Estimating the natural rate of unemployment in euro-area countries withcointegrated systems

Estimating the natural rate of unemployment in
euro-area countries with cointegrated systems∗
by Sven Schreiber†
this version: February 2009
Running title: “The natural rate of unemployment in euro-area countries”
Abstract
Given that for France, Germany, Italy, and the Netherlands the unemployment
rates are best classified as I(1), we apply permanent-transitory decompositions based
on cointegrated VARs with relevant variables (labor productivity, wages, tax wedges,
foreign relative prices) to estimate the time-varying natural unemployment rates. Our
implied unemployment gaps are better than the OECD gaps in predicting unemploy-
ment changes and inflation gaps, although they perform as badly as the OECD gaps
in forecasting inflation changes. Productivity represents one of the driving factors,
while foreign relative prices also drive French and German unemployment, whereas
it is the tax wedge in the Netherlands.
Keywords: euro-area unemployment, VECM, permanent-transitory decomposition
JEL: C51 (model construction), E24 (employment), J23 (employment determina-
tion)
∗Work on this project started during a visit of the author at the Deutsche Bundesbank, and the hospitality
I enjoyed there is gratefully acknowledged. This paper has benefited from comments of seminar participants
at the Bundesbank, at Humboldt University Berlin, at the ESEM and VfS meetings, and at DIW Berlin; I
wish to thank especially Dieter Nautz, Ragnar Nymoen, Christian Schumacher, and Harald Uhlig for useful
suggestions.
†Macroeconomic Policy Institute (IMK) at Hans Boeckler Foundation, and Goethe University Frank-
furt. E-mail: svetosch@gmx.net, fax: +49(0)211-7778-4332, address: Hans-Böckler-Stiftung/IMK, Hans-
Böckler-Str. 39, D-40476 Düsseldorf, Germany.
1

---

1
Introduction
In this paper we use cointegrated multivariate systems to estimate the time-varying natural
unemployment rates of important euro-area countries. The most common measures of the
natural rate of unemployment are Nairus, especially those from the OECD (see Richard-
son et al., 2000, sometimes in the guise of wage-inflation-based “Nawrus”). Those are
built around a single-equation Phillips-curve framework, often with some further exoge-
nous variables in the spirit of the “triangle model” of inflation (see Gordon, 1997). Be-
cause of the smooth and rising paths of many unemployment rates, a random walk is a
popular and hard-to-beat model for the unobserved Nairu component, even though it is
clear that such a model can only serve as an empirical approximation in a given finite
sample.1 Given this widespread empirical I(1) assumption for unemployment it seems
obvious that the corresponding permanent components should be estimated in a coin-
tegrated multivariate system, by using the duality between cointegration and common
stochastic trends. However, to the best of our knowledge this natural cointegrated-VAR
step has not been taken in the literature so far. (For example, Fabiani and Mestre, 2004,
use a VAR-based method, but they did not find cointegration, and thus could not extract
any non-trivial permanent components from the data.) In this paper we therefore spec-
ify cointegrated systems and then extract the time-varying natural unemployment rates
as the permanent components based on the estimated common trends (Stock and Wat-
son, 1988), in the form suggested by Proietti (1997). As an alternative measure, we also
use the Gonzalo-Granger decomposition (Gonzalo and Granger, 1995). A case for using
permanent-transitory decompositions has recently been made by Garratt et al. (2007).
By choosing this approach we take into account that “the determination of the joint
steady state of real wages, the real exchange rate, and the rate of unemployment requires
a full dynamic model rather than wage and price setting equations alone” (Bårdsen and
Nymoen, 2003). It is also interesting because the non-stationary development of un-
employment in many European countries is still a puzzle, i.e., we lack “a satisfactory
empirical explanation” for it (Nickell, 1998, p. 802). Our method adequately reflects the
implicit equilibrium relations between the variables, and in contrast to typical unobserved-
components models such as those of the OECD there is no need for ad-hoc smoothness
restrictions.2
1Sometimes even a double random walk specification is used (i.e., random walk with stochastic drift,
where the drift itself is again a random walk), see for example Laubach (2001). However, such an I(2)
specification would imply that unemployment changes themselves are non-stationary and therefore would
seem quite extreme.
2It must be acknowledged, however, that there is also a recent literature using more complex multivariate
unobserved-components models to estimate potential output and/or the Nairu without the need to impose
a-priori smoothness restrictions. See for example Apel and Jansson (1999, using Sweden for illustration),
Proietti et al. (2002, dealing with aggregate euro-area data), and Basistha and Startz (2004, for the US). But
2

