 (11) and v = R( 0) u. (12) To consruc R(1) from he VAR, firs se L=1 in (11): 5
b( 1) = R( 0) R( 1) 1 (13) and use (12) o find Σ v = R( 0) R( 0). (14) Then solve (13) for R(0), eliminae R(0) from (14) and simplify o obain: 1 1 v b( 1) Σ [ b( 1) ] = R( 1) R( 1) (15) A convenien way o calculae R(1) in (15) is by he Cholesky decomposiion. (15): Coefficiens from he recursive model are relaed o he srucural parameers by equaing (9) and R( 1) R( 1) = θ( 1) θ( 1). (16) In he mos general case, 2(1) is no lower riangular, and herefore each coefficien in R(1) is a nonlinear funcion of he 2(1) srucural parameers. However, if he economic srucure is recursive in he long run, 2(1) is lower riangular, and herefore R( 1) = θ( 1) because he riangular facor is unique. In oher words, if he srucural sysem is long-run recursive and he economis chooses he correc ordering, he marix of long-run srucural mulipliers is idenified. The relaionship beween he srucural MAR in (1) and he MAR obained by he long-run recursive model in (10) is of primary ineres. Empirical researchers consruc he MAR in (10) by premuliplying (3) by b(l) -1 using (12) o eliminae VAR innovaions: To relae R(L) o 2(L), use (4) o eliminae b(l) in (18): (17) 1 1 y. (18) = b( L) v = b( L) R( 0) u 1 y = θ( L) θ( 0) R( 0) u = R( L) u (19) If R(0)=2(0), hen R(L)=2(L). However, equaion (19) should insead be pu in erms of he long-run srucural parameers and coefficiens from he long-run recursive model. Solving (8) for 2(0) and (13) 6
for R(0) yields θ( 0) 1 ( 0) θ( 1) 1 ( 1) 1 R = b b( 1) R( 1) = θ( 1) 1 R( 1). (20) Subsiuing he resul from (20) ino (19) gives 1 R( L) = θ( L) θ( 1) R( 1). (21) Clearly if he long-run srucure is recursive and he economerician selecs he correc recursive ordering, R(1)=2(1) and he empirical model will idenify all he srucural responses because R(L)=2(L). However, if he srucure is no recursive in he long run, he MAR associaed wih u will generally be a complicaed funcion of he srucural MAR. 3. Srucural Inference using VAR Models wih Differenced Daa The objecive is o deermine general condiions under which long-run recursive orderings are able o idenify srucure. Firs I define a class of srucures. I hen show when VARs wih differenced daa are idenified by cerain long-run recursive orderings, some srucural impulse response funcions can be obained for economies from his general class. Definiion 1: A long-run parially recursive srucure consiss of: (i) a srucural sysem wih a block-recursive marix of long-run mulipliers; (ii) equaions in one of hese blocks can be ordered recursively; (iii) uncorrelaed srucural shocks. 7
The model of Secion 2 is modified o consider a long-run parially recursive srucure. Le he vecor of n variables in he srucural sysem be broken ino hree groups: The firs n 1 variables are in y, he nex n 2 variables are in y2, he final n 3 variables are in y3 and n 1 + n 2 + n 3 = n. Le he vecor of srucural shocks be similarly pariioned so ha ε has he firs n 1 shocks, ε 2 holds he nex n 2 shocks and ε 3 conains he final n 3 shocks. Once again he covariance marix for uncorrelaed srucural shocks is normalized o he ideniy marix. 10 Hence, he srucural MAR is y y y 2 3 θ 11( L) θ12( L) θ13( L) ε = θ21( L) θ 22( L) θ23( L) ε 31 L 32 L 33 L θ ( ) θ ( ) θ ( ) ε 2 3 (22) where θ ij ( L) = θ ij0 + θ ij1 L+ θ ij2l 2 +..., wih 2 ijk an n i n j marix for all non-negaive ineger k, i=1,2,3 and j=1,2,3. The long-run muliplier marix for shocks from equaion (22) can generally be wrien as θ θ θ θ( 1) = θ θ θ θ θ θ 11 12 13 21 22 23 31 32 33 (23) where θ = θ ( 1). 11 ij ij Assumpion 1: The srucure is long-run parially recursive from he following resricions: θ 12 = 0 12, θ 13 = 0 13, θ 23 = 0 23 where 0 ij is an n i n j marix of zeros: and 2 22 is a lower riangular n 2 n 2 marix. θ 0 0 11 12 13 θ( 1) = θ 21 θ 22 023, (24) θ 31 θ 32 θ 33 8
