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How Big Is the Random Walk in GNP?
John H. Cochrane
The Journal ofPolitical Economy, Volume 96, Issue 5 (Oct., 1988), 893-920.
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How Big Is the Random Walk in GNP?
John H. Cochrane
University of Chicago
This paper presents a measure of the persistence of fluctuations in
GNP based on the variance of its long differences. That measure
finds l i t t le long-term persistence in GNP. Previous research on this
question found a great deal ofpersistence in GNP, suggesting mod
els such as a random walk. A reconciliation of this paper's results
with previous research shows that conventional criteria for time-
series model building can produce misleading estimates of per
sistence.
I . Introduct ion
Macroeconomists once viewed fluctuations in gross nat ional p roduct
as temporary deviat ions f r o m a tr en d. T h e economi c the ory o f busi
ness cycles described temporary deviations f r o m "po ten t ia l GNP,"
w h i c h was assumed to evolve smoothly over t ime, and data were
rou t ine ly d e t r e n d e d p r i o r to analysis. A bo dy o f rece nt em pi ri ca l
w o r k (described belo w) has ques tion ed this t i me -h on or ed view. By
us in g a va ri et y o f tim e-seri es mo de ls , i t finds tha t fluctuations i n G N P
are pe rm an en t th at a decli ne i n G N P today lowers forecasts o f G N P
in to the i n f i n i t e f u t u r e .
T h i s paper reexamines the lo ng -r un proper t ies of G N P an d argues
tha t GNP does, in fact , rever t toward a " t rend" f o l l o w i n g a shock.
However , that reve rsi on occurs over a t i me ho ri zo n character ist ic of
business cyclesseveral years at least. The re for e , the shor t - ru n prop
erties o f G N P are consis tent w i t h a model w i t h ve ry pers ist ent shocks,
I thank Eugene Fama, Lars Hansen, John Huizinga, Robert Lucas, James Stock,
Robert Shiller, an anonymous referee, and the editors of thisJournal for many helpful
comments and suggestions.
[Journal of Political Economy, 1988, vol. 96, no. 5]
1988 by Th e Universit y of Chic ago. Al l rights reserved. 0022-3808/88/9605-0003$01.50
893
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94 JOURNAL OF POLITICAL ECONOMY
and one can incorrectly i n fe r a grea t deal o f lo ng -h or iz on persistence
by f i t t i n g a t ime-series model to this short-run behavior.
T h e class o f t ime-series mo d el mos t co m mo n ly used to describe
temporary devia t ions about t rend is
oo
yt = b t + X ajet-P (1)
7 = 0
whe r e yt stands for log GN P, btdescribes the t rend, and et is a r a n do m
dis turbance .1
Fluc tua t ions inyt a re te mpo ra r y i f ^ - is a s ta t ionary
stochastic process (yt is then called "trend stationary") . For Xajet-j to
be stat ionary, the aj mu st ap pr oa ch zero fo r large j . As a result, a
decl ine i n G N P bel ow t r e n d today has no effect o n forecasts o f the
level of GNP , Et(yt+J), i n the far fut ur e, an d i t imp lie s that gr o w t h
rates o f G N P mu st r ise above the ir hist ori cal average for a few periods
u n t i l the t r end l ine is reestablished.
The simplest t ime-series model that captures permanent fluctua
tions in GNP is a random walk w i t h d r i f t :
yt = + + (2)
F luc tua t ions in a r a nd om walk are pe rm ane nt i n the f o l l o w i n g sense:
suppose tha t et = - 1, so th a t yt falls one u n i t bel ow last p eri od' sexpect ed value. T h e n , since yt+j yt + j\x, + e, + 1 + . . . + et+j,
forecasts Et{yt+j) fal l by one u n i t fo r the indef i ni t e fut ure . Also, a low
o r nega t ive growth rate today impl ies no th in g about g r ow th rates i n
the fut ur e, an d the re is no tende ncy fo r fu tu re levels o f G N P to rev ert
to a t r end l ine. T h e r a n d o m walk is also n on st ati on ary .
T h e d is t in c t ion be tween a ra nd om walk (2) an d a trend -st ati onar y
series (1) is ext rem e. L on g- ra ng e forecasts o f a r a n d o m walk move
one for one w i t h shocks at each date, whi le lo ng -ra ng e forecasts o f a
t rend-s ta t ionary series do not change at all . T h e r e are tw o related
ways to t h i n k about a series that lies between these two extremes.
First, one can ask ho w m u c h l o ng -t er m forecasts re sp on d to shocks.
I n one in te r pre t a t io n , the measure of this paper asks the question,
H o w m u c h does a on e- u ni t sho ck to G N P affect forecasts i n th e f ar
futur e? I f by one un i t , it finds a random walk; i f by zero , i t finds a
t rend-s ta t ionary process l i k e (1) . I t can also find numbers be tween
zero and one, characterizing a series tha t r e turn s t owa rd a " t re nd " in
the far future, but does no t get all th e way th er e, o r it can find anu mb er grea ter th an one , charac ter iz ing a series tha t w i l l cont inue to
1
Simple univariate time-series models like (1 ) should be thought of as a way of
capturing the dynamic behavior ofyt that results from a rich multivariate world. They
are not "structural" in any way.
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RANDOM WALK IN GNP 895
diverge f r o m its previously forecast value f o l l o w i n g a shock. Cam pb el l
a nd M a n k i w (1987) or i gi na te d an d emphasize th is in t erp ret at ion .
Second, one can model a series wh os e fluctuations are pa r t l y t em
p o r a r y a n d pa r t l y p e r m a n e n t as a c o m b i n a t i o n o f a stat ionary series
and a r a nd om walk. T h e r a n d o m walk carr ies the pe rm an en t pa rt o f a
change and the stationary series carr ies the te mp or ar y pa rt o f a
change. T h e n , one can ask ho w im po r t an t the pe rm an en t or r a nd om
walk co mp on en t is to the beha vio r o f the series. I n a second in t erp re
ta t ion, the measure of this paper asks the ques ti on, H o w lar ge is the
variance o f shocks to the ra n d o m walk or perm ane nt compo nen t o f
G N P c o m p a r e d w i t h the var iance of year ly G N P gr ow th rates? O r ,
equivalent ly , H o w bi g is the r a n d o m walk in GNP?
I f the variance o f the shocks to the ra n d o m walk component is zero ,the series is tre nd- sta tio nar y, and lo ng -t er m forecasts do no t change
i n response to shocks. I f the vari ance o f the shocks to the r a n d o m
walkco mp on en t is equa l to the variance of first differences, the series
is a pu re r a n d om walk. As befor e, t her e is a co nt inu ous range o f
possibilities be tween zero a n d one a n d be y ond one.
A model cons is t ing o f a r andom walk plus a s tat ionary co mp on en t
may seem quite special. However, I show below that we can t h i n k o f
any series whose growth rates o r first differences are stationary (any
series w i t h a u n i t root) as a combination of a stationary series plus a
r a n d o m walk. T h e decompo s i t i on in t o s ta t ionary and ra nd om walk
co mpo ne nts is a co nve nie nt way o f t h i n k i n g abou t the prop ert ies o f a
t ime series, but i t adds no structure. I also show that the response to
innovat ions is pr op or t i on al to the square ro ot o f the variance o f
shocks to a random walk co mp on en t , so we can f reely t r an sf or m
between these two in terpre ta t ions .
T h e idea that G N P may con tai n a r a n d o m walk goes back to I r v i n g
Fisher' s "M on te Car lo hypothesis ," exa mi ne d by McCul loch (1975).
