Common stochastic trends, common cycles, and asymmetry in economic fluctuations

Abstract

This paper investigates the nature of U.S. business cycle asymmetry using a dynamic factor model of output, investment, and consumption. We identify a common stochastic trend and common transitory component by embedding the permanent income hypothesis within a simple growth model. Markov-switching in each component captures two types of asymmetry: Shifts in the growth rate of the common stochastic trend, having permanent effects, and “plucking” deviations from the common stochastic trend, having only transitory effects. Statistical tests suggest both asymmetries were present in post-war recessions, although the shifts in trend are less severe than found in the received literature.

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WORKING PAPER SERIES
Common Stochastic Trends, Common Cycles, and Asymmetry in
Economic Fluctuations
Chang-Jin Kim and Jeremy M. Piger
Working Paper 2001-014A
http://research.stlouisfed.org/wp/2001/2001-014.pdf
October 2001
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Common stochastic trends, common cycles, and asymmetry in economic
fluctuations
Chang-Jin Kim,a Jeremy Piger*,b
aKorea University, Anam-Dong, Seongbuk-ku, Seoul, 136-701, KOREA
*,bFederal Reserve Bank of St. Louis, 411 Locust St., St. Louis, MO 63102
Abstract
This paper investigates the nature of U.S. business cycle asymmetry using a dynamic factor model of
output, investment, and consumption. We identify a common stochastic trend and common transitory
component by embedding the permanent income hypothesis within a simple growth model. Markov-
switching in each component captures two types of asymmetry: Shifts in the growth rate of the common
stochastic trend, having permanent effects, and "plucking" deviations from the common stochastic trend,
having only transitory effects. Statistical tests suggest both asymmetries were present in post-war
recessions, although the shifts in trend are less severe than found in the received literature.
Key words: Asymmetry, Business Cycles, Common Shocks, Markov-Switching, Productivity Slowdown
J.E.L classification: C32, E32
* Corresponding Author: Jeremy M. Piger, Federal Reserve Bank of St. Louis, 411 Locust St., St. Louis,
MO 63102; Telephone: 314-444-8718; Fax: 314-444-8731; E-mail: piger@stls.frb.org
Kim acknowledges financial support from the National Science Foundation under grant SES9818789 and
Piger from the Grover and Creta Ensley Fellowship in Economic Policy. This paper is based on chapter 2
of Piger’s Ph.D. dissertation at the University of Washington. We thank, without implicating, an
anonymous referee, James Morley, Charles Nelson, Jessica Rutledge, Dick Startz, Eric Zivot and seminar
participants at the Federal Reserve Board and the Federal Reserve Banks of Dallas, St. Louis and Kansas
City for useful comments. The opinions expressed in this paper are those of the authors and do not
necessarily reflect those of the Federal Reserve Bank of St. Louis or the Federal Reserve System.
Running Headline: Common Stochastic Trends

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The question of whether the dynamics of recessions are different from those of
expansions has a long history. Early students of the business cycle, including Mitchell (1927),
Keynes (1936), and Burns and Mitchell (1946) noted that declines in economic activity take hold
quicker, are steeper, and last for a shorter amount of time than expansions. To these observers,
recessions appeared to come from a different regime than booms. Recent interest in this type of
asymmetry was sparked by Salih Neftci (1984), who presented evidence that increases in the
unemployment rate are sharper and shorter than declines.
Since that time, two parametric time-series models of U.S. output were proposed that are
capable of capturing steep, short recessions. However, they are fundamentally different in their
implications for the effects of recessions on the long run level of output. In other words, the
hypothesized persistence of shocks that lead to recessions is very different in the two models.
The first model, due to Hamilton (1989), divides the business cycle into two phases, negative
trend growth and positive trend growth, with the economy switching back and forth according to
a latent state variable. This two phase business cycle implies that following the trough of a
recession, output switches back to the expansion growth phase, never regaining the ground lost
during the downturn. Recessions will therefore have permanent effects on the level of output.
The second model, having its roots in work by Friedman (1964, 1993) and recently formalized in
an econometric model by Kim and Nelson (1999a), suggests that recessions are periods where
output is hit by large negative transitory shocks, labeled “plucks” by Friedman. Following the
trough, output enters a high growth recovery phase, returning to the trend. This “bounce-back
effect” or “peak-reversion” is the critical phase of Friedman’s model. Output then begins a
normal, slower growth, expansion phase. Thus, Friedman’s view is that recessions are entirely
transitory deviations from trend, not movements in the trend itself.

