We study the impact of alternative detrending techniques on the distributional properties of U.S. output time series. We detrend GDP and industrial production time series employing first-differencing, Hodrick-Prescott and bandpass filters. We show that the resulting distributions can be approximated by symmetric Exponential-Power densities, with tails fatter than those of a Gaussian. We also employ frequency-band decomposition procedures finding that fat tails occur more likely at high and medium business-cycle frequencies. These results confirm the robustness of the fat-tail property of detrended output time-series distributions and suggest that business-cycle models should take into account this empirical regularity.Statistical Distributions, Detrending, HP Filter, Bandpass Filter, Normality, Fat Tails, Time Series, Exponential-Power Density, Business Cycles Dynamics

Andrea Roventini

Giorgio Fagiolo

Marco Piazza

Mauro Napoletano

English

Research Papers in Economics

           
￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿
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￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿Detrending and the Distributional Properties of U.S. Output
Time Series∗
Giorgio Fagiolo† Mauro Napoletano‡ Marco Piazza§ Andrea Roventini¶ .
Abstract
We study the impact of alternative detrending techniques on the distributional properties of
U.S. output time series. We detrend GDP and industrial production time series employing
ﬁrst-diﬀerencing, Hodrick-Prescott and bandpass ﬁlters. We show that the resulting distri-
butions can be approximated by symmetric Exponential-Power densities, with tails fatter
than those of a Gaussian. We also employ frequency-band decomposition procedures ﬁnding
that fat tails occur more likely at high and medium business-cycle frequencies. These results
conﬁrm the robustness of the fat-tail property of detrended output time-series distributions
and suggest that business-cycle models should take into account this empirical regularity.
Keywords: Statistical Distributions, Detrending, HP Filter, Bandpass Filter, Normality,
Fat Tails, Time Series, Exponential-Power Density, Business Cycles Dynamics.
JEL Classiﬁcation: C1, E3.
∗Thanks to Marco Lippi, Sandro Sapio, Andrea Vaona. All usual disclaimers apply.
†Sant’Anna School of Advanced Studies, Pisa, Italy. Mail address: Sant’Anna School of Advanced Studies,
Piazza Martiri della Libert` a 33, I-56127 Pisa, Italy. Email: giorgio.fagiolo@sssup.it
‡Observatoire Fran¸ cais des Conjonctures Economiques, Department for Research on Innovation and Competi-
tion, Sophia-Antipolis, France, and Sant’Anna School of Advanced Studies, Pisa, Italy. Mail Address: Observa-
toire Fran¸ cais des Conjonctures Economiques, Department for Research on Innovation and Competition, 250 rue
Albert Einstein, 06560 Valbonne, France. Email: mauro.napoletano@ofce.sciences-po.fr
§Sant’Anna School of Advanced Studies, Pisa, Italy. Mail address: Sant’Anna School of Advanced Studies,
Piazza Martiri della Libert` a 33, I-56127 Pisa, Italy Email: m.piazza@sssup.it
¶University of Verona, Italy, Department of Economics, and Sant’Anna School of Advanced Studies, Pisa,
Italy. Mail Address: Universit` a di Verona, Dipartimento di Scienze Economiche, Via dell’Universit` a, 3 - I-37129
Verona. Email: andrea.roventini@univr.it
11 Introduction
In the recent years, diﬀerent detrending techniques (e.g. linear trend, ﬁrst-diﬀerencing, Hodrick-
Prescott and bandpass ﬁlters) have been proposed in macroeconomics to study the business-cycle
properties of aggregate time series (see e.g. Stock and Watson, 1999; Christiano and Fitzgerald,
2003). However, the choice of the detrending procedure is anything but neutral: diﬀerent ﬁltering
methods can aﬀect both the qualitative and the quantitative stylized facts of the business cycle
(Canova, 1998) and can bias the determination of business-cycle turning points (Canova, 1999).
Furthermore, the choice of the ﬁlter has also signiﬁcant implications for macroeconomic theory:
Delle Chiaie (2009) shows that diﬀerent detrending techniques aﬀect the estimation of structural
parameters in Dynamic Stochastic General Equilibrium (DSGE) models (e.g., Woodford, 2003;
Gal´ ı and Gertler, 2007), altering in turn the magnitude and persistence of model responses to
shocks.
Following this line of research, this work explores whether detrending has any relevant impact