Th er e is no w a large l i te rat ure f o l l o w i n g the first half o f Nels on an d
Plosser (1982 ) th at appli es the Dic ke y an d Fu ll er (1979 , 1981) an d
subsequent tests f o r u n i t roots to aggregate t i m e series. Since a series
w i t h a u n i t root is equivalent to a series that is compo sed o f a r a n do m
walk and a s tat ionary component , tests for a u n i t ro ot are atte mpts to
dis t ingui sh between series tha t have n o r a n d o m walk component (o r
for w h ich the variance o f shocks to the ra n d o m walk component i s
zero) and series tha t have a r a n d o m walk com pon ent (o r fo r w h ich the
variance o f shocks to the r a n d o m walk co mp on en t is between zero
an d i n f i n i t y ) . Stat ed thi s way, it is clea r w h y tests for a u n i t r o o t have
low po we r: i t is h a r d to t e l l a stationary series f r o m a stationary series
plus a very smal l r a n d o m walk . T h i s paper and the related l i terature
cited i n i t go bey on d tes ting fo r the presence o r absence o f a u n i t r oo t
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8 9 6 JOURNAL OF POLITICAL ECONOMY
o r r a n d o m walk c o m p o n e n t a n d measure how impo r t a n t t he u n i t r o o t
o r r a n d o m walk co mp on en t is to the behav ior o f a series.
Implications of the Random Walk in GNP
T h e s ize o f a r a n d o m walk i n G N P is im po r t an t f r o m a purely statist i
cal v iewpoin t . M an y s tatis t ical proce dure s rely critically on the dist inc
t ion between series tha t do no t con ta in a r andom walk component (1 ) ,
w h i c h we can a n d s h ou l d d e t r e n d , a n d first-difference st at io na ry
series(3) be low , o r series tha t do con ta in a r andom walk c o m p o
n e n t w h i c h we s h o u l d first-difference p r i o r to analysis. Hyp oth esi s
tests tha t rely on asympto t ic d i s t r i b u t io n theory are an impor tan t
example because t ha t d i s t r i b u t io n th eo ry is of te n qu it e sensitive to t hepresence o f a r a n d o m walk co mp on en t . A mea sure men t o f the size o f
t h e r a n d o m walk co mp on en t can be a bet ter guid e to the pr op er
p r o c e d u r e t h a n a u n i t r o o t test because i f t he r a nd om walk c o m p o
ne nt is sma ll bu t s t i l l nonzero, then an asymptot ic d i s t r i b u t io n t heo r y
based on t r en d s ta t ionar i ty may pro v id e a be t te r ap pr ox im at io n i n a
given small sample th an the th eo ry based o n a u n i t r oo t .
T h e s ize o f a r a n d o m walk i n G N P has been cast as a direct test
be tween c o m p e t i n g mode l s o f the economy. Fo r example , N e l son a n d
Plosser (1982) in te rp re te d th ei r resul t that G N P has a large ra n d o m
walk c om po ne nt as evide nce fo r stochastic eq u i l i b r iu m models over
t r ad i t i o n a l monetary o r Keynes ian business cycle mode ls. T h e y ar
gued tha t t r ad i t io nal models p r odu ce on ly t emporary dev ia t ions f r o m
t r e n d , whi le models that find the ul t ima te source o f G N P var iabi l i ty i n
te ch no lo gy shocks ca n p r o d u c e p e r m a n e n t fluctuations.
W i t h the advantages of h inds igh t , i t now seems that the size or
existence o f a r a n d o m walk com pon en t i n G N P canno t direct ly dis t in
guish broad classes o f eco nomi c theor ies o f the business cycle at theirpresent stage o f deve lop men t . T h e K y d l a n d and Prescott (1982) and
L o n g a n d Plosser (1983) stochastic e q u i l i b r i u m models were con
structed precisely to generate t e m p o r a r y fluctuations ab ou t t re n d . O n
the other hand, K i n g et al. (1987) show that one can mo d i fy these
models to p roduce a r andom walk com pon en t by in t r odu c i ng a r an
d o m walk i n the tec hnol ogy shocks or a l ine ar tech nolo gy fo r h u m a n
or physical capi tal ac cu mul at i on. Presumably, the same modif ica
t ions w o u l d i n t r o d u c e a r a n d o m walk c o m p o n e n t i n to mone ta r y o r
"Keynesian" models as w e l l .
Fu r t he rm or e , the resul t s o f th i s paper are compat ib le w i t h a variety
o f r a n d o m walk com pon ent s . I show below that an AR (2 ) about a
de termin is t ic t r end , w h i c h has no random walk component , and a
m o d e l w i t h a r a n d o m walk who se vari anc e is 0.18 time s th e vari anc e
o f first dif ferences o f lo g G N P accou nt equal ly w e l l for the results of
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RANDOM WALK IN GNP 897
this paper . Also, the standa rd err ors i n this paper are large, and I
argue that this is una voi da ble . I co nc lu de th at the existence o r size o f
a r andom walk co mp on en t i n G N P is no t a precisely measured "s ty l
ized fact" that we should require any reasonable mod el to r eproduce .
T h e most pr om is in g dir ect use for the poi nt estimates of the size o f
a r a n d o m walk com pone nt i n th is paper may be the cali bra ti on o f a
given mo de l rath er tha n a test that can dis t ingui sh co mpe t i ng classes
o f models . I f a mo de l ( l i k e the ones cited above) produces a r a n d o m
walk i n G NP , the results o f this paper suggest that the parameters o f
that model should be picked to also generate in teres t ing shor t - run
dyn ami cs o f GN P , so tha t the variance o f year ly changes in GNP is
much larger than the variance of shocks to i ts random walk compo
nent .
Other Estimates
Several authors have estimated the persistence o f fluctuations i n
G N P , and their estimates vary greatly. Nelson and Plosser (1982)
matc hed a mo de l consis t ing o f per man ent and temp ora ry comp o
nents to a s ty lized auto corr ela t ion f unc t i on for gro wt h rates of G N P
and conc lude d that the pe rma ne nt c omp on en t was mo re imp or ta nt
th an the te mp or ar y co mp on en t. Wat son (1986) an d Cl ar k (1987) esti
mate d dif fe ren t unobse rved compon ents models and fo un d a smal l
p e r ma ne n t c omponen t . C a mpb e l l a nd M a n k i w (1987) esti mate d the
effect o f a shock o n lo ng -t er m forecasts of G N P f r o m the parameters
o f low -or der autoregress ive, mo vi ng average ( A R M A ) representa-
t ions o f postwar G N P an d fo u n d a large r a n d o m walk componen t .
Several authors have examined the persistence o f fluctuations i n
other t ime series using a variety of methods. Rose (1986) presents a
survey ofpapers that find l a rge r andom walk comp one nts i n variousmacroeconomic t ime series. I n finance, co nv en ti on al wi sd om fa vor ed
the r andom walk m o d e l while macroec onomist s favor ed the tr en d-
stationary mo de l. Poterba an d Su mme rs (1987), Fama an d Fre nch
(1988), an d Lo an d MacKin lay (1988) use varia nce rat io estimators
s imilar to the one used in this paper and related estimators to docu
me nt a te mp or ar y co mp on en t i n stock prices. Hu iz in ga (1987) uses a
closely relat ed estimator to do cu me nt a te mp or ar y co mp on en t i n real
exchange rates. Coc hra ne and Sbo rdon e (1988) present a mul t ivar
iate extension.
This Paper's Technique
I n this paper, I measure the size of a r a nd om walk comp onen t in GN P
f r o m the variance o f its l on g diffe renc es. T h e i n t u i t i o n behind th is
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8 9 8 JOURNAL OF P O L I T I C A L ECONOMY
measure comes f r o m the f o l l o w i n g arg ume nt : Ima gin e tha t log GN P,
deno ted yt, is a pu re r a n d om walk( m o d e l [2 ] ) . T h e n the variance o f i ts
^-differences grows l inear ly w i t h the dif ference k: \ar(yt yt-k)=
kcrl. O n the ot he r ha nd , i f lo g G N P is station ary abou t a tr e nd (m od el
[1]) , the variance of its ^-differences approaches a constant, twice the
u n co n d i t i o n a l vari anc e o f the series: var(yt yt-k) ~^ 2o^. Now plot
{\lk)\zx(yt - yt-k) as a fu n c t i o n o f k. I fyt i s a random walk, the plot
should be constant at ai. I f yt is trend-stationary, the plo t shou ld
decline toward zero.
