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Some New Estimates of Returns to Scale for EU Regional Manufacturing, 19862002
Alvaro Angeriz
1
, John McCombie
2
, and Mark Roberts
3
Cambridge Centre for Economic and Public Policy
Department of Land Economy
University of Cambridge
19 Silver Street
Cambridge, CB3 9EP
Abstract: Recent theoretical advances have emphasised the importance of localised increasing
returns to scale in understanding both the regional growth and agglomeration processes.
However, considerable empirical controversy still exists over whether returns to scale are
constant or increasing. Consequently, this study aims to provide some new estimates of the
degree of returns to scale for EU regional manufacturing. It does so within the framework of
the Verdoorn law. Unlike previous studies, issues of specification of fundamental importance to
recent theoretical developments are brought to attention. Overall, the paper concludes that
localised increasing returns in EU regional manufacturing are substantial.
JEL codes: O18, O33, R11
Keywords: increasing returns, Verdoorn law, manufacturing, productivity growth, spatial
econometrics.
1
Wolfson College, Cambridge.
2
Downing College, Cambridge.
3
Also New Hall and Girton College, Cambridge.
1
1. Introduction
An understanding of whether or not regional production within the European Union (EU) is
subject to increasing returns to scale is crucial for policymakers and economists alike. For both,
it is crucial for an assessment of the potential impact of continuing integration on the economic
geography of the EU, with implications both for social cohesion and the future evolution of
regional policy. For economists, it is further important from a theoretical viewpoint, especially
in view of developments in growth theory over the past two decades and the emergence of a
new field of "geographical economics" that aims to model the centripetal and centrifugal forces
that underlie the spatial distribution of economic activity.
To elaborate, it has traditionally been the case that economists' models of growth at both the
national and regional levels have been based upon the assumption of constant returns to scale,
as have models of the spatial distribution of activity based upon the static concept of
comparative advantage. Thus, the neoclassical model of Solow (1956) and Swan (1956)
provides the traditional growth model and, in this model, the existence of constant returns to
scale, combined with an associated pure public good treatment of technology, implies a stable
process in which all regions should converge to the same steadystate growth path.
Furthermore, by increasing the mobility of both capital and labour, regional integration should
eliminate not only growth rate differences between regions, but also longrun differences in
levels of income per capita and productivity, so that a process of absolute convergence results.
Likewise, when its assumption of no factor mobility is relaxed, the classic HeckscherOhlin
model implies that interregional differences in underlying factor endowments should disappear,
thereby engendering an associated convergence in regional production structures.
2
However, since the mid1980s, there has been a sustained theoretical effort to replace the
assumption of constant returns to scale with that of increasing returns to scale. In growth
theory, this effort has been driven by a recognition that growth is endogenous rather than
exogenous, and, in particular, by the argument that it is the result of decisions made by
economic agents rather than technological progress arriving as "manna from heaven." Thus,
technological progress, and, therefore, economic growth, has been modelled as both the
accidental, and indirect, outcome of decisions to invest in capital accumulation (Romer, 1986)
and the intentional outcome of decisions to invest in the production of new technologies
(Romer, 1990; Grossman and Helpman, 1991; Aghion and Howitt, 1998). In both cases,
fundamental to the story of endogenous growth is the existence of knowledge spillovers, leading
to the existence of increasing returns, as, without increasing returns, growth would dryup in the
absence of an exogenous driving force. Meanwhile, with respect to the spatial distribution of
economic activity, it has been realised that this is often difficult to explain using the static
notion of comparative advantage. Thus, without increasing returns, it is difficult to explain why
dense agglomerations of economic activity continue to exist even when the historical reasons
that led to their establishment have disappeared, and this is the case at many different spatial
levels (Krugman, 1991). Indeed, a "problem of backyard capitalism" arises by which it would
be expected that every household would produce a fully diversified range of commodities in its
own backyard so that the distribution of economic activity was uniform across geographic
space. It is this realisation of the importance of increasing returns in explaining the spatial
distribution of activity that has led to the development of geographical economics by, inter alia,
Fujita, Krugman and Venables (see, for example, Fujita, Krugman, and Venables, 1999). Along
with this have gone policy implications shared with endogenous growth theory, for the
existence of regional increasing returns implies that integration brings, at least the potential of,
intensified forces for divergence in regional production structures, growth rates, income per
capita and productivity levels.
3
However, although the above developments mean that the potential role of increasing returns in
driving spatial processes of growth and distribution is now the subject of widespread research
interest, it is important to point out that the modern emphasis on increasing returns in such
processes is actually considerably predated by a related literature. Thus, there is a notable
literature inspired by Myrdal (1957) and Kaldor (1966, 1970) that pinpoints increasing returns
to scale as the source of "circular and cumulative" processes in space, where increasing returns
are given a wide interpretation so as to incorporate not only conventional static sources, but also
the dynamic sources of knowledge spillovers and induced technological progress that
mainstream endogenous growth theory has pickedup on. In this literature, increasing returns
are captured by the socalled Verdoorn law, which asserts the existence of a positive
relationship between either labour productivity or total factor productivity (TFP) growth and
output growth.
4
The estimation of this relationship then provides an explicit means of testing
for increasing returns to scale, be it at the national level (Kaldor, 1966) or the regional level
(McCombie and de Ridder, 1984).
Despite the sustained theoretical efforts to replace the assumption of constant returns to scale
with that of increasing returns to scale that have been outlined above, it is interesting to note
that the empirical subject of whether or not returns to scale are constant or increasing at the
regional level is far from being resolved. Hence, whilst many geographical economists have
been content to refer to the “backyard capitalism” argument as providing sufficient proof of the
existence of localised increasing returns, actual empirical work on the subject is far from
arriving at a consensus. Studies estimating regional production functions, for example, have
traditionally found either constant returns or very small increasing returns (Moroney, 1970;
Griliches and Ringstad, 1971; Douglas, 1976), whilst findings of crossregional convergence are
4
For a collection of some of the latest developments in this literature see McCombie et al
(2002).
4
often interpreted as being consistent with the traditional SolowSwan model and, therefore,
constant returns to scale (see, inter alia, Barro and SalaiMartin, 1991, 1992; Mankiw, Romer
and Weil, 1992). Furthermore, whilst timeseries estimation of industry production functions
(expressed in terms of growth rates) have been found to indicate the existence of substantial
externalities in production (Caballero and Lyons, 1992), these have been subject to criticism by,
for example, Basu (1995). Finally, there has, at the national level, been a distinct absence of
“scale effects”, whereby, even if increasing returns were only small at this level, we would
expect a distinctive positive relationship in the data between country population sizes and
productivity levels for countries at the same level of development. There is no evidence of
support of this conjecture (Jones, 2002).
In this context, the Verdoorn law literature mentioned above is important. This is because, as
indicated, the law provides an interesting means of testing for significant increasing returns at
the regional level. Indeed, previous work estimating this law for the European regions has
found evidence of substantial increasing returns (Fingleton and McCombie, 1998; PonsNovell
and ViladecansMarsal, 1999).
5
However, at the same time as providing support for the key
assumption of both endogenous growth theory and geographical economics, these studies
provide a challenge to both of these theoretical literatures. This is because by specifying the
Verdoorn law with output growth as the regressor, they hold true to the Kaldorian origins of the
law in seeing the regional growth and agglomeration processes as being fundamentally demand
driven. By contrast, both endogenous growth theory and geographical economics are
neoclassical approaches and, therefore, much more supplyoriented in their focus (Roberts and
Setterfield, 2006). Still, this does not mean that Fingleton and McCombie (1998) and Pons
Novell and ViladecansMarsal (1999) are necessarily correct in their Kaldorian specification of
the law. Indeed, in this context, there is an old controversy surrounding the issue of
5
Evidence of substantial increasing returns have also been found in other regional samples, not
to mention in crosscountry and crossindustry data (McCombie et al, 2002).
5
endogeneity and the proper specification of the law that both sets of authors abstract from (see
Kaldor, 1975; Rowthorn, 1975a, 1975b). There is, furthermore, a paradox in the specification
of the law, confirmed by Fingleton and McCombie (1998), which raises doubt over the findings
of substantial increasing returns for the European regions. This is the socalled “staticdynamic
Verdoorn law paradox” of McCombie (1981). In particular, it has previously been found that
when the law is respecified from being in terms of growth rates (the dynamic version of the law)
to being in terms of log levels (the static version of the law), constant or decreasing rather than
increasing returns to scale are found. This is despite both versions of the law being estimated
using a common dataset.