on the distributional properties of ﬁltered output time series. More speciﬁcally, we ﬁlter U.S.
GDP and industrial production (IP) time series employing a number of diﬀerent detrending
techniques and we study the shape of the resulting time-series distributions using a parametric
approach.
Our results show that: (i) detrended U.S. output time-series distributions can be well proxied
by Expontial-Power densities with tails much fatter than those of a Normal distribution; (ii)
fat-tailed distributions emerge irrespectively of the particular ﬁltering technique employed to
detrend the series (e.g. ﬁrst-diﬀerencing, Hodrick-Prescott, bandpass ﬁlters).
Furthermore, we investigate whether there exist frequencies that are more conducive to fat-
tail behaviors. Our exercises suggest that fat tails do not seem to be a distinctive feature of
all classes of output ﬂuctuations, as they can be more likely associated with high and medium
business-cycle frequencies.
The foregoing results have implications for both theoretical and empirical research. First,
they add further support to the widespread emergence of fat tails in the distributions of country
output ﬂuctuations and growth shocks. Indeed, Fagiolo, Napoletano, and Roventini (2008)
show that fat-tailed, Exponential-Power densities can well proxy time-series distributions of
GDP and IP growth rates – computed employing a ﬁrst-diﬀerencing ﬁlter – for the U.S. and
several other OECD countries1. Therefore, the ﬁndings presented in this note complement such
previous results suggesting that non-Gaussian fat tails are a distinctive and robust feature of all
distributions that proxy the deviations of business ﬂuctuations around the trend.
Second, the fact that fat tails are more likely to appear at relatively high frequency bands
associated to business-cycle periodicities implies that short-run models such as the DSGE should,
on the one hand, try to replicate this additional stylized fact and, on the other hand, aim at
delivering implications that are robust to non Gaussian shocks. Finally, our results indicate
that short-run phenomena (e.g. business inventory management) might be possible candidates
to provide convincing theoretical explanations of output fat-tail distributions.
1Fagiolo, Napoletano, and Roventini (2008) also show that fat tails in output growth-rate distributions emerge
independently of the measures of the aggregate output used, the densities employed in the estimation, the length
of time lags used to compute growth rates, and the presence of outliers, autocorrelation and heteroschedasticity.
See also Castaldi and Dosi (2008) for similar results from a cross-country perspective.
2The paper is organized as follows: Section 2 describes the data and the methodology em-
ployed. Section 3 presents the results. Section 4 concludes.
2 Data and Methodology
The main objects of investigation are U.S. GDP and industrial production (IP) time series.
Quarterly GDP ranges from 1947Q1 to 2005Q3 (234 observations), monthly IP ranges from
1921M1 to 2005M10 (1017 observations). The series are provided by the St. Louis Federal
Reserve Economic Data (FRED) database2.
In order to observe the behavior of the series at business cycle frequencies it is common
practice to detrend the series employing diﬀerent ﬁlters. In this paper we take an agnostic
approach to this problem: none of the ﬁltering methods employed is supposed to be the correct
one. Instead, following Canova (1998), we assume that all procedures are approximations that
isolate diﬀerent aspects of the secular and cyclical components of the series. So, the question
is not which method is the more appropriate, but whether the statistical properties of the
distributions of ﬁltered time series are invariant to the ﬁlter employed. In what follows, we
shall employ four ﬁlters methods: ﬁrst-diﬀerencing ﬁlter (FD), the Hodrick and Prescott (1981)
ﬁlter (HP), the Baxter and King (1999) bandpass ﬁlter (BK), and the Christiano and Fitzgerald
(2003) bandpass ﬁlter (CF).
After having removed the trend from output time series, we take a parametric approach to
the detection of fat-tail behavior by ﬁtting detrended output time series with the Exponential-
Power (EP) family of densities, also known as Subbotin distribution (for details, see Bottazzi
and Secchi, 2003a,b). The EP probability density function reads:
