deciles below the median in the comparison distribution come before those of the reference distribution along the consumption scale, while cut-off points for deciles above the median come after.

# **3 Results**

## **3.1 Changes in the Ghanaian consumption distribution**

To introduce the results obtained from using the methods and data described in previous sections, in Figure 1(a) we present two probability density functions of the Ghanaian distribution of total consumption expenditure.<sup>13</sup> The solid line is the distribution of household consumption in 1991/92, taken as the baseline throughout the analysis. The density drawn with the dotted line, which we will treat as the comparison, is the distribution in 2012/13.<sup>14</sup> Examining these two distributions, we see that the reference or 1991/92 distribution has a slight right skewness, while the comparison distribution has a larger median and variance.

However, the graphical display above does not provide much information on the relative impact that location and shape changes had on the differences in the two distributions at every point of the expenditure scale. It also does not convey whether the upper and lower tails of the consumption distribution were growing at the same rate and for what reasons (i.e. location and/or shape driven). As already pointed out in Subsection 2.2, this is exactly what relative distribution methods are particularly good at pulling out of the data.

The relative density of total consumption expenditure of Ghanaian households between 1991/92 and 2012/13 is examined in Figure 1(b), showing the fraction of households in 2012/13 that fall into each decile of the 1991/92 distribution.<sup>15</sup> The graph offers the immediate impression that the proportion of households in the upper deciles increased dramatically throughout the two decades, while the

> 13 To handle data sparseness, the two densities have been obtained by using an adaptive kernel estimator with a Silverman’s plug-in estimate for the pilot bandwidth (see e.g. Van Kerm, 2003). The advantage of this estimator is that it does not over-smooth the distribution in zones of high expenditure concentration, while keeping the variability of the estimates low where data are scarce – as, for example, in the highest expenditure ranges. 14 Obviously, reversing the reference and comparison distribution designation will change the view provided by the relative distribution graph and the displays of the estimated effects of location and shape shifts, because these are defined in terms of the reference distribution scale. However, designating which distribution will serve as the reference is a decision that must be made by the analyst, and in our application the natural choice was suggested by time ordering. In addition, the relative polarization indices (measurements of the degree to which a comparison distribution is more polarized than a reference distribution, and defined in terms of the relative

> distribution of the comparison relative to the median-adjusted reference) are _symmetric_ , meaning that they are effectively invariant to whether the 1991/92 or 2012/13 consumption distribution is chosen as the reference – in fact, swapping the comparison and reference distributions yields indices of the same magnitude and opposite sign (Handcock and Morris, 1999, pp. 71-72; Hao and Naiman, 2010, pp. 88-89). Thus, reversing the reference and comparison distributions designation will not alter our findings in a substantive way – if not for the fact that polarization would now be analyzed in the reverse direction of time.

> 15 Throughout, we rely on the `R` statistical package `reldist` (Handcock, 2015) to implement the relative distribution method.

14