❊♥s❛✐♦s ❊❝♦♥ô♠✐❝♦s
❊s❝♦❧❛ ❞❡
Pós✲●r❛❞✉❛çã♦
❡♠ ❊❝♦♥♦♠✐❛
❞❛ ❋✉♥❞❛çã♦
●❡t✉❧✐♦ ❱❛r❣❛s
◆◦ ✸✸✺ ■❙❙◆ ✵✶✵✹✲✽✾✶✵
❈♦♠♠♦♥ ❈②❝❧❡s ❛♥❞ t❤❡ ■♠♣♦rt❛♥❝❡ ♦❢ ❚r❛♥✲
s✐t♦r② ❙❤♦❝❦s t♦ ▼❛❝r♦❡❝♦♥♦♠✐❝ ❆❣❣r❡❣❛t❡s
✭❘❡✈✐s❡❞ ❱❡rs✐♦♥✮
❋❛rs❤✐❞ ❱❛❤✐❞✱ ❏♦ã♦ ❱✐❝t♦r ■ss❧❡r
❖s ❛rt✐❣♦s ♣✉❜❧✐❝❛❞♦s sã♦ ❞❡ ✐♥t❡✐r❛ r❡s♣♦♥s❛❜✐❧✐❞❛❞❡ ❞❡ s❡✉s ❛✉t♦r❡s✳ ❆s
♦♣✐♥✐õ❡s ♥❡❧❡s ❡♠✐t✐❞❛s ♥ã♦ ❡①♣r✐♠❡♠✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ ♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛
❋✉♥❞❛çã♦ ●❡t✉❧✐♦ ❱❛r❣❛s✳
❊❙❈❖▲❆ ❉❊ PÓ❙✲●❘❆❉❯❆➬➹❖ ❊▼ ❊❈❖◆❖▼■❆ ❉✐r❡t♦r ●❡r❛❧✿ ❘❡♥❛t♦ ❋r❛❣❡❧❧✐ ❈❛r❞♦s♦
❉✐r❡t♦r ❞❡ ❊♥s✐♥♦✿ ▲✉✐s ❍❡♥r✐q✉❡ ❇❡rt♦❧✐♥♦ ❇r❛✐❞♦ ❉✐r❡t♦r ❞❡ P❡sq✉✐s❛✿ ❏♦ã♦ ❱✐❝t♦r ■ss❧❡r
❉✐r❡t♦r ❞❡ P✉❜❧✐❝❛çõ❡s ❈✐❡♥tí✜❝❛s✿ ❘✐❝❛r❞♦ ❞❡ ❖❧✐✈❡✐r❛ ❈❛✈❛❧❝❛♥t✐
❱❛❤✐❞✱ ❋❛rs❤✐❞
❈♦♠♠♦♥ ❈②❝❧❡s ❛♥❞ t❤❡ ■♠♣♦rt❛♥❝❡ ♦❢ ❚r❛♥s✐t♦r② ❙❤♦❝❦s t♦ ▼❛❝r♦❡❝♦♥♦♠✐❝ ❆❣❣r❡❣❛t❡s ✭❘❡✈✐s❡❞ ❱❡rs✐♦♥✮✴ ❋❛rs❤✐❞ ❱❛❤✐❞✱ ❏♦ã♦ ❱✐❝t♦r ■ss❧❡r ✕ ❘✐♦ ❞❡ ❏❛♥❡✐r♦ ✿ ❋●❱✱❊P●❊✱ ✷✵✶✵
✭❊♥s❛✐♦s ❊❝♦♥ô♠✐❝♦s❀ ✸✸✺✮ ■♥❝❧✉✐ ❜✐❜❧✐♦❣r❛❢✐❛✳
Common Cycles and the Import ance of
Transitory Shocks t o Macroeconomic Aggregates
¤
Jo~
ao Vict or Issler
Graduat e School of Economics - EPGE
Get ulio Vargas Foundat ion
Praia de Bot afogo 190 s. 1125
Rio de Janeiro, RJ 22253-900, Brazil
jissler@fgv.br
Farshid Vahid
Texas A& M University
Depart ment of Economics
College Station, T X 77843-4228, USA
fv@econ4.t amu.edu
T his version: Sept ember 1998
A bst r act
Alt hough t here has been subst ant ial research using long-run co-movement (coint egrat ion) rest rict ions in t he empirical macroeconomics lit erat ure, lit -t le or no work has been done inves-t iga-t ing -the exis-t ence of shor-t -run co-movement (common cycles) rest rictions and discussing t heir implicat ions.
¤We grat efully acknowledge comment s from Heat her Anderson, Wout er den Haan, Robert
In this paper we ¯ rst invest igat e the existence of common cycles in a aggre-gate data set comprising per-capit a out put , consumption, and invest ment. Lat er we discuss their usefulness in measuring t he relat ive import ance of t ransit ory shocks. We show that , t aking int o account common-cycle re-st rict ions, t ransit ory shocks are more important than previously thought at business-cycle horizons. T he cent ral argument relies on e± ciency gains from imposing these short -run rest rict ions on the est imat ion of t he dynamic model. Finally, we discuss how t he evidence here and elsewhere can be in-t erprein-t ed in-to supporin-t in-t he view in-thain-t nominal shocks may be imporin-t anin-t in in-the short run.
1. I nt r oduct i on
It is a well known stylized fact in macroeconomics t hat economic dat a display co-movement . For example, Lucas(1977, sect ion 2) report s t hat out put co-movement s across broadly de¯ ned sect ors have high coherence. On t he ot her hand, K osobud and K lein(1961) document t hat aggregat e consumpt ion, invest ment and out put follow balanced growt h pat hs. While t he ¯ rst observat ion is a st at ement about short -run co-movement , which imposes rest rict ions on t ransit ional dynamics of sect oral out put s, t he second is a st at ement about long-run co-movement , impos-ing t he rest rict ion t hat macroeconomic aggregat es cannot drift apart over t ime. T hese two types of rest rict ions play an import ant role in det ermining t he dynamic behavior of macroeconomic t ime series.
Gal¶i(1996), who uses long-run rest rict ions t o ident ify \ product ivity" shocks and examine how t he predict ions of a class of Real-Business-Cycle models ¯ t t he dat a. Alt hough t he use of low-frequency (coint egrat ion) rest rict ions t o decompose economic series int o t rends and cycles is widely used, t his is by no means t he only type of rest rict ions t hat can be employed t o t hat end. High-frequency rest rict ions (common cycles) can also be used in conjunct ion wit h coint egrat ion rest rict ions, whenever t he lat t er exist . T his was init ially shown by Vahid and Engle(1993), who proposed t he use of t hecommon trends and common cycles met hod, lat er applied t o a sect oral out put dat a set by Engle and Issler(1995); see also Engle and K ozicki(1993) for a broader view of t hese \ common feat ures."