Given the controversy surrounding the empirical question of whether or not returns to scale at
the regional level are constant or increasing and the theoretical and policy importance of this
question, this paper aims to provide some new estimates of the degree of returns to the scale for
European regional manufacturing. Although it does so in the context of the Verdoorn law
framework that has been previously been used by both Fingleton and McCombie (1998) and
PonsNovell and ViladecansMarsal (1999), the paper represents a considerable advance on the
work of both of these sets of authors. First, the paper explicitly considers both the Verdoorn
law controversy concerning endogeneity in the specification of the law and the staticdynamic
Verdoorn law paradox on the grounds of the relevance of both to modern theoretical and policy
debates. In particular, with respect to the former, both specifications of the law with output
growth and factor inputs are estimated, and instrumental variable (IV) techniques are employed.
Meanwhile, with respect to the latter, a possible resolution to the paradox suggested by
McCombie and Roberts (2006) is tested. This suggestion implies that the static, but not the
dynamic, version of the Verdoorn law is misspecified because of the existence of a spatial
aggregation bias. Secondly, rather than just estimating the simple Verdoorn law with a single
regressor and no consideration of capital accumulation, an augmented specification is estimated
6
in which total factor productivity growth is the dependent variable and in which the independent
influence of both technological diffusion and agglomeration economies arising from the density
of production in a region are taken into account. Thirdly, and finally, estimation of the
Verdoorn law is conducted within a spatial econometric framework. Although the studies of
both Fingleton and McCombie (1998) and PonsNovell and ViladecansMarsal (1999) are also
set within such a framework, the spatial econometric approach adopted in this study is both
more sophisticated and theoretically driven. Indeed, in itself, it represents a contribution to the
spatial econometric literature with the estimation of a new spatial specification presented in the
results reported.
The structure of the rest of this paper is as follows. The next section introduces the Verdoorn
law as a means of testing for increasing returns to scale. In so doing, it examines both the
theoretical basis of the law and its augmentation. It also discusses both the Verdoorn law
controversy regarding the question of endogeneity and the staticdynamic Verdoorn law
paradox. Following this, spatial econometric issues in the estimation of the law are considered
and our preferred spatial econometric model introduced. The econometric results obtained are
then presented and discussed. The final section offers some concluding thoughts.
2. The Verdoorn law  theoretical framework, controversy and a paradox
2.1. The Verdoorn law and its theoretical framework
The traditional Verdoorn law
Traditionally, the Verdoorn law has been estimated as a linear relationship between labour
productivity growth and output growth (Kaldor, 1966):
7
p
j
= c
1
+ b
1
q
j
(1)
where p and q are the growth rates of manufacturing labour productivity and output respectively
of region, or country, j. The coefficient b
1
is the Verdoorn coefficient and it traditionally takes a
value of 0.5 (Kaldor, 1966), with Fingleton and McCombie (1998) and PonsNovell and
ViladecansMarsal (1999) obtaining estimates of 0.575 and 0.628 respectively for their samples
of European regions.
6
Notoriously absent from the above the specification of the Verdoorn law, however, is the
growth of the capital stock (McCombie and de Ridder, 1984). Neither Fingleton and McCombie
(1998) nor PonsNovell and ViladecansMarsal (1999) include this because of an absence of
data on gross investment, relying instead on the explicit or implicit hypothesis that the capital
output ratio is constant. To examine the consequences of this absence, assume that the
Verdoorn law is derived from a CobbDouglas production function of the form:
7
(2)
)a1(
j
ta
jj
)LAe(KQ
−
=
λ
where Q, K, and L are the levels of output, capital, and labour respectively. Meanwhile,
λ
is the
rate of technological progress and a and (1a) are production function parameters, which under
the assumption of constant returns to scale, equal the shares of K and L in Q respectively.
6
Fingleton and McCombie (1998) use a sample of 178 NUTS2 regions for the period 1979
1999, whilst PonsNovell and ViladecansMarsal (1999) use a sample of 74 NUTS1 regions for
the period 19841992.
7
The assumption that the Verdoorn law is derived from a CobbDouglas production function is
not innocuous, as we shall see in Section 2.3.
8
A key assumption of the Verdoorn law is that the rate of technological progress is largely
endogenously determined.
8
This can occur, for example, through localised knowledge spillovers
emanating from learningbydoing or induced technological change. To capture this, specify
λ
as:
(
)
(
)
laak −++= 1
~
πλλ
(3)
where the lower case variables denote exponential growth rates, so that a faster growth of the
(weighted) factor inputs leads to faster TFP growth.
Substituting equation (3) into equation (2) gives:
(4)
βλα
)LAe(KQ
j
t
jj
′
=
where
α
and
β
are the observed output elasticities of capital and labour respectively, (
α
+
β
>
1
); and
α
=(1+(1a)π)a = va and
β
=(1+(1a)π)(1a) = v(1a) where v is the degree of returns
to scale. Note, however, that
v is more encompassing that the traditional definition of returns to
scale as it also includes the effect of the induced rate of technological change,
φ
(ak + (1a)
l
).
The rate of exogenous technological change is given by:
v
λ
λ
~
'= .
Consequently, taking logarithms of equation (4), differentiating with respect to time, and
rearranging gives:
8
This assumption is shared with endogenous growth theory.
9
jjj
kq
1
p
β
α
β
β
λ
+
−
+
′
= (5)
Augmenting the Verdoorn Law
Since the work of Fingleton and McCombie (1998) and PonsNovell and ViladecansMarsal
(1999), data on gross investment has become available for the European regions, allowing for
the construction of a measure of
k.
9
Given this, the use of OLS to estimate equation (5) seems
inappropriate because it is likely that
k is endogenous, being largely determined by the growth
of output (Kaldor, 1970). To tackle this, the Verdoorn law can be respecified as:
jj
q
vv
tfp
⎟
⎠
⎞
⎜
⎝
⎛
−+
′
=
1
1
λβ
(7)
where tfp = q – a k – (1a)
l
is the growth of total factor productivity (TFP).
Equation (7) is a more flexible form of the production relationship than that derived from the
CobbDouglas production function, as it allows for the factor shares, and, therefore, the
underlying production technology, to vary both between regions and over time.
However, even this “simple” Verdoorn law attributes all of the crosssectional variation in
productivity growth to induced knowledge spillovers and technological change resulting from
increasing returns, broadly defined. Yet, consistent with endogenous growth theory (see, for
example, Barro and SalaiMartin, 2004, chapter 8), part of the variation in
tfp
j
could equally be
due to the diffusion of innovations from hightechnology to low technology regions.
9
For a description of the data used in this paper and the construction of the capital stock
estimates, see the data appendix.
10
Furthermore, in equation (7), the realisation of increasing returns is clearly demanddriven
through output growth, but recent theoretical advances looking to combine insights from
endogenous growth theory with those from geographical economics (Baldwin, 1999; Baldwin
and Martin, 2003), suggest that the density of production within a region might be a source of
dynamic agglomeration economies and, therefore, increasing returns. It follows that variation in
the density of production might also help to explain the variation in
tfp
j
.
To capture the above possibilities, the Verdoorn law is augmented as follows:
011
lnln
1
1 TFPDq
vv
tfp
jjj
θζ
λβ
++
⎟
⎠
⎞
⎜
⎝
⎛
−+
′
= (8)
where ln
TFP
0
is the log of the initial level of TFP for region j and is intended as a proxy for the
initial level of technology. Of course, if the diffusion hypothesis is correct then
θ
< 0 should
hold.
10
Meanwhile, lnD
j
is the log of region j’s output density (D
j
), where D
j
= Q
j
/H
j
with H
j
being the area of region j in sq. km.
11
Equation (8) implies that the density of production within region
j has an effect on its steady
state growth path, and, because it is specified as a relationship between TFP growth and output
growth, we label it the
dynamic Verdoorn law. However, an alternative is to define D
j
as only
having a “level effect” as is done in, for example, the empirical work of Ciccone (2002) and
Ciccone and Hall (1996). In this case,
D
j
only affects the level, and not the longrun growth
10
Fingleton and McCombie (1998) also attempt to proxy for the initial level of technology.
However, given their lack of capital stock estimates, they make use of the initial level of labour
productivity. This is less satisfactory than the initial level of TFP because variations in labour
productivity might equally be attributable to variations in the capitallabour ratio, thereby
leading to SolowSwan style conditional convergence rather than technological diffusion.
11
The level of output is taken to be that of the initial level to avoid the possibility of reverse
causation from TFP growth to
lnD
j
. Using the average density of production over the period
made little difference to the results obtained.
