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working paper
department
of economics
WHAT DO WE LEARN FROM UNIT ROOTS
IN MACROECONOMIC TIME SERIES?
by
Danny Quah
Number 469
October 198 7
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
WHAT DO WE LEARN FROM UNIT ROOTS
IN MACROECONOMIC TIME SERIES?
by
Danny Qa ah
Niimber 469
October 1987
i'5,s. sfiSf.
MP^ 23 1988
LIBRARIES
What Do We Learn from
in
Unit Roots
Macroeconomic Time
Series?
Danny Quah*
MIT
October 1987.
Economics Department
and NBER.
Department of Economics, MIT, Cambridge MA 02139. I thank Olivier Blanchard, Stanley Fischer,
James Poterba and Julio Rotemberg for useful comments on earlier versions of this paper. I also thank John
Taylor, Mark Watson and an anonymous referee for suggestions and criticisms that have helped to focus the
*
discussion.
The
hospitality of the
MIT
Statistics
,C
Center
is
008618
gratefully acknowledged.
Abstract
It is
oRen asgued
tba,t
'business cycle': the
the presence of a unit root in hggregaXe output implies that there
economy does not return
to trend following b disturbance. This
this notion precise, but then develops a simple aggregative
model where
this relation
is
is
no
paper makes
contradicted.
Id the model, output both has a unit root, and displays repeated short-run fluctuations around a
deterministic trend.
Some summary
described in the paper
is
statistical evidence is presented that suggests the
not without empirical basis.
phenomena
1
Much macroeconomic research
has focused on the evidence for and the implicatione of unit roots in aggre-
gate output. Well-known examples of this work include the contributions of John Campbell and N. Gregory
Mankiw
(1987), Peter Clark (1987),
John Cochrane (1987), Steven Durlauf and Peter Phillips (1986), Charles
Nelson and Charles Plosser (1982), and Mark Watson (1986).
view that business
Many have
cycles, relative to secular changes, are small
and
taken away from this discussion the
insignificant:
output does not fluctuate
around trend.
This paper has two main objectives. The
first is
pedagogical;
between the presence of unit roots and the phenomenon of
than
explicit
is
it
makes technically
(lack of) cycle
currently available in the macroeconomic literature.
around trend
Most studies almost
the key characteristic to be the implied path of an impulse response function.
and Mankiw, 1987. In the empirical
1987.)
literature, a notable exception
The impulse response may be
The reason
is
a property of the sample path behavior of a time
is
at best a conditional expectation,
is
a precise sense
about trend: average cycle length
The
first
is
in
in general not
We
when
(See for instance
purposes, but
not of
its
it is
ill-suited
impulse response function. The latter
is
inconsistent with fluctuations
what exactly
are important
is
are the implications of a
illustrated
by the second goal
construct a simple model of employment growth and leaming-by-doing.
is
Campbell
a linear time series process has a unit root.
In the model,
at least as explosive as that for a unit root process;
sample path for output aJways displays well-behaved
are
instinctually take
even that, but only a linear least-squares predictor.
That the points made here
the conditional expectation of output
estimated
many
which the presence of unit roots
infinite
way more
that the presence of fluctuations about trend
objective therefore serves to sharpen discussion on
process having a unit root.
of the paper.
and
series,
is
in a
Francis Diebold and Glenn Rudebusch,
a sensible statistic to examine for
to represent the absence of cycle about trend.
Nevertheless, there
is
precise the relation
finite
length cycles about trend.
Thus the
however the
finding that
ARMA representations for output have a unit root should not be identified with a view that there
no business cycles about trend. Some summary
also presented that suggests that the distinction here
The plan
of the
paper
is
as follows. Section I
is
to observable sample path behavior. This contains
statistical evidence for
is
aggregate output in the
US
is
not empirically irrelevant.
a technical section that relates the presence of unit roots
arguments that would normally be omitted, except
for
2
the fact that impulse responBe functions have been so convincing to so many. Section
economic model where unit roots are consistent with well-behaved
necessary technical results are also developed here.
Section
III
suggest some empirical basis for the main message of Section
II.
finite
11
presents a simple
length cycles about trend. Further
reports informal
summary
The paper concludes with
statbtics that
a brief Section
IV. Technical proofs are presented in an Appendix.
I.
Unit Roots and Sample Paths.
For the most part,
pure random walk with
this section will formally consider a
carry over to where the
first difi'erence is
the reader should keep in
mind the
drift.
a serially correlated stationary sequence.
distinction between difference stationary
The main
Throughout the
conclusions
discussion,
and trend stationary sequences
emphasized by Nelson and Ploaser (1982) and Campbell and Mankiw (1987).