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Both forms of asymmetry have received substantial attention in the empirical literature,
with conflicting conclusions. Using classical likelihood based tests, Hansen (1992) and
Garcia (1998) both fail to reject a linear autoregressive model in favor of Hamilton’s model for
U.S. GNP. Kim and Nelson (2001) reach a similar conclusion using Bayesian methods. On the
other hand, both Chib (1995) and Koop and Potter (1999) find evidence in favor of Hamilton’s
model using Bayesian techniques. Support for the peak-reversion implication of Friedman’s
model is given by Wynne and Balke (1992, 1996), Sichel (1993, 1994), and Beaudry and
Koop (1993). However, Elwood (1998) argues that the evidence in favor of peak-reversion has
been overstated. Specifically, Elwood presents evidence that negative shocks are not
significantly less persistent than positive ones for U.S. GNP. A shortcoming of this empirical
literature is that most authors have analyzed the two forms of asymmetry separately from one
another. That is, little attention is paid to evaluating the marginal significance of the two forms of
asymmetry.1 An additional shortcoming is the literature’s domination by univariate analysis. As
pointed out by Kim and Nelson (2001), tests based on univariate models have low power in
detecting a specific form of asymmetry in the business cycle as the data may be obscured by
idiosyncratic variation.
In this paper, we estimate a dynamic two-factor model of real private GNP, fixed
investment, and consumption of non-durables and services that incorporates the common
stochastic trend suggested by neoclassical growth theory and a common transitory component.
Building on work by Cochrane (1994) and Fama (1992) we define consumption as the common

1 An exception is Kim and Murray (1999), who estimate an experimental coincident index of economic activity
which incorporates both types of asymmetry discussed above. However, their investigation employs economic
indicators that are not cointegrated. Also, they do not investigate the implications of their model for the dynamics of
real GNP.

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stochastic trend. As we discuss below, this assumption can help to eliminate bias that may arise
when using Hamilton’s model to capture shifts in trend growth rate. We model the Hamilton and
Friedman types of asymmetry through regime switching in the permanent and transitory
components respectively. This method allows tests of the marginal significance of one type of
asymmetry while the other is allowed to be present. As a byproduct of the estimation we
consider the possibility of a one-time structural break in the growth rate of the common
stochastic trend (a productivity slowdown). We search for the date of this break using a
multivariate version of a technique suggested by Chib (1998).
Section 1 of the paper presents a review of the Hamilton and Friedman types of
asymmetry in business cycle dynamics. Section 2 discusses the theory supporting a common
stochastic trend and a common cyclical component in output, investment and consumption, and
presents the formal empirical model. Section 3 presents estimation results and statistical tests of
the importance of the two types of asymmetry. Such tests suggest that both types of asymmetry
have played a significant role in post-war recessions, although the nature of shifts in the growth
rate of trend is different than the received literature suggests. In particular, we find evidence of
reduced, but still positive, growth rates in trend during recessions, not the negative trend growth
suggested by Hamilton (1989). We present some simulation evidence that this discrepancy may
be caused by a potential bias in applying Hamilton’s model to data which displays “plucking”
type recessions. The investigation of a one-time structural break in the average growth rate of
trend is suggestive of a productivity slowdown, the estimated date of which is centered around
1971. Section 4 summarizes and concludes.