f(x;b,a,m) =
1
2ab
1
bΓ
￿
1 + 1
b
￿e− 1
b|
x−m
a |
b
, (1)
where Γ(·) is the Gamma function.
The exponential-power density is a ﬂexible statistical tool, which is characterized by a scale
parameter a, a location parameter m, and a shape parameter b. The latter determines the
fatness of the tails. Since the EP encompasses both Gaussian (b = 2) and Laplace (b = 1)
densities, the estimate of b (together with its standard deviation) can be employed to measure
how far the empirical distribution is from these benchmarks: the lower the estimate for b, the
fatter the tails. We jointly estimate the three parameters of the Subbotin density via maximum
likelihood (ML), employing the package SUBBOTOOLS3. In what follows, we indicate by ˆ x the
ML estimate of the parameter x ∈ {a,m,b}.
3 Results
We start presenting the ﬁrst four moments of U.S. output time series for the diﬀerent ﬁlters
employed (cf. Table 1). The mean levels are near zero, with the exception of FD ﬁlter. More-
2Freely available online at http://research.stlouisfed.org/fred2/.
3Available online at http://cafim.sssup.it/giulio/software/subbotools/. See Bottazzi (2004) for details.
For other theoretical and computational issues concerning this procedure, we refer to Bottazzi and Secchi (2006).
3Series Filter Mean Std. Dev. Skewness Kurtosis
GDP FD 0.0083 0.0098 -0.0500 4.3614
GDP HP 0.0000 0.0165 -0.6071 3.8007
GDP CF -0.0001 0.0158 -0.2505 3.4900
GDP BK 0.0007 0.0156 -0.3896 3.0602
IP FD 0.0031 0.0191 0.3594 14.6537
IP HP 0.0000 0.0707 -0.4691 6.4696
IP CF 0.0005 0.0716 -0.1029 6.0671
IP BK 0.0001 0.0655 -0.5437 7.2340
Table 1: Summary Statistics of U.S. Output Time Series
over, the variance levels of HP, BK and CF ﬁlters are very large, relatively to the FD ﬁlter.
Furthermore, skewness levels are generally small, implying a marked symmetry of the underly-
ing distributions. Finally, the relatively large levels of kurtosis (especially for IP) suggest that
output distributions exhibit fat tails.
In order to better explore the leptokurtosis present in the data, we ﬁt U.S. output distribu-
tions with the Exponential-Power density. Maximum likelihood estimates, standard errors and
Cramer-Rao conﬁdence intervals (for ˆ b only) are reported in Table 24. We ﬁnd that ﬁltered GDP
and IP series display tails fatter than Gaussian ones. The only exception is BK ﬁltered GDP
series. Note also that for all ﬁltered IP time series ˆ b is smaller than 1, implying tails even fatter
than those of a Laplace distribution5. These results are statistically conﬁrmed by Cramer-Rao
conﬁdence intervals (CI), which show that all CIs are below 2 (again, BK ﬁltered GDP is the
only exception). The CI for IP series remains entirely to the left of 1. The foregoing results seem
to conﬁrm the ﬁndings of non-normality presented in Fagiolo, Napoletano, and Roventini (2008)
for the case of ﬁrst-diﬀerencing ﬁlter (i.e., output growth rates). In addition, the considerable
fatter tails exhibited by IP distributions as compared to GDP ones suggest that there might
exist smoothing mechanisms at work (related to e.g. services industry, automatic stabilizers,
etc., cf. Zarnowitz, 1992) dampening the eﬀects of idiosyncratic ﬁrm shocks.
Note also that estimates for the shape parameters of BK, CF and HP time series are greater
than those of the FD ﬁlter. A possible explanation is that HP, BK and CF ﬁlters wash away
high-frequency ﬂuctuations, which could be responsible for the emergence of fat tails. In order
to explore this hypothesis, we try to understand which frequencies in the spectrum of the series
are more conducive to the leptokurtosis observed in the data. We address this issue by isolating
diﬀerent frequency bands in the spectrum of the series and estimating the parameters of the
Exponential-Power density in each single band. More precisely, we bandpass ﬁlter output growth
rates employing rolling bandwidths of 1 and 6.5 years and we plot the ensuing ˆ b together with
their CI in Figure 1.
Let us begin with GDP. For the wider bandwidth of 6.5 years, ˆ b are initially almost constant
and then start growing as the band moves towards lower frequencies. In particular, ﬁltered series
display tails fatter than Gaussian ones until the shifting band takes on board periodicities higher
than 35 quarters. The picture changes if we employ a one-year bandwidth. In this case, ˆ b remains
