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E U R O P E A NC O M M I S S I O N
An assessment of
multivariate output gap estimates in the Euro area
20
04
ED
ITIO
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4TH EUROSTAT AND DG ECFIN COLLOQUIUM ON MODERN TOOLS FOR BUSINESS CYCLE ANALYSIS
"GROWTH AND CYCLE IN THE EURO-ZONE"
20 TO 22 OCTOBER 2003
Luxembourg, European Parliament Hémicycle, Schuman building
An assessment of multivariate output gap estimates in the Euro area
Odile Chagny, Observatoire Français de la Conjoncture Economique Matthieu Lemoine, Observatoire Français de la Conjoncture Economique
Florian Pelgrin, Bank of Canada
Aa assessment of multivariate output gap estimates in theEuro Area1
Odile CHAGNY
O.F.C.E
Matthieu LEMOINE
O.F.C.E
PELGRIN Florian
Bank of Canada, EUREQua, Universite de Paris I and O.F.C.E
Abstract
This paper assesses the statistical reliability of different measures of the out-
put gap - the multivariate Hodrick-Prescott Filter, the multivariate unobserved
components method and the structural vector autoregressive model - in the Euro
area. Three criteria are used: the consistency of descriptive statistics, the fore-
casting performance in terms of inflation and some measures of uncertainty. Our
results indicate that these multivariate methods provide some useful information
relative to standard univariate models.
JEL Classification: C32, E32
Keywords: Output gap estimates; Times series models
1The views in this paper are those of the authors and not necessarily those of the Bank ofCanada.
1
1. Introduction
The concepts of potential GDP and output gap are widely used in macroeco-
nomics even though their definition and estimation raise a number of theoretical
and empirical questions, which reflect the ongoing controversy over the origins
of economic fluctuations. Indeed, since the seminal contribution of Nelson and
Plosser (1982) suggesting that output series are better described as integrated
stochastic processes, measuring the permanent component of output, e.g. poten-
tial output, with any degree of accuracy has proved to be difficult. Recent research
has outlined many problems associated with univariate methods. Specifically, the
question of the reliability (and accuracy) of outgap estimates has been extensively
studied in the literature (Orphanides and Van Norden, 2003). Also, for example,
recent research on optimal monetary policy under incomplete information has
stressed the importance of the output gap measure and its revisions (Orphanides,
1999, 2003). However, most of these studies focus on univariate methodology.
In contrast, this paper consider different multivariate methodologies. We apply
these methods to estimate output gaps in the Euro area and assess the reliabil-
ity of these measures using different criteria - descriptive statistics, forecasting
performances and uncertainty measures.
An output gap is defined as the difference between - unobservable - potential
and actual GDP. The precise understanding of the output gap concept therefore
requires a precise definition of potential output. Potential output is commonly
defined as the ”maximum output an economy can sustain without generating a rise
in inflation” (De Masi, 1997) or equivalently as the level of the output consistent
with a stable inflation rate given the productivity shock of capital. Hence, a level of
real output above potential output, i.e. a sustained positive output gap, indicates
demand pressures and signals to the monetary authority that inflation pressures
are increasing and that policy tightening may be required. This implies, in turn,
that an accurate measure of potential output presents an important challenge to
policy makers. In this respect, since the contribution of Okun (1962), numerous
methodologies have been proposed in the literature.
One approach consists of using non-structural univariate methods, which sep-
arates a time series into a permanent and cyclical component. In this context,
one may distinguish between deterministic detrending, mechanical filters and un-
2
observed components models. A first approach includes standard linear, phase
average or robust detrending estimations of potential output. Such methods are
obviously overly simplistic and do not capture the overall data generating process.
A second methodology uses mechanical filters such as the Hodrick-Prescott (HP)
filter or the band-pass filter (BK) developed by Baxter and King (1995). Never-
theless, the limits of such filters are now well-known. Indeed mechanical filters
do not accurately decompose time series into their trend and cyclical compo-
nents when the spectrum or pseudo-spectrum of the data has a Granger’s typical
shape (Guay and St-Amant, 1996). In addition, as is noted by Cogley and Nason
(1995), spurious cyclicality may be induced with nearly integrated processes in
the case of the HP filter. Finally, Baxter and King (1995) pointed out that both
filters show instability of estimates at the end and beginning of sample periods,
which limits the interest of policymakers (Van Norden, 1995) A third method-
ology assumes that output contains an unobserved component and a temporary
component consisting of a random walk with drift and a stationary autoregres-
sive process. However, Quah (1992) has shown that ”without additional ad hoc
restrictions those (univariate) characterisations are completely uninformative for
the relative importance of the underlying permanent and transitory components”.
Another strategy for identifying the potential output involves non-statistical struc-
tural methods. In this case, the detrending method relies on a specific economic
theory. For instance, the production function-based approach is often used in the
literature. This has the advantage of identifying explicitly the sources of growth
(labour, capital, intermediate inputs) but raises a number of issues, such as the
choice of an appropriate production function or the measurement of unobservable
variables (total factor productivity).
Partly in response to criticisms of the previous methodologies, a variety of
multivariate methodologies have been proposed as an alternative. Three different
methods are considered here. First, the multivariate Hodrick-Prescott (HPMV)
filter is a middle ground between univariate statistical methods and full, simul-
taneous estimations of potential output. It consists merely of using additional
economic relationships and/or terminal constraints in the HP maximisation pro-
gram. Second, the multivariate unobserved components (UCM) method estimates
the unobserved potential output level using information from observable variables
and other latent variables. The explicit relationships are written in a state-space
3
form, that is, as a dynamic system where the observed variables are specificied
as a function of the unobserved state variables and/or exogenous variables in the
measurement equation and separate transitions equations specify the stochastic
processes for the state variables. Within this context, unobserved state vector,
including the potential output, can be estimated with a Kalman filter. Third, the
structural vector autoregressive (SVAR) methodology assumes that the dynamics
of macroeconomic variables are defined by an equivalent number of shocks, as for
instance supply, demand or nominal disturbances. Estimating a reduced-form and
imposing some appropriate restrictions on the long-run variance-covariance ma-
trix based on economic theory (Shapiro and Watson, 1988; Blanchard and Quah,
1989) lead to the definition of potential output as the sum of the deterministic
and permanent components.
In this respect, the focus of this paper is to estimate the output gap using
multivariate detrending methods and to assess the statistitical reliability of these
measures in the Euro area. Therefore, our approach is complementary to the
methodology proposed by Camba-Mendez and Rodriguez-Palenzuela (2003) and
to a less extent of the works of Fabiani and Mestre (2002) and Runstler (2002).
Specifically, Camba-Mendez and Rodriguez-Palenzuela (2003) propose three cri-
teria for assessing output gap estimates based on the multivariate unobserved
component model, the multivariate Beveridge and Nelson decompositon and the
SVAR models: forecasting power, temporal consistency of the initial estimates
and their subsequent revisions and the statistical correlation with known cyclical
indicators. In contrast, we propose to estimate also the multivariate HP filter and
to use some additional discriminating criteria.
The rest of the paper is organised as follows. The second section describes
the multivariate methods: multivariate HP filter, multivariate unobserved compo-
nents models and SVAR. Section 3 discusses the respective advantages/disadvantages
of these three multivariate detrending methods. Section 4 describes the techni-
cal details of the methodology followed to compare and evaluate the different
measures. Section 5 presents the results and section 6 concludes.
4
2. Multivariate methods of estimating the out-
put gaps
The model we implement build on different variable vector series x1t = (yt,πt)0,
x2t = (yt,πt, ut)0, x3t = (yt,πt, ut, cut)
0; where yt denotes the log of real GDP, πt
is the inflation rate defined as the first difference of the consumer price deflator,
ut is the unemployment rate and cut is the capicity utilisation rate. In order
to model xt, three alternative multivariate time series techniques are used: the
multivariate Hodrick-Prescott filter (HPMV), the unobserved component (UC)
model and structural vector autoregressive (VAR) models.
2.1 The multivariate Hodrick-Prescott Method
This method stems from the use of the standard HP filter augmented by relevant
economic information (Laxton and Tetlow, 1992).2 Indeed, the HPMV filter seeks
to estimate the unobserved variable as the solution to the following minimisation
problem3:
(τ y,t)1..T = argmin (PT
t=1 (yt − τ y,t)2
+λPT−1
t=2 [(τ y,t+1 − τ y,t)− (τ y,t − τ y,t−1)]2
+Pn
i=1
PTt=1 µ
i(εit)2)
(1)
where εit is the residual from the ith regression and the smoothing constants, λ
and (µi) , reflect the weights attached to the different elements of the minimisation
problem.
The following additional equations are considered4:
2See annex 1 for a survey of HPMV studies.3It is to be noted that some terminal conditions can be added in order to improve the end-of-
sample performances of the filter. Butler (1996) proposed a steady-state condition regarding theend-of-sample estimates. To this end, an additional term penalises the variations of the trendcomponent or the difference between the growth rate of the trend and a steady-state growth rate.Butler (1996) also proposed to use a recursive updating restriction that penalises the deviationbetween the trend and its previous quarter’s estimates. Therefore, this restriction intends toprevent large revisions to the level of the trend due to new data.
4These relations have been already used in Laxton and Tetlow (1992), Conway and Hunt(1997), De Brower, Haltmaier (1996) and Cote and Hostland (1993).
5
πt = α(L)πt−1 + απ(L) (yt − τ y,t) + γ(L)st + επ,t (2)
ut − u∗t = αu(L) (yt − τ y,t) + εu,t, (3)
uct − uc∗t = αuc(L) (yt − τ y,t) + εuc,t, (4)
where s denotes the relative import price growth rate and αk(L) are lagged
polynomial.
Equation (2) is the well-known Phillips curve, in which απ(1) is constrained
to be equal to one (non-accelerating inflation). Equation (3) draws on Okun’s
law, with the unemployment rate below the non-accelerating inflation rate of
unemployment (NAIRU) when output is below potential. Equation (4) states
that the capicity utilisation rate is above trend when output is above potential
(Conway and Hunt, 1997).
In this respect, the estimated unobserved variable is not only a simple moving
average going through the observed series but it is also modelled to fit better to
economic relationships.
Four alternative models are implemented: model HP is the standard Hodrick-
Prescott filter which can be written as a special case of the HPMV filter. Model
HPMV1 only incorporates a Phillips curve (Eq.2). Model HPMV2 takes into
account the Okun’s law (Eq.3) and the Phillips curve. Model HMPV3 considers
the three additional equations (Eq. 2 to 4).
The minimisation problem can be rewritten as a state-space model5 (Boone,
2000). For example, model HPMV3 can be represented by the following state-
space system6:
5For a comprehensive presentation of state-space models, see Harvey (1989), Durbin andKoopman (2001).
6See annex 2 for a survey of estimation of state-space models.
6
(Yt = AYt−1 +BX
1t + Ut
Zt = CYt +DX2t + Vt
(state equation)
(measurement equation)(5)
where U ∼ NID(0, Q), V ∼ NID(0, R), Cov(U, V ) = 0, Yt = (y∗t , y∗t−1, yt −
τ y,t, yt−1− τ y,t−1, u∗t , u
∗t−1, uc
∗t )0 is the state vector, Zt = (yt,πt, ut, uct)
0 is the mea-
surement vector andX2t = (πt−1,πt−2,πt−3, st−1, st−2)
0 is a vector of predetermined
variables.
