Tracking the business cycle of the Euro area:
a multivariate model-based band-pass filter
João Valle e Azevedo, Siem Jan Koopman ∗ and António Rua
Abstract
This paper proposes a multivariate band-pass filter that is based on the trend plus
cycle decomposition model. The underlying multivariate dynamic factor model relies on
specific formulations for trend and cycle components and produces smooth business cycle
indicators with band-pass filter properties. Furthermore, cycle shifts for individual time
series are incorporated as part of the multivariate model and estimated simultaneously
with the remaining parameters. The inclusion of leading, coincident and lagging variables
for the measurement of the business cycle is therefore possible without a prior analysis of
lead-lag relationships between economic variables. This method also permits the inclusion
of time series recorded with mixed frequencies. For example, quarterly and monthly time
series can be considered simultaneously without ad hoc interpolations. The multivariate
approach leads to a business cycle indicator that is less subject to revisions than the ones
produced by univariate filters. The reduction of revisions is a key feature in the real-time
assessment of the economy. Finally, the proposed method computes a growth indicator
as a by-product. The new approach of tracking business cycle and growth indicators is
illustrated in detail for the Euro area. The analysis is based on nine key economic time
series.
Keywords:
Band-pass filter; Business cycle; Dynamic factor model; Kalman filter;
Unobserved components time series model; Phase shift; Revisions.
JEL classification:
∗
C13, C32, E32.
Corresponding author: Department of Econometrics, Vrije Universiteit Amsterdam, De Boelelaan 1105,
1081 HV Amsterdam, The Netherlands.
[email protected].
Emails: [email protected]
[email protected]
1
Introduction
Undertaking fiscal and monetary policies requires information about the state of the economy.
Given the importance of this knowledge for the policymaker, a clear economic picture needs to
be present at any time. However, the assessment of the economic situation can be a challenging
task when one faces noisy data giving mixed signals about the overall state of the economy.
In order to avoid a solely judgemental based procedure in the economic analysis one should
be able to extract the relevant information through a statistical rigorous method capable of
providing a clear signal regarding current economic developments.
A business cycle indicator aims to present deviations from a long term trend in economic
activity. Such deviations typically last between 1.5 and 8 years. The time series of real Gross
Domestic Product (GDP) is undoubtedly one of the most important variables for business cycle
assessment. For example, Stock and Watson (1999) consider that “... fluctuations in aggregate
output are at the core of the business cycle so the cyclical component of real GDP is a useful
proxy for the overall business cycle ...”. Common practice is to signal extract the cycle from
real GDP using nonparametric filters such as the ones of Hodrick and Prescott (1997) and
Baxter and King (1999). These filters are based on symmetric weighting kernels and provide
consistent methods of extracting trends and cycles in the middle of the time series but encounter
problems at the beginning and end of the series. The Baxter and King (1999) filter is referred to
as a band-pass filter since it aims to extract the dynamics that are strictly associated with the
business cycle frequencies 2π/p with p ranging from 1.5 to 8 years. The optimality of band-pass
filters is evaluated in the frequency domain by investigating gain functions that should only
affect the business cycle frequencies. Recently, Christiano and Fitzgerald (2003) have developed
an almost optimal filter in this respect together with a solution to the endpoints problem.
The signal extraction of a business cycle can also be based on formulating an unobserved
components time series models for real GDP as in Clark (1987) and Harvey and Jaeger (1993).
Recently, Harvey and Trimbur (2003) have presented a new class of model-based filters for
extracting trends and cycles in economic time series. The main feature of these model-based
filters is that it is possible to extract a smooth cycle from economic time series. It is shown
that the implied filters from this class of models possess band-pass filter properties similar
2
to those of the Baxter and King (1999) filter. Further, model-based filters provide a mutually
consistent way to extract trends and cycles even at the beginning and end of the series. However,
the univariate approaches of band-pass and model-based filters do not exploit the information
stemming from several economic time series. Recently, the work of Forni, Hallin, Lippi, and
Reichlin (2000) and Stock and Watson (2002) have shown that business cycle and growth
indicators can be obtained from hundreds of economic time series. Such an approach is in
sharp contrast with the common practice of using univariate band-pass filters and univariate
model-based methods for obtaining business cycle indicators. In this paper we aim to take a
position between the ”all or one” strategy by considering a moderate set of time series instead
of considering real GDP only or considering a vast amount of time series. Our approach requires
the maintenance of a moderate database of, say, nine time series.
When considering trends and cycles in multivariate time series, we also need to focus on
possible lead-lag relationships between variables and the business cycle. We therefore incorporate the phase shift mechanism of Rünstler (2004) for individual cycles with respect to the base
cycle that is modeled as the generalised cycle component of Harvey and Trimbur (2003). This
development enables the simultaneous modeling of leading, coincident and lagging indicators
for the business cycle without their a priori classification. As a result, phase shifts are estimated as parameters of the model. This contrasts with the multivariate model-based analysis
of Stock and Watson (1989) where the coincident indicator is defined as the common factor
of a small set of coincident variables which are identified prior to estimation. The treatment
of phase shifts in a model-based approach allows the exploitation of information contained in
both lags and leads of variables for computing the business cycle indicator and it overcomes
the restriction of only using coincident variables.
Economic and financial policy decision makers require a business cycle indicator at a high
frequency despite the fact that time series information is usually not available at the desired
frequency. On the other hand, information recorded at a lower frequency must not be discarded.
The crucial example refers to the time series of real GDP that is disregarded by Stock and
Watson (1989) for the construction of its composite indicator presumably because it is only
available on a quarterly basis. Others, such as Altissimo, Bassanetti, Cristadoro, Forni, Lippi,
Reichlin, and Veronese (2001), handled this shortcoming by considering a monthly interpolated
3
GDP series. In the multivariate analysis of this paper observations of different frequencies
can be incorporated since missing observations can be treated by the implemented state space
methods. The ability to deal with missing observations also ensures the business cycle indicator
to be up-to-date even when long delays in data releases are present. This feature is important
given the significant data release delays in the Euro area.
In this paper we develop a feasible procedure for the computation of business cycle indicators. The resulting business cycle filter is based on a multivariate dynamic factor model for a
vector of economic and monetary time series recorded at quarterly and monthly frequencies.
The dynamic specification of the model allows the latent factors to be interpreted as trend
and cycle components. The common cycle component is formulated as in Harvey and Trimbur
(2003) and can be shifted for individual variables as in Rünstler (2004). The signal extraction
of the cycle component can be regarded as the result of a multivariate model-based approximate
band-pass filter. The multivariate procedure is shown to produce a cycle indicator that is subject to smaller revisions compared to univariate band-pass filters and univariate model-based
filters. Hence the real-time signal extraction has been improved against commonly used filters
in business cycle analysis. This key feature is important for macroeconomic stabilisation policy
decisions since reliable estimates of the business cycle position in real-time are crucial in this
respect.
We implement the multivariate model-based band-pass filter for the business cycle tracking
of the Euro area. By pooling information from nine key economic variables (including GDP,
industrial production, retail sales, confidence indicators, among others) it obtains a business
cycle indicator in line with common wisdom regarding Euro area business cycle developments.
The business cycle indicator can be regarded as a monthly proxy for the output gap. A growth
indicator can also be constructed from our analysis as a by-product and is compared with the
EuroCOIN indicator. This growth indicator of Altissimo, Bassanetti, Cristadoro, Forni, Lippi,
Reichlin, and Veronese (2001) is based on a generalised dynamic factor model with almost a
thousand economic series as input. The parsimonious approach proposed in this paper leads
to a similar growth indicator and overcomes some drawbacks of EuroCOIN.
The remaining of the paper is organised as follows. In section 2 we present the underlying
dynamic factor model and its extensions. In section 3 we proceed to the empirical application
4
for the Euro area. Section 4 concludes.
2
Model for multivariate band-pass filtering
Assume that a panel of N economic time series are collected in the N × 1 vector yt and that
we observe n data points over time, that is t = 1, . . . , n. The i-th element of the observation
vector at time t is denoted by yit .
2.1
The basic trend-cycle decomposition model
The basic unobserved components time series model for measuring the business cycle from a
panel of economic time series is based on the decomposition model
yit = µit + δi ψt + εit ,
i = 1, . . . , N,
t = 1, . . . , n,
(1)
where µit is the individual trend component for the ith time series and the business cycle
component (common to all time series) is denoted by ψt . For each series the contribution of
the cycle is measured by the coefficient δi for i = 1, . . . , N. The idiosyncratic disturbance εit is
assumed normal and independent from εjs for i 6= j and/or t 6= s. The variance of the irregular
disturbance varies for the different individual series, that is
i.i.d.
2
),
εit ∼ N(0, σi,ε
i = 1, . . . , N.
(2)
Each trend component from the panel time series is specified as the local linear trend
component (see Harvey, 1989), that is
i.i.d.
µi,t+1 = µit + βit + ηit ,
2
ηit ∼ N(0, σi,η
),
βi,t+1 =
2
),
ζit ∼ N(0, σi,ζ
βit + ζit ,
i.i.d.
(3)
for i = 1, . . . , N. The disturbances ηit and ζit are serially and mutually independent of each
other, and other disturbances, at all time points and across the N equations. The trends are
therefore independent of each other within the panel. The cycle component is common to
all time series in the panel and can be specified as an autoregressive model with polynomial
5
coefficients that have complex roots. Specifically, we enforce this restriction by formulating the
cycle time series process as a trigonometric process, that is




