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Assessing U.S. Aggregate
Fluctuations Across Time and
Frequencies

WP 19-06

Thomas A. Lubik
Federal Reserve Bank of Richmond
Christian Matthes
Federal Reserve Bank of Richmond
Fabio Verona
Bank of Finland

Assessing U.S. Aggregate Fluctuations Across Time and
Frequencies
Thomas A. Lubik
Federal Reserve Bank of Richmondy

Christian Matthes
Federal Reserve Bank of Richmondz

Fabio Verona
Bank of Finlandx
February 2019
Working Paper No. 19-06

Abstract
We study the behavior of key macroeconomic variables in the time and frequency
domain. For this purpose, we decompose U.S. time series into various frequency components. This allows us to identify a set of stylized facts: GDP growth is largely a
high-frequency phenomenon whereby in‡ation and nominal interest rates are characterized largely by low-frequency components. In contrast, unemployment is a mediumterm phenomenon. We use these decompositions jointly in a structural VAR where
we identify monetary policy shocks using a sign restriction approach. We …nd that
monetary policy shocks a¤ect these key variables in a broadly similar manner across
all frequency bands. Finally, we assess the ability of standard DSGE models to replicate these …ndings. While the models generally capture low-frequency movements via
stochastic trends and business-cycle ‡uctuations through various frictions, they fail at
capturing the medium-term cycle.
JEL Classification: C32, C51, E32
Key Words: Wavelets, bandpass …lter, SVAR, sign restrictions, DSGE model

We wish to thank Bob King, Annika Lindblad, and Denis Tkachenko for useful comments. We are
also grateful to participants at the 2018 CEF Conference in Milan, the University of Adelaide, Carleton
University, the 2019 Meeting of the Finnish Economic Association and the Bank of Finland. James Geary
provided exceptional research assistance. The views expressed in this paper are those of the authors and
should not be interpreted as those of the Federal Reserve Bank of Richmond, the Federal Reserve System,
or the Bank of Finland.
y
Research Department, P.O. Box 27622, Richmond, VA 23261. Email: thomas.lubik@rich.frb.org.
z
Research Department, P.O. Box 27622, Richmond, VA 23261. Email: christian.matthes@rich.frb.org
x
Monetary Policy and Research Department. Snellmaninaukio, PO Box 160, 00101 Helsinki. Email:
fabio.verona@bof.….

1

1

Introduction

Economists have often found it useful to separate long-run trends from business-cycle ‡uctuations, which generally are considered those that occur with a cycle length of between
two and eight years. On the statistical side, this approach is probably best characterized
by the idea of a trend-cycle decomposition as in Beveridge and Nelson (1981), where the
trend is associated with permanent movements in a time series as opposed to a business
cycle being driven by transitory shocks. Conceptually, this idea is also inherent in …ltering
methods such as the Hodrick-Prescott (HP) …lter, which has been the dominant approach in
business-cycle modeling to extract a trend from aggregate times and render them stationary.
Such decompositions are convenient since they align with the idea of economic ‡uctuations
as being driven by either permanent or temporary shocks that do not necessarily interact.
In addition, monetary policy is often framed in terms of stabilizing the ‡uctuations of key
variables around a trend that is una¤ected by policy.
However, there is a growing awareness in the macroeconomics literature that this common view of economic ‡uctuations is no longer adequate to characterize the behavior of
economic activity over time. For instance, Comin and Gertler (2006) argue that a substantial part of economic ‡uctuations is located in what they label a ‘medium-term cycle’, that
is, ‡uctuations that are beyond a length of eight years, but fall short of a trend. Moreover,
these medium-term ‡uctuations cannot be thought of in isolation from other frequency
bands. Using a theoretical model, Comin and Gertler (2006) show that business cycles and
medium-term cycles are intimately connected since they are driven by the same underlying
temporary shock. Speci…cally, a temporary innovation to, say, productivity or the policy
rate can reverberate throughout several frequency bands as they get propagated over time.1
Against this background, we aim to provide a somewhat more encompassing view of
cyclical behavior across all frequencies. In particular, we study three issues. First, we
compute a decomposition of key macroeconomic time series using wavelet-based …ltering.
That is, we decompose a time series into several time series components, each of them
‡uctuating within a speci…c frequency band. We …nd the use of wavelets advantageous for
our purposes since this …ltering approach is more ‡exible than standard Fourier analysis and
more traditional bandpass …ltering. In particular, it allows di¤erent frequency movements
to be more pronounced in some parts of the sample than others and thereby reveals time
1

Cogley (2001) makes a similar point for trend speci…cations where he shows the e¤ects of trend speci…cation errors are not con…ned to low frequencies, but are spread across the entire frequency domain.
Researchers, therefore, have to have a clear understanding of the inter-relatedness of frequency bands for
which a wavelet approach o¤ers a covenient tool.

2

variation in the importance of di¤erent frequency components. The second question looks
at the e¤ects of identi…ed monetary policy shocks across di¤erent frequency bands to assess
the plausibility of medium-term cycles as being generated by temporary shocks. The third
question asks whether standard dynamic stochastic general equilibrium (DSGE) models
that are used in monetary policy analysis can replicate the volatility of di¤erent cycles of
each macroeconomic variable under consideration and are thereby useful in addressing the
policy questions raised.
We establish three main …ndings. First, the wavelet decomposition of key macroeconomic variables shows that the bulk of ‡uctuations in GDP growth, unemployment, and
in‡ation occurs across di¤erent frequency bands. More than half of real GDP growth is
explained by short-term, high-frequency components with only a third of ‡uctuations attributable to business-cycle frequencies between two and eight years. Unemployment is
dominated by medium-term ‡uctuations between eight and 32 years and, to a lesser extent,
by low-frequency movements while close to three-quarters of in‡ation and short-term interest rate ‡uctuations fall into the slow-moving trend component. The corollary to these
results is that business cycles play only a secondary role in explaining overall aggregate
‡uctuations as real GDP growth is very much a high-frequency phenomenon, while the
behavior of in‡ation is all trend.
Since these variables are central to thinking about monetary policy, both in terms of
target variables as well as their information content for the state of the economy, we next
assess the e¤ects of monetary policy shocks on the individual frequency components. Using
identi…ed structural VARs with sign restrictions, we …nd that across all frequency bands
the results from an aggregate VAR carry over to individual components and short-term,
business-cycle, medium-term, and long-term components. In a baseline speci…cation that
includes only the overall data series, a contractionary policy shock, that is, an increase in
the federal funds rate, lowers in‡ation, raises the unemployment rate, and decreases real
GDP growth. We …nd similar patterns across most frequency bands, but as we increase the
cycle length, the peak response moves further out, while precision of the impulse response
estimates worsens and the quantitative importance declines. We take this as somewhat
tentative evidence that monetary policy has an impact across all frequency bands and that
a mechanism in line with interaction of endogenous growth and cycles as in Comin and
Gertler (2006) is at play. In addition, we …nd that in the long run the relationship between
the nominal interest rate and the in‡ation rate is positive, whereas in the short run an
interest-rate increase lowers in‡ation. This relationship weakens or is non-existent over the

