Benchmark averaging and the measurement of changes in international income inequality

Abstract

Using data for 113 countries covering the period 1980–2005 we show how international comparisons of income inequality and the way it changes over time are inherently sensitive to (i) the choice of multilateral price index formula used to convert per capita incomes into units of the same currency, (ii) the approach (if any) used for reconciling spatial benchmarks with national growth rates, and (iii) the way inequality is measured. We then consider how best to deal with these issues and highlight some distortions that can arise in such comparisons. Based on our preferred methods we observe convergence when countries are population weighted and divergence when they are not.

Appendices

Appendix 1: Weighted least squares estimation of our active benchmark averaging model

We assume for simplicity that we have complete data (i.e., that is all countries are present in all benchmarks and have complete temporal data). The extension to the case where there is incomplete data is immediate and just involves removing the relevant rows of the various vectors and matrices.

The full model can be written in matrix notation as follows:

$$ \mathbf{y}= {} \mathbf{X} {\alpha } + \mathbf{v}, \quad \mathrm{E} ( \mathbf{v} \mathbf{v}^T ) = \varvec{\Sigma };$$

(28)

where

$$\begin{aligned}&\mathbf{y} = \left( \begin{array}{c} \mathbf{\ln P_{jk|80}} \\ \mathbf{\ln P_{jk|85}} \\ \mathbf{\ln P_{jk|96}} \\ \mathbf{\ln P_{jk|05}} \\ \mathbf{\ln P_{k|80,85}} - \mathbf{\ln P_{j|80,85}} \\ \mathbf{\ln P_{k|85,96}} - \mathbf{\ln P_{j|85,96}} \\ \mathbf{\ln P_{k|96,05}} - \mathbf{\ln P_{j|96,05}} \end{array} \right) ; \quad \mathbf{X} = \left( \begin{array}{cccc} \mathbf{D} &{} \mathbf{0} &{} \mathbf{0} &{} \mathbf{0} \\ \mathbf{0} &{} \mathbf{D} &{} \mathbf{0} &{} \mathbf{0} \\ \mathbf{0} &{} \mathbf{0} &{} \mathbf{D} &{} \mathbf{0} \\ \mathbf{0} &{} \mathbf{0} &{} \mathbf{0} &{} \mathbf{D} \\ -\mathbf{D} &{} \mathbf{D} &{} \mathbf{0} &{} \mathbf{0} \\ \mathbf{0} &{} -\mathbf{D} &{} \mathbf{D} &{} \mathbf{0} \\ \mathbf{0} &{} \mathbf{0} &{} -\mathbf{D} &{} \mathbf{D} \end{array} \right) ; \nonumber \\&{\alpha } = \left( \begin{array}{c} { \alpha _{jk|80}} \\ { \alpha _{jk|85}} \\ { \alpha _{jk|96}} \\ {\alpha _{jk|05}} \end{array} \right) ; \quad \mathbf{v} = \left( \begin{array}{c} { \varepsilon _{jk|80}} \\ { \varepsilon _{jk|85}} \\ { \varepsilon _{jk|96}} \\ {\varepsilon _{jk|05}} \\ \mathbf{u_{k|80,85}} - \mathbf{u_{j|80,85}} \\ \mathbf{u_{k|85,96}} - \mathbf{u_{j|85,96}} \\ \mathbf{u_{k|96,05}} - \mathbf{u_{j|96,05}} \end{array} \right) . \end{aligned}$$

(29)

Here $\mathbf{\ln P_{jk|t}}$ is a column vector of spatial price indexes with country j as the base for benchmark t and $\mathbf{\ln P_{k|st}} - \mathbf{\ln P_{j|st}}$ is a column vector of differences between the temporal price indexes of countries j and k from year s to t, again with j as the base. These price indexes are related to the spatial parameters of interest by the matrix $\mathbf{X}$. The $\mathbf{X}$ matrix is itself constructed from component $\mathbf{D}$ matrices, each of which is a diagonal matrix of ones with dimension $K \times K$ (where K is the number of countries in the benchmark). The first four rows in the $\mathbf{X}$ matrix as written above relate to the spatial price index Eq. (9) for the four benchmarks, while the last three rows relate to the Eqs. (16)–(18) which measure the differences across countries in temporal price indexes calculated over the time interval between adjacent benchmarks.

Given this model we use the following weighted-least-squares estimator for the spatial parameter coefficients:

$$ \hat{{\alpha }}= {} \left( \mathbf{X} \hat{\varvec{\Sigma }}^{-1} \mathbf{X} \right) ^{-1} \mathbf{X}^T \hat{\varvec{\Sigma }}^{-1} \mathbf{y}.$$

(30)

The error covariance matrix $\hat{\varvec{\Sigma }}$ is diagonal with the terms on the lead diagonal obtained from the spatial and temporal variance Eqs. (11) and (12). We iterate between the estimation of (30) and using the regression residuals, conditional on the current value of $\hat{{\alpha }}$, to model the variance function shown in Eqs. (11) and (12). The predictions from the variance model are inverted and used as weights in $\hat{\varvec{\Sigma }}$. Convergence occurs when there is a suitably small change in $\hat{{\alpha }}$.