 

ψ
cos λ sin λ
κ
ψ
 t+1  = φ 
 t  +  t ,
+
ψt+1
− sin λ cos λ
κ+
ψt+
t
κt
i.i.d.
∼
N(0, σκ2 ),
κ+
t
i.i.d.
N(0, σκ2 ),
∼
(4)
where κt and κ+
t are white noise disturbances and mutually independent of each other at all
time points for t = 1, . . . , n. The damping term 0 < φ ≤ 1 ensures that the stochastic process
ψt is stationary or fixed as the cycle variance is specified as σκ2 = (1 − φ2 )σψ2 where σψ2 is the
variance of the cycle. The period of the stochastic cycle is defined as 2π/λ where 0 ≤ λ ≤ π
is approximately the frequency at which the spectrum of the stochastic cycle has the largest
mass. The initial distribution of the cycle is given by
i.i.d.
i.i.d.
ψ1 ∼ N(0, σψ2 ),
ψ1+ ∼ N(0, σψ2 ),
(5)
where σψ2 = σκ2 /(1 − φ2 ). The dynamic properties of the cycle can be characteristed via the
autocovariance function which is given by
γ(τ ) = E(ψt ψt−τ ) = σκ2 φτ (1 − φ2 )−1 cos(λτ ),
for τ = 0, 1, 2, . . ..
2.2
Signal extraction and approximate band-pass filter properties
The basic trend-cycle decomposition model falls within the class of multivariate unobserved
components time series models or multivariate structural time series models which can be casted
in state space form. The details of this general class of multivariate models are discussed by
Harvey and Koopman (1997) including its relations with other multivariate approaches such as
the common feature models of Engle and Kozicki (1993). Further, Harvey and Koopman (2000)
discuss vector autoregressive model representations of unobserved component models which are
related to the canonical correlation analyses of Tiao and Tsay (1985) and the multiple time series
analyses of Ahn and Reinsel (1988). The unknown parameters such as variances associated
with the disturbances of the model and loading factor coefficients δi for the common cycle
component are estimated by numerically maximising the likelihood function that is evaluated
6
by the Kalman filter. The signal extraction of unobserved trend and cycle components is
carried out by the Kalman filter and associated smoothing methods. An introduction to such
techniques is found in, for example, Harvey (1989) and Durbin and Koopman (2001). The
parameters are identified
It may be of interest to know how observations are weighted when, for example, the business
cycle estimate at time t is constructed. Such weights can be obtained as discussed in Harvey
and Koopman (2000). The implications of the filter in the frequency domain can be assessed
via the spectral gain function, see Harvey (1993, Chapter 6). A filter is said to have band-pass
filter properties when the gain function has sharp cut-offs at both ends of the range of business
cycle frequencies.
2.3
Generalised components
The basic trend plus cycle decomposition model (1) can be modified in such a way that the
model-based filtering and smoothing method possesses band-pass filter properties, see Gomez
(2001) and Harvey and Trimbur (2003). For example, the m-th order stochastic trend for series
(m)
i can be considered as given by µit = µit
where
i.i.d.
(m)
∆m µi,t+1 = ζit ,
2
ζit ∼ N(0, σi,ζ
),
(6)
j = m, m − 1, . . . , 1,
(7)
or
(j)
(j)
(j−1)
µi,t+1 = µit + µit
(0)
,
i.i.d.
2
with µit = ζit ∼ (0, σi,ζ
) for i = 1, . . . , N and t = 1, . . . , n. The trend specification (6) can be
taken as the model representation of the class of Butterworth filters that are commonly used in
engineering, see Gomez (2001). The special case of (6) with m = 2 reduces to the local linear
(1)
2
trend component (3) with σi,η
= 0 from which it follows that βit = µit . The effect of a higher
value for m is pronounced in the frequency domain since the low-pass gain function will have
a sharper cut-off downwards at a certain low frequency point. In the time domain the effect
(m)
is evident since ∆m µi,t+1 = ζit suggests that the time series yit implied by model (1) and (6)
needs to be differenced m times to become stationary. From this perspective, the choice of a
relatively large choice for m may therefore not be realistic for many economic time series.
7
The generalised specification for the k-th order common cycle component is given by ψt =
(k)
ψt
where