Placing he block of recursive equaions in he inerior of his block-recursive sysem yields a fairly general long-run parially recursive form ha can laer be used o discuss ineresing special cases. All remaining parameer marices in equaion (24) are unconsrained. If boh 2 11 and 2 33 were also lower riangular, he analysis in Secion 2 shows ha he appropriae long-run recursive ordering would idenify dynamic responses associaed wih each srucural shock. Proposiion 1: If he economic srucure is long-run parially recursive, each srucural shock has a permanen effec on a leas one variable and he daa are no coinegraed, hen VARs wih differenced daa are able o recover some srucural impulse responses wih longrun recursive idenifying resricions, as long as he recursive model is consisen wih he underlying srucure. Consider he following linear combinaions of srucural disurbances: λ = θ ε 11 where each linear combinaion is wrien as: n1 i ij j λ = θ11ε for i j= 1 = 1, 2,..., n1 (25) where superscrips indicae paricular elemens in he vecors g and 8 and in he marix 2 11. Linear projecion equaions wih 8 are used o map his srucure ino he long-run recursive empirical model. Firs projec he second elemen in 8 ono he firs elemen: 2 1 21 1 1 2 λ = P λ + ρ 9
where 21 is he projecion coefficien and 2 P 1 ρ is he projecion error. Coninue projecing each variable in 8 ono all preceding variables: 3 1 31 1 1 32 2 1 3 λ1 = P λ + P λ + ρ.. n n 1 1 1 1 n, n 1 n 1 1 1 λ = P λ +... + P λ + ρ n 1 1 1 1 1 1 where projecion errors are indexed by he dependen variable and projecion coefficiens are indexed by he dependen variable and he explanaory variable, respecively. Along wih he ideniy which ses λ 1 1 equal o iself, his sysem of projecion equaions can be wrien as: or more convenienly as: 1 0 0. 0 1 1 λ 1 21 λ1 1 0 0 2 1 31 1 32 1 1 2. P λ ρ 1. 0 3 1 1 3 P P λ = ρ....... n11 n1, n1 1 1.. 1 1 1 1 P P n n λ ρ1 P 1 λ = ρ, (26) where P 1 is he lower riangular marix of projecion coefficiens and D is he vecor of projecion errors. The covariance marix for D is given by Eρ ρ = D 1 where D 1 is a diagonal marix by consrucion. 1/ 2 Using D 1 as he square-roo of he diagonal covariance marix, he vecor of projecion errors can be wrien as: 1/ 2 ρ = D 1 u (27) where u has an ideniy covariance marix, Eu u = I, wih I j an n j n j ideniy marix. Equaions (26) 1 and (27) combine o yield he following expression for 8 : 1 1 2 λ = P 1 D1 / u (28) 10
where P1 1 D1 1/ 2, being he produc of a lower riangular marix and a diagonal marix, is lower riangular. Nex define λ = θ ε 3 33 3, (29) and using a sequence of recursive linear projecions similar o ha which was used wih 8, obain P3 λ 3 = ρ3 where P 3 is lower riangular wih ones along is main diagonal and he covariance marix for D 3 is given by Eρ ρ = D 3 3 3 1/ 2 ρ 3 = D 3 u 3 where D 3 is a diagonal marix. Projecion errors can hen be wrien as where u 3 has an ideniy covariance marix: Eu u = I and P3 1 D3 1/ 2 is a lower riangular marix. ε From (25) and (28) solve for ε = θ P D / u 3 33 1 3 1 1 3 2 3 1 1 2 λ 3 = P 3 D3 / u 3 = θ P D / u 11 1 1 1 1 1 2 3 3 3, hen inser boh of hese expressions ino (22):. Hence,, from (29) and (30) solve for y 11 L 11 1 P1 1 D1 1 / 2 12 L 13 L 33 1 P3 1 D 1 / θ ( ) θ θ ( ) θ ( ) θ 3 2 u y L P D L L P D. (31) 2 21 11 1 1 1 1 1 2 22 23 33 1 3 1 1 3 2 / / = θ ( ) θ θ ( ) θ ( ) θ ε y3 31 L 11 1 P1 1 D1 1 2 32 L 33 L 33 1 P3 1 D 1 3 2 2 / / θ ( ) θ θ ( ) θ ( ) θ u3 The covariance marix for (u, g 2, u 3 ) is by consrucion he ideniy marix. Se L=1 in he marix lag polynomial of equaion (31), and use he resricions from Assumpion 1 o obain he marix of long-run mulipliers: P1 1 D1 1 / 2 012 0 13 P D. (32) 21 11 1 1 1 1 1 2 / θ θ θ22 023 31 11 1 P1 1 D1 1 / 2 32 P3 1 D 1 / 3 2 θ θ θ Since each block along he main diagonal is a lower riangular marix and each block above he main diagonal consiss of zeros, equaion (32) is he marix of long-run mulipliers for a paricular long-run recursive model, and herefore, (31) is he moving average represenaion for his long-run recursive (30) 11
