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Increasing returns and economic growth: some evidence for manufacturing from the European Union regions.

1. Introduction

It is somewhat ironical that the recent development of the new or endogenous growth theory, stemming from the initial work of Romer (1986), has led to a revival of interest in the old growth theory of Solow (1956) and Swan (1956). The new growth theory developed its impetus from the observation that for the world as a whole, there seemed to be little evidence of convergence as the traditional neoclassical growth model, with its assumptions of constant returns to scale and a common technology, would imply. However, Barro (1991) and Mankiw et al. (1992) have shown that once an allowance is made for human capital, there is evidence for conditional convergence using a large sample of less-developed and advanced countries. Moreover, Barro and Sala-i-Martin (1992, 1995) find evidence of unconditional convergence using data for regional total output for the USA, the European Union, and the Japanese Prefectures.

The fact that, especially, the Barro and Sala-i-Martin studies find evidence that is compatible with the assumption of constant returns to scale is somewhat surprising, because it is almost an article of faith of regional economists that production is characterised by substantial internal and external (agglomeration) economies of scale (Krugman, 1992). McCombie and de Ridder (1984), McCombie (1985), and Bernat (1996), for example, found substantial increasing returns to scale using US state data.

The purpose of this paper is to shed some further light on whether or not there are increasing returns to scale, broadly defined, for manufacturing at the European regional level. We undertake this by estimating the Verdoorn law (Verdoorn, 1949), which is the relationship between the growth of labour productivity and the growth of output. The Verdoorn law is augmented to allow for the possibility of the diffusion of innovations from the relatively technologically advanced to the more backward regions. Other studies have found what may be termed the 'static-dynamic Verdoorn law paradox' (McCombie, 1982). This is to say, estimation of the Verdoorn law specified with the logarithmic values of the various variables using cross-country or cross-regional data gives an estimate of the degree of returns to scale that are considerably smaller than those given by the traditional Verdoorn law using growth rates. Consequently, we examine whether or not this occurs with our data. We also undertake Barro and Sala-i-Martin's test (1992) for convergence.

We find that the Verdoorn law provides evidence of large increasing returns to scale while, at the same time, the Barro and Sala-i-Martin test suggests rapid convergence. We conclude by discussing this apparent contradiction.

2. Increasing returns to scale and the Verdoorn law

The Verdoorn law has been traditionally estimated as a linear relationship between the exponential growth rate of labour productivity (p) and of output (q), viz(1)

p = [b.sub.0] + [b.sub.1]q (1)

When the Verdoorn law has been estimated using cross-country data or cross-industry data for manufacturing over the growth cycle (i.e. using growth rates calculated from peak-to-peak over periods of about five years or longer) a Verdoorn coefficient ([b.sub.1]) of about one-half has usually been found. This implies that a one percentage point increase in output growth induces an increase in the growth of employment of one-half of a percentage point and an equivalent increase in the growth of productivity. In the terminology of the new growth theory (which the Verdoorn law considerably pre-dates), a substantial proportion of technical change is endogenous, in the sense that it is induced by the growth of output (see Thirlwall, 1983).

The Verdoorn law may be interpreted as a production relationship with the Verdoorn coefficient being a technological parameter. Indeed, the law is a version of the linear technical progress function, which Kaldor (1957) advanced to overcome his own criticism that 'any sharp or dear cut distinction between the movement along a 'production function' with a given state of knowledge and a shift in the 'production function' caused by a change in the state of knowledge is arbitrary and artificial' (see Bairam, 1987, and McCombie and Thirlwall, 1994, Ch. 2, for comprehensive surveys of the literature relating to the law).