The matrices are given by
A =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
2 −1 0 0 0 0 0
1 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 0 2 −1 0
0 0 0 0 1 0 0
0 0 0 0 0 0 1
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, B =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
0
0
0
0
0
0
tuc
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,
C =
⎡⎢⎢⎢⎢⎣1 0 1 0 0 0
0 0 0 απ 0 0
0 0 0 αu 1 0
0 0 αuc 0 0 1
⎤⎥⎥⎥⎥⎦ ,D =⎡⎢⎢⎢⎢⎣0 0 0 0 0
α1 α2 α3 γ1 γ2
0 0 0 0 0
0 0 0 0 0
⎤⎥⎥⎥⎥⎦
Ut =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
ε∆∆y∗,t
0
εgap,t
0
ε∆∆u∗,t
0
εuc∗,t
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, Vt =
⎡⎢⎢⎢⎢⎣0
επ,t
εu,t
εuc,t
⎤⎥⎥⎥⎥⎦
(6)
and the weights in eq. 1 can be expressed as:
λ = σ2gap/σ2∆∆y∗, µπ = σ2gap/σ
2π, µu = σ2gap/σ
2u, µcap = σ2gap/σ
2cap,
7
There are two advantages to be gained from reproducing an HPMV filter with
the Kalman filter. On the one hand, the estimation is done in one step (while
the procedure proposed by Laxton and Tetlow (1992) is a multi-step procedure).
On the other hand, it allows the estimation of the hyper-parameters, that is, it is
possible to freely estimate the different weights without imposing some arbitrary
values. Apart the issues of state-space estimation, two implementation issues are
of particular importance. First, equations 2-4 incorporate unoberservable latent
variables, as for instance the NAIRU. A preliminary estimate of these variables is
thus necessary. However, it induce the so-called ”generated regressor” problem.
Moreover, it has been shown that reliable measures of these variables are difficult
to obtain and affect the other estimates (Stock and Watson, 1997; Orphanides
and Williams, 2003). Second, increasing the number of hyperparameters turns
to reduce the acccuracy of estimates. Therefore, the fact that the weights are
estimated allows comparing the values with the existing literature but it further
complicates the estimation procedure.
2.2 Unobserved component models
The multivariate UC model is specified as follows7:
yt = ty,t + cy,t + εy,t (7)
ut = tu,t + δ(L)cy,t + εu,t (8)
∆ti,t = µi,t + ηi,t, µi,t = µi,t−1 + ζi,t, with i = y or u
cy,t = ρ(cosλ.cy,t−1 + sinλ.c∗y,t−1) + κt
c∗y,t = ρ(− sinλ.cy,t−1 + cosλ.c∗y,t−1) + κ∗t (9)
πt = πet + βcy,t−1 + γSt + vt, with πet = α(L)πt−i (10)
Equation (7) is the decomposition of yt into a trend ty,t, a cycle cy,t. and an
irregular component εy,t. Equation (8) states that the unemployment rate ut can
be integrated with a specific trend and a common cycle. Trends are assumed to
be I(2), although they can be restricted as random walks or smooth trends in par-
ticular cases. Following the decomposition proposed in Harvey (1989), the output
7See annex 3 for a survey of recent multivariate UC models.
8
gap cy,t (8) is modelled, with a dual variable c∗y,t as an AR(1) with constrained
parameters, ρ and λ. In other words, the model is equivalent to a constrained
ARMA(2,1), which can produce some cyclical oscillations8. Equation (8) is the
standard Phillips curve. Finally, εi,t, ηi,t, ζ i,t, κt and vt are uncorrelated i.i.d. white
noise processes with standard deviations σε,ση,σζ and σκ .
The major difference between the standard HPMV filter presented above and
the multivariate UC model is that there is an explicit modelling of the fluctuations
around the trend. Specifically, as is argued by Durbin and Koopman (2001), the
key advantage of unobserved components models and the underlying state-space
approach is that individual peaces like the trend, cycle and possible exogenous and
endogenous explanatory variables can be modeled seperately and subsequently
combined in the state-space model. In this paper, we use the local linear trends
specifications. The choice of this specification is mainly motivated by the robust-
ness of the results and the fact that Camba-Mendez and Rodriguez-Palenzuela
(2003) show that, among the unobervable components models, the best performer
in terms of forecasting corresponds to the one which specifies the trends as local
linear trends. However, imposing the restriction σζ = 0 (respectively ση = 0), the
trends simplify to random walks with drifts (respectively smooth trend compo-
nents).
Three models are considered: model UC is the standard univariate unoberved
component model. Model UC1 incorporates the dynamics of inflation while model
UC2 takes into account the inflation and unemployment dynamics.
The model can also be written in state-space form9:
(Yt+1 = A.SVt + Ut
Zt = B.Yt + C.X2t + Vt
(11)
where Yt = (ty,t, µi,t, tu,t, µu,t, cy,t, c∗y,t)
0, Zt = (yt, ut,πt)0 ,X2
t = (st,πt−1)0
8The autocorrelation function exhibits the cut-off at lag one characteristic of the MA(1)process. The reduced form is therefore ARMA(2,1).
9The model is observable unless ρ is zero or λ is zero or π. However, it is always detectable.Controllability and stabilisality require that σ2κ and σ2ζ be strictly positive.
9
and
A =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
1 1 0 0 0 0
0 1 0 0 0 0
0 0 1 1 0 0
0 0 0 1 0 0
0 0 0 0 ρ cosλ ρ sinλ
0 0 0 0 −ρ sinλ ρ cosλ
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(12)
B =
⎡⎢⎣ 1 0 0 0 1 0
0 0 1 0 δ 0
0 0 0 0 β 0
⎤⎥⎦ , C =⎡⎢⎣ 0 0 0 0 0
0 0 0 0 0
γ1 γ2 α1 α2 α3
⎤⎥⎦ .We also assume that the trend component and the cycle component are uncor-
related. This assumption has been discussed in recent papers by Morley, Nelson
and Zivott (2002), Proietti (2002).
2.3 Structural vector autoregressive models
Structural VAR models (Blanchard and Quah, 1989) are also implemented to
extract the trend and cycles components. Given the vector series of non-stationary
variables xt described above, we consider the following reduced form (using the
Wold decomposition):
∆xt = δ(t) + C(L)εt (13)
where δ(t) is deterministic, C(L) is a matrix of polynomial lags, C0 = I is the
identity matrix and the vector εt is the one-step ahead forecast errors in xt given
information on lagged values of xt.10
The SVAR approach assumes that xt has the following structural representa-
tion:
∆xt = δ(t) + T (L)ηt (14)
10Non-fundamental representations are ruled-out by assuming that the determinantal polyno-mial |C(L)| has all its roots on or outside the unit circle (Lippi and Reichlin, 1993). See annex4 for a discussion of non-fundamental representations.
10
where ηt is the vector of structural shocks, E[ηt] = 0, and E[ηtη0t] = In.
Therefore, the structural form equation (14) can be retrieved from the esti-
mated reduced form by using the relationships
T0T00 = Ω, εt = T0ηt and C(L) = T (L)T
−10 . (15)
The long-run covariance matrix of the reduced form is equal to C(1)ΩC(1)0.
From equations 13-14, the following conditions can be derived:
C(1)ΩC(1)0 = T (1)T (1)0. (16)
This relation suggests that the matrix T0 can be identify with the appropriate
number of restrictions on the long-run covariance matrix of the structural form.11
Using the notation of vector xt, the log of real output is then equal to
∆yt = µy + Tp1 (L)η
pt + T
c1 (L)η
ct (17)
where ηpt is the vector of permanent shocks affecting output, ηct is the vector
of shocks having only transitory effects on output, and (T p1 (L), Tc1 (L)) represent
the dynamics of these shocks.
Potential output growth based on the SVAR method can then be defined as:
∆yt = µy + Tp1 (L)η
pt (18)
Thus, ”potential output” corresponds to the permanent component of output
and the ”output gap” is the part due to purely transitory shocks.
For our application of the SVAR methodology to Euro-area data, we assume
that the growth rate of real output follows a stationary stochastic process respond-
ing to two types of structural shocks: permanent (εp) and transitory (εt). Also
included in the estimated VARs are either the first differences of: inflation and the
unemployment rate (model SVAR2) or the first difference of the unemployment
rate (model SVAR1). We assume that these series are stationary and that there
is no cointegration involved.12
11See annex 5 for the different representations of VAR models.12Adding the exchange rate to the estimated VARs would have little impact on the results.
11
To identify the models, we introduce the following long-term relation between
variables in ∆xt and the vector of structural shocks in models SVAR2 and SVAR1
respectively: "∆y
∆u
#=
"T11(1) 0
T12(1) T22(1)
#"εs
εd
#. (19)
and ⎡⎢⎣ ∆π
∆y
∆u
⎤⎥⎦ =⎡⎢⎣ T11(1) 0 0
T12(1) T22(1) 0
T13(1) T23(1) T13(1)
⎤⎥⎦⎡⎢⎣ εm
εs
εd
⎤⎥⎦ (20)
The first scheme represents the standard Blanchard-Quah long term restric-
tions, i.e. the demand shock does not affect output in the long run while the supply
shock affects in the long run both output and unemployment. The second scheme
of identification is in line with the work of Bullard and Keating (1995) and is used
by Camba-Mendez and Rodriguez-Palenzuela (2003).13 Inflation is determined
in the long run by one structural shock labelled inflation shock. The output is
affected by two structural shocks in the long term: the inflation shock and produc-
tivity or technology shock. Finally, the last shock, which affects unemployment
but not inflation and output, is labelled a demand shock. This representation has
the advantage that is does not require the detrending of unemployment but has
the disadvantages that it introdiuces hysterisis in unemployment dynamics and
that it may depend on the ordering of the variables.
3. A General Assessment of Multivariate De-
trending Methods
This section discusses three main issues about these multivariate detrending meth-
ods : the economic content of the methods, the main assumptions concerning the
trend and the cycle processes, the implementation problems.
13Camba-Mendez and Rodriguez-Palenzuela also use a multivariate Beveridge-Nelson decom-position. See annex 6 for a description of related SVAR methods.
12
3.1 Economic content of the methods
All these methodologies rely on the assumption that adding economic content
can improve the trend-cycle decomposition, through a better disentanglement of
supply and demand shocks. This is, however, achieved in quite different means.
The HPMV and the UCM methodologies do not assume an economic a priori
about the nature of the shocks underlying the trend or the cycle. Trend and cycle
are then assumed to follow some specific stochastic processes and the economic
relationships play merely the role of additional information. But the choice of
these economic relationships is directly related to an explicit representation of
excess supply or excess demand in specific markets, which can be helpful for the
interpretation of business cycle movements.
Regarding specification issues, the SVAR methodology has advantages over
other popular multivariate methods such as filter-based methods, the Beveridge-
Nelson decomposition or the state-space representation proposed by Kuttner (1994).
First, unlike these alternatives methods, the components of output that the SVAR
identifies can be given an economic interpretation. For example, fluctuations in
potential output may be caused by certain types of shocks even with a degree of
uncertainty. Second, the SVAR methodology does not impose undue restrictions
on the short-run dynamics of the permanent component of output, i.e. the short-
run dynamics are taken into account. Specifically, the estimated potential output
is allow to differ from a strict random walk. At the same time, the specification
of SVAR models has some drawbacks. On the one hand, one needs to identify
at most only as many types of shocks as there are variables. This is often hard
to model in the case of larger VAR. In addition, the SVAR methodology assumes
that the orthogonality constructed exogenous innovations correspond to pure un-
correlated different shocks, as for instance supply or demand shocks. However,
many types of shocks may have some varying supply or demand characteristics.
Accordingly, composite pure shocks are often difficult to relate to some specific
economic variables.
13
3.2 Main assumptions concerning the trend and cycle pro-
cesses
One common assumption in HPM and multivariate unobserved components meth-
ods is that the trend component in output can be characterised as an integrated
random walk - an assumption that is not maintained in the SVAR approach14.
This hypothesis has been mainly discussed in the literature. Specifically, what is
the order of integration of the output? Do standard tests have enough power?
and do the trend innovations really not have any auto-correlation ?