For a given dat a set , t he joint use of common-t rend and common-cycle re-st rict ions t o ident ify permanent and t ransit ory shocks has a clear advant age over t he use of common-t rend rest rict ions alone. First , t here is t he econometric is-sue of relat ive e± ciency. Obviously, if common-cycle rest rict ions are correct ly imposed, est imates of t he dynamic model (usually a Vect or Aut oregression) are more precise, leading t o a more precise measurement of t he relative import ance of permanent and t ransit ory shocks. Indeed, if two series have a common cycle t heir impulse-response funct ions are exact ly colinear (Vahid and Engle(1997)). T hus, variance-decomposit ion and impulse-response calculat ions will be based on a re-duced set of paramet ers. Second, t here is t he issue of t he horizon when measuring t he relat ive import ance of shocks. For bot h met hods, t he relat ive import ance of permanent and t ransit ory shocks should not di®er much for long horizons, since bot h impose t he same long-run rest rict ions. However, t hey have t he pot ent ial t o be di®erent for short horizons, because rest rict ions on short -run dynamics are imposed by only one of t hem.
T he goal of t his paper is t o examine whet her U.S. per-capit a out put , consump-t ion, and invesconsump-t menconsump-t share common cycles using consump-t hecommon trends and common cycles met hod discussed in Vahid and Engle(1993). T here, long-run co-movement is charact erized as common st ochastic t rends and short -run co-movement is char-act erized as common cyclical component s t hat are synchronized in phase but may have di®erent amplit udes. Using e± cient est imat es of a vect or aut oregres-sive model which is rest rict ed t o produce common t rends and common cycles, we calculat e t he relat ive import ance of permanent and t ransit ory shocks. We lat er con¯ rm t he e± ciency gains for our dat a set in an out -of-sample forecast ing exercise.
Invest igat ing t he exist ence of common cycles is int erest ing in it s own right , given t hat t his is a t heoret ical implicat ion of several dynamic macroeconomic mod-els. Examples of t hese models are provided below. Using a more e± cient t rend-cycledecomposit ion of t hedat a weareable t o answer more precisely a key quest ion in macroeconomics, considered, among ot hers, by Nelson and Plosser(1982), Wat -son(1986), Campbell and Mankiw(1987), Cochrane(1988, 1994), K ing et al.(1991), and Gal¶i(1996): what is t he relat ive import ance of permanent and t ransit ory shocks in explaining t he variat ion of macroeconomic dat a?
Our empirical ¯ ndings con¯ rm t he presence of common cycles in t he dat a. T he variance decomposit ion result s show t hat , for out put and invest ment , t ran-sit ory shocks are import ant in explaining t heir variat ion. On t he ot her hand, permanent shocks are t he most import ant source of variat ion for consumpt ion, whose behavior is close to a martingale (Hall(1978) and Flavin(1981)). Cont rast -ing our variance decomposit ion result s wit h t hose of K -ing et al.(1991), who only considered coint egrat ion rest rict ions t o ident ify permanent and t ransit ory shocks, illust rat es t hat ignoring common-cycle rest rict ions leads t o non-t rivial di®erences in t he relat ive import ance of t ransitory shocks. Indeed, we ¯ nd t ransit ory shocks t o be more import ant t han previous research has found. T his is t rue especially at business-cycle horizons, illust rat ing t he advant age of t he met hod used here. T hese ¯ ndings are consist ent wit h t he recent result s in Gal¶i(1996) - showing t hat nominal (t ransit ory) shocks are crit ical in explaining several feat ures of aggregat e dat a for G-7 count ries, and of den Haan(1996) - showing t hat demand shocks may be relevant for explaining t he condit ional correlat ions of out put and prices and of hours and real wages in t he short run.
t est ing for common t rends and cycles as well as t he est imat ion of dynamic sys-t ems under long- and shorsys-t -run co-movemensys-t ressys-t ricsys-t ions. A desys-t ailed explanasys-t ion of t he met hodology used is present ed in t he Appendix. Sect ion 4 present s empir-ical result s and Sect ion 5 concludes.
2. T heor y and Test abl e I m pl i cat i ons
Common t rends and common cycles appear in t he macroeconomics lit erat ure in several t heoret ical models. Here we discuss a few examples. In t he dynamic st ochast ic general equilibrium model of K ing, Plosser, and Rebelo(1988), out put , consumpt ion and invest ment have a common trend1and a common cycle as a re-sult of t he opt imizing behavior of t he represent at ive agent . Coint egrat ion comes from having a common forcing variable (product ivity) and a common cycle arises from t he fact t hat t he t ransit ional dynamics of t he syst em is a (linear) funct ion of a unique fact or - t he deviat ion of t he capit al st ock from it s st eady st at e value. Alt hough t hese result s are obt ained under log-ut ility, full-depreciat ion of t he cap-it al st ock, and Cobb-Douglas t echnology, t hey can be generalized for a variety of paramet erizat ions under a quadrat ic approximat ion of t he value funct ion. T he closed-form solut ions for t he logarithms of out put , consumpt ion and invest ment are respect ively:
log (Yt) = log (Xtp) + y + ¼ykkbt
log (Ct) = log (Xtp) + c + ¼ckkbt
log (It) = log (Xtp) + i + ¼i kkbt; (2.1)
where log (Xtp) = ¹ + log (X p t ¡ 1) + ²
p
t is t he random-walk product ivity process in
product ion, y, c and i are t he st eady-st ate values of log (Yt=Xtp); log (Ct=Xtp),
and log (It=Xtp) respect ively, and¼j k; j = y; c; i is t he elast icity of variable j wit h
respect t o deviat ions of t he capital st ock from it s st at ionary value
³ b
kt
´
2. From
1T he same result is t rue using t he endogenous growt h model of Romer(1986). T his makes
t hese two models observat ionally equivalent in coint egrat ing t est s.
2As not ed by K ing, Plosser and Rebelo(1988), t he above t heoret ical model is t oo simplist ic
t o be t aken as a full charact erizat ion of t he dat a-generat ing process, since it s only source of randomness is t he product ivity shock, making t he syst em in (2.1) st ochast ically singular. T here is an imbedded ident ity in t his syst em:
(¼i k¡ ¼ck) (log (Yt) ¡ y) = (¼i k ¡ ¼y k) (log (Ct) ¡ c) + (¼y k¡ ¼ck)
¡
log (It) ¡ i
(2.1) it is st raight forward t o verify t hat t hese variables have a common t rend (log (Xtp)), and a common cycle
³ b
kt
´
. T he following linear combinat ions have no t rend:
log (Yt) ¡ log (Ct)
log (Yt) ¡ log (It); (2.2)
and log (Yt), log (Ct), and log (It) are coint egrat ed in t he sense of Engle and
Granger(1987), wit h (2.2) showing two (linearly) independent coint egrat ing re-lat ionships. T he following linear combinat ions have no cycle:
¼cklog (Yt) ¡ ¼yk log (Ct)
¼i klog (Yt) ¡ ¼yk log (It); (2.3)
and log (Yt), log (Ct), and log (It) have a common cycle in t he sense of Vahid and
Engle(1993), wit h (2.3) showing two (linearly) independent cofeat ure combina-t ions.