A potential problem in the specification of the Verdoorn law is the omission of the contribution of the growth of the capital stock. Kaldor (1967) tested for the effect of capital growth by including as a regressor the investment-output ratio as a crude proxy for the growth of capital services. However, he found that the interpretation of the results and the value of the Verdoorn coefficient were essentially unaltered. In the study here, we are faced with the problem that there are no estimates available for the gross or net capital stock at the European regional level, and even regional gross investment data are absent. However, while ideally the growth of the capital stock (k) should be explicitly included in the regression, its omission is unlikely to be serious in this study. To see this, consider first the conventional Cobb-Douglas production function, with increasing returns to scale, expressed in terms of levels

Q = [A.sub.0]exp ([Lambda]t)[K.sup.[Alpha]][E.sup.[Beta]] ([Alpha] + [Beta] [successor] 1) (2)

where [Lambda] is the growth of total factor productivity or exogenous technical change. Q, K, and E are the levels of output, capital, and employment. Taking natural logarithms and differentiating with respect to time, eq. (2), after some rearranging of the variables, may be expressed as

p = [Lambda]/[Beta] + [([Beta] - 1)/[Beta]]q + ([Alpha]/[Beta])k (3)

It is a stylised fact that for the advanced countries the capital stock grows either at approximately the same rate as that of output or slightly slower. McCombie and Thirlwall (1994, footnote 5, p.557) found that regressing k on q, for a sample of advanced countries, gave a slope coefficient that was not significantly different from unity.

Generally

k = [Omega] + [Gamma]q (4)

and setting [Omega] = 0 and [Gamma] = 1 and substituting eq. (4) into (3), we obtain

p = [Gamma]/[Beta] + [([Alpha] + [Beta] - 1)/[Beta]]q (5)

It may be seen that if the estimate of the Verdoorn coefficient is statistically significantly greater than zero, it follows that there are increasing returns to scale since [Alpha] + [Beta] [greater than] 1. If we assume that the elasticity of output with respect to capital and labour are roughly equal so that [Alpha] = [Beta], then a Verdoorn coefficient of one-half implies that [Alpha] and [Beta] sum to 1.33.

The Verdoorn law is a dynamic relationship 'because technical progress enters into it' (Kaldor, 1966). We may model this by assuming that technical progress is partially induced by the growth of output

[Lambda] = [Lambda][prime] + [Xi]q (6)

Substituting eq. (6) into eq. (5), we obtain

p = [Lambda][prime]/[Beta] + [([Xi] + [Alpha] + [Beta] - 1)/[Beta]]q (7)] (7)

It can be seen that it is now no longer possible to distinguish between the separate effects of conventional static returns to scale ([Alpha] + [Beta]) and those that are dynamic ([Xi]). However, a statistically significant Verdoorn coefficient is still evidence of returns to scale, broadly defined.

3. The data and regression results

The data, Gross Value Added (GVA), and employment were taken from Cambridge Econometrics' European regional database, which in turn is based upon the European REGIO database together with national sources. The advantage of using Cambridge Econometrics' data is that considerable care has been taken to ensure that the data are consistent both across space and over time; this is not true of the REGIO statistics. Data for 178 regions were used for 13 countries (Germany, France, Italy, Netherlands, Belgium, Luxembourg, the United Kingdom, Ireland, Denmark, Greece, Spain, Portugal, and Austria). The regions are at the NUTS 2 level, which is generally smaller than the UK standard regions - there are 35 NUTS 2 regions for the UK compared with the 11 standard regions. While traditionally the Verdoorn law has been found to hold for manufacturing and public utilities, the data used in this study comprise manufacturing plus energy - it is not possible to obtain data for the separate components. Care was therefore taken to examine, by using Cook's d statistic, the regression results for influential cases, especially as these may be due to a high proportion of energy production in a region (e.g. Groningen). It was found, however, that there were no regions with exceptional influence, meaning that the vector of regression coefficients was not simply the result of a handful of extreme observations.

The regressions were run using growth rates for the various regions from 1979 to 1989, which represents a peak-to-peak of the growth cycle. A number of different specifications and estimation techniques were used and the results are reported in Table 1. Equation (a) reports the OLS results when national dummies are introduced to capture national specific differences in exogenous productivity growth. The Verdoorn coefficient is 0.575, which is highly statistically significant. These results are remarkably close to those found in Kaldor's (1966) original study and suggest the presence of substantial increasing returns to scale.

One of the problems in modelling contiguous spatial regions is that there may be significant spatial autocorrelation and this is confirmed by Moran's I statistic (see the Appendix) which is statistically significant for the OLS equation.

