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What’s Real About the Business Cycle?
James D. Hamilton
This paper argues that a linear statistical model with homoskedastic errors cannot capture the
nineteenth-century notion of a recurring cyclical pattern in key economic aggregates. A simple
nonlinear alternative is proposed and used to illustrate that the dynamic behavior of unemployment
seems to change over the business cycle, with the unemployment rate rising more quickly than
it falls. Furthermore, many but not all economic downturns are also accompanied by a dramatic
change in the dynamic behavior of short-term interest rates. It is suggested that these nonlinearities
are most naturally interpreted as resulting from short-run failures in the employment and credit
markets and that understanding these short-run failures is the key to understanding the nature of
the business cycle.
Federal Reserve Bank of St. Louis Review, July/August 2005, 87(4), pp. 435-52.
WHATIS THE BUSINESS CYCLE? In part, this shift in the profession’s concep-
he term “cycle” is used to describe a tion of what needs to be explained about business
process that moves sequentially between fluctuations reflects a desire to integrate the deter-
T minants of long-run economic growth and the
a series of clearly identifiable phases in a causes of short-run economic downturns within
recurrent or periodic fashion. Economists of the asingle unified theory of aggregate economic per-
nineteenth and early twentieth centuries were formance. Since improvements in overall produc-
persuaded that they saw such a pattern exhib- tivity are widely acknowledged to be one of the
ited in the overall level of economic activity key factors driving long-run growth, and since
and enthusiastically sought to characterize the such improvements cannot reasonably be expected
observed regularities of what came to be known to occur at a constant rate over time, it is natural
as the “business cycle.” The most systematic to explore the possibility that variation over time
and still-enduring summaries of what seems to in the rate of technological progress could be a
happen during the respective phases were pro-
vided by Mitchell (1927, 1951) and Burns and primary cause of variation over time in the level
Mitchell (1946). of economic activity. Brock and Mirman (1972)
The expression “business cycle theory” were the first to incorporate stochastic variation
remains in common usage today, even though, in in the rate of technical progress into a neoclassical
most of the modern models that wear the label, growth model, though they clearly intended this
there in fact is no business cycle in the sense just as a model of long-run growth rather than a realis-
described. These are models of economic fluctu- tic description of short-run fluctuations. Kydland
ations, to be sure, but they do not exhibit clearly and Prescott (1982) later took the much bolder
articulated phases through which the economy step of proposing that this class of models might
could be said to pass in a recurrent pattern. explain variations in economic activity at all fre-
James D. Hamilton is a professor of economics at the University of California, San Diego. This research was supported by the National Science
Foundation under grant No. SES-0215754.
©2005, The Federal Reserve Bank of St. Louis. Articles may be reprinted, reproduced, published, distributed, displayed, and transmitted in
their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses, and other derivative works may be made
only with prior written permission of the Federal Reserve Bank of St. Louis.
FEDERAL RESERVE BANK OF ST. LOUIS REVIEW JULY/AUGUST 2005 435
Hamilton
quencies, in what has come to be known as “real whether the nineteenth-century economists were
business cycle models.” on to something that their modern descendants
Although unifying growth and business cycle may have forgotten. Is there really a business cycle,
theory holds tremendous aesthetic appeal, this or is the expression an unfortunate linguistic
particular solution is not without its detractors. vestige of a less-informed era? I will argue that
Indeed, the reasons that Irving Fisher gave in 1932 indeed there is a recurring pattern in the level of
for rejecting such an approach have in the opinion economic activity that needs to be explained, but
of many yet to receive a satisfying response from that a statistical characterization of this pattern
modern real business cycle theorists: requires a nonlinear dynamic representation and
[I]n times of depression, is the soil less fertile? calls for an asymmetric interpretation of the forces
Not at all. Does it lack rain? Not at all. Are the that cause employment to rise and fall. I further
mines exhausted? No, they can perhaps pour observe that one element of this pattern has often
out even more than the old volume of ore, if been a related cyclical behavior of interest rates.
anyone will buy. Are the factories, then, lamed To the question, “Is the business cycle real?”