To elaborat e more on t his issue, consider t he ¯ rst di®erences of t he logarit hm (i.e. growt h rat e) of t he syst em (2.1):
¢ log (Yt) = ²pt + ¼yk¢kbt
¢ log (Ct) = ²pt + ¼ck¢kbt
¢ log (It) = ²pt + ¼i k¢kbt: (2.4)
Given t hat ²pt is whit e noise, equat ions (2.4) show t hat all t he (short -run)
se-rial correlat ion of macroeconomic aggregat es is due t o a single common fact or
³
¢kbt
´
. T hus, \ cycles" in ¢ log (Yt), ¢ log (Ct), and ¢ log (It) are synchronized,
but amplit udes may di®er since t he¼j k's may be di®erent .
In t he class of part ial equilibrium models, Campbell(1987) shows t hat saving (St) can be writ t en as a funct ion of expect ed fut ure-income variat ionsEt(¢Yt + s):
St = ¡ 1
X
s= 1
½sEt(¢Yt+ s); (2.5)
whereEt is t he condit ional expect at ion operat or using informat ion up t o period
t, and ½is t he one-period discount fact or for fut ure income (Yt + s). Since saving
is st at ionary, consumpt ion and disposable income must coint egrat e if Yt is an
int egrat ed series.
For part ial equilibrium models in t he t radit ion of Hall(1978) and Flavin(1981), consumpt ion is a mart ingale. T hus, it s ¯ rst di®erence has no serial correlat ion even if t he (di®erenced) income process is serially correlat ed. T hus, consumpt ion and income fail t o have common cycles. However, if a proport ion of t he populat ion follows t he \ rule of t humb" of consuming t heir income ent irely in every period, Campbell and Mankiw(1989) show t hat aggregate consumpt ion and aggregat e income will have a common cycle as a result of t his myopic behavior.
Let t ing (C1t; Y1t) and (C2t; Y2t) be t he consumpt ion-income pairs of \
perma-nent income" and \ myopic" agent s respect ively, and let t ing (Ct; Yt) denot e t he
aggregat e consumpt ion-income pair, wit h ¸ = Y2t
Yt measuring t he income propor-t ion of myopic agenpropor-t s, Campbell and Mankiw show propor-t hapropor-t :
¢Ct = ¸ ¢ Yt + (1¡ ¸ ) ¹t; (2.6)
where¹t is proport ional t o t he innovat ion in Y1t. Since¹t is unpredict able, and
since ¢Yt is usually serially correlat ed, equat ion (2.6) shows t hat all t he serial
correlat ion of ¢Ct comes from ¢Yt. In t his case, t he cycles of Yt andCt are
syn-chronized. Not ice t hat t he amplit ude of t he cycle in consumpt ion is an increasing funct ion of t he import ance of myopic agent s (¸ ).
T he examples discussed above su± ce t o show t hat common t rends and com-mon cycles for macroeconomic aggregat es can be t he result of eit her opt imal or myopic behavior, and are a feat ure of part ial and general equilibrium models. T his in it self mot ivat es t heir invest igat ion. T here is an addit ional reason t o st udy t hem: common t rends and common cycles represent respect ively low- and high-frequency rest rict ions on mult ivariat e dat a set s. Whenever t hese rest rict ions are present , imposing t hem can considerably reduce t he number of est imat ed param-et ers in t ime series models, leading t o e± ciency gains in est imat ion.
3. Est im at i on and Test ing
Following K ing et al.(1991), it is useful t o not e t hat t he reduced form for a (lin-ear) syst em cont aining (t he log of ) out put , consumpt ion and invest ment is nest ed in t he general dynamic framework of Vect or Aut oregressions (VAR's). If t here are common trends in t he dat a, t he VAR has cross-equat ion rest rict ions as shown by Engle and Granger(1987). T hus, we can reduce t he number of paramet ers of t he dynamic represent at ion by est imat ing a Vect or Error-Correct ion Model (VECM) which t akes t hese rest rict ions int o account . Common cycles impose ext ra rest ric-t ions on ric-t his dynamic sysric-t em (Vahid and Engle(1993)). In ric-t his case iric-t is possible t o furt her reduce t he number of paramet ers in t he VECM. E± cient est imat ion requires est imating a rest rict ed VECM, which t akes int o account t he common cyclical dynamics present in t he ¯ rst di®erences of t he dat a; see equat ions (2.4) and (2.6) for example.
Test s for coint egrat ion are not discussed in any lengt h here, since t his lit erat ure is now well known. We employ Johansen's(1988, 1991) t echnique, which est imat es t he number of linearly independent coint egrating vect ors (r ). Test ing for common cycles amount s t o searching for independent linear combinat ions of t he (level of t he) variables t hat are random walks, t hus cycle free. T he t est t herefore is a search for linear combinat ions of t he ¯ rst di®erences of t he variables whose correlat ion wit h t he element s of t he past informat ion set in t he right -hand side of t he VECM will be zero. T his can be done by t est ing for zero canonical correlat ions between t he ¯ rst di®erences of t he variables and t he element s of t he past informat ion set . T his t est is in e®ect t est ing for cross-equat ion rest rict ions on t he paramet ers of t he VECM; see Vahid and Engle(1993).
Assuming t hat out put , consumpt ion and invest ment haveonest ochast ic t rend3, and that t he two coint egrat ing relat ionships are respect ively (log (Ct=Yt)) and
(log (It=Yt)), t he t est for common cycles and t he fully e± cient est imat ion of t he
rest rict ed VECM ent ails the following st eps:
1. Det erminep, t he required number of lags in t he VECM that adequat ely capt ures t he dynamics of t he syst em.
2. Comput e4 t he sample squared canonical correlat ions between
3T his is not t o say t hat t he common cycles t est is not applicable when variables have
de-t erminisde-t ic rade-t her de-t han sde-t ochasde-t ic de-t rends. In facde-t , common cycles are aboude-t co-movemende-t in det rended series, regardless of t he form of t he t rend. We explain t he procedure for t he case of st ochast ic t rends because it is more relevant t o t he present paper. T he case of det erminist ic t rend is st raight forward.
f¢ log (Yt); ¢ log (Ct); ¢ log (It)gand
flog (Ct ¡ 1=Yt ¡ 1); log (It ¡ 1=Yt¡ 1); ¢ log (Yt ¡ 1); ¢ log (Ct ¡ 1); ¢ log (It ¡ 1);
:::; ¢ log (Yt ¡ p); ¢ log (Ct ¡ p); ¢ log (It ¡ p)g, labelled ¸i, i = 1;¢¢¢; n, where
n is the number of variables in t he syst em (n equals t hree in t his case).
3. Test whet her t he¯ rst smallests canonical correlat ions are zero by comput ing t he st at ist ic:
¡ T
s
X
i = 1
log (1¡ ¸i);
which has a limit ing Â2 dist ribut ion wit h s (np + r )¡ s (n¡ s) degrees of
freedom under t he null, wherer is t he number of coint egrat ing relat ionships (r equals two in our example). In t he absence of ident it ies in t he syst em, t he maximum number of zero canonical correlat ions t hat can possibly exist isn¡ r (n¡ r is one in our example, since we assumed two coint egrat ing vect ors).