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1. A Review of the Hamilton and Friedman Models
1.1 Hamilton’s (1989) Model
In an influential 1989 Econometrica paper, James Hamilton proposed a model in which
the growth rate of the trend function of U.S. GNP switches between two different states
according to a first order Markov process. Hamilton’s results suggest the two states correspond
to business cycle dynamics, the first being normal growth and the second recession. Figure 1
contains a stylized graph of a business cycle characterized by Hamilton type asymmetry. Note
that following the recession output does not rebound back to its level had the recession not
occurred. Instead, because recessions are movements in the trend of the series, output is
permanently lower. Specifically, Hamilton’s results suggest that a typical economic recession is
characterized by a 3% permanent drop in the level of GNP. Thus, while the Hamilton model is
capable of explaining a business cycle in which recessions are quick, steep drops in economic
activity, it also has a dire implication for the welfare effects of recessions.
Evaluation of Hamilton’s model is complicated by the fact that standard distribution
theory for hypothesis testing does not apply to Markov-switching models. Testing the Markov-
switching model vs. linear alternatives is troubled by the familiar Davies’ (1977) problem, in
which a nuisance parameter is not identified under the null hypothesis. Hamilton’s original paper
offers suggestive evidence that the two state Markov-switching model outperforms linear models
in terms of forecasts, but no statistical tests. Hansen (1992) and Garcia (1998) use classical
likelihood based test procedures designed to deal with the Davies’ problem and find that linear
autoregressive models cannot be rejected for real GNP. Kim and Nelson (2001) confirm this
result using Bayesian techniques. Also using Bayesian techniques, Chib (1995) and Koop and

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Potter (1999) find evidence that the Markov-switching model outperforms linear models. Thus,
the empirical evidence regarding Hamilton’s model is mixed and incomplete.
Hamilton’s model has been followed by a growing volume of theoretical work in which
the economy undergoes endogenous switching between “good” and “bad” states. Specifically,
Howitt and McAfee (1992) employ a model of switching consumer confidence which leads to
multiple equilibria with statistical properties well characterized by Markov-switching. In
Cooper (1994), agents choose between multiple equilibria and then remain in the chosen
equilibrium until a large shock induces a switch. Acemoglu and Scott (1997) and Startz (1998)
also employ models in which shocks generate endogenous switching between growth states.
However, negative growth states, such as those found by Hamilton (1989) during recessions, are
not generated by these models in general. For example, in Startz (1998) the economy switches
between two positive growth states.
1.2 Friedman’s (1964, 1993) “Plucking” Model
Friedman (1964, 1993), argued for a type of business cycle asymmetry that, while
yielding steep recessions, has very different implications for the long run effects of recessions
than Hamilton’s model.2 Specifically, in Friedman’s “plucking” model, recessions are caused by
large negative transitory shocks that yield steep recessions. Following these shocks output
“bounces back” or “peak reverts” to trend. This is commonly referred to as the high growth
recovery phase. Finally, output begins a normal, slower growth, expansion.3 Figure 2 contains a
stylized graph of a business cycle characterized by “plucking”.

2 The behavior described in this paragraph is also consistent with De Long and Summers (1988).
3 Friedman’s “plucking” model has another strong implication - that deviations from trend are only negative,
meaning increases in output are permanent. In this paper we do not attempt to model this feature. Instead we focus
on the peak reversion of recessions, or the tendency of output to “bounce back”.

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The literature contains many statistical tests of various implications of Friedman’s model.
Here we focus on the literature surrounding the peak-reverting nature of recessions. Wynne and
Balke (1992, 1996) find that the deeper the recession the stronger the ensuing recovery while
Sichel (1994) finds evidence of a high growth recovery phase following recessions, both
implications of peak reversion. Another implication of peak reversion is that negative shocks are
less persistent than positive shocks.4 Beaudry and Koop (1993) showed that a variable measuring
the depth of real GNP below its historic high was useful for predicting changes in output. They
use this variable to investigate impulse response functions for negative vs. positive shocks, and
show that negative shocks are much less persistent. Elwood (1998) took issue with Beaudry and
Koop’s techniques, arguing that by considering only shocks which reduce the level of GNP they
ignore a large number of negative shocks that fail to reduce the level of the series. Elwood uses
an unobserved components model capable of identifying all negative and positive shocks and
finds that negative shocks are not statistically significantly less persistent than positive shocks.
This controversy is suggestive of two kinds of negative shocks to the economy: large,
asymmetric, recession causing shocks and smaller shocks that come from a symmetric process.
Beaudry and Koop’s analysis proxies for the large negative shocks by considering only shocks
that actually reduce the level of GNP. On the other hand, Elwood’s analysis smears the effects of
large and small negative shocks together by assuming all negative shocks have the same
variance. In this paper we take the approach of Kim and Nelson (1999a) and allow for both
continuous, symmetric transitory shocks and infrequent, asymmetric transitory shocks, which we
model as coming from a Markov-switching process.