 

(j)
(j)
(j−1)
ψ
cos λ sin λ
ψ
ψ
 t+1  = φ 
 t
+ t
,
+(j)
+(j)
+(j−1)
ψt+1
− sin λ cos λ
ψt
ψt
j = 1, . . . , k,
(8)
with
i.i.d.
i.i.d.
ψt0 = κt ∼ N(0, σκ2 ),
2
ψt+0 = κ+
t ∼ N(0, σκ ),
for t = 1, . . . , n. A higher value for k leads to more pronounced cut-offs of the band-pass gain
function at both ends of the range of business cycle frequencies centered at λ. For example,
the model-based filter with k = 6 leads to a similar gain function as the one of Baxter and
King (1999), see Harvey and Trimbur (2003). However, by increasing k, there is a trade-off
between the sharpness of the implicit band-pass filter and its behaviour at the endpoints of the
sample. A sharper cut-off implies larger revisions of the estimates near the endpoints since more
weight is given to the (implicitly forecasted) observations in the far future. In the empirical
study of this paper, a good compromise is found at k = 6 although smaller values of k did not
lead to completely different estimates of the key parameters in the model. Different variations
within this class of generalised cycles are discussed in Harvey and Trimbur (2003) who refer
to specification (8) as the balanced cycle model. They prefer this specification since the timedomain properties of cycle ψt are more straightforward to derive and it tends to give a slightly
better fit to a selection of U.S. economic time series. The number of unknown parameters
remain three for cycle specification (4), that is 0 < φ ≤ 1, 0 ≤ λ ≤ π and σκ2 > 0, for any order
k.
2.4
Phase shifts of business cycle
In the multivariate case, with N > 1, multiple time series are considered for the construction
of a business cycle indicator. The contribution of the ith time series to cycle ψt is determined
by loading δi , for i = 1, . . . , N. In the current model (1) with generalised trend and cycle components only economic time series with coincident cycles are viable candidates to be included
in the model. Prior analysis may determine whether leads or lags of economic variables are
appropriate for its inclusion. To avoid such a prior analysis, the model is modified to allow
8
the base cycle ψt to be shifted for each time series. For this purpose, we adopt the phase
shift mechanism for a multivariate cycle specification as proposed by Rünstler (2004). For a
given cycle component given by (4), it follows from a standard trigonometric identity that the
autocovariance function of cycle ψt is shifted ξ0 time periods to the right (when ξ0 > 0) or to
the left (when ξ0 < 0) by considering
cos(ξ0 λ)ψt + sin(ξ0 λ)ψt+ ,
t = 1, . . . , n.
The shift ξ0 is measured in real-time so that ξ0 λ is measured in radians and due to the periodicity
of trigonometric functions the parameter space of ξ0 is restricted within the range − 12 π < ξ0 λ <
1
π.
2
For example, it follows that when a cosine is shifted by xλ, its correlation with the base
cosine is the same as its correlation with a cosine that is shifted by (π − x)λ but with an
opposite sign, for 21 π < xλ < π. The shifted cycle is introduced in the decomposition model by
replacing equation (1) with the equation
o
n
(m)
(k)
+(k)
+ εit ,
yit = µit + δi cos(ξiλ)ψt + sin(ξiλ)ψt
i = 1, . . . , N,
t = 1, . . . , n,
(9)
where the specifications in (6), (8) and (2) remain applicable. Rünstler (2004) shows that the
phase shift appears directly as a shift in the autocovariance function of the stochastic cycle.
This remains to hold for model specification (9), see Appendix A for more details. We further
restrict attention to a common cycle specification. It is assumed that the first equation identifies
the base cycle and therefore ξ1 = 0 and δ1 = 1 so that
y1t = µ1t + ψt + ε1t ,
t = 1, . . . , n.
The common cycle ψt coincides contemporaneously with the first variable in the vector yt . The
restriction δ1 = 1 can be replaced by the constraint of a fixed value for σκ2 . The restriction
ξ1 = 0 allows the cycles of other variables to shift with respect to the cycle of the first variable.
The general multivariate trend-cycle decomposition model can be formulated in state space
with partially diffuse initial conditions. The details are given in Appendix B.
2.5
Some further issues
The multivariate trend-cycle decomposition model contains a common cycle that is specified as
a generalised cycle and is allowed to shift for every individual series with respect to a base cycle.
9
The generalised cycle specification is taken from Harvey and Trimbur (2003) and the phase shift
mechanism is taken from Rünstler (2004). The trend and irregular components in the model
are idiosyncratic factors that account for the dynamics in the non-cyclical range of frequencies
in the time series. The model is casted in the state space framework. Therefore, variables
with different observation frequencies in the panel can be handled straightforwardly using the
Kalman filter, see Durbin and Koopman (2001, Section 4.6). For example, the treatment of
quarterly time series using a monthly time index in the state space framework amounts to
having two consecutive monthly observations as missing followed by the quarterly observation.
There are alternative ways to account for mixed frequencies, see, for example, in Mariano and
Murasawa (2003). Such solutions require nonlinear techniques since macroeconomic time series
are often modeled in logs. Our main interest is to capture the cycle and growth dynamics.
Therefore, the different solutions will have minor impact on the extracted cycle.
The specification of the dynamic factor model (9) can be extended to a more general setting.
Nevertheless, the key features of business cycle analysis are incorporated in the model. The
generalised cycle enables us to interpret the extracted cycle as the result of an approximate
band-pass filter to the reference series in the spirit of Harvey and Trimbur (2003). As a result,
the model can be viewed partly as a filtering device and partly as a specification of the true
data generating process. When we take the cycle of a particular series as the base cycle and
we enforce the base cycle on the other series (adjusted for scale and phase) as in the proposed
model specification, we effectively use information from the cyclical fluctuations of the other
series only to the extent that they match those of the series that are originally attached to the
base cycle. Subject to phase shift adjustments, the cyclical components of the various series
are perfectly correlated. The band-pass filter interpretation is therefore warranted. We should
note that cross-correlations (adjusted for phase) can only be equal to unity if the phase shift
implies an integer shift in the cross-correlation function, see Appendix A. It is unlikely that an
integer shift occurs in practice since the shift parameter is a typical non-integer value and needs
to be estimated. These insights of the model indicate that the model is appropriate for the
purpose of the analysis: to obtain a business cycle indicator that is close to the one obtained
from a univariate band-pass filter but one that has better properties in terms of revision and
availability at a higher frequency.
10
The identification of the parameters in the model is guaranteed as they all appear in the
autocovariance function of the model in its reduced form. These results are standard for
the multivariate unobserved components time series model of which our model without cycle