4. Suppose t hat s zero canonical correlat ions were found in t he previous st ep. Use t heses cont emporaneous relat ionships between t he ¯ rst di®erences as s pseudo-st ruct ural equat ions in a syst em of simult aneous equat ions. Aug-ment t hem wit h n¡ s equat ions from t he VECM and est imat e t he syst em using full informat ion maximum likelihood (FIML). T he rest rict ed VECM will be t he reduced form of t his pseudo-st ruct ural syst em.
In addit ion t o leading t o a parsimonious model, t he exist ence of unpredict able linear combinat ions of t he ¯ rst di®erences may allow us t o readily ident ify t he shocks wit h permanent e®ect s. Take, for example, t he model of K ing, Plosser, and Rebelo(1988) discussed above. T here,
¼ck¢ log (Yt) ¡ ¼yk¢ log (Ct) = (¼ck ¡ ¼yk)²pt
¼i k¢ log (Yt) ¡ ¼yk¢ log (It) = (¼i k ¡ ¼yk)²pt: (3.1)
Eit her of t hese two equat ions ident ify ²pt, given knowledge of ¼yk, ¼ck, and ¼i k.
Along t he same lines, for t he model of Campbell and Mankiw(1989), we have:
¢Ct ¡ ¸ ¢ Yt = (1¡ ¸ ) ¹t; (3.2)
which, given ¸ , ident i¯ es ¹t, t he shock t o permanent income of t he type one
consumer.
T his form of shock ident i¯ cat ion is much simpler t han t he met hod employed by K ing at al.(1991), since it only requires knowledge of cointegrat ing and cofea-t ure veccofea-t ors; see cofea-t he Appendix. T he lacofea-t cofea-t er requires invercofea-t ing cofea-t he aucofea-t oregressive represent at ion. Also, since we impose t est able long- and short -run rest rict ions to t he VAR, we achieve more e± cient est imat es of VAR coe± cient s. T his leads to more e± cient est imat es of impulse responses and variance decomposit ions, since t hese are based on a more parsimonious model.
It is import ant t o underst and t he possible di®erences in result s from using t he two met hods. If t he long-run rest rict ions imposed by bot h met hods are t he same, impulse-response and variance-decomposit ion result s will be similar for long horizons (being exactly t he same in t he in¯ nit e horizon). However, t hey have t he pot ent ial t o di®er at business-cycle horizons, since only t he met hod used here imposes short -run co-movement rest rict ions. Given t he e± ciency gains discussed above, our variance-decomposit ion and impulse-response result s will be more pre-cise at short horizons. As argued above, t his is import ant for economic agent s who discount fut ure event s more heavily, and t hus care more about what happens at business-cycle horizons. It is also import ant as a t ool in evaluat ing t heoret ical models in short horizons, especially regarding t he import ance of nominal shocks. Since t he e± ciency-gains result is t heoret ical, we compare out -of-sample fore-cast s between t he rest rict ed and the unrest rict ed VECM to subst ant iat e t hat t hese gains are relevant for t he present dat a set . We ¯ nd t hat t he former performs bet -t er.
4. Em pi r ical Ev idence
T he dat a being analyzed consist of (log) real U.S. per-capit a privat e out put - y, personal consumpt ion per-capit a - c, and ¯ xed invest ment per-capit a - i . T he dat a were ext ract ed from Cit ibase on a quart erly frequency5. Alt hough Cit ibase
has dat a available from 1947:1 t o 1994:2, we used only 1947:1 t hrough 1988:4 in est imat ion, in order t o mat ch t he sample period used in K ing et al.(1991), t hus making result s direct ly comparable.
5Using t he Cit ibase mnemonics (1995) for t he series, t he precise de¯ nit ions are: GCQ
T he plot of (logged) per-capit a real out put , consumpt ion and invest ment is present ed in Figure 1. T here are two st riking charact erist ics. First , t he data are ext remely smoot h (typical of I (1) dat a) and appear t o be t rending t oget her in t he long run. Second, t he dat a show similar short -run behavior: during recessions all t hree aggregat es drop. However, invest ment drops much more t han consumpt ion and out put , and t he lat t er drops more t han consumpt ion - t he most insensit ive series t o recessions.
Test s for coint egrat ion were performed using Johansen's(1988, 1991) t echnique and are present ed in Table 1. Crit ical values were ext ract ed from Ost erwald-Lenum(1992). We conclude t hat t he coint egrat ing rank r is two. T his implies t he exist ence of a common st ochastic t rend for out put , consumpt ion and invest ment . Table 1 also present s the point est imat es of a normalized version of t hese two vect ors. T hey are very close t o (¡ 1; 1; 0)0and (¡ 1; 0; 1)0respect ively, which implies t hat t he consumpt ion-out put and invest ment -out put rat ios areI (0). In order t o joint ly t est t hese hypot heses, we use t he likelihood ratio t est proposed in Johansen(1991). T he result s of t his t est do not reject t hat (¡ 1; 1; 0)0and (¡ 1; 0; 1)0are a basis for t he coint egrat ing space. T hese ¯ ndings are consistent wit h t he t heoret ical models ment ioned above, and with t he result s in K ing et al.(1991), who used a di®erent t est for coint egrat ion. T hey imply t hat t he \ great ratios" areI (0) processes.
T henext st ep of t he t est ing procedureis t o use canonical correlat ion analysis to examine whet her t he dat a have common cycles. For t he common-cycle t est , we use t he VECM wit h (¡ 1; 1; 0)0and (¡ 1; 0; 1)0as t he two coint egrat ing vect ors. We follow K ing et al.(1991) in condit ioning on eight lags of t he dependent variables in t he VECM, which is enough t o capt ure t he dynamics of t he syst em. Table 2 shows t he results of t he common-cycle t est s: t he p-values for t heÂ2 t est and it s F-t est
approximat ion6 of t he null hypot heses t hat t he current and all smaller canonical
correlat ions are st at ist ically zero. As not ed before, t he cofeat ure rank s is t he number of st at ist ically zero canonical correlat ions. At t he 5% level, bot h t est s cannot reject t he hypot hesis t hat t he smallest canonical correlat ion is st at ist ically zero, which implies t hat s is one. T hus, out put , consumpt ion and invest ment share two independent cycles and do have similar short -run ° uctuat ions.
As discussed in t he Appendix, t he fact t hat n = r + s allows a special t rend-cycle decomposit ion of t he dat a. Since we found a common st ochast ic t rend for our syst em, t he t rend component of out put , consumpt ion and invest ment
is t he same, being generat ed by t he linear combinat ion of t he dat a t hat uses t he cofeature vect or (t he maximum likelihood est imat e of t he common t rend is 0:42¢yt + 0:87¢ct ¡ 0:29¢it). On t he ot her hand, t hese variables will have cycles
t hat combine two distinct I (0) serially correlated component s, which are in t urn generat ed by t he linear combinat ions of t he dat a using t he coint egrat ing vect ors (t he great rat ios); see Table 3. Plot s of t he t rends and cycles of t he dat a are given in Figures 2 t hrough 4 and 5 t hrough 7 respect ively. T he t rend is very smoot h compared t o t he dat a in levels, and t he cycles show a dist inct pat t ern of serial correlat ion.