3

medium term, which arguably re‡ects a contrast between a demand e¤ect in the short run
and the Fisher e¤ect in the long term.
Our third …nding shows that standard DSGE models are in principle capable of replicating the behavior of macroeconomic variables in di¤erent frequency bands. We simulate
arti…cial time series from three canonical DSGE models (Smets and Wouters, 2007; del
Negro et al., 2015; and Christiano et al., 2016) and apply our wavelet decomposition to
the same set of variables as before. Generally, all three models perform reasonably well
for business-cycle frequencies and for long-term ‡uctuations. In a sense, this is perhaps
not surprising in that the models are built as business-cycle models around the idea that
such ‡uctuations are the outcome of stochastic shocks and endogenous propagation. These
DSGE models also include elements such as habit formation, investment adjustment costs,
and wage and price indexation to impart persistence on the variables, which helps match
behavior at business-cycle frequencies.2 Long-run behavior is captured by stochastic trends
and time-varying in‡ation targets, which have been introduced successively over the course
of model development to capture trends. We show, however, that these models largely fail
in capturing behavior at medium-term frequencies, which is particularly prevalent in the
case of unemployment and a monetary DSGE model with search and matching frictions
in the labor market. We interpret these …ndings as a challenge for modelers to develop
frameworks capable of capturing medium-term cycles.
This paper touches upon various literatures in macroeconomics and time series analysis.
There has been a long-standing debate as to whether a frequency-based view of economic
‡uctuations is useful for analyzing and understanding policy. Perhaps emblematic of a
critical viewpoint is Watson (1993), who argues that policy analysis at di¤erent frequencies is not relevant for policymakers and that the close relationship between a time series
representation of a variable and its counterpart in the frequency domain, such as the spectrogram, invalidates the need for a separate analysis of frequency-speci…c considerations.
This viewpoint is implicitly questioned by Onatski and Williams (2003), who study the
e¤ects of uncertainty, broadly understood, on monetary policy decisions. They show that
when uncertainty enters a policymaker’s decision problem at di¤erent frequencies it may
have substantially di¤erent e¤ects on outcomes. This criticism of the Watson-critique is
taken up by Brock et al. (2007), who analyze the di¤erential e¤ects of various policy rules
on outcomes across frequencies. In a follow-up paper, Brock et al. (2013) demonstrate how
2
Tkachenko and Qu (2012) and Sala (2015) estimate medium-size DSGE models in the frequency domain
with a focus on business-cycle frequencies. They report similar …ndings as to the ability of such models to
replicate observed behavior over the cycle.

4

reductions of variance at some frequencies lead to increases in variance at others, which
then creates a policy trade-o¤. Our paper informs this debate by showing empirically the
contributions of di¤erent frequency bands to the overall volatility of key macroeconomic
variables and how they are impacted by monetary policy shocks.
Our paper also continues and contributes to the debate about the use of detrending
methods in macroeconomics. Many empirical methods require the underlying data series
to be stationary and thereby necessitate the use of a …lter to remove trending components.
However, as Canova (1998) has demonstrated, di¤erent detrending methods extract di¤erent
information from the underlying data series. This implies that the thus derived stylized facts
can di¤er substantially qualitatively and quantitatively across di¤erent …ltering methods.3
This insight is extended by Gorodnichenko and Ng (2010) and to the estimation of DSGE
models. When researchers apply standard data transformations, this induces biases in
structural estimates and distortions in the policy conclusions. In order to address this issue
Canova (2014) proposes joint modeling of the cycle and the trend within the model and the
raw data.
We add to this literature by establishing a set of stylized facts based on the timefrequency decomposition inherent in wavelet analysis that has certain advantages over more
traditional methods. Thereby, we also highlight the importance of joint theoretical modeling
of economic behavior across all frequency bands and especially the medium term as an
important component of economic ‡uctuations. While the importance of the medium run
has been on economists’ minds for a long time (e.g., Blanchard, 1997), there has been a
‡urry of recent research in the wake of Comin and Gertler’s (2006) contribution that studies
the origin and e¤ects of medium-term cycles (e.g., Beaudry et al., 2017; Cao and Huillier,
2018).
In this paper, we exploit the bene…ts of wavelet analysis as a complementary approach to
classical time series and spectral analysis. We …rst use the univariate wavelet transform for
exploratory data analysis of US macroeconomic variables. In addition, we use the wavelet
power spectrum to analyze the evolution over time of the variance of the variable at di¤erent
frequencies. We then use this approach to isolate speci…c frequency components from each
variable and use those frequency components in a standard VAR regression setup. Our
paper thus contributes to a growing literature on the use of alternative …ltering methods in
economics and …nance, such as Aguiar-Conraria et al. (2012) and Bandi et al. (2019).
The remainder of the paper is structured as follows. In the next section, we present
3

This observation is also in line with the recent criticism in Hamilton (2018) on the use and application
of the HP-…lter in macroeconomic modeling.

5

our …rst set of results, namely new stylized facts based on a wavelet decomposition of
aggregate data. In Section 3, we use the decomposition to assess the e¤ects and importance
of monetary policy shocks across di¤erent frequency bands in a structural VAR framework.
Section 4 considers the question, whether existing DSGE models are able to capture these
regularities. Section 5 concludes.

2

A Frequency-Band Decomposition of Aggregate Time Series

We use the wavelet methodology to decompose standard US macroeconomic time series
into di¤erent components that can be associated with the scale of the underlying cycles.
We regard this time-frequency decomposition, that is, a decomposition of a variable into
components in the time domain with precise counterparts in the frequency domain, as a
useful and informative alternative to typical trend-cycle decompositions that provides a
more encompassing view of the nature of economic ‡uctuations. In what follows, we brie‡y
discuss the methodology and detail the data used in our empirical exercise. We then present
our baseline results, followed by an extensive robustness analysis with respect to alternative
…ltering methods and choices.

2.1

Methodology and Data

The analysis in this paper is based on a time-frequency decomposition of key economic time
series. Our basic objective is to decompose a time series into individual components that can
be cleanly and clearly associated with ‡uctuations at di¤erent frequencies or di¤erent lengths
of a cycle, but are represented in the time domain. For this purpose, we employ wavelet
multiresolution analysis (MRA), which performs such decomposition in a way similar to
the traditional time series trend-cycle decomposition approach (e.g., Beveridge and Nelson,
1981; Watson, 1986), or other …ltering methods like the Hodrick and Prescott (1997) or
the Baxter and King (1999) band-pass …lter. However, a wavelet approach aims for a more
…ne-grained understanding of the di¤erent components of a time series that make up what
is considered a ‘cycle’as opposed to a ‘trend’.4 Speci…cally, we employ a particular version
of a wavelet transformation of a time series called the Maximal Overlap Discrete Wavelet
4

Conceptually, our line of reasoning is informed by the notion of medium-term cycles as advocated by
Comin and Gertler (2006). There is a growing understanding that the neat trend-cycle view of economic
‡uctuations is inadequate to capture the nature of economic activity.

6

Transform (MODWT).5
As an example, by using the speci…c form of a Haar wavelet …lter, any time series Xt
can be decomposed into a scale component SJ;t and J detail components Dj;t :
J
X

Xt =

Dj;t + SJ;t ;

(1)

j=1

where these coe¢ cients are given by:
0
2j 1 1
1 @ X
Dj;t =
Xt
2j
i=0

2J

SJ;t =

1
1 X
Xt i :
2J

i

j 1
2X

i=2j

1

1

Xt i A ;

(2)

(3)

i=0

Intuitively, the wavelet …lter separates the original series Xt , which is de…ned in the time
domain, into di¤erent time series components. These represent the ‡uctuations of Xt in
a speci…c frequency band, that is, a range of frequencies, or length of cycles, that are
grouped together.6 In this example, the smooth scale component SJ;t at time t is computed
as the weighted average of lagged values of Xt at scale J, while the detail components
Dj;t are overlapping weighted moving averages up to scale J. The bands are associated
with di¤erent details j such that for small j, the wavelet component Dj;t captures the
higher-frequency characteristics of the time series, that is, its short-term ‡uctuations. As
j increases, the components represent lower frequency movements of the series. Finally,
the smooth component SJ;t captures the lowest frequency dynamics, that is, the long-term
behavior.7
The key parameter for the economic interpretation of the wavelet decomposition is the
scale J, which determines how …ne-grained or detailed the decomposition is. For J large
enough, the scale component SJ;t approximates the true underlying trend of the series. If
J is small, then the scale component includes ‡uctuations of shorter duration, which one
may not normally associate with a trend.8 An alternative interpretation is that SJ;t is the
5

The MODWT version of the wavelet …lter has become the standard in the empirical …nance and forecasting literature, e.g. Berger (2016) or Faria and Verona (2018).
6
The individual components, or wavelets, thus make up the overall wave in a prescribed manner.
7
As in the Beveridge and Nelson (1981) time-series decomposition into stochastic trends and transitory
components, the wavelet coe¢ cients Dj;t can be viewed as components with di¤erent levels of calendartime persistence operating at di¤erent frequencies, whereas the scaling component SJ;t can be seen as the
low-frequency trend of the time series under analysis.
8
The Appendix contains a simple example how the scale parameter J is related to the idea of taking
various di¤erences of time series.