---

Since a pre-requisite for applying permanent-transitory decompositions is a develop-
ment of unemployment which is best classified as I(1), all countries had to be excluded
where the unemployment rate is stationary, possibly after the removal of some breaks. In-
formation on unit roots in OECD unemployment rates is given in Papell et al. (2000), who
find evidence against I(1)-ness in several countries after allowing for mean shifts. The
remaining countries where the unit root cannot be rejected are Germany, France, Italy,
and the Netherlands (apart from Japan). The four European countries that we include in
our study represent the largest part of the euro area in terms of output and population.
The structure of this paper is as follows: The next section 2 points to the theory that
guides our choice of relevant variables, and clarifies the details of the our econometric
framework, including an illustrative example in a stripped-down bivariate system. After-
wards, section 3 provides descriptive evidence of the variables in the four countries and
tests for the number of common trends in each country. Section 4 presents the estimated
permanent unemployment components as the main results of this study, compares them to
the OECD Nairus, and provides an evaluation of their predictive powers. Finally, section
5 offers conclusions.
2
General modeling and specification issues3
2.1
Relevant literature and variables
We base the choice of variables on the established literature, of which we only mention
a small selection here: First there is the influential book by Layard et al. (1991), mainly
based on bargaining theories of the labor market. Another important reference is the dis-
cussion of the wage curve and of differences with respect to Phillips curve formulations
in Blanchflower and Oswald (1995). Blanchard and Katz (1999) provide a short but very
useful discussion of similar issues, and clarify how a wage curve formulation also covers
more modern search and matching frameworks. The issue of labor taxes affecting unem-
ployment or not is discussed, e.g., in Pissarides (1998), and a modern formulation of the
Phillips curve framework in an open economy is given in Batini et al. (2005). From the
empirical modeling side we highlight the comprehensive book by Bårdsen et al. (2005)
that deal with wage-price-unemployment systems in great detail. Having been guided by
handling the estimation of these complex models is far from trivial, and problems like non-convergence of
the algorithms are common. Our approach does not require iterative numerical algorithms because closed-
form algebraic formulae are available.
3Empirical results in this paper have been produced with NumPy (the numerical extension of the Python
language) programs written by the authors (for the common-trends tests, tests on α and α⊥, and the decom-
positions), JMulTi (for cointegration and stability tests, see Krätzig, 2004), and PcGive 10 (for estimates
of cointegration restrictions, see Doornik and Hendry, 2001). The NumPy code is publicly available at:
http://econ.schreiberlin.de/schreibersoftware.html
3