For
>1,
t
let
Y(t) be an observed sequence generated by
r(i)
where
£ is
sequence
a zero
Y
is
For any
mean
=
/s
+ r(f-i) +
£(f),
covariance stationary sequence, with spectral density bounded away from zero.
therefore difference-stationary, in the terminology of Nelson
and Plosser
The
(1982).
tiine trend coefficient a, the deviations process
f
W{t; a)
rem.aiiiE
=
Y{t)
-at =
Y{D]
+
(^
-
a)
•
f
+ [^ e(t)
a difference stationary sequence:
W{0-a]
=
Y{0).
Thus, removing any deterministic time trend from a difference stationary sequence only produces yet another
difference stationary sequence.
unchanged, except possibly
with, there
is
in
The
probability law for the first-differenced sequence remains completely
mean. The
a presumption tha.t
some
first
lesson can therefore be stated as follows:
difference sta.tionasy sequence should display cycUcaJity, arbitrary
linear detrending cannot induce spurious cyclica.lity.
continually being re-chosen
when
unJess to begin
a given finite sample
(This
is
may
of course fail
growing through time.)
if
the "trend" coefficient
is
Next, write
f
Y{t)
=
fit + ^cU) +
Y(0).
The
Notice that the right hand side contains terms of different asymptotic order.
^
different
from
zero,
is
of
mean square asymptotic
order 0{n^) whereas the stochastic component
more
mean square asymptotic
order 0{n), where n
from cyclical behavior
the dominant deterministic trend part. However, this
is
by trend-stationary sequences (data that
this part in the
is
the sample size. Thus, the
likely
^
is
for
t,
only of
avenue for deviation
component
covariance stationary about a linear trend).
is
shared as well
Thus we
will ignore
subsequent discussion, and concentrate on the purely stochastic component. But then this
exactly a rero drift unit root sequence.
To
1
is
is
deterministic part
and
fix
-1
ideas
and to
simplifj' the calculations,
with equal probability.
suppose that
it
once the
origin in one step.
sequence, which takes the values
sum
gets sufSciently distant from an
takes a special sequence of events for the process to return to
to have infinite support (such as for a normally distributed
its
iid
anything, one would imagine that this assumption severely restricts
If
the possible cyclical behavior of the accumulation J3^(y)'
arbitrary starting point,
an
e is
The assumption
of serial
random
its origin.
If e
were
variable), X^£(y) can always return to
independence makes the calculations
easier; relaxing
it
would
change none of the conclusions.
Let W[t; P)
We
study
its
=
W{t]
=
y^^-i
£(y):
this
may
be interpreted as the deviation of the economy from trend.
sample path properties through the following
behavior of the sequence of conditional expectations
the
economy has deviated from
what
is
trend,
the probability that at time
set of questions. First, for
E [W[t -f m]
|
^''(t)]
what are expectations regarding
t -r
m, the economj'
,
its
for
m>
W[t]
=
what
0,
is
1? In words, given that
future path? Next, for W{t)
will return to trend,
W[t
the
-i-
m) =
0?
What
=
is
0,
the
average waiting time untD such a return to trend? The average cycle length would then be twice this average
waiting time.
For the process here, the conditional expectation E\W{i-r m) W{t)]
\
to the point,
forever.
when W[t]
=^ 0,
the conditional expectation of future
If c is serially correlated,
somewhat. However, provided
e
is
just W[t), for all
W[t+ m)
is
m>
1.
More
bounded away from zero
the exact pattern for the conditional expectations of future
W
changes
does not have a spectral density that vanishes at frequency zero, the main
4
conclusion that the process
in this expectation sense, only
sequence
serially correlated.
is
has to do with whether
this probability
to trend at time
i
W[t + 2n) = W[t)
is
of different,
^)
(
=
is
not
has a unit root, not whether the fijst-difference
interpret this to
mean
that there should be no
natural rate.
difficult to calculate.
+ m=f+2nif W[t + m) =
(1/2)"
from the distribution
coefficient
its
reversion, or trend reversion,
the likelihood that the economy does return to trend at some future time t-'rTnl For the simple
is
example here,
number
W
Campbell and Mankiw (1987)
presumption that an economy should return to
What
Mean
never expected to return to trend holds.
is
of
e.
•
is
and therefore mutually
(2n!)/(n!)^.