The first issue has been addressed by Harvey & Jaeger (1993): because of their
low power rates, DF tests can not distinguish a second unit root in the presence of
MA dynamics, which is the case for multivariate UC models, allowing to choose
an I(2) trend. For pure auto-regressive models like SVAR, a single unit root
seems better adapted, although the auto-correlation of innovations also allows for
a smooth trend under certain conditions on the spectral density, as shown by
Quah (1992). Taking into account possible instantaneous structural breaks, we
might even assume that output is trend I(0) by detecting these breaks with the
Bai-Perron methodology. In conclusion, the integration order we have to assume
depends on the retained modelling framework.
Concerning the innovation process, it may appear difficult to reconcile the
white noise hypothesis on trend innovations with the widely held view that the
permanent component of output is at least in part driven by technological inno-
vations. As is noted by King et al. (1991), ”productivity shocks set off transi-
tional dynamics, as capital is accumulated and the economy moves towards a new
steady-state”. In addition, Lippi and Reichlin (1994) argue that such an hypoth-
esis is inconsistent with the dynamics of productivity shocks when one introduces
learning-by-doing or time-to-build constraints. The SVAR methodology does not
impose undue restrictions, that is, the estimated potential output is allowed to
differ from a strict random walk, having auto-correlated innovations. UCM and
HPMV also differ from a random walk, as their trend are twice integrated.
A more striking difference concerns the nature of the cycle. The HPMV speci-
fication of the output gap is quite constrained compared to the one given by UCM
or SVAR models. Indeed, the HPMV gap is defined only as a medium frequency
14It should be noted that this assumption is also present in the univariate context.
14
component15 of the output, which follows some economic relations with other eco-
nomic variables, e.g. the gap can be related to inflation by a Phillips curve etc.
In contrast, UCM models provide more flexibility in order to specify the output
gap when one may estimate persistence or frequency parameters. Multivariate
UC models also allow adding some exogenous variables like real interest rates
in the definition of the cycle or even considering a correlation between supply
and demand shocks. For example, Morley et al. (2002) estimated a correlation
between supply and demand shocks, which is particularly interesting for taking
into account a possible transmission mechanism of productivity shocks from the
trend to the cycle. Although SVAR models can also incorporate most of these
properties of the output gap, they can not identify this correlation because of the
assumption of orthogonality between supply and demand shocks.
3.3 Estimation issues
Finally, the different methodologies give rise to important implementation issues.
Specifically, in the case of the HPMV filter or the UCM, the estimation requires
a prior setting of parameters and criteria to choose between different alternatives
are not straightforward. As the calibration issues become more accurate when
a great number of economic variables are integrated, a possible way to choose
between different models would be to arbitrate between a parsimony principle
and the precision of the estimation.
Focusing on the precision of the estimation, two sources of uncertainty around
the estimated output gap have been identified, the filter uncertainty and the
parameter uncertainty. The filter uncertainty can be significantly reduced adding
exogenous variables like supply shocks in the inflation equation.
In contrast to other multivariate methods, the SVAR approach requires nei-
ther the imposition of arbitrary smoothing parameters,as for instance the HPMV
method, nor the imposition of arbitrary calibrated smoothing parameters, as for
instance in the HPMV method or unoberved components models. In addition, the
output gap estimates are not subject to any end-of-sample biases and are insensi-
15Looking at the HP filter as a low-pass filter, the frequency threshold is defined with thesmoothing parameter λ. For example, in the case of quarterly data, a smoothing parameterequals to 1600 corresponds to a cycle with periods lower than 8 years.
15
tive to the initial guesses for the parameters. At the same time, inference of SVAR
models induces some problems. First, as we discussed earlier, it is crucial that the
estimated VAR includes a sufficient number of lags. Moreover, DeSerres and Guay
(1995) show that information-based criteria (Akaike or Schwarz) tend to select an
insufficient number of lags whereas Wald or likelihood ratio tests perform much
better. This means that the practitionner must take care of the retained lag struc-
ture. Second, Faust and Lepper (1997) noted that the long-run effect of shocks
would be imprecisely estimated in finite samples, leading to imprecise estimates of
other parameters in the model. In effect, results of SVAR model crucially depend
on the identification process and thus the choice of exclusion restrictions. Even if
these restrictions are based on theory, it gives rise to a number of issues. First, the
restrictions chosen may not be appropriate in all circumstances. For example, in
the Blanchard-Quah model, this is true when changes in the unemployment rate
do not provide good indications of the cyclical components in output. Second, as
is shown by Faust and Lepper (1997), there is no consistent test of the long-run re-
strictions. A by-product is that the estimates of the long-run restrictions are very
uncertain, i.e. the long-run restrictions transfer this uncertainty to all the coeffi-
cients of the impulse response function.16 In addition, as is noted by Cooley and
Dwyer (1998), the SVAR approach aims to provide robust inference by imposing
only weak theoretical restrictions. However, these restrictions can not be tested
as models are just-identified. Indeed, these exclusion restrictions only become
testabe when they allow an over-identified structure to be obtained.17 Third,
16For example, in the Blanchard-Quah model, the long-run restriction is based on the esti-mated long-run effects matrix bC. Therefore, the reliability of the resulting general conclusionsrests on the quality of the VAR estimate of C(1). At the same time, any test of H0 : c
d12,k = 0
has significance level greater than or equal to maximum power. In practice, this means that atest rejecting H0 when it is in fact true 5 per cent of the time will reject that same hypothesis atmost in no more than 5 per cent of cases when it is false. This result stems from the null measureof the spectrum at zero frequency, i.e. the convergence to normality of the point estimate is notuniform.17The main point is that when the underlying process is characterised by an infinite parametri-
sation, convergence of the estimates of the model’s parameters is not sufficient to guarantee theconvergence of some functions of those parameters. To resolve the problem, Faust and Lepper(1997) proposed two solutions. The first solution is to assume that the data generating processis a VAR with known maximum lag order, i.e. one imposes a priori that the true model isexactly a VAR with an order that it is small relative to the sample size. A second approachis to identify the model using short-run exclusion restrictions and then to use the long-horizonresponses as an informal diagnostic.
16
Faust and Leeper (1997) point out that reliable results depend on familiar idenfi-
cation problems inherent in models that aggregate across variables and/or across
time. These arguments can however be applied to a wide class of econometric
applications.
4. Criteria of assessment
The statistical reliability of the output gap measures is assessed by analysing
standard descriptive statistics, by the relative forecasting performances and some
measures of uncertainty.
4.1 Measures of forecasting performance
We test the forecasting performance of the different measures of the output gap
by the root mean squared error (RMSE) for multi-step ahead inflation forecasts.18
The RMSE for any forecast is the square root of the arithmetic average of the
squared differences between the actual inflation rate and the predicted inflation
rate over one-period for which simulated forecasts are constructed. As it is com-
mon in the literature, we use two reference models: a ”naive” random walk and
an AR(3) process.19 We compare the accuracy of the inflation forecasts to our
reference model by comparing the RMSE of the two sets of forecasts. Specifically,
we form the ratio (Theil statistics) of the RMSE for each model of output gap
to the RMSE for the naive model (or AR(3) model). A ratio greater than one
thus indicates that output gap model’s forecast is less accurate than the ”naive”
or AR(3) models.20
We use the test of predictive performance proposed by Diebold and Mariano
(1995) to introduce a formal statistical procedure. The procedure is designed to
test the null hypothesis of equality of expected forecast performance by considering
the mean of the difference of the RMSE of a pair of two competing models. Let
18Other criteria can be used, as for instance the mean absolute error. Results are robust toalternative measures of forecasting performances.19In the case of SVAR models, we use an AR(3) in first differences in order to be consistent.20It is to to be noted that the significance of the relative mean squared error (RRMSE) is not
directly tested. Indeed, the one-sided test H0 :RRMSE=1 againt H0 :RRMSE < 1 can be usedand the corresponding p-values measure type-I error associated with the test.
17
us suppose that a pair of h-step ahead forecast have produced forecasts errors
(e1t, e2t), t = 1, 2, ..., T and the RMSE is the specified function of the forecast
error. Then the null hypothesis of the Diebold-Mariano test is:
E[RMSE(e1t)−RMSE(e2t)] = 0.
Defining dt = RMSE(e1t)− RMSE(e2t), t = 1, 2, ..., T , the Diebold-Marianotest statistic is then:
SDM =hbV (d)i−1/2 d d→ N(0, 1)
where bV (d) ≈ T−1 ∙bγ0 + 2 h−1Pk=1
¡1− k
h
¢ bγk¸ and bγk = T−1 TPt=k+1
(dt − d)(dt−k −
d).21
However, as is noted by Diebold and Mariano (1995), simulation evidence sug-
gests that their test could be seriously over-sized in the case of two-steps ahead pre-
diction and the problem become more acute as the forecasting horizon increases.
In this respect, Harvey, Leybourne and Newbold (1997) proposed employing an
approximately unbiased estimator of the variance of d, which leads to a modified
Diebold-Mariano test statistic:
S∗ =
∙T + 1− 2h+ T−1h(h− 1)
T
¸SDM .
Harvey et al. (1997), Clark and McCracken (2001) show that this modified test
statistic performs better than the DM test statistic even if it still performs poorly
in finite samples (Clark and McCracken, 2001). They also show that the power
of the test is improved, when p-values are computed with a Student distribution.
21Diebold and Mariano (1995) initially proposed an estimate of V (d) that was not always pos-itive semi-definite. In this paper, a consistent estimate of the standard deviation is constructedfrom a weighted sum of the available sample autocovariances of the loss differential vector - thedifference between the squared forecast error of the models and that of the reference model.Chosen weights insure that the matrix V (d) is positive semi-definite (Newey and West, 1987).
18
4.2 Measures of uncertainty and consistency
The statistical reliability of the output gap is also assessed by distinguishing the
filter uncertainty and the parameter uncertainty in the HPMV method and the
multivariate UC models. As it is common in the literature, we use the decompo-
sition of Hamilton (1986):
E³Cyt,t(bθ)− Ct´2 = E ³Cyt,t(bθ)− Ct(θ)´2 + Σt,t(θ)
where Cyt,t(bθ) is the estimate of the output gap given (yt)1..T and the parameterestimate bθ.The left-hand side term is the parameter uncertainty, directly obtained from
the Kalman filter at each date t. The second term of the right-hand side expres-
sion is the filter uncertainty (ΣPt,t). It needs to be computed by a Monte-Carlo
simulation procedure as follows:
• Draw K parameters (θk)k=1..K from a normal distribution N(bθ, bσ2θ);• Compute the corresponding K cycles Ct(θk) with the Kalman filter;
• For each date t, the parameter uncertainty is defined by a mean square errorcriterion:
ΣPt,t =1
K
KXk=1
hCt(θk)− Ct(bθ)i2 .
In addition, we also compare the one-sided/filtered estimates and the two-
sided/smoothed estimates by computing the parameter and filter uncertainty of
smoothed estimates¡ΣT,t,Σ
PT,t
¢.
Finally, we compute estimates of the output gap using quasi-real time data
(Orphanides and Van Norden, 1999). Both sources of uncertainty are also charac-
terised. Orphanides and Van Norden (1999), Cayen and Van Norden (2002) pro-
posed a battery of descriptive statistics but no formal testing in order to evaluate
the estimates of the output gap using real time data, final data or quasi-real time
data. Specifically, Orphanides and Van Norden (2003) find large discrepancies
between the sequentially estimated measures of the output gap when compared
19
with final estimates. The disparities are explained by the unreliability of the mod-
els in estimating end of sample values and to a lower extent by data revisions.
However, Orphanides and Van Norden (2003) mainly compare univariate meth-
ods. Camba-Mendez and Rodriguez-Palenzuela (2003) assess the consistency of
output gap estimates by implementing the Pesaran and Timmermann (1992) test
of directional change and a Fisher test to compare the variances of recursively es-
timated output gap sequence and the finally estimated sequence. Camba-Mendez
and Rodriguez-Palenzuela (2003) show that the concern of reliability of estimates
may be to some extent overdone in the Euro-area. However, Van Norden (2003)
casts some doubts on their conclusion.