T hree feat ures are wort h not ing from t hese graphs: ¯ rst , it seems t hat t here is lit t le di®erence between consumpt ion and t he common t rend, which result s in a very small cycle for t hat variable (Cochrane(1994)). Alt hough consumpt ion cannot be charact erized as a random walk, t hus failing Hall's(1978) t est of t he permanent -income hypot hesis, it seems t hat it s cyclical component is very small7.
Similar result s are achieved by Fama(1992) and Cochrane(1994), alt hough wit h di®erent t echniques. Second, invest ment has a much more volat ile cycle t han out put and consumpt ion, which t ranslat es int o t he investment cycle having t he highest amplit ude of t hem all. T his is one of t he stylized fact s of business cycles cit ed in Lucas(1977). T hird, all t hree cycles drop during NBER recessions8.
T he est imat es of t rends and cycles of t he dat a allow us t o answer a quest ion considered import ant by most aut hors in t he applied macroeconomics lit erat ure: do permanent or t ransit ory shocks explain t he bulk of t he variance of aggregat e dat a? At t empt s t o answer t his quest ion can be found, among ot hers, in t he work of Nelson and Plosser(1982), Wat son(1986), Campbell and Mankiw(1987), Cochrane(1988, 1994), K ing et al.(1991), and Gal¶i(1996). Nelson and Plosser ¯ nd t his issue import ant because t hey associat e t rends t o permanent fact ors in° u-encing out put - such as product ivity, and cycles t o t ransit ory fact ors - such as monet ary policy.
Since t he t rend-innovat ion e®ect s are permanent while t he cycle-innovat ion e®ect s are only t ransit ory, it seems reasonable t o at t ach import ance t o t he t rend component if t he t rend innovat ion explains a signi¯ cant proport ion of t ot al fore-cast errors at business-cycle horizons. T he result s of t he variance decomposit ion of
7We ret urn t o t his issue lat er, aft er present ing t he result s of t he variance decomposit ion of
innovat ions of t he dat a set .
8Some minor except ions are veri¯ ed. T he t rend somet imes drops during recessions as well,
out put , consumpt ion and invest ment are present ed in Table 4. T his t able shows t he percent age of t he variance of t ot al forecast errors explained by permanent shocks at di®erent horizons9. For out put , t ransit ory shocks explain about 50% of t he forecast error variance (FEV) at t he two year horizon, more t han 40% at t he three-year horizon, and more t han 30% at t he four-year horizon. For short er horizons, t ransit ory shocks explain more t han half of t he FEV. In t hat sense, we cannot discard t he import ance of t he t ransit ory component of out put . T he result s for invest ment are much st ronger. Transit ory shocks explain most of t he FEV for all t abulat ed ¯ nit e horizons. For t he one-year horizon, t hese shocks explain more t han 90% of t he FEV, and for t he two-year horizon almost 80%.
T he result s of t he variance decomposit ion for consumpt ion allow discussing t he permanent -income hypot hesis. At t he one-year horizon, the permanent com-ponent explains almost 80% of t he FEV, and at t he two-year horizon about 85%. T he pict ure t hat emerges is t hat of an all-import ant permanent component . T he plot of Figure 3 corroborat es t his evidence. On t he one hand, if one is willing to label t he permanent part of consumpt ion as permanent income10, t he
variance-decomposit ion analysis allows one t o conclude t hat permanent income is by far t he most import ant component of consumpt ion. Moreover, consumpt ion and per-manent income obey long-run proport ionality, an import ant t heoret ical result . On t he ot her hand, although we found excess sensit ivity in consumpt ion, e.g., Hall(1978) and Flavin(1981), consumpt ion's t ransit ory part has very lit t le impor-t ance.
At t his point , it is useful t o compare t he result of our variance decomposit ion wit h t hose of K ing et al.(1991). Since bot h st udies used t he same dat a and sample period, t he result s are direct ly comparable and any di®erences in result s can be at tribut ed t o short -run co-movement rest rict ions being imposed by our met hod.
9Trend innovat ions are t he ¯ rst di®erences of t he common t rend. Cycle innovat ions are
t he residuals obt ained by regressing cycles on t he right -hand-side variables in t he VECM (t he lagged error-correct ion t erms and t he eight ¯ rst lags of t he dependent variables). Since t he two innovat ions are correlat ed, we ort hogonalize t hem, and we denot e t he port ion of t he cycle innovat ion ort hogonal t o t he t rend innovat ion as t he \ t ransit ory shock" . T he \ permanent shock" will t hen comprise t he original t rend innovat ion and t he port ion of t he cycle innovat ion explained by t he t rend innovat ion.
10According t o one view of t he permanent -income hypot hesis, consumpt ion should be equal t o
K ing et al. result s are reported in parent heses in Table 4. As expect ed, t he major di®erences in result s occur for short horizons (1-12 quart ers): t heir met hod under-est imat es t he cont ribut ion of t he t ransit ory port ion of out put and invunder-est ment . T he same happens for consumpt ion for t he 1-t o-4-quart er horizon, being reversed, however, for longer horizons. T he biggest discrepancy happens for out put : in t heir met hod, t he permanent port ion of out put explains almost 60% (70%) of out put variance for t he one-(two-)year horizon, whereas our result assigns about 40% (50%) t o it . T hese di®erences are enough t o change t he emphasis of t he variance decomposit ion result s for out put . It is clear t hat a much more considerable role should be at t ribut ed t o sources of t ransit ory noise. It is import ant t o not e t hat t his result was obt ained in a framework where only real variables were considered, which in it self pot ent ially limit s t he role of some sources of t ransit ory shocks, e.g., monet ary policy; see t he result s in K ing et al.(1991) when monet ary variables are included in t he VAR and also t he discussion in Hansen and Heckman(1996, foot not e 9).
T he import ance of t ransit ory component s of out put was also report ed by Cochrane(1994), and a similar general result is achieved by King et al.(1991), aft er augment ing t heir VAR wit h monet ary-sector variables. More recent ly, Gal¶i(1996) has shown t hat nominal (non-permanent ) shocks are critical for explaining t he condit ional correlat ion pat t ern of labor product ivity and employment (or hours). Moreover, t hese nominal shocks are responsible for t he synchronized cyclical be-havior of GDP and hours: when t he nominal-shock component s of GDP and hours are ignored, t hese two series fail t o show any dist inct business-cycle pat t ern (see Gal¶i's Figure 6). Along t he same lines, den Haan(1996) ¯ nds t hat out put and prices are positively correlat ed at business-cycle horizons, alt hough negat ively correlat ed in t he long run. T his support s t he conclusion t hat nominal shocks are import ant in t he short run while real shocks are import ant in t he long run. It also illust rat es t hat t he horizon mat t ers for invest igat ing t he plausibility of di®erent t heories, a point also st ressed by Hansen and Heckman(1996).
