POVERTY-SPECIFIC PURCHASING POWER PARITIES (PPPs) IN AFRICA recall periods that vary depending on the type of goods and services. Finally, the third step focuses on detecting and fixing top outliers in consumption values to further ensure the reliability of the survey data. For more details on the standardization process, see Dupriez (2007).

# **4.** **_Poverty-specific PPPs: Reweighting of consumption patterns_**

## _4.1 Method and approach_

The usage of national accounts instead of poverty relevant weights in calculating consumption PPPs is a typical critique of the IPL used for global poverty measurements. However, as mentioned earlier, this was not a problem as found by Deaton and Dupriez (2011) using 2005 ICP data.

In revisiting this topic, we follow up on Deaton and Dupriez (2011) and calculate poverty-reweighted PPPs using 2011 ICP data. The reweighting procedure involves substituting the conventional national accounts-based basic heading expenditure weights with poverty-relevant weights from household expenditure surveys. Implicit in this process is the requirement to identify poor households consistently across countries, which in turn leads to a well-recognized circularity issue: PPPs determine the IPL below which the poor live, whose expenditure weights in turn affect the PPPs. We solve this issue using an iterative procedure to arrive at a final set of poverty-weighted PPPs. The procedure can be described as follows: the **n** _th_ iteration would involve estimating a new set of PPPs— **PPP** **_n_** —with some previous set of poverty weights **w** **_n-1_** and then use **PPP** **_n_** to estimate a new set of poverty weights **w** **_n_** to be used in the next iteration **_n+1_** .

In this context, we estimated a variety of poverty-weighted PPPs using different methods to identify poor housing across household expenditures surveys. In particular, we used the uniform and bi-weight kernels to obtain the consumption patterns from households around the IPL. We then derived average basic heading level expenditure weights using a _democratic_ method.<sup>10</sup> As a control, we also ran the computation for poverty-weighted PPPs using expenditures from households below the IPL and averaged the basic heading weights using a _plutocratic_ method.

In discussing the poverty-weighted PPPs based on iterative procedures, the issue of uniqueness of the solution deserves special attention. Deaton and Dupriez (2011) note that there is no guarantee that a unique solution exists in the general case. They report that uniqueness is guaranteed though for log-linear budget shares and the Törnqvist index.

However, it seems that uniqueness is also guaranteed in the general monotonic case for budget shares, not only log-linear, and even for a non-monotonic relationship without sharp oscillations. In practice, given actual data, there seems to be no problem with convergence to a unique solution, or, at least, cases of multiple solutions have not been discovered.

## _4.2 Convergence speed of iterations and kernels_

In general, the convergence to a unique solution was found to be extremely fast, but it did depend on the type of filter (i.e., kernel) and the bandwidth (bw) employed. On the latter, we explore three different bandwidths for each of the two kernel shapes.<sup>11</sup> Understandably, the uniform kernel is less stable as it is affected by weight irregularities and distribution lumpiness at both ends of the band, whereas for the bi-weight kernel the discrepancies at the ends have almost no effect on the result. As the uniform kernel is actually a band with no re-weighting,

> 10 Plutocratic weights are the kind obtained from the national accounts, whereby all households are treated as one unit. Weights using this method are derived from the total expenditures of all households on a given basic heading, so richer households exert a greater influence on the computation. Democratic weights represent all households equally and are derived by taking the average of the expenditure shares of each household on a given basic heading, so all households exert an equal influence on the computation.

> 11 Kernel bandwidths used are from widest to most narrow: 1.0, 0.5, and 0.25. A wider bandwidth increases the width of the shape around the IPL so that more households are included in the sample when extracting expenditures from a household expenditure survey.

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