Mark W. Watson

English

Research Papers in Economics

FEDERAL RESERVE BANK OF ST. LOUIS REVIEW JULY/AUGUST 2005 453
Commentary
Mark W. Watson
value in 1948. Evident in the figure is sustained
growth that is occasionally interrupted by one of
the ten recessions in the postwar period. These
alternating periods of expansion and recession are
the “business cycle,” and in his paper, Hamilton
uses a Markov-switching model for the unemploy-
ment rate to delineate these expansions and
recessions. As Hamilton shows, the dynamics of
the unemployment rate are quite different in the
expansion and recession states. In this sense, the
business cycle is real; that is, unemployment
dynamics are significantly different during expan-
sions and recessions.
The remaining panels of the figure show
results for three other economies, and I often
begin my time-series course by asking students
to identify the economies that I have plotted. All
three countries show periods of expansions and
recession like the United States. Country 1 expe-
rienced a sharp and severe recession in 1975 along
with several other less severe recessions. Country 2
suffered a minor recession in 1953, but then grew
more or less steadily until its 1970 recession; it
weathered 1975 without a recession, but has
suffered four recessions since the late 1970s.
Country 3 grew rapidly from 1948 until 1963,
when it experienced its first postwar recession,
then suffered a prolonged recession from 1967 to
1970, a mild recession in 1980, but has been per-
forming well since then. Foreign students typically
have a better idea of international business cycles
than domestic students and often recognize these
business cycle patterns.
H
amilton’s paper (2005) asks two
provocative questions. First, are
business cycles “real” in the sense
that recession/expansion phases
represent fundamental shifts in the dynamic
model characterizing the macroeconomy? Second,
are business cycles “real” in the sense of being
caused by real shocks such as productivity or
labor supply? Hamilton’s careful empirical analy-
sis of the postwar unemployment rate and of
interest rates in nineteenth century and postwar
periods leads him to answer yes to the first ques-
tion; on the basis of this analysis he conjectures
that the answer to the second question is no.
My comments will address the first of
Hamilton’s questions. I first ask whether the non-
linear switching models of the sort estimated by
Hamilton for the unemployment rate are necessary
to explain business cycles of the sort experienced
by the United States in the postwar period. My
answer, like the answer given by Slutsky (1937),
is no. I next ask whether nonlinear models provide
a better fit and produce more-accurate forecasts
than linear models for postwar U.S. macroeco-
nomic data. My answer is a cautious maybe. 
DO WE NEED “REAL” BUSINESS
CYCLES TO EXPLAIN THE CYCLE?
The first panel of Figure 1 plots quarterly
values of the logarithm of real gross domestic
product (GDP) for the United States relative to its
Mark W. Watson is a professor of economics and public affairs at Princeton University and a research associate at the National Bureau of
Economic Research. This research was funded in part by the National Science Foundation under grant No. SBR-0214131.
Federal Reserve Bank of St. Louis Review, July/August 2005, 87(4), pp. 453-58.
© 2005, The Federal Reserve Bank of St. Louis. Articles may be reprinted, reproduced, published, distributed, displayed, and transmitted in
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            Most students seem surprised when I
announce that I produced the plots for countries
1, 2, and 3 using a random number generator.
More precisely, these plots were produced using
three realizations from a linear AR(2) model with
i.i.d. Gaussian innovations that was calibrated to
the U.S. data. Of course, this is just an updated
version of the remarkable simulations shown in
the classic paper by Slutsky (1937).
Slutsky’s simulations have important implica-
tions for business cycle analysis. They show that
simple time-invariant linear time-series models
are capable of generating realizations that have
the important cyclical properties that we have
come to call the business cycle. Evidently, non-
linear switching models are not required to gener-
ate business cycles.
This discussion highlights an important differ-
ence in empirical characterizations of the business
cycle. One characterization—Hamilton’s—is that
recessions and expansions represent fundamental
shifts in the stochastic process characterizing
the macroeconomy. Another characterization—
Slutsky’s—is that recessions and expansions are
features of the realization of the stochastic process;
the process doesn’t shift, but sometimes it pro-
duces data that decline (recessions) and some-
times it produces data that grow (expansions).
EVIDENCE ON “REAL” BUSINESS
CYCLES
While nonlinear models are not required to
generate time series with business cycle charac-
teristics, nonlinear models may provide better
Watson
454 JULY/AUGUST 2005 FEDERAL RESERVE BANK OF ST. LOUIS REVIEW
1940 1950 1960 1970 1980 1990 2000 2010 1940 1950 1960 1970 1980 1990 2000 2010
1940 1950 1960 1970 1980 1990 2000 2010 1940 1950 1960 1970 1980 1990 2000 2010
2.5
2.0
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Log GDP Log GDP
Log GDP Log GDP
A. Quarterly U.S. Data B. Country 1
C. Country 2 D. Country 3
Figure 1
Logarithm of GDP for the United States and Three Other “Countries”descriptions of the stochastic processes charac-
terizing typical macroeconomic series than simple
linear models. A series of papers building on
Hamilton’s (1989) original contribution have
shown that Markov-switching models provide
an improvement on the fit of linear models for
several important macroeconomic time series.