---

these strands of the literature, the variables that enter the multivariate systems in this pa-
per are given by real wages and labor productivity, the domestic inflation rate, some price
wedges (especially those between domestic and imported manufactured goods), and a tax
wedge between gross and net wages.
2.2
Framework
Our baseline models deal with the following variable vector of dimension N = 6, sepa-
rately for all countries:
yi,t = (ui,t, πi,t, f xi,t,twi,t, wi,t, qi,t) ,
(1)
where i ∈ {DE, FR, IT, NL} indexes the countries, u is the unemployment rate, π is the
annualized quarterly inflation rate, f x are foreign relative prices, and tw is a tax wedge.
Details on the all variables are given in section 3. The last two variables w and q are hourly
labor costs and labor productivity which are the only variables that are a priori known to
contain a deterministic trend. The statistical framework is the well-known cointegrated
VAR as described by Johansen (1995), which can be represented as the following vector
error correction models (VECM) for the different countries:
ki−1
∆yi,t = αiβi yi,t−1 + ∑ Γi, j∆yi,t− j + µi + Diδi,t + εi,t
t = 1, ..., Ti
(2)
j=1
In this standard notation βi is the (country-specific) cointegration matrix of dimension
6 × ri with full column rank, where ri < 6 is the cointegration rank. The full-rank matrix
αi also has dimension 6 × ri, and thus the long-run matrix αiβ is of reduced rank. Note
i
that the individual cointegration vectors need not be identified in any particular way for
the purpose of our study, because only the complete cointegration space spanned by the
columns of βi matters, and that space is always identified by the Johansen procedure.
The number of I(1) trends driving each system is 6 − ri. The constant term µi is left
unrestricted, and thus it will cumulate to a linear trend in the non-stationary directions
of the process yi,t. The additional regressor δi,t (with associated coefficients Di) contains
further deterministic terms such as outlier-removing impulse dummies, and a level shift
dummy in the cases of Germany and Italy.4 However, no linear trend term is included
there, because we wish to model the equilbrium relationships between the variables with-
4However, the shift will be restricted to the stationary directions by forcing its coefficient to be of
the form αiτi, where only the part τi is unrestricted. Put differently, a broken trend in yi,t will be ruled
out a priori. Another way of formulating that restriction is to replace yi,t and βi by y∗ = (y , s
i,t
i,t
i,t ) and
∗
∗
β
= (
,
) in the VECM.
i
βi τi
4

---

out resorting to unexplained exogenous terms as much as possible. Note that this specifi-
cation renders a cointegration rank of ri = N − 1 = 5 implausible a priori, because such a
result would imply pairwise cointegration between all variables, including pairs such as
inflation and deterministically trending productivity levels.
Another nested specification would be to impose long-run homogeneity between pro-
ductivity and wages, based on the prior belief that the other four variables do not contain
a deterministic trend. Then there would be a unique proportion of wages and productivity
canceling the linear trend. This assumption is a testable restriction, formulated as
βi,51
βi,5 j = βi,6 j
,
j = 2, ..., ri,
(3)
βi,61
where it is assumed without loss of generality that productivity enters the first cointe-
grating vector and thus βi,61 = 0. Furthermore, the stronger restriction of a known 1:1-
proportionality between wages and productivity in the long run can be formulated as the
following linear hypothesis:
 0

4×1
I4
βi = 
1
 φi,
(4)

0

2×4
−1
where φi is unrestricted. This restriction means that the cointegration space could be
simplified by using the (log) labor share wi,t − qi,t instead of wages and productivity sep-
arately. If hypothesis (4) were accepted, one could also consider testing additional short-
run homogeneity restrictions, but unreported test results provide clearcut evidence against
the homogeneity restrictions (3) and (4).
Our approach in this paper is to follow modern macroeconomic theory by specifying
only few common stochastic trends driving the data in the long run – for the extreme
case of only a single trend see King et al. (1991). To check whether this assumption is
supported by the data we directly apply the appropriate common-trends test suggested by
Nyblom and Harvey (2000), complementing the popular Johansen test procedure. The
Nyblom-Harvey test is a multivariate generalization of the better-known KPSS test for
stationarity of a univariate series (see Kwiatkowski et al., 1992). In contrast to the Jo-
hansen test, it tests the hypothesis of few stochastic trends against the alternative of a
higher number, i.e., the null and alternative hypotheses are reversed.
Our general strategy for the choice of the number of stochastic trends (or equiva-
lently of the cointegrating rank) is as follows: First, we apply the common-trends and
cointegration tests. However, if the tests produce an ambiguous outcome,5 we will apply
5For example, one that is very sensitive with respect to the lag truncation parameter – the lag truncation
5