phase shifts is a special case, see Harvey and Koopman (1997). The identification of parameters
associated with cycle shifts is discussed by Rünstler (2004, Proposition 2) and requires an unique
decomposition for the autocovariance function of the cycle component. The specification that
we have adopted is a special and simplified case of the Choleski decomposition.
3
3.1
Empirical results for the Euro Area
Data
The multivariate framework presented in the previous section allows the simultaneous modeling of leading, coincident and lagging indicators for the business cycle with phase shifts being
estimated as an integrated part of the modeling process. However, as a preliminary step, one
has to define the base cycle. We follow Stock and Watson (1999) who consider the cyclical
component of real GDP as a useful proxy for the overall business cycle. Therefore, we impose
a unit common cycle loading and a zero phase shift for Euro area real GDP. The remaining
series included in the multivariate analysis are the most commonly used monthly indicators
on Euro area economic monitoring. This includes industrial production, retail sales, unemployment, several confidence indicators (consumers, industrial, retail trade and construction)
and interest rate spread. As it is well known, industrial production is highly correlated with
the business cycle and it is coincident or even slightly leading. On the other hand, retail sales
provide information about private consumption, which accounts for a large proportion of GDP.
Unemployment is a countercyclical and lagging variable but it also displays a high correlation
with the business cycle. Confidence indicators provide information regarding economic agents’
assessment of current and future economic situation so they are expected to present coincident
or leading properties. Finally, the interest rate spread has been widely recognised as a forwardlooking indicator for real activity. These time series are available for the period between 1986
and 2002 and graphically presented in figure 1. The resulting dataset covers different economic
11
sectors and it is based on both quantitative and qualitative data as well as on both real and
monetary variables. GDP is available on a quarterly basis whereas the other indicators are
monthly. Appendix C provides a detailed description of the data.
14.30
14.25
GDP
IPI
Interest rate spread
Construction confidence indicator
Consumer confidence indicator
Retail sales
Unemployment (inverted sign)
Industrial confidence indicator
Retail trade confidence indicator
14.20
14.15
14.10
14.05
14.00
13.95
13.90
1990
1995
2000
Figure 1: Nine key economic variables for the Euro area between 1986 and 2002.
3.2
Details of model for multivariate band-pass filtering
The empirical study of the Euro area business cycle will be based on the multivariate model (9)
with generalised trend µit , given by specification (6) with m = 2, and common cycle ψt , given
by specification (8) with k = 6. The first time series in the panel is real GDP. The time index
refers to months and applies to all series in the panel except for GDP that is observed every
quarter. This implies that the monthly series of GDP has two consecutive monthly observations
as missing followed by its quarterly observation.
A preliminary analysis of the panel of time series using band-pass filters reveals that cyclical
correlations of the various series with GDP ranges from 0.70 to 0.95. This range of correlation
12
values is relatively high and we do not expect the model to capture this perfectly. However, it
justifies, to some extent, the decomposition model with a common base cycle and with trend
and irregular components that account for the other dynamics in the series.
Harvey and Trimbur (2003) show that the choice of m = 2 and k = 6 leads to estimated
trend and cycle components with band-pass filtering properties similar to the ones of Baxter
and King (1999). The role of the trend component µit in model (9) is to capture the persistence
of the time series excluding the cycle. The order m of the trend (6) applies to all time series in
the panel.
The frequency of the common cycle component is fixed to assure that the cycle captures
the typical business cycle period in the time series. The frequency is therefore chosen as
λ = 0.06545 which implies a period of 96 months (8 years) and is consistent with Stock and
Watson (1999) for the U.S. and the ECB (2001) for the Euro area. The frequency parameter is
fixed to guarantee the extraction of typical business cycle fluctuations. When applying bandpass filters to a typical integrated series, the peak in the spectrum of the filtered series is
reached at the lowest frequency of the band of interest. The value of λ is therefore chosen to
correspond approximately with the location of the peak in the spectrum. The choice is not
based on minimizing the difference of the gain function between an ideal filter and a filter that
is implicitly given by the model.
Further, we note that δ1 and ξ1 , the load and shift parameters associated with the base
series, GDP, are restricted to one and zero, respectively. Apart from these restrictions, the
2
2
number of parameters to estimate for each equation is four (σi,ζ
, δi , ξi and σi,ε
) and for the
common cycle is two (φ and σκ2 ). The total number therefore is 4N (4 for each equation minus
two restrictions plus two for the common cycle).
The maximum likelihood estimates of the parameters are obtained by maximising the loglikelihood function that is evaluated through the Kalman filter. This is a standard exercise
although computational complexities may arise because the parameter dimension rises quickly
as N increases. For example, with N = 9 the number of parameters to be estimated is 36. In
principle, there is no difficulty of being able to empirically identify the parameters from the data
since the number of observations increases accordingly with N. It can however be computationally and numerically cumbersome to obtain the maximum value of the log-likelihood function in
13
a 36 dimensional space. A practical and feasible approach to overcome some numerical difficulties is to obtain starting values from univariate and bivariate models. The model dimension is
then increased step by step and for each step realistic starting values will be available from the
earlier low-dimensional models. This approach can be fully automated and different orderings
in the panel of time series can be considered in order to be assured (at least, to some extent)
that the likelihood maximum is obtained. Further, the maximisation process is stopped when
specific parameter values become unrealistic. The maximisation is restarted at a different set of
(realistic) values in order to safeguard the estimation process from producing unsettling model
estimates. It should be noted that this has occurred a few times when the dimension of the
model increases.
Once the model parameters are estimated, the trends and the common business cycle can
be extracted from the data using the Kalman filter and associating smoothing algorithms. The
Kalman filter is sometimes known to be slow for multivariate models, especially when specific
modifications are considered for the diffuse initialisations of nonstationary components such
as trends. However, we would like to stress that the implementation of the Kalman filter for