4.3 Correlation with capacity utilisation
Following Camba-Mendez and Rodriguez-Palenzuela (2003), an additional assess-
ment of reliability is to compute the correlation between indicators of economic
activity directly obtained from survey data and the estimated output gap series.
Specifically, we report the correlation between the alternative measures of output
gap and the capicity utilisation (OECD main economic indicators database). We
expect a positive correlation even if this measure should not be too overemphasised
sine it only represents fluctuations in the manufacturing sector.
5. Empirical results
5.1 Data and estimation procedure
Our sample extends from the first quarter of 1970 to the fourth quarter of 2002.
The sources and definitions of the variables are described in appendix 1.
In order to apply the Kalman filter and the maximum likelihood estimation
in the HPMV model and the multivariate UC model, we need to make some
assumptions on the initial values of the state variables and their matrix of variance-
covariance. Different methods have been proposed in the literature. Assuming
a diffuse prior is a standard procedure for the initialisation of integrated state
variables. In this case, high variances denote the degree of uncertainty on the
20
initial values of the state vector. On the other hand, the initial values of the state
variables can also be treated as a parameter in a first step and then replace by their
estimated values in a second step. Also, a pre-filtering technique can be used by
replacing the initial state values by the values generated by an Hodrick-Prescott
filter (Guarda, 2002). In the latter case, the steady-state solution is then derived
analytically and used to initialise the Kalman filter. In this paper, a diffuse prior
is used22 and the initial coefficients are computed by estimating separately each
additional equation. Overall, our results display some robustness to the choice of
the initialisation procedure.
Once the values of the parameters have been set, and given initial values of
the state variables, and their corresponding covariance matrix, optimal estimates
of the potential output, output gap, NAIRU based on the information available
at time t (refereed to as filtered estimates) and on the information available from
the full sample of observations to time T (referred to as smoothed estimates) are
obtained from the Kalman filter. More precisely, estimates of the parameters of
the model and the unobserved state variables can be obtained by maximising the
likelihood function:
LogL = −TN2log(2π)− 1
2
SXt=1
log |Ft|−1
2
SXt=1
v0tF−1t vt
using the Kalman filter, where N is the number of observed variables, T is
the sample size, v is the prediction error matrix and F is the mean square error
matrix for prediction errors.
To identify the cycle in UCM models, the level disturbance variance is fixed
equal to zero (σ2η = 0). Indeed, our empirical evidence shows persistent cycles -
the autoregressive coefficient ρ is higher than 0.9. Therefore, the cycle is nearly
non-stationary. In this respect, an identification problems occurs since two local
maxima may co-exist: the cyclical or level disturbance variance could be equal
to zero (σ2κ = 0 or σ2η = 0). Moreover, the period is fixed in models UCM and
UCM1 to the value obtained in the model UCM2 (P = 10.6 years). As is shown
by Bentoglio, Fayolle and Lemoine (2001), the GDP can be decomposed into a
22As is shown by Shepard and Harvey (1990), the diffuse initialisation of integrated statevariables allow to avoid the ”pile-up problem”, i.e. a bias of variance ratios estimates towardzero. Another possibility would have been to use the asympotically median unbiased estimator.
21
trend and two cycles, which have the same weights. The duration of these two
cycles are 3 and 10.6 years.23 In order to avoid the local maximum trap, the
period parameter needs to be constrained. The best value can be estimated in a
multivariate model, which incorporates unemployment (UCM2), as unemployment
does not fluctuate with the short cycle.
In order to obtain the SVAR output gap estimates, the autoregressive reduced
form is first estimated by OLS using a state-space representation (with the com-
panion matrix)24:
∆xt =kXi=1
Ci∆xt−i + εt
where k is the number of lags and εt is a vector of estimated residuals.
It is crucial that the estimated VARs included a sufficient number of lags.
Indeed, DeSerres and Guay (1995) show that imposing a too parsimonious lag
structure can significantly bias the estimation of the structural components in a
structural VAR. However, using the standard information criteria (SBC, Akaike,
BIC) to obtain the number of lags, our results are not sensitive to the choice of
the lags. At the same time, one important issue is the sample size. The finite
sample properties of information criteria are rather weak. In a second step, we
impose the long run restrictions presented in the second section as well as the
standard orthonormalisation of the variance-covariance matrix.
All models are recursively estimated from 1970Q1 to i=1996Q4...2002Q3. In
the case of SVAR models, the number of lags is selected recursively and recal-
culated at each stage of the out-of-sample forecasting process. Finally, out-of-
samples forecasts from 1 to 10 quarters ahead are computed over the period
1997Q1-2002Q4.
5.2 Descriptive statistics and estimations
5.2.1 HPMV filter
23In the first case, the cycle is driven by the inventories dynamics whereas the investmentdynamics explains the cycle in the second case.24See annex 7.
22
Table (1a) displays the descriptive statistics of the HMPV filter. First, the profile
of the output gap is very similar to the HP filter when only the Phillips curve is
used in the specification of the HPMV filter. It confirms earlier results in the lit-
erature and the difficulty to take into account inflation dynamics in the HP filter.
Second, the statistical properties of the different models vary with the information
considered. Therefore, the output gap describes much more fluctuations (i.e. the
standard deviation increases) when the Okun’s Law and the capicity utilisation
equations enter in the minimisation problem. Therefore, model HPMV2 (respec-
tively HPMV3) displays fluctuations of the output gap between -2.43 (-3.46) and
2.86 (2.74) and the standard deviation is equal to 1.32 (1.54) whereas the output
gap in HPMV1 fluctuates around its mean value (-0.03) with a standard deviation
of 1.03.
(Insert Table 1a)
Third, it is to be noted that the fluctuations of the output gap in the model
HPMV3 cycle are very close to the ones of the capicity utilisation cycle (see Figure
1).
(Insert Figure 1)
This result is already present in Proeitti, Musso and Westermann (2002) when
they consider a multivariate production function approach based on UC. It casts
some doubts on the usefulness of this equation in the HPMV filter and, more
generally, the usefulness of more sophisticated models if the estimated output gap
only reproduces the capicity utilisation cycle. In addition, the capicity utilisation
only incorporates partial information since only the manufacturing sector is taken
into consideration. Fourth, all models show more excess demand at the end of the
80s and more excess suply in the first half of the 90s than the HP filter except
for the model HPMV1. Fifth, the end-of-sample estimates differ substantially
when the models HPMV2, HPMV3 are used. In effect, one of the motivation of
the HPMV filter is to improve the performances at the end of samples of the HP
filter. Recall that the HP filter solves the trade-off between the size of deviations
from trend and the smoothness of that trend. In face of a transitory shock, the
23
trend is not changed so much since this implies raising the trend before the shock
and lowering it afterwards. However, the latter penalty is absent at the end of
the sample. It turns that the optimal trend will be more responsive to transitory
shocks at the end-of-sample than in mid-sample. Here, the two HPMV display
two different patterns. The model HPMV2 conducts to less excess demand than
the standard HP filter while the reverse is true for the model HPMV3.25
Table (2a) presents the estimates of the different equations and the relative
weight on the inflation dynamics. Coefficients have the expected signs and are
significant at standard levels.
(Insert Table 2a)
The estimate of the relative weight on the Phillips curve is close to the value
used in the literature.26 Interestingly, the relative weight is much larger when the
unemployment rate and the capacity utilisation dynamics are taken into consid-
eration in the derivation of the HPMV filter. At the same time, the mean square
error on the inflation dynamics does not improve significantly when adding eq. 3
or 4.
Overall, the results (descriptive statistics and estimations) indicate that the
use of additional economic relationships provides subtantial differences in the es-
25The end-of-sample performances of HPMV filter have been already adressed in the literature.Indeed, in order to to assess the impact of the addition of the economic relationships andof the additional constraints aimed at improving the end of sample properties of the filter,Conway and Hunt (1997) and Butler (1996) compare rolling estimates of the output gap withthe estimates obtained by running the filter over the full sample period. In both cases, revisionsin response to new data are much smaller. In the Conway-Hunt model, the average absolutedifference is 50 percent less between actual and HPMV rolling estimates are always better thanwith the univariate filter. Butler (1996) attributes these improvements both to the economicrelationships and to the additional constraints. An evaluation of the additional constraints ofthe filter proposed by Butler is also made by Saint Amant and van Norden (1997). They showthat the additional constraints cause the filter to behave at the end of the sample much morelike a two-sided filter, as much less weight is put on the last few observations. However, theynote that imposing an a priori judgement on potential output (as for the steady-state growthrate) depends on arbitrary choices and contradicts the stochastic nature of the trend in HPfilter. Moreover, they show that these additional constraints cause the filter to pass more onthe trend component (low frequencies) than other HP filters.
26Results are robust to different values of the weights.
24
timation of the output gap and the potential output. Specifically, results are sig-
nificantly different when both the inflation and unemployment dynamics (model
HPMV2) are considered in the HPMV filter since the HPMV3 results are ruled
out. However, one of main issue remains the estimation of one important latent
variable - the NAIRU. Here, the NAIRU is estimated as the result of an HP fil-
ter. But the NAIRU could be estimated with various other techniques (Stock
and Watson, 1997), which lead to different estimates. These NAIRU estimates
in turn affect the relative weight on each equation in the minimisation problem
(information content) and the estimates of the output gap.
5.2.2 Multivariate UC models
Table (1b) reports the results of the descriptive statistics of the Multivariate UC
models. First, the profile of the output gap looks very similar to the univariate
unobserved component model when only the Phillips curve is used in the state-
space specification. Results in Table (2b) suggest, however, that the standard
deviations of the trend are different while the charateristics of the cycle (am-
plitude and period) are similar. Second, the output gap describes much more
fluctuations (i.e. the standard deviation increases) when the Okun’s Law enter
in the minimisation problem. Therefore, model UC2 displays fluctuations of the
output gap between -2.74 and 3.61, and the standard deviation is equal to 1.77
whereas the output gap in UC1 fluctuates around its mean value (-0.04) with a
standard deviation of 1.44.
(Insert Table 1b)
Third, the two multivariate UC models display different implications in terms
of the pattern of the output gap. The model UC2 exhibits more excess demand
at the end of the 80s and more excess suply in the first half of the 90s than
the UC model. At the same time, the model UC1 implies less excess demand
at the end of the 80s and less excess suply in the first half of the 90s than the
UC model. These results are consistent with those of the HPMV filter. In this
respect, the multivariate UC models provide some additional information. Fourth,
the end-of-sample estimates differ significantly betwen the UC model and the two
multivariate UC models. Indeed, the model UC1 indicates a negative output gap
25
while the model UC2 shows a large positive output gaps. It raises some important
questions on the reliability of these measures since the policy implications differ
completely depending on the measure used.
(Insert Figure 2)
Table (2b) reports the estimates of the different equations.
(Insert Table 2b)
Overall, the results (descriptive statistics and estimations) indicate that the
use of multivariate UC models provides some differences in the estimation of the
output gap and the potential output.
5.2.3 SVAR models
Table (1c) displays the descriptive statistics of the SVAR models.27
(Insert Table 1c)
In contrast to previous multivariate methods, it is difficult to compare the
SVAR with an ”equivalent” univariate method. However, the two different models
give the same results in terms of the output gap and results are in line with those
of the HPMV models and the multivariate UC models. One significant point
is that there is a high degree of uncertainty surrounding the estimation of the
output gap. Some of these uncertainty is attribuable to the number of lags and
the length of the sample. Indeed, DeSerres and Guay (1995) show that many
lags have to be used to provide an unbiased decomposition into permanent and
transitory components with structural VAR. To some extent, the purpose of these
lags is to approximate the moving-average part of the underlying DGP using the
Wold decomposition.
Overall, these output gaps appear reasonnable, in that positive output gaps
are associated with episodes of accelerating inflation, while negative output gaps
correspond to episodes of decelerating inflation. The output gap describes much
27Estimations are not reported here but are available on requests upon authors.