11
rate, of TFP. This does not, however, affect the specification of equation (7), i.e. the non
augmented dynamic Verdoorn law, merely its interpretation. In fact, in this case, it is not
possible to directly test for the independent influence of agglomeration economies arising from
the density of production. To see this, assume, as before, that the underlying functional form is
provided by a CobbDouglas production function. However, this time, it takes the form:
12
β
λ
α
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
′
j
j
t
j
j
j
j
H
L
Ae
H
K
H
Q
(9)
Consequently, the Verdoorn law in loglevel form (which, it will be recalled we term the
static
Verdoorn law
) is now given by:
jjj
H
v
QtATFP ln
1
ln
1
1'lnln
⎟
⎠
⎞
⎜
⎝
⎛
−−
+
⎟
⎠
⎞
⎜
⎝
⎛
−+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+=
βα
ν
λ
ν
β
(10)
Consequently, if there are increasing returns to scale, the greater the density of production is
(i.e., the lower
H
j
is, ceteris paribus), the higher the level of TFP will be. With constant returns
to scale, though, the density of production has no effect.
As
H is constant over time, however, note that, when expressed in growth rate (i.e.: dynamic)
form, the Verdoorn law given by equation (10) is the same as equation (7). This has a relevant
consequence when
D
j
only has a level effect, as in this case the dynamic Verdoorn Law does not
allow the separate influence of agglomeration economies to be disentangled from that of
increasing returns, interpreted more generally.
12
For expositional ease, any possible technological diffusion effect is ignored.
12
2.2. Endogeneity and the Verdoorn law controversy
The specification of the augmented dynamic Verdoorn law in equation (8) holds true to the
Kaldorian origins of the law. Thus,
q
j
is specified as an exogenous and independent determinant
of
tfp
j
, so that demand growth is seen as the fundamental driving force behind the regional
growth and agglomeration processes. However, the specification by Kaldor (1966) of the law
(in its original guise of equation (1)), was criticised by Rowthorn (1975a). In particular,
Rowthorn argued that, in the context of the argument that Kaldor was using the law,
q
j
was
endogenous to employment growth, which implies that, in our augmented specification,
q
j
is
endogenous to
tfp
j
. On these grounds, Rowthorn (1975a) advocated respecifying equation (1) as
p
j
= c
2
+ b
2
l
j
where
l
is the growth of employment in region j. In terms of our augmented
dynamic Verdoorn law, this is equivalent to respecifying equation (8) as:
()
j
jjj
TFPDtfitfp
,022
lnln1'
θ
ζ
ν
βλ
+
+
−+= (11)
where
tfi = a k – (1a)
l
denotes the growth of total factor inputs in region j.
In respecifying the law, Rowthorn found that, using the same dataset as Kaldor (1966), he could
not reject the hypothesis of constant returns to scale. This is equivalent to finding a coefficient
on
tfi
j
in equation (11) that is not significantly different from zero Kaldor (1966), on the other
hand, found significant increasing returns to scale using equation (1), which is identical to
finding a coefficient on
q
j
in equation (8) that is significantly greater than zero.
13
13
Kaldor's original sample consisted of 12 advanced countries for the early postSecond World
War period. Rowthorn used the same sample as Kaldor with the exception that he dropped
Japan on the grounds that it was an outlier.
13
The reason for the divergence in the implied estimates of
ν
obtained by Kaldor and Rowthorn
can, however, be easily understood. It occurs because the relationship between the two slope
coefficients in the original Kaldorian and Rowthorn specifications of the dynamic Verdoorn law
is given by
= R)1)(1(
21
bb +−
ˆˆ
ˆ
ˆ
ˆ
2
. Given that Kaldor (1966) and many subsequent studies have
found = 0.5 (implying increasing returns) and that, in crosssectional data,
R
1
b
2
usually
presents a reasonably good fit of 0.5, it follows that ≈ 1 ⇒ (implying constant
returns). In the case of the augmented specifications, this indicates that the true estimate of
ν
will lie between the (lower bound) estimate obtained from equation (11) and the (upper bound)
estimate obtained from equation (8).
)b1(
2
+ 0
2
=b
Although Kaldor (1975) argued that Rowthorn (1975a) had misinterpreted his original argument
behind the use of the Verdoorn law and although there are persuasive reasons for believing that
regional growth is demand driven (Thirlwall, 1980), the above discussion is clearly of modern
relevance. In particular, given that they build on conventional production functions,
endogenous growth models suggest that causation runs from the growth of factor inputs to
output growth, i.e., from the supplyside of the economy to the demandside. By contrast, we
know that the Kaldorian origins of the Verdoorn law suggest the opposite. However, even here,
there is acknowledgement that the regional growth and agglomeration processes are circular and
cumulative with feedback from productivity growth to output growth (Dixon and Thirlwall,
1975). This being the case, the use of OLS to estimate either equation (8) or equation (11) will
be subject to simultaneity bias. Consequently, an instrumental variable (IV) estimator should be
used (Rowthorn, 1975b) and, ideally, this should help to bring about a convergence of the
estimates of
ν
obtained from the two specifications.
Even the use of an IV estimator, however, has not proved to be without its problems. Thus, in a
previous study with nonregional data, McCombie (1981) advocated using Durbin’s ranking
14
method where the instrument is the rank of the regressor. This raises two problems. First, it
implies that whereas in equation (8) the instrument is the rank of
q
j
, in equation (11) it is the
rank of
tfi
j
. Second, if more than one instrument is used, the model is overidentified. In both
cases, the method of normalisation, i.e., whether
q
j
or tfi
j
is chosen as the regressor, affects the
estimates. Hence, the difference in the estimates of the degree of returns to scale still remains.
McCombie and de Ridder (1984) found that, for the US states, both specifications gave
estimates of substantial increasing returns to scale, although the Kaldorian specification of the
Verdoorn law gave a larger figure. Here the explicit inclusion of the growth of the capital stock
meant that the
R
2
was sufficiently good that, consistent with Wold’s proximity theorem (Wold
and Faxer, 1957), the estimates of
v converged.
2.3. The StaticDynamic Verdoorn law Paradox
Equations (8) and (10) give, what have been referred to as, the
dynamic Verdoorn law and the
static Verdoorn law respectively. In particular, ignoring both the possibility of technological
diffusion and agglomeration economies arising from the density of production, the dynamic
Verdoorn law can be derived from its static counterpart by differentiating with respect to time.
This being the case, it might be expected that the estimation of the following two equations
would give identical estimates of
ν
:
jj
q
vv
tfp
⎟
⎠
⎞
⎜
⎝
⎛
−+
′
=
1
1
λβ
(12)
jj
QtATFP ln
1
1'lnln
⎟
⎠
⎞
⎜
⎝
⎛
−+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+=
ν
λ
ν
β
(13)
15
However, this has not been found to be the case in previous studies, including those for the
European regions (Fingleton and McCombie, 1998).
14
In particular, it has been found that
whereas dynamic specifications of the Verdoorn law give estimates of ν significantly greater
than unity, static specifications do not. This is a puzzle, notwithstanding the different
assumptions underlying any error terms appended to static and dynamic specifications of the
law. Consequently, there has previously been found to exist a paradox in the estimation of the
Verdoorn law, namely, the staticdynamic Verdoorn law paradox (McCombie, 1981).
A possible explanation for the above paradox is provided by McCombie and Roberts (2006)
through the concept of
spatial aggregation bias. They argue that the ideal unit of observation is
not the (administrative) region (of which the NUTS1 regions used in this paper are examples),
but, what they term, the Functional Economic Area (FEA). The FEA is the area over which
substantial agglomeration economies occur and is likely to be determined by various factors,
such as journey to work patterns, for instance. These authors suggest that any particular region
is likely to consist of a number of FEAs, with the larger regions containing proportionately
more. The spatial aggregation error occurs because the data for each region are the values of
output, employment, and capital for each constituent FEA summed
arithmetically. This
potentially biases (the static) estimates of
ν
obtained from equation (13) towards constant
returns to scale. To see this, assume that the true specification of the Verdoorn law for an FEA
is given in static form by where
i denotes the particular FEA, j is the region in
which it is located.
γ
ijij
BQL =
15,16
The underlying assumption is that
γ
= 0.5 and so at the FEA level it is
immaterial whether the law is estimated in static or dynamic form. For expositional ease,
14
Although note that Fingleton and McCombie (1998) do not estimate versions of the Verdoorn
law that allow for capital accumulation.
15
To simplify the argument, we ignore the possibility of capital accumulation. Allowing for
such accumulation does not affect the nature of the argument
16
This is formally equivalent to: , as by definition
)1(
1
γ
−
−
=
ijij
QBP
L
Q
P ≡
and
(
)
γ
−1 is the Verdoorn
coefficient.
16
assume that all of the FEAs are the same size and that the smallest region contains one FEA, the
second smallest region two FEAs, and so on. Given these assumptions, the recorded levels of
employment and output will take the form reported in Table 1.