---

a “litmus” test as an additional guide for choosing the cointegration rank, based on the
following considerations: Since the whole idea of the present paper is to estimate the non-
stationary component of unemployment rates, a requirement of the chosen model is that
it should classify unemployment as non-stationary. Given a certain cointegration rank r,
the stationarity of unemployment is a standard, and testable, restriction on the cointegra-
tion space, namely that one of the stationary relations is formed by unemployment alone.
Now if a model with a certain rank ri,0 were unable to reject this test hypothesis of sta-
tionary unemployment even at moderate significance levels, we would conclude that such
a model is useless for our purposes, and we would choose a different cointegrating rank.
2.3
Extracting permanent components
Finally, after having determined the number of common stochastic trends in each system,
the main goal of this study is to estimate the permanent component of each unemployment
rate. We use two established permanent-transitory decompositions for that, first the mul-
tivariate generalization of the Beveridge-Nelson decomposition based on the estimated
common trends, and secondly the one by proposed by Gonzalo and Granger (1995). Usu-
ally the common trends are represented in the moving-average form used by Stock and
Watson (1988), but for reasons to be explained below we will use a different approach
suggested by Proietti (1997).
Starting with the Gonzalo-Granger (GG) decomposition for country i, it is defined as
yi,t = βi⊥(αi⊥βi⊥)−1αi⊥yi,t + αi(βi αi)−1βi yi,t,
(5)
where the orthogonal complements αi⊥ and βi⊥are N ×(N −ri)-matrices such that αi⊥αi =
0 = βi⊥βi. Note that although the choice of the orthogonal complements is not unique,
the concrete choice is irrelevant for the decomposition in (5). Given that β y
i i,t are the
stationary error-correction terms, the transitory component of yi,t is obviously given by
αi(β
y
i αi)−1βi i,t , whereas the permanent part consists of the common factors αi⊥yi,t that
drive the system and are loaded into the variables by the coefficients βi⊥(αi⊥βi⊥)−1.
Given the additive nature of the identity (5) and the fact that no deterministic and/or
exogenous terms appear there, it is clear that its transitory part is only stationary around
the path of any restricted terms of the model (2), and that the absolute level of the perma-
nent part does not automatically correspond to the levels of yi,t. For our purposes this is
determines the lag window that is used to calculate the estimated long-run variance of the time series in the
nonparametric correction of the test statistic for autocorrelated series. It is a well-known problem that the
choice of that bandwidth parameter often affects the results substantially in real-world samples. Typically,
too small a lag truncation value implies an oversized test, whereas a value that is too large reduces power
considerably. Unfortunately, in contrast to the univariate case, in a multivariate setting the literature does
not provide a data-based procedure to choose an optimal bandwidth.
6