26
more fluctuations (i.e. the standard deviation increases) when inflation enters in
the dynamics. Therefore, model SVAR2 displays fluctuations of the output gap
between -2.81 and 2.61, and the standard deviation is equal to 1.07 whereas the
output gap in SVAR2 fluctuates around its mean value (-0.02) with a standard
deviation of 1.38. In addition, the decomposition of the variance indicates that the
output gaps account for a large fraction of fluctuations in the trend of inflation.
However, a part of that gap may be well unrelated to the trend since it may
include very-high-frequency cycles that have little to do with that trend. In this
respect, another method suggested by Dupasquier, Guay and St-Amant would be
to impose restrictions on both real output and inflation in order to produce an
output gap that is constrained to be associated with movements in the trend of
inflation.
(Insert Figure 3)
5.3 Forecasting power
Tables 3a-3c display the results on the forecasting performances of all the models
under study. The different models display significant differences over the fore-
casting horizon. Overall, SVAR-based models have an advantage with respect to
a naive random walk to forecast inflation at short and long horizons. These re-
sults are confirmed by the modified DM statistic proposed by Harvey, Leybourne
and Newbold (1997). At a 10 % level, UCM models perform better than a naive
random walk only over short-term horizons. In spite of RRMSE values similar to
UCM ones, the modified DM test can generally not reject, over short and long-run
horizons, the equality of HPMV prediction mean squared errors and naive ones
(p-values are generally higher than 10 %)28.
(Insert Table 3a-3c)
Overall, the two multivariate UC models have the same forecasting properties.
This result also holds among the HPMV models. It is to be noted that the poor
28This is related to different error auto-covariances.
27
performances in the long run may be explained by the retained forecasting period
and thus the finite sample problems. In addition, the performances of the SVAR
models are not strictly comparable to the other models since the random walk and
the AR(3) are estimated in first differences. For short forecasting horizon, both
VAR models perform similarly. However, over long-term horizons, model SVAR2
performs better than model SVAR1.
In contrast, when the reference model is an autoregressive process of order
three (AR(3)), conclusions are different. Generally, none of the multivariate mod-
els performs better than the AR(3) process at long-term horizons (p-values are
lower than 10 %). Only UCM models perform better at short run horizons (the
p-value is equal to 4 % for an horizon 2).
Overall, clear evidence is difficult to assess regarding the forecasting perfor-
mance. Camba-Mendez and Rodriguez-Palenzuela (2002) show that the forecast-
ing performances of the SVAR models are better than those of the multivariate
Beveridge-Nelson model or the multivariate UC model. Nevertheless, Cristadoro
et al. (2001) indicate that a random walk is more difficult to out-perform than
the AR model, i.e. the best linear optimal linear predictor of h-ahead inflation is
current inflation. In contrast, Bruneau, De Bandt and Flageollet (2003) show that
the two models are indistinguihable from a statistical point of view. For example,
they find that the performance of the random walk model relative to the AR(3)
model is different whether the core HIPC or total HICP is considered. In addi-
tion, the out-of-sample forecasting performances are different when considering
what they called the ”disinflationary period” (1996:Q1-1998:Q4) or the ”inflation
upturn” (1999Q1-2002:Q1). One possible explanation is the lower variability of
inflation in the euro area due to the ”averaging effect” of euro area aggregates as
compared to national developments that are more erractic.
5.4 Revisions
Table 4a-4b report the measures of uncertainty on filtered, smoothed and quasi-
real time data in the HPMV models. First, the standard deviation of the output
gap estimates is larger than the standard deviation of the HP model even when
filtered or smoothed estimates are considered. Therefore, the HPMV do not sig-
nificantly reduce the uncertainty surrounding the HP filter except for the model
28
HPMV3.
(Insert Tables 4a, 4b)
Moreover, the multivariate HPMV models do not significantly reduce the stan-
dard deviation of the quasi-real time data revisions except in the case of the model
HPMV3. However, we have already discussed the limits of this model.
Results are different for multivariate UC models. Table 4 shows that (i) the
total uncertainty and the filtered uncertainty decrease when additional endoge-
nous variables entered in the model, (ii) the parametric uncertainty increases as
more information is considered. Overall, the standard deviation of the filtered or
smoothed output gap (UC) is larger to the corresponding values of the multivari-
ate UCM1 and UCM2 models. In simple UCM models of output and inflation,
moving to the smoothed estimates can strongly decrease the overall RMSE. In
contrast, filtered estimates would not allow to distinguish a boom or a recession
from the balanced growth path! However, Runstler (2002) has shown that adding
endogenous variables as for instance the unemployment rate and the capacity uti-
lization ratio, can diminish the filtered uncertainty to a level closed to the usual
smoothed uncertainty, rendering boom or recession detection possible.
(Insert Table 4c)
However, this result may be explained by the presence of the capacity utilisation in
the set of variables. In addition, the multivariate UC models significantly decrease
the standard deviations of the quasi-real time data revisions, i.e.quasi-real time
revisions are much more important for univariate UC model.
Overall, the two methods exhibit some important differences in terms of un-
certainty and quasi-real time revisions. The multivariate UC models appear more
robust to these issues than the HPMV models.
Finally, we also compute recursively the standard deviation of revisions (quasi-
real time estimates of the output gap) in the case of SVAR models (Table 4c).
Unemployment increase the size of revisions: SVAR2 standard revisions (0.682)
29
are higher than SVAR1 ones (0.485). The standard revisions in SVAR1 are smaller
than in other multivariate detrending methods.
5.5 Capacity utilisation coincidence
In order to explore the extent to which the cyclical component of the output is
reliable, the correlation between the capacity utilisation and the estimated output
gap series. Table 5 report the results.
(Insert Table 5)
The correlation coefficient are significant at standard levels and is positive as
expected for both periods 1970Q1-2002Q4 and 1980Q1-2002Q4. This result is
consistent with those of Camba-Mendez and Rodriguez-Palenzuela (2003).
6. Conclusion
In this paper, we apply three different multivariate detrending methods - multi-
variate HP filter, multivariate UC models and SVAR models - in order to estimate
the output gaps in the Euro-area. These methods have different theoretical im-
plications as well as some specific underlying assumptions. When we assess the
statistical reliability of these methods, three criteria are used: the consistency of
descriptive statistics, the forecasting performance in terms of inflation and some
measures of uncertainty. Our results show that (1) additional economic informa-
tion may provide some useful information for the estimation of the output gap,
(2) the economic interpretation may differ across the different methods and within
a given method (when different specifications are used), (3) all the multivariate
detrending models performs less than an autoregressive process when we test the
prediction in terms of inflation and (4) multivariate UC models performs better
than HPMV models in relative terms in order to reduce the filtered, smoothed
uncertainty or quasi-real time estimates. However, it is difficult to conclude that
a multivariate detrending method outperforms the others.
30
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37
7. Annexes
7.1 Description of the main HPMV empirical studies
The main HPMV empirical studies are summarised below.
• LT: Laxton and Tetlow (1992) : potential output of Canada, over the period1960-1990. HPMV with an Okun’s law and a Phillips curve.
• BU: Butler (1996) : potential output of Canada, over the period 1970-1992.HPMV using a production function decomposition.
• HA: Haltmaier (1996) : potential output of the G7, over the period 1970-1996. Aggregate HPMV with a Phillips curve, production function decom-
position with a Phillips curve.
• COHU: Conway and Hunt (1997) : potential output of New-Zealand, overthe period 1985-1998, HPMV with a Phillips curve, an Okun’s law and a
capacity utilisation constraint.
• DB: De Brouwer (1998) : potential output of Australia, over the period 1980-1999, HPMV with a Phillips curve, an Okun’s law and a capacity utilisation
equation.
• COHO: Cote and Hostland (1993) : potential output and NAIRU of Canada,over the period 1960-1992. HPMV Filter with four economic equations :
price consumer equation, GDP price equation, wage equation, unemploy-
ment equation.
38
Studyσ2gapσ2∆∆y
Inflation equation weights
LT 1600 1
BU 1600 1
16 000 for the participation rate
10 000 for the labour-output
elasticity(univariate HP filter).
HA 1600 25 (values between 12-50 tested)
CO-HU 1600 2
DB 1600 Ratio of variances.
CO- alues between 10 to 100 000 tested,
HO 500 for output, 100 for unemployment.
Note : weights are coherent with year on year inflation rates.
7.2 Estimation of state-space models
UCM models have to be formulated in a state-space framework, for implementing
their estimation. Therefore, we present here within a general state-space model
the solution to the filtering problem, i.e. the research of the best approximate of
the state vector given current and past observations Y0, ..., Yt. Then, we explain
how parameters can be estimated with an EM algorithm.29
Let begin with a general definition and some preliminary notations.
Definition 1 State-space model
We call ”state-space model” of a process Yt the system (I) described by equa-
tions (21) :
Zt+1 = AtZt +BtX1,t + εt (21)
Yt = CtZt +DtX2,t + ηt (22)
where
Ãεt
ηt
!is a Gaussian white noise with a variance-covariance matrix
29For more details, smoothing and forecasting techniques, you can refer to Lemoine & Pelgrin(2002).
39
Ωt =
ÃQt St
S0t Rt
!, matrices At, Ct having sizes K ×K , n×K. Bt and Dt are
deterministic matrices having sizes K1×K and K2×K. and Z0 is a random vectorfollowing a normal distribution, N(m,P ), and independent from the Gaussian
white noise.
Let note :
- Yt the output, observation or measure variable;
- Zt the state variable at date t;
- εt the vector of innovations or inputs at date t;
- ηt the vector of measurement errors or the noise at date t;
- At the transition matrix ;
- Ct the measure matrix
- X1,t, X2,t the exogenous variables, known a priori ;
- CtZt the signal at date t.
7.2.1 The Kalman filter
In this first estimation step, we assume that matrices At, Ct, Rt, Qt, P and the
vector m, which define the state-space model (I), are known.
Notations Let note :
- bZt,t the current estimation of the state vector ;- Zt,t the estimated error on the state vector ;
- Σt,t = V (Zt,t) the mean quadratic error on Zt ;
- bZt−1,t the forecasted state vector at date t-1 ;- Zt−1,t the forecasted error on the state vector ;
- Σt−1,t = V (Zt−1,t) the mean quadratic error of the associated forecast;
- Kt the Kalman gain matrix, which will be defined in the following algorithm.
Algorithm The Kalman filter is an iterative algorithm allowing the optimal
estimation of a state Z hidden behind a measured variable Y, in the framework
of a state-space model. We now present and comment the Kalman filter. Each
iteration can be summarised by equations 1 to 5 :
1. bZt,t = bZt−1,t +Kt(Yt −DtX2,t − Ct bZt−1,t)40
2. Σt,t = (I −KtCt)Σt−1,t
3. bZt,t+1 = At bZt,t +BtX1,t4. Σt,t+1 = AtΣt,tA
0t +Qt
5. Kt = Σt−1,tC0t(CtΣt−1,tC
0t +Rt)
−1
6. bZ−1,0 = m et Σ−1,0 = P.
Equation (1) computes the current state vector estimate bZt,t as the sum of twocomponents: - on the one hand, the forecast of Zt at the date t − 1 and, on theother hand, - the expression Kt(Yt−Ct bZt−1,t) which depends on the gain matrix,updated by the equation (5) and on the observations at the date t. Equation
(2) defines the a posteriori covariance matrix Σt,t, which depends on the a priori
covariance matrix Σt−1,t and on the expression KtCtΣt−1,t generally associated
with a precision gain. Equation (3) allows to compute the forecast of the state
vector at the date t + 1, bZt,t+1, as the projection of Zt+1 on its past. Equation(4) explains the a priori covariance matrix at t+ 1, Σt,t+1, which can be decom-
posed, on the one hand, in a factor, AtΣt,tA0t, increasing (decreasing) Σt,t+1 if the
filter converges (diverges) and, on the other hand, in a positive definite factor Qt
(covariance matrix of the vector of innovations). Finally, equation (5) defines the
Kalman gain matrix. Matrices Kt weight the adjustment of the last estimate with
the last forecasting error. Finally, the Kalman filter needs the initial conditions
(6).