The
when
there are
n +l's and n
=
more than one such path where W[t-\- 2n)
exclusive, paths with
=
•
a return
2n independent draws
-I's in
W[t +
2n)
=
W[t)
probability of a return to trend in 2n periods
P2„
is
W[t), The probability of each possible sample path with
(1/2)": this happens
However there
Starting on trend at W[t) say, there
(1/2)=",
for
n >
is
is
W[t). The total
simply the binomial
therefore:
1.
(^J")
=
Defining Pq
1,
the probability generating function for a return to trend
p[s)
is:
= f; P2„.^
n=0
where
f is
William
a
dummy
Feller, 1968,
variable.
(The reader who
Ch.
But then by Newton's binomial formula,
13.)
is
unfamiliar with these kinds of calculations
is
referred to
^w=i:Cr)-(i/2)=^"-^"'=(i-^')"'^'This function diverges to infinity as the variable
can
now
often witi probabilirj-
probability
But
why
this
Even
state the following:
is
1.
In
>
in the presence of a vnit root, the
any time period, no matter how
we turn
weak sense
in
far the
is
significant because
economy returns
we
to trend infinitely
economy has wandered from
trend, the
which shocks to a unit root economy are not permanent. To
to the final question posed above.
Let the probability of a
{Qzn, «
tends to unity. This divergence
always positive of a return to trend some time in the future.
this is a relatively
is,
s
first
What
is
the average time between returns to trend?
return to trend in 2n periods be denoted Q^n-
1} satisfies:
n
P2n
= 53
Q2:P2{n-j),
see
for
n >
1.
Then
the sequence
.
5
'XL* right
hand
side
is
representative term Qay P2(n-})
return in 2(n
— j)
Bum
Bimply the
periods.
of probabilities of n mutualiy exclusive events.
'he probability of a
>*
Taken together,
first
this collection of disjoint events
tions, the
by e^", summing over n
>
1,
j,
the
return to trend in j periods, followed by another
a return to trend in 2n periods, the probability of which has been
Multiplj'ing both sides
For each
is
simply the event that there
is
computed above.
and utOizing a fundamental property of convolu-
equation above becomes:
P{a)-l=Q{s)P{s),
where Q{s) has been defined as
Q2nS^"- The average waiting time
l^^j
for a return to trend
is
then
OO
E \2n \W[i + 2n) =
W(t)
=
W{t +
0,
;) 7^ 0,
1
<
;
<
2n]
= X]
2n
•
Qzn
n=l
=
But,
we
lim
——
»— 1 ds
also have:
(?(.)
= l-P(.)-^ = l-(l-.^)^/=,
which implies
^=
s(l
-
«2)-^/=
- +00 as a
1.
ds
Thus, tie averag-e length of a cycie
Therefore,
if
in a unit root
economy
is infinite.
business cycles are defined to be fluctuations about trend, a unit root
economy
will
not
have business cycles of average length equal to the 50 months given by conventional wisdom.
In summarj', this section has
be %'iewed as inconsistent.
As
made
far as
I
precise the sense in
know,
this
is
the
first
which unit roots and business cycles
may
discussion where these difi'erent notions of
persistence and business cycles have been explicitly related to the presence of unit roots. (Although again,
in the empirical literature,
it is
is
Diebold and Rudebusch, 19S7, have taken steps in this direction.)
Typicallj',
taken for granted that the conditional expectation, or more accurately, the impulse response function
a sufficient measure for persistence. That the distinction between sample path behavior and that of the
conditional expectation sequence
is
important
is
highlighted in the analysis of the next section.
6
n.
A
Model with Unit Roots and
(Finite Length) BusineeB Cycles.
This section develops a simple model where aggregate output
is
linearly represented
by a unit
(or explosive)
root process, but nevertheless output fluctuations about a deterministic trend have finite average length.
The model
that
may
itself
not complicated; however
is
be unfamiliar. Thus we
first
its
dynamic properties
empirical investigation of stochastic nonlinearities
in
Technically,
this paper, although
we
will
in
section.
James Hamilton (1987) presents a careful
aggregate output. His model
both share the feature that there
show below that a random sequence can be
average length, even
need to be studied by using ideas
analyze a sequence of examples to build intuition.
There are two papers related to the discussion of this
from that
will
is
is
substantively different
a discreteness in output growth.
strictly stationary
with fluctuations of
finite
has the conditional expectations behavior of a unit root process. Daniel Nelson
if it
(1987), in a different context, has also constructed strictly stationary unit root sequences.
The mechanism
that produces stationarity in his work are quite different from that used here. Taken together however, this
suggests that there
remain
may
be other examples of stochastic sequences that have a unit root but nevertheless
strictly stationarj'.