4 If recessions are entirely transitory, as Friedman’s model suggests, while expansions, being driven in part by a
stochastic trend, have a permanent component, negative shocks will have less persistence than positive shocks.

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The bounce-back effect in Friedman’s model is consistent with a wide variety of
economic models. In demand side models, output might be driven into recession by a large
infrequent demand shock. Following the recession, output grows faster than when at trend
because resources are underutilized. Walrasian models can also generate a high growth recovery
phase if recessions are partially absorbed by running down the capital stock. Then, just as in the
Solow growth model, the economy will experience faster growth until the capital stock is
restored to its new steady state value.
1.3 Do Both Types of Asymmetry Matter?
Empirical work has focused on either the Hamilton or Friedman type of asymmetry
separately, a consequence of the prevalence of univariate techniques. However, since the two
types of asymmetry both capture the steep, sharp nature of recessions, both might provide
improvement over linear models if considered alone. For example, in Section 3 we present
simulation evidence that Hamilton’s model will fit data generated with Friedman type
asymmetry with a positive and negative growth state in the stochastic trend, even though all
recessionary shocks are transitory. To evaluate whether both types of asymmetry are important
one must employ a model that separates the two types of asymmetry from one another. In the
following section we present a model capable of achieving this separation and test the marginal
significance of each type of asymmetry when the other is present.
2. Model Motivation and Specification
2.1 Common Permanent and Transitory Components - Theory
The concept of trend vs. cycle plays an important role in defining the Hamilton and
Friedman types of asymmetry. One advantage of our multivariate model of output, investment

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and consumption is its natural interpretation of trend provided by neoclassical growth theory. To
see this, consider a basic one-sector model of capital accumulation based on that in King,
Plosser, and Rebelo (1988). Output is produced by two factors, capital and labor, and is subject
to exogenous growth in labor augmenting technology,
tA :
),(
tttt
LAKFY =
(1)
Each representative agent in the economy has identical preferences over the consumption of
goods,
t C and leisure,
tR given by:
?
=
t
∞
=
0
),(
tt
t
RCuU
β
(2)
where utility is increasing in both consumption and leisure. Finally, the capital accumulation
process is:
ttt
IKdK
+−=
+
) 1 (
1
(3)
where d is the rate of depreciation on capital and
tI is investment. The economy is also subject
to constraints on the amount of time a worker has to allocate between work and leisure and the
amount of consumption and investment possible for a given level of output. If a steady state
exists in this model it will be one in which the logarithms of output, investment, and
consumption grow at a rate determined by labor augmenting technological progress. In the case
where there are permanent technology shocks, as is the case if the logarithm of
tA follows a
random walk, these three quantities share a common stochastic trend.5 Each series is then

5 A steady state under random walk productivity growth, called a stochastic steady state, will obtain under
restrictions on preferences and production technology, (Cobb-Douglas production is not required). The interested
reader is referred to King, Plosser and Rebelo (1988) and King and Rebelo (1987) for details.