Moran's I and its standardised value are reported in Table 1. The criterion adopted for statistically significant spatial autocorrelation is if Z is greater than 1.64, corresponding to a one-tailed probability of 5%. It can be seen that in eq. (a), the value of Z (2.418) exceeds the critical value. This implies the presence of significant spatial autocorrelation among the regression residuals. We estimated the Verdoorn law without the national dummies and found that this gave an even greater degree of spatial autocorrelation. The existence of significant spatial autocorrelation in eq. (a) indicates that the model is in some way misspecified (even though the RESET test does not suggest misspecification errors). In fact, the model, as it stands, ignores cross-regional spillovers, whereas it is reasonable to assume that productivity growth in region i depends on causal factors in region i ([q.sub.i] and [u.sub.i]) and region j ([q.sub.j] and [u.sub.j], j = 1 ... n, i [not equal to] j), depending on the distance between region i and j, namely, [d.sub.ij]. The sum total of region j's effects is [p.sub.j], and so productivity [TABULAR DATA FOR TABLE 1 OMITTED] growth rates are likely to interact over space simply because neither the q-effect nor the u-effect is contained by administrative boundaries.

In order to test this proposition, we constructed a variable denoted slp which is a measure of spatially lagged productivity growth. This was calculated as a weighted average of the productivity growth rates in the surrounding regions, using the weighting matrix W, which is defined in the Appendix. Hence

slp = [summation over j] [W.sub.ij][p.sub.j] (8)

The adoption of slp as an additional explanatory variable poses problems for estimation, since slp is endogenous. In contrast to time series analysis, in which lagged variables are treated as predetermined and the residuals are not autocorrelated, OLS estimators of models with spatially lagged dependent variables are not consistent (Whittle 1954; Ord, 1975), and we have to use maximum likelihood (ML) or instrumental variable (IV) estimation(2) (see Haining, 1978; Bivand, 1984; Upton and Fingleton, 1985; Anselin, 1988; and, for a Bayesian approach, Hepple, 1979).

The estimation of a model by the ML procedure which incorporates a spatially lagged dependent variable provides consistent estimates. In general terms, the model outlined above can be written as

p = [b.sub.0] + [b.sub.1]q + [b.sub.2]slp + u

where u denotes spherical normal errors. Table 1 shows that the ML results (eq. (b)) are similar to that of the OLS, with a highly significant Verdoorn coefficient. The externality effect (slp) is also found to be significant.

Consequently, in the presence of the national dummies, a 1 percentage point change in productivity growth in neighbouring regions induces about 0.2 percentage point change in the productivity growth of the region under consideration. There is also an absence of the error correlation. As Moran's I ceases 'to be valid when the model contains an autoregressive component' (Cliff and Ord, 1981), we used Anselin's (1988) LM test statistic which does not suffer from this problem. Traditionally Moran's I has been used in this context to provide an informal assessment of the presence of residual autocorrelation. Anselin's LM test statistic, denoted by H, is referred to the [[Chi].sup.2] distribution with one degree of freedom. Table 1 contains the values obtained using this diagnostic test, with a critical value of 3.84 corresponding to a two-tailed probability of 5%, together with the estimated Moran's I on which it is based. It can be seen that there is no statistically significant spatial autocorrelation present in eq. (b). A comparison of eqs (a) and (b) shows that, in fact, there is very little difference between the OLS and the ML estimates of both the Verdoorn coefficient and its standard error.

4. The effect of the diffusion of technology

Implicit in the specification of the Verdoorn law is the assumption that all regions have access to the same technology. However, it is possible that some regions are more technologically backward than others and hence part of their growth of productivity may be due to a catch-up phenomenon. It may be thought that with the dominance of manufacturing by a small number of firms that are often multinational, it would be unlikely that there would be significant regional differences in technology. (For example, Borts and Stein, 1964, in their classic study of US regional growth simply assume that all regions have access to the same blueprint of technology.) Moreover, it may be considered that any spatial differences in technology would be largely at the international level and that these should be captured by the national dummies. Nevertheless, the diffusion hypothesis should be tested explicitly. The standard approach for testing the technological catch-up hypothesis has been to include some proxy for the initial level of technology in the regression equation. This proxy has usually been some function of GDP per capita when the aim has been to explain disparities in the growth of total output per capita. The exact specifications of the proxy have included the level of GDP per capita or per employee, its reciprocal, and the ratio of the level of GDP per capita to the technological leader. These variables are sometimes expressed in terms of logarithms. Since we are concerned with the manufacturing sector, a suitable candidate is the logarithm of manufacturing productivity (ln P) in the base year, viz 1979.