multivariate models as documented in Koopman and Durbin (2000) and implemented using the
library of state space functions SsfPack was not slow nor numerically unstable, see Appendix
B for further details.
3.3
The business cycle indicator
The estimated components for the Euro area real GDP series are presented in figure 2. Several
interesting features become apparent. First, although GDP is recorded on a quarterly basis
the estimated components are monthly. These components can be seen as the outcome of the
underlying monthly GDP decomposition which can be recovered resorting to the information
contained in the remaining dataset. Second, by analysing the trend it seems that Euro area
potential growth has declined after the major recession that occurred in 1993. This feature
becomes clear by plotting the trend slope against the time axis. Before 1993 recession monthly
potential growth was around 0.3 per cent (3.7 per cent in annualised terms) while afterwards
it was closer to 0.2 per cent. This is 2.4 per cent in annualised terms and falls within the
14
2.0 − 2.5 per cent GDP potential growth range underlying the ECB monetary policy. Third,
the resulting GDP cyclical component, that is our Euro area business cycle indicator, is in line
with the common wisdom regarding Euro area business cycle developments (see ECB (2002)).
In particular, a major recession was recorded in 1993 (as German unification boost delayed
the impact of the recessions in the United States and the United Kingdom in the early 1990s)
while there was an economic slowdown in the late 1990s following the crisis in Asian emerging
market economies and the financial instability in Russia.
(i)
(ii)
0.003
14.2
14.1
0.002
14.0
0.001
13.9
(iii)
1990
1995
2000
0.01
(iv)
1990
1995
2000
1990
1995
2000
0.0050
0.0025
0.00
0.0000
−0.0025
−0.01
−0.0050
1990
1995
2000
Figure 2: Decomposition of GDP: (i) GDP and trend; (ii) slope; (iii) cycle; (iv) irregular.
The variances of the trends and the damping factor and standard deviation of the common
cycle are estimated. For the latter parameters we obtained estimates φ̂ = 0.57 and σ̂κ = 0.0066,
respectively. The variances of the individual trends are also estimated and they vary from very
small values (suggesting that the trend is fixed and may coincide with the fact that the trend
does not exist) to relatively high values (suggesting that trend and its corresponding slope
change throughout the series).
The estimated business cycle component is computed by the multivariate Kalman filter and
15
smoother which effectively weight the observations in an optimal way for signal extraction. The
associating gain function is also multivariate but is difficult to interpret from a practical point
of view. Therefore, figure 3 presents the gain function of the univariate filter as in Harvey and
Trimbur (2003, section IV). The gain function is for the extraction of the monthly business cycle
associated with GDP and is based on coefficients that are obtained from parameter estimates
of the multivariate model. The cut-off at the lower frequencies is sharp while the cut-off at
the higher frequencies is more smooth. The latter is partly due to the estimate φ̂ = 0.57 as it
implies a somewhat less persistent cycle component.
G(λ)
1.0
0.8
0.6
0.4
0.2
0.0
0.5
1.0
1.5
2.0
2.5
λ
3.0
Figure 3: Gain function of the implied univariate filter for extracting the cycle for GDP at a
monthly frequency. The gain coefficients are from parameter estimates of multivariate model.
The approach of this paper can be taken as an attempt to extract a business cycle that is
similar to the one obtained from a filter implied by the gain function of figure 3 but that is
actually based on a dataset of monthly economic indicators as well as quarterly GDP. The motivations of this approach are to circumvent the non-availability of GDP at a monthly frequency
and to obtain a clearer signal at the end of the sample. These objectives are of importance
16
for the policy maker. It should be stressed that figure 3 gives a distorted picture of the gain
function at the end of the sample. However, by using multiple sources of information at a
monthly frequency, this distortion is diminished. Furthermore, it may lead to smaller revisions
of the signal at the end of the sample when new observations become available.
Some key estimation results are presented in table 1. It is noted that we do not aim to
develop a model that captures all dynamic characteristics in the available panel of time series.
Therefore we do not present a detailed discussion of the estimation results for each individual
equation of the multivariate model. The table aims to provide information about the adequacy
of our choice of economic variables and their roles in the construction of the multivariate bandpass filter.
The individual time series are decomposed by trend, common cycle and irregular components. The relative importance of each component for a particular time series i is determined
by the scalars σi,ζ (trend), δi σκ (cycle) and σi,ε (irregular). In table 1 the relative importance
of the trend and the cycle is presented based on the maximum likelihood estimates of these
scalars. Trend variations appear most prominently in the unemployment series and, to a lesser
extent, in the series interest rate spread, consumer confidence and industrial confidence. It is
interesting to observe that industrial confidence tops the list of contributors to the construction
of the business cycle while GDP, industrial production, consumer confidence and construction
confidence are important too. Series related to retail trade contribute relatively little to the
estimated business cycle.
The ability of simultaneously identifying and estimating cycle shifts within the model for
a given set of time series is a key development that is important for the current analysis since
it avoids the pre-selection of coincident variables, whether or not by lagging or leading the
variable. The estimation results reveal that unemployment lags the business cycle by more
than one year (almost 16 months) while interest rate spread clearly leads the business cycle
by almost 17 months. The common cycle component for the other series displays relatively
small shifts, confirming the coincident character of these series. These results are in line with
what was expected and according to what was found elsewhere, see for example Altissimo,
Bassanetti, Cristadoro, Forni, Lippi, Reichlin, and Veronese (2001). An indication whether the
2
multivariate trend-cycle model fits the individual series successfully is the goodness-of-fit RD
17
Table 1: Selected estimation results
time series
rel trd
rel cyc
shift
2
RD
Gross domestic product (GDP) 0.10
0.074
0
0.31
Industrial production
0.068
0.064
6.85
0.67
Unemployment (inverted sign)
0.55
0.033
-15.9
0.78
Industrial confidence
0.26
0.14
7.84
0.47
Construction confidence
0.17
0.036
1.86
0.51
Retail trade confidence
0.098
0.013
-0.22
0.67
Consumer confidence
0.33
0.049
3.76
0.33
Retail sales
0.027
0.0064
-4.70
0.86
Interest rate spread
0.34
0.023
16.8
0.22
The following values are based on maximum likelihood estimates (denoted by hats): rel trd measures the relative
importance of trend component and is computed by σ̂i,ζ /(σ̂i,ζ + δ̂i σ̂κ + σ̂i,ε ); rel cyc is for the cycle component
2
and is computed by δ̂i σ̂κ /(σ̂i,ζ + δ̂i σ̂κ + σ̂i,ε ); shift reports estimate ξˆi . The coefficient of determination RD