T his body of evidence goes against t he claim in Nelson and Plosser(1982) t hat permanent shocks are t he most import ant source of variat ion for out put . Alt hough t here is no doubt t hat t his is t rue for long horizons, result s for business-cycle horizons point t owards t he import ance of t ransit ory shocks. If t he horizon mat t ers, it would be int erest ing t o consider t he most precise t echnique at short horizons, and t his is exact ly what we have t ried t o provide in t his paper.
nominal shocks at business-cycle horizons, it is not obvious how t o int erpret t he evidence. T heoret ical models are rarely built in t erms of permanent or t ransi-t ory shocks. Raransi-t her, ransi-t hey are builransi-t in ransi-t erms of real (producransi-t iviransi-ty) or nominal (monet ary) shocks. T hus, t o t est t heories, one has usually t o impose ident ifying assumpt ions. For example, Blanchard and Quah(1989), K ing et al. (t ri-variat e syst em), and Gal¶i assume t hat t he only source of permanent shocks is produc-t iviproduc-ty. Non-permanenproduc-t shocks are labelled \ demand" shocks by Blanchard and Quah and Gal¶i. However, one can cert ainly t hink of permanent demand shocks (a permanent change in preferences, for example) or of t ransit ory supply shocks (t ransit ory t echnology shocks, for example). In these cases t hose ident ifying as-sumpt ions will fail11.
Alt hough answering t hese issues is out of t he scope of t his paper, we argue t hat t he result s here and elsewhere are st ill useful. We rely on t he reasonable assumpt ion t hat permanent shocks are more likely to be real, whereas t ransit ory shocks are more likely t o be nominal. T his \ axiom," as far as we know, has never been openly challenged, despit e t he fact t hat more t han 15 years have now elapsed since Nelson and Plosser ¯ rst used it .
4.1. Post -Sample For ecast s of Per -C api t a Out put , C onsum pt ion and I nvest m ent
T his last sect ion compares post -sample forecast s of two economet ric represent a-t ions of our a-t ri-variaa-t e sysa-t em. T he ¯ rsa-t is a-t he Unresa-t rica-t ed VECM (UVECM), which does not t ake int o account short -run rest rict ions implied by t he exist ence of common cycles. T he second is t he Restrict ed VECM (RVECM), which t akes t hose rest rict ions int o account . Sample est imat es used dat a for out put , consump-t ion, and invesconsump-t menconsump-t from 1947:1 consump-t o 1988:4. Posconsump-t -sample one-sconsump-t ep-ahead forecasconsump-t s for each represent at ion were t hen calculat ed from 1989:1 t o 1994:2, comprising 22 quart erly observat ions.
Est imat ion of t he UVECM used eight lags for all lagged dependent variables and one lag for t he error-correct ion t erms. Since it is a reduced-form, it was est imat ed by Least Squares. T he RVECM was estimat ed using t he same lag st ruct ure as the UVECM, but imposing common-cycle rest rict ions on t he syst em.
11T his, however, is not t he only problem. In an economy wit h dist ort ions, Basu and
As discussed in t he Appendix, t hese rest rict ions can be convenient ly formulat ed in t erms of exclusion rest rict ions in a syst em of \ st ruct ural"12 equat ions. T hus, t he
syst em was est imat ed using FIML, a suit able met hod for est imat ing st ruct ural forms, t o ¯ nd t he maximum likelihood est imat es of t he paramet ers.
Forecast ing result s are reported in Table 5, which cont ains t he mean squared error (MSE) in forecast for each equat ion separat ely, and t he determinant of t he mean squared forecast error mat rix, a measure of t he overall forecast ing perfor-mance for t he syst em.
For t he overall syst em, it is clear t hat t he RVECM represent at ion does bet t er, wit h a di®erence in t he det erminant of t he mean squared forecast error mat rix of more t han 25%. For individual equat ions, t he RVECM out performs t he UVECM in forecast ing for all t hree series. T he forecast ing improvement is most remark-able for out put and consumpt ion. T he empirical result s achieved here con¯ rms t he t heoret ical predict ion t hat rest rict ed est imat ion reduces MSE whenever \ t rue rest rict ions" are imposed on est imat ion.
5. Concl usi ons
T his paper con¯ rms t he predict ion of several t heoret ical models t hat out put , consumpt ion and invest ment share bot h a common t rend and common cycles. Alt hough common t rends have been invest igat ed and con¯ rmed before, ¯ nding common cycles const it ut es new evidence regarding t his aggregat e dat a set . As discussed above, t his ¯ nding is relevant for calculat ing more precisely t he relat ive import ance of permanent and t ransit ory shocks at business-cycle horizons, for which t his issue is st ill cont roversial and which economic agent s ¯ nd more relevant for welfare considerat ions.
T he result s show t hat t ransit ory shocks are more import ant t han previously t hought . T hey explain about 50% of t he variat ion of out put at t he 2-year horizon, more t han 40% at t he 3-year horizon, and more t han 30% at t he 4-year horizon. T he result s for invest ment are even st ronger: more t han 50% up t o t he 5-year horizon. Despit e t hese result s, t he permanent shock explains a very large pro-port ion of consumpt ion variat ion, providing evidence of consumpt ion smoot hing over t ime. T he import ance of t ransit ory shocks document ed here ¯ nds support in t he recent research by Gal¶i(1996) and den Haan(1996), as well as in t he work
of Cochrane(1994). It may be a sign t hat nominal shocks are relevant for t he short -run variat ions of macroeconomic dat a.
Finally, by using a post-sample forecast ing exercise, t his paper est ablishes t hat ignoring common-cycle rest rict ions in t his mult ivariat e dat a set can lead t o a non-t rivial loss of e± ciency in est imat ing reduced-form VECM's. T his can a®ect t he precision of est imat es of t rends and cycles, as well as t he precision of impulse-response and variance-decomposit ion exercises. T herefore, t est ing for common cycles should always precede economet ric est imat ion whenever short -run co-movement rest rict ions are likely t o be present .
R efer ences
[1] Basu, S. and Fernald, J.(1997), \ Aggregat e Product ivity and Aggregat eTech-nology," Working Paper: University of Michigan.
[2] Beveridge, S. and Nelson, C.R.(1981), \ A New Approach t o Decomposit ion of Economic T ime Series int o a Permanent and Transit ory Component s wit h Part icular At t ent ion t o Measurement of t he \ Business Cycle" , Journal of Monetary Economics, vol. 7, pp. 151-174.
[3] Blanchard, O.J. and Quah, D.(1989), \ T he Dynamic E®ect s of Aggregat e Supply and Demand Dist urbances" , American Economic Review, vol. 79, pp. 655-673.
[4] Campbell, J.Y.(1987), \ Does Saving Ant icipat e Declining Labor Income? An Alternat ive Test of t he Permanent Income Hypot hesis," Econometrica, vol. 55(6), pp. 1249-73.
[5] Campbell, J.Y. and Mankiw, N.G.(1987), \ Permanent and Transit ory Com-ponent s in Macroeconomic Fluct uat ions" , American Economic Review.
[6] Campbell, J.Y. and Mankiw, N.G.(1989), \ Consumpt ion, Income and Int er-est Rat es: Reint erpret ing the T ime Series Evidence," NBER Macroeconomics Annual.