Appendix 2: Derivation of the underlying dissimilarity indexes of dispersion measures A, D and E

The variance of log per capita income can be written as follows:

$$\begin{aligned} (\sigma _t)^2= \,&{} \frac{1}{K-1} \sum _{j=1}^K \left( \ln y_{jt} - \frac{1}{K} \sum _{k=1}^K \ln y_{kt} \right) ^2 = \frac{1}{K-1} \sum _{j=1}^K \left[ \frac{1}{K} \sum _{k=1}^K \left( \ln y_{jt} - \ln y_{kt}\right) \right] ^2 \\= \,&{} \frac{1}{K(K-1)} \sum _{j=1}^K \sum _{k=1}^K \left[ \left( \ln y_{jt} - \ln y_{kt}\right) \left( \ln y_{jt} - \frac{1}{K} \sum _{l=1}^K \ln y_{lt}\right) \right] . \end{aligned}$$

The implicit dissimilarity index underlying the standard deviation of log per capita income is therefore

$$ d(y_{jt},y_{kt}) = \left( \ln y_{jt} - \ln y_{kt}\right) \left( \ln y_{jt} - \frac{1}{K} \sum _{l=1}^K \ln y_{lt}\right) .$$

(31)

As it stands, this dissimilarity index violates the symmetry axiom A4. However, taking the sum of $d(y_{jt},y_{kt})$ and $d(y_{kt},y_{jt})$ as defined in (31) we obtain the log quadratic dissimilarity index $d^A$ defined in (21). It follows that

$$(\sigma _t)^2 = \frac{\sum _{j=1}^K \sum _{k=1}^K d^{\sigma }(y_{jt},y_{kt})}{K(K-1)} = \frac{1}{2} \frac{\sum _{j=1}^K \sum _{k=1}^K d^A(y_{jt},y_{kt})}{K(K-1)} \equiv \frac{{\delta }^A_t}{2}. $$

Hence $\sigma _t = \sqrt{{\delta }^A_t/2}$. In other words, the standard deviation of log per capita income is an increasing monotonic function of ${\delta }^A$.

The square of the coefficient of variation of per capita income can likewise be rewritten as follows:

$$\begin{aligned} \left( \frac{\sigma _t}{\mu _t}\right) ^2= \,& {} \frac{1}{(K-1)} \frac{1}{\bar{y}_t^2}\left[ \sum _{j=1}^{K} y_{jt}^2 - \frac{(\sum _{k=1}^{K} y_{kt})^2}{K} \right] \\= \,& {} \frac{1}{K(K-1)} \sum _{j=1}^K \sum _{k=1}^K \left( \frac{y_{jt}^2 - y_{jt} y_{kt}}{\bar{y}_t^2} \right) . \end{aligned}$$

The implicit dissimilarity index underlying the coefficient of variation of per capita income therefore is

$$ d^{D}(y_{jt},y_{kt}) = \frac{y_{jt}^2 - y_{jt} y_{kt}}{\bar{y}_t^2}. $$

(32)

Taking the sum of $d(y_{jt},y_{kt})$ and $d(y_{kt},y_{jt})$ as defined in (32) we obtain the dissimilarity index $d^D$ defined in (24). It follows that:

$$ \left( \frac{\sigma _t}{\mu _t}\right) ^2 = \frac{\sum _{j=1}^K \sum _{k=1}^K d^{\sigma /\mu }(y_{jt},y_{kt})}{K(K-1)} = \frac{1}{2} \frac{\sum _{j=1}^K \sum _{k=1}^K d^D(y_{jt},y_{kt})}{K(K-1)} \equiv \frac{\delta _t^D}{2}. $$

Hence $\sigma _t/\mu _t = \sqrt{{\delta }^D_t/2}$. In other words, the coefficient of variation of per capita income is an increasing monotonic function of ${\delta }^D$. Similarly, given that GE(2) = $(\sigma /\mu )^2/2$ it follows that GE(2) = $\delta _t^D/4$, where GE(2) is the generalized entropy index with $\alpha =2$.

The Gini index can be written as follows:

$$ \mathrm{Gini}_t = \frac{1}{2 K^2} \sum _{j=1}^K \sum _{k=1}^K \left| \frac{y_{jt}}{\bar{y}_t} - \frac{y_{kt}}{\bar{y}_t} \right| = \frac{1}{K^2} \sum _{j=1}^K \sum _{k=1}^K d^E(y_{jt}/\bar{y}_t,y_{kt}/\bar{y}_t) = \delta _t^E. $$

Appendix 3

See Table 4.

Table 4 List of ICP participating countries