model. Thus, when an economy has he long-run parially recursive srucure given by Assumpion 1, a long-run recursive empirical model wih he ( y, y, y ) 1 2 3 ordering will idenify srucural responses for g 2. The oher MARs from his recursive model are linear combinaions of srucural effecs: The dynamic effecs obained for u are a funcion of he srucural responses o g and he effecs for u 3 are a funcion of he srucural responses o g 3. 4. Examples Much empirical macroeconomic research idenifies VAR models by using long-run resricions. 12 I employ he economic srucure developed in Amed e al. (1993) o illusrae he idenificaion resuls from Secion 3. One purpose of his secion is o provide specific examples of long-run parially recursive srucures. A second purpose is o presen some of he ways he Secion 3 resul can be uilized by empirical researchers. This discussion raises an imporan poin: Economiss wih differen views abou he appropriae economic heory, may neverheless agree ha a paricular long-run recursive empirical model is informaive abou some srucural issues. Amed e al. consruc a 6 variable model ha includes growh in labor hours for he home counry ( n h ), he home counry's oupu growh ( y h ), he foreign counry's oupu growh ( yf ), p p he difference beween privae oupu growh raes beween he wo counries ( yh yf ), he change in he log of he erms of rade ( q f ), and he difference in growh raes of nominal money beween he wo counries ( m h m f ). These variables are driven by 6 srucural disurbances wih shocks arising from labor supply in he home counry ( τ h ), world-wide echnology ( τ ), labor supply in he 12
* * f h foreign counry ( τ f ), he cross-counry difference in exogenous shocks o fiscal policy ( η η ), he difference in preference shocks ( ε f ε h ), and he cross-counry difference in exogenous shocks o money supply ( ν * * h ν f ). 13 Amed e al. idenify heir model by assuming srucural shocks are uncorrelaed and ha he long-run muliplier marix is compleely recursive: 14 n Ψ h 11 0 0 0 0 0 τ h yh Ψ21 Ψ22 0 0 0 0 τ yf Ψ Ψ Ψ f. (33) p p = 31 32 33 0 0 0 τ yh y f Ψ Ψ Ψ Ψ f h 41 42 43 44 0 0 * * η η qf Ψ51 Ψ52 Ψ53 Ψ54 Ψ55 0 ε f ε h m h m * * f Ψ Ψ Ψ Ψ Ψ Ψ νh ν 61 62 63 64 65 66 f This model is moivaed by a se of plausible srucural assumpions. Amed e al. assume ha long-run movemens in hours worked for a paricular counry are caused exclusively by shocks o ha counry's labor supply and ha a counry's oupu is driven exclusively by domesic labor supply shocks and worldwide echnology shocks in he long run. These resricions yield he firs hree equaions in (33). 15 Then hey assume fiscal expendiures affec he composiion of oupu beween privae and governmen spending, bu can have no effec on aggregae oupu or hours worked in he long run. Their fourh equaion allows shocks o home counry labor supply, foreign labor supply, echnology and fiscal policy o affec privae spending. The fifh equaion permis he erms of rade o respond in he long run o all he srucural shocks from he firs four equaions and also o preference shocks. Preference shocks are no, however, allowed o affec privae spending, aggregae spending, hours worked or fiscal policy in he long run. The sixh equaion les he relaive growh of he money supply respond o all shocks in he model. This specificaion is based on he assumpion ha moneary policy may reac o a wide variey of macroeconomic informaion. Money is assumed o be long-run neural, and herefore money supply 13
shocks have no long-run impac on any of he oher variables in his sysem. Parameers for his purely recursive long-run srucural model are esimaed by means of Blanchard and Quah s (1989) echnique. Many alernaive srucural assumpions o hose employed by Amed e al. exis ha some economiss migh consider equally plausible. This poin is no mean o derac from he imporance of heir research, bu is insead a reflecion on he field of macroeconomics which is currenly wihou a unifying paradigm and undergoing rapid ransformaion. I illusrae he idenificaion resuls from Secion 3 by proposing wo reasonable modificaions o heir assumpions: Assumpion A: Permanen echnological improvemen reduces labor supply by a wealh effec; Assumpion B: Fiscal policymakers reac o he same informaion as cenral bankers. Assumpion A is quie plausible from economic heory. Assumpion B is moivaed by he fac ha moneary and fiscal auhoriies have similar policy goals and frequenly aemp o coordinae policies. If boh assumpions are added o Amed e al., he long-run srucure becomes: n Ψ Ψ h 11 12 0 0 0 0 τ h yh Ψ21 Ψ22 0 0 0 0 τ yf Ψ Ψ Ψ f. (34) p p = 31 32 33 0 0 0 τ yh y f Ψ Ψ Ψ Ψ Ψ Ψ * * f h 41 42 43 44 45 46 η η qf Ψ51 Ψ52 Ψ53 Ψ54 Ψ55 0 ε f ε h m h m f Ψ61 Ψ62 Ψ63 Ψ64 Ψ65 Ψ 66 ν * h ν * f These addiional assumpions yield a 3 block sysem in which he long-run srucural muliplier marix is block recursive. The firs block conains he firs wo srucural equaions, he second block has he foreign oupu growh equaion and he final block is composed of he las hree equaions. Since here is only one way o order a single iem, he equaion in he middle block has a unique recursive ordering. 14