Nex t , suppose th at fluctuations i n G N P are pa rt ly p e r ma ne nt an d
par t ly t e m p o r a r y , w h i c h we can m od e l as a co mb in at io n o f a station
ary series and a r a nd om walk. Now the plot o f ( l /&)var ( ;y , yt-k)
versus ksh ou ld settle d o w n to the varian ce o f the shock to the r a n d o mwalk componen t .
I f fluctuations i n G N P are pa r t l y t e m p o r a r y i f t he r andom walk
co mp on en t is small an d a shock today w i l l be par t ia l ly reversed in the
l o n g runthat reversal is l i k e l y to be slow, loosely st ru ct ur ed , an d no t
easily ca pt ur ed i n a simp le pa ram etr ic mo de l. T h e variance o f k-
differences can find such loosely st ru ct ur ed reve rsion , whereas ma ny
o the r approaches ca nno t. I show in Secti on I V tha t this dif fe ren ce can
reconcile the results of this paper w i t h o the r measures o f the per ma
nence o f fluctuations i n G N P .
Results
Figure 1 a n d tab le 1 pre se nt (l/k)\ar(yt yt-k) fo r lo g rea l pe r capita
GNP , 1869-1986. Pre-1939 data are taken f r o m Fr i edman and
Schwart z (1982). I use real per capi ta G N P to el im in at e possible no n-
stat ionari ty induced by i n f l a t i o n or popu la t ion g ro w t h . ( H ence f o r th , I
w i l l ref er to lo g rea l pe r capit a G N P as j us t "G NP ." ) Fi gu re 1 a nd table1 also in cl ude asymp toti c sta nda rd err ors , discussed below. Ta bl e 1
also presents \lk time s the vari anc e o f ^-differe nces d i v i d e d by the
variance of first differe nces ( the variance ra t io) . T h e uni ts i n table 1
a nd figure 1 are an nu al pe rcentage g r o w t h .
Since l/k time s the va rian ce o f A-differences settles d o w n to about
o n e - th i r d o f the variance o f first dif ferences , figure 1 a n d tab le 1
suggest that the in no va t i on var iance o f the ra nd om walk c o m p o n e n t
is abou t on e- th ir d o f the variance o f year- to-year changes: annua l
g r o w th rates o f G N P cont ain a large te mp or ar y com pon ent . I n fact , I
show belo w that the pa tt er n o f figure 1 is consistent w i t h a de t e r min
istic t r e n d , w h i c h has no per ma nen t o r r a nd om walkcomponent , and
whose fluctuations are en ti re ly t e m p o ra ry .
Figure 2 presents the l og o f rea l per ca pita GN P . No ti ce tha t this
data set loo ks as i f it has a t r e n d i n it . Fl uct uat ion s occu r, bu t the level
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R A N D O M WALK INGNP 899
1 1 1 f r T..._ T 1"i 1
1/k var k-di f f erences
0 / \ - - - - standard errorsin
/ \/ \' r^.
4
0 / \ ^^^- / \ \
\ 4" / \ 4
4 -
4 X \ \ \ \
N \\
0CN
\_ .
1 1 1 l 1 l_. _j 1 1 1 1I I I I I I I I I I I 1 1
0. 5 10 15 20 25 30F I G . 1.\lk times the varianceof -difTerences of log real per capita G N P , 1 8 6 9
1986, with asymptotic standard errors.
o f t h e series always r e t u r n s t o t h e " t r e n d l i n e . " F u r t h e r m o r e , t h a t
t r e n d l i n e is l i n e a r : t h e r e a r e n o "w av es" o f l o w - f r e q u e n c y m o v e m e n t .
These c h a r a c t e r i s t i c s d r i v e t h e f i n d i n g o f a s m a l l r a n d o m w a l k c o m -
p o n e n t . ( N o t e t h a t l o w - f r e q u e n c y m o v e m e n t g e n e r a t e d b y a n o n -
l i n e a r t r e n d , a s h i ft , etc. w o u l d s h o w u p as a l a r g e r a n d o m w a l k
c o m p o n e n t i n t h i s a n d m o s t o t h e r e s t i m a t i o n t e c h n i q u e s based o n
l i n e a r time- ser ies m o d e l s . )
P r e w a r G N P data a r e m o r e v a r i a b l e t h a n p o s t w a r d a t a , a n d o n em i g h t suspect t h a t t h i s c h a r a c t e r i s t i c d r i v e s t h e r e s u l t . H o w e v e r ,
f i g u r e 3 a n d t a b l e 1present \lk t i m e s t h e v a r i a n c e o f/ - d i f f e r e n c e s f o r
p o s t w a r G N P , a n d t h e same p a t t e r n is e v i d e n t . B o t h t h e v a r i a n c e o f
f i r s t d i f f e r e n c e s a n d t h e v a r i a n c e o f t h e r a n d o m w a l k c o m p o n e n t ar e
l o w e r , b u t t h e i r p r o p o r t i o n s d o n o t change m u c h . 2
2
The pattern of fig. 2 is sensitive to the precise specification of the variables. First,
thevarianceofquarterlydifferences ofseasonally adjustedG N Pislessthan one-fourth
the varianceof yearlydifferences, so the varianceratio ishigher ifone usesquarterlyrather thanannualdifferences in the denominator. This observation explainsmost of
thedifference between fig.2andthe resultsreportedbyCampbellandMankiw ( 1 9 8 8 ) ,
who use asimilartechnique on quarterlydata.Second, takingthe varianceofoverlap-
pingA-yeardifferences ofquarterlydatavs.thevarianceof /t-differences ofannual
averages, including or excluding population growth, taking logs or not, and even
changing the sample by a few years can all change the variance ratio by about one
standard error.
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Z
1^ '
^^
00)
G4 1 000> ^ 'm
00 ^ )in 00 CD ' ^
> ^ 00
00 - ^
g o o
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F I G . 2.Log real per capita GNP, 1869-1986
00
' . , , , , ,
1/k var k-differences
1 \/
- - - standard errors
/
/
\ ^
\
\
\
\
\
^ ^ ^ ^ \
\ -------
\
1 l_ ^ 1 ' 1\ I I ' I I
0. 5 10 15 20
F I G . 3. IIktimes the variance of ^-differences of log real per capita GNP, 1 9 4 7 - 8 6 ,
with asymptotic standard errors .
QOl
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go2 JOURNAL OF P O L I T I C A L ECONOMY
Ro me r (1986) ar gu ed that pr ew ar G N P data overstate the actual
cyclical varia bili ty of GNP. This possibi l i ty w i l l n o t bias th e esti mate o f
the variance o f the r a n d o m walk component . Taking ^-d i f fe rences
acts as a f i l t e r tha t ignores cyclical fluctuations an d concentrates o n
the variabil i ty o f lon ge r "run s," so a di ff er en t G N P data set w i l l have a
di f feren t var ian ce o f A-differ ences i f th e early G N P has a significantly
d i f f e r en t and mo re var iable t r en d l ine , no t i f its cyclical fluctuations
are diff er en t. A gr a ph si mil ar to fi gu re 1, us in g Rome r's adjusted
early GNP series, produce s a variance o f a r a n d o m walk c o m p o n e n t
very sim ila r to tha t o f fi gu re 1. I t sh ou ld because Romer kept the
decade t rends the same i n he r correct ions fo r cyclical v o l a t i l i t y . H e r
cri t icism, or the seasonal adjus tmen t o f qua rte r l y data , w i l l affect the
variance of f i r s t differences, so the variance ratio can be biased byexcessive v o l a t i l i t y o r smoothness of the f i r s t differences.