---

inconvenient, and therefore we have adjusted the components by the following two steps.
First of all, we correct for the mean of the stationary directions, which is known to be
(Johansen, 1995)
(αi αi)−1αi (ΨiCi − I)µi,
(6)
where
k−1
Ψi = I − ∑ j=1 Γi, j, and Ci = βi⊥(αi⊥Ψiβi⊥)−1αi⊥ is the long-run impact matrix of
reduced rank N − ri. Thus in the first step we subtract the estimate of (6) premultiplied by
the relevant loadings αi(βi αi)−1 from the transitory component and add that to the per-
manent component to preserve the additivity of the decomposition. Secondly, to account
for any exogenous terms restricted to the cointegration space –i.e., the step dummies for
Germany and Italy– we compare the raw error correction terms β y
i i,t to the augmented
ones including the restricted terms
∗
β
y∗ , and again subtract that difference times the
i
i,t
relevant loadings from the transitory component and add it to the permanent component.
Altogether, the resulting permanent GG-component of yi,t looks as follows:
yGG
∗
i,t
= βi⊥(αi⊥βi⊥)−1αi⊥yi,t + αi(βi αi)−1 (αi αi)−1αi (ΨiCi − I)µi + βi yi,t − βi y∗i,t ,
(7)
where the estimate of the natural unemployment rates is simply the first element of the
vector.
The main identifying feature of the GG decomposition within the class of permanent-
transitory decompositions is that the permanent component is a linear combination of the
contemporaneous observables, which renders the interpretation of the permanent factors
easier. However, this implies that in general the GG permanent component is autocorre-
lated in differences, which means that the permanent component will change even if no
new shocks hit the system. This property makes it appear less useful as an equilibrium
measure.
Another popular approach is therefore to find the multivariate random-walk compo-
nent of the variable set, based on the Stock-Watson (SW) common trends. The random-
walk property of this multivariate Beveridge-Nelson decomposition means that changes
of its permanent component are not forecastable. Thus it provides the long-run forecast
for the included variables, given the information of the current period. Of course, the GG
and SW permanent components only differ by stationary terms and must be cointegrated,
therefore they share the same long-run features. Note that the GG and SW decomposi-
tions actually coincide if there are common cycles in the data, and if their number is equal
to the cointegration rank (Proietti, 1997).
Given the estimated coefficients of the VECM (2) as before, and assuming a fixed
7

---

initial value, the permanent SW components are given by
t
ySW
i,t
= yi,0 +Ciµit +Ci ∑ εi,s.
(8)
s=1
However, we do not apply the formula (8) directly, because the common trends must
be estimated from the residuals of the model, and they are therefore heavily influenced
by outliers and other features that the empirical model may not have fully captured. We
therefore follow Proietti (1997) and instead determine the common-trends based perma-
nent components as a distributed lag of the observable variables:
ySW P
∗
∗
i,t
= (I − Pi)(Γi (1) + αiβi )−1Γi (L)yi,t
(9)
where P
∗
∗
∗
i = (Γ (1) +
)−1
(
(1) +
)−1
, with
=
, and
i
αiβi
αi[βi Γi
αiβi
αi]−1βi
Γi, j
Γi, j + αiβi
∗
k
∗
Γ (L) = I −
i−1
L j. For any further terms such as the German level shift dummy,
i
∑ j=1 Γi, j
we apply the same adjustment as for the GG decomposition. Note that if the true data-
generating process exactly obeys (2), then the SW and the SWP components are iden-
tical. But the important feature of the SWP component is that any outliers and other
non-modelled features in the data can affect the estimated permanent components at most
for ki periods, not permanently as with the pure SW representation.
2.4
A bivariate illustration of the approach
Before dealing with the full six-dimensional systems we demonstrate our approach in a
small bivariate system. For this purely illustrative purpose consider the Dutch unemploy-
ment rate and the tax wedge with k = 2 lags, such that in addition to the cointegration and
adjustment coefficients only one matrix of short-run coefficients needs to be estimated.
For the sake of the illustration we will ignore the remaining residual autocorrelation, and
also simply impose the cointegration rank of one here. The resulting system estimate with
β normalized on the unemployment rate is as follows:

u

NL,t−1
uNL,t
−0.022
∆
=
1 −0.111 3.736
 twNL,t−1 
tw


NL,t
0.114
1
0.958 0.001
uNL,t−1
+
∆
+ εt
1.072 0.166
twNL,t−1
Note that here the constant is restricted to the cointegration relations, in contrast to the
specification of the larger systems below. The needed quantities for the SWP components
8