7

underlying scale of the time series upon which ‡uctuations of higher frequencies and shorter
cycle durations are built. In that sense, our analysis falls in line with a more standard trendcycle decomposition. On a …nal note, the …lter discussed above is one-sided since future
values are not needed to compute the wavelet coe¢ cients of the transform of Xt at time
t. This implies that that the Dj;t and SJ;t lag Xt . Moreover, since the length of the …lters
increases with j, so does the delay. Hence, the coarser the scale, the more the Dj;t and SJ;t
are lagging Xt . We use this fact in our VAR analysis below.
What distinguishes the wavelet decomposition is that the choice of the scale allows
the researcher to hone in on and isolate speci…c frequency bands that are the objects of
interest. While other …ltering methods, such as Fourier analysis, also allow a researcher
to focus on speci…c frequencies, a wavelet approach has some key advantages. Traditional
decomposition techniques, such as spectral analysis of a time series, tend to impose strong
assumptions about the data-generating process. Speci…cally, they often require data to
be stationary or pre-…ltered. However, many economic and …nancial time series are hardly
stationary as they exhibit trends and patterns such as structural breaks, volatility clustering,
and long memory, which the wavelet approach can handle with ease.
Unlike Fourier analysis, wavelets are de…ned over a …nite window in the time domain,
which is automatically and optimally resized according to the frequency of interest and the
choice of the scale J. Wavelets and standard Fourier analysis are essentially approximations
with basis functions, but Fourier basis functions are non-zero almost everywhere, making
it harder for them to capture local phenomena. Using a short time window isolates the
high-frequency features of a time series, while treating the same signal with a large time
window reveals its low-frequency features. By varying the size of the time window, we
can therefore capture time-varying and frequency-varying features of the time series at the
same time. Wavelets are, thus, very useful when dealing with non-stationary time series,
irrespective of whether the non-stationarity comes from the level of the time series (that is,
from a long-term trend or jumps) or from higher-order moments (that is, from changes in
volatility).
Wavelet …ltering methods are similar to …ltering by a set of band-pass …lters so as to
capture the ‡uctuations of a time series in di¤erent frequency bands, e.g., Christiano and
Fitzgerald (2003). The band-pass …lter is a combination of a Fourier decomposition in the
frequency domain with a moving average in the time domain. It applies optimal Fourier
…ltering to a sliding window in the time domain with constant length regardless of the
frequency being isolated. Wavelet …ltering, in contrast, provides better resolution in the

8

time domain as the wavelet basis functions are both time-localized and frequency-localized.
In this paper, we use the MODWT to compute the decomposition. This version is
not restricted to a particular sample size: if the data are discrete, the standard wavelet
decomposition requires a sample of length 2J for the decomposition to be exact; that is,
it imposes a tight restriction on which and how many frequency bands can be considered
and might require dropping observations. The MODWT avoids this problem and is also
translation-invariant, that is, it is not sensitive to the choice of a starting point for the
examined time series. Finally, implementation of the wavelet decomposition requires choice
of a speci…c functional form for the …lter that maps the original series into its components.
We follow the literature and choose as a benchmark the Haar …lter, but we also consider
the Daubechies …lter as an alternative. Speci…cally, we employ the …lter to decompose
our time series of interest into seven individual series, labeled D1 , ..., D6 for the detail
components and S6 for the scale component; that is, we choose J = 6. The individual
components are such that they add up to the underlying series. Given the scale of the
decomposition as powers of two we can associate the components with individual frequency
bands. Speci…cally, D1 captures ‡uctuations up to four quarters, D2 between four and
eight quarters, up to D6 , which covers the band between 64 and 128 quarters. The scale
component S6 is associated with movements above 128 quarters.
We collect quarterly data on US macroeconomic aggregates, interest rates, and prices.
Speci…cally, we report results for real GDP, the unemployment rate, the in‡ation rate for
the overall personal consumption price index (PCE), the federal funds rate (FFR), and a
3-month and 10-year interest rate.9 The data are described in more detail in the Appendix.
The full range of our sample covers 1954Q3 to 2017Q3. We utilize data in levels and in
growth rates, where growth rates are computed as quarter-over-quarter values. Although
not required for the wavelet …ltering, we report results for GDP growth as it is the focus
of policymakers’decisions. For our baseline decomposition, we use a one-sided Haar …lter,
which is then employed in the VAR analysis. In a sense, the di¤erent scale components are
generated regressors where we do not want to impart information onto the econometrician
running the VAR that he could not possibly possess; that is, knowledge of the data at the
end of sample should not be used to produce a decomposition for periods in the middle.
For informative purposes and as a robustness check, we also provide results for two-sided
9

The 3-month Treasury rate at constant maturity is only available from 1981Q4 on. We use the 3-month
Treasury rate from secondary market instead since it is available from 1947Q1. Preliminary analysis for the
two series shows that they co-move extremely closely and that there is at most a level di¤erence of up to 50
basis points.

9

(smoothed) wavelet …lters, for alternative kernels, and for alternative …lters, such as the
Hodrick-Prescott and the Christiano-Fitzgerald bandpass …lters.

2.2

Baseline Results

We report two sets of results. For purposes of exposition, we group the seven series into
four categories that we label ‘Short Term’ (D1 , D2 ), ‘Business Cycle’ (D3 , D4 ), ‘Medium
Term’(D5 , D6 ), and ‘Long Term’(S6 ). The short-term category captures high-frequency
‡uctuations under two years, which in macroeconomic applications are often discarded as
noise, but may contain useful information about the incidence of shocks. The business-cycle
category covers ‡uctuations at frequencies between 8 and 32 quarters (2-8 years), which most
macroeconomic research on the sources of aggregate movements focuses on. This frequency
band is, for instance, designed to be isolated by the application of the Hodrick-Prescott
…lter with a smoothing parameter of

= 1; 600.

We maintain this terminology for clarity, although one aspect of our paper is to argue
for less rigid classi…cations in the standard trend-cycle methodology. Components D5 and
D6 are grouped under ‘Medium Term’‡uctuations and cover frequencies up to 128 quarters
(32 years). We note that this scale is shorter than the medium-term cycle adopted in Comin
and Gertler (2006), de…ned as movements between 8 and 50 years. Finally, we associate
S6 with the ‘Long Term’ or, loosely speaking, the trend. We report the grouped wavelet
decompositions for real GDP growth, the unemployment rate, the in‡ation rate, the federal
funds rate, the 3-month and 10-year rate, and the di¤erence between the latter two series,
namely the term spread, in Figures 1-6. The decompositions into the individual wavelets
are collected in the Appendix. Table 1 reports the variance decompositions by frequency.
We …nd that more than 50 percent of overall ‡uctuations in real GDP growth are
explained by the short-term components D1 and D2 , roughly one-third by the businesscycle components D3 and D4 , with the rest by medium- to long-term components.10 This
raises the question whether and to what extent macroeconomic stabilization policy can
a¤ect this short-term component, especially since it is likely to contain measurement error.
At the same time, the low-frequency component S6 declines from above 4 percent to below
2 percent over the course of the sample (see Figure 1). This is in line with the secular
decline in trend growth that has been found in numerous studies. However, this is not
the full picture behind the recent lower growth rates, as the two medium-term components
D5 and D6 essentially o¤set each other since 2000 and thereby do not contribute to the
10

The medium-frequency components as de…ned by Comin and Gertler (2006) thus make up only 12.5
percent of the overall ‡uctuations, with half falling on the band between 32-64 quarters.