[8] Cochrane, J.H.(1994), \ Permanent and Transit ory Component s of GNP and Stock Prices," Quarterly Journal of Economics, vol. 30, pp. 241-265.
[9] den Haan, W.(1996), \ T he Comovements Between Real Act ivity and Prices at Di®erent Business Cycle Frequencies," Working Paper: University of Cal-ifornia, San Diego.
[10] Engle, R.F. and Granger, C.W.J.(1987), \ Coint egrat ion and Error Correc-t ion: RepresenCorrec-t aCorrec-t ion, EsCorrec-t imaCorrec-t ion and TesCorrec-t ing," Econometrica, vol. 55, pp. 251-276.
[11] Engle, R.F. and K ozicki, S.(1993), \ Test ing for Common Feat ures," Journal of Business and Economic St at ist ics, vol. 11, pp. 369-395, wit h discussions.
[12] Engle, R.F. and Issler, J.V.(1995), \ Est imating Common Sect oral Cycles" , Journal of Monetary Economics, vol. 35, pp. 83-113.
[13] Engle, R.F. and Yoo, B.S.(1987), \ Forecast ing and Test ing in Coint egrat ed Syst ems," Journal of Econometrics, vol. 35, pp. 143-159.
[14] Fama, E.(1992), \ Transit ory Variat ion in Invest ment and Out put " , Journal of Monetary Economics, vol. 30, pp. 467-480.
[15] Flavin, M.A.(1981), \ T he adjust ment of Consumpt ion t o Changing Expec-t aExpec-t ion abouExpec-t FuExpec-t ure Income" , Journal of Political Economy, vol. 89(5), pp. 974-1009.
[16] Gal¶i, J.(1996), \ Technology, Employment and t he Business Cycle: Do Tech-nology Shocks Explain Aggregate Fluct uat ions?" NBER Working Paper # 5721.
[17] Hall, R.E.(1978), \ St ochast ic Implicat ions of t he Life Cycle-Permanent In-come Hypot hesis: T heory and Evidence," Journal of Political Economy, 86, pp. 971-987.
[18] Hansen, L.P. and Heckman, J.J.(1996), \ T he Empirical Foundat ions of Cal-ibrat ion" Journal of Economic Perspectives, vol., 10(1), pp. 87-104.
[20] Johansen, S.(1991), \ Est imation and Hypot hesis Test ing of Coint egrat ion Vect ors in Gaussian Vect or Aut oregressive Models," Econometrica, vol. 59, pp. 1551-1580.
[21] K ing, R.G., Plosser, C.I. and Rebelo, S.(1988), \ Product ion, Growt h and Business Cycles. I I. New Direct ions," Journal of Monetary Economics, vol. 21, pp. 309-341.
[22] K ing, R.G., Plosser, C.I., St ock, J.H. and Wat son, M.W.(1991), \ St ochast ic Trends and Economic Fluct uat ions," American Economic Review, vol. 81, pp. 819-840.
[23] K osobud, R. and K lein, L.(1961), \ Some Economet rics of Growt h: Great Rat ios of Economics," Quarterly Journal of Economics, 25, 173-198.
[24] Lucas, R.E.,Jr.(1977), \ Underst anding Business Cycles," Carnegie-Rochest er Conference Series on Public Policy, vol. 5, pp. 7-29. Amst erdam: Nort h Hol-land.
[25] Nelson, C.R. and Plosser, C.I.(1982), \ Trends and Random Walks in Macroe-conomic T ime Series," Journal of Monetary Economics, vol. 10, pp. 139-162.
[26] Ost erwald-Lenum, M.(1992), Quant iles of t he Asympt otic Dist ribut ion of t he Maximum Likelihood Coint egration Rank Test St at ist ics," Oxford Bulletin of Economics and Statistics, vol. 54, 461-472.
[27] Rao, C.R.(1973), \Linear Statistical I nference" , New York: Wiley.
[28] Romer, P. (1986), \ Increasing Ret urns and Long Run Growt h," Journal of Political Economy, vol. 94, pp. 1002-1037.
[29] Vahid, F. and Engle, R.F.(1993), \ Common Trends and Common Cycles," Journal of Applied Econometrics, vol. 8, pp. 341-360.
[30] Vahid, F. and Engle, R.F.(1997), \ Codependent Cycles," Journal Economet-rics, vol. 80, pp. 199-121.
A . C o-M ovem ent R est r i ct i ons in D ynam i c M odel s
Before discussing t he dynamic represent at ion of t he dat a, and t he t rend-cycle de-composit ion met hod we have used, we present t he de¯ nit ions of common t rends and common cycles. For a full discussion see Engle and Granger(1987) and Vahid and Engle(1993) respect ively. First , we assume t hat yt is an-vector of I (1)
vari-ables, wit h t he st at ionary (M A (1 )) Wold represent at ion given by:
¢ yt = C (L ) ²t; (A.1)
where C (L ) is a mat rix polynomial in t he lag operat or, L , wit h C (0) = In, 1
P
j = 1
kCjk< 1 . T he vect or ²t is an£ 1 vect or of st at ionary one-st ep-ahead linear
forecast errors inyt; given informat ion on t he lagged values of yt. We can rewrit e
equat ion (A.1) as:
¢ yt = C (1) ²t + ¢C¤(L ) ²t (A.2)
whereC¤ i =
P
j > i
¡ Cj for all i : In part icular C0¤= In¡ C (1).
If we int egrat e bot h sides of equat ion (A.2) we get :
yt = C (1) 1
X
s= 0
²t ¡ s+ C¤(L ) ²t
= Tt + Ct (A.3)
Equat ion (A.3) is t he mult ivariat e version of t he Beveridge-Nelson t rend-cycle represent at ion (Beveridge and Nelson(1981)). T he series yt are represent ed as
sum of a random walk part Tt which is called t he \ t rend" and a st at ionary part
Ct which is called t he \ cycle" .
D e¯ ni t i on A .1. T he variables in yt are said t o have common t rends (or
coin-t egracoin-t e) if coin-t here arer linearly independent vect ors, r < n, st acked in an r £ n mat rix ®0, wit h t he property t hat :
®0
r £ n C (1) = 0:
D e¯ ni t i on A .2. T he variables in yt are said to have common cycles if t here are
s linearly independent vect ors, s· n¡ r , st acked in an s£ n mat rix ~®0, wit h t he
property t hat :
~ ®0
s£ n C
T hus, coint egrat ion and common cycles represent rest rict ions on t he element s of C (1) and C¤(L ) respect ively.