The results when ln P is included as a regressor are reported as eq. (c). The coefficient of the logarithm of the level of productivity is negative and highly statistically significant, indicating that, ceteris paribus, the lower a region's initial level of productivity, the faster has been its productivity growth in the subsequent decade. The Verdoorn coefficient still takes its traditional value of about one-half when national dummies are excluded (eq. (c)), but it falls to 0.275 when they are included (eq. (d)). Nevertheless, the estimate of the increasing returns to scale is still statistically significant. It should be noted that the national dummies are likely to be also capturing the disparities in the national levels of technology, with In P now capturing the differences at the regional level.

There is a problem with all of the above proxies, including our use of ha P, because they implicitly assume that all differences in productivity levels are due to disparities in technology. This is only true if there are constant returns to scale. To see this, consider the case where there are increasing returns to scale, namely the traditional Verdoorn law but with employment growth as the regressand, viz

e = -[b.sub.0] + (1 - [b.sub.1])q (10)

Integrating eq. (10) with respect to time gives, at the base period t = 0

[Mathematical Expression Omitted] (11)

or

Q = A[E.sup.1/(1-[b.sub.1])] (12)

where [Mathematical Expression Omitted] and is a shift factor which is a function of the level of technology.

Thus, after taking logarithms and rearranging, eq. (12) may be expressed, at t = 0, as

ln A = ln [P.sup.*] = ln Q - (1/(1 - [b.sub.1])) ln E (13)

If, alternatively, there are constant return to scale, [b.sub.1] equals zero and lnA is simply equal to lnP.

Consequently, including hi [P.sup.*] in the Verdoorn law, to capture the effects of the diffusion of innovation, gives

p = [b.sub.0] + [b.sub.1]q + [b.sub.2]slp + [b.sub.3][ln [Q.sub.0] - (1/(1 - [b.sub.1]) ln [E.sub.0]] + u (14)

where ln [Q.sub.0] and ln [E.sub.0] are the values for the base year 1979.

It follows that the results obtained by including In P, rather than In [P.sup.*], as a regressor (reported as eqs (c) and (d)) are misleading since the Verdoorn coefficient implies increasing returns to scale ([b.sub.1] [greater than] 0) while ln P is based on the assumption of constant returns to scale ([b.sub.1] = 0).

Equation (14) was estimated by ML methods outlined above with the estimates of [b.sub.1] (the coefficient of q and the coefficient used in the construction of In [P.sup.*]) being constrained to take the same value. From Table 1, eq. (e), it can be seen that when national dummies are included the coefficient of In [P.sup.*] is negative and statistically significant. However, the quantitative impact of this variable is considerably reduced when compared with In P. The value of the coefficient of ln [P.sup.*] is -0.029 (with a t-value of 2.83), whereas that of ln P is 0.496 (-9.48). This suggests that most of the differences in the levels of regional productivity are due to increasing returns to scale rather than disparities in regional levels of technology, although the latter still have a small, but significant, effect on productivity growth rates.

5. The Verdoorn law and the static-dynamic paradox

It has been argued that the most plausible interpretation of the Verdoorn law is a linear technical progress function. However, just as Black (1962) has shown that the latter may be derived from a conventional Cobb-Douglas production function, so one possible underlying structure of the Verdoorn law in log-linear form is

ln P = (1/[Beta]) ln A[prime] + [([Xi] + [Alpha] + [Beta] - 1)/[Beta]] ln Q (15)

where P and Q are the levels of productivity and output, and A[prime] = [A.sub.0] exp ([Lambda]t). Indeed, it is not coincidental that eq. (2), from which the Verdoorn law is derived above, is a conventional Cobb-Douglas production function.