TABLE 1 HERE
It can be seen that when the aggregate crossregional data is used to estimate the static Verdoorn
law, it will suggest constant returns to scale. Thus, if this is the correct explanation, the dynamic
Verdoorn law (i.e. equation (12)) is the correct specification. Using simulation analysis,
McCombie and Roberts (2006) show that this result is robust even when the sizes of the
individual FEAs are allowed to vary, provided that they are relatively small compared with the
size of the average region. They also show that timeseries estimation of the static Verdoorn
law will give an unbiased estimate of
ν
, provided the inherent problem of variations in capacity
utilisation is solved. Furthermore, they demonstrate that a oneway fixedeffects (FE) estimation
of the static Verdoorn law will lead to a biased estimate of
ν
, i.e., it will suggest constant returns
to scale, by picking up the crosssection variation. However, the twoway FE estimator gives an
unbiased estimate, as it also employs the timeseries variation in the data.
Clearly, assuming that the staticdynamic Verdoorn law paradox is found to hold for the data set
in this paper, it is important to test the above hypothesis concerning its explanation. This is
especially the case, because the aforementioned hypothesis indicates significant increasing
returns to scale do exist at an economically meaningful level of spatial aggregation, but it is
only possible to pickup correctly these increasing returns by the Verdoorn law estimated in
dynamic form or by the law estimated in static form using an appropriate panel data estimator.
Obviously, if this is found to be the case, it resolves some of the confusion in the empirical
literature as to whether returns to scale are constant or increasing in a manner that favours
17
endogenous growth theory and geographical economics, as well as the Kaldorian approach to
growth.
This does, however, raise problems concerning the appropriate measure of the level of TFP to
use as a proxy for the level of technology in equation (8). Clearly, if there are significant
localised increasing returns, then TFP levels could differ because of this factor and so the
variable should be adjusted to take account of this. If McCombie and Roberts (2006) are correct,
then the corrected index for the initial period for any region should be:
ij0
i
ij0ij0
ij0
j0
LK
Q
PTF
ω
βα
∑
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
′
(14)
where
i denotes the FEA, j the region and ω is the appropriate weight of FEA i.
17
The difficulty,
of course, is that the data for the FEAs are not available. Consequently, the procedure adopted
below is to use two alternative proxies for
TFP
0j
. First, the initial TFP was calculated under the
assumption that the returns to scale apply to the
whole of region j’s output, namely,
where
α
and
β
are the estimates implicit in the estimated Verdoorn
coefficient. However, the use of the data for the whole region will bias the estimates of TFP
downwards (if α + β >1) compared with the correct measure given by equation (14).
Consequently, the initial TFP, calculated under the assumption of constant returns to scale
(
), was also used in the regressions. The assumption is that these two
measures provide the limits of the true measure of TFP.
βα
=
jjjj
LKQTFP
000
*
0
/
)1(
0000
/
a
j
a
jjj
LKQTFP
−
=
17
Such as so that > .
βαβα
ij0ij0
i
ij0ij0
LK/LK ∑
βα
ij0ij0
i
ij0
i
j0
LK/QPTF ∑∑=
′
βα
j0
j0j0
*
ij0
LK/QTFP =
18
3. Spatial econometric issues
3.1. The standard approach to spatial autocorrelation
As noted, both Fingleton and McCombie (1998) and PonsNovell and ViladecansMarsal
(1999) make use of a spatial econometric framework in their estimation of the Verdoorn law for
the European regions. Indeed, more generally in the use of regional data,
it is now becoming
standard to explicitly test for spatial effects in regression models.
18
In this context, it is useful
to divide the econometric problems arising from the use of spatial data into two categories. First
are problems of
spatial heterogeneity, which reflect the fact that the parameters of interest may
vary over space. Such problems are normally dealt with by the use of panel data style
estimators. Second, there are problems posed by the existence of
spatial autocorrelation,
whereby the assumption of independently distributed error terms breaks down across spatial
units. With the latter set of problems, a "testingup" approach is normally adopted as a solution.
Thus, it is fast becoming standard to estimate the model under consideration by OLS and then to
test for spatial autocorrelation through the use of an appropriate diagnostic test such as one
based on Moran’s
I. If spatial autocorrelation is found to be present then two alternative spatial
specifications are considered with Lagrange Multiplier (LM) diagnostics being used to choose
between them. Specifically, the two alternative spatial specifications are the spatial
autoregressive model (SAR), otherwise known as the spatial lag model, and the spatial error
model (SEM). However, contrary to the impression given by this testingup procedure, it has
been argued that the SAR and SEM are not mutually exclusive specifications. Rather, both are
special cases of a more general set of equations in which they are nested (Florax and Folmer,
1992). Thus, consider the following general functional form for a spatial crosssection
regression:
18
Abreu, De Groot and Florax (2005) provide a survey of empirical growth work employing a
spatial econometric framework.
19
y = X
δ
+
η
Wy + WX
ρ
+
ε
(15)
where
ε
=
ξ
W
ε
+
μ
and
y is the dependent variable, X is a matrix of nonstochastic regressors,
δ
the associated
vector of coefficients, and
ε
is the error term. W is an a priori specified matrix of exogenous
weights and is often either a contiguity matrix (with a value of 1 if the regions have adjoining
boundaries, 0 otherwise) or is based on a distance decay function from the region under
consideration to the other regions.
η
is the spatial autoregressive parameter,
ρ
is a vector of
crosscorrelation coefficients, and
μ
is vector of random errors with E(
μ
) =0 and E(
μμ′
) = I.
Note that when the spatial weights matrix, W, is applied to a variable, this is referred to as the
spatial lag of the variable.
2
μ
σ
From this general specification, at least five restricted specifications can be identified:
(i)
Ordinaryleast squares
This is appropriate when the constraints
η
= 0,
ρ
= (0,….0)
′
and
ξ
= 0 hold:
y =
X
δ
+
μ
(16)
This is the correct specification when there is no spatial autocorrelation, providing the standard
assumptions underlying OLS hold.
(ii)
The spatial autoregressive or spatial lag model (SAR)
20
Used when the constraints
ρ
= (0,….0)
′
and
ξ
= 0 hold:
y =
X
δ
+
η
Wy +
μ
(17)
(iii)
The spatial crossregressive model
Occurs when the restrictions
η
= 0 and
ξ
= 0 are imposed:
y = X
δ
+ WX
ρ
+
μ
(18)
Note that in this model, the spatially lagged variables are the regressors and, in contrast to the
SAR model, the spatial crossregressive model can be estimated using OLS. In the case of our
augmented dynamic Verdoorn law given by equation (8), this translates into including
Wq
j
,
WlnTFP
0,j
and WlnD
j
as additional explanatory variables. Consequently, this can be interpreted
as crossregional spillovers to region
j occurring and/or being affected by output growth,
technology levels and levels of agglomeration in neighbouring regions. This contrasts with the
SAR model, which is often interpreted as saying that spillovers occur directly through
productivity growth. However, in this sense, the spatial crossregressive model would seem
preferable because it enables us to identify and estimate the separate contributions of the
different independent variables to crossregional spillovers. This allows, for instance, for testing
of the hypothesis that higher productivity growth is more likely to be observed in region
j if that
region is surrounded by technologically advanced regions. From an economic theory
perspective, this seems very plausible.
(iv)
The spatial Durbin model
This combines the SAR and spatial crossregressive specifications by using the single restriction
ξ
= 0:
21
y =
X
δ
+
η
Wy + WX
ρ
+
μ
(19)
hence, the spatially lagged variables are both the independent variables and the dependent
variable. This specification, however, is likely to suffer from severe multicollinearity between
Wy and WX .
(v)
The spatial error model (SEM)
This model results when
η
= 0 and
ρ
=( 0…0)
′
:
y = X
δ
+
ξ
W
ε
+
μ
(20)
Note that
(
ξ
Wy 
ξ
WX
δ
) =
ξ
W
ε
; therefore:
y =
X
δ
+ (
ξ
Wy 
ξ
WX
δ
) +
μ
(21)
And then, equation (20) can also be expressed as:
y = X
δ
+ (I 
ξ
W)
1
μ
(22)
As mentioned, the SAR and SEM specifications are the most commonly applied in spatial
econometric studies and, indeed, these are the specifications that Fingleton and McCombie
(1998) and PonsNovell and ViladecansMarsal (1999) restrict themselves to in their estimation
of the Verdoorn law. These specifications have been interpreted as capturing spatial
autocorrelation of the "substantive" and "nuisance" variety respectively. Thus, whilst, as
22
indicated,
η
in the SAR model has been given the economic interpretation of capturing the
strength of crossregional spillovers,
ξ
in the SEM model has been seen as capturing the spatial
correlation of any omitted variable, such as human capital, for instance (Bernat, 1996).