(Examples include Chauvet, 1998, Diebold and
Rudebusch, 1996, and Filardo, 1994.) Stock and
Watson (1999) compared the forecast performance
of various linear and nonlinear univariate fore-
casting models for 215 monthly macroeconomic
time series using a pseudo out-of-sample forecast-
ing experiment. Table 1 contains a summary of
their findings. For several categories of series
(production, employment, construction, invento-
ries, orders, interest rates, and wages), the non-
linear models outperformed linear models.
Does the three-state Markov-switching model
that Hamilton proposes in this paper produce
more precise forecasts of the state of the business
cycle than linear models? To investigate this, I
computed the one-sided (“filtered”) estimates of
the state probabilities from Hamilton’s model.
These filtered probabilities are plotted in Figure 2
and are the one-sided versions of the probabilities
plotted in Hamilton’s Figure 4. I considered six
different monthly series that serve as coincident
indicators of the business cycle: the unemploy-
ment rate, the index of industrial production
(logarithm), real personal income (logarithm),
manufacturing and trade sales (logarithm), employ-
ment (logarithm), and an index of coincident
indicators constructed as a weighted average of
the last four series. Using data from 1959-2003, 
I estimated regression models of the form
(1)
yt+h– yt = β0 + φ(L)∆yt + γ(L)ut + β2p2t/t + β3p3t/t + εt+h
where yt denotes the indicator being forecast, ut
denotes the unemployment rate, and p2t/t and p3t/t
denote the filtered state probabilities (that is, the
nonlinear functions of the unemployment rate
plotted in Figure 2). Results for this regression
are shown in Table 2 for 1-month-ahead (h = 1)
Watson
FEDERAL RESERVE BANK OF ST. LOUIS REVIEW JULY/AUGUST 2005 455
Table 1
Performance of Linear and Nonlinear Univariate Forecasting Models for 215 Series
Percentage best
Series category (No. of series) Linear Nonlinear
Production (24) 21 79
Employment (29) 21 79
Wages (7) 29 71
Construction (21) 5 95
Trade (10) 50 50
Inventories (10) 30 70
Orders (14) 7 93
Money and credit (21) 57 43
Interest rates (11) 45 55
Producer prices (16) 50 50
Consumer prices (15) 63 37
Consumption (5) 80 20
Other (31) 52 48
NOTE: This table summarizes the forecasting experiment in Stock and Watson (1999) involving linear and nonlinear methods for
forecasting 215 series in the 13 categories, shown in the first column. The second and third columns show the percentage of series
for which the linear model outperformed the nonlinear model (column 2) or the reverse (column 3).and 3-month-ahead (h =3) forecasts. There does
seem to be evidence that p2t/t and p3t/t help pre-
dict the indicators, particularly at the 1-month
horizon.
An alternative, and arguably more compelling
test of the predictive power of p2t/t and p3t/t, comes
from using recursive estimates of the parameters
of (1) to compute pseudo out-of-sample forecasts.
Table 3 summarizes results from these calculations
over the 1970-2003 out-of-sample forecast period.
The results presented in the table are the mean-
squared forecast errors for various versions of (1)
relative to a simple AR model. The results shown
in the column labeled “P” are for the model that
includes p2t/t and p3t/t in addition to lags of ∆yt
(so that γ (L) = 0); the results in the column labeled
Watson
456 JULY/AUGUST 2005 FEDERAL RESERVE BANK OF ST. LOUIS REVIEW
1.0
0.5
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1.0
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1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
A. Probability of State 1
B. Probability of State 2
C. Probability of State 3
Figure 2
Filtered Probabilities of Unemployment Rate StatesWatson
FEDERAL RESERVE BANK OF ST. LOUIS REVIEW JULY/AUGUST 2005 457
Table 2
Granger-Casualty Tests for the Model
yt+h – yt = β0 + φ(L)∆yt + γ (L)ut + β2 p2t/t + β3 p3t/t + εt+h
Forecast horizon h = 1 Forecast horizon h = 3
F-statistic, F-statistic,
Series forecast β2 β3 p-value β2 β3 p-value 
Unemployment rate 0.09 (0.04) 0.34 (0.12) 0.01 0.15 (0.10) 0.35 (0.27) 0.33 
Industrial production  –1.08 (2.20) –11.56 (4.89) 0.05 –1.33 (2.47) –6.71 (4.00) 0.18 
Personal income  –1.39 (1.14) –2.43 (2.83) 0.47 –1.09 (1.06) –1.11 (1.85) 0.58 
Manufacturing and  1.15 (2.69) –11.98 (6.30) 0.05 –1.53 (2.06) –5.33 (4.60) 0.50 
trade sales
Employment –0.55 (0.54) –3.17 (1.38) 0.07 –0.58 (0.56) –1.47 (1.06) 0.38 
Coincident index –0.94 (1.41) –6.47 (3.25) 0.13 –1.19 (1.51) –3.87 (2.74) 0.37 
NOTE: The table shows OLS estimates of β2 and β3, HAC standard errors, and p-values for the F-test that β2 = β3 = 0.