If eq. (15), which is the so-called static Verdoorn law (McCombie, 1982), is the correct underlying specification of what may be termed the dynamic Verdoorn law (estimated using exponential growth rates), there should be no significant difference in the degree of returns to scale. However, when the static Verdoorn law is estimated using cross-country data for the advanced countries for the post-war period, a paradox is found. The estimates of the Verdoorn coefficient were found to be either not significantly different from zero or to be small, with typical values around 0.10. If we assume that the capital-output ratio does not greatly vary across nations, this implies either constant returns to scale, or very small increasing returns to scale. (If there is no induced technical progress and [Alpha] = [Beta], a Verdoorn coefficient of 0.10 implies returns of scale of 1.05.) Using the same data and estimating the dynamic Verdoorn law gave the traditional value of 0.5.

We therefore estimated the static Verdoorn law using the European regional data for 1979. The results are reported in Table 2. Without national dummies, the Verdoorn coefficient is 0.115, but there is significant spatial autocorrelation (eq. (h)). With national dummies, the spatial autocorrelation becomes Insignificant but the Verdoorn coefficient falls to a value of 0.057 (eq. (i)). These results suggest very much smaller Increasing returns to scale than those given by the dynamic Verdoorn law and are very similar to those found using international data for the advanced countries, noted above (McCombie, 1982). In both the regressions, there is a very dose statistical fit as evidenced by the [Mathematical Expression Omitted], with over 90% of the variation of the regressand explained. Either the static or the dynamic law, or possibly both could be misspecified (see McCombie (1982) for a discussion of these issues).
Table 2 The static Verdoorn law

Equation                                 (h)           (i)
Method                                    ML            ML

Dummies                                   No           Yes
In Q                                     0.115        0.057
                                        (5.84)       (3.06)
In SLP                                   0.245        0.020
                                        (5.38)       (0.48)
Moran's I                                0.243        0.083
Z                                        5.280        3.147
[Mathematical Expression Omitted]        0.309        0.235
[Mathematical Expression Omitted]        0.379        0.640
[Mathematical Expression Omitted]        0.922        0.953
H                                       31.14         3.35


Consequently, we re-estimated eq. (14), but with the level of productivity calculated using the estimate of returns to scale from the static specification (with national dummies), namely pt. Because the estimate of increasing returns to scale is so small (see Table 1 and compare eqs (d) and (g)), the use of lnP(*) as a regressor gives virtually identical results to the use simply of In P.

6. The Keynesian versus neoclassical explanations of European economic growth

The Verdoorn law forms a linchpin in the cumulative causation explanation of economic growth (Kaldor, 1970, Myrdal, 1957). Most versions of the cumulative causation models stress that growth is essentially demand driven, especially by the performance of regional (or national) exports (Thirlwall, 1980; McCombie and Thirlwall, 1994). A faster growth of productivity enhances output growth through either increasing the price competitiveness of the region's exports or enhancing the development of new technology industries which have high income elasticities of demand, or both (Fingleton, 1992, 1994). Unlike the Solow-Swan neoclassical approach, growth is not determined by the growth of the labour supply and exogenous technical progress. Whether or not growth becomes explosive (which, of course, is not observed in reality), converges to different equilibrium regional growth rates (with regions' per capita incomes diverging over time), or eventually converges, depends on the exact specification of the model (see Dixon and Thirlwall, 1975, 1978; and Guccione and Gillen, 1977). Faini's (1984) essentially neoclassical model with increasing returns in the sector whose products do not enter into international trade has the property that the region with the initial growth advantage continuously grows at the expense of the lagging region, which perpetually declines. (The same result occurs if increasing returns to scale are found in the traded goods sector.) Of course, there is nothing to prevent the supply side from eventually acting as a constraint on growth. Diseconomies of scale (through for example, increasing land prices and congestion in the more prosperous regions) and a shortage of labour (since there are limits to the rate of migration) may eventually limit the growth of output. Nevertheless, in contrast to the neoclassical approach, the growth of labour is not an autonomous determinant of growth.

Recently, however, Barro and Sala-i-Martin (1992) and Mankiw et al. (1992) have argued that the one-sector neoclassical model, with the assumptions of constant returns to scale and a spatially common technology, can explain the growth performance of the European regions and a set of 98 advanced and less-developed countries, respectively. Assuming that all regions have the same steady-state growth of output per worker, differences in productivity growth will reflect the fact that regions are not necessarily on their equilibrium growth path. In particular, the smaller is the capital-labour ratio compared with the steady-state value, the faster the capital-labour ratio, and hence productivity, will be growing.