Given that both the SAR and SEM specifications are nested within the general spatial
specification, equation (15), it follows that the standard testingup procedure, which is used by
PonsNovell and ViladecansMarsal (1999) is powerful when either LM
SAR
or LM
SEM
is
significant. However, when both are significant, it is not necessarily legitimate to choose the
one with the highest value for the LM statistic, as is generally done as part of the procedure.
This is because the results would seem to suggest that
both Wy and W
ε
are statistically
significant and are likely to be highly collinear. In other words, the appropriate restrictions
discussed above are not met. It would thus seem unwise to base model selection on this
criterion.
Ideally, the more appropriate statistical procedure would be the Hendrystyle one of estimating
the more general specification and “testing down”. There are, however, two drawbacks with this
strategy. First, if
Wy and W
ε
are highly collinear then the standard errors will be inflated and the
presence of multicollinearity should be tested for. Second, and more seriously, using the same
weights matrices in the general specification means that the estimated equation is not identified
(Anselin, 1988). Yet, it is difficult to determine on theoretical grounds why the weights
matrices should differ between
Wy and W
ε
. The upshot of this is that we are sceptical about
distinguishing between the quantitative impact of the two variables and attaching different
economic interpretations to them, unless one is statistically insignificant. Thus, we would
hesitate to interpret
η
≠ 0 as capturing a crossregional spillover effect unless the estimate of
ξ
is statistically insignificant.
23
3.2. A new spatial specification
Notwithstanding the above conventional approach to the modelling of spatial effects, there is a
further spatial specification nested within equation (15). This specification seems to have been
ignored in the spatial econometrics literature. However, for reasons discussed below, it seems
the
a priori preferable specification.
(vi)
The spatial crossregressive error model (the spatial hybrid model)
This model involves the single restriction
η
= 0 and therefore takes the form:
y = X
δ
+ WX
ρ
+
ξ
W
ε
+
μ
(23)
This specification presents the advantage of explicitly modelling both the "substantive" and
"nuisance" components of any possible spatial autocorrelation. In particular, whilst
WX models
the substantive component, the nuisance component is captured by
W
ε
.
4. The results for total manufacturing
19
4.1 Estimation of the Kaldorian version of the augmented dynamic Verdoorn law
Table 2 starts by presenting crosssectional results for the fullsample period of 19862002 for
the Kaldorian version of the augmented dynamic Verdoorn law given in equation (8). It does so
19
All estimations reported in this section were carried out using MatLab v 7.0 with the
assistance of James Le Sage's spatial econometrics toolbox. With the exception of column 1 in
table 2, all estimates were obtained using Maximum Likelihood (ML) procedures. For details
of the ML procedures in this toolbox see Le Sage (1999), although note that in employing these
procedures, estimates of both
η
and
ξ
were constrained to the range (0.999, 0.999) instead of
the default of (1, 1). This was necessary to allow for the proper numerical evaluation of the
concentrated loglikelihood function given the structure of the weights matrices used in
particular, given that a small number of regions in the sample were only contiguous with one
other region.
24
for all six of the specifications discussed in the previous section. In all cases, the measure of
initial TFP adopted is that which makes no correction for increasing returns.
20
We later discuss
the consequences of correcting for increasing returns in this measure, as well as presenting
panel data results for our preferred spatial specification. In subsequent sections, we progress to
results for the Rowthornstyle specification of the augmented dynamic Verdoorn law and to
associated IV results. We also present results from estimation of the static Verdoorn law,
considering both the staticdynamic Verdoorn law paradox and the resolution of this paradox
suggested by McCombie and Roberts (2006).
From Table 2, it can be seen that all five spatial specifications gave similar results, which were
close to the OLS estimates. The coefficient on
q
j
(i.e. the Verdoorn coefficient) ranged from
0.502 to 0.673, implying that
ν
ˆ
(the composite measure of returns to scale) varied from 2.199
to 3.060. The diffusion of innovations from the more to the relatively less advanced regions is
an important source of TFP growth, as indicated by the significant negative coefficient on
lnTFP
0
with the (conditional) speed of diffusion,
φ
, estimated as being between 1.42 and 2.24%
per annum.
21
The density variable is also significant with a positive coefficient, suggesting that
agglomeration economies produce dynamic intraregional knowledge spillovers and therefore
also have a role to play in explaining TFP growth. Moran’s
I confirms the presence of spatial
autocorrelation in the OLS specification and therefore justifies our additional use of spatial
econometric methods.
It is interesting to note the virtually identical coefficients and
tvalues associated with Wtfp and
W
ε
in the spatial autoregressive model (SAR) and the spatial error model (SEM) specifications
20
The estimates using increasing returns to scale are available on request. There was little
difference in the estimates of increasing returns to scale, but the speed of diffusion was much
slower.
21
The estimate of
φ
is given by
φ
= (ln(1 
θ
1
T))/T.
25
respectively (Table 2, equations (ii) and (v)) This makes it very difficult to discriminate
between the two specifications on statistical grounds. As noted, the normal "testing up"
procedure has been to compare LM
SAR
with LM
SEM
, which would suggest that the SEM
specification is to be preferred over the SAR in Table 2. We know, however, that this is not an
appropriate test procedure if we accept that equation (15) is the general functional form.
As discussed above, we prefer, on theoretical grounds, either the spatial Durbin model, the
spatial cross regressive error model, or what we have termed the “spatial hybrid model” for
short, to the SAR and SEM. This is because these specifications allow for the breaking down
of the substantive component of any spatial autocorrelation, allowing an assessment of the
channels through which crossregional spillovers might occur. In particular, the different
channels are captured by the coefficients on the spatially lagged independent variables.
Nevertheless, the spatial Durbin and spatial hybrid specifications lead to a number of different
conclusions. In the spatial Durbin model, a faster growth of output of the surrounding regions
has no statistically significant effect on the region under consideration. However, there is a
large and statistically significant effect in the spatial crossregressive and spatial hybrid models.
Thus, in these specifications, the gains from the Verdoorn effect through learningbydoing and
induced technical change are not completely localised to the region in question, but directly
spillover into surrounding regions. This does not, however, occur in the spatial Durbin
specification.
Moreover, the spatial Durbin model suggests that if a region is surrounded by regions with high
levels of TFP, i.e., advanced levels of technology, this has the effect of raising the region’s rate
of TFP growth. Thus, there is a spatially lagged diffusion of innovations effect that is more
important, the more advanced is the technology in the neighbouring regions. In the spatial
crossregressive and spatial hybrid models, this effect is not statistically significant. There is,
26
however, evidence in all three models of a crossregional spillover effect from agglomeration
economies as evidenced by the positive coefficient of
WlnD, although the effect is larger in the
spatial crossregressive and spatial hybrid models.
We are now in a position to understand why the estimated coefficients (and their standard
errors) of
Wtfp and Wε are so similar in the SAR and SEM specifications (see Table 2,
equations (ii) and (v)). If we estimate the side relationship
Wtfp
j
= c + bWq
j
+ ρW
ε
j
then
ρW
ε
j
≡
Wtfp
j
– c  bWq
j.
Suppose that Wq
j
is either not statistically significant or barely so. It
follows then that the estimate of
ρ tends to be close to 1 and Wtfp
j
and W
ε
j
will be virtually
identical. Consequently, the specifications of the SAR model (equation (17)) and the SEM
(equation (20)) are virtually identical. The problem is that in the literature, the SAR model has
often been given an economic interpretation (“the control of economic phenomena representing
spatial autocorrelation of the ‘substantive’ variety”) while the SEM specification is viewed
merely as correcting for “spatial autocorrelation of the ‘nuisance’ variety”. However, given that
in this case,
Wtfp
j
and W
ε
j
are virtually identical, it seems difficult to make this distinction in
the interpretation of the results.
As explained, our preferred model in these circumstances is the spatial hybrid model where the
true spillovers come from the economic variables, i.e.
WX, and not Wy. This specification
explicitly tests for the substantive component of any spatial autocorrelation through the
inclusion of
WX while correcting for the nuisance component through W
ε
. Meanwhile, the
spatial Durbin model gives similar results, but is misspecified as
Wy is, in effect, capturing the
joint effect of
WX and W
ε
.
27
The reported
sR
2
of all the specifications are subject to an element of spurious regression due
to the fact that
q appears on both sides of the regression (it will be recalled that tfp
≡
q – tfi).
The
2
adj
R in Table 2 is the
2
R
adjusted to remove this spurious correlation and is obtained by
running the regression with
tfi as the dependent variable. In the specification of the Verdoorn
law simply as
tfp
j
≡
c
3
+ b
3
q
j
and tfi
j
≡
c
4
+ b
4
q
j
, the choice of the dependent variable makes no
difference to either the estimate of the degree of returns to scale or its statistical significance
(i.e. ) and the two specifications are mirror images of each other.
