Regional convergence and spatial dependence: a worldwide perspective

Abstract

This paper incorporates technological interdependence into a neoclassical regional growth framework with imperfect factor mobility, leading to a convergence equation with spatial effects. The empirical analysis is based on the estimation of a spatial Durbin panel data model and the implementation of multiple imputation techniques. Our results show that taking into account both unobserved heterogeneity and spatial dependence increases the estimated regional convergence rate. This provides an explanation for puzzling findings in the related literature. We also obtain evidence of heterogeneity across country groups regarding the regional speed of convergence and the degree of diffusion of technology.

Appendices

Appendix 1: steady-state uniqueness

The existence of a unique steady state is demonstrated by finding the values $h_{i}^{*}$ and $h^{*}=s\sum ^{N}_{i=1} (h_i^{*})^{\alpha }$ such that the capital $h_{l}^{*}$ in any region l entails that there is no migration:

$$\begin{aligned} 1=v^{*} (s A_{l} )^\tau (h_{l}^{*})^{\alpha \tau -1} \left[ \bar{A}_{l} s \sum ^{N}_{i=1} A_{i} (h_{i}^{*}) ^{\alpha } \right] ^{1-\tau } \end{aligned}$$

(18)

where $v^{*}$ is the steady-state value of the normalization factor in expression (10).

The previous equation can be rewritten as

$$\begin{aligned} (h_{l}^{*})^{1-\alpha \tau }=v^{*} s A_{l}^{\tau } \left\{ {\frac{\tilde{A}_{l}^{\frac{1}{1-\alpha }}{\sum ^{N}_{i=1} A_{i} (h_{i}^{*})^{\alpha }}}{\sum ^{N}_{j=1} \frac{A_{j}^{\frac{1}{1-\alpha }}}{\left[ \prod _{j \ne i} \tilde{A}_{j}^{\sum _{r=1}^{\infty } (\rho w_{ji})^{r}} \right] ^{\frac{1}{1-\alpha }}}}} \right\} ^{1-\tau } \end{aligned}$$

(19)

which is equivalent to saying that there is always an equilibrium with $h_{i}^{*}=0, \forall i$. Once this possibility is discarded, there is an equilibrium given by $h_{l}^{*} = b \left[ A_{l}^{\tau } \tilde{A}_{l}^{\frac{1-\tau }{1-\alpha }} \right] ^{\frac{1}{1-\alpha \tau }}$, where b is a positive constant with the same value for every region. Even though this constant depends on the entire profile of capital employment levels, each $h_{i}^{*}$ has a negligible impact on it. This implies that the steady-state equilibrium will be interior and unique.

Appendix 2: multiple-imputation techniques

Multiple imputation is a simulation-based statistical technique that consists of replacing missing values by multiple sets of plausible values. The imputed values are not intended to represent the ‘real’ values, but rather to reproduce the variance–covariance structure that would have been observed if the missing information had not been missing. In doing so, multiple imputation handles missing data in a way that results in valid statistical inference. Although the theoretical foundations of this method were derived under the Bayesian paradigm, they are valid from a frequentist point of view. The Bayesian approach is used to create the imputations and underlies the combination of the estimated parameters. Once an imputation model is selected, M complete data sets are generated on which the analysis of interest is performed. Finally, the results from these M analyses are combined into a single multiple-imputation result. At this stage of the estimation, the coefficients and standard errors are adjusted for the variability between imputations, following the combination rules proposed by Rubin (1987).

Multiple imputation is preferable to alternative methods like listwise deletion, pairwise deletion, mean imputation and single imputations, because it only requires the missing data mechanism to be ignorable. In other words, it only requires that it is possible to disregard the process that caused the information to be missing, assuming that the data is missing at random (MAR). Another possibility is to assume that the data are missing completely at random (MCAR). In this case, the probability that information is missing does not depend on observed or unobserved variables, implying that the missing values are a random sample of all data values. If the missing data were not at random (MNAR), the reasons for being missing should be taken into account in the imputation model to obtain valid results.

Under the MAR assumption, iterative Markov chain Monte Carlo (MCMC) methods are used to simulate the imputed values from the posterior predictive distribution of the missing data given the observed data. The simulation error will decrease when the number of imputations increases, especially with high fractions of missing information. In the present paper, the simulations have been carried out running multiple independent chained equations (MICE), which obtain univariate conditional distributions for each variable from a fully conditional specification of the prediction equations. Even though this technique lacks a rigorous theoretical justification, its flexibility has made it a frequent choice in practice. More specifically, MICE is similar to the Gibbs sampler, a popular MCMC method for simulating data from complicated multivariate distributions.