Table 3
Out-of-Sample Mean-Squared Error Relative to Univariate Autoregressive Model
yt+h – yt = β0 + φ(L)∆yt + γ (L)ut + β2 p2t/t + β3 p3t/t + εt+h
All Recessions Expansions
Series P U U-and-P P U U-and-P P U U-and-P
A. Forecast horizon h = 1
Unemployment 0.98 0.97 0.98
Industrial production 0.95 0.97 0.97 0.87 0.88 0.88 1.00 1.03 1.02
Personal income 0.95 0.93 0.93 0.92 0.92 0.92 0.95 0.93 0.94
Manufacturing and trade sales 0.99 0.99 1.01 1.01 0.95 1.07 0.99 1.00 0.99
Employment 0.96 0.96 0.97 0.89 0.87 0.86 1.00 1.02 1.02
Coincident index 0.97 0.97 0.97 0.94 0.94 0.95 1.02 1.01 1.02
B. Forecast horizon h = 3
Unemployment 1.01 1.00 1.01
Industrial production 0.98 0.97 0.98 0.94 0.94 0.95 1.02 1.01 1.02
Personal income 0.92 0.89 0.90 0.90 0.90 0.90 0.93 0.89 0.90
Manufacturing and trade sales 0.99 0.97 0.99 1.00 0.96 1.00 0.99 0.97 0.99
Employment 0.96 0.94 0.96 0.89 0.91 0.90 1.04 0.98 1.02
Coincident index 0.96 0.94 0.95 0.93 0.92 0.93 0.99 0.96 0.98
NOTE: The table shows the mean-squared forecast error for each model relative to that for the univariate AR model. The model labeled
“P” imposes the constraint that γ(L) = 0; the model labeled “U” imposes the constraint that β2 = β3 = 0, and the model labeled “U-and-P”
imposes no constraints. Results are shown for the full 1970-2003 out-of-sample and for the recession and expansion subsamples.“U” are for the model that includes lags of ut and
∆yt (so that β2 = β3 = 0), and the results in the col-
umn labeled “U and P” include all of the terms in
(1). Relative mean-squared errors are shown for
the entire out-of-sample period and for recessions
and expansions separately.
Most of the table entries are less than 1.0,
indicating an improvement on the univariate AR
model. However, it is less clear whether the model
with nonlinear functions of the unemployment
rate (p2t/t and p3t/t) outperform the linear model
that includes the unemployment rate. There are
few entries in which the P or U-and-P models
outperform the U model.
My interpretation of the evidence in Tables 2
and 3 is that they provide some additional (albeit
weak) evidence supporting the “real” switching
model proposed by Hamilton.
REFERENCES
Chauvet, Marcelle. “An Econometric Characterization
of Business Cycle Dynamics with Factor Structure
and Regime Switches.” International Economic
Review, November 1998, 39(4), pp. 969-96.
Diebold, Francis X. and Rudebusch, Glenn D.
“Measuring Business Cycles: A Modern Perspective.”
Review of Economics and Statistics, February 1996,
78(1), pp. 67-77.
Filardo, Andrew. “Business-Cycle Phases and Their
Transitional Dynamics.” Journal of Business and
Economic Statistics, July 1994, 12(3), pp. 299-308.
Hamilton, James D. “A New Approach to the Economic
Analysis of Nonstationary Time Series and the
Business Cycle.” Econometrica, 1989, 57(2), pp.
357-84.
Hamilton, James D. “What’s Real About the Business
Cycle?” Federal Reserve Bank of St. Louis Review,
July/August 2005, 87(4), pp. 435-52.
Slutsky, Eugen. “The Summation of Random Causes
as the Source of Cyclic Processes.” Econometrica,
April 1937, 5(2), pp. 105-46.
Stock, James H. and Watson, Mark W. “A Comparison
of Linear and Nonlinear Univariate Models for
Forecasting Macroeconomic Time Series,” in
Robert F. Engle and Halbert White, eds., Festschrift
in Honor of Clive W.J. Granger. Oxford: Oxford
University Press, 1999, pp. 1-44.
Watson
458 JULY/AUGUST 2005 FEDERAL RESERVE BANK OF ST. LOUIS REVIEW

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