Given he srucure in (34), consider wha happens when Amed e al. esimae a model based on he long-run muliplier marix in (33). While heir empirical model would now be mispecified, resuls from Secion 3 show ha heir long-run recursive ordering would sill idenify srucural effecs for he foreign labor supply shock. 16 The MAR for he firs wo shocks in heir recursive ordering would, however, confound he effecs of echnology and home counry labor supply shocks because of Assumpion A. Similarly, he effecs for he las hree shocks in heir empirical model will confound he dynamic effecs of shocks o fiscal policy, preferences and money supply because of Assumpion B. Clearly, an appropriae long-run recursive ordering mus be employed o idenify srucural effecs. In he general case of Secion 3, he middle n 2 variables mus be ordered in a paricular sequence o obain srucural resuls. However, he firs n 1 variables can be arbirarily ordered and so can he las n 3 variables. For example, given Assumpions A and B, Amed e al. could use a long-run recursive model in which hey inerchange he firs 2 variables and/or selec some oher ordering of he las 3 variables o idenify he effecs of foreign labor supply shocks. In general, here are n 1!n 3! differen orderings for he sysem ha will idenify srucural effecs for he n 2 shocks in he inerior block. Special cases of he general specificaion of long-run parially recursive srucure from Secion 3 illusrae how he subse of recursive equaions may occur in he firs or las block of a sysem. Suppose only wo ses of block-recursive srucural equaions exis and he second group consiss of recursively ordered equaions. This amouns o seing n 3 =0 in he general case from Secion 3. Under his assumpion, an appropriae long-run recursive ordering will idenify he srucural MAR for shocks o he las n 2 equaions. The only consrain placed on he iniial n 1 equaions is ha hey mus be block recursive in he long run wih respec o he remaining n 2 equaions. For example, suppose Assumpion A, bu no Assumpion B, is added o he srucural assumpions in Amed e al. In his case, he block-recursive sysem consiss of only wo blocks. The firs block conains he firs wo equaions and he second block 15
includes he las four. Adding only Assumpion A o he srucural model of Amed e al. implies ha heir recursive ordering will idenify srucural effecs for he shocks o foreign labor supply, fiscal policy, preferences and he supply of money. The firs wo shocks from heir recursive ordering, however, would no idenify srucural responses because echnology shocks have a wealh effec. A second special case is when he block recursive sysem has wo blocks and he firs block consiss of recursively ordered srucural equaions. This example is handled by seing n 1 =0 in he general case from Secion 3. Hence, he MAR associaed wih each of he firs n 2 shocks can be idenified by an appropriae long-run recursive ordering. In his case, he remaining n 3 equaions are lef unconsrained. For example, suppose ha only Assumpion B is added o he se of assumpions in Amed e al. In his case, heir firs hree equaions would form one block and heir las hree would form he oher block. These wo blocks are block-recursive and he equaions in he firs block are equaion-by-equaion recursive in he long run. Therefore, he recursive ordering used by Amed e al. would idenify he effecs of shocks o home counry labor supply, echnology and foreign labor supply. The las hree shocks from heir empirical model would no idenify srucural responses because moneary and fiscal policymakers respond o a common se of informaion. I is also worh noing ha he basic resul from Secion 3 is easily exended o a long-run blockrecursive sysem wih more han hree blocks in which muliple blocks consis of long-run recursive equaions. Each block can be handled individually using he mehods in Secion 3. Hence, long-run recursive orderings consisen wih his more complex block-recursive srucure would idenify srucural responses for each block of long-run recursive equaions. 5. Srucural Inference using Models of Coinegraion 16