As the difference between the steady state level of productivity (denoted by [Mathematical Expression Omitted]) and the actual level of productivity tends to zero, the growth of productivity tends to its steady state value [Lambda], which is equal to the rate of exogenous technical progress.

Consequently

[Mathematical Expression Omitted] (16)

where the approximation is due to a Taylor series linearization around the steady state. The speed of convergence is given by [Theta] = (1 - [Alpha])(e + [Delta] + [Lambda]), where [Alpha] is the output elasticity with respect to capital, e is the growth of employment, and [Delta] is the rate of depreciation.

Integrating and rearranging eq. (16) gives an expression for the logarithm of productivity at time [t.sup.*] as

[Mathematical Expression Omitted] (17)

where T is the length of time under consideration, i.e. the length of time between [t.sup.*] and 0. Simplifying and adding normal errors gives

ln [P.sub.[t.sup.*]] = [b.sub.0] + [b.sub.4] ln [P.sub.0] + u (18)

where [Mathematical Expression Omitted] is also approximately normally distributed (Fingleton, 1995).

In practice, the OLS estimation of eq. (18) could produce a biased estimate of the speed of convergence, [Theta], as the level of productivity at time T is also dependent on other variables, as discussed by, for example, Levine and Renelt (1992). Unfortunately, variables such as human capital, R&D, and investment-output ratio are not available on a comprehensive basis at the NUTS 2 regional level. Therefore, we use the autocorrelated errors specification to minimise any possible bias. Haining (1990) outlines the rationale for this approach, arguing that when a set of explanatory variables is not exhaustive and extra regressors are not available or are incapable of removing the residual correlation, it is preferable to use the autocorrelated errors model.

The autocorrelated errors model is the specification given by eq. (18), but with additional regressors defined by spatial lagging. In contrast to eq. (9), the spatially lagged regressors involve both the regressand and all the regressors and the regression is subject to a common factor constraint (Ord, 1975; Burridge, 1981). The model is therefore

ln [P.sub.[t.sup.*]] = [b.sub.0] + [b.sub.4] ln [P.sub.0] + [u.sub.1] (19)

sl[u.sub.1] = [summation over j] [W.sub.ij][u.sub.1j] (20)

[u.sub.1] = [Rho]sl[u.sub.1] + [u.sub.2] (21)

[u.sub.2] [similar to] N(0, [[Sigma].sup.2]) (22)

The assumed normal distribution for [u.sub.2] governs the form of the likelihood function used for ML estimation. As with the ML estimation of eq. (9), the model is estimated using an iterative search routine over the range 1/[i.sub.min] to 1/[i.sub.max], where [i.sub.max] and [i.sub.max], are the minimum and maximum eigenvalues of W. This defines the feasible range for a stable model and within this the likelihood function is consistently unimodal and local minima are never encountered. The eigenvalues are also used to calculate the asymptotic variance-covariance matrix (the Cramer-Rao lower bound). The standardisation of W to row sums of unity means that/[i.sub.max] is equal to one.

Table 3 shows the results of OLS and ML estimation of the neoclassical model. OLS estimation without national dummies, eq. (j), indicates that the unconditional estimate of [Theta] is statistically significantly different from zero, and that convergence has proceeded at approximately 4% per annum, although this estimate is undoubtedly biased. Bias would occur if significant regressors were present in the residuals which correlated with the start-of-period productivity level. Moran's I test of residual autocorrelation points to significant residual spatial autocorrelation and therefore the possible presence of missing (spatially autocorrelated) regressors. The autocorrelated errors specification, eq. (l), which attempts to capture these bias-inducing residual effects, suggests even faster catch-up, with the ML estimate suggesting a rate of about 5% per annum. This faster rate of convergence indicates that the autoregressive error terms are proxying for variables which are retarding the growth of some regions, leading to slower unconditional convergence. Nevertheless, the rate of convergence of manufacturing productivity is much faster than is apparent for total GVA per capita. Comparable unconditional OLS and ML estimates by Fingleton (1995) for total GVA per capita are 0.3% and 0.7% per annum.