The examples here draw on a
by Olivier Blanchard
(1979).
generalization of a process that
He does not
any properties
establish
first
appeared
(as far as
for the special case;
I
we do
know)
so
generalization.
A.
Some
TechDica.! Results.
Let X{t) evolve as
X{t)
where
[-i,rj) is iid,
with t independent of
tj,
= n{t)X{t-i)~v[t)
E')
=
T, Er]
=
=
0.
0)
>
0.
[(7(i)
-
r)
Pr(7(l)
Suppose
also that |r|
This can be revkTitten as follows:
X[t)
=>A'(t)
= YX{t =
1)
rA'(f-i)
+
+ £(t)
X[t
-
1)
+
r?(t)]
>
1,
while
now
in
work
for the
7
Notice that while not independent,
€{t)
turns out to be uncorrelated with
X{t—
l),
to all lagged A"8. This fact allows us to conclude that the equation immediately
that
r
is
when r
Thus X(t)
the population regression coefficient.
=
1,
When
in fact,
above
is
is
orthogonal
a regression,
and
a stochastic process that has explosive roots;
is
A' has a unit root.
Pr(7(l)
reason for this
Theorem
and
1
is
=
>
0)
0,
that no matter
(Stationarit>').
we assume
as
how
A
here,
it
will
be convenient to
small this probability
clinging process
is
is,
we have
X
call
a clinging process.
The
the following property:
strictly stationary,
and has mean zero and inBnite
variance.
Proof: Appendix.
While
X
has a linear regression representation that appears explosive
ARM A
does not have a stationarj'
X displays
mean
its
inside the unit circle).
and
7(i),
lag, A'
Thus a sample path
X{t
The
X, where the
resulting
-f 1)
is
clear
from the arguments
lag distribution
is
is
again strictly stationary,
clinging sequence
well.
displays
analysis
much
may
effects of initial conditions vanish eventually,
sharper returns to
its
mean, than movements away from
uncover evidence of asymjnetry. However, conditional on X[t)
rj
is
symmetrically distributed.
provided that the accumulated sequence
in
t?
is
Since the
dominant,
have a symmetric distribution. Next, a clinging sequence will display cycles
sense of returns to a neighborhood of
its
mean) persistently and
cycles turn out to have finite average length. If a time trend
would reconcile a number
in the
stable (Le., the lag
But now the transformed
sj'mmetrically distributed provided that
will also unconditionally
it
joint unconditional distributions
random sequence
'explosive' linear autoregressive representation.
Without a distributed
is
Box-Jenkins terminology,
that have finite average length.
lag function of
wiU display much richer dynamics as
the mean.
its
fluctuations about
no ceroes
and again, has an
actually stable:
of initial conditions. Further, as
Next consider any distributed
distribution has
it is
and are independent
are time-invariant
Proof,
representation),
(in
is
infinitely often.
added
of seemingly contradictor}' findings: J.
(in the
As mentioned above,
to a clinging sequence, this
X
these
example
Bradford DeLong and LavTence Summers
(1985) find that after detrending, business cycles are actually 'symmetric' in unconditional distribution. In
8
opposition to this, Salih Neft(i (1984) concludes that certain features of economic fluctuations do display a
kind of asymmetry. Campbell and
Mankiw
in linear representations for output;
recurring events.
the clinging
If
The conclusion from
well-behaved
(1987) and Nelson and Plosser (1982) find evidence for a unit root
however
model
is
correct, these statements are not inconsistent.
this discussion
that the presence of unit roots alone
is
of initial conditions, disturbances
The question that remains
is
do not
the following:
B.
A
of labor comprises skiDed
We
Skilled labor A^o
and
is
there a plausible economic
mechanism that
will
is
turn to this next.
of production: capital
K
and labor N. In any time period
Output
t,
is
the measured stock
and unskilled employees:
= No{t] + Ni{t).
labor that has been employed for at least one period.
We make
produce
Simple Economic Model.
Nit)
training
Is
to be described here relates employment, learning-by-doing, and output growth.
produced with two factors
entrants.
not evidence against
persist indefinitely even in the presence of unit roots.
a clinging sequence for output in equilibrium?
new
is
length business cycles. Further, in the sense that the joint distributions are independent
finite
The model
also a wide-spread belief that fluctuations are persistently
it
the extreme assumption that only skilled labor
completely unproductive
in
is
Unskilled labor
Ni comprises
productive; unskilled labor
is
in
the apprenticeship period.