1b
ˆ
b
ˆ
43
=+
However, this is not the case in equation (ii), the SAR model, and in equation (iv), the spatial
Durbin model. This is because these equations include the spatially lagged dependent variable,
which is
Wtfp and Wtfi depending on the choice of dependent variable. This means that in each
of these cases, the degree of returns to scale differs, depending upon whether
tfp or tfi is the
dependent variable. The two estimates of the degree of returns to scale for equation (ii) are
2.415 and 1.758 and for equation (iv) are 2.495 and 1.758.
22
Unfortunately, there does not seem
to be any reason, either statistical or theoretical, for preferring either
tfp or tfi as the dependent
variable, but the disparities in the estimates are not large.
A commonly neglected shortcoming of traditional spatial models, including the spatial
specifications of the Verdoorn law of Fingleton and McCombie (1998) and PonsNovell and
ViladecansMarsal (1999), concerns the weights matrix. Consider a particular region,
j. The
weight given to the crossregional spillover effects of the neighbours of
j in the above approach
are equal, regardless of their absolute size in terms of output. The inclusion of
Wq
j
, for
example, implies that the impact on region
j of a neighbouring region's output growth is not
independent of the size of that region, which is rather implausible. It is likely that the impact of
22
The full results with tfi as the dependent variable are available on request from the authors.
28
the growth of a neighbouring region that is several times larger than that of another bordering
region will have a greater effect on
j, even if both neighbours are growing at the same rate. To
allow for this, an alternative specification is to weight the relevant variable by
Q
i
/Q
j
, where Q
j
and Q
i
are the outputs of region j and neighbouring region i respectively. Thus, we constructed
a nonrow standardised weights matrix
W
1
where w
j,i
= Q
i
/Q
j
if j and i are contiguous regions
(and
j ≠ i) and w
j,i
= 0 otherwise. This matrix was used to weight the growth of output, but not
the other variables, in the spatial hybrid model.
23
TABLE 2 HERE
Using
W
1
for output growth and the rowstandardised contiguous weights matrix for the other
regressors, Table 3 reports two different specifications of the spatial hybrid model. This is done
for both crosssectional and panel data.
24
In particular, the panel consists of three periods,
19861991, 19911996, 19962002. Whilst the crosssectional data has the advantage of
minimising bias attributable to cyclical fluctuations by allowing growth rates to be calculated
over a longer period, the panel data has the advantage of permitting control for fixed effects.
In
Table 3, equations (i), (iii) and (v) impose constant returns to scale on
lnTFP
0
, while equations
(ii), (iv) and (vi) correct the measure of initial TFP for increasing returns in the manner
discussed in section 2.3.
25
23
An alternative is to row standardise W
1
so that the relative sizes of the surrounding regions,
rather than their absolute sizes are taken into account. This alternative procedure yielded similar
results to those reported.
24
For the panel data estimation we used the spatial FE estimators of Elhorst (2003), which are
ML estimators. We appreciate J.Paul Elhorst’s kind help, especially in making available his
MatLab routines for the implementation of these estimators.
25
In particular, an iterative procedure was employed whereby the equation under consideration
was first estimated with the initial level of TFP calculated under the assumption of constant
returns to scale. The estimate of
ν
obtained was then used to recalculate the initial level of TFP
under the alternative assumption of increasing returns. This procedure was repeated until
successive estimates of
ν
converged. In most cases, the convergence required a maximum of
eight iterations, and, in many, less than four.
29
Table 3 presents the results obtained with these specifications, which regarding the main
variables, are very similar to the ones estimated so far, both when applying crosssectional data
sets and when panel data sets are employed. Thus, all equations show very large composite
increasing returns, a statistically significant diffusion effect and a statistically significant, but
quantitatively small, agglomeration effect, except for equation (vi), in which the latter is non
significant.. As we would expect, the quantitative impact of the diffusion effect is substantially
smaller when we adjust
lnTFP
0
for increasing returns. This can be seen by comparing equations
(ii), (iv) and (vi) with equations (i), (iii) and (v) respectively. Note, however, that when using a
contiguity matrix to assess the effect from the surrounding area due to increments in the output,
we collect a significantly positive impact, whereas positive but nonsignificant coefficients
result when this effect is weighted by the size of the adjoining region. Other differences
resulting after this change are that while the weighted impact of initial total factor productivity
and density, as reported in Table 2, showed positive and stable results, in Table 3 these effects
show different signs according to distinct specifications. We also have an anomaly in the
estimation of the intercept in the crosssectional data equation (ii), whose large negative value is
difficult to explain.
TABLE 3 HERE
4.2. Estimation of Rowthorn’s specification of the dynamic Verdoorn law
Equation (11), it will be recalled, gives the Rowthornstyle specification of the dynamic
Verdoorn law in which causation is hypothesised to run from the growth of inputs to the growth
of output and demand rather than
vice versa. Although a Rowthornstyle specification has not
previously been estimated for the European regions, when such specifications have been
30
estimated for other samples, they have been found to suggest constant returns to scale
(McCombie and Thirlwall, 1994, Chapter 2).
26
Consequently, we estimated equation (11) for
each of the specifications discussed in section 3. Although the results are not explicitly reported
for reasons of space, they were, in all cases, found to suggest either constant (
ν
= 1) or
decreasing returns to scale (
ν
< 1), despite the corresponding Kaldorian specifications reported
in Table 2 all suggesting increasing returns. Table 4 reports the results of equation (11) for the
preferred spatial hybrid model.
TABLE 4 HERE
It can be seen that all the results also suggest
v < 1, although in the case of equation (i) in Table
4 the null hypothesis of
v = 1 cannot be rejected. If this is the correct specification of the law,
the estimate of decreasing returns to scale could be due to a relatively fixed factor of production
such as land. It can also be seen that, with the exception of the estimations which correct for
decreasing returns to scale, both ln
TFP
0
and the density variable take on the expected signs in
all of the specifications. Not all of the models, however, provide significant estimations of these
parameters. Of the spatially lagged variables,
Wtfi is non significant except for the case of
model (v). By contrast,
WlnTFP
0
is (except in equation (v)) and takes on the expected sign,
showing that regions benefit from close neighbours with higher levels of TFP. Finally,
WlnD
0
implies that there are negative agglomeration effects from the surrounding regions, although the
variable is not statistically significant in equations (iii) and (v).
4.3. IV estimation of the dynamic Verdoorn law
26
The sole exception seems to be provided by McCombie and de Ridder (1984) for the US
states, who found increasing returns using both Kaldorian and Rowthornstyle specifications of
the Verdoorn law.
31
Given the dramatic differences obtained from the Kaldorian and Rowthornstyle specifications
of the dynamic Verdoorn law, the assumption of exogeneity is clearly crucial in the estimation
of
v. In section 2.2, it was seen that, although arguing that the dynamic Verdoorn law is better
specified with
q
j
as the regressor, the Kaldorian position does accept the possibility of
simultaneity bias given the hypothesised circular and cumulative nature of the regional growth
and agglomeration processes. Furthermore, even if a SolowSwan style neoclassical perspective
is adopted with no postulated feedback between
tfp
j
and tfi
j
, it is doubtful if tfi
j
can be assumed
to be strictly exogenous. This is because TFP growth will affect factor returns and, in a system
of open regional economies, this will stimulate interregional capital and labour flows.
Consequently, from either a neoclassical or a Kaldorian perspective, both equations (8) and (11)
should be estimated by methods that take endogeneity into account and ideally the implied
estimates of
v should converge.
Given the above, we report estimates for the spatial hybrid specification for both equations (8)
and (11) using the IV approach. In particular, following McCombie (1981), we adopted
Durbin’s ranking method, which uses the ranks of the endogenous variables as instruments.
27
Another problem, however, is the difficulty of estimating the spatial hybrid model using IVs, as
it was originally estimated using ML techniques (see footnote 18). We therefore report IV
results for the spatial Durbin specifications of equations (8) and (11), but regard
Wtfp as a proxy
for the spatial error term. It is therefore interpreted as merely capturing any residual spatial
27
Using an IV approach also has the advantage of not requiring the assumption of a normally
distributed error term, as is the case with the ML results reported in preceding tables. This is
significant because, using the JarqueBera test, the null of normality was frequently rejected for
these results. More generally, however, the reestimation of all the spatial specifications
reported using IV techniques did not significantly alter any of the results obtained.
32
autocorrelation of the nuisance variety and as having no substantive economic meaning. The
results are reported in Table 5.