10

underlying growth trend. This comes largely from the business cycle components during the
recovery from the Great Recession. The Great Moderation is most visible in the short-term
components and to a lesser extent in the business-cyle band.11 The wavelet decomposition
shows that it is more of a higher frequency phenomenon. This observation lends support
to the argument that the Great Moderation came about because of an improvement in
the way monetary stabilization policy was conducted rather than a change in, for instance,
inventory management.
The unemployment rate decomposition in Figure 2 and Table 1 reveals a slightly different pattern. Roughly one-third of unemployment ‡uctuations are due to short-term and
business-cycle movements, while medium- and longer-term frequencies (D5 -S6 ) each explain
around 20 percent. Fluctuations in the unemployment rate can therefore be described as a
medium-term cycle. What dominates the level of the unemployment rate is its long-term
component S6 , which could be interpreted loosely as a natural rate of unemployment. A
focus of the next section is the extent to which the trend components are a¤ected by monetary policy. What is striking is that the di¤erent components do not seem to comove
closely. For instance, the unemployment rate is at 5.4 percent in 1990, while the long-term
component S6 is at 7.2 percent, the di¤erence being made up by components D4 -D6 . In
other words, the business-cycle peak produces a negative unemployment gap relative to a
very high natural rate on account of strong medium-term components that might be tied
to labor force participation peaking in the late 1990s. Finally, the Great Moderation is
considerably less visible in the unemployment rate, if at all.
We now turn to the nominal side of the economy. Figure 3 contains the results from the
decomposition of the PCE in‡ation rate. 40 percent of in‡ation movements can be traced
back to the long-term component S6 . The business cycle component explains around one…fth of the overall variability, while medium-term components cover 25 percent. About 15
percent of in‡ation variability can be traced back to very short-term or noise components.
As in the case of the unemployment rate, the scale of the decomposition is dominated by the
trend S6 . The monetary policy literature often interprets this component as the in‡ation
target or the perception thereof. It can also be seen as a measure of the extent to which
in‡ation expectations are anchored. In our decomposition, it shows a gradual rise from
almost zero in the late 1960s to a peak of 6.2 percent in the early 1980s followed by a
gradual decline until reaching the 2 percent target in the 2000s.
11

Figure C.1 in the Appendix shows that this largely due to the D3 component, indicating that the Great
Moderation is essentially a high-frequency event. To this point, see also Aguiar-Conraria et al. (2012) and
Pancrazi (2015).

11

A similar pattern in terms of the Volcker disin‡ation can be found in the medium-term
components D5 and D6 . What is striking is the run-up in trend in‡ation over the course
of the 1970s and the drawn-out, three-decade struggle to return it to 2 percent. Since the
Federal Reserve arguably did not change its implicit in‡ation target over that time, this
component may therefore be better described as the public’s perceived target. Our results
then depict a striking loss of central bank credibility.12 In light of this aspect, it is perhaps
surprising that there is not much of the Great Moderation visible when interpreted as a
binary event, that is, a break in policy or a structural change before or after the early
1980s. Instead, in the graphs in Figure 3 it is possible to discern the high volatility of the
1970s, preceded and followed by the more stable 1960s and 1980s, respectively. Interestingly,
in‡ation volatility seems to have gone up again in the 2000s, especially around the Great
Recession.
We report decompositions for the FFR and the 10-year rate in Figures 4 and 5.13 They
show similar patterns as the in‡ation decompositions, whereby volatility in the 10-year rate
can be attributed to almost 70 percent to the long-term component S6 , ten percentage points
more than for the short rates. Presumably, this re‡ects that longer rates are less subject
to the vagaries of higher-frequency ‡uctuations. Since the interest rates share common
components, especially in the medium and longer run, it is therefore often instructive to
consider the term spread, in our case the di¤erence between the 10-year and the 3-month
rate. The term spread decomposition in Figure 6 puts most weight, almost 45 percent,
on the business-cycle components. This supports the idea that at frequencies commonly
associated with the business cycle the spread is a useful indicator of economic and …nancial
conditions. Interestingly, the long-term component has gone up considerably since the early
1980s to a level of above 2%, implying that the di¤erence between the short and long rates
has become more persistent.
As a …nal exercise, we produce the power spectra from the wavelet decomposition for real
GDP growth, unemployment, and the federal funds rate in Figures 7-9.14 The wavelet-based
spectra are akin to classical spectra in that they decompose a time series into frequencies
and measure the contribution of each frequency to the overall behavior of a time series.15
12

This interpretation is consistent both with the learning and inherent in‡ation persistence story in Primiceri (2006) or Sargent, Williams, and Zha (2006), the in‡ation misperception argument in Lubik and Matthes
(2016), as well as a number of recent papers on evolving private sector beliefs, for instance, in Bianchi (2013).
13
The results for the 3-month rate are almost identical to those for the FFR. The respective decompositions
are shown in the Appendix.
14
We brie‡y describe and discuss the concept behind the wavelet power spectra in the Appendix.
15
As discussed before, the wavelet power spectrum does not require stationarity.

12

Ordered by frequency, this spectral density is depicted in the right column of each …gure. In
addition, we show the time-frequency decomposition in the left column. Using the wavelet
…lter, the graph reports the wavelet power spectrum (WPS), a decomposition by time (on
the horizontal axis) and frequency (on the vertical axis). The graph is coded as a heat map
such that areas of higher activity are depicted as redder on the color spectrum. The solid
black lines delineate 95 percent con…dence regions. Note that the wavelet-based spectral
density can be obtained by integrating the WPS over time.
The spectral peak of GDP growth in Figure 7 is just above two years, which con…rms our
prior …nding that output ‡uctuations are highly concentrated among the highest frequencies.
The WPS, however, shows that this observation is largely driven by the late 1950s and
1970s, which show considerably higher concentration of activity in the short term than what
occurred during later periods. In the same vein, the Great Moderation is quite visible from
the graph.16 The unemployment rate in Figure 8 shows two local spectral peaks associated
with cycles of around 8 and 32 years. This is in line with our previous …ndings, but sharpens
the observation of unemployment being subject to medium-term cycles towards the edges of
that frequency band. Notably, the Great Moderation is not apparent from the WPS, while
deep recessions in the mid-1970s and the Great Recession impart some higher frequency
components on the decomposition. Finally, Figure 9 reveals that the FFR has a spectral
peak at a very low frequency, which we associate with the presence of an explicit or implicit
in‡ation target. However, the WPS shows a local peak at a frequency of 8 years, which is
largely driven by the period from 1968 until the early 2000s.
Overall, what emerges from these decompositions is a multifaceted picture of macroeconomic ‡uctuations. Across all variables, the business-cycle components D3 - D4 , that is,
cycles between two and eight years, explain about one-third of overall ‡uctuations. There
is considerable heterogeneity across variables as far as the other components are concerned:
50 percent of real GDP growth is captured by high frequency components (cycles of less
than 2 years). Essentially, the majority of quarterly GDP movements occur at very high
frequencies.17 In turn, short-run ‡uctuations do not seem to play much of a role for the
other variables. The behavior of unemployment is dominated by medium-term movements
with a cycle length of between 8 and 32 years and to a lesser extent by longer-term move16

This is in line with related research by Pancrazi (2015), who argues that the reduction of volatility of
GDP after the mid-1980s is mainly a high-frequency phenomenon of cycles up to 4 years and that it is much
milder, or absent, for other frequencies.
17
We use …nal data in our empirical study, that is, the last data vintage available. In contrast, policymakers
operate in a real-time environment where initial data releases are subject to sometimes large measurement
error. Lubik and Matthes (2016) show that this can lead to what looks like policy mistakes ex post.