in some way—down at the heel? No; machinery these findings suggest that, yes, the business cycle
and invention may be at the very peak. is real in the sense that it is a feature of the data
(Fisher, 1932, p. 5) that needs to be explained. In the other meaning
of the term “real,” however—the sense from which
Continuing along the lines of Fisher’s reason- springs the label “real business cycle,” namely, a
ing, the size of the population places an obvious cycle unrelated to monetary developments—the
physical limit on how much a given nation can evidence adduced here for the importance of
produce and is certainly a key reason that aggre- comovements between financial and real variables
gate output increases over time. But just as surely, suggests that the cycle is not “real” at all or, at the
a decrease in population is not the cause of the least, not completely divorced from monetary
decrease in employment that we observe in times developments.
when the unemployment rate is shooting up dra-
matically. There is in this respect an obvious THE BEHAVIOR OF
inherent asymmetry in fluctuations in the number
of workers employed—the measure must go up UNEMPLOYMENT
for different reasons than it goes down. A parallel Figure 1 plots the monthly unemployment rate
argument can be made in terms of the capital 1
stock, another key factor determining long-run in the United States from 1948:01 to 2004:03. I
growth, which again places an upper limit on would suggest that someone looking at such a
how much a country can produce. Yet in times graph for the first time would indeed be inclined
when we see all measures of capacity utilization to identify a repeated sequence of ups and downs,
falling, the natural inference is that some forces with each of the obvious sharp upswings in the
other than the quantity or quality of available unemployment rate occurring during periods that
manufacturing facilities account for the drop in the National Bureau of Economic Research (NBER)
aggregate output. has classified as economic recessions (indicated
If we agree that these three factors—technol- by shaded regions on the graph).
ogy, labor force, and the capital stock—are the Although one’s eye is sympathetic to the claim
three main determinants of long-run economic that these data display a recurrent pattern, it does
growth, we might greet with considerable skepti- not appear to be cyclical in the sense of exhibiting
cism the suggestion that the same three factors strict periodicity. For example, the two consecu-
are in a parallel way responsible for producing tive unemployment peaks in 1958:07 and 1961:05
the drop in real GDP that we observe during a are separated by less than three years, whereas
business downturn. 1 This is the seasonally adjusted civilian unemployment rate from
The purpose of this paper is to explore the Bureau of Labor Statistics; http://stats.bls.gov.
436 JULY/AUGUST 2005 FEDERAL RESERVE BANK OF ST. LOUIS REVIEW
Hamilton
Figure 1 Figure 2
U.S. Monthly Civilian Unemployment Rate Estimated Spectrum of U.S. Monthly Civilian
and U.S. Recessions, 1948:01–2004:03 Unemployment Rate, 1948:01–2004:03
11 14
10 12
10
9
8
8 6
7 4
2
6
0
5 0 10 20 30
Period of Cycle (years)
4
3 NOTE:Plotted as a function of the period of the cycle in years.
2
1948 1953 1958 1963 1968 1973 1978 1983 1988 1993 1998 2003
(2) uy=−c−φφy−y
tt 11t−−2t2
k =+log ΓΓνν12/ −log / 2
() []
{}
{}
(3)
those of 1982:11 and 1992:06 are separated by a 2
− 12/logσνππ
()
decade. More formally, one can look for any sort ()
of periodic pattern by examining the spectrum of 2
with respect to θ = (c,φ ,φ ,σ ,ν)′ subject to the
the unemployment rate, an estimate of which is 1 2
3 2
constraints that σ > 0 and ν> 0. These maximum
plotted in Figure 2 as a function of the period of likelihood estimates (MLEs) (with asymptotic
2
the cycle. If one tries to decompose the unem- standard errors in parentheses) imply that the
ployment series in Figure 1 into a series of strictly unemployment rate y for month t could be mod-
periodic cycles, by far the most important of these t
eled as follows:
are those with the longest period, as opposed to (4) yy++0..060 1 117 −0.128yv+0.,158
tt−1 tt−2
0..028 0037 0.037 0.007
something regularly repeating every 3 to 5 years. ()() (() ()
Let y denote the unemployment rate. Consider
t where v is distributed Student twith 4.42 degrees
an AR(2) representation of these data with Student t
t innovations, obtained by maximizing the log of freedom, with the standard error for the degrees-
likelihood function of-freedom parameter ν being estimated at 0.74.