Skilled labor evolves as follows:
A'o(f)=minlA-(J-l}-A2(;),
where N^it]
of
is
market-wide
the labor withdrawn in period
skills
This would include factors such as retirements, a reduction
due to the introduction of new technologj', or possibly sectoral
so that the average skill level
last
t.
Oj,
falls.
shifts
This equation states that skilled labor this period
is
period minus any decumulation due to labor withdrawals; at worst, skilled labor
For
simplicitj',
we assume K^ and Ni have
entrants just equals the
number
between
equal to skilled labor
falls
to zero.
the same expectation, so that on average, the
of withdrawals. Also
assume that both
A'2
and
industries,
number
A^i are serially
of
new
independent
9
and to
rule out uninteresting degeneracies,
arguments below become a
conclusions can be
We
>
=
No{t)
A'j
and
more complicated and subtle
A'^i
have strictly positive variances.
main substantive
these don't hold, but the
if
(The
to remain intact.)
then have the following result:
Theorem
No[0)
made
little
we assume
2 (Eventual Lobs of all Skilled Labor). For an &Tbitr&ry initial quantity of skiUed labor
tJie
0,
probability
some period
positive that at
is
t
>
0,
there
is
a totai loss of skilled labor:
0.
Proof: Appendix.
Thb
is
is
not a deep result:
it
says that
it
will
be unusual to come upon situations where the stock of labor
always productive. One suspects that small deviations from the assumptions here will leave the major
conclusion unmodified. Notice that the result allows the variation in withdrawals ^'2 to be arbitrarily small.
If
labor entrants on average exceed withdrawals, then the result maj' not hold, as there
the stock of labor. However for population growth rate not too different from zero, the difference
drift in
must be
small, so that equality of the
two means
aggregate demographic factors cannot be too
labor force
The
output
is
is
rest of the
model economy
=
described by Y{t)
is trivial.
No[t) K[t)
however there
Thus the model
force,
is
much
a
good approximation. Further, most would agree that
related to economic fluctuations.
Thus
a
nongrowing
probably the natural assumption to use here.
after one period;
work
then a positive
is
is
Let the production function be such that (the logarithm
-r T7(t),
where
no consumption, and
rj
all
displays high persistence in output
is
a technology shock. Capital decays completely
output
when
is
invested, so that K[t)
A*o(t) exceeds 1.
output evolves as a highly positively correlated sequence.
output this period,
>
Tj[t)
0,
tend to increase output persistently
from the labor
force, at
the accumulated capital
By Theorem
some
is
stage, the
I'(i
—
Given a technicaUy
is
near future.
1).
skilled
Increased output
transmitted to increased output
2 however,
no matter how small the fluctuations
economy
sure to lose
is
=
Productivity shocks that increase
in the
implies a higher capital stock; with skilled labor, this increase in output
in future periods as well.
of)
all its skilled labor.
"useless", except to train labor for the next period.
in
When
withdrawals
this
happens,
In this case, the effects of
10
productivity shocks are completely transitory.
The model
The
therefore quite easily produces a process for output that
random
crucial property that the
coefficient
is
(close to) a clinging sequence.
on lagged output periodically realizes as tero
is
borne by
our assumptions on the process followed by skilled labor. The model has quite reasonable assumptions; we
no very special feature
see that
needed to produce persistently recurring short-lived fluctuations despite
is
the possible existence of a unit root linear representation. Slight variations in the assumptions on capital
decay can be accommodated by the fact that distributed lag transformations of a clinging sequence leave
key properties unchanged. The fact that the loss of
in
all
of the skilled labor force
producing the result should not detract from the plausibility of the
smaUer
and the
that
loss
field
occurs only for a short period of time,
that the loss
In Eummarj",
we
is felt
it
may
if
(As an example on a
he or she
sufficiently slow the flow of
is
one among a few
graduate students in
for several generations hence.)
Productivity shocks both
deEcribe again informally the characteristics of the model.
have a persistent long-run
technically a key factor
effects here.
a department's loss of a senior econometrician. Even
scale, consider
is
effect (on average),
and also an eventuaUy transitory
effect.
The
comes from a feedback between capital and output. Increased output leads to a high capital
a highly skilled labor force, this produces even higher output in the next period.
from the
its
fact that the slight (exogenous) variation in labor
The
persistence
stock.