13

ments with a lower frequency than that. In‡ation and interest rates have sizeable long-term
components, too. These components can be interpreted as “trends”and natural or potential
rates. Their behavior arguably conforms to conventional wisdom, that is, in‡ation seems
to be all trend, driven by the Federal Reserve’s implicit, and then later, explicit in‡ation
target.
This naturally raises the question, whether stabilization policy aimed at the business
cycle is misdirected or misses important aspects that policymakers should focus on.18 An
immediate follow-up question is whether models that are being used to describe and analyze
monetary policy are consistent with the heterogeneity in ‡uctuations. We address these
two questions in turn in the following two sections. First, we investigate whether identi…ed
monetary policy shocks have di¤erential e¤ects on key variables for di¤erent frequencies;
and second, we study whether some standard DSGE models are capable of replicating the
wavelet-based variance decompositions in this section.

2.3

A Comparison of Alternative Filters

We assess the robustness of our baseline …ndings for the one-sided Haar …lter along several
dimensions. First, we consider a two-sided version of the Haar …lter. The second exercise
considers an alternative kernel for the wavelet decomposition, namely the Daubechies …lter.
The third robustness check uses …lters that are more common in the macroeconomics literature, speci…cally the Christiano-Fitzgerald bandpass …lter and the Hodrick-Prescott …lter.
As before we focus on four broad frequency bands for exposition. The decompositions are
reported in Figures 10-12.
Figures 10 and 11 contain the decompositions of, respectively, real GDP growth and
unemployment for the one- and two-sided Haar …lter and the Daubechies …lter. By de…nition the two-sided …lter is smoother than a one-sided …lter since it uses all available
information over the whole span of the sample and not just up to the data point at which
the …lter is applied. This is evident by comparing the one-sided Haar …lter with its two-side
counterpart in the …gures. Generally, there are no large di¤erences in terms of the overall
direction and volatility for both unemployment and real GDP growth, but the one-sided
…lter imparts more volatility to the short-term and business-cycle components than the
other …lters. Moreover, the one-sided Haar …lter lags the other …lters in picking up general
directional movements. This is especially visible in the medium- and long-term components
of unemployment in Figure 11.19 The fact that the one-sided Haar is slow in picking up the
18
19

This argument has been made most succinctly by Brock et al. (2008, 2013).
We …nd similar patterns in the decompositions for in‡ation and the interest rates. These results are

14

rise and subsequent fall in trend unemployment in the 1970s and 1980s is simply a feature
of how it is constructed. As discussed before, we prefer a one-sided …lter since we use the
individual components as variables in a VAR, which rests on the idea that the innovations
are one-step-ahead forecast errors and thereby do not re‡ect the full information in the
sample.
The …gures also report results for the Debauchies …lter as an alternative to the twosided Haar …lter. The Haar …lter produces less volatile components than the Daubechies,
but the di¤erence seems minor. There are a few episodes where the two …lters do not
overlap each other. For instance, the medium-term components of in‡ation in the mid1970s di¤er noticeably, but these occurrences are the exception. We prefer the Haar over
the Debauchies implementation of the wavelet decomposition since the former has a more
intuitive interpretation (see the discussion in the Appendix). The di¤erences between the
various implementations of the decomposition are small enough, however, not to a¤ect the
conclusions drawn in the next two sections.20
In contrast, the decompositions based on two widely used …lters in macroeconomic analyses are materially di¤erent. Figure 12 compares our baseline …lter with the corresponding
bandpass …lter of Christiano and Fitzgerald (2003) (CF) and the canonical Hodrick-Prescott
(HP) …lter. In a sense, the CF …lter and our Haar …lter are conceptually similar in that
they explicitly isolate speci…c frequency bands and represent them in the time domain. This
is evident from comparing the two …lters in the …gure for unemployment rate decompositions as an illustrative example. At business-cycle frequencies, the CF …lter extracts more
volatile components, but is arguably not that di¤erent from the wavelet-based …lter. The
exception are the longer-term components, especially D6 and S6 , where the two …lters pick
out di¤erent peaks and are generally not that well aligned.21
In contrast, the HP …lter produces quite di¤erent series. For a smoothing parameter
of

= 1; 600, it extracts the business-cycle frequencies corresponding to our components

D3 and D4 . It is considerably more volatile than the wavelet decomposition. More striking
is the pattern for lower frequencies. The …gure reports the HP trend, which is computed as
the di¤erence between the original series and the business-cycle component obtained with
= 1; 600. It is akin to the S6 component with wavelets, which is the “residual”part of the
included in the Appendix.
20
We performed the empirical excercises in Sections 3 and 4 using alternative wavelet decompositions.
The results are available on request.
21
However, recall that the CF …lter is optimized to extract business-cycle frequencies and not low frequencies. At the same time, this is another argument in favor of wavelet …lters since it treats all frequencies the
same way.

15

series. This slow-moving component is quite di¤erent from the other series and thus raises
concerns as to whether the HP …lter introduces spurious dynamics (see Hamilton, 2017).

3

The Frequency-Speci…c E¤ects of Monetary Policy Shocks

We now study whether and to what extent monetary policy shocks a¤ect key aggregate
variables across di¤erent frequencies. The previous section demonstrated that the behavior
of GDP growth, unemployment, and the in‡ation rate di¤ers in terms of the contribution of
various frequency bands to overall volatility. Whereas the majority of ‡uctuations in GDP
growth are located among the highest frequencies, that is, the short-term components, the
unemployment rate is more evenly split between a large medium-term and lower-frequency
components. In turn, most of the movements in the in‡ation rate are driven by the long
term, which we might associate with the in‡ation target. As we think of monetary policy
as trying to stabilize movements in GDP growth and unemployment against a background
of stable prices or constant in‡ation, the question is whether policy is successful in a¤ecting
these variable at frequencies that are the main drivers of their overall volatility.
Our approach is as follows. We assess the e¤ects of monetary policy shocks on individual
frequency bands by using the …ltered series as explanatory variables in a VAR. Given a
plausible identi…cation of policy shocks, we then compute impulse response functions to
these shocks for the various decompositions. We begin by assessing the plausibility of our
preferred identi…cation scheme in a standard model. To this end, we estimate a threevariable VAR in an activity variable, that is, either the unemployment rate or real GDP
growth, in‡ation, and the federal funds rate. We then identify a structural monetary policy
shock using a sign restriction approach where we assume that all restrictions are imposed
only on impact. Speci…cally, we assume that a contractionary monetary policy shock - one
that raises the federal funds rate on impact - lowers output, increases unemployment, and
lowers in‡ation.
Figure 13 reports impulse responses to an identi…ed policy shock from the two VARs.
The left column shows the responses in the model with unemployment, the right those of
the model with GDP growth. In this baseline speci…cation, a rise in the interest rate by 25
basis points increases unemployment by 10bp with a hump-shaped peak after three to four
quarters of 20bp. It lowers in‡ation by 60bp on impact before gradually returning to its longrun level. Similarly, a contractionary monetary policy shock lowers GDP growth by almost
1.5 percentage points and in‡ation shows a similar decline as in the other speci…cation.22
22

Both the unemployment rate and GDP overshoot their long-run level in their adjustment path after