7.2.2 Estimation
In the last section, we have assumed that matrices At, Ct, Qt, Rt, P and the vec-
tor m were known. In practice, these matrices are unknown and, after a brief
description of the likelihood computation, we present the EM algorithm used for
estimating the parameters.
The likelihood function Let define the density of Y0, Y1, ..., YT as follows:
lT (y0, y1, ..., yT ; θ) = f(y0; θ)f(y1/y0; θ)f(y2/y1, y0; θ)...f(yT/y0:T−1; θ).
41
All conditional densities are Gaussian, i.e. the general term f(yt/y0:t−1; θ) is de-
fined by N(byt−1,t(θ), V (eyt−1,t)) where V (eyt−1,t) = Mt−1,t(θ) = CΣt−1,tC0 + R. For
a fixed and unknown vector θ, Kalman filter can be used to compute the log-
likelihood:
LT (y0:T ; θ) = Cste − 1
2
TXt=0
log det(Mt−1,t(θ))−T 0Xt=0
ey0t−1,t(Mt−1,t(θ))−1eyt−1,t (23)
with ey−1,0(θ) = y0−by−1,0 = Eθ(y0) andM−1,0 = V (ey−1,0) = Vθ(y0). This expression(23) is called the decomposition formula of the likelihood forecasting error.
The EM algorithm The EM algorithm is usually used for finding Maximum
Likelihood Estimators (MLE) of parameters in a state-space model. This iterative
algorithm is quite simple, although it converges more slowly than other compli-
cated algorithms. It was first introduced in 1977 by Dempster (1977) in order to
estimate the maximum likelihood of Hidden Markov Models (HMM).
For proceeding the maximum likelihood estimation, the likelihood has to be
defined and maximised. Both steps constitute the kernel of the EM algorithm, as
shown further.
The EM algorithm is therefore an iterative algorithm, which creates a sequence
of estimates (θi)i=1,2,... from an initial condition θ0. Each iteration is decomposed
into two following steps :
• Step E : Q(θ, θi) = E(L(y|θ))
• Step M : θi+1 = argmaxθQ(θ, θi)
The first step E, as ”Expectation”, computes the likelihood from the last sec-
tion formula. This formula mainly uses a Kalman filter for knowing the state
of the system, when the observations and the parameters are fixed. The second
step M, as ”Maximisation”, consists in searching a set of parameters, which would
maximise the likelihood estimated in the step E. This maximisation can be pro-
ceeded analytically or numerically, according to the complexity of the problem.
After one iteration, θi+1 is deduced so that L(Yt|θi+1) > L(Yt|θi). Iterating thesetwo steps, the likelihood function should converge towards its maximum value.
42
Otherwise, the EM step can be approximated with the following expression :
θi+1 = θi + F−1(θi)∂L(Y1:T |θi) (24)
with F = E(∂L(Y1:T |θi)∂L(Y1:T |θi)) = −E(∂i∂jL(Y1:T |θi))
The equation (24) shows that the EM algorithm updates parameters, according
to the stronger slope of the likelihood function L, with the distance defined by
the Fischer matrix F .
Description of the main UCM empirical studies We have defined in our
paper a general UCM output gap estimation methodology and we have described
some modelling issues, which have been developed and raised in the papers: Kut-
tner (KU, 1994), Gerlach & Smets (GS, 1999), Appel & Jansson (AJ, 1999),
Kichian (KI,1999), Orlandi & Pichelmann (OP, 2000), Scott (SC, 2000), Butter
& Koopman (BK, 2001) and Proietti, Musso & Westermann (PMW, 2002). We
report in the following table countries, estimation periods and considered variables
used in these papers for estimating UCM output gaps.
Papers Country Period Variables
KU US 1954-1992 GDP,CPI
GS EMU5, EMU10 1975-1997 GDP,CPI
AJ US, CAN, UK 1970-1998 GDP,CPI,U
KI CAN 1961-1997 GDP,CPI
OP EU11 1960-1998 CPI,U
SC NZ 1970-1999 GDP,CU,U
BK US, GE, NDL, UK 1970-1994 IPI,L
PMW EU11 1970-2001 GDP,CPI,U,CU...
7.3 Non-fundamental representations
In order to simplify the presentation, we consider an univariate process, as fo
instance the following MA(1) process:
Xt = (1− θL)εt (1)
43
where εt ,→WN(0,σ2ε) and |θ| < 1.The above process has exactly the same correlation function of the following
process:
Xt =
µ1− L
θ
¶ut (2)
where ut ,→WN(0, θ2σ2ε).
Proof: By definition, γ(k) = cov(Xt,Xt−k) = γ(−k) = 0 ∀ |k| > 1. Therefore,it remains to show that both process have the same variance and covariance of or-
der one. Using equation (1), it follows that V (Xt) =¡1 + θ2
¢σ2εand γ(1) = −θσ2ε.
Similarly, using equation (2), one obtains V (Xt) =¡1 + 1
θ2
¢σ2u =
¡1 + 1
θ2
¢θ2σ2ε =¡
1 + θ2¢σ2ε and γ(1) = −1
θσ2u = −θσ2ε.
Hence, it is not possible to decide which model is the most appropriate for a
given stochastic process. In addition, the two equations have different properties
in terms of AR(∞) representation. Before analysing their respective properties,the transformation from equation (1) to equation (2) is explained:
Xt = (1− θL)L− θ
1− θL
∙L− θ
1− θL
¸−1εt
=
µ1− L
θ
¶vt
where θ is the (conjuguate) reciprocal of θ.
More generally, one may find a rational function, f(z), such that:
Xt = (1− θL)f(z) [f(z)]−1 εt (3)
=
µ1− L
θ
¶vt.
Comparing the two representations (1) and (2), two considerations can be
drawn. On the one hand, the first representation is invertible (|L∗| =¯1θ
¯> 1), i.e.
there exists an infinite autoregressive form. Therefore, by observing a sufficiently
large sample of Xτ , τ ≤ t, it is possible to make inference on the shocks εt andon their dynamic effects on the observable variable Xt. In contrast, the second
representation is not invertible (|L∗| = |θ| < 1), i.e. the vector error space does
44
not belong to the space spanned by current and past values of Xt. On the other
hand, the dynamic response of Xt regarding εt and vt are different.
Therefore, starting from a given linear univariate process, it is possible to gen-
erate a non-fundamental representation which might have radically different prop-
erties with respect to their fundamental counterparts because the disturbances are
qualitatively very different (table 1).
Fundamental representation εt² Xt = Xt,Xt−1, ...,Xt−k, ...Non fundamental representation εt /∈ Xt
When the agents’ information set is larger than the space spanned by current
and past values of the observable variable, then the moving average representation
is non fundamental.
These results can be easily extended to the case of multivariate moving average
process (Lippi and Reichlin, 1993).
7.4 Different representations of VAR models
Different reduced-forms of VAR models can be proposed.
Sims (1980) proposed to move from a non-orthogonal vector moving average
representation to an orthogonalised representation using Choleski factorisation
of the variance-covariance matrix. This amounts to modelling contemporaneous
relationships among the endogenous variables in a triangular form. Given the
variance-covariance matrix, the Choleski-based decomposition is uniquely deter-
mined. However, if the elements of the endogenous vector are permutted differ-
ently, then the variance-covariance matrix would have a different Choleski-factor,
i.e. a different orthogonalised representation. Thus, Choleski’s representation is
unique only given a particular ordering of the observable variables included in Xt.
Three additional representations may be used.
Type 1-model consists in finding a n× n invertible matrix K such that30:
KB(L)Xt = Ket (1)
Ket = εt
30Equation (17) is used to represent the reduced form model. The constant vector is droppedwithout loss of generality.
45
where εt is a vector of orthonormalised disturbances, E[εt] = 0n and E[ε0tεt] =
In.
The contemporaneous correlations among the elements of Xt are therefore
modelled through the specification of the matrix K.
Type 2-model, as for instance the Blanchard-Quah model, aims at finding a
n× n matrix C such that:
B(L)Xt = et (2)
et = Cεt
where et is regarded as being generated by a linear combination of independant
(orthonormal) disturbances.
In this context, there is no instantaneous relationship among endogenous vari-
ables. Each variable in the system is affected by a set of orthonormal disturbances
whose impact effect is explicitely modelled in the C matrix. In other words, co-
movements of Xt are indirectly organised since they result as linear combinations
of a vector of independant shocks.
Finally, type 3-model is the most general representation and consists in finding
two matrix K and B such that:
KB(L)Xt = Ket (3)
Ket = Cεt
Therefore, it is possible to model explicitely the instantaneous links among
the endogenous variables and the impact effect of the orthonormal random shocks
hitting the system. However, the number of parameters to estimate is larger than
the two previous models.
Finally, independently of the reduced-form being specified, after imposing the
n(n+1)/2 restrictions in the variance-covariance matrix of each model, it remains
the problem of estimating n2 parameters for the K matrix (respectively C-matrix)
in the K-model (respectively C-model) and 2n2 parameters in the KC model.
The exclusion restrictions and the need for exact identification may reduce the
practical meaning of the structural VAR approach.
Three methods of identication have been generally used in the literature. First,
46
as explained in section 4, Blanchard and Quah (1989) proposed an exact identica-
tion scheme by introducing homogenous long-run restrictions on the parameters of
the C matrix, through an infinite-horizon theoretical constraint. Second, Keating
(1990) introduced a set of non-linear restrictions on the off-diagonal elements of
the K matrix. These restrictions are derived from a variant of the rational expecta-
tions model of Taylor (1988). Third, Doan (1989) proposes a complete solution of
over-identified and exactly identified type 3-model, with C diagonal and exclusion
restrictions in the off-diagonal elements of the K matrix.
7.5 The Beveridge-Nelson Decomposition
Beveridge and Nelson (1981) showed that any single integrated ARIMA process
has an exactly identified trend plus a transitory representation. The trend is thus
defined as a random walk (eventually with a drift) and the transitory component
is covariance stationary.
Let assume that the first difference of a stochastic process Xt is defined by the
following moving average representation:
∆Xt = µ+A(L)εt (1)
where A(L) is a lagged polynome with A(0) = 1, εt is i.i.d. with mean zero
and variance σ2.
Equation (1) can be rewritten as:
∆Xt = µ+ [A(1) + (1− L)A∗(L)] εt (2)
Proof: A(L) =∞Pk=0
akLk =
∞Pk=0
ak(Lk− 1)+
∞Pk=0
ak = A(1)+ (L− 1)∞Pk=0
ak(−1−
...− Lk−1)
It follows that A(L) = A(1) − (1 − L)∞Pk=0
̰P
j=k+1
aj
!Lk = A(1) + (1 −
L)∞Pj=0
αkLk where αk = −
∞Pj=k+1
aj.
Let denote A∗(L) =∞Pj=0
αkLk, expression (2) is thus obtained.
Following expression (2), the trend component is defined by µ+A(1)εt and the
47
transitory component is (1−L)A∗(L)εt.One can show that the two components arecorrelated and that it does not exist a decomposition where the trend component
and the cyclical component are independant.
Fourtis and Dickey (1986) extended the decomposition to VAR models with
one stochastic component. Cochrane (1994) considered the case where all vari-
ables are I(1) whereas Evans and Reichlin (1994) discussed a representation with
I(1) and stationary variables. Stock and Watson (1992) provided a general rep-
resentation for any stochastic trend. We present here their general methodology.
Two equivalent representations can be given.
First, let define the following moving average representation of a multivariate
process Xt :
∆Xt = µ+A(L)εt
where Xt is a n× 1 vector, A(L) is lagged polynome with A(0) = In and εt is
i.i.d. white noise with mean zero and covariance matrix Σ.