is
with respect to the total sum of squared differences.
18
statistic as reported in table 1. The sum of squared prediction errors is compared to the sample
variance of the differenced time series for each equation of the model. These statistics give
evidence that the fit of the model for the individual time series is adequate to good.
0.010
0.005
0.000
−0.005
−0.010
−0.015
1990
1995
2000
Figure 4: Euro area business cycle indicator with ECB datings.
Figure 4 presents the smoothed estimate of the business cycle indicator together with the
Euro area business cycle turning points settled by the European Central Bank, see ECB (2001).
Since the ECB dating was made on a quarterly basis, the quarters where presumably occured a
turning point are represented by a shaded area due to the monthly nature of the time axis. One
can see that the suggested business cycle indicator tracks, in general, quite well the turning
points. Only at the end of the sample it seems to perform less satisfactorily. However, one
should note that the ECB dating was made using data only up to 2000. In particular, this can
explain why the last peak dated by ECB is not confirmed by our business cycle indicator since
it suggests that the peak only occurred at the beginning of 2001. The historical minimum value
of the business cycle is observed in August 1993, which falls within the most severe recession
period recorded in the Euro area, while the maximum value is at January 2001.
19
0.020
0.015
0.010
0.005
0.000
H−C
A−K−R
C−F
−0.005
−0.010
H−P
−0.015
1985
1990
1995
2000
Figure 5: Comparison of four different business cycle indicators: H-P is the Hodrick-Prescott
filter; C-F is the Christiano-Fitzgerald filter; H-C is the model-based decomposition of Harvey
and Clark; A-K-R is our business cycle indicator. The business cycle indicators are presented
at a quarterly frequency.
20
In Figure 5 the business cycle obtained from our multivariate filter is compared with the
ones resulting from some univariate approaches to business cycle assessment based on quarterly
real GDP. In particular, we consider the Hodrick and Prescott (1997) filter, the Christiano and
Fitzgerald (2003) band-pass filter and the Harvey-Clark model-based filter, see Clark (1987)
and Harvey (1989). The latter model-based filter is based on the univariate trend-cycle decomposition model (1) with N = 1. As it is standard in the literature for quarterly data, we
set λ = 1600 for Hodrick-Prescott filter and considered cycles of periodicity between 6 and
32 quarters for Christiano-Fitzgerald filter. All cycle indicators appear to move upwards or
downwards at roughly the same time. However, our multivariate filter provides a smooth business cycle similar to the one from Christiano and Fitzgerald (2003) band-pass filter while the
other detrending approaches are more affected by irregular movements. The cycle indicators
are presented at a quarterly frequency whereas our multivariate filter obtains business cycle
estimates at the higher monthly frequency and is able to recover the latent monthly output gap
which is highly relevant for the policy maker. The underlying methods used in this paper also
overcome the release lags that end up delaying the assessment of the current economic situation and are noteworthy in the Euro area. Since the state space framework can deal with the
missing observations problem, it ensures an up-to-date high-frequency business cycle estimate
and allows for a timely analysis of current economic developments.
3.4
Revisions
A major issue regarding business cycle estimates is their real-time reliability, see Orphanides
and van Norden (2002). In practice, business cycle estimates are subject to revisions over time
due to data revision and to re-computation when additional data becomes available. Although
it is not possible to evaluate the consequences of the first source of revision for the Euro area due
to lack of data vintages, the second source of revision can be assessed by comparing smoothed
and filtered estimates of the business cycle. The filtered estimate is the one conditional on the
information set available up to each time point. The smoothed estimate is the one based on
the full set of data. Their comparison reveals the magnitude of the revision. In figure 6 final
(smoothed) and real-time (filtered) estimates are presented for different filters. In line with the
21
0.02
0.01
0.01
0.00
0.00
−0.01
−0.01
−0.02
H−P
1985
1990
1995
2000
C−F
1985
1990
1995
2000
0.01
0.01
0.00
0.00
−0.01
−0.01
1985
A−K−R
H−C
1990
1995
2000
1985
1990
1995
2000
Figure 6: Comparison of filtered (dotted line) and smoothed (solid line) estimates related to
four different business cycle indicators: H-P is the Hodrick-Prescott filter; C-F is the ChristianoFitzgerald filter; H-C is the model-based decomposition of Harvey and Clark; A-K-R is our
business cycle indicator. The business cycle indicators are presented at a quarterly frequency.
22
Table 2: Reliability statistics for different business cycle indicators
Method
correlation
noise-to-signal
86 − 02 93 − 02 86 − 02
sign concordance
93 − 02
86 − 02
93 − 02
Hodrick-Prescott
0.33
0.70
1.23
0.99
0.47
0.55
Christiano-Fitzgerald
0.54
0.76
0.90
0.67
0.55
0.65
Harvey-Clark
0.42
0.69
0.91
0.75
0.61
0.63
Azevedo-Koopman-Rua
0.58
0.80
0.81
0.61
0.69
0.83
Correlation is the contemporaneous correlation between the real time (filtered) estimates and the final
(smoothed) estimates of the business cycle. Noise-to-signal ratio is the ratio of the standard deviation of
the revisions against the standard deviation of the final estimates. Sign concordance is the percentage of times
that the sign of the real time and final estimates are equal. The statistics are reported for the sample periods
of 1986-2002 (86-02) and 1993-2002 (93-02).
findings of Orphanides and van Norden (2002) for the U.S., the difficulty of assessing the cyclical
position in real-time for the Euro area is clear. In fact, Orphanides and van Norden (2002)
show that the reliability of real-time estimates of U.S. output gap is low whatever method is
used. In addition to a graphical inspection, the importance of the revisions can be measured
by the cross-correlation between final and real-time estimates, by the noise-to-signal ratio (the
ratio of the standard deviation of the revisions to the standard deviation of the final estimate)
and by the sign concordance (the proportion of time that final and real-time estimates share
the same sign). These measures are presented in table 2 for the whole sample and for a more
recent period. It should be beared in mind that it is extremely hard to estimate the output gap
in real-time since only with the advance of time one can be more accurate about the cyclical
position at a given period. From table 2, one can see that the Hodrick-Prescott filter presents
the worst results while the multivariate band-pass filter method outperforms considerably all
univariate filters in real-time. Therefore, it seems that a multivariate setting allied with the use
of leading, coincident and lagging variables can overcome, to some extent, the shortcomings of
the currently used filters for real-time analysis.
23
3.5
A growth indicator
The focus of this paper is on the business cycle indicator and on how the information content
of an economy-wide dataset can be exploited for the assessment of the cyclical position of the
