GROWTH AND BUSINESS CYCLE COMPONENTS (PEC11_11) 1
Riccardo Fiorito
November 2011
1. Detrending via the HP filter
In modern macroeconomics, business cycles are defined as persistent, stationary, deviations from
a steady growth path, displaying at different phases comovements among aggregate variables
(Lucas, 1977). When the business cycle component is evaluated via the Hodrick-Prescott (1980)
filter, the trend component is stochastic though it does not change too much and too often.
Namely, the trend component () changes slowly over time since it is intended to capture the
long-run productivity pattern which is assumed to change at the low frequencies only.
Thus, business cycles are made observable removing from the data the long-run, smooth, trend
component. In the real business cycle models (RBC), productivity is the main instigator of the
business cycle which can be understood (Kydland-Prescott, 1982) by using the neoclassical,
stochastic, growth model in a general equilibrium and intertemporal setting (Prescott, 1986).
However, the HP filter is used nowadays by most researchers since it provides a simple way for
detrending the macroeconomic data without implying too many assumptions and without requiring
a different ad hoc procedure for each of the series to be investigated.
A full explanation of the HP filter goes beyond this course requirements and the interested reader
should be redirected to the specialized literature (King-Rebelo, 1993). In a nutshell, the HP filter
can be seen as the solution of the following problem:
Min with respect to t (t=1,…,T) the following expression:
(1)
(yt - t)2 ,
subject to:
(2)
[(t - t-1) – (t-1 - t-2)]2  ,   R+.
This means that the smooth trend component is obtained by minimizing the sum of squared
differences from the data (1) subject to the constraint (2) that the sum of the squared second
differences be less than some appropriate bound . The smaller is , the smoother is the trend so
that the HP trend converges to the deterministic (TS), trend when  = 0.
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 Riccardo Fiorito, March 2006 (Revised: November 2011).
Problems (1) and (2) can be treated in terms of the following unconstrained problem:
(3)
(yt - t)2 + [(t - t-1) – (t-1 - t-2)]2,
  R +,
where  is a smoothing parameter. As implied above, the smaller  is, the smoother the trend will
be so that when  goes to zero the smoothed trend coincides with the deterministic TS trend. For
quarterly data, the widely accepted value for  is 1600. Of course, this value will be smaller for
yearly data - since Backus and Kehoe (1992)  is typically 100 - and much bigger for monthly
data (a standard value for  is about 14.000!) since low frequency data have much more variability
to be removed through an appropriate value of .2
However, the cyclical results are not too sensitive to the choice of  so that the stylized facts of
the business cycles (Kydland-Prescott, 1990) are relatively immune from the narrow choice of the
smoothing parameter and also from the detrending procedure too. Actually, this last statement
holds true for the comovements only (Fiorito-Kollintzas, 1994), it being obvious that the
univariate properties of the relevant time series are affected if a unit root (DS) instead – say - of a
deterministic procedure (TS) is used to remove the stochastic trend. (Section 2, Table 1).
Figure 1- Italy's real gdp (quarterly data, ISTAT)
Hodrick-Prescott Filter (lambda=1600)
7.2
7.1
7.0
6.9
.03
6.8
.02
6.7
.01
6.6
.00
-.01
-.02
-.03
80 82 84 86 88 90 92 94 96 98 00 02 04 06
LY
Trend
Cycle
2
Ravn and Uhlig (2002) address the issue of the relation between the smoothing parameter and the data frequency. They
suggest for annual data a smaller value (6.25) on the basis of the band-pass filter properties outlined by Baxter and King
(1999).
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2. Cyclical components
To show how detrending might affect time-series properties, I display in the following Table 1
the autocorrelations of the logged Italy’s real GDP series (LY) and the detrended data obtained by
applying either the HP filter or the unit-root, first difference filter (see Figure 1). The real GDP
log-level data show a very long decay process, i.e. a strong persistence that could be compatible
with the presence of a unit root. This case can be evaluated – though some caveat is required – via
a Dickey-Fuller (DF) or an augmented DF (ADF) test which is available in most econometric
packages for time-series data.3
An obvious difference between the HP and the unit root (DS) detrending is that the HP filter
removes the slowly moving stochastic trend only. Thus, the remaining cyclical component is still
persistent and generally able to comply with the observed duration (2-6 years) of the historical
business cycle as measured by the NBER chronology (see Section 3) [PEC11_10].
The unit-root detrended data (DLY) show – even at the quarterly frequencies - that the % change
series has no persistence at all since the autocorrelations decay in about one year as it was first
shown for the US by the seminal Nelson and Plosser (1982) article [PEC11_13].
Thus, while unit roots detrending reveals that most of the variability in macroeconomic series
rests on low-frequency components, it also reveals that the DLY series does not convey any useful
information on Italy’s business cycle, given that real GDP quarterly changes are not too different
from a non-informative, white noise, process: the quarterly autocorrelations decay quickly and,
statistically, are not different from zero at the usual confidence levels. Hence, it seems high the
risk that unit-rot detrending
eliminates more than the trend only since this procedure cannot
disentangle between trend and cyclical components unless some additional information is provided
or assumed.