In addition, A(1) is assumed to have rank k < n, so that Xt is cointegrated,
i.e. there is a n×r matrix Γ (where r = n−k) such that Γ0A(1) = 0n and Γ0µ = 0.Let vt = Σ−1εt and ut =
tPs=1
vs (cumulative function) and εs = 0 ∀s ≤ 0, andassume X0 is a non-random initial value.
Then,
Xt = X0 + µt+A(1)Σ1/2ut +A
∗(L)Σ1/2vt (3)
where (1− L)A∗(L) = A(L)−A(1).Given the propertires of A(1), it follows that equation (3) can be re-written
as follows:
Γ0Xt = Γ0£X0 + µt+A(1)Σ
1/2ut +A∗(L)Σ1/2vt
¤(4)
= Γ0X0 + Γ0A∗(L)Σ1/2vt.
Second, the stochastic process Xt has also an alternative representation. Be-
cause A(1) has rank k < n, there is a n × r matrix Π1 with rank r such that
A(1)Π = 0. Furthermore, if Π2 is an n × k matrix with rank k and columns
48
orthogonal to the columns of Π1, then C = A(1)Π2 has rank k. It immediately
follows that Π = Π1 Π2 is then non singular and A(1)Π = (0 C) = CSk where Sk is
the k×n selection matrix£0k×(n−k), Ik
¤. In addition, µ ² V ecA(1), i.e µ = A(1)eµ.
In this respect, equation (3) is then given by:
Xt = X0 +A(1)eµt+A(1)Σ1/2ut +A∗(L)Σ1/2vt (5)
= X0 +A(1)Π£Π−1eµt+Π−1Σ1/2ut
¤+A∗(L)Σ1/2vt
= X0 + CSk£Π−1eµt+Π−1Σ1/2ut
¤+A∗(L)Σ1/2vt
= X0 + CTt +A∗(L)Σ1/2vt
where Tt = β + Tt−1 + ζt, β = SkΠ−1eµ is a drift and ζt = Π−1Σ1/2ut.
This representation expresses Xt as a linear combination of k random variables
with drift β plus some transitory components (A∗(L)Σ1/2ut) that are integrated
of order 0.
7.6 The companion form and related issues
Any VAR(p) can be written as a VAR(1). This is called the companion form. It
is particularly useful for computing impulse response functions for a VAR(p).
Define the np× 1 vector z0t = (x0t, ..., x
0t−p) then
zt = Azt−1 + ut
where
A =
⎡⎢⎢⎢⎢⎢⎢⎣−A1 −A2 ... −Ap−2 −Ap−1In 0 ... 0 0
0 In ... 0 0
0 0 ... ... ...
0 0 ... In 0
⎤⎥⎥⎥⎥⎥⎥⎦ , ut =⎡⎢⎢⎢⎢⎢⎢⎣et
0
...
0
0
⎤⎥⎥⎥⎥⎥⎥⎦ .
In order to obtain the impulse response functions or forecasts for a VAR(p):
(i) we write the VAR(p) in its companion form, (ii) we then proceed as though
it were a VAR(1) to obtain impulse response functions or forecasts and (iii) we
obtain the relevant element of xt by extracting it form zt using xit = l0izt (where
49
li is an np× 1 zero vector, except for the ith, which is one).In this respect, forecastings can be computed recursively. Hence, the best
forecast conditional on information at time t is
Etzt+s = Aszt.
The forecast error variance decomposition is then
zt+s −Ezt+s =s−1Xi=0
Aiet+s−i.
The variance of the forecast error is thus
V (zt+s) =s−1Xi=0
AiΩAi0.
7.7 Aggregate versus disaggregate estimation of the UE5
output gap
In this appendix, we propose to examine the impact of estimating the ouput gap
as the aggregation of country-specific models instead of area wide models. This
issue is of some importance in the euro area, where aggregate analysis is gaining
in importance, although it is recognised that many structural differences remain
across countries. A number of recent papers have been focused on the aggregation
issue (Fagan and Henry 1998, Dedola et al. 2001, Marcellino et al. 2002, Fabiani
and Morgan 2003). These papers highlight the bias which can emerge when using
aggregate approaches and examine whether some advantages can be gained to
adopt a national approach.
In order to compare the respective output gap estimates and forecasting per-
formance of country-specific and aggregate approaches, the same HPMV model
has been estimated for each of the five largest countries of the euro area in a first
step (Germany, France, Italy, Spain and the Netherlands, accounting for slightly
more than 85 percent of the euro area GDP in 2002), and for the UE5 in a second
step. Out of sample inflation forecasts are in a second step computed from the
aggregation of the country-specific models and compared to the UE5 wide model,
50
considered here as the naive model.
For reasons of simplicity, the model incorporates as additional economic equa-
tion only the inflation equation, and the weight associated to this equation has
been imposed to a relatively high value (corresponding to a weight of 30 for annual
data). The same specification is used for each country model and for the aggregate
UE5 model , and has the following state space representation (estimation period
covers the period 1970:1 2002:4.) :
(Y it = AY
it−1 + U
it
Zit = CYit +DX
it + Vt
i
(state equation)
(measurement equation)(25)
for i = Germany, France, Italy, Spain, the Netherlands and UE5 , where
U ∼ NID(0, Q), V ∼ NID(0, R), Cov(U, V ) = 0, Y it = (y∗it , y∗it−1, yit − τ iy,t, yit−1 −
τ iy,t−1)0 is the state vector, Zit = (yit,π
it)0 is the measurement vector and X i
t =
(πit−1,πit−2,π
it−3)
0 is the vector of predetermined variables.
The matrices are given by :
A =
⎡⎢⎢⎢⎢⎣2 −1 0 0 0 0 0
1 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 1 0 0 0 0
⎤⎥⎥⎥⎥⎦ ,Ci =
"1 0 1 0 0 0
0 0 0 αiπ 0 0
#, Di =
"0 0 0
αi1 αi2 αi3
#
U it =
⎡⎢⎢⎢⎢⎣εi∆∆y∗,t
0
εigap,t
0
⎤⎥⎥⎥⎥⎦ , Vt ="0
εiπ,t
#(26)
and the weights can be expressed as:
λi = σ2gap/σ2∆∆y∗ = 1600, µ
iπ = σ2igap/σ
2iπ = 30
51
UE5 aggregate variables used for the estimation of the UE5 wide model (GDP
in constant prices, private consumption in current and constant prices) are con-
structed as the sum of each country variables. UE5 inflation forecast of the aggre-
gation of country specific models is constructed using for each quarter the weight
associated to each country in the private consumption UE5 aggregate in current
prices of the precedent quarter. This method provides a UE5 wide consumer price
inflation wich is coherent with the implicit weights of the UE5 aggregate.
Results are displayed in tables 6a, 6b, 6c and Figure 4. Parameters are rel-
atively heterogenous among countries. Specifically, the output gap coefficient is
not significative for Spain and the Netherlands, and parameters for the UE5 ag-
gregate are similar to those obtained for France and Germany, these two countries
acccounting for 66 percent of the UE5 GDP. However, results suggest that output
gap descriptive statistics and profiles are very similar for the UE5 wide model and
for the aggregation of country-specific models . When the forecasting performance
of the aggregation of country models is compared to the UE5 wide model , results
point to a better performance of the country-specific approach, wich corroborates
other empirical studies (Fabiani and Morgan 2003, Marcellino et al. 2003). How-
ever, the improvement is obtained only for short horizons and the null hypothesis
of equal prodictive power of both models cannot be rejected.
52
Table 1a: Descriptive Statistics, HPMV
HP HPMV1 HPMV2 HPMV3 Mean -0.01 -0.03 -0.04 -0.09
Standard deviation 1.02 1.03 1.32 1.54 Minimum -2.55 -2.55 -2.43 -3.46 Maximum 2.40 2.39 2.86 2.77
Actual potential output growth rate
2.11 2.12 2.70 1.93
Standard deviation of potential output
growth rate
0.17 0.17 0.15 0.27
Table 1b: Descriptive Statistics, UCM
UC UC1 UC2 Mean 0.00 -0.04 0.06
Standard deviation 1.16 1.41 1.77 Minimum -2.99 -2.70 -2.74 Maximum 2.77 2.82 3.61
Actual potential output growth rate
1.64 1.93 2.27
Standard deviation of potential output growth
rate
0.20 0.14 0.22
Table 1c: Descriptive Statistics, SVAR
SVAR1 SVAR2 Mean -0.02 -0.02
Standard deviation 1.06 1.38 Minimum -2.81 -2.59 Maximum 2.61 2.83
Actual potential output growth rate
2.04 1.81
Standard deviation of potential output growth
rate
0.16 0.19
53
Table 2a: HPMV estimation
HP HPMV1 HPMV2 HPMV3 Inflation dynamics1
πt-1
0.37 (0.00)
0.37 (0.00)
0.38 (0.00)
πt-2
0.43 (0.00)
0.43 (0.00)
0.43 (0.00)
yt-1-y*t-1
0.07 (0.03)
0.03 (0.00)
0.05 (0.02)
σ
0.281 (0.00)
0.280 (0.00)
0.280 (0.00)
Okun’s law
yt-1-y*t-1
-0.30 (0.00)
-0.27 (0.00)
σ
0.14 (0.00)
0.13 (0.00)
Capacity utilization dynamics
yt-1-y*t-1
2.06 (0.00)
σ
0.92 (0.00)
Trend
σ
0.030 (0.00)
0.021 (0.00)
0.043 (0.00)
0.065 (0.00)
Note : Standard deviation in parentheses. 1 : Implicit weights associated to the inflation equation are given by the ratio of the variance of the gap to the mean squared error of the inflation equation. Computed for year on year inflation, they amount respectively 1.2 (HPMV1), 2.4 (HPMV2) and 5.4 (HPMV3), to compare with values ranging from 1 to 25 in the literature.
54
Table 2b: UCM estimation
UC UCM1 UCM2 Inflation dynamics
πt-1
0.37 (0.08)
0.40 (0.07)
πt-2
0.41 (0.08)
0.40 (0.07)
yt-1-y*t-1
0.06 (0.02)
0.03 (0.01)
σ
0.28 (0.02)
0.27 (0.01)
Okun’s law
yt-1-y*t-1
-0.43 (0.03)
σ
-
Trend
σ
0.08 (0.03)
0.05 (0.02)
0.05 (0.01)
Cycle
ρ
0.96 (0.03)
0.95 (0.03)
0.99 (0.01)
P
10.6 (-)
10.6 (-)
10.6 (1.01)
σ
0.48 (0.03)
0.48 (0.05)
0.30 (0.03)
Irregular
σ
0.48 (0.04)
Note: Standard deviation in parentheses.