While VARs wih differenced daa are quie common, even more economic research is conduced wih models of coinegraed imes series. I is, herefore, imporan o deermine if he previous resuls exend o his class of models. To address his issue, I use he riangular represenaion 17 of Philips (1991) which wries he coinegraed sysem as a VAR wih inegraed and saionary variables. Le s be linear combinaions of ime series ha are saionary o allow for all coinegraing relaionships. 18 Assuming deerminisic elemens are removed from all variables, augmen he model from Secion 3 wih variables s and shocks µ, wih he number of hese addiional shocks maching he number of saionary variables: y θ11( L) θ12( L) θ13( L) θ14( L) ε y 2 21 L 22 L 23 L 24 L 2 = θ ( ) θ ( ) θ ( ) θ ( ) ε. (35) y 3 θ31( L) θ32( L) θ33( L) θ34( L) ε3 s θ41( L) θ42( L) θ43( L) θ44( L) µ These µ shocks are assumed o have no permanen effec on any variables. This is equivalen o having he number of independen permanen shocks mach he number of differenced series in (35). The basis for his assumpion is he common sochasic rends represenaion for coinegraed sysems developed by Sock and Wason (1988). Transiory shocks are assumed uncorrelaed wih he permanen shocks, and his assumpion is crucial for idenifying permanen shocks. The ransiory shocks may, however, be correlaed wih one anoher. Since ransiory shocks will have a mos a emporary effec on inegraed variables, he following resricions mus hold: 2 14 (1)=0 14, 2 24 (1)=0 24 and 2 34 (1)=0 34. These resricions provide a srucural basis for he riangular represenaion. The sum of parameers given by 2 41 (1), 2 42 (1), 2 43 (1) and 2 44 (1) are unconsrained because saionary variables, by definiion, can no experience a permanen response o any shock. Assume he srucure is long-run parially recursive wih he form given in Assumpion 1. 17
Precisely he same mehod from Secion 3 can be used here o map he long-run parially recursive srucure from equaion (35) ino he long-run recursive model: y y y s 2 3 11 11 1 1 1 1 1 / 2 12 13 33 1 3 1 1 / θ ( L) θ P D θ ( L) θ ( L) θ P D3 2 θ14( L) Ω u 21 L 11 1 P1 1 D1 1 2 22 L 23 L 33 1 P3 1 D 1 3 2 / / θ ( ) θ θ ( ) θ ( ) θ θ24( L) Ωε = 31 L 11 1 θ ( ) θ P1 1 D1 1 / 2 θ32 ( L) θ33( L) θ33 1 3 1 1 / 3 2 P D θ34( L) Ωu 41 11 1 1 1 1 1 / 2 42 43 33 1 L P D L L P3 1 D 1 / θ ( ) θ θ ( ) θ ( ) θ 3 2 θ 44( L) Ω u along wih an arbirary se of idenificaion resricions, µ =Su 4, which map he ransiory srucural shocks ino a se of orhogonal shocks u 4, each wih uni variance and no correlaion wih he oher shocks in he model. Seing L=1 in (36) shows ha he long-run muliplier marix for he effecs of 2 3 4, (36) permanen shocks (u, g 2, u 3 ) on he inegraed variables ( y, y, y ) is given by he lower riangular 2 3 marix in equaion (32). Since (36) is he MAR obained from his paricular ordering of long-run mulipliers, he effecs of g 2 are idenified wih his empirical model. Hence, all he resuls for long-run parially recursive srucures found in Secions 3 and 4 naurally exend o models of coinegraion. While he resricions associaed wih u 4 would ypically come from conemporaneous idenificaion assumpions, S=I 4 is also a possibiliy. In oher words, excep for he assumpion ha he ransiory srucural disurbances are uncorrelaed wih he permanen srucural disurbances, assumpions abou he ransiory shocks are irrelevan. 6. Concluding Commens This paper proves ha a long-run parially recursive srucure is sufficien for long-run recursive orderings o idenify some srucural effecs, as long as he ordering is consisen wih he underlying srucure. One can also show ha wo-block versions of he long-run parially recursive srucure are necessary for he iniial block or he final block of shocks from a long-run recursive ordering o yield 18