T h e presence o f a spli ce i n 1947 also does no t dr iv e the result .
Every l o n g series o f G N P data contains at least one splice. T h e wi de
surveys used to co ns tr uc t lat erdata are s imp ly no t available fo r earl ier
pe riods, so some pr oj ec ti on us in g a res tric ted set o f indu stri es is un
avoidable . However, fo r c i ng the levels o f the "o l d" an d "new" G N P
series to match at a certain date does no t bias the varia nce o f k-
diff ere nce s. I t is biased only i f the o ld series has d i f fe ren t gr ow th
rates over long hor izons .
T h e bod y o f this paper consists o f an i nves tiga tion o f 1 Ik times the
variance o f ^-differences as an estimate o f the ra n d o m walk compo
nen t in G NP . Sect ion I I provi des severa l inte rpr e ta t ion s o f a r a n d o m
walk c o m p o n e n t . Se ct io n I I I discusses est ima t ion. Sect ion I V recon
ciles these results w i t h previous research tha t f ou nd a la rge r a nd om
walk co mp on en t by sh ow in g ho w con ven t io nal t ime-series es t ima t ion
techniques can pr ov id e mis lea din g estimates o f a r a n d o m walk com
ponent . Section V conta ins a s u m m a r y a n d c o n c l u d i n g remarks .
I I . Unit Roots and Random Walk Components
T h i s section discusses an d docum ent s severa l c la ims i n the In tr od uc
t i o n about the repr ese nta t ion o f t im e series. It shows that f i r s t -
di f fe rence sta t ionary t ime series or t ime series w i t h a u n i t root are
equivalent to t ime series tha t are comp ose d o f a stat ion ary and a
r a n d o m walk componen t . I t argues th at the vari an ce o f shocks to a
r a n d o m walk co mp on en t is ju s t a conveni ent in t e rp re t a t i on o f the
parameters of an arbi t rary f i rs t -di fference s ta t ionary series, but i t
requires no addi t i ona l s t ru ctur e . I t shows ho w to t r an sf or m between
the variance of a r a n d o m walk component and the response o f long-
t e r m forecast s t o a shock .
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R A N D O M W A L K I N G N P 903
Assume tha t log G N P fol lows a f i rs t - di ffer ence s ta t ionary l in ear
process; that is, g r o w t h rates o f G N P are s ta t ionary. I n this case, l o g
GNP has a m o v i n g average repre senta t ion o f the f o r m
00
byt = ( 1 - L)yt = - + A(L)et = + ^ ajet-j9 (3)j= 0
w h i c h I take as the s tart ing point; L is the la g op er at or , Lyt = yt- \. T h e
first equal i ty defines the nota t ion kyt and (1 L)yt f o r first differences
o f yt. T h e last eq ua li ty defines t he lag p o l y n o m i a l n o t a t i o n A ( L ) . The e,
a r e i ndependen t iden t ical ly d i s t r i b u t e d ( i . i . d . ) e r ro r t e rms w i t h c o m
mo n variance af .
T h e r a n d o m walk process ( 2 ) obvi ous ly has a rep rese nta t ion o f the
f o r m (3) . T h e t re nd-s ta t io nary process ( 1 ) is a l i m i t i n g case o f (3): i f\i
= b an d i f the lag p o l y n o m i a l A(L) i n (3) has a u n i t roottha t i s , we
can express A(L) = ( 1 - L)B(L)we recover ( 1 ) by can cel ing the
t e rm s ( 1 L ) . M a n y unob serve d com pone nts models a re first-
d i f fe rence s ta t ionary and hence have a rep rese nta t ion (3) . Nel son a nd
Plosser ( 1 9 8 2 ) and Wat s on ( 1 9 8 6 ) are exampl es . O n the oth er ha nd ,
(3 ) does no t i nc l ude non l i nea r processes such as Quah ( 1 9 8 6 ) , a pro
cess w i t h a no nl in ea r t r en d, or second-di fference s ta t ionary processes
(the g r o w t h rates o f G N P fo l low a r a n d o m walk) as i n C l a r k ( 1 9 8 7 ) .
Gi v en the rep re sen ta t io n (3), we have the f o l l o w i n g fact.
FACT 1 . A n y first-difference st at io nar y processes can be re pr e
sented as the sum of s ta t ionary and r a n d o m walk componen t s .
T o show that a rep res ent a t i on as s ta t ionary plus r a n d o m walk c o m
ponents exists, we s imp ly const ruct i t f r o m the representa t ion (3) .
T h i s d e c o m p o s i t i o n comes f r o m Bever i dge an d Nelson ( 1 9 8 1 ) . L e t
yt = zt + ctf (4)
w h e r eoo
zt = + zt-i + ( ]T aMtf\j = o '
oo oo oo
- c t = ( Xfl;)* + ( Z fl;)'-i + ( X
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904 J O U R N A L O F P O L I T I C A L E C O N O M Y
F r o m the def in i t ion (4) we can wr i t e the var iance o f the r a n d o m walk
c o m p o n e n t v \ z in terms of the m o v i n g average representat ion (3) :
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R A N D O M WA L K I N G N P 9 5
bt is i r r e le van t fo r this ar gu me nt , so the i nn ov at io n var iance o f every
decompos i t i on (6) o f th e same m o v i n g average representat ion (3)
must have the same vari ance o f shocks to the ra n d o m walk c o m p o
nent . Th i s ar gu me nt demonstra te s fact 2.
Th er e is one mo re in te r pre ta t ion , w h ich w i l l be use ful in the ne xtsect ion. The spectra l dens i ty
5 o(kyt is, by (1), SAy(e~lUi) = | A ( e " " ) | 2 a 2 .
Th er ef or e , we have the f o l l o w i n g fact.
FACT 3. T h e in no va t i on var iance o f the ra n d om walk component i s
equa l to the spectral densi ty o fAyt at frequency zero, that is,
Equ ati ons (8) a nd (8') su mm ar iz e thre e equi val ent ways o f l o o k in g
at the lo n g - r u n pr op er ti es o f a series: we can break it in to p e r m a n e n t
( r a n d o m walk) an d te mp or ar y (s ta t ionary) comp one nts , we can exam
ine the response o f lo n g -t er m forecasts to an in no va ti on , or we can
ex ami ne the spectral densi ty at fre que ncy zero o f i ts f i r s t differences .
A l l three in te rpre ta t ions allow us to t h i n k o f the perm anen ce o f the
fluctuations i n a series as a con ti nu ous ph en om en on ra the r th an a
discrete choice. Fu r t h er mo re , equati ons (8) an d (8') show tha t the
quant i ty a | 2 o r (j\j(j\y de f ined f r o m the Bev eri dg e an d Nelso n de
compos i t ion (3) is no m or e th an a usef ul int er pr et at io n o f the su m o f
the m o v i n g average coefficients Say. The decomposition in to s ta t ion
ary a n d r a n d o m walk c o m p o n e n t s adds no s tructure .
