The contribution of physical and human capital accumulation to Italian regional growth: a nonparametric perspective

Abstract

This paper examines changes in the labor productivity, efficiency, technology, and physical and human capital experienced by different regions in Italy between 1980 and 2006. Cobb-Douglas and translog production specifications are not supported by the data. Thus, non-parametric methods are used to compute the Malmquist indices and their components. Moreover, the bootstrap technique allows us to determine the confidence intervals of all components of the labor productivity decomposition. The results suggest that the contributions of efficiency, technology, and physical and human capital accumulation to labor productivity growth differ significantly between Southern Italy and the remainder of the country.

Appendix A

Given the estimates ${\widehat{{\mathcal{M}}}(t_1,t_2)}$ of the unknown true values of ${{\mathcal{M}}(t_1,t_2)}$, we generate, through the DGP process, a series of pseudo-datasets to obtain a bootstrap estimate ${\widehat{{\mathcal{M}}}^*(t_1,t_2)}$. Simar and Wilson (1998) discussed the problems that arise for bootstrapping in DEA models, and they suggested the use of a smooth bootstrap procedure. In addition, the Malmquist index uses panel data, with the possibility of temporal correlation. For this reason, Simar and Wilson (1999) modified the bootstrap algorithm for efficiency scores to preserve any temporal correlation present in the data by applying a bivariate smoothing procedure. The procedure can be summarized as follows:

• 1. Compute the Malmquist productivity index ${\widehat{{\mathcal{M}}}_i(t_1,t_2)}$, for each region $i=1,\ldots,20$, by solving the DEA models and using Eq. (1), as described in Färe et al. (1994).

• 2. Calculate the pseudo-dataset $\left\{\left(\user2{X}^*_{it},Y^*_{it} \right);i=1,\ldots,20; t=1,2 \right\}$ to obtain the reference bootstrap technology using bivariate kernel density where the bandwidth was selected following the normal reference rule.

• 3. Compute the bootstrap estimate of the Malmquist index ${\widehat{{\mathcal{M}}}^*_{i,b}(t_1,t_2)}$ for each region through the pseudo-sample obtained in step 2.

• 4. Repeat steps 2 and 3, B times (number of bootstrap replications) to obtain the bootstrap sample ${\left\{\widehat{{\mathcal{M}}}^*_{i,1}(t_1,t_2),\ldots ,\widehat{{\mathcal{M}}}^*_{i,B}(t_1,t_2)\right\}}$.

• 5. From the bootstrap sample, compute the confidence intervals for the Malmquist index by selecting the appropriate percentiles.

The construction of the confidence intervals is obtained by sorting the values ${\left\{\widehat{{\mathcal{M}}}^*_{i,b}(t_1,t_2)- \widehat{{\mathcal{M}}}_{i}(t_1,t_2)\right\}^{B}_{b=1}}$ in increasing order and deleting the $\left(\frac{\alpha}{2}\cdot 100 \right)$-percent of the elements at either end of the sorted list. Then, for setting $-\widehat{a}^*_\alpha$ and $-\widehat{b}^*_\alpha$ (with $\widehat{a}^*_\alpha <\widehat{b}^*_\alpha$), which is equal to the endpoints of the sorted array, the estimated (1 − α)-percent confidence interval for the productivity index is:

$$ \widehat{{{\mathcal{M}}}}_{i}(t_1,t_2) + \widehat{a}^*_\alpha \leq {{\mathcal{M}}}_{i}(t_1,t_2) \leq \widehat{{{\mathcal{M}}}}_{i}(t_1,t_2) + \widehat{b}^*_\alpha $$

(4)

The relation (4) is similarly computed for the other components of the labor productivity decomposition: efficiency change (EFF), technological change (TECH), capital (KACC) and human capital accumulation (HACC).