With
transitoriness
comes
withdrawals and entrants leads periodically to a
completely unskilled labor force that needs some length of time to become re-trained. Given an economy-wide
low-skilled labor force, no
output.
amount
of beneficial productivity shocks
Under these circumstances, productivity shocks have a
again only temporary for the skiDed labor force
low output period, the capital stock
for
subsequent periods.
economic
setting:
The next
we
is
come
immediately persistent
transitorj" effect.
online again
only apparently unproductive;
sometime
it
Of
in increased
course, this effect
in the future.
serves to tool
is
Thiis in a
up the labor
force
This example has therefore embedded the djTiamics of a clinging sequence in an
see that
section
is
v.-ill
is
it is
in fact quite plausible to
empirical:
it
model output
as a clinging sequence.
attempts to obtain some evidence on whether the kinds
described here are present in actual aggregate measures of output.
of effects
11
TTT.
Some
The
clinging sequence above produces a meaningful distinction between 'hard'
first
kind of unit root describes the processes of Section
Empirical Evidence.
present some
summary
statistical evidence
Let Y(t) be (the natural logarithm
Y
of)
and
the second describes those of Section
I;
The
unit roots.
'soft'
II.
We now
on distinguishing the two.
some measure
The
of aggregate output.
null hypothesis
is
that
contains a hard unit root:
Y(t+1) = fi-\-Y[t)+A{L)-\(t +
'
where
rj
Ud
is
A'(0,o-^). Defining
W{t] l'(0],^)
W{t+1; Y{0)J) =
Next,
let
X[t)
=
A[L)W{t), so
alternative hj'pothesis that
the definition of
W,
Y{t)
-
Y(0)
W(t- Y{Q),fi)
7,
r;
Y
and
can be rewritten:
+ A(Lr'r,{t+
contains a soft unit root
substitute ^o ui place of y(0), where
are pairwise
this
1)
= X{t-l;Y{0),P)-^v{t).
;So is
X{t)='j{t)X{t-l)
where
- ^ t,
that:
X{t;Y[0),^)
The
=
fort>0,
l),
independent,
serially
rj
is
most concisely expressed
is
a free parameter.
Then
as follows.
In
write
+ v{t),
A'(0, r^), E'l
=
1,
=
Pr(7
0)
=
p
>
0,
and
X
is
denned
exactly as above.
—
In woroE. a filtered version of deviations from trend Y{D)
a pure
random walk with no
drift.
Under the
alternative, the
;?t is
same
hj-pothesized to be a clinging sequence with a soft unit root.
The
hj-pothesized under the null to be
filtered version of
trend deviations
difference stationary
model
is
is
strictly
nested within the clinging alternative: that null hypothesis model obtains by setting the variajice of 7 to
zero and
^60
to Y{0).
Estimation under the null proceeds as follows. First difference (the log
of)
observed output, and estimate
the regression
Y{i)
-
Y{t
-
1)
=
fiA{l)
- [L-'A{L)]
[Y{t
-
1)
-
Y[t
-
2))
+ v[t).
12
The model
is
Under our assumptions, ordinary
just identified.
likelihood. Estimation
under the alternative
is
a
little
least squares
and pairwise independent Bernoulli and normal random
of serially
We
more complicated.
equivalent to
is
maximum
parametrize 7 as the product
variables:
'y(t)=fc(t).c(i),
where
Bernoulli with probability p of equalling unity,
b is
recovers the property implied by the null hypothesis that 7
alternative
model can be written
in a
vector of parameters in the model 6
;
way
close to that of
M[p~^, c^). Setting
e is
=
1 for all
The
t.
trf
to eero
and p to unity
likelihood function for this
an unobserved switching regression model Call the
X[t—
the likelihood of the i—th observation on X[t) conditional on
1)
is
t{X(t)\X{t-l);e)=p-J
where
(f>
the density of a standard
is
^^l'^-i'f^'-%
normal The
likelihood of the entire
to be simply the product of these conditional likelihoods. Since
the impact of the initial condition
nuU
A''(0)
are used as starting points for the
Tables
1
and
A[L)
=
maximum
is
always
1982 dollars and industrial production
is
is
seen
stationary under the alternative,
two measures
for
1
and k
for
varies
of aggregate output, quarterly
A[L) chosen
from
1
to be Erst
through
3.
The
GNP
and
through third order; that
GNP
measured
in
Thus, twice the difierence
in
series
is
seasonally adjusted.
There are three restrictions under the
log-likelihoods
is strictlj'
A''(0) is
likelihood observation.
model
2 display estimates of the
y^ _o ajL' where ao
X
sample conditional on
vanishes asymptotically. Parameter estimates obtained under the
monthly industrial production. Results are presented
is,
]+{l-p).4>[X[t)/a,.).
null: ^0
asymptotically distributed x^
=
'*'i*'^
5'(0), p
=
1,
and rf
=
0.
three degrees of freedom. For
GNP,
this test-statistic
takes values 2.614, 3.S42, and 1.532 with marginal significance levels of 0.46, 0.28, and 0.67, respective!}',
when
A
is
set to first, second,
difi'erence stationarj',
and third order
rather than trend stationary cannot be rejected.
