## 1. INTRODUCTION

Why have some countries grown rich while others have remained poor? This is a recurrent question in the literature on theoretical and empirical economic growth. One of the traditional stylized facts about growth over the last 50 years is that national growth rates appear to depend critically on the growth rates and income levels of other countries, rather than just on any one country's own domestic investment rates in physical and human capital. For example, Easterly and Levine (2001) present as a stylized fact the concentration of economic activity at different scales: world, countries, regions, cities. More recently, Klenow and Rodriguez-Clare (2005) present stylized facts reflecting worldwide interdependence, which could be explained by cross-country externalities.

Knowledge accumulated in one country depends on knowledge accumulated in other countries. These spatial externalities involve technological interdependence among countries. Therefore, in this paper we argue that a model needs to include this global interdependence phenomenon in order to explain development and growth. Several models of economic growth emphasize the importance of international spillovers as a major engine of technological progress. These international spillovers result from foreign knowledge through international trade and foreign direct investment (Coe and Helpman, 1995; Eaton and Kortum, 1996; Caves, 1996), or technology transfers (Barro and Sala-i-Martin, 1997; Howitt, 2000) or human capital externalities (Lucas, 1988, 1993). In addition, Keller (2002) suggests that the international diffusion of technology is geographically localized, in the sense that the productivity effects of R&D decline with the geographic distance between countries.

Moreover, in the recent literature, several papers provide empirical evidence of spatial effects, spatial autocorrelation and heterogeneity, on growth. As noted by De Long and Summers (1991, p. 487): ‘It is difficult to believe that Belgian and Dutch or US and Canadian economic growth would ever significantly diverge, or that substantial productivity gaps would appear within Scandinavia.’ Temple (1999), in his survey of the new growth evidence, draws attention to error correlation and regional spillovers, though he interprets these effects as mainly reflecting an omitted variable problem. In empirical papers, Conley and Ligon (2002), Ertur *et al.* (2006) and Moreno and Trehan (1997) use geographic and economic distance to underline the impact of cross-country spillovers on growth processes.

In theoretical work, international spillovers have been studied mostly in the framework of endogenous growth models (Aghion and Howitt, 1992; Grossman and Helpman, 1991; Rivera-Batiz and Romer, 1991; Howitt, 2000). Nevertheless, in this paper we consider the more tractable neoclassical growth model (Solow, 1956) as augmented, for example, by Mankiw *et al.* (henceforth MRW, 1992) and we propose an estimate of the magnitude of worldwide technological interdependence. More precisely, this paper presents an augmented Solow model that includes both physical capital externalities as suggested by the Frankel–Arrow–Romer model (Arrow, 1962; Frankel, 1962; Romer, 1986) and spatial externalities in knowledge to model technological interdependence.

In Section 2, we suppose that technical progress depends on the stock of physical capital per worker, which is complementary with the stock of knowledge in the home country as in Romer (1986). It also depends on the stock of knowledge in other countries which affects the technical progress of the home country. The intensity of this spillover effect is assumed to be related to some concept of socio-economic or institutional proximity, which we capture by exogenous geographical proximity. Our model provides an equation for the steady-state income level as well as a conditional convergence equation characterized by parameter heterogeneity. Therefore, in Section 3, after presenting the database and the spatial weight matrix which is used to model the spatial connections between all the countries in the sample, we estimate these equations and test the qualitative and quantitative predictions of the model.

In Section 4, we estimate the effects of investment rate, population growth and neighborhood on real income per worker at steady state using a spatial autoregressive specification. This estimation can be used to assess the values of the structural parameters in the model. First, we estimate the share of physical capital to be close to one-third, as expected. Indeed the estimated value of the capital share of GDP in the textbook Solow model is overestimated (about 0.7). Two approaches are suggested in the literature to explain this value.

First, as proposed by MRW (1992), human capital should be taken into consideration together with physical capital to achieve the commonly accepted value of one-third for the capital share. This first approach has been largely developed in the theoretical as well as the empirical literature underlining the important role for some measure of human capital in explaining cross-country income differences. Nevertheless, many recent empirical studies point out that human capital growth has an insignificant, and even negative effect on per capita income growth (Benhabib and Spiegel, 1994; Bils and Kleenow, 2000; Pritchett, 2001). These results cast some doubt on the real role played by human capital in growth processes and draw attention to the so-called human capital puzzle. Does only the rate at which human capital is accumulated matter as argued in the neoclassical framework (MRW, 1992)? Or, is it only once a threshold level of human capital is reached that economies can be expected to escape the poverty trap (see, for example, Azariadis and Drazen, 1990; Romer, 1990)? A deeper investigation of the role of human capital in growth processes is therefore needed, but this is beyond the scope of the present paper.

Second, as suggested by Romer (1986, 1987) among others, an alternative approach to raise the capital share from one-third to two-thirds is to argue that there are positive externalities to physical capital. However, he is unable to identify and hence estimate the value of physical capital externalities in the model he develops. In this paper, following the latter approach, we show that in our model we can indeed identify the parameter associated with physical capital externalities at steady state and estimate it. We find evidence in favor of physical capital externalities but these externalities are not strong enough to generate endogenous growth. Moreover, the estimated capital share is close to 1/3 without adding human capital as a production factor. Finally, we assess the effect of technological interdependence in growth processes by estimating the parameter describing spatial externalities, which is also significant. Therefore, in our opinion, taking into account technological interdependence is fundamental to understanding differences between income levels and growth rates in a worldwide economy.

In Section 5, we estimate our spatial version of the conditional convergence model. In fact, several empirical papers have found evidence of β-convergence between economies after controlling for differences in steady states (MRW, 1992; Barro and Sala-i-Martin, 1992, 2004). However, as underlined by Brock and Durlauf (2001) and Durlauf *et al.* (2005), the typical cross-country growth regressions used in the literature raise different kinds of problem both from the theoretical and methodological points of view. More precisely, they categorize these problems in three groups: open-endedness of theories or model uncertainty, parameter heterogeneity, correlation and causality. They subsume these problems within the concept of exchangeability, which can be loosely defined as interchangeability of the standard growth regression errors across observations: ‘different patterns of realized errors are equally likely to occur if the realizations are permuted across countries. In other words, the information available to a researcher about the countries is not informative about the error terms’ (Durlauf *et al.*, 2005, p. 36). Most of the criticisms of standard growth regressions can be interpreted as a violation of the implicit exchangeability hypothesis traditionally made to estimate growth regressions. It is the case of omitted variables and parameter heterogeneity problems often raised in the literature. Presence of cross-section correlation, more specifically spatial autocorrelation, in growth regressions also constitutes a major violation of the exchangeability hypothesis.

Therefore, we show in this paper that spatial autocorrelation often detected in empirical cross-country growth regressions has to be explained at the theoretical level and included in the structural model. In addition, Durlauf and Johnson (1995) have directly tested and rejected the hypothesis that the coefficients in these cross-country regressions are the same in different subsets of the sample of countries, highlighting the heterogeneity problem. Moreover, Durlauf *et al.* (2001) account for country-specific heterogeneity in the Solow growth model using varying coefficients. The model we propose takes into account both of these problems simultaneously. We first estimate, as a benchmark, the homogeneous version of our spatially augmented conditional convergence model. We show that the technological interdependence generated by spatial externalities is important in explaining the conditional convergence process. Finally, we estimate the spatial heterogeneous version of our local convergence model, which is precisely the reduced form derived from our structural model, using the spatial autoregressive local estimation method developed by Pace and LeSage (2004).