T h e var iance o f shocks to the ra n d o m walk component or spectra l
density at frequency zero of f i r s t differences also captures all the
effects o f a u n i t ro ot o n the beh avi or of a series i n a f i n i t e sample. As asample ofT observati ons o f a series is com pl et ely chara cteri zed by its
T 1 auto cova rian ces, it is also co mp le te ly cha rac ter ize d by T 1
p e r i o d o g r a m ord ina tes . B y c h a n g i n g the p e r i o d o g r a m o rd i n a t e at
f requen cy zero o f f i r s t differences w i t h o u t changing the others , we
can make a s tationary series in to a series w i t h a u n i t roo t o r r a nd om
walk component and vice versa .6
Since th e size o f a r a n d o m walk co mp on en t is a con tin uou s choice,
any tes t fo r t re nd s ta t ionar i ty ( a | 2 = 0 o r SAy(e~t0) = 0) mu st have
arb i t r a r i ly low po we r against the alte rnat ive o f a small en ou gh ran -
5
I use the notation S(e~tbi) for the spectral density at frequency u> and, hence, S(e
l
)
for the spectral density at u> = 0.6
With an infinite sample, or in population, this proposition does not hold. The
spectral density is defined only almost everywhere; and in some cases we can bound the
variation of the population spectral density function with very weak assumptions.
or , d i v i d i n g by the varian ce o f first differences ,
(8 ' )
(8)
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go6 JOURNAL OF POLIT ICAL ECONOMY
d om walk c o m p o n e n t v \ z. As a result, efforts to categorize series as
trend-sta t ionary or difference-sta t ionary and read great things i n t o
the difference between the two w i l l not be very f r u i t f u l .
I I I . Est imat ion
I cla im ed in the I n t r o d u c t i o n tha t the variance o f ^-differences cou l d
be used to estimate the i n n o v a t i o n var iance o f a r a n d o m walk com
ponent . T o d o c u m e n t that cla im an d to pr ov id e s tand ard erro rs ,
this section discusses the statist ical pro per tie s o f the variance o f k-
differences .
Asymptotic Properties
Le t
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RANDOM WALK IN GNP 907
T h e prope r t ies o f the Bar t le t t es t ima tor are w e l l k n o w n , so we can
establish the asymptotic properties of l/k times the variance of k-
dif fere nces by ref ere nce t o those o f the Bar t le t t es t imator . I n par t i cu
lar , (1) i f //> 0 as T > 00, w h e r e T is the sample size, l/k times the
sample variance of / -d i f f e r ences is a cons ist ent esti mate o f th e spect r a l dens ity at fre qu enc y z ero; (2) the as ymp tot ic vari ance o f 2 is
4kS2
(e-l0
)/3T (A nd er so n 1971 , p . 531) .
The equivalence between l/k t imes the variance o f / -d i f fe rences
an d the Bar t l e t t es t im ator prov ides a useful in te rpr e t a t io n o f the
vari ance o f /^-differences fo r readers famil iar w i t h spectral density
es t imat ion; i n t u r n , the variance of / -d i f fe rences is a useful and i n t u i
tive t i me do ma in co un te rp ar t to the Bar t le t t spectra l densi ty es
t imator . T o use the B ar tl et t est ima tor , we have to deci de wh at k to
use: ho w ma ny autocovariances or autoco rrel ati ons to in cl ud e i n (9)
or how many per iodogram ordinates to smooth. The choice of k re
quires a tr ad e- of f bet ween bias an d efficiency, an d it is usua lly ma de
a rb i t ra r i ly . I n this con tex t, a plo t o f l/k times the variance of k-
differences versus k is an expe r im ent a l de t e rm in a t i on o f the pro per k
or wi n d o w w i d t h .
Small-Sample Properties
I n sma ll samples, l/k t imes the variance of / -d i f fe rences and the Bar t
lett esti mat or can be biased, an d the asymp toti c sta nda rd e rro rs may
be a p o o r a p p r o x i m a t i o n t o the actual s t andar d er ro r s . I n this subsec
t i o n , I discuss corrections for small-sample bias, and I present some
M o n t e Ca rlo exp er i men ts to evaluate s tan dard er ror s .
I correc ted for two sources o f smal l-sa mpl e bias i n the sam ple v a r i
ance o f / - d i f f e r e n c e s . These corre ctio ns pr od uc e an esti mato r o f cr 2
tha t is unbiased w he n ap pl ie d to a pu re r an d o m walk w i t h d r i f t . First ,
I used the sample m ea n o f the first dif fere nces to estima te the d r i f t
t e r m p, at all k ra t her th an es t imate a di f f ere nt d r i f t term at each k
f r o m the mea n o f the / - d i f f e r e n c e s . Second, I in cl ud ed a degrees o f
f r e e d o m c o r r e c t ion TI(T k + 1) . W i t h o u t th i s cor rec t ion , l/k t imes
the variance of / -d i f fe rences declines toward zero as k> T for any
process because you cannot take a variance w i t h one data point.
I w i l l use the not at io n d2
to denote l/k t imes the bias-corrected
samp le var ianc e o f /^-differences. T h e f o r m u l a for d2
is pre sen ted i n
the A p p e n d i x as equ at i on (A 3) . T h e A p p e n d i x a lso conta ins a p r o o f
that d2
is unbi ased w h en yt is a random walk w i t h d r i f t .
Table 2 presents s tandard e r ror s f r o m a Monte Car lo exper iment
us ing 100 observations o f a r a n d o m walk w i t h d r i f t . I picked the
i nnova t ion var iance of th is random walk cr2
= a | 2 = 1. T h e me an o f
d2
was very close to one at all kin th i s exp er i men t , c o n f i r m i n g the bias
correc t ions for a pure random walk. The table presents the s tandard
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9 8 J O U R N A L OFP O L I T I C A L E C O N O M Y
T A B L E 2
M O N T E C A R L O S T A N D A R D E R R O R S FOR \lk T I M E S T H E V A R I A N C E OFk D I F F E R E N C E S
Model: =1 + yt-1 + /; erf = \(T = 100, 500 trials)
\00k/T
1 2 3 4 5 10 20 30 40 50
Monte Carlo .137 .160 .200 .231 .263 .409 .607 .772 .888 .896
Bartlett* .115 .163 .200 .231 .258 .365 .516 .632 .730 .816
*Thisrowgives(4k/3T);).
e r r o r s f r o m t h e M o n t e C a r l o e x p e r i m e n t a n d t h e c o r r e s p o n d i n g
B a r t l e t t s t a n d a r d e r r o r s f o r c o m p a r i s o n . T h e B a r t l e t t e r r o r s s l i g h t l y
u n d e r s t a t e t h e M o n t e C a r l o e r r o r s at l a r g e kIT, b u t t h e d i f f e r e n c e is
s m a l l c o m p a r e d t o t h e size o f t h e s t a n d a r d e r r o r s . M o n t eC a r l o e x p e r -
i m e n t s w i t h d i f f e r e n t s a m p l e sizes a n d r a n d o m w a l k v a r i a n c e c o n f i r m
t h a t t h e s t a n d a r d e r r o r s o f t a b l e 2 scale w i t h kIT a n d t h e i n n o v a t i o n
v a r i a n c e o f t h e r a n d o m w a l k .
W h a t a b o u t processes t h a t ar e m o r e c o m p l i c a t e d t h a n a p u r e r a n -
d o m w a l k ? T h e A p p e n d i xpresents a d e r i v a t i o n o f ( d2
) f o r a first-
o r d e r m o v i n g a v e r a g e : (1 L)yt = p + (1 + 6L)e,. I t s how s t h a t
( d2
) approaches 2 f o r l a r g e k, so cr2
c a n r e c o v e r t h e v a r i a n c e o f t h e
r a n d o m w a l k c o m p o n e n t f o r t h i s process as w e l l .