economy using a multivariate approximate band-pass filter. Nevertheless, the policymaker may
also be interested in tracking economic activity growth. As a by-product a growth indicator can
be obtained as part of our model-based approach. This can be done by restoring the smoothed
estimate of the business cycle together with the corresponding smoothed estimate of the trend
component of a particular time series and by computing its growth rate. The three-monthly
change of the trend restored business cycle indicator is presented in figure 7. The resulting
growth indicator can be interpreted as the latent monthly measure of GDP quarterly growth.
Figure 7 shows that the growth indicator tracks GDP quarterly growth quite well. Moreover, it
is available on a monthly basis and avoids the erratic behaviour of GDP quarterly growth rate
which complicates the assessment of the current economic situation. Figure 7 also presents our
growth indicator together with EuroCOIN, see Altissimo, Bassanetti, Cristadoro, Forni, Lippi,
Reichlin, and Veronese (2001). The EuroCOIN is based on the generalised dynamic factor model
of Forni, Hallin, Lippi, and Reichlin (2000) and resorts to a huge dataset (almost a thousand
series that refer to the six major Euro area countries). The model underlying EuroCOIN can
be described briefly in the following way. Each series in the dataset is modeled as the sum of
two uncorrelated components, the common and the idiosyncratic. The common component is
driven by a small number of factors common to all variables but possibly loaded with different
lag structures, while the idiosyncratic component is specific to each variable. EuroCOIN is the
common component of the GDP growth rate after discarding its high frequency component.
The EuroCOIN indicator presented in panel (ii) of figure 7 has been subject to revisions each
time new observations become available. Note that the GDP monthly series used as input was
obtained through the linear interpolation of quarterly figures. Although this approach allows
for more general cyclical dynamics when compared with our model, it also presents additional
drawbacks. For instance, it requires the detrending of the series prior to dynamic factor analysis
and one cannot use simultaneously series of different periodicity. Moreover, with the estimation
of only two parameters for each series, the common cycle factor load and the shift, we are able
24
to get a quite similar outcome.
0.015
(i)
0.010
0.005
0.000
−0.005
(ii)
1990
1995
2000
1990
1995
2000
0.015
0.010
0.005
0.000
−0.005
Figure 7: Comparisons of (i) Euro area quarterly GDP growth (dotted line) versus our growth
indicator (solid line) and (ii) EuroCOIN indicator (dotted line) versus our growth indicator
(solid line).
4
Conclusion
In this paper we propose a multivariate approximate band-pass filter for obtaining a business
cycle indicator. The filter is based on a multivariate unobserved components time series model
that incorporates two recent and relevant contributions: (i) the inclusion of a particular cycle
component that has properties similar to those of band-pass filtered series, see Harvey and
Trimbur (2003); (ii) the incorporation of phase shifts within the cycle component, see Rünstler
(2004). The resulting multivariate filter is used to extract trend and cycle components using
several economic time series for the Euro area. The innovations to the cycle component of each
time series are common but their contribution is derived from a model-based filter that depends
25
only on three parameters. One parameter is the frequency of the cycle, which we fix so as to
assure that the typical business cycle frequencies are captured. Another parameter accounts
for phase shifts while the third parameter accounts for the relative contribution of the cycle
to the dynamics of the individual time series. The common cycle component resumes to our
business cycle indicator. These generalisations lead to a smooth business cycle indicator and
the suggested procedure can be seen as a multivariate band-pass filter. Besides the multivariate
setting, we are not constrained to the use of only coincident variables to identify the business
cycle, as in Stock and Watson (1989), since we account simultaneously for phase shifts between
the cyclical components of the series. Thus, one can also exploit the information contained in
both leading and lagging variables and this allows for a richer information set when obtaining the
business cycle. This multivariate approach is shown to provide business cycle estimates that are
less subject to revisions than the ones resulting from the commonly used filters. This feature
represents a valuable improvement for policymaking since reliable estimates of the current
cyclical position are required in real-time when policy decisions are made. Moreover, we are
also able to reconcile information available at different frequencies. It becomes possible to obtain
a high frequency business cycle indicator without discarding series such as GDP despite of its
quarterly frequency as in Stock and Watson (1989). The use of quarterly GDP together with
monthly indicators can be dealt with straightforwardly by exploiting the state space framework
in a consistent manner. Furthermore, the ability to deal with missing obervations also allows
timely business cycle estimates regardless of data release lags that can be substantial in the
Euro area. This feature is also crucial for the policymaker since it permits an evaluation of the
current state of the economy without delay. The new approach is applied to Euro area data
and the resulting business cycle indicator appears to be in line with common wisdom regarding
Euro area economic developments. As a by-product, one can also obtain a growth indicator.
In particular, our more parsimonious approach leads to a growth indicator for the Euro area
similar to EuroCOIN. The latter is based on a more involved approach by any standard and
uses hundreds of time series from individual countries belonging to the Euro area. Furthermore,
the suggested procedure also overcomes some of its drawbacks.
26
Appendix A: Phase shifts for generalised cycle
The cycle shifts are modeled as shifts in the cross autocovariance function for different cycles. In
case of the common cycle component model (1), (3) and (4), the cross autocovariance function
for the cycle component is given by
γij (τ ) = E(ψit ψj,t−τ )
= σκ2 φτ (1 − φ2 )−1 δi δj cos{λ(τ + ξi − ξj )},
(10)
i, j = 1, . . . , N,
for τ = 0, 1, . . . and where
ψit = δi cos(ξiλ)ψt + sin(ξi λ)ψt+ ,
t = 1, . . . , n,
where ψt is modeled by (4). This result is a direct consequence of the general results given in
Rünstler (2004, Proposition 1). The contemporaneous correlation between elements i and j in
p
ψt is γij (0)/ γii (0)γjj (0) = cos{λ(ξi − ξj )}. However, when cycles are shifted accordingly, the
correlation is cos(0) = 1 as expected from a common cycle specification. It is important to
realise that a real-time shift ξi is rarely estimated as an integer time point and since observed
time series data can only be shifted at discrete time intervals the contemporaneous crosscorrelation of cycles associated with shifted time series will usually be smaller than one.
The autocovariance function of the generalised cycle (8) is reported as
γ(τ ) = σκ2 φτ cos(λτ )g(φ, τ ),
(11)
where
g(φ, τ ) =
τ
X
i=k−τ