Turning to the industrial production data
to be soundly rejected. For second
50,
lag polynomials. Therefore, the
in
Table
2,
the
GNP
nuD hypothesis
and third order polynomials, the
nuU hypothesis
that
GNP
is
has a hard unit root.
of a hard unit root
is
differences in log-likelihoods
with associated marginal significance levels well below the standard cutoff points.
now
is
seen
around
Table
(In,
1
^0
6962.7
*
B
7.909
(0.61)
P
^.'
1.0
^^
0.0
109.29
•
Mag
2-la2
OLS
.
MLE
OLS
6963.64 (21.27)
6962.7
•
6980.22 (22.84) 6962.7
8.784
0.991
(034)
7.946
(0.47)
7.886
1.0
7.884
0.990
(0.27)
(0.11)
(0.12)
1.0
100.07
(121)
107.70
9438
(1.20)
105.56
0.0
(0.09)
0.0
0.00
(0.14)
0.0
-0.408
(0.07)
»
•
•
(0.72)
W
»
696337
(21.34)
8.795
(035)
0.991
(0.11)
97.21
(120)
0.00
(0.10)
-0315
(0.08)
-0369
(0.08)
-0338
(0.08)
-0376
(0.08)
-0.138
(O.OS)
-0.113
(0.08)
-0.189
(0.08)
-0.176
(O.OS)
-
4-0.161
(0.08)
+0.140
(O.OS)
^32.548
^30.104
^3
—
-
-
InL
^38.439
-437.132
-434.469
• Fixed in the Rcsnictcd Estimadon.
MLE
OLS
MLE
^366 (0.08)
—
—
—
°7
1947,1 to 1986,1
X 103)
Mag
°\
GKP82.
Numbers
Tabb
in Parentheses are
2
-429338
numerical standard eiron.
19E63
Induscial Producdon, 1947,1 to
X 10^)
(In,
OLS
hC-E
33534.C
£
31.12
P
1.0
^^
11032
C.^
0.0
°1
-0.437
^
-
InL
-241220
3-l2£
2-las
1-lag
1
(0.40)
*
nis
MLE
33534.0 •
35141.8 (2413) 33534.0 •
30.94
30.13
0.990
(0.99)
31.57
1.0
29.80
0.9E9
(053)
(0.72)
6171.0
(7.14)
0.01
(O.W)
-0.497
(0.04)
(036)
1.0
^
-
(034)
KG-E
35278.0 (237)
(0.73)
6272.9
(72S)
10904
0.01
(0.03)
0.0
»
-0397
(0.05)
-0.499
(0.O4)
-0393
(0.C5)
-0.093
(0.05)
-0.099
(0.04)
-0.075
(0.05)
-0.073
(O.W)
-0.W6
(0.05)
-0.071
(0.O4)
0.0
(0.&4)
.
»
10909
»
OLS
-
—
-24&4.^
-235325
• Fixed in the RssrirtscEsdriadon. Niimbsrs in Par-entheses
t Un-h gged frszi Rssrictrd Essnaas.
;
>.x/
ru errors.
;.7o
13
Examining the
from
1,
and that
resulte
cr^ is
more
closely,
we
see that individually the probability
the individual f-statistics
may
not significantly different
p and a^
parameters are correlated and so
for these three
be misleading. Testing for the significance of
to examining the likelihood-ratio statistic, which
—
is
not significantly different from «ero. Thus the reason for rejection appears to come
mainly from the ^o intercept term. However, the estimates
possibility that 1
p
may
But
it is
line
changes for every different
differ
from
is
is
what we have done
sufficient to pin
down
three
all
here.
is
Evidently, allowing for the
the intercept term quite precisely.
one of the important implications of the hard unit root hj^pothesis that the intercept
initial condition.
More
precisely,
OLS
that, contrary to this, for our
line, is in fact
weD
Our
results indicate
quite precisely estimated.
in
p at
0.99.
This implies a realization of 7(i)
approximately twice the consensus length of a business
for cyclical
"trend"
sample of industrial production data, the intercept, and consequently the trend
plot of the likelihood function for industrial production (not presented here)
unique peak
in the
estimates for this intercept diverge as
the sample size grows arbitrarily large (see for example Durlauf and Phillips, 1986).
A
essentially equivalent
cycle.
=
everj' 91
shows a
distinct
months on average.