4Goodness-of-ﬁt analysis, performed through several tests, do not typically reject the null hypothesis that
ﬁltered output time series are exponenial-power distributed.
5The comparison between GDP and IP ˆ b can be biased by the diﬀerent econometric sample sizes employed.
Nevertheless, in line with Fagiolo, Napoletano, and Roventini (2008), we ﬁnd that (not shown) estimates of b rise
but are still lower than GDP ones if the same sample period is employed.
4Series Filter ˆ b Std.Err. Cramer-Rao CIs ˆ a Std.Err. ˆ m Std.Err.
GDP FD 1.1757 0.1454 0.8849 , 1.4665 0.0076 0.0005 0.0079 0.0005
GDP HP 1.3956 0.1798 1.0360 , 1.7552 0.0141 0.0010 0.0008 0.0011
GDP CF 1.5189 0.2001 1.1186 , 1.9191 0.0140 0.0010 0.0002 0.0012
GDP BK 1.8180 0.2656 1.2868 , 2.3492 0.0149 0.0011 0.0006 0.0017
IP FD 0.7026 0.0376 0.6274 , 0.7778 0.0095 0.0004 0.0032 0.0002
IP HP 0.7839 0.0428 0.6983 , 0.8695 0.0413 0.0017 0.0023 0.0011
IP CF 0.7210 0.0388 0.6434 , 0.7986 0.0402 0.0017 0.0025 0.0011
IP BK 0.7765 0.0438 0.6889 , 0.8641 0.0371 0.0016 0.0019 0.0010
Table 2: Estimated Exponential-Power Parameters
initially constant, then starts growing and ﬁnally, after a slight decrease, stays constant again.
Anyway, leptokurtosis is present in the data up to 13 quarters. More generally, the analysis of
ﬁltered GDP seems to point out that relatively higher frequencies in the business cycle spectrum
are more conducive to fat-tail distributions. Filtered IP series distributions appear to be non
Gaussian whatever band and bandwidth we consider. In particular, shape parameters appear
to be lower than one. Moreover, similarly to GDP, fat tails are more marked at frequencies
belonging to the relatively more irregular part of the spectrum.
4 Concluding Remarks
In this paper, we have investigated the statistical properties of the distributions of U.S. ﬁltered
output time series. We have shown that fat tails in the distributions of output are robust to
diﬀerent ﬁltering methods employed for isolating the cyclical component in the series. Moreover,
we have performed a detailed frequency analysis ﬁnding that high and medium frequencies in
the spectrum (associated to relatively short business cycle ﬂuctuations) are also the ones more
conducive to fat tails in the data.
Our results bring further support to the claim that fat tails are a very robust feature of
detrended output distributions at the aggregate level, as they emerge also independently from
the particular ﬁlter applied to data. In addition, they suggest that business-cycle models such
as those belonging to the DSGE family, should try to reproduce this empirical regularity, but
their empirical performance should also be robust to the inclusion of shocks drawn from fat-tail
distributions. Finally, they appear to indicate that possible theoretical explanations of output
fat-tail distributions could be obtained studying short-run phenomena such as business inventory
management.
The present work could be extended in many ways. To begin with, one could perform the
same exercises in other countries taking into account diﬀerent business cycle chronologies in the
speciﬁcation of bandpass ﬁlter parameters. Moreover, one could study the distributional prop-
erties (and the relatively robustness as to diﬀerent detrending techniques) of other important
macroeconomic time series, such as consumption, investment, monetary aggregates, etc. In par-
ticular, the presence of fat tails in ﬁltered inﬂation and unemployment time-series distributions
may be of particular interest for policy makers. Finally, one could examine the distribution of
disaggregated industrial production time series in order to try to understand which sectors are
more responsible for the fat-tail property observed in IP series.
527 28 29 30 31 32 33 34 35 36 37 38
0
0.5
1
1.5
2
2.5
3
q
ˆ
b
 
 
Estimated b
Conf. Int.
(a) GDP, 6.5 year band
5 10 15 20 25 30
0
0.5
1
1.5
2
2.5
3
q
ˆ
b
 
 
Estimated b
Conf. Int.
(b) GDP, 1 year band
85 90 95 100 105 110
0
0.5
1
1.5
2
2.5
3
q
ˆ
b
 
 
Estimated b
Conf. Int.
(c) IP, 6.5 year band
20 30 40 50 60 70 80 90
0
0.5
1
1.5
2
2.5
3
q
ˆ
b
 
 
Estimated b
Conf. Int.
(d) IP, 1 year band
Figure 1: Estimated Exponential-Power ˆ b Parameters vs. Shifting Frequency Bands (q)
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Detrending and the Distributional Properties of U.S. Output Time Series

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