We now discuss rest rict ions on t he dynamic autoregressive represent at ion of economic t ime series arising from coint egrat ion (common t rends) and common cycles. First, we assume t hat yt is generat ed by a Vect or Aut oregression (VAR):
yt = ¡1yt¡ 1+ : : : + ¡pyt ¡ p+ ²t (A.4)
If element s of yt coint egrat e, t hen t he mat rix I ¡ p
P
i = 1¡i must have less t han
full rank, which imposes cross-equat ion rest rict ions on t he VAR. In t his case, Engle and Granger(1987) show t hat t he syst em (A.4) can be writ t en as a Vect or Error-Correct ion model (VECM):
¢ yt = ¡ ¤1¢ yt¡ 1+ : : : + ¡p¡ 1¤ ¢ yt ¡ p+ 1+ ° ®0yt ¡ 1+ ²t (A.5)
where° and ® are full rank mat rices of order n£ r , r is t he rank of t he coint e-grat ing space, I ¡ Pp
i = 1
¡i = ° ®0, and ¡¤j = ¡ p
P
i = j + 1
¡i , j = 1; : : : ; p¡ 1. Given t he
cointegrat ing vect ors st acked in ®0, it can be seen t hat (A.5) parsimoniously
en-compasses (A.4). Conditional on knowledge of coint egrat ing vect ors, t he VECM hasn2(p¡ 1) + n¢r paramet ers in t he condit ional mean, while t he VAR has n2¢p
paramet ers. T hus, the former hasn¢(n¡ r ) fewer paramet ers, since r < n. If we t ake into account t he free parameters in t he coint egrat ing vect or, t he VECM has n2(p¡ 1) + 2n¢r ¡ r2 mean paramet ers, (n¡ r )2
fewer t han t he VAR.
Vahid and Engle(1993) show t hat t he dynamic represent at ion of t he dat a yt
may have additional cross-equat ion rest rict ions if t here are common cycles. To see t his, recall t hat t he cofeat ure vect ors ~®0i, st acked in an s£ n mat rix ~®0,
elim-inat e all serial correlat ion in ¢ yt; i.e. ~®0¢ yt = ~®0²t. Since t he cofeat ure vect ors
are ident i¯ ed only up t o an invert ible t ransformat ion, we can, wit hout loss of generality, rot at e ~® t o have an s dimensional ident ity sub-mat rix:
~ ® = " In ~ ®¤
(n¡ s)£ s
#
Considering ~®0¢ y
t = ~®0²t ass equat ions in a syst em, and complet ing t he syst em
of ¢ yt; we obt ain,
2
4 Is ®~
¤0
0
(n ¡ s)£ s In¡ s
3
5¢ yt = 2
4 s£ (n p+ r )0
¡¤¤
1 : : : ¡¤¤p¡ 1 °¤
3 5 2 6 6 6 6 4
¢ yt¡ 1
.. . ¢ yt ¡ p+ 1
®0yt¡ 1
3 7 7 7 7
5 + vt; (A.6)
where ¡ ¤¤
i and°¤ represent t he part it ions of ¡¤i and° respect ively, corresponding
t o t he bot t omn¡ s reduced form VECM equat ions, and vt =
2
4 Is ®~
¤0
0
(n ¡ s)£ s In¡ s
3 5 ²
t:
It is easy t o show t hat (A.6) parsimoniously encompasses (A.5). Since
2
4 Is ®~
¤0
0
(n ¡ s)£ s In¡ s
3
5 is invert ible, it is possible t o recover (A.5) from (A.6). Not ice
however t hat t he lat t er hass ¢(np + r )¡ s¢(n¡ s) fewer paramet ers.
B . Tr end-Cy cl e D ecom p osi t i on
We now discuss t he t rend-cycle decomposit ion used here. For a full discussion see Vahid and Engle(1993). From equat ion(A.3):
yt = C (1) 1
X
s= 0
²t ¡ s + C¤(L ) ²t (B.1)
= Tt + Ct
Consider t he special case of n = r + s; and st ack t he cofeat ure and t he coin-t egracoin-t ing combinacoin-t ions coin-t o obcoin-t ain,
"
~ ®0y
t
®0y t
#
=
"
~ ®0T
t
®0C t
#
: (B.2)
T hen£ n mat rix A =
"
~ ®0
®0
#
has full rank and t herefore is invert ible. Part
i-t ion i-t he columns of i-t he inverse accordingly asA¡ 1 = [ ~®¡ ®¡ ] and recover t he common-t rend common-cycle decomposit ion by pre-mult iplying t he cofeat ure and cointegrat ing combinat ions by A¡ 1:
T his implies t hat Tt = ~®¡ ®~0yt andCt = ®¡ ®0yt, i.e. t rends and cycles are simple
linear combinat ions of t he dat ayt.
TABLE 1
COINTEGRATING RESULTS USING JOHANSEN’S(1988) TECHNIQUE
EIGENVALUES
( )
µiTRACE TEST
(
)
− − ≤∑
T j j iln 1 µ
CRITICAL VALUE
AT 5%
NULL HYPOTHESES
0.01886 3.06 3.76 ∃ at most 2
cointegrating vectors
0.07726 16.01 15.41 ∃ at most 1
cointegrating vectors
0.13944 40.18 29.68 ∃ at most 0
cointegrating vectors
Estimated Normalized Cointegrating Space:
$ . . ′ =−−
α 1 06 1 0
101 0 1
Test of Restrictions in the Cointegrating Space:
H0 1 1 0
1 0 1
: ′ = − − α
( )
χ2TABLE 2
CANONICAL CORRELATION ANALYSIS COMMON-CYCLE TESTS
SQUARED CANONICAL
CORRELATIONS
( )
λi2
Prob> χ2
( )
d (d)
Prob.>F NULL HYPOTHESES
0.4892 >0.0001
(78)
0.0001 Current and all smaller
( )
λi are zero
0.2860 0.004
(50)
0.0226 Current and all smaller
( )
λi are zero
0.1544 0.3200
(24)
0.4651 Current and all smaller
( )
λi are zero
TABLE 3
TRENDS AND CYCLES AS LINEAR COMBINATIONS OF THE DATAa
TRENDS
VARIABLE Y c i
y 0.42 0.87 -0.29
c 0.42 0.87 -0.29
i 0.42 0.87 -0.29
CYCLES
VARIABLE y c i
y 0.58 -0.87 0.30
c -0.42 0.13 0.30
i -0.42 -0.87 1.30
TABLE 4
COMPARING RESULTS OF VARIANCE DECOMPOSITION OF INNOVATIONS:
% OF THE VARIANCE ATTRIBUTED TO THE PERMANENT INNOVATION FOR: (KING ET AL. RESULTS IN PARENTHESIS) HORIZON
(QUARTERS)
OUTPUT CONSUMPTION INVESTMENT
1 32.1 (45.0) 64.8 (88.0) 0.0 (12.0) 4 37.4 (58.0) 77.1 (89.0) 9.0 (31.0) 8 50.7 (68.0) 85.3 (83.0) 20.8 (40.0) 12 57.5 (73.0) 88.2 (83.0) 24.9 (43.0) 16 66.7 (77.0) 90.4 (85.0) 35.0 (44.0) 20 76.1 (79.0) 92.5 (87.0) 44.6 (46.0) ∞ 100.0 (100.0) 100.0 (100.0) 100.0 (100.0)
TABLE 5
POST-SAMPLE FORECASTING RESULTS FOR THE RVECM AND UVECM REPRESENTATIONS
MEAN SQUARED ERROR (MSE)
DEPENDENT VARIABLE
RVECM UVECM
y 0.5079 0.6099
c 0.5000 0.5889
i 3.6503 3.6772