16

In the next step, we add the frequency components to the baseline speci…cation, either in
terms of unemployment or real GDP growth, as an activity variable. For each speci…cation
we identify the policy shock separately, which allows for the possibility that there could
be di¤erences across models. We consider two alternative speci…cations. First, we add
the seven frequency bands, D1 -S6 , of each variable included in the VAR one by one to
the baseline speci…cation. This results in a six-variable VAR, estimated separately for
each band. We report selected impulse responses for GDP growth in Figures 14-16, where
the left column shows the responses of the aggregate variables and the right column the
corresponding responses for a frequency band. The respective responses with unemployment
as the activity variable can be found in the Appendix.
We …nd that the responses of the high-frequency components D2 are signi…cant, and
are in line with the baseline results and what theory would suggest; however, the responses
are not large quantitatively and economically small. Nevertheless, this indicates that the
monetary transmission mechanism works as theoretical reasoning and practical experience
would indicate. The response of the business-cycle component D4 is not signi…cant on impact but becomes more sharply estimated a few quarters out. As before, the direction of the
responses is consistent with the identi…cation scheme on the overall series. Contractionary
shocks are thus likely to have their strongest impact in a few quarters, which is in line with
the idea that monetary policy stabilizes business cycles with a lag.
Finally, the response for the long-term component S6 is drawn out and not signi…cant
over the business-cycle horizon, but it exhibits comovement between the federal funds rate
response and in‡ation. In other words, at longer horizons and cycles, the Fisher e¤ect,
namely that interest rates and in‡ation rates are positively correlated, comes through;
whereas at higher frequencies this correlation moves in the opposite direction as the demandconstricting e¤ect of higher rates reduces in‡ation. It is clear from these …ndings that in the
transition between high frequency and low frequency movements the comovement patterns
for these two variables switch.
The second VAR speci…cation adds the …ltered series in groups that represent broader
frequency bands. Since the wavelet decomposition is fully additive, we cannot include all
individual series. We therefore focus on a speci…cation that looks at the business-cycle
components (D3 + D4 ), the medium-term cycles (D5 + D6 ), and the long term (S6 ). This
results in a 12-variable VAR, where we identify the policy shock by imposing sign restrictions
on impact on the aggregate variables only. Figure 17 contains the respective responses
the shock in line with the interest rate responses. That is, monetary policy responds endogenously to the
worsening economic conditions due to the unanticipated contraction by loosening policy.

17

for the model with GDP growth, while the corresponding responses for unemployment are
reported in Figure 18. In each graph, the top row contains the aggregate responses, followed
by the short-term, medium-term, and long-term components in separate rows.
A contractionary monetary policy shock has a negative impact on real activity in each
frequency band whereby the largest response is for the business-cycle component. If we just
look at the short-term frequencies (not reported), the impact e¤ect is larger. This possibly
re‡ects the dominant role of high-frequency movements in GDP growth (see Table 1). The
responses of the business-cycle and medium-term components return to zero after 20 and
30 quarters, respectively. The response of the long-term component on the other hand
remains negative over the full projection horizon of 10 years. This indicates that monetary
policy shocks can have long-lasting e¤ects even on GDP growth. We …nd a similar pattern for
the unemployment rate, with oscillating behavior of the higher-frequency components and a
more drawn out response of the trend. In terms of the size of the policy-induced movements,
the business-cycle and medium-term components are roughly similar, in contrast to the
variance decompositions in Table 1. This suggests that there are other shocks that drive
movements in the unemployment rate in these frequency bands.
The response of the FFR and in‡ation components for both VAR speci…cations is very
similar. At higher frequencies, the FFR rises and in‡ation falls, where especially the
business-cycle components move together closely. The response of the respective trend
components is di¤erent, however. In‡ation and the FFR do not react much on impact and
in the near term, but they move together positively over the longer horizon. A contractionary policy shock thus has a long-lasting negative e¤ect on the long-term component of
the FFR and in‡ation. These results con…rm the existence of a Fisher e¤ect in the long-term
component, whereas in the short term the demand-constricting e¤ect of an interest rate hike
dominates as in standard monetary policy models. Moreover, the results also show that
contractionary policy lowers the long-term component persistently presumably through an
expectations e¤ect: tightening policy gains credibility, anchors in‡ation expectations, and
lowers in‡ation overall.

4

Assessing DSGE Models

We now investigate whether several medium-scale DSGE models can replicate the stylized
facts identi…ed above. Such models have been developed explicitly with an eye on replicating
the performance for business-cycle movements and the long run. This raises the question,
whether they can, in fact, capture behavior along all frequency bands identi…ed by our
18

wavelet decomposition. In a preview of the results, we …nd that the models generally
do well for business-cycle frequencies and in the long term as these are frequency bands
that the models are designed to replicate. However, the models generally fail at capturing
medium-term frequencies.

4.1

DSGE Models and Simulation

In the DSGE literature, it is well-known that various modeling devices are useful in matching
persistence in the data, at least over the business cycle (see the programmatic papers by
Christiano et al., 2005, and Christiano et al., 2010, and also the seminal DSGE models
by Smets and Wouters, 2003, 2007). Examples are modi…cations to utility, such as habits
in consumption; production, such as investment adjustment costs; and highly persistent
shock processes. At the same time, stochastic trends have proved to be a ‡exible modeling
component to capture drifting behavior over time. This section studies whether these
modeling elements are useful across all frequencies.
We select models based on their widespread use in monetary policy analysis and their
consistency with the speci…c data that we have considered so far. Moreover, we want
to give the chosen models a fair chance at capturing the patterns found in the wavelet
decomposition. We therefore require that one of the underlying drivers of business cycles
is a stochastic trend in productivity that can smoothly vary over time. This speci…cation
is well-known for matching the movements in the GDP trend. We thus focus on three
canonical models in the literature: Smets and Wouters (2007), del Negro et al. (2015), and
Christiano et al. (2016).23
Smets and Wouters (2007) (SW) is a further development of the canonical Smets and
Wouters (2003) New Keynesian DSGE model. It is the prototype of a medium-sized,
optimization-based model designed to jointly capture the evolution of output and in‡ation and the monetary policy process. To this end, the model contains a variety of shocks
and frictions that have come to be accepted as central to understanding aggregate ‡uctuations. The basic setup involves a representative household that makes consumption choices
and supplies labor to a competitive labor market. On the production side, there are monopolistically competitive …rms that employ labor and capital to generate output, make
investment decisions, and set prices. The third type of agent in this model is a policymaker
who sets interest rates based on given feedback rules.
The model features nominal price stickiness and sticky wages with backward in‡ation
23

We use computer codes for these models available at Volker Wieland’s Macroeconomic Model Data Base
(MMB): https://www.macromodelbase.com/ and from the journal websites of the published articles.

19

indexation to capture slow-moving aspects of these variables. On the real side, there is
habit formation in consumption and investment adjustment costs designed to create humpshaped responses of these aggregate demand components. The model is driven by seven
structural shocks including a monetary policy disturbance. One key distinguishing feature of
Smets and Wouters (2007) as opposed to Smets and Wouters (2003) is that the former does
not have a time-varying in‡ation target. The model is estimated using Bayesian methods
over the period 1966-2004 for seven key aggregate variables, but the set of observables
does not include the unemployment rate. We can therefore not compare their model with
our decomposition along this margin.24 We take their parameters estimates as given and
simulate the model under this speci…cation.
The second model that we consider, del Negro et al. (2015) (dNGS), is an extension of
the SW model. It introduces a time-varying target in‡ation rate and incorporates …nancial
frictions in the vein of Christiano et al. (2014). The model is estimated for a slightly larger
dataset than the SW model and over the period 1964-2008. The key …nding of the paper is
that the model is compatible with Great Recession outcomes in that it successfully predicts
a sharp contraction in economic activity along with a drawn-out but modest decline in
in‡ation. The third model is Christiano et al. (2016) (CET). While it is built around the
same nominal structure as SW, CET introduces a much richer labor market setting governed
by search and matching frictions and various wage determination mechanism. We report
results both for a benchmark speci…cation with Nash bargaining and an alternative, namely
alternative o¤er bargaining. What is important for our purposes is that the framework
models the unemployment rate in contrast to the previous two DSGE models. Christiano
et al. (2016) estimate the model over the sample period 1951-2008, with the same end date
as del Negro et al. (2015).
Our simulation procedure is as follows. We take the estimated models as given and …x
the parameter values at the reported posterior medians. The models are then simulated
by drawing from the innovation distributions over 10,000 periods. This is repeated 1,000
times, whereby we record the last observations to coincide with the length of our sample.
From this sampling distribution we then compute the variance decomposition from the
wavelet …lter as in section 2 and report 90-percent con…dence regions for the mean estimate
of the variance decomposition. We group the individual wavelet decompositions into the
categories ‘Short Term’(D1 -D2 ), ‘Business Cycle’(D3 -D4 ), ‘Medium Term’(D5 -D6 ), and
‘Long Term’(S6 ).
24

They also use the GDP de‡ator to measure in‡ation, whereas we report results for PCE in‡ation. The
di¤erences between these two in‡ation measures are minor.