srucural responses. These resuls follow from a simple exension of necessary condiions found in he proof from Secion 5 of Keaing (1996). To be specific, assume he series are inegraed and replace he conemporaneous srucural parameer marix and he conemporaneous idenificaion resricions wih he long-run parially recursive srucural parameer marix and he long-run recursive idenificaion resricions, respecively. 19 A key implicaion of his paper is ha long-run recursive empirical models yield srucural responses for a class of economic sysems much larger han he se of srucures which are fully recursive in he long run. A ypical applicaion is when heory provides a paricular long-run parially recursive srucure, and an economis esimaes an appropriae long-run recursive empirical model o address specific economic quesions. In a second applicaion, exising empirical sudies ha use long-run recursive models can be examined under alernaive heoreical assumpions. If hese alernaive assumpions recas he srucure as a long-run parially recursive sysem, hen paricular long-run recursive models are able o provide informaion abou he economy. If he empirical model happens o be consisen wih his alernaive srucure, i will sill obain specific srucural informaion under his differen se of assumpions. Researchers who wan o aach srucural inerpreaions o empirical models of inegraed or coinegraed ime series may find hese applicaions useful and may develop addiional ways o use his paper s resuls abou idenificaion. 19
Noes 1. See Sroz and Wold (1960) for classic references o his lieraure. 2. I should be noed, however, ha economic heory ofen generaes plausible long-run recursive srucures. For example, following Blanchard and Quah (1989) much of his lieraure uses he assumpion ha aggregae demand has a long-run neural effec on oupu, and his assumpion ypically yields a longrun recursive economic srucure. Long-run recursive srucures have also been developed by Bullard and Keaing (1995), King and Wason (1997) and Robers (1993) o address he superneuraliy of money. 3. Their resuls are derived for long-run srucures ha are no necessarily recursive. Some of heir resuls will also apply o conemporaneous srucures. 4. The srucure in Shapiro and Wason (1988) has a lower block riangular marix of long-run mulipliers. The resuls shown here hold for heir sysem and for more complicaed examples from he class of long-run parially recursive srucures. 5. Equaion (2) is consruced by inegraing equaion (1). 6. Lippi and Reichlin (1993) and Blanchard and Quah (1993) presen alernaive views on inveribiliy. Non-inverible srucures can be consruced from he VAR s fundamenal represenaion. 7. Equaion (4) assumes R lags will adequaely approximae he infinie order srucural MAR in (1). 8. Hamilon (1994,p.91) proves ha he riangular facorizaion is unique. 9. Bernanke (1986), Blanchard and Wason (1986) and Sims (1986) are foundaional works on srucural VAR idenificaion wih conemporaneous resricions. Keaing (1990) shows ha exclusion resricions on conemporaneous coefficiens are generally invalid if agens are forward looking and have raional expecaions. Wes (1990) also uses raional expecaions in a srucural VAR. Gali (1992) combines conemporaneous and long-run resricions. 10. Technically speaking, if he covariance marix of srucural shocks is block diagonal and if each shock in g 2 is uncorrelaed wih every oher shock, hen he resuls will sill go hrough. This assumpion allows shocks in he firs block o be correlaed wih one anoher and shocks in he hird block o be correlaed wih one anoher, bu no correlaion beween shocks from differen blocks. 11. In conras, Keaing (1996) uses 2 ij noaion o denoe conemporaneous srucural parameers. 12. See Amed, Ickes, Wang and Yoo (1993), Bullard and Keaing (1995), Gamber and Jouz (1993), Keaing and Nye (1998) and Lasrapes and Selgin (1994) and heir references o addiional research. 13. The aserisks indicae ha srucural parameers muliply he moneary and fiscal shocks in heir model. These parameers are irrelevan for he purposes of his paper, and herefore, o simplify noaion hey are no deal wih explicily. 20