Using Student t innovations instead of Normal
T innovations increases the log likelihood by 52.04,
L θθ= ℓ
()∑ t() a huge gain from estimating the single parameter
t=3 ν(see Table 1).
2
u
t As further evidence against a cycle with reg-
(1) ℓ θν=−k +12/log1+
t () ()
νσ2 ular periodicity, it is interesting to note that the
roots of the second-order difference equation in
2 This was calculated by smoothing the sample periodogram with a (4) are both positive and real, meaning that this
Bartlett window (e.g., Hamilton, 1994, eq. [6.3.15]) with lag q = 13, system does not exhibit any oscillatory behavior
as calculated using the RATS fft procedure with window (type = in response to a shock to v .
tent, width = 25). See the procedure hamp167.prg available at t
www.estima.com/procs_hamilton.shtml for details. The resulting
ˆ
estimate sY(ωj) for ωj + 2πj/T is plotted in Figure 2 for given j as a 3
function of T/j, which is the variable measured on the horizontal See, for example, Hamilton (1994, Section 5.9) on numerical
axis. maximization subject to inequality constraints.
FEDERAL RESERVE BANK OF ST. LOUIS REVIEW JULY/AUGUST 2005 437
Hamilton
Table 1
Comparison of Selected Models of Postwar Unemployment Rates
Model No. of parameters Log likelihood Schwarz criterion
Gaussian AR(2) 4 75.59 62.57
Student t AR(2) 5 127.63 111.35
Student t AR(2) with MS intercept 11 174.58 138.77
NOTE:Schwarz criterion calculated as L – (k/2)log(T) for L the log likelihood, k the number of parameters, and T = 673 the sample size.
good and bad values of the innovations v, and
Figure 3 t
perhaps we could make up some rule for categoriz-
Simulated Unemployment ing a relatively unlikely string of mostly negative
innovations as a “recession.” But any such rule
7.0 would be completely arbitrary and tell us more
about our imagination or quest for patterns and
6.5 labels than about anything in the objective reality.
6.0 There is nothing qualitatively different about a
value of v that puts us within the arbitrary reces-
5.5 t
sion category and one that leaves us just short of it.
5.0 I would argue that this inability to define a
4.5 business cycle as a fundamental attribute of the
data-generating process (4) is in fact inherent in
4.0 any time-series model that describes y as a linear
t
3.5 function of its lagged values plus an i.i.d. inno-
3.0 vation. Even if the linear difference equation did
1948 1953 1958 1963 1968 1973 1978 1983 1988 1993 1998 2003 exhibit an oscillatory impulse-response function
or imply more power in the spectrum at periods
NOTE:Simulated sample generated from equation (4). of 3 to 5 years, it seems to be some other feature
of the data in Figure 1 that constitutes the “busi-
ness cycle.”
Is the appearance of a repeated cycle in I would suggest instead that what we have
Figure 1 just a figment of our imagination, then? in mind is that there is something in common
Another interesting exercise is to simulate a time- between the rapid run-ups in unemployment
series realization from (4), which is displayed in that occurred in each of the postwar recessions,
Figure 3. These simulated data have the same even though the length of time it takes for unem-
mean, variance, and serial correlation as the real ployment to spike up varies from episode to
data in Figure 1, as of course they should. Even so, episode, and the timing separating such events
one has little of the sense of a recurrent cycle in is irregular. Indeed, the idea of looking for com-
these simulated data that seemed compelling in monality across recessions whose elapsed calendar
the actual data. If one were to label some of the
episodes in this simulated data set as “recessions,” time is different for different episodes was pre-
where would they be? Indeed, expression (4) cisely the methodology that Burns and Mitchell
characterizes the true process from which these used to create their graphs summarizing typical
artificial data were simulated. What in terms of business cycle patterns. Stock (1987, 1988) showed
the qualities of this data-generating process would that such a way of thinking about data necessarily
one characterize as a “business cycle?” There are implies a nonlinear data-generating process.
438 JULY/AUGUST 2005 FEDERAL RESERVE BANK OF ST. LOUIS REVIEW
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