I r a n s everal f u r t h e r M o n t e C a r l o s i m u l a t i o n s t o e x a m i n e w h e t h e r
t h e v a r i a n c e o f / - d i f f e r e n c e s is r o b u s t w h e n a p p l i e d t o m o r e c o m -
p l i c a t e d processes f o r G N P . I fit a v a r i e t y o f A R M A processes t o first
d i f f e r e n c e s o f l o g r e a l p e r c a p i t a G N P , s i m u l a t e d 118 o b s e r v a t i o n s o f
each process, a n d c o m p u t e d d2
i n 100 t ri a l s . I n each case, t h e m e a n o f
d2
at k = 3 0 wa s e q u a l t o t h e v a r i a n c e o f t h e r a n d o m w a l k c o m p o n e n t
i m p l i e d b y t h e e s t i m a t e d A R M A processesk = 3 0 wa s l a r g e e n o u g ht o i d e n t i f y t h e r a n d o m w a l k f r o m t h e s t a t i o n a r y c o m p o n e n t s a n d
t h e s t a n d a r d e r r o r s at l a r g e k w e r e close t o those i m p l i e d b y t a b l e 2,
scaled t o t h e v a r i a n c e o f t h e r a n d o m w a l k c o m p o n e n t .
A l l t h e l o w - o r d e r A R M A processesp r o d u c e d d 2 l i n e s t h a t rise f o rk
f r o m 1 t o 5 a n d t h e n a re flat at t h e v a r i a n c e o f t h e r a n d o m w a l k
c o m p o n e n t f r o m k = 10 o n , u n l i k e figure 1.T h e y i m p l i e d a | 2 > |
T w o processes t h a t d o c a p t u r e t h e b e h a v i o r o f figure 1a re a n A R ( 1 5 ) ,
figure 4, a n d A R ( 2 )a b o u t a d e t e r m i n i s t i c t r e n d , figure 5. I n t h e n e x t
s e c t i o n , I w i l l discuss w h y t h e l o w - o r d e r A R M A m o d e l s f a i l e d t o
c a p t u r e t h e b e h a v i o r o f figure 1. F o r n o w , n o t e t h a t since t h e y r e p l i -
cate t h e b e h a v i o r o f d2
f o r G N P , figures 4 a n d 5 c a n p r o v i d e s m a l l -
s a m p l e s t a n d a r d e r r o r s . These s t a n d a r d e r r o r s are s i m i l a r t o t h e
a s y m p t o t i c s t a n d a r d e r r o r s u s e d i n figure 1.
F i g u r e s 4 a n d 5 also i n c l u d e d2
f o r G N P f r o m figure 1, m a r k e d
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F I G . 4.Monte Carlo simulation of an AR(15 )
F I G . 5.Monte Carlo simulation of an A R ( 2 ) with a linear trend
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g i O JOURNAL OF POL IT ICAL ECONOMY
d f ( G N P ) . Since the d f ( G N P ) line falls ins ide the one-s tandar d-erro r
bands, neither model can be rejected for real GNP. However, the
standard er rors f r o m t h e r a n d o m walk (table 2) o r any o f th e ot he r
l o w- o r d e r A R M A processes are lar ge en o ug h that we can no t reject
t h e m at 5 perc ent ei ther . (No te tha t the stan dar d er ror s scale w i t h the
size of the random walk com po ne nt . Un d e r the hypothesis o f a ran
d o m walk, the sta nd ar d e rr or s are big ge r th an in di ca te d i n fig. 1.) A
conf idence in te rva l inc ludes bo th v \ z l a \ y = 0 a n d 1.
W hi le th is is unfor tunate , I w i l l argue below that estimates o f a
r a n d o m walk component a re l i m i t e d by the number of nonover lap-
p i n g " l o ng runs" in the data set, so that large efficiency gains are not
possible w i t h o u t im po si ng ad di t i on al s t ru c tur e on the t ime-ser ies pr o
cess for GNP. As a result, this and related exercises can provide apo i n t estimate o f th e size o f a r a n d o m walk c o m p o n e n t w i t h associ
ated standard errors but w i l l not provide useful tests to d i sc r imina te
be tween models tha t i m p l y var ious sizes o f the r a nd om walk c om
ponent .
T h e parameters of the A R ( 1 5 ) m o d e l i m p l y that the variance ratio
.18, while the AR (2 ) about a t r en d impli es v\z/v\y = 0.
He nce , the si mul ati ons b e h i n d figures 4 and 5 also reveal an upward
bias i n d f as an estimate o f the r a n d o m walk component when the
series has a small r a n d o m walk component or is t rend-sta t ionary.
I n s u m m a r y , Ilk times the variance of/ -d i f ferences d2
provides an
upw ard -bi ase d po in t estimate of the variance rati o | 2 / | of about
.34, and two models w i t h a ^ / a ^ = .18 an d 0 replicate the behavio r o f
the variance of / - d i f f e r ences o f GN P. How eve r , s tan dard er rors are
larg e e n ou g h tha t we ca nn ot statistically reject vari ance ratios be twe en
zero an d one at co nv en ti on al levels o f signifi cance.
I V . Reconci l ia t ion wi th Prev i ous Est imates
Given the def in i t ion o f the r a nd om walk co mp on en t in te rms o f the
parameters o f a m o v i n g average rep re se nt at io n, (4) or (8) above, the
obvious t h i n g to d o is ei th er to estimat e a pa rs im on io us time-series
m o d e l f or Ay, an d c alcu late 2 a 7 or to ident i fy and estimate a simple
pa ramet r i c unobse rved com ponen t s m o d e l l i k e (4) . Ca mp be l l a nd
M a n k i w (1987) and Nelson and Plosser (1982) di d ju st that, respec
t ive ly , a nd bo th f o u n d l a r ge r a ndom walk components . W h y do N e l
son and Plosser an d Campb e l l and M a n k i w find l a r ge r a ndom walk
components , whi le Wa ts on (1986 ), Cl ar k (1987) , an d I find small
ones? T h o u g h there are small d if ferences in d e f i n i t i o n w h i c h qua n
tities we look at to measure the impor tance of u n i t roo ts o r r a n do m
walk co mp on en ts t he maj or dif fer enc e is i n est ima tio n strategies.
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RANDOM WALK IN GNP
N e l s o n a n d Plosser spec i f ied an unobse rved components m o d e l o f
the f o r m
yt = ut + vt,
(1 - L)ut = + A ( L ) e e, i . i . d . , (11)
vt = B(L)bt, bt i . i . d .
(et a n d bt may be cor r e la ted) . They i d e n t i f i e d the two components
f r o m a s t yl i ze d a u toc o r r e l a t i o n f u nc t i o n o f G N P g r ow th rates. I f t h e
first a u t o c o r r e l a t i o n o f Ayt is posit ive but the others are zero, th en the
on ly m o de l o f t he f o r m (11) that works is A(L) = 1 an d B(L) = (1 +
0L) . B y e xa m in ing p l a us ib l e parameter values fo r th is res t r ic ted
m o d e l , Ne l s on a nd Plosser co nc l ud ed tha t c r 2 > o"!-8
C a m p b e l l a n d M a n k i w (1987) e s t ima ted pa r s imo nio us A R M A r e p
resentations o f l og G NP , u s ing seasonally ad jus ted quar te r ly pos twar
data. T h e y measured t he im p or t a nc e o f t he r a nd om walk c o m p o n e n t
by Sa 7 = A ( l ) , the change i n zt ( the lo ng - t er m forecas t ) in response to
a un i t un iva r i a t e i nn ov a t i on i n GN P . T h e y f ou nd values f or A ( l )
equal to or la rge r th an one , w h i c h i m p l y an in no va t i on va r iance of the
r a n d o m walk c o m p o n e n t greater th an the var iance o f first di f f e r ences
o f G N P . 9
8
This measure of the importance of a random walk component has the conceptual
disadvantage that it depends on which arbitrary unobserved components decomposi
tion we choose. For example, since every series of the form (11) has a unique moving
average representation, we could rewrite (11) as (1 - L)yt = fx + C{L)vt, vt i.i.d., and
eliminate the stationary component. Alternatively, we could use the Beveridge and
Nelson decomposition ofSec. I I to make the component with a unit root into a pure
random walk:
yt = zt + Ct,
(1 L)zt = | i + vt, vt i.i.d.,
ct = C{L)ih it i.i.d.