Considering that the complete data X is a partially observed random sample from a multivariate distribution $P\left( X|\theta \right)$, it has been assumed that the latter is completely specified by the unknown vector of parameters $\theta$ (Buuren and Groothuis-Oudshoorn 2011). In order to obtain its multivariate distribution, the MICE algorithm samples iteratively, on a variable-by-variable basis, from the p univariate conditional distributions $P\left( X_{1}|X_{-1},\theta _{-1}\right)$, ..., $P\left( X_{p}|X_{-p},\theta _{-p}\right)$. It is worth noting that $\theta _{1}$, ..., $\theta _{p}$ are specific to their corresponding conditional densities and not necessarily the product of a factorization of $P\left( X|\theta \right)$. The variables will be imputed from the most observed to the least observed. Taking a sample drawn from the observed marginal distributions as the starting point, the $\kappa$th iteration is a Gibbs sampler that draws

$$\begin{aligned} \begin{gathered} \theta _{1}^{*(\kappa )} \thicksim P\left( \theta _{1} | X_{1}^\mathrm{obs},X_{2}^{\kappa -1},\ldots ,X_{p}^{\kappa -1}\right) \\ X_{1}^{*(\kappa )} \thicksim P\left( X_{1} | X_{1}^\mathrm{obs},X_{2}^{\kappa -1},\ldots ,X_{p}^{\kappa -1},\theta _{1}^{*(\kappa )}\right) \\ \vdots \\ \theta _{p}^{*(\kappa )} \thicksim P\left( \theta _{p} | X_{p}^\mathrm{obs},X_{1}^{\kappa },\ldots ,X_{p -1}^{\kappa }\right) \\ X_{p}^{*(\kappa )} \thicksim P\left( X_{p} | X_{p}^\mathrm{obs},X_{1}^{\kappa },\ldots ,X_{p}^{\kappa },\theta _{p}^{*(\kappa )}\right) \end{gathered} \end{aligned}$$

(20)

where $X_{m}^{(\kappa )} = \left( X_{m}^\mathrm{obs},X_{m}^{*(\kappa )}\right)$ denotes the mth imputed variable.

Appendix 3: preliminary analysis

This appendix studies the extent to which changes in data handling and the sample period analysed and the use of the multiple-imputation technique, briefly described in “Appendix 2”, alter standard regression results. With this aim, Table 7 presents pooled OLS estimates of the growth equation (15) without spatial effects (i.e., imposing $\rho =0$) in four alternative samples. The first column of results reproduce those reported by GLLS—see specification (4) in Table 5 (page 282)—in order to compare them to the estimated parameters under our proposed treatment of the data, displayed in the second column. Although the explanatory power is now slightly higher, the number of observations is smaller and different coefficients are obtained.

Table 7 Determinants of regional growth. Pooled OLS estimation

The most important disparities are found for national GDP per capita and the indicator reflecting whether the region includes the capital city, whose estimated parameters are negative and statistically significant. Although coefficients of a similar magnitude are obtained for latitude, average distance to the coast, the ‘malaria index’ and years of schooling, this is not the case for oil and gas production and population density. These variables also change their statistical significance, which is now favourable for population density. Last, but not least, the parameter for the initial level of regional GDP per capita implies an even lower convergence speed than that reported by GLLS. The explanatory power and estimated coefficients are stable across samples. The greatest difference is in the variable that reflects the regional disease environment, as its corresponding parameter is not only higher but also statistically significant. In absolute values, the coefficients for the capital city indicator and the proxy for human capital are higher in the most recent period. These findings allow us to conclude that restricting the analysis to $1980{-}2010$ does not alter the underlying relation between regional growth and its determinants in this empirical framework.

For each sample period analysed, the last two columns show the estimated parameters when the multiple-imputation technique is implemented, fixing $M=20$. The difference between these pairs of sets of results depends on whether or not the panel structure underlying the data has been explicitly taken into account. Together with the growth determinants not affected by the missing information problem, the longitude of the regional centroid and country and time dummies have also been considered as explanatory variables in the imputation step. It can be stated that, in general, the use of this method does not modify either the sign or the statistical significance of the estimated parameters. This is especially the case when the panel nature of the data is controlled for, but at the cost of a higher sampling variance due to the missing information, as measured by the average relative variance increase (RVI) and the largest fraction of missing information (FMI).

Fig. 3

Kernel density estimation: observed (black) and imputed values (light grey) of variables with missing information

In order to give a visual impression of how the multiple-imputation method works in practice, and for the sample covering the years from 1980 to 2010, Fig. 3 plots kernel estimates of the density function for the regional growth determinants with missing data. While the black lines correspond to the distributions of the observed data, the grey lines also include the imputed values. The density functions of both the observed and imputed data sets are similar for population density, which might explain why the estimated coefficient for this variable does not change when it is taken into account that some information is missing. More interestingly, both national and regional GDP per capita and years of education display higher frequencies in their lower values when imputed values are considered. This reflects the fact that the unbalanced character of the panel is mainly driven by the missing information problems suffered by the regions in countries with lower levels of development.

Appendix 4: panel-correlated random effects estimations—coefficients of region-specific averages

Mundlak (1978) proposed an empirical framework that permits the estimation of both time-varying and time-invariant variables with panel data. This method, known as correlated random effects (CRE) estimation, consists of including individual-specific means of time-varying covariates to account for their correlation with the individual effects. The CRE approach was extended to the panel version of the spatial Durbin model by Debarsy (2012). Table 8 reports the estimated coefficients in the empirical analysis carried out in the present paper for the region-specific averages of lagged regional and national income per capita levels, population density and years of schooling, as well as their spatial lags.

Table 8 Panel-correlated random effects estimations: coefficients of region-specific averages