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APPENDIX
Assessing U.S. Aggregate Fluctuations Across Time and
Frequencies
Christian Matthes
Federal Reserve Bank of Richmondz
Thomas A. Lubik
Federal Reserve Bank of Richmondy
Fabio Verona
Bank of Finlandx
February 2019
Working Paper No. 19-06
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
A
A.1
Some Background on Wavelets
Continuous Wavelet Transform
A wavelet
(t) is a function of …nite length that oscillates around the time axis. The name
wavelet (small wave) derives from the admissibility condition, which requires the mother
wavelet to be of …nite support (i.e., small) and of oscillatory (wavy) behavior. The most
commonly used mother wavelet in economic applications - and the one we use in this paper
- is the Morlet wavelet de…ned by
(t) =
1
4
e6it e
t2
2
. The continuous wavelet transform of
a time series x(t) with respect to a given mother wavelet is:
Z +1
1
t
x(t)
Wx ( ; s) = p
dt,
s
s 1
where
denotes the complex conjugate of , and
(A.1)
and s are the two control parameters of
the continuous wavelet transform (CWT). The location parameter
determines the position
of the wavelet along the time axis, while the scale parameter s de…nes how the mother
wavelet is stretched. The scale is inversely related to frequency f , with f
1=s. A lower
(higher) scale means a more (less) compressed wavelet which allows to detect higher (lower)
frequencies of the time series x(t). The ability and ‡exibility to endogenously change the
length of the wavelets is one of the main advantages of the wavelet transform when compared
with the most common alternative, the short-time Fourier transform. The wavelet power
spectrum (WPS) of x(t) is de…ned as (W P S)x ( ; s) = jWx ( ; s)j2 . It measures the local
variance distribution of the time series x(t) around each time and scale/frequency. The
WPS can be averaged over time so that it can be compared to classical spectral methods.
In particular, the global wavelet power spectrum (GWPS) can be obtained by integrating
R +1
the WPS over time: (GW P S)x (s) = 1 Wx ( ; s)d .
A.2
Maximal Overlap Discrete Wavelet Transform and Wavelet Multiresolution Analysis
Wavelet multiresolution analysis (MRA) allows decomposition of any variable into a trend,
a cycle, and a noise component, irrespective of its time series properties. This is similar
to the traditional time series trend-cycle decomposition approach (Beveridge and Nelson,
1981, and Watson, 1986) or other …ltering methods like the Hodrick and Prescott (1997) or
the Baxter and King (1999) band-pass …lter. We employ a particular version of the wavelet
transform called the Maximal Overlap Discrete Wavelet Transform (MODWT). To perform
the MODWT of a given time series, we need to apply an appropriate cascade of wavelet
2
…lters, which is similar to …ltering by a set of band-pass …lters. This procedure allows us to
capture ‡uctuations from di¤erent frequency bands. By using the Haar wavelet …lter, any
variable Xt , regardless of its time series properties, can be decomposed as:
J
X
Xt =
Dj;t + SJ;t ;
(A.2)
j=1
where the Dj;t are the wavelet coe¢ cients at scale j, and SJ;t is the scaling coe¢ cient. These
coe¢ cients are given by:
0
1 1
2jX
1 @
2j
Dj;t =
Xt
j 1
2X
i
i=0
i=2j
J
2 1
1 X
Xt i :
2J
SJ;t =
1
1
Xt i A ;
(A.3)
(A.4)
i=0
Equations (2) - (4) illustrate how the original series Xt , exclusively de…ned in the time
domain, can be decomposed into di¤erent time series components, each de…ned in the time
domain and representing the ‡uctuation of the original time series in a speci…c frequency
band. 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 calendar-time persistence operating at di¤erent frequencies; whereas
the scaling coe¢ cient SJ;t can be interpreted as the low-frequency trend of the time series
under analysis. In particular, when j is small, the j wavelet coe¢ cients represent the higher
frequency characteristics of the time series (i.e., its short-term dynamics). As j increases,
the j wavelet coe¢ cients represent lower frequencies movements of the series.