TABLE 5 HERE
While there is some slight convergence in the estimates of
v, it can be seen that the Kaldorian
specification still gives an estimate of large increasing returns to scale, while the Rowthorn
style specification cannot reject the null of constant returns. As might be expected from the
discussion of section 2.2, the problem is that two different instruments are being used in the
above estimations and so the direction of normalisation (i.e. whether
q
j
or tfi
j
is used as a
regressor) still matters.
As an alternative procedure, we used the ranks of
both q
j
and tfi
j
together in the IV estimation of
the Kaldorian and Rowthornstyle specifications, but this made no significant difference to the
estimates of returns to scale given by the two methods of normalisation. Consequently, the IV
estimates are close to those previously obtained and do not resolve the problem. However,
a
priori,
in the regional case, it seems more plausible to agree with the Kaldorian position that
output growth, which is determined by the type of good that the region produces and other
demand factors, is the more appropriate regressor. In this case, the results suggest large
increasing returns to scale.
4.4. Estimation of the static Verdoorn law and resolution of the staticdynamic Verdoorn law
paradox
Given the variables ln
TFP
0
and lnD
0
, there is no static specification of the Verdoorn law that
corresponds to our augmented dynamic Verdoorn law. Therefore, for our augmented law, we
33
cannot test for the existence of the staticdynamic Verdoorn law paradox in the European
regional data. However, whilst ln
TFP
0
and lnD
0
have been found to be statistically significant
and give economically meaningful results, their inclusion has not dramatically altered the
implied estimates of
v obtained. Consequently, we estimated static versions of the Kaldorian
and Rowthornstyle specifications of the Verdoorn law excluding these variables, using our
preferred spatial hybrid specification to control for spatial autocorrelation. The results are
reported in Table 6 for the panel data.
From Table 6, it can be seen that the estimate of
v using time effects (which has the effect of
allowing for shifts in the production relationship) is, in both cases, consistent with constant
returns to scale. This is equivalent to the use of pooled data, with a dummy variable to allow for
exogenous technical change. Consequently, the estimates of the static Verdoorn law stand in
marked contrast to the dynamic specification. In the case of the Rowthornstyle specification,
both the static and dynamic estimates are in accord.
As we have seen, McCombie and Roberts (2006) have suggested that the most likely
explanation for the staticdynamic paradox is the existence of spatial aggregation bias in the
static estimates. According to this hypothesis, the use of a twoway estimator that captures both
time and regional effects should give unbiased estimates of
ν
similar to those obtained from the
dynamic Verdoorn law. This is confirmed for the European data set under consideration.
Estimates of the static Kaldorian specification presented in Table 6 exhibit substantial
increasing returns to scale of a magnitude comparable to the estimates from the corresponding
dynamic law.
TABLE 6 HERE
34
However, interestingly, the static Rowthornstyle specification estimated using both oneway
and twoway fixed effects gives an estimate of constant returns to scale. This is not surprising,
though. The twoway fixed effects gives a correct estimate of
ν
, by capturing the within region
variation of the data which is not subject to the aggregation problem. In the case here, we know
that the within variation will approximate to the results using growth rates, and in the case of the
Rowthornstyle specification, this gives constant or decreasing returns to scale. It is worth
noting, however, that LeónLedesma (1999) found, using postwar Spanish regional data for
manufacturing and twoway random effects estimators, that Rowthorn’s static specification
gave an estimate of increasing returns.
In the dynamic specification of both the Kaldorian and Rowthornstyle specifications, ln
D
0
was,
with one exception, found to have a significant effect on the growth of TFP, suggesting
significant (intraregional)
dynamic knowledge spillover effects from agglomeration. It is not, of
course, possible to derive an estimable static specification of this model. However, an
alternative hypothesis that is discussed in section 2.1 is that of agglomeration economies of the
static variety which only have a "level effect".
Consequently, we estimated both the static Kaldorian and Rowthornstyle specifications of the
Verdoorn law using output and inputs expressed in per sq. km. terms, (i.e., with these variables
divided by
H) with panel data and time effects. The theoretical rationale for this was discussed
above with respect to equations (9) and (10).
However, the estimated results in both cases did not differ greatly from the “conventional”
specification of the static laws using the unadjusted log levels and both sets of results did not
refute the hypothesis of constant returns to scale.
28
In retrospect, this is not surprising, as, with
constant returns to scale, the results will not be affected if all the variables are divided
H. See,
28
The results are not reported here but are available on request from the authors.
35
for example, the equivalent specification of the static Verdoorn law as equation (10) when
α +β
= 1 holds. A similar argument holds for Rowthorn’s specification. Using the density data and
now also including regional effects, as well as time effects, (so the model is estimated using the
twoway estimator) washes out the effect of
H, which is due to its interregional variation.
Consequently, the results obtained are the same as those reported in Table 6 and obtained using
the twoway estimator and the log levels of the variables. The Kaldorian specification exhibits
substantial returns to scale and the Rowthornstyle representation, decreasing returns to scale.
Consequently, the results using density variables do not shed any further light on
whether or not there are increasing returns to scale when levels are used.
36
5. Conclusion
This paper has revisited the estimation of the Verdoorn law by spatial economic techniques
using EU regional data manufacturing. On theoretical grounds, the spatial hybrid model (or the
spatial crossregressive error model) was preferred as this enables the sources of crossregional
spillovers to be more closely investigated and tested. Unlike the previous studies for the EU of
Fingleton and McCombie (1998) and PonsNovell and ViladecansMarsal (1999), estimates of
the capital stock were calculated and used in the specification of the law. Our results with the
Kaldorian specification of the Verdoorn law gave estimates of substantial increasing returns to
scale, where the estimates also included the effect of induced technical change. It was also
found that the coefficient of the logarithm of the initial level of TFP was negative and
statistically significant. This suggests that the diffusion of innovations from the relatively more
to the relatively less advanced regions was a significant explanatory factor in accounting for
disparities in TFP growth. A density variable that was introduced to capture the effect of
agglomeration economies on TFP
growth also proved to be statistically significant, although its
quantitative effect was small. These variables, when spatially lagged, often turned out to be
significant suggesting significant crossregional spillover effects, although, interestingly,
spatially lagged output growth was not significant.
The alternative Rowthornstyle specification using the weighted growth of the total factor inputs
as a regressor always suggested either decreasing or constant returns to scale and the use of an
IV approach was not able to resolve the discrepancy between the two specifications. On
theoretical grounds, the method of normalisation of the Kaldorian specification seems preferable
and this is the specification we prefer.
37
It was also found that the EU regional data also gives rise to the staticdynamic Verdoorn law
paradox. In particular, estimation of the Kaldorian specification of the Verdoorn law in static
form suggests constant returns to scale prevail, whilst estimation in dynamic form suggests
substantial increasing returns to scale. The conjecture of McCombie and Roberts (2006) that
this is due to spatial aggregation bias is given support by the finding that the twoway
estimation of the static relationships finds, as predicted, increasing returns in accord with the
dynamic estimates. The static Rowthornstyle specification still did not refute the hypothesis of
either constant or decreasing returns to scale.
38
Data Appendix
The data were taken from the Cambridge Econometrics regional database, supplemented and
amended where necessary from national sources. Output is gross value added in constant 1995
prices and measured using a purchasing power standard exchange rate, whilst employment is
the total number of hours worked. The analysis was confined to the NUTS1 regions, as this is
the lowest level of spatial aggregation for which independent gross investment figures are
available – at the NUTS2 level the gross investment measures for a large number of regions in
the database are interpolated. This gave 59 regions, which are reported in Table A1.
The capital stocks were calculated by the perpetual inventory method, where the increase in the
capital stock of a particular region in time
t is given by
Δ
K
t
= I
t

δ
K
t1
, where I is gross
investment and
δ
is the proportion of the capital stock lost through scrapping and depreciation.
Hence
K
t
=
Δ
K
t
+ K
t1.
This method requires a value for the baseyear capital stock. In order to
provide an estimate of this, we used the relationship
05.0k
I
K
t
1t
+
=
−
where
k is growth of the capital stock. However, as there is no data available for the growth of
the capital stock, we approximated it by the growth of investment over the period 19811985.
The sampleperiod was consequently 19862002 for the crosssectional estimations, and for the
panel data estimation, 19861991, 19911996, and 19962002. Similarly, the level of investment
in the numerator is the average value over 19811985. There are no reliable data for wages at
the regional level and so we used the national manufacturing factor shares in the calculation of
the regional indices of total factor inputs and total factor productivity.