---

are
∗
0.936 0.003
Γ (L)
= I −
L,
1.186 0.153
−1
1.129 −0.125
24.561 0.029
(I − P)
∗
Γ (1) + α β
=
I −
×
1.164 −0.129
31.570 1.237
0.784 0.151
=
,
7.060 1.362
such that the SWP random-walk component for the unemployment rate would be given
by uSW P = 0.784u
NL,t
NL,t + 0.151t wNL,t − 0.913uNL,t−1 − 0.026t wNL,t−1, if we abstract from
the restricted constant term for simplicity here. This could alternatively be expressed
as uSW P = −0.129u
NL,t
NL,t + 0.125t wNL,t + 0.913∆uNL,t−1 + 0.026∆t wNL,t−1.
Taking into
account the different scales and variations of the two variables (see section 3) we see that
the bulk of the long-run variations in the unemployment rate is attributed to the tax wedge
in this (merely illustrative) case.
Indeed, the adjustment coefficient in the twNL-equation is insignificant with a t-ratio
of 1.12, which suggests that the tax wedge here is weakly exogenous. Imposing this
zero-restriction on the second row of α would imply that α⊥ = (0; 1) , and hence the
permanent shocks would be given by the innovations in the tax wedge equation alone.
Equivalently, the permanent factor in the Gonzalo-Granger sense would be given by the
tax wedge, since here α (
⊥ uNL,t , twNL,t ) = twNL,t . Furthermore, in a bivariate system with
r = n − r = 1 there is an algebraic symmetry between
2
α and α⊥ which implies that the χ1
tests of weak exogeneity mirror the tests of exclusion from the Gonzalo-Granger factors.
Thus the p-values of the weak exogeneity test on the tax wedge would be identical here to
the ones of the factor exclusion test on unemployment. However, in a higher-dimensional
system there are in general more possibilities.
3
Descriptive evidence and initial tests
3.1
Time series developments
Unemployment: The developments of unemployment in France and Italy are quite sim-
ilar in the longer run (see figure 1), although recently (since 2002) the movements are
in opposite directions. Due to steep increases during the 1990s, Germany has recently
“managed” to catch up to French and Italian unemployment levels. The Dutch unemploy-
ment rate presents a completely different picture, showing a downward trend for roughly
two decades now. Although it looks as if unemployment in the Netherlands could well
9

---

12
DE
FR
10
IT
NL
8
6
4
2
1975
1980
1985
1990
1995
2000
2005
Figure 1: Unemployment rates
be stationary, standard ADF tests confirm the findings of Papell et al. (2000) that the unit
root cannot be rejected, even in this longer sample (not reported to save space). Therefore
the Dutch case will be interesting as a cross-check for other countries’ findings because
the unemployment development there has been completely different.
Wage and productivity dynamics: Labor productivity and real hourly labor costs them-
selves are not shown because they are in general smoothly growing and therefore they are
difficult to analyze visually. As a closely related substitute graph consider the labor shares
in the four countries in figure 2, which have been calculated as the gap between (log) labor
costs and (log) labor productivity. From the downward developments we see that on av-
erage labor productivity has grown faster than real labor costs. The Italian share displays
somewhat different behavior in that it has been especially low since the mid 1990s, which
is partially due to reforms that lowered the social security contributions.
Inflation dynamics: All inflation series except perhaps the Dutch data exhibit the typi-
cal gradual disinflation process that implies non-stationarity of inflation within the sample
(see figure 3). The degree of disinflation is highest for Italy and lowest for Germany. For
the Netherlands, the overall situation is again a little different, and its inflation rate may
well be empirically I(0). However, ADF tests are unable to reject at the 5% level.6
Relative external prices: Interestingly, the external price wedges as measured by the
6The nominal (hourly) wage inflation displays roughly the analogous behavior, albeit with more short-
term volatility. The inflation of imported raw materials (in local currency) is virtually identical for all four
countries, since the price volatility dwarfs any exchange rate discrepancies. Also, it is clearly stationary.
The inflation series of non-commodity imports is much more volatile than the domestic inflation series and
contains a less pronounced downward tendency over the longer run. (All those series and results are not
shown to save space.)
10

---