Table 1- Italy’s real GDP autocorrelations: actual and detrended quarterly data
Autocorrelations
Log levels
HP-Filtered Cycles
Log differences: unit
root differences
LY
CY= LY-LYHP
DLY=LY(t)-LY(t-1)
(k)
1
.974
.833
.344
2
.946
.610
.171
3
.918
.397
.226
4
.890
.150
-.034
8
.768
-.195
.043
12
.638
-.175
-.009
16
.510
-.258
-.132
20
.386
-.056
-.028
3
A reference for an excellent, introductory level, textbook is: Johnston-DiNardo, 1997.
3
30
.115
-.223
-.107
3. Classical (NBER) Business Cycles
The classical approach to business cycle was popular among the early the NBER scholars (Burns
and Mitchell, 1946) who first tried to evaluate the business cycle properties using data inspection
rather than the deterministic math prevailing at those times among business cycle scholars.
These early studies started in the interwar period before econometric methods were fully
developed and widely used in the research community. They tried to predict the real GDP pattern
and the major business cycles comovements without attempting to separate business cycle from
trend components. The main concern was finding some rule-of-thumb association between a single
macroeconomic variable at time and the overall activity level in order to anticipate future real
GDP changes. This effort was aiming at reducing the cost of recessions that was economically and
socially destructive in the US during the long-lasting Great Depression.
The candidate variables were labeled as leading indicators and contributed anyway to develop
the empirical knowledge of the US economy in the pre-war2 period. Today, the leading indicators
are becoming popular again and are used by several economists for understanding business cycles,
though modern studies reveal a much higher degree of technical sophistication (Stock-Watson,
2002). The ECB and the OECD too use leading indicators for predicting the economy, though
mixing sometimes this approach with a liberal use of large econometric models that were typically
designed for releasing conditional forecasting and wide-sense policy analysis.
Basically, the NBER approach established a quarterly chronology for the US [PEC11_10] in
which stationary business cycles are replaced by recessions, i.e. by situations in which the
economy reduces its absolute level with respect to the previous period or else to the case in which
real GDP falls in two consecutive quarters.4 Recessions are characterized by turning points
(peaks and troughs in the NBER terminology) whose evaluation can also be judgemental at the
higher frequencies where noisy components mostly occur.5
In modern economies, however, recessions are much less frequent than growth and this is why
economic data are non-stationary and generally rising. Conversely, cyclical swings around the
trend generate a number of upward and downward swings that eventually compensate over the
4
This textbook definition aims at avoiding that an occasional contraction be wrongly considered a true recession episode.
This point was first stated by Shiskin (1974).
5
This is why for monthly data is frequently used the Bry and Boschan (1971) procedure which also led to a number of
algorithms trying to replicate the NBER turning point selection.
4
sample because the cyclical components are by definition stationary, zero-mean, variables
extracted from a non-stationary trend.
References
D. Backus-P. Kehoe (1992), International Evidence on the Histprocal Properties of Business Cycles,
“American Economic Review”, 82(4), 864-88.
M. Baxter-R. King (1999), Measuring Business Cycles: Approximate Band-Pass Filters for Economic Time
Series, “Review of Economics and Statistics”.
G. Bry-C. Boschen (1971), Cyclical Analysis of Time Series: Selected Procedures and Computer Programs,
NBER.
A.F. Burns-W.C. Mitchell (1946), Measuring Business Cycles, NBER, New York.
R.Fiorito-T.Kollintzas (1994), Stylized Facts of the Business Cycles from a Real Business Cycles
Perspective, “European Economic Review”.
R. Hodrick-E.C. Prescott (1980), “Post-War US Business Cycles”, Carnegie-Mellon University
Working Paper.
J. Hohnston-J. DiNardo (1997) Econometric Methods, McGraw Hill.
R. King- S.Rebelo, Low Frequency Filtering and Real Business Cycles, “Journal of Economic
Dynamics and Control”, 1993.
F.Kydland-E.C.Prescott (1982), Time to Build and Aggregate Fluctuations, “Econometrica”, 1345-70.
F. Kydland-E. Prescott E. (1990), Business Cycles: Real Facts and a Monetary Myth, Federal Reserve
Bank of Minneapolis Quarterly Review, 3-18.
R.E. Lucas (1977), Understanding Business Cycles in “Carnegie-Rochester Conference on Public Policy”,
North Holland, vol. 5, 7-29 [reprinted in: R.E. Lucas (1981), Studies in Business Cycle Theory, The MIT Press,
215-39].
C. Nelson-C. Plosser (1982), Trends and Random Walks in Macroeconomic Time Series, “Journal of Monetary
Economics”, September, 139-62.
E.C. Prescott (1986), Theory Ahead of Business-Cycle Measurement, Carnegie-Rochester Conference Series
on Public Policy, 25, 11-44.
M.Ravn-H. Uhlig (2002), On Adjusting the HP-filter for the Frequency of Oservations, “Review of Economics
and Statistics”, 84, 371-80.
J. Shiskin (1974), The Changing Business Cycle, New York Times, December 12.
J. Stock-M.Watson (2002), Has the Business Cycle Changed and Why? In M. Gertler- K. Rogoff (eds.),
Macroeconomics Annual 2002, The MIT Press, 159-218.
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