A.3
The Wavelet Transform: A Simple Example
The wavelet coe¢ cients resulting from the MODWT with Haar …lter are fairly straightforward to interpret as they are simply di¤erences of moving averages. Consider the case
of J = 1. A time series Xt is then decomposed into a transitory component D1 and a
persistent scale component S1 as:
Xt =
Xt
|
Xt 1 Xt + Xt 1
+
:
2
2
{z } | {z
}
D1;t
(A.5)
S1;t
When J = 2, the decomposition results in two detail components D1 and D2 and a scale
component D1 :
Xt Xt 1 Xt + Xt
Xt =
+
2
| {z
} |
D1;t
1
(Xt
4
{z
2
+ Xt
D2;t
3
3)
}
+
Xt + Xt
|
1
+ Xt
4
{z
S2;t
2
+ Xt
3
:
}
(A.6)
While the …rst component D1 remains unchanged at the now higher scale J = 2, the prior
persistent component S1 is divided into an additional transitory component D2 and a new
persistent one S2 . The length Kj of the …lter, that is, the number of observations needed
to compute the coe¢ cients, increases with j: Kj = 2j . Hence, the coarser the scale, the
longer the …lters. Intuitively, the lower the frequencies a researcher wants to capture, the
wider the time window to be considered. Alternatively, the lower the frequencies targeted,
the longer the data sample required. The equations also show that this is a one-sided …lter,
as future values of Xt are not needed to compute the coe¢ cients of the wavelet transform
of Xt at time t. This implies that the Dj;t and SJ;t lag Xt . In other words, they re‡ect the
changes in Xt with some delay. Moreover, since the length of the …lters increases with j, so
does the delay. Hence, the coarser the scale, the more the wavelet components are lagging
behind Xt . Finally, the scale of the decomposition is releated to the frequency at which
activity in the time series occurs. For example, with annual or quarterly time series, Table
A.1 shows the interpretation of the di¤erent scales.
Table A.1: Scales and Cycle Length
Scale j
Period Length
Annual Data
Quarterly Data
1
2y-4y
2q-4q
2
4y-8y
4q-8q=1y-2y
3
8y-16y
8q-16q=2y-4y
4
16y-32y
16q-32q=4y-8y
5
32y-64y
32q-64q=8y-16y
6
64y-128y
64q-128q=16y-32y
...
>128y
>128q=32y
4
B
Data
We extract aggregate time series from the Haver database. The data are collected quarterly
and cover the period from 1954Q3 to 2017Q3, which is the longest available time span for
the variables we consider. Table B1 reports further details on the data, while Figure B1
shows the raw data series. We report results for GDP growth, which we compute as the
quarter-over-quarter rate. Similarly, our measure of in‡ation is the quarter-over-quarter
growth rate of the PCE price index. We also construct a time series for the spread between
the long and the short bond rate, computed as the simple di¤erence.
Table B1: Data
Variable
Mnemonic
Comment
Real GDP
GDPH@USECON
Seasonally Adjusted
Unemployment
LR@USECON
Seasonally Adjusted, 16 and over
PCE Price Index
JC@USECON
Seasonally Adjusted
Federal Funds Rate
FFED@USECON
Monthly Average of Daily Data
3-Month Treasury Rate
FTBS3@USECON
Monthly Average of Daily Data
10-Year Treasury Rate
FCM10@USECON
Monthly Average of Daily Data
5
Figure B1: Macroeconomic Time Series Data
6
C
C.1
Additional Wavelet Decompositions
One-Sided Haar Filter
The …gures in this section report the wavelet components D1 - D6 and the scale component
S6 individually for each of the six macroeconomic time series and for the term spread, the
di¤erence between the 10-year and the 3-month rate. In the …gures, we show the respective
component in dark blue against the overall underlying data series in grey.
7
8
Figure C.1: One-Sided Haar Filter Wavelet Decomposition: Real GDP Growth
9
Figure C.2: One-Sided Haar Filter Wavelet Decomposition: Unemployment
10
Figure C.3: One-Sided Haar Filter Wavelet Decomposition: In‡ation
11
Figure C.4: One-Sided Haar Filter Wavelet Decomposition: Federal Funds Rate
12
Figure C.5: One-Sided Haar Filter Wavelet Decomposition: 3-Month Treasury Rate
13
Figure C.6: One-Sided Haar Filter Wavelet Decomposition: 10-Year Treasury Rate
14
Figure C.7: One-Sided Haar Filter Wavelet Decomposition: Term Spread
C.2
One-Sided Daubechies Filter
The …gures in this section report the wavelet decompositions for the four categories ‘Short
Term’(D1 , D2 ), ‘Business Cycle’(D3 , D4 ), ‘Medium Term’(D5 , D6 ), and ‘Long Term’(S6 )
from a decomposition that uses the Daubechies wavelet …lter. We report the decompositions
for in‡ation, the federal funds rate, and the 10-year rate. For comparison purposes, the
…gures also report the corresponding Haar-…lter decompositions.