14. This way of represening a long-run srucure comes from aking firs differences of equaion (2) and ignoring saionary effecs. 15. These assumpions imply Q 31 =0. Since Amed e al. do no impose any of he overidenifying resricions generaed by heir assumpions, I will ignore overidenificaion issues. 16. Having Q 56 = 0 in (34) does no imply anyhing else is srucural from any recursive ordering. 17. The Appendix shows ha he vecor error correcion model can also be used o show his resul. 18. Inherenly saionary variables yield a paricularly simple coinegraing vecor. 19. Equaion (20) from Secion 3 in his paper can be used o raionalize hese subsiuions. 21
Appendix: Srucural Inference using Vecor Error Correcions Models The vecor error correcions model (VECM) of Engle and Granger (1987) may be he mos popular model for a sysem of coinegraed ime series. In a VECM, dependen variables are differenced enough imes o become saionary and he se of regressors includes lagged dependen variables along wih saionary linear combinaions of he series. Assume X is a vecor of m variables ha are inegraed of order 1. This order of inegraion is arbirary. Le β X be he saionary linear combinaions of X. If here are m-n coinegraing vecors, hen $ is an m (m-n) marix where 0<n<m. Jus like he riangular represenaion, he number of ransiory shocks is equal o he number of coinegraing vecors, and hus he number of sochasic rends in he sysem mus equal n. King, Plosser, Sock and Wason (1991) developed a mehod o idenify permanen shocks in a VECM based on a recursive ordering of permanen componens. They wrie he moving average represenaion for a coinegraed sysem as VECM and η X = Γ( L)η, where (L) is obained by invering he P η =, where 0 P are he permanen shocks and 0 T are he ransiory shocks. Permanen T η shocks are assumed o be uncorrelaed wih one anoher. Transiory shocks are no necessarily idenified, alhough he assumpion ha ransiory shocks are uncorrelaed wih permanen shocks is crucial for idenifying permanen shocks. The long-run effecs of permanen shocks are consrained by he coinegraing relaionships and by he economerician s inerpreaion of he sources of he permanen ~ componens. King e al. show ha he long-run mulipliers can be wrien as Γ( 1) = [ Aπ, 0m, m n ] where ~ A is an m n marix of coefficiens derived from he parameers in $ and B is a full rank n n marix. The moving average represenaion can also be wrien as * X = Γ( 1) η + Γ ( L) η o separae permanen effecs from ransiory dynamics. From previous expressions, permanen effecs are 22
given by ~ Γ( 1) η = Aπη P. King e al. assume B is lower riangular, and show how o obain B from he Cholesky decomposiion of a marix consruced from parameers of he VECM. The permanen P P P componens from heir recursive model can be wrien as e = πη where elemens of e are associaed wih permanen shocks o variables. In heir 6 variable model, King e al. find hree coinegraing vecors which implies ha he sysem has hree independen permanen shocks. They associae he permanen componens wih real oupu (calling his a balanced growh shock), inflaion and he real ineres rae. Esimaion of B allows hem o decompose he permanen componens for he hree variables ino hree orhogonal permanen shocks 0 P. If he economic srucure happens o be long-run recursive and he correc ordering of variables is used, hen clearly each permanen shock will idenify srucural effecs. Le hese permanen movemens in variables be relaed o srucural disurbances by he following sysem of equaions: e P = θ( 1) ε. Suppose ha his srucural sysem is parially recursive in he long run wih he paricular form given in Assumpion 1, implying he srucure is given as e e e P P 2 P 3 θ 0 0 = θ θ 0 θ θ θ 11 12 13 21 22 23 31 32 33 wih 2 22 a lower riangular marix. To conform wih hese srucural blocks, he long-run recursive empirical model can be wrien as: P e P π11 012 013 η P P e2 = 21 22 0 23 2. π π η P P e 3 π31 π32 π33 η3 Using exacly he same ransformaions found in Secion 3 i is easy o map he long-run parially ε ε ε recursive srucure ino his recursive model, and find ha B 22 =2 22, B 32 =2 32 and idenifies he effecs of g 2. 2 3 P η 2 = g 2. Hence P η 2 23
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