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individually integrated but the ratio of any two is stationary. In the terminology of Engle and
Granger (1987), the logarithms of output, fixed investment and consumption are cointegrated
with cointegrating vectors (1, -1, 0)’ and (1, 0, -1)’.
In this paper we employ consumption as a proxy for the common stochastic trend in the
system. The recent literature, for example Fama (1992) and Cochrane (1994), suggests that while
consumption does seem to contain a statistically significant transitory component, it is so small
as to be economically insignificant. Based on this result, Fama (1992) chooses to define
consumption as the common stochastic trend in output, investment, and consumption.
Cochrane (1994) argues that consumption is an effective measure of the trend in output by
presenting evidence that shocks to GNP holding consumption constant are almost entirely
transitory, a result consistent with simple versions of the permanent income hypothesis. Defining
consumption as the trend serves a useful purpose in this paper. As we argued above, because
both the Hamilton and Friedman models are capable of capturing the steep nature of recessions,
either may fit the data well even if the data exhibits only one type of asymmetry. Simulation
evidence in Section 3 support this conclusion. Thus, in order to separate the two forms of
asymmetry, we would like to define the Hamilton type of asymmetry on a series that proxies for
only the trend and does not undergo the transitory Friedman type asymmetry. Consumption is a
useful proxy for this trend.
In the neoclassical growth model, movements in the stochastic trend account for all of the
movement in output, investment, and consumption in the long-run. However, at business cycle
horizons transitory deviations from this stochastic trend are likely to be important. For example,
many real business cycle models, such as Kydland and Prescott (1982), extend the model

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presented above in ways that allow technology shocks to induce transitory dynamics as the
economy moves towards the new steady state. Transitory deviations from a long
run stochastic trend might also come from more traditional demand-side nominal shocks.
Regardless of whether transitory shocks stem from Walrasian or Keynesian sources however, it
is likely that some portion of the shocks will come from sources that are common to output,
investment, and consumption. For example, if shocks to the money supply have real, albeit
transitory, effects, one would expect that these effects would be pervasive across macroeconomic
time series. Likewise, if general productivity shocks induce transitory dynamics, these dynamics
should be felt economy-wide. Thus, in addition to the common stochastic trend suggested by
neoclassical growth theory, we would also expect common sources of transitory dynamics at
business cycle horizons.
2.2 A Dynamic Two-Factor Regime Switching Model
The above discussion is suggestive of a general empirical model in which the logarithms
of output,
ty , and investment, ti , are influenced by shocks to a common stochastic trend, defined
as the logarithm of consumption,
tc , a common transitory component, and idiosyncratic
transitory shocks. The common stochastic trend and common transitory component are captured
by two dynamic factors, labeled
tx and
tz :



tcct
ittitiit
yttytyyt
xac
ezxai
ezxay
γ
λγ
λγ
+=
+++=
+++=
(4)
The
iyjejt
,, =
are stationary residuals that capture idiosyncratic transitory variation in
ty and
ti .
j γ and
j λ are factor loadings on the common stochastic trend and the common transitory
component respectively. For identification,
y
γ and
y
λ are normalized to one. Consistent with our

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specification of log consumption as the random walk trend in the system, tc does not contain
any transitory component, common or idiosyncratic. In addition to the reasons for this modeling
choice provided in Section 2.1, it is worth pointing out that any transitory dynamics in tc that do
exist are small enough to be difficult to identify in this already highly parameterized model.
We are now ready to discuss how the two types of asymmetry are incorporated in the
model. We begin with the Hamilton type asymmetry, which we incorporate as in
Hamilton (1989). Recall, the Hamilton type asymmetry involves shifts in the growth rate of the
trend function between two different states. Thus, we allow the common stochastic trend,
tx , to
follow a random walk with a switching drift term:
ttttt
vxSx
+++=
−1
*
01
µµ
(5)
where
), 0 (
N
~
2
vt
v
σ
, and } 1 , 0{
=
tS
indicates the state of the economy. We assume that
tS is
driven by a first order Markov process with transition probabilities given by:
001
11
p
1
) 0 | 0(
) 1
=
| 1
=
(
SSP
pSSP
tt
tt
=
===
−
−
(6)
To incorporate the Friedman type asymmetry we allow the idiosyncratic transitory component of
output and investment to undergo regime switching as in Kim and Nelson (1999a). Formally:
iyjSeL
tj jtjtj
, ,)(
=+=τεψ
(7)
where ), 0 (
N
~
2
ε
j jt
σε
,
)(L
j
ψ
has all roots outside the unit circle, and 0
<
j
τ
is a term which
“plucks” output and investment down when 1
=
tS
. When the economy returns to normal times