39
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45
Table 1. The Verdoorn Law and Spatial Aggregation Bias
_______________________________________________
Region
Employment Output
1
L
1
γ
1
BQ
2 2L
1
γ
1
BQ2
… … …
j jL
1
γ
1
jBQ
_____________________________________________
46
Table 2: The augmented dynamic Verdoorn Law (Kaldorian version): Cross Sectional Data, 19862002
(i) OLS
(ii) SAR
(iii) SCM
(iv) Durbin
Model
(
v) SEM
(vi) SHM
Constant
0.001
(0.47)
0.005
(2.37)
0.003
(0.94)
0.003
(1.05)
0.001
(0.38)
0.003
(0.73)
q
0.664
(7.95)
0.586
(8.54)
0.673
(7.96)
0.599
(8.06)
0.502
(6.81)
0.673
(8.73)
lnTFP
0
0.016
(2.61)
0.016
(3.17)
0.026
(3.40)
0.027
(4.20)
0.022
(3.34)
0.026
(4.55)
lnD
0
0.006
(4.76)
0.005
(5.61)
0.005
(4.60)
0.005
(4.48)
0.004
(3.44)
0.006
(5.60)
Wq
0.277
(2.79)
0.046
(0.40)
0.297
(3.40)
WlnTFP
0
0.014
(1.36)
0.018
(2.09)
0.011
(1.11)
WlnD 0.005
(2.93)
0.003
(1.69)
0.006
(3.95)
Wtfp
0.406
(4.77)
0.387
(3.16)
W
ε
0.499
(4.62)
0.499
(4.63)
2
R
0.582
0.663
0.666
0.671
0.675
0.757
2
adj
R
0.454
0.593
0.540
0.634
0.552
0.666
Moran’s I
3.02
[0.003]
LM
BP
11.70
[0.009]
LM
SAR
6.29
[0.012]
LM
SEM
14.18
[0.000]
v
2.975 2.415 3.058 2.495 2.119 3.060
φ
0.0143 0.0143 0.0216 0.0224 0.0189 0.0217
Notes: The figures in parentheses are tratios and the figures in square brackets probability values.
SAR is the spatial autoregressive model, SCM is the spatial crossregressive model, SEM is the
spatial error model, SHM is the spatial hybrid model.
47
Table 3: The augmented dynamic Verdoorn Law (Kaldorian version) –
The Spatial Hybrid Model
Cross sectional
(i) (ii)
Panel Data:
Time Effects
(iii) (iv)
Panel Data:
Spatial and Time
Effects
(v) (vi)
Constant
0.002
(0.495)
0.070
(2.76)
n.a. n.a. n.a n.a.
q
0.554
(7.45)
0.648
(8.40)
0.647
(13.14)
0.651
(13.48)
0.491
(10.93)
0.747
(16.52)
lnTFP
0
*
ln
o
TFP
0.026
(4.24)
0.002
(2.21)
0.015
(2.82)
0.001
(1.48)
0.085
(9.49)
0.022
(8.78)
lnD
0
0.006
(4.84)
0.004
(3.65)
0.005
(5.03)
0.004
(4.01)
0.047
(5.09)
0.005
(0.69)
W
1
q
0.024
(1.66)
0.023
(1.15)
0.008
(0.88)
0.001
(0.09)
0.007
(1.00)
0.0003
(0.04)
WlnTFP
0
W
*
o
TFPln
0.010
(0.90)
0.001
(1.38)
0.004
(0.47)
0.003
(2.85)
0.037
(2.81)
0.001
(0.24)
WlnD
0
0.005
(2.88)
0.003
(2.07)
0.004
(3.38)
0.003
(2.25)
0.039
(3.71)
0.026
(2.70)
W
ε
0.532
(5.15)
0.299
(2.31)
0.401
(5.80)
0.300
(4.01)
0.487
(7.71)
0.417
(6.13)
2
R
0.726 0.670 0.597 0.593 0.829 0.794
2
adj
R
0.623 0.641 0.522 0.518 0.797 0.767
v
2.244
2.839
2.832 2.864 1.968 3.9449
φ
0.022 0.002 0.014 0.001 0.053 0.019
Note: n.a. denotes not applicable
48
Table 4: The augmented dynamic Verdoorn Law (Rowthornstyle version) 
The Spatial Hybrid Model
Pooled Data
(i) (ii)
Panel Data: Time
Effects
(iii) (iv)
Panel Data: Time
Effects and Spatial
effects
(v) (vi)
Constant
0.016
(2.85)
0.099
(2.55)
n.a. n.a. n.a. n.a.
tfi
0.169
(1.09)
0.354
(2.33)
0.362
(4.04)
0.413
(4.85)
0.175
(2.40)
0.903
(10.19)
lnTFP
0
*
ln
o
TFP
0.031
(3.43)
0.004
(0.71)
0.013
(1.72)
0.015
(3.72)
0.124
(11.83)
0.219
(9.01)
lnD
0
0.003
(1.58)
0.001
(0.40)
0.003
(1.82)
0.001
(0.54)
0.087
(8.04)
0.200
(7.75)
W
1
tfi
0.002
(0.08)
0.031
(1.00)
0.006
(0.364)
0.011
(0.74)
0.021
(2.04)
0.018
(1.46)
WlnTFP
0
W
*
o
TFPln
0.037
(2.61)
0.038
(3.42)
0.022
(2.09)
0.013
(2.02)
0.002
(0.12)
0.067
(2.22)
WlnD
0
0.001
(0.28)
0.007
(2.76)
0.003
(1.66)
0.005
(2.71)
0.020
(1.46)
0.086
(2.49)
W
ε
0.562
(5.67)
0.259
(1.94)
0.381
(5.42)
0.251
(3.25)
0.502
(8.11)
0.391
(5.61)
2
R
0.477 0.372 0.264 0.301 0.725 0.658
2
adj
R
0.505 0.407 0.431 0.412 0.787 0.618
v
0.831
0.646 0.638 0.587 0.825 0.097
φ
0.025 0.004 0.012 n.a 0.068 n.a.
Note: n.a. denotes not applicable.
49
Table 5: Dynamic Verdoorn Law; Durbin Model, Instrumental Variable
Estimates: CrossSectional Data, 19862002
Kaldorian specification
(i) (ii)
Rowthornstyle specification
(iii) (iv)
Constant
0.002
(0.75)
0.047
(2.24)
0.003
(0.61)
0.002
(0.56)
q
tfi
0.416
(3.88)
0.540
(5.20)
0.010
(0.05)
0.009
(0.05)
lnTFP
0
0.031
(4.26)
0.003
(1.61)
0.040
(3.84)
0.040
(3.87)
lnD
0.005
(3.86)
0.004
(3.09)
0.004
(2.08)
0.004
(2.10)
W
1
q
W
1
tfi
0.008
(0.40)
0.006
(0.25)
0.003
(0.08)
0.004
(0.10)
WlnTFP
0
0.028
(2.66)
0.002
(0.61)
0.044
(3.48)
0.921
(4.57)
WlnD
0.001
(0.41)
0.002
(1.06)
0.003
(1.25)
0.003
(1.29)
Wtfp
0.635
(4.37)
0.387
(2.46)
0.877
(4.14)
0.921
(4.57)
v
φ
1.713
0.025
2.176
0.003
0.990
0.031
0.991
0.031
50
Table 6: The Static Laws
Kaldorian specification
(i) Time Effects (ii) Time and
Regional Effects
Rowthornstyle specification
(i) Time Effects (ii) Time and
Regional Effects
lnQ 0.034
(1.88)
0.623
(13.66)
lnTFI 0.040
(2.23)
0.380
(5.21)
WlnQ 0.100
(3.00)
0.038
(0.68)
WlnTFI 0.038
(1.19)
0.232
(2.46)
W
ε
0.651
(15.33)
0.451
(7.91)
0.669
(16.37)
0.361
(5.82)
v 1.035
2.653 0.960 0.620
2
R
0.527 0.909 0.548 0.845
2
adj
R
0.950 0.990 0.952 0.983
51
TABLE A1. The EU Regions
Austria
Ostosterreich
Greece
Voreia Ellada
Sudosterreich
Kentriki Ellada
Westosterreich
Attiki
Belgium
Vlaams Gewest
Nisia Aigaiou, Kriti
Region Walonne
Ireland
Switzerland
Italy
Nord Ovest
Germany
BadenWurttemberg
Nord Est
Bayern
Centro
Bremen
Sud
Hamburg
Isole
Hessen
Netherlands
OostNederland
Niedersachsen
WestNederland
NordrheinWestfalen
ZuidNederland
RheinlandPfalz
Portugal
Continental
Saarland
Sweden
SchleswigHolstein
UK
North East
Denmark
North West
Spain
Noroeste
Yorkshire and the Humber
Noreste
East Midlands
Madrid
West Midlands
Centro
Eastern
Este
London
Sur
South East
Finland
South West
France
Ile de France
Wales
Bassin Parisien
Scotland
NordPas de Calais
Northern Ireland
Est
Ouest
SudOuest
CentreEst
Mediterranee
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