55
Table 2c: SVAR estimation (unconstrained)
∆y ∆u ∆yt-1 0.25
(0.09) -5.09 (2.18)
∆yt-2 0.12 (0.09)
-0.89 (2.16)
∆y t-3 0.145 (0.07)
-0.84 (1.85)
∆y t-4 -0.07 (0.09)
-1.87 (1.20)
∆y t-5 -0.08 (0.09)
6.97 (2.01)
∆u t-1 -0.010
(0.00) 0.45
(0.09) ∆u t-2 0.012
(0.00) 0.19
(0.10) ∆u t-3 0.001
(0.004) 0.22
(0.08) ∆u t-4 -0.006
(0.004) 0.11
(0.07) ∆u t-5 0.003
(0.004) -0.09 (0.08)
Log-likelihood 437.07 88.04
Akaike -6.52 -1.29 Schwarz -7.24 -1.07
56
Table 2c: SVAR estimation (unconstrained)
∆y ∆u ∆inf ∆yt-1 0.19
(0.10) -0.61 (0.30)
0.156 (0.06)
∆yt-2 0.17 (0.09)
-0.27 (0.24)
-0.280 (0.07)
∆y t-3 0.16 (0.11)
-0.01 (0.02)
0.291 (0.67)
∆y t-4 -0.01 (0.09)
-0.00 (0.09)
-0.142 (0.63)
∆y t-5 -0.17 (0.08)
-0.16 (0.08)
0.261 (0.61)
∆u t-1 -0.01
(0.004) -0.01
(0.004) -0.129 (0.277)
∆u t-2 0.02 (0.00)
0.01 (0.002)
-0.163 (0.292)
∆u t-3 0.003 (0.004)
0.03 (0.04)
-0.056 (0.302)
∆u t-4 -0.007 (0.004)
-0.007 (0.004)
0.029 (0.286)
∆u t-5 0.00 (0.00)
0.001 (0.004)
0.155 (0.269)
∆inf t-1 0.62
(0.09) 0.013 (0.03)
0.62 (0.09)
∆inf t-2 0.02 (0.09)
0.05 (0.02)
0.02 (0.08)
∆inf t-3 0.48 (0.09)
0.01 (0.03)
0.48 (0.09)
∆inf t-4 -0.47 (0.10)
-0.06 (0.03)
-0.49 (0.10)
∆inf t-5 0.38 (0.09)
-0.00 (0.03)
0.38 (0.07)
Log-likelihood -26.38 458.01 86.45
Akaike 0.61 -7.19 -1.23 Schwarz 0.94 -7.14 -0.89
57
Table 3a: Out-of-sample inflation forecasting performances, HPMV
Horizon AR(3) Random walk Naïve model
RMSE RRMSE DM P-value RMSE RRMSE DM P-value
1 0.198 1.000 0.000 1.000 0.231 1.000 .000 1.000 2 0.200 1.000 0.000 1.000 0.247 1.000 .000 1.000 3 0.203 1.000 0.000 1.000 0.219 1.000 .000 1.000 4 0.199 1.000 0.000 1.000 0.301 1.000 .000 1.000 5 0.202 1.000 0.000 1.000 0.313 1.000 .000 1.000 10 0.225 1.000 0.000 1.000 0.282 1.000 .000 1.000
HPMV1 1 0.197 0.991 0.063 1.000 0.197 0.853 -1.036 0.318 2 0.197 0.988 -0.016 0.988 0.197 0.799 -2.091 0.055 3 0.202 0.992 0.099 1.000 0.202 0.923 -.301 0.768 4 0.199 0.999 0.154 1.000 0.199 0.661 -1.145 0.271 5 0.198 0.980 0.016 1.000 0.198 0.634 -1.302 0.214 10 0.210 0.932 -0.135 0.895 0.210 0.744 -0.244 0.810
HPMV2 1 0.191 0.963 0.154 1.000 0.191 0.828 -1.790 0.095 2 0.190 0.954 0.064 1.000 0.190 0.772 -1.655 0.120 3 0.197 0.967 0.250 1.000 0.197 0.900 -0.172 0.768 4 0.191 0.960 0.219 1.000 0.191 0.635 -1.145 0.866 5 0.192 0.947 0.162 1.000 0.192 0.612 -1.067 0.304 10 0.208 0.927 -0.119 0.907 0.208 0.740 -0.239 0.815
HPMV3 1 0.184 0.928 -0.354 0.729 0.184 0.798 -1.225 0.241 2 0.187 0.936 -0.200 0.844 0.187 0.757 -1.600 0.132 3 0.211 01.04 0.658 1.000 0.211 0.967 0.053 1.000 4 0.191 0.962 0.269 1.000 0.191 0.637 -1.038 0.317 5 0.187 0.925 0.118 1.000 0.187 0.598 -1.092 0.293 10 0.214 0.951 -0.135 0.894 0.214 0.759 -0.216 0.832
Table 3b: Out-of-sample inflation forecasting performances, UCM
Horizon AR(3) Random walk RMSE RRMSE1 DM P-value RMSE RRMSE1 DM P-
value UCM1 1 0.192 0.936 -1.464 0.165 .192 .833 -1.664 0.118
2 0.192 0.926 -2.258 0.040 .192 .777 -2.646 0.019 3 0.199 0.943 -0.776 0.451 .199 .912 -0.625 0.542 4 0.191 0.930 -1.111 0.285 .191 .636 -2.235 0.042 5 0.197 0.950 -0.915 0.375 .197 .629 -2.616 0.020 10 0.221 0.929 -0.864 0.402 .221 .783 -1.550 0.144
UCM2 1 0.184 0.896 -1.273 0.224 .184 .798 -1.997 0.066 2 0.181 0.875 -2.209 0.044 .181 .734 -2.536 0.024 3 0.200 0.945 -0.215 0.833 .200 .914 -0.519 0.612 4 0.180 0.873 -1.120 0.282 .180 .597 -2.316 0.036 5 0.194 0.935 -0.157 0.878 .194 .619 -2.235 0.042 10 0.224 0.945 0.033 1.000 .224 .797 -1.507 0.154
1. : Statistics reported for the Theil and, DM statistics and the associated P value refer to the same naïve models as in the table 3a.
58
Table 3c: Out-of-sample inflation forecasting performances, SVAR
AR(3) Random walk RRMSE1 DM P-value RRMSE1 DM P-
value SVAR1 1 1.252 2.649 0.995 0.751 -2.221 0.013
2 1.138 2.287 0.988 0.854 -1.808 0.035 3 1.012 1.686 0.957 0.742 -2.178 0.015 4 1.075 1.898 0.971 0.872 -1.727 0.042 5 1.089 1.844 0.967 0.652 -2.352 0.009 10 1.112 1.882 0.970 0.734 -1.967 0.024
SVAR2 1 1.117 2.363 0.990 0.814 -2.408 0.008 2 1.058 2.124 0.982 0.828 -1.751 0.039 3 1.032 1.718 0.957 0.814 -2.389 0.008 4 1.061 1.869 0.969 0.895 -1.772 0.038 5 1.078 1.816 0.965 0.712 -2.566 0.005 10 1.215 1.964 0.975 0.765 -2.051 0.020
1. : Statistics reported for the Theil and, DM statistics and the associated P value refer to the same naïve models as in the table 3a.
59
Table 4a: Accuracy and revisions of output gaps, HPMV
HP HPMV1 HPMV2 HPMV3 Smoothed estimation Total uncertainty .305 .296 .394 .280
Filtered uncertainty
.305 .295 .394 .259
Parametric uncertainty
- .002 .061 .102
Filtered estimation Total uncertainty .554 .544 .757 .514
Filtered uncertainty
.554 .541 .773 .478
Parametric uncertainty
- .048 .105 .238
Output gap standard deviation Smoothed1 1.021
(3.35) 1.031 (3.48)
1.321 (3.35)
1.536 (5.49)
Filtered1 1.085 (1.96)
1.08 (1.99)
1.43 (1.89)
1.676 (3.26)
Standard deviation of revisions Quasi-real time .820 .781 1.024 0.407
Note: t-student in parentheses.
Table 4b: Accuracy and revisions of output gaps, HPMV
UC UCM1 UCM2 Smoothed estimation Total uncertainty 1.02 1.210 1.780
Filtered uncertainty
.933 .750 .480
Parametric uncertainty
.392 .890 1.700
Filtered estimation Total uncertainty 1.515 1.470 1.780
Filtered uncertainty
1.430 1.260 .920
Parametric uncertainty
.487 .690 1.480
Output gap standard deviation Smoothed1 1.300
(1.27) 1.449 (1.19)
1.664 (.94)
Filtered1 .732 (.48)
1.135 (.77)
1.562 (.87)
Standard deviation of revisions Quasi-real time .730 .540 .520
Note: t-student in parentheses.
60
Table 4c: Revisions of output gaps, SVAR
SVAR1 SVAR2 Standard deviation of revisions
Quasi-real time .485 .682
Table 5: Correlation with capacity utilization
1970q1-2002q4 1980q1-2002q4
HP 0.552 0.583 HPMV1 0.566 0.605 HPMV2 0.573 0.644 HPMV3 - -
UC 0.567 0.571 UCM1 0.642 0.665 UCM2 0.533 0.673 SVAR1 0.612 0.589 SVAR2 0.641 0.661
61
Table 6a: Descriptive Statistics for UE5 output gap
HPMV aggregate HPMV national
approach Mean -0.19 -0.15
Standard deviation 1.26 1.27 Minimum -2.80 -2.79 Maximum 2.68 2.79
Actual potential output growth rate
2.18 2.20
Standard deviation of potential output
growth rate
0.16 0.17
Table 6b: HPMV estimations
UE5
aggregate Germany France Italy Spain Netherlands
Inflation dynamics πt-1
0.37
(0.04) 0.26
(0.05) 0.44
(0.04) 1.02
(0.04) 0.57
(0.04) 0.14
(0.03) πt-2
0.36
(0.05) 0.52
(0.04) 0.3
(0.03) -0.06 (0.07)
0.25 (0.06)
0.64 (0.03)
yt-1-y*t-1 0.05 (0.01)
0.05 (0.01)
0.05 (0.01)
0.02 (0.01)
0.01 (0.01)
0.007 (0.02)
Trend σ
0.11 (0.00)
0.17 (0.00)
0.17 (0.00)
0.17 (0.00)
0.22 (0.00)
0.25 (0.00)
Note: Standard deviation in parentheses.
62
Table 6c: Out-of-sample inflation forecasting performances, HPMV models
Horizon RMSE RRMSE DM P-value Naive model : UE5 aggregate model
1 0.206 1.00 0.000 1.00 2 0.204 1.00 0.000 1.00 3 0.214 1.00 0.000 1.00 4 0.207 1.00 0.000 1.00 5 0.208 1.00 0.000 1.00
10 0.234 1.00 0.000 1.00 National approach model
1 0.198 0.96 -0.822 0.425 2 0.197 0.97 -0.593 0.563 3 0.208 0.97 -0.666 0.516 4 0.207 1.00 -0.036 0.972 5 0.214 1.03 0.236 1.000
10 0.238 1.02 0.474 1.000
63
Figure 1: HPMV results
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
1970
Q119
71Q1
1972
Q119
73Q1
1974
Q119
75Q1
1976
Q119
77Q1
1978
Q119
79Q1
1980
Q119
81Q1
1982
Q119
83Q1
1984
Q119
85Q1
1986
Q119
87Q1
1988
Q119
89Q1
1990
Q119
91Q1
1992
Q119
93Q1
1994
Q119
95Q1
1996
Q119
97Q1
1998
Q119
99Q1
2000
Q120
01Q1
2002
Q1
HPHPMV1HPMV2HPMV3Capacity utilisation
Figure 2: UCM results
64
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
1970
Q119
71Q1
1972
Q119
73Q1
1974
Q119
75Q1
1976
Q119
77Q1
1978
Q119
79Q1
1980
Q119
81Q1
1982
Q119
83Q1
1984
Q119
85Q1
1986
Q119
87Q1
1988
Q119
89Q1
1990
Q119
91Q1
1992
Q119
93Q1
1994
Q119
95Q1
1996
Q119
97Q1
1998
Q119
99Q1
2000
Q120
01Q1
2002
Q1
UC UCM1 UCM2
Figure 3: SVAR results
65
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
1970
Q119
71Q2
1972
Q319
73Q4
1975
Q119
76Q2
1977
Q319
78Q4
1980
Q119
81Q2
1982
Q319
83Q4
1985
Q119
86Q2
1987
Q319
88Q4
1990
Q119
91Q2
1992
Q319
93Q4
1995
Q119
96Q2
1997
Q319
98Q4
2000
Q120
01Q2
2002
Q3
SVAR1 SVAR2
66
Figure 4 : UE5 HPMV aggregate and country approach output gaps
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
1970 1974 1978 1982 1986 1990 1994 1998 2002
Multivariate filter (UE5 ag.)Univariate HO filter (UE5 ag.)Multivariate filter (UE5 disag.)
In %
67