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the economy reverts back to the stochastic trend. The farther the economy is plucked down, the
faster the growth of the economy as it “bounces back” to trend.6
To complete the model we must specify the dynamics of the common transitory
component
tz :
ttzL
ωφ=
)((8)
where
) , 0 (
N
~
2
ω
σω
t
, and
)(L
φ
is a lag polynomial with roots that lie outside the unit circle.
For identification we assume that
tv ,
t
ω ,
yt
ε , and
it
ε are uncorrelated at all leads and lags.
The model presented above is closely related to a recent literature discussing models
which simultaneously capture comovement and asymmetry in business cycle indicator variables.
Diebold and Rudebusch (1996) discuss this idea in detail, while Kim and Yoo (1995),
Chauvet (1998) and Kim and Murray (1999) all estimate such models. However, this literature is
exclusively concerned with the development of a new coincident index of economic activity and
not with the dynamics of real GNP. In addition, these papers consider economic variables that
are not cointegrated. Finally, with the exception of Kim and Murray (1999), only the Hamilton
type regime switching is used to capture asymmetry. Here, by analyzing a cointegrated system
with a precise definition of trend we hope to gain a clearer look at the nature of both the
Hamilton and Friedman types of asymmetry in the dynamics of U.S. GNP.7

6 The “plucking” parameter is incorporated in the idiosyncratic transitory component of output and investment to
allow for the possibility that the magnitude of the pluck might be different across economic series. However, in
interpreting the model the plucks are better characterized as common shocks because they are driven by the same
state variable. In other words, when output is plucked down, so is investment.
7 Our model is also similar to the “common trends” representation suggested by King, Plosser, Stock and
Watson (1991). There, the effects of the common and idiosyncratic transitory components above are combined into

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Notice that the two types of regime switching are driven by the same state variable,
tS .
In essence, this assumption forces all recessions to have the same relative importance of
permanent vs. transitory Markov-switching shocks and can be motivated as an extension of
Hamilton (1989) and Kim and Nelson (1999a). In these papers, the authors also force all
recessions to have the same relative importance. In Hamilton’s paper, recessions are entirely
permanent while in Kim and Nelson’s they are entirely transitory. Here we extend these results
to allow recessions to have both a permanent and transitory component. The assumption is
important in that it allows us to test the null hypothesis that one type of asymmetry is marginally
statistically insignificant when the other is present. If the two types of asymmetry were driven by
separate state variables, testing this null hypothesis would be complicated by the familiar Davies’
problem, or the fact that one set of Markov-switching parameters would be unidentified under
the null hypothesis.
2.3 A One Time Permanent Structural Break in Average Growth Rate
There is a large literature suggesting that the growth rate of productivity has slowed at
some point in the postwar sample, with the predominant view being that this slowdown roughly
coincides with the first OPEC oil shock. For example, Perron (1989) identifies 1973 as the date
of a break in the trend growth of U.S. quarterly real GNP.8 In a recent paper, Bai, Lumsdaine and

an I(0) disturbance which may be correlated across indicators. Their empirical analysis employs a VECM
framework to investigate the relative importance of the common stochastic trend in real GNP, fixed investment, and
consumption. While a VECM lends itself easily to impulse analysis, incorporation of asymmetry is difficult.
Identification of asymmetry in a dynamic factor model is natural, motivating our choice of empirical model.
8 Preliminary estimation of our model suggested that if a productivity slowdown is not incorporated the
autoregressive dynamics of
yt
e ,
it e , and
tz are very persistent. This is consistent with Perron’s (1989) finding that
unit root tests are biased towards non-rejection if a break in mean growth is not accounted for.