20

4.2

DSGE Models and Frequency-Band Decompositions

We report the results of the simulation exercise in Table 2. Our focus is on the results
for four key variables, namely real GDP growth, in‡ation, the federal funds rate, and the
unemployment rate. We compare the simulation results to two di¤erent sets of underlying
data: …rst, our original sample that covers 1954Q3 to 2017Q3; and second, the actual sample
period over which the respective model was originally estimated. For all three models this
excludes the Great Recession period and its aftermath. The latter results are reported
separately in Table 3.
The SW model is remarkably successful in replicating the overall volatility of real GDP
components across all frequency groups, essentially matching the data exactly: around 60
percent is attributed to the short-term component, 30 percent to the business-cycle component and a much smaller percentage to the medium and long term. The same pattern is
found for the dNGS model and with some minor di¤erences for the Nash-bargaining speci…cation of the CET model. A key driver for this …nding is the speci…cation of the exogenous
productivity process as a stochastic trend, which is now standard modeling device in DSGE
models. The wavelet decomposition thus con…rms the importance of this assumption.
Turning to the nominal variables, in‡ation and interest rates, the performance of the
SW model notably deteriorates. While the model is consistent with the short-term and
medium-term components in in‡ation, the contribution of the business-cycle and long-term
‡uctuations is essentially ‡ipped. The SW model attributes only 15 percent to the long
term and more than one-third to the business cycle. In contrast, the dNGS model comes
much closer to the patterns in the data, although it underpredicts the contribution of the
long-term component by almost 10 percentage points. The key di¤erence between the two
models is that del Negro et al. (2015) incorporate a time-varying in‡ation target that
is stationary but highly persistent. Over the sample period it e¤ectively pins down the
trend movements in the in‡ation rate. As discussed before, trend in‡ation in the data
might simply re‡ect the changing implicit or explicit in‡ation target, which in the DSGE
modeling sense can be captured by such an exogenous process.
Interestingly enough, the Nash-bargaining speci…cation of Christiano et al. (2016) has
di¢ culty with this pattern, as it attributes considerable variability to the short-term and
business-cycle components and not enough to the long-term component. Notably, their
model does not feature a time-varying in‡ation target, which reinforces the point raised
above. However, the alternative speci…cation of the CET model with wage determination
based on an alternative o¤er bargaining mechanism is on point in capturing the behavior
21

of in‡ation across all frequency bands. It is well-known, e.g., Krause et al. (2008), that
a Nash-bargaining nominal wage mechanism does not impart enough in‡ation persistence
in a New Keynesian search and matching framework which these results con…rm. On the
other hand, the alternative o¤er bargaining mechanism implies endogenous wage inertia
that then translates into inertial prices (see the discussion in Christiano et al., 2016).
The frequency-speci…c patterns of the FFR in the data do not di¤er much from that of
the in‡ation rate, although the wavelet decomposition attributes 80 percent of its movements
to medium-term and long-term components, as opposed to two thirds in the case of in‡ation.
A similar pattern is discernible for the three DSGE models in that they cannot replicate
the importance of the long-term component and the relative lack thereof in the businesscycle frequencies. Most strikingly, the trend in the interest rate is associated with almost
60 percent of movements in the data, only half of which the dNGS model can capture. As
before, the alternative speci…cation of the CET model does remarkably well for the behavior
of the FFR.
We …nally consider the decompositions of the unemployment rate, which, of our three
DSGE models, only Christiano et al. (2016) can address. In the data, half of the ‡uctuations in unemployment are captured by the medium-term component with the remainder roughly equally attributed to the business-cycle and long-term components. Under the
Nash-bargaining speci…cation, the CET model attributes half of unemployment ‡uctuations
to the long-term component, one third to the medium-term component and the remainder
to the business cycle. The model gets the broad pattern of ‡uctuations at di¤erent frequencies right: what matters for explaining the unemployment rate are the medium- to
long-term components, but not those that are arguably more directly shaped by monetary
policy.
At the same time, the alternative bargaining speci…cation of CET results in a considerably worse performance for the unemployment rate. It also has problems with the
decompositions of GDP growth where it attributes too much volatility to the long-term
and medium-term components. Yet, its performance for the two nominal variables, the
in‡ation rate and the interest rate, is spot on, where the baseline speci…cation with Nash
bargaining put too much weight on the business-cycle components. Comparing the two
approaches to modeling wage determination, these …ndings indicate that alternative o¤er
bargaining generates more persistence in the model than Nash bargaining does.25 The ‡ip
25
This is, of course, related to the Shimer (2005) puzzle who argues that the standard search and matching
model cannot replicate the observed volatility and also persistence of the unemployment rate and vacancies,
that is, open positions. Alternative o¤er bargaining therefore presents an attractive solution to the Shimer

22

side of this …nding is that the former imparts too much persistence, which hurts the model’s
performance with respect to the medium- and long-term components of GDP growth and
unemployment.
Our …nal exercise considers the importance of the sampling period for the assessment
of the models. We subject each of the four model speci…cations to the same test, namely
whether they could replicate the behavior of the wavelet decompositions for the full length
of our empirical sample from 1954Q3 to 2017Q3. However, in their published versions the
estimation periods of the three model frameworks di¤er. Speci…cally, the estimation period
for the SW model is 1966-2004, for the dNGS model it is 1964Q1-2008Q3, and for the
CET framework the sample period is 1954Q1-2008Q4. The former two periods are similar,
they both miss 10 years at the beginning of our sample and then the Great Recession and
its aftermath; whereas the CET sample di¤ers from ours in that it ends at the onset of
the Great Recession. Although the underlying idea of structural DSGE modeling is that
structural parameters are generally invariant over these sample periods, it is also well-known
that sample size and sample period can a¤ect structural parameter estimates (e.g., Canova
and Ferroni, 2012).
In Table 3 we therefore contrast select decompositions for the actual estimation sample
with the simulated sample for the same number of observations. The decompositions for
the SW and dNGS sample periods are very similar to each other for real GDP growth and
in‡ation. The biggest di¤erence is the long-term in‡ation component in the dGNS sample,
which includes an additional four years before the onset of the Great Recession. In our full
sample, this component explains 41 percent of in‡ation movements, in the shorter sample
only 34 percent. For both models, the biggest discrepancy can be found in the behavior
of the FFR. The shorter sample attributes much more volatility to the business-cycle and
medium-term components for the policy rate, that is, 60 percent compared to 40 percent
in the full sample. The long-term component explains about one third of the volatility in
the SW sample, but almost 60 percent in the full sample. This discrepancy is arguably due
to di¤erences in policy across the two periods, either at the beginning of the full sample
period between 1954 and 1964 or during the Great Recession. In any case, the long-term
FFR component is more pronounced over the longer period. The results in Table 3 do show,
however, that the SW model struggles to match these facts, while the dNGS model is closer
to the data.
This pattern is also evident from the CET model, where we …nd that the long-term
puzzle, which does not have to rely on exogenous wage stickiness.