These representations are observationally equivalent to the first form (11), but the
measure
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Q12 JOURNAL OF POLIT ICAL ECONOMY
I n pe r f or mi ng the M on te Car lo s imula t ions o f Sec tion I I , I a l so
f o u n d t ha t l ow - o r de r A R M A models o f G N P i m p l y that cr2 should r ise
w i t h k, and they i m p l y a la rge r andom walk c o m p o n e n t , whi le in fact
cr2 decl ines and the es t imated random walk co mp on en t is smal l . T o
replic ate the beh avi or o f cr 2 for GNP, I had to estimate an AR(15) o r
impose a de termin is t ic t r end .
T o inves tigat e this fact fur t he r, I fit a var iet y o fA R M A processes to
G N P g r o w t h rates, r a n g i n g f r o m wh it e noise ou t to an AR(15) (see
table 3) . 1 0 All r epresen ta t ions past white noise are adequate by usual
stan dard s: the Du r b i n - W a t s o n statistics are close to 2, the signifi cance
levels o f the Q-statistic are a r o u n d .5, the pa ram et ers o foverfi t models
are statistically insignif icant, and so f o r t h . B u t the variance rat io an d
2a 7 start at about 1.2 for second-order processes and decline steadily
to a variance ratio of .18 and 2a 7 = .5 f o r a n A R (1 5 ) . L o w - o r d e r
A R M A mode ls systematically overe stima te the r a n d o m walk c o m p o
nent o f GN P , even t ho ug h they adequately represen t the series by all
the usual diagnostic tests. T h e ques tio n is , why?
T h e in nov at i on var iance o f a r a nd om walkco mpo ne nt is a p ro pe r ty
o f the very lo ng -r un beha vior o f a series alon e. I t is th e spect ral
densi ty at the fre que ncy o> = 0 co rr es po nd in g to a pe ri od or "r u n " o f
i n f i n i t y , it is related to the i n f i n i t e su m o f the m o v i n g average co
efficients | 2 a 7 | 2 or the aut oc or re la ti on coeff icients (1 + 2Sp 7 ), and it
co rre spo nds to the effect o f a shock tod ay on forecasts in to the i n f i n i t e
f u tu r e . I n the ory , th en , we sh ou ld have to wait an infinite a m o u n t o f
t ime to get ju st one obs erv ati on on the size o f the r a n d o m walk com
ponent !
I n practice, we typical ly believe that the dynamic response o f G N P
to a shock is flat after a suitable l o n g r u n has a r r i ve d . 1 1 T h i s belief is
i m p l i c i t above: the graphs stop after the t h i r t i e t h dif ference, ref lect
i ng a belief that after 30 years the temporary ef fects of business cycles
are over . T h e nu mb er o f no no ve r l ap p i ng l o n g ru ns is a r o u g h gu id e
to the number o f degrees o f fr ee do m (precisely, the n um be r o f pe
r i o d o g r a m ordi nate s) i n this exercise. W i t h a 1020-year l o n g r u n
there are no more than five to 10 in de pe nd en t observations in 100
years o f data an d tw o to f o u r observatio ns i n postw ar data. Obv iou sl y,
us ing mo re f re que nt l y samp led data does not help .
Es t imat ing an uno bse rve d com pon ent s mo de l or a pars imo niou s
1 0
I used the RATS program to perform the estimation. Autoregressive models are
estimated by ordinary least squares and moving average models by conditional max
imum likelihood. The unreported moving average models did not converge.1 1
Precisely, if the coefficients of the moving average representation (1) are zero past
a long-run value M < o, then the derivative of the spectral density of Ay at zero is
bounded. Ifyt is in fact trend-stationary and the spectral density ofA3; at frequency zero
is in fact zero, then the slope of the spectral density of A3; at zero is also zero.
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*
CM
CM 11 11
1 1 C^
oo"CMCO 00
cr>
CM
CM00i n i n CO cr>CD IT) CO CM
^CM
cSCOCM
CD00CM CO
(U> ^
~
I 3 CJ
2 5-" !_ _^
>^ CD
) N N 00 i n CM
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9 4 JOURNAL OF POL IT ICAL ECONOMY
A R M A mo de l is an a t te mpt to c i rc um ve nt this pr ob le m. These m o d
els make i den t i fy ing res t r ic t ions across frequencies: they dr aw i n
ferences about the lo ng - r un (h ig h-or der au toc or re la t ion or low-
frequency) dynamics f r o m a mo de l f i t to the sho rt - r un ( low -or der
autocorre la t ion or high-frequency) dynamics . For an example that
demonstra tes how "effect ive" these procedures are , Campbel l and
M a n k i w (1987) repor t estimates suc h as A ( l ) = 1.306 .073 fo r th e
20-year forecast o f G N P . Since th er e are only t w o n o n o v e r l a p p i n g
20-year forecasts i n th ei r data set, it is clear how heavily their esti
mates o f A ( l ) dep end on t he i den t i fy ing assumpt ion that the series
f o l l o w a g iven low-order A R M A m o d e l .
I f the short - an d lo n g- ru n dynam ics o f G N P can bo th be cap tur ed
by the assumed t ime-series model, these pro cedu res can he lp estimat i o n because we have m u c h m o r e data o n h ig h - fr eq uen cy fluctuations.
Howeve r , i f the lo n g- ru n dyna mics cannot be cap tur ed i n the m od el
used to s tudy the sho rt ru n , these ident i f ica t ion pro ced ure s bias con
clus ions about long-run behavior .
I offer t wo ways to see th is fact. Firs t, rec all th at th e vari anc e o f the
shock to the random walk co mp on en t is rela ted to the su m o f the
autocorre la t ions by
W h e n we model short - run dynamics , we safe ly ignore high-order
statistically ins igni f icant autoco rre la t ions or we slightly misspecify
them by fitting a sim ple mo de l. B u t al l auto cor rel at io ns ente r i nt o (12)
equally, so a large nu mb er o f smal l hi gh -o rd er autocorre la t io ns can
offset a few large low-order autocorre la t ions .
Second, G N P g r o w t h has a posit ive au to co rr el at io n at shor t lags
and a smal l r a n d o m walk co mp on en t at lo n g lags. A sim ple t im e-series mo d el may no t be able to cap tur e b ot h kin ds o f beha vior . For
exa mpl e, i f (1 - L)yt = |x + (1 + 0L)e ,, we nee d 0 > 0 to captu re
posi tive first-order a ut oc o r re l at io n b u t 0 < 0 to ca pt ur e a sm al l ra n
d o m walk component . Faced w i t h a cho i ce , max i mum l i ke l ihood esti
mates ma tc h the sh or t- ru n be hav ior (they fit 0 > 0 i n the exa mpl e)
and mis represent the lo ng - r un behavior .
T h e Ap p e nd i x contains a dem ons t ra t io n of th is pr op er ty o f max
i m u m l i ke l ihood es timates. I t shows that ma x i m u m l i ke l ihood esti
mates o f a mo de l such as a lo w- or de r A R M A or a s imple para metr i c
unobserved components model pick parameters that match the mod
el's an d the actu al spect ral den sity ov er the en ti re fre qu enc y ran ge.
T h e r e f o r e , m a x i m u m l i ke l ihood w i l l sacrifice accur acy i n th e smal l
r e g i o n a r o u n d = 0 to bet ter m at ch spectral densities at hi gh er
frequencies.
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RANDOM WALK IN GNP 915
I n summary , the low-order A R M A appro ach o f Cam pbe l l and
M a n k i w an d the unob serv ed compo nen ts app roa ch of Nels on an d
Plosser cannot match the shor t- run dynamics and the small random
walk co mp on en t i n the lo ng -r un dynami cs a t the same t ime . Faced
w i t h the choice , they capture the shor t- run dynamics and incorrec t ly
i m p l y l a r ge r a ndom walk c ompone n t s .