(1 − φ2 )1−i−k 

τ
k−i

i−1
X
j=0

φ2j 
i−1
j


k−1
j +k−i

,
for a k-th order cycle and for τ = 0, 1, 2, . . .; see Trimbur (2002, Proof of Proposition 2). The
cross autocovariance function of the generalised shifted cycle component is then given by
γij (τ ) = σκ2 φτ δi δj cos{λ(τ + ξi − ξj )}g(φ, τ ),
(12)
for τ = 0, 1, 2, . . .. This result follows straightforwardly by applying Proposition 1 of Rünstler
(2004) to the proof of Proposition 2 in Trimbur (2002). Since the autocovariance function for
27
generalised cycles also applies to negative discrete values for τ and it is symmetric in τ = 0,
the shift mechanism has the desired effect on the autocovariance function of the shifted cycle.
It follows that the shift mechanism of Rünstler (2004) can also be applied to correlated
generalised cycles whether the correlation is perfect or not. This means that cycles with bandpass filter properties can be subject to shifts within a multivariate cycle model based on the
generalised cycle component (8). In the case of model (9), the properties of the cycle component
for the first series in the panel will be closely related to those of the univariate band-pass filtered
series. However, the information from the other series is also exploited in the estimation of the
cycle component.
Appendix B: State space representation of model
A state space representation consists of two equations, the observation equation for yt and the
transition equation that describes the dynamic process through the state vector. The model
(9) requires the following state vector,

αt = 
(j)
(j)
αtµ
αtψ
(j)

,




µ
αt = 



(m)
µt
(m−1)
µt
..
.
(1)
µt




,










ψ
αt = 






(k)
ψt
(k)+
ψt
(k−1)
ψt
(k−1)+
ψt
..
.
(1)+
ψt







,






(j)
with µt = (µ1t , . . . , µN t )′ and µit as in (7) for i = 1, . . . , N, j = 1, . . . , m and t = 1, . . . , n.
The observation equation is a linear combination of the state together with observation noise,
that is
yit =
n
ei 01×(m−1)N δi cos(λξi) δi sin(λξi ) 01×2(k−1)
o
αt + εit ,
i = 1, . . . , N,
where ei is the i-th row of the N × N identity matrix IN and 0p×q is the p × q matrix of zeros.
Define Ap as a p × p matrix with zero elements except for elements (i, i + 1) that are equal
28
to unity for i = 1, . . . , p − 1. The transition equation for the state vector is then given by


µ
αt+1
 = (diag [Im ⊗ IN Ik ⊗ Tλ ] + diag [Am ⊗ IN Ak ⊗ I2 ]) αt + ηt ,
αt+1 = 
ψ
αt+1
where ηt′ = (01×(m−1)N , ζ1t . . . ζN,t , 01×2(k−1) , κt , κ+
t ),


h
i
cos λ sin λ
,
Tλ = 
Var(ηt ) = diag 0(m−1)N ×(m−1)N Σζ 02(k−1)×2(k−1) σκ2 I2 ,
− sin λ cos λ
2
and Σζ is a diagonal N × N matrix with σi,ζ
of (6) as its i-th diagonal element. We note that
diag [· ·] refers to a block diagonal matrix. The initial state vector α1µ will be treated as a
partially diffuse vector and appropriate amendments to the Kalman filter and related algorithms
need to be made, see Durbin and Koopman (2001, Chapter 4) for a discussion. The stochastic
properties of the initial state vector α1ψ are implied by mean zero and variance matrix P which
is the solution of
P = [Ik ⊗ Tλ + Ak ⊗ I2 ]P [Ik ⊗ Tλ + Ak ⊗ I2 ]′ + diag 02(k−1)×2(k−1)
σκ2 I2 .
We note that the solution of A = BAB ′ + C is vec(A) = (I − B ⊗ B)−1 vec(C), see Magnus and
Neudecker (1988, Theorem 2, p.30). Explicit formulations for P are in Trimbur (2002).
The multivariate state space model under consideration is linear and Gaussian. Standard
Kalman filter and smoothing techniques can therefore be employed for maximum likelihood estimation and signal extraction. The computational effort for a multivariate model can be severe
when the dimension of the observation vector N is relatively large. However, the computational
load can be reduced considerably when the elements of the observation vector yt are treated
element by element when filtering and smoothing updates are carried out at each time point t.
Since the irregular variance matrix is diagonal, we can adopt this approach straightforwardly,
see Durbin and Koopman (2001, Section 6.4). The actual reduction of computing time in our
case has been in the neighbourhood of 50%.
All computations for this paper are done using the object-oriented matrix language Ox of
Doornik (2001) together with the state space functions in version 3 of SsfPack as developed by
Koopman, Shephard, and Doornik (1999) (for later versions, see http://www.ssfpack.com).
29
Appendix C: Data details
As far as possible, we used series from Euro area official statistical sources, i.e., Eurostat, the
European Commission and the European Central Bank. However, the available data regarding
the Euro area as a whole is rather limited, in particular, in terms of time span. Therefore, in
some cases, we had to chain the official time series with an Euro area aggregate proxy, obtained
by weighting country level data with 1995 (PPP) GDP weights, which were rescaled whenever
there was any missing data for a country. The sample runs from the beginning of 1986 up to the
end of 2002. Regarding real GDP, the seasonally adjusted series provided by the Eurostat only
starts in the first quarter of 1991. Therefore, from 1991 backwards we used the series provided
by Fagan, Henry, and Mestre (2001) to obtain a quarterly series for the above mentioned
sample period. All confidence indicators are seasonally adjusted and provided by the European
Commission. The industrial production volume index is seasonally adjusted and provided by
the Eurostat. The unemployment level series is also seasonally adjusted and provided by the
Eurostat. Since there is no official series before June 1991, an aggregate series was obtained, in
this particular case, by summing country level data. The retail sales volume index is seasonally
adjusted and provided by the Eurostat since 1996. To obtain a longer time span series, we
resorted to country level data provided by the OECD. Regarding nominal interest rates, since
both short and long-term interest rates are made available by the ECB only for the period after
January 1994, we had also to use OECD country level data. The nominal short-term interest
rate refers to the 3-month deposit money market rate and the nominal long-term interest rate
refers to the 10-year government bond yield. The yield curve spread is given by the difference
between the long and the short-term interest rates. Excluding confidence indicators and interest
rate spread, all data are in logarithms.
30
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