The models incorporating national dummy variables provide estimates of within-country convergence. In this case convergence is estimated to be occurring at a dramatically high rate of about 9% per annum using both OLS and ML [TABULAR DATA FOR TABLE 3 OMITTED] methods (eqs (k) and (m)), a figure far higher than any of the reported figures based on total GDP or GVA. Barro and Sala-i-Martin (1995) give estimates around 1% per annum for European regions and Armstrong (1995) gives roughly comparable figures. Fingleton (1995) estimates within-country convergence also at the rate of roughly 1% per annum. Of course, there is considerable variation around these approximate figures depending on the time period considered, the data set analysed and the conditioning regressors, but nevertheless there is a noticeable difference from those of the present paper which can only be attributed to the fact that the current analysis is focused on manufacturing productivity growth.

The autocorrelated errors model recognises that eq. (18) is subject to misspecification, notwithstanding that this equation (or a rearrangement of it) has been widely used in the literature to estimate the rate of convergence. The problem is that parameter [Mathematical Expression Omitted], which equals (1 - [Alpha])([Lambda] + [Delta] + e) and is derived from [Mathematical Expression Omitted], in fact contains a variable, namely the growth of employment, which shows a considerable amount of spatial variation. (It is possible that this might also be true of the rate of technical progress and depreciation, but the assumption that these are spatially constant is more defensible.) To avoid this problem, eq. (18) was respecified as

ln [P.sub.t] = [b.sub.0] + [b.sub.5][ln [P.sub.0] exp (-(1 - [Alpha])et)] + u (23)

where [b.sub.5] = exp (-[Theta][prime]T)and [Theta][prime] = (1 - [Alpha])([Lambda] + [Delta]) which is assumed constant.

A further advantage of eq. (23) is that it is now possible to obtain estimates of the convergence rate of each individual region, i, and this is given by [Mathematical Expression Omitted]. In addition, the value of the unweighted mean estimator may be derived as ([Sigma][[Theta].sub.i])/n, where n is the number of regions.

[TABULAR DATA FOR TABLE 4 OMITTED]

Table 4 reports the results of these alternative estimates of the rates of convergence. Equation (n) is the result of a non-linear least squares regression which estimated both [b.sub.5] and (1 - [Alpha]), the latter being the coefficient of eT. (Computational difficulties precluded the inclusion of national dummies in this regression.) The mean convergence rate was 3.2% per annum with a range from 5.6% to -1.6%. However, the estimate of [Alpha], while highly statistically significant, took an implausibly large value of 0.709 (which implies an output elasticity with respect to labour of 0.291. Consequently, we constrained the value of [Alpha] to one-third and the regression results are reported in eqs (o) and (p) (with and without national dummies, respectively.) Confining our attention to eq. (p), it may be seen that the mean convergence rate increases dramatically to 11.7% per annum. The values for the individual regions range from 17.3% to 0.7% per annum. It should be noted, however, that both equations fail the RESET test, casting doubt on the appropriateness of the neoclassical specification. Equations (q) and (r) report the results of an autocorrelated errors specification (see eqs (19)-(23) above), and the results are very similar to the non-linear and ordinary least-squares estimates.(3)

7. Conclusions

In this paper, we have shown that the preferred specification of the Verdoorn law exhibits strong increasing returns to scale and that there is a significant, but weak, diffusion of innovation effect. On the other hand, the results of the convergence analysis suggest a very rapid convergence and the question arises as to how this is to be reconciled with the findings above of substantial increasing returns and a weak effect of the diffusion of technology. The difficulty is that the hypotheses to be tested are not nested and are derived from different underlying assumptions. In other words, there is problem of equifinality, with different models producing the same results.

The major difficulty in making any comparison is that in the estimation of the Verdoorn law, the initial level of per capita output was taken as a proxy for the level of technology which is allowed to differ between regions. Consequently, the statistically significant coefficient reflects the contribution to productivity growth of the diffusion of technology from the more to less technologically advanced regions and as such is perfectly compatible with increasing returns and cumulative causation. The Solow growth model assumes that all regions have the same level of technology and attributes the significance of the coefficient to the more rapid productivity growth of those regions with capital-labour ratios that are initially lower than their equilibrium values. (Why this misallocation of resources occurred in the first place is never satisfactorily explained.) For example, Cheshire and Carbonaro (1995) observe that the detection of 'beta convergence' is merely a sign that the data are not inconsistent with neoclassical theory, and not a direct test of diminishing returns to capital or of the income equalising consequences of factor mobility. They see the estimated [Theta] as the net effect of a number of processes, some, such as regional policy, labour mobility, or technological diffusion, perhaps inducing convergence, and others, such as increasing returns, responsible for divergent behaviour. Moreover, they also find that for GDP per capita (using the functional urban region as their spatial unit of measurement), once the percentage national growth rate is included as a regressor in the regression, the coefficient of the logarithm of the initial level of per capita income ceases to be statistically significant. Clearly, the question as to 'why growth rates differ' is still, to a large extent, unresolved.