O n the ot he r han d, Clark' s (1987) an d Watson's (1986) deco mpo si
t ions can accomm odate the behavior o f G N P i n bo th f requency
ranges. (See, e.g., Watson's fig. lb, i n whic h he shows how his model
can represent a large nu mb er o f small h i gh -o rd er autoco rre la t ion s
that a low-order A R M A cannot match. ) B o t h Watson and Clark find a
small r a n d o m walk com po nen t . How eve r , the ir decompos i t ions a lso
i m p l y i den t i fy ing res tr ic t ions to es t imate l on g- ru n behavio r f r o ms h o r t - r u n dynamics. Since these restr ic tions are no mo re or less p la u
sible than Nelson and Plosser's or Camp be l l and Mankiw ' s , they m i g h t
no t be able to cap tu re the pat te rn of hi gh -o rd er corre lati ons i n ot her
data sets as they seem to do for GNP.
Since the size o f th e ra n d o m walk co mp on en t is a pr op er ty o f the
p e r i o d o g r a m ordinate at frequency zero alone, any estimation tech
nique must make some i den t i fy ing re s t r ic t ion across the frequency
rang e. T h e vari ance o f /^-differences assumes tha t past a certain k the
r a n d o m walk co mp on en t is adequate ly ident i f ied , empir ica l ly deter
m i n e d as the point in whic h th e g r a p h (fi g. 1) flattens o ut . T h e r e f o r e ,
the variance of / - d i f f e r ences (or any other spectral w i n d o w est imator)
uses 1020-year period i n f o r m a t i o n to iden t i fy the i n f in i t e - run p r o p
erty, t h e r a n d o m walkco mp on en t. T h e variance o f /^-differences does
no t use i n f o r m a t i o n about dynamics at business cycle freq uenc ies to
iden t i fy l on g- ru n move ment s , an d this is i ts im po r t an t advantage .
V . C o n c l u s i o n
T h e variance o f ^-diff erences (fi g. 1 an d table 1) pr od u ce d a po in t
est imate tha t the inn ov at io n var iance o f the ra n d o m walk c o m p o n e n t
o f G N P is abo ut on e- th ir d the variance o f yearly G N P g r o w t h rates.
T h a t estimate is u p w a r d biased for small r a n d o m walk c ompone n t s :
the parameters o f tw o mode ls that repl ica ted the behavi or o f the
variance o f ^-differences o f G N P i m p l i ed variance ratios of .18
(AR(15)) and 0 (AR(2) about a de te rmin is t i c t re nd) . I concl ude tha t i f
ther e is a r a n d o m walk co mp on en t i n G N P at all , i t is small .
A n o t h e r way to characterize these results, w i t h o u t reference to ran
d o m walk comp onen ts , is tha t GN P g r o w t h is posit ively autocor-
rel ate d at sh ort lags, b ut th er e are ma ny small negat ive a ut oco rre la
t ions at l o n g lags. These b r i n g fu tu re G N P back towa rd , i f not a l l the
way back to, its pr ev io us ly forecast val ue f o l l o w i n g a shock.
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gi6 JOURNAL OF P O L I T I C A L ECONOMY
These results do not mea n tha t "G NP follows an AR (2 ) about a
determi nis t i c t r en d. " O u r forecasts o f the fut ur e may qui t e r igh t ly be
m u c h mo re var iable th an the " t r en d" i n G N P we have seen in the
recent 118-year past m i g h t sugges t . 1 2 T h e s e results do mean that an
AR (2 ) about a det erm ini s t i c t re n d or a d if ference -s tat iona ry A R M A
process w i t h a very smal l r a n do m walkco mp on en t is a go od in-sample
charac ter iz at ion o f the beha vior of GN P .
I n r econc i l i ng these results w i t h pre vio us research, I ar gu ed that
c onve n t i ona l cr i t er ia fo r t ime-ser ies m od el ident i f icat ion and estima
t i on can pro duc e mis lea din g es timates o f the r a n d o m walk compo
nen t o f a series l ike G N P . T h e r a n d o m walk com pon ent i s a p ro per ty
o f al l autoco rrelat ions tak en togeth er , but con ven t io nal procedure s
conce ntrat e on the first few autoc orrel at ion s in or de r to par s im oni
ously cap tur e s ho rt -r un dyn ami cs . W h e n used to estimate the size o f a
r a n d o m walk component , they impose iden t i fy ing res t r ic t ions across
the frequency range to i n f e r the lon g- ru n proper t ies o f a series f r o m
it s sh or t - ru n dynami cs . I ar gu ed that , i n the absence of credible iden
t i f y i n g restrictions, it is best to leave the sho rt r u n ou t alt oge the r, as
the variance o f -differences or some other spectral w i n d o w es t imator
does.
H ow ever , th is v i ew t ha t we shoul d use only l ong - r un p r oper t i e so f G N P data to es t imate the lon g- ru n behav ior of GN Pimp l i e s t ha t
s tan dard er ror s of uni var iat e es timates o f the ra n d om walk c o m p o
nen t w i l l r e ma in la rge in ce n tu ry- l ong macroeco nomic da ta and
larger s t i l l i n postwar macr oec onom ic data because there are inher
ent ly f ew n o n o v e r l a p p i n g l o n g runs available. These observations ar
gue against the research strategy that says that the presence o f a u n i t
roo t an d the size o f a r a n d o m walk co mpo nen t a re c ruc ia l and w e l l -
do cu me nt ed st ylized facts tha t any the oret ica l mo de l mus t replicate.
Appendix
A. Derivation of Equation (9)
Start with00
( 1 - L)yt = jx + A (L ) e , = + XaJ*'-r
7 = 0
1 2
A plausible model for G NP should have some random walk component. I fG N P is
truly stationary about a linear trend, then the variance of the forecast error of the level
of G N P is the same for all dates in the far future. As long as there is some random walk
component , the variance of forecast errors will grow unboundedly over the forecast
horizon. However, only a very small random walk component is required to achieve this
desirable property.
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R A N D O M WALK I NG N P
Us ing
(1 - L*)( l L)~l = (1 + L + L 2 + . . . + Lk~1)yk - 1 J J
yt - yt-k = k\L + X ( X FL
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9 i 8 J O U R N A L OF POL IT ICAL E C O N O M Y
Note that (1)as E(&2k) [(1+ 8)2 - (28/)]cr
2
; (2)as ,k. (A7)The first term isjust (A5).SinceA
1
= B, the second term is 2,and the
t h i r d and f o u r t h are
* (BA +B*A*)do>.
Under the assumption thatAandare one-sided and that A(0) = B(0) = 1,
00 00
j " BAdu = j " + X V " ^1
+ X aje-^dm;
since J*% ~
> = 0, BAdo> = B*A*do>= 2.Therefore, (A6)
reduces to (A5)plus constants.
Equation (A6)isanalogous toSims's (1972)approximationfo rmu l a , repro-
duced in Sargent (1979, p. 293). The message of (A6) is that maximum
likelihood attempts to match the frequency response of the autoregressive
representation across the entire frequency range, weightedby the true spec-
tral density o fxt. Themethod ofmaximumlikelihood w i l l sacrifice accuracy
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RANDOM WALK IN GNP 919
of the estimated (~
) at a point in the frequency range ( = 0) in order to
achieve a better fit over an interval. Similarly, it wil l sacrifice accuracy in a
small window (20 years to infinity is /10 wide) to gain accuracy in a large
window (24 years is /2 wide). I fSx(e~lt,>
) is smaller near = 0 th an else
where, as the variance of ^-differences suggests for GNP, then (A6) showsthat maximum likelihood further deemphasizes accuracy in windows about
zero.
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