23

FFR component in the full sample explains more of the overall volatility. This suggests
that the Great Recession period has had a noticeable e¤ect on the behavior of the FFR which may not be surprising since during this period the Federal Reserve held its policy rate
essentially …xed at its e¤ective lower bound of zero. This, by itself, imparts persistence onto
the FFR. Interestingly, such di¤erences are not visible in any of the other variables, GDP
growth and in‡ation, with the exception of the unemployment rate. It therefore seems that
the behavior of the policy rate is largely disentangled from that of other macroeconomic
aggregates. Table 3 also shows that the behavior of the unemployment rate is di¤erent across
the samples and that the CET model under alternative o¤er bargaining cannot capture the
behavior in the di¤erent sample either.
In summary, we conclude that the three canonical DSGE models are able to replicate
the wavelet decomposition we found in the data. We identify a stochastic trend in productivity and a time-varying in‡ation target as the key modeling elements. The random
walk component in the former and a highly persistent in‡ation target capture the long-term
components in real GDP and in‡ation exceptionally well. Replicating the frequency-speci…c
components of the unemployment rate proves to be more di¢ cult. While the decompositions
from the simulated data go broadly in the same direction, the challenge is that the variance
decomposition is more evenly distributed among frequency bands than for the other variables. Based on these …ndings, we advocate wavelet decompositions as a straightforward
tool to assess the validity of a DSGE model as a data-generating process, especially with
respect to the contribution of individual modeling elements.26

5

Conclusion

This paper advances three main …ndings. First, we show that more than two thirds of
in‡ation and unemployment ‡uctuations in the US occur at low frequencies, whereas at most
a quarter are attributable to business-cycle frequencies. However, it is mainly these latter
‡uctuations that are the focus of monetary policymakers and researchers: policy objectives
are normally phrased in terms of stabilizing ‡uctuations around trends or potential. This
dichotomy is generally re‡ected in the DSGE models that are used to study monetary
policy and its e¤ects. Frequency-speci…c decompositions, such as the one we performed
26
In a sense, we are simply con…rming the results of Sala (2015), who estimates the SW-model in the
frequency domain using likelihood-based methods and working o¤ the counterpart of the time-series respresentation of a state-space model. He …nds that this DSGE model broadly performs well and matches the
data at various frequencies, but fails to capture labor market data and the interactions between real and
nominal variables. However, he uses stationary, thus pre-…ltered data, and can therefore not speak to the
overall decomposition into the several frequency bands.

24

using wavelet methodology, thus produce information relevant for policymakers.
Our second …nding shows that several standard DSGE models do a credible job of
replicating behavior at business-cycle frequencies that we identi…ed in the data. However,
the models need to be suitably modi…ed to account for long-term movements via stochastic trends or time-varying in‡ation targets. They generally fail to capture behavior at
medium-term cycles of between 8 and 32 years. We demonstrate in a third set of results
that monetary policy shocks exert in‡uence over all frequency bands and in a broadly similar manner with the exception of the relationship between short-term interest rates and
in‡ation, where the Fisher e¤ect prevails in the long run.
Our paper thus contributes to a growing area of research that suggests that the notion
of a cycle relevant for stabilization policy should be extended to include at least the medium
term. Speci…cally, the analysis in the paper indicates that temporary shocks can have longlasting e¤ects that traditional business-cycle modeling largely abstracts from. Future work
could therefore study time-frequency decompositions in models with such a transmission
mechanism as in, for instance, Comin and Gertler (2006). Similarly, the …ndings in this
paper also support the idea that what matters for monetary policy is less the short-term
response of policy rates to deviations of economic activity from some target, but rather the
credible anchoring of expectations.27 Typical analyses of optimal monetary policy focus on
weighted averages of the unconditional variances of policy targets. It is common to compare
policies by considering, for example, a weighted average of the unconditional variances of
in‡ation and unemployment. However, such computations mask the e¤ects of policies on
the variance of ‡uctuations at di¤erent frequencies. Frequency-based optimal policy in the
vein of Brock et al. (2013) would thus be an interesting extension based on the analysis in
this paper.

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27

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25

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28

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29

30

32.7
1.3
8.6
1.5
1.3
0.6
5.5

GDP Growth

Unemployment

In‡ation

Federal Funds

3-Month Rate

10-Year Rate

Term Spread

9.7

1.3

2.4

2.8

7.4

3.8

22.8

Short Term
D1: 2-4Q D2: 4-8Q

17.1

2.5

4.8

5.6

7.7

9.6

19.0

26.7

4.0

8.9

10.2

9.9

18.5

12.6

Business Cycle
D3: 8-16Q D4: 16-32Q

19.1

7.5

10.1

10.5

9.3

25.4

6.1

5.4

15.5

13.1

12.8

15.7

20.3

2.5

Medium Term
D5: 32-64Q D6: 64-128Q

Table 1: Variance Decomposition US 1954 - 2017

16.5

68.5

59.3

56.6

41.3

21.1

4.3

Long Term
S6: >128Q

Table 2: Variance Decomposition for Simulated Data
Short Term

Business Cycle

Medium Term

Long Term

56
59

32
29

8
10

4
2

(48-71)

(21-38)

(5-18)

(1-5)

RGDP
Data
SW
dNGS
CET (Nash)
CET (AOB)

60

28

10

2

(50-70)

(20-35)

(5-17)

(1-3)

65

27

6

2

(56-74)

(21-34)

(3-9)

(1-5)

27

40

23

10

(20-35)

(29-50)

(14-37)

(3-22)

16
20

18
35

25
29

41
15

(13-31)

(22-48)

(14-36)

(3-36)

In‡ation
Data
SW
dNGS
CET (Nash)
CET (AOB)

17

20

29

34

(7-31)

(8-32)

(14-47)

(9-66)

27

39

18

16

(18-37)

(24-50)

(9-29)

(3-42)

13

22

20

44

(4-25)

(6-41)

(8-36)

(10-80)

4
16

16
36

24
33

57
17

(9-23)

(23-50)

(16-51)

(3-38)

12

24

33

31

(5-21)

(10-39)

(17-53)

(8-61)

FFR
Data
SW
dNGS
CET (Nash)
CET (AOB)

Unemployment
Data
CET (Nash)
CET (AOB)

25

38

19

16

(16-34)

(25-50)

(11-31)

(3-41)

10

23

25

43

(3-17)

(7-42)

(11-44)

(9-79)

5
6

29
18

45
30

21
46

(2-11)

(5-33)

(12-52)

(13-79)

1

10

30

59

(0-3)

(2-21)

(10-57)

(23-86)

31

Table 3: Variance Decomposition for Simulated Data - Alternative Sample
Short Term

Business Cycle

Medium Term

Long Term

Smets-Wouters
FFR
Full Sample
SW Sample
Simulated

4
8
16

16
25
36

24
35
33

57
32
17

dNGS
FFR
Full Sample
dNGS Sample
Simulated

4
7
12

16
25
24

24
32
33

57
36
31

CET (AOB)
FFR
Full Sample
CET Sample
Simulated

4
6
10

16
21
23

24
28
25

57
45
43

Unemployment
Full Sample
CET Sample
Simulated

5
6
1

29
27
10

45
37
30

21
30
59

32

Figure 1: Wavelet Decompositions: Real GDP Growth

Figure 2: Wavelet Decompositions: Unemployment

33

Figure 3: Wavelet Decompositions: In‡ation

Figure 4: Wavelet Decompositions: Federal Funds Rate

34

Figure 5: Wavelet Decompositions: 10-Year Treasury Rate

Figure 6: Wavelet Decompositions: Term Spread

35

Figure 7: Wavelet Power Spectra: Real GDP Growth

Figure 8: Wavelet Power Spectra: Unemployment

Figure 9: Wavelet Power Spectra: Federal Funds Rate

36

Figure 10: Wavelet Decompositions for Alternative Filters: Real GDP Growth

Figure 11: Wavelet Decompositions for Alternative Filters: Unemployment

37

38
Figure 12: Decompositions with HP and CF Filters: Unemployment

39
Figure 13: Impulse Response Functions: 3-Variable Baseline VAR with Sign Restrictions.

40
Figure 14: Impulse Response Functions with D2 Components

41
Figure 15: Impulse Response Functions with D4 Components

42
Figure 16: Impulse Response Functions with D7 Components

43
Figure 17: Impulse Response Functions with All Components: GDP Growth

44
Figure 18: Impulse Response Functions with All Components: Unemployment