Acknowledgements

An earlier version of this paper, Fingleton and McCombie (1995), was presented to the ESRC Urban and Regional Economics Study Group conference at Downing College, Cambridge, January 1996. We are grateful to the participants of this meeting for their comments.

1 Because p [equivalent to] q - e, where e is the growth of employment, the [Mathematical Expression Omitted] obtained by estimating eq. (1) will have an element of spurious correlation. This is avoided ire is regressed on q, and we routinely report the corrected [Mathematical Expression Omitted] (denoted by [Mathematical Expression Omitted]) from this regression when reporting the estimates of the Verdoorn law.

2 Because the estimates of the IV regressions are very similar to those of the ML estimations (but less efficient), they are not reported here. The instrument chosen was spatially lagged output growth, (constructed in the same way as slp), and the national dummies were used as additional instruments when they featured in the model.

There is the further problem as to whether q should be treated as exogenous or endogenous (Rowthorn, 1975a, b; Kaldor, 1975). Consequently, we also estimated the equation including as an instrument either the rank or the lagged growth of q (over the period 1975-79). Since the use of these instruments made no significant difference to the estimates, the results are also not reported here.

3 A referee suggested that one possible explanation for the coexistence of increasing returns and convergence is that those regions with below-average levels of productivity also had larger increasing returns to scale. We tested this proposition by creating a dummy taking the value 0 or 1 depending upon whether the initial level of regional productivity was above or below the mean. Multiplying this by q creates an interaction term and provides a test of the hypothesis that the degree of increasing returns to scale varies with the initial level of productivity. In fact, the interaction term was statistically insignificant and its inclusion had virtually no effect on the other regression estimates.

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Appendix: Moran's I

The definition of Moran's I for OLS regression residuals is

[Mathematical Expression Omitted] (24)

where [Mathematical Expression Omitted] is the vector of observed values of the dependent variable, [Mathematical Expression Omitted] is the vector of fitted values and W is the so-called weights matrix involving n(n - 1) inter-locational distances {[d.sub.ij]}. W is specified as

[Mathematical Expression Omitted] (25)

[Mathematical Expression Omitted] (26)

and the matrix is standardised to sum to 1.0 across rows. Hence

[Mathematical Expression Omitted] (27)

The standardisation has the effect that the maximum eigenvalue of W is equal to 1.0. This facilitates the maximum likelihood estimation and it also simplifies the formulae for [I.sub.k] and its moments.

In the definition of Moran's I, the subscript k is a reminder that the vector of residuals [Mathematical Expression Omitted] is subject to k linear constraints as a result of the estimation of k regression coefficients. In interpreting [I.sub.k], it is usual to assume asymptotic normality for the [I.sub.k] sampling distribution, with

E([I.sub.k]) = tr(MW)/(n - k)(28)

Vsr([I.sub.k]) = tr(MWMW[prime]) + tr(MWMW) + [(tr(NW)).sup.2] / (n - k)(n - k + 2) - E[([I.sub.k]).sup.2] (29)

and

M = I - X[(X[prime]X).sup.-1]X[prime] (30)

where I is an identity matrix and x is the n x k matrix of regressors, and to refer

Z = ([I.sub.k] - E([I.sub.k]))/[square root of Var]([I.sub.k]) (31)

to a N(0, 1) distribution. These moments are derived under the normality assumption, as described by Cliff and Ord (1981). The exact distribution is given by Tiefelsdorf and Boots (1995).
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Author:Fingleton, B.; McCombie, J.S.L.
Publication:Oxford Economic Papers
Date:Jan 1, 1998
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