Figure C.8: Wavelet Decompositions for Alternative Filters: In‡ation
15
Figure C.9: Wavelet Decompositions for Alternative Filters: Federal Funds Rate
Figure C.10: Wavelet Decompositions for Alternative Filters: 10-Year Treasury Rate
16
D
D.1
The Frequency-Speci…c E¤ects of Monetary Policy Shocks
VAR Speci…cation
We closely follow Arias et al. (2018) in the speci…cation and estimation of a structural
VAR (SVAR) to identify the e¤ects of a monetary policy shock. Speci…cally, we estimate
an SVAR of the following form:
yt0 A0 = c +
L
X
yt0 l Al + "0t :
(A.7)
l=1
yt is a column vector that collects the obervable variables, and "t collects the structural
innovations; c is a vector of constants, while L is the number of lags in the VAR. Our
focus is on determining the elements in the structural impact matrix A0 . Since we do not
impose overidentifying restrictions, we can estimate the reduced-form VAR and impose our
identi…cation restrictions after estimation. To do so, we post-multiply the previous equation
by A0 1 to arrive at:
yt0 = x0t B + u0t ;
(A.8)
where xt also contains the intercept term. We use conjugate Normal-inverse Wishart priors
of the form used in Arias et al. (2018). We assume four lags and a loose, but proper,
prior throughout. Once we have parameter estimates for B and the covariance matrix
of ut , we follow the algorithm outlined in Rubio-Ramirez et al. (2010) to impose sign
restrictions on impact. With respect to the latter, we assume that the level of the nominal
rate increases on impact after a monetary policy shock, in‡ation decreases, and either that
(i) the unemployment rate increases or (ii) that real GDP growth decreases, given the
activity variable used in the estimation.
D.2
Impulse Response Functions
In this seection we report the impulse response functions based on unemployment as the
macroeconomic activity variable in the VAR. Speci…cally, we report results from a speci…cation where we add the short-term D2 , the business-cycle D4 , and the long-term S6
component.
17
Figure D.1: Impulse Response Functions with D2 Components
Figure D.2: Impulse Response Functions with D4 Components
18
Figure D.3: Impulse Response Functions with S6 Components
19
References
[1] Arias, Jonas E., Juan F. Rubio-Ramírez, and Daniel F. Waggoner (2018): “Inference
Based on Structural Vector Autoregressions Identi…ed With Sign and Zero Restrictions:
Theory and Applications”. Econometrica, 86(2), pp. 685-720.
[2] Baxter, Marianne, and Robert G. King (1999): “Measuring Business Cycles: Approximate Band-Pass Filters for Economic Time Series”. The Review of Economics and
Statistics, 81(4), pp. 575-593.
[3] Beveridge, Stephen, and Charles R. Nelson (1981): “A New Approach to Decomposition
of Economic Time Series into Permanent and Transitory Components with Particular
Attention to Measurement of the ‘Business Cycle’”. Journal of Monetary Economics,
7(2), pp. 151-174.
[4] Hodrick, Robert, and Edward C. Prescott (1997): “Postwar U.S. Business Cycles: An
Empirical Investigation”. Journal of Money, Credit, and Banking, 29(1), pp. 1-16.
[5] Rubio-Ramírez, Juan F., Daniel F. Waggoner, and Tao Zha (2010): “Structural Vector Autoregressions: Theory of Identi…cation and Algorithms for Inference”. Review of
Economic Studies, 77(2), pp. 665-696.
[6] Watson, Mark W. (1986): “Univariate Detrending Methods with Stochastic Trends”.
Journal of Monetary Economics, 18, pp. 49-75.
20
@FedFRASER