However, notice that A{L)
is
and
This
is
a further source
dynamics. In the second order case, this polynomial has roots of 0.65 and 0.16. These roots are
inside the unit circle,
and serve
to enrich the estimated cyclicality.
rV. Conclufiion
This paper has undertaken two tasks. The
first
and pedagogical objective
in
Section
I
makes
clear
why
root processes and the notion of persistently recurring business cycles are at odds with each other.
analysis in terms of the relation between sample path behavior
to have been
That
made
it is
explicit thus far in the
important to make
macroeconomic
this connection
becomes
and the presence
unit
The
of unit roots appears not
literature.
clear in Section 11
where a simple model
is
con-
structed that displays persistently recurring fluctuations in the sample paths of output despite the presence
of a unit root.
the
model
Summarj"
statistical evidence presented in Section III suggests that the effects described in
are not without empirical basis for measures of aggregate output in the US.
1
14
Appendix.
This appendix proves the technical resultB
Proof of Theorem
Fork >
shifts.
We
We
1.
1, Jet ti, <2,
wish to show that for
•
•
verify that the joint distribution o/ subsequences of
•
all
,
tfc
F
estabL'sii
M>
X{m) = 'i(m)X{m —
(m
')[]')
case,
-i-anishes for
X[m]
m. Setting
l)
0<
j
From
r7{m),
i{j)\
= m-rl-M
and
+
fi
< M, we
have 7(fi
—
\
—
M
we
=
=
0),
'^(tl -r s
>
witi p
(7,
— /) =
=
0,
1
^rn.r.
to
)
show
that
EX
be this conditional expectation:
product of random variables. Since 7
\
n
Yi
iWjvu).
\k=}-^-l
J
X[m —
and
M and m.
A'(ti
More
M).
m+
partial sequence between epochs
tj)
+
1
In that
—
M and
s) are identical,
explicitly, if for
some
tien
for
—
and
some y between
(1
is zero,
—
p^)^.
write for
But
r.
>
is iid
it is
a well-de£ned
M — 1}
this tends to 1 as
m
>
f
DeEne
rn
I
F[X[t,^s),X[t,-rs),...,X(i^~s)).
occurs with probability bounded from below by
establishing strict stationarity. To
be integer with
Tie condizioning event
0.
7(^1
M
Let m,
s).
suffices to
it
iiave
some j between m-rl —
0,
— y) =
0).
and m, X[m) becomes independent of
~*~y =
{^(-i
+
see that the distributions 0/ A''(ti)
F{X{t,],X{i^],...,Xit,))
Let p denote Pr(-r[0)
and X(ti
j=m + l-M
m~
turn,
s in
;)
we
J
some j between
to tj
A''(ti)
m—
can be written entirely in terms of the
m
invariant across time
the form of the clinging sequence,
x{m-M)+vH+
conditional on a zero realisation for 7(y), for
J,
+
+ s], X{t, + s),..., X{t, +
F{X{t,
\
n
If
=
X[t,))
...,
equaiity of the unconditional distributions of
j
is
integer a:
denotes the joint d'lstribution function.
Iterating on
1.
X
^e some collection of integers, assumed increasing without loss of generality.
F{Xit,),X{t2),
wJiere
paper.
in the
random
M
—• oc, hence
0;
"
^
-vBxiabie as
and places positive probability on
0,
it is
for fixed
simply a
m
the
finite
random
15
sequence {fm.m i
^
rn) converges almost Burely to the degenerate ra.ndom variable at
itera.ted expectations,
property
is
obvious.
Proof of Theorem
then
N
is
a tero drift
£(lim„_oo lm,n)
=
uniformly in
m which implies E{X[t)) =
0.
0.
The
By
the iaw of
infinite
variance
QED
2.
Suppose
random
tha.t
contrary to the conclusion of the theorem, N{t)
walk. Thus, from
inEnitely often with probability
1.
This
is
any starting point,
a contradiction as
N
is
N
will take
>
for all
t.
But
on arbitrarily negative values
restricted to be always nonnegative.
QED
16
FOOTNOTES.
^
Department
of Economics,
MIT, Cambridge
MA 02139.
Poterba and Julio Rotemberg for useful comments on
Taylor,
Mark Watson and an anonymous
discussion.
The
hospitality of the
MIT
I
thank Olivier Blanchard, Stanley Fischer, James
earlier versions of this paper.
referee for suggestions
Statistics
Center
is
and
I
also thank
John
criticisms that have helped to focus the
gratefuDy acknowledged.
bt
17
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361
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i
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3 TDfiD
QD5 130 EBl