Lanjouw (2003). A recent review article considers the issue of transformations for small area estimation in detail (Tzavidis et al, 2018). It reports design-based simulation results using Mexican data that considered alternative transformations, including log-shift and Box-Cox transformations (Box and Cox, 1964). A log-shift transformation involves adding a constant to welfare prior to taking the log, while the Box-cox transformation can be written as: <u>y</u> −1 , and log y for λ=0 . Tzavidis et al (2018)<sup>λλ</sup> conclude on the basis of the simulation results that log-shift transformations tend to perform well. Stata λλ for λλ≠0 has implemented two commands that can quickly estimate the parameters for the log shift and Box-Cox transformations that minimize the skewness of the welfare distribution. These commands, however, only go part of the way towards normalization, because they cannot guarantee that the resulting distribution will have a kurtosis of three, which is characteristic of a normal distribution. Furthermore, the log shift transformations is not monotonic when the underlying distribution is left-skewed.<sup>18</sup> Finding an appropriate monotonic transformation for the welfare variable is important, as failure to satisfy the normality assumption can lead to systematic bias in EBP models. While benchmarking can be used to at least partially correct this bias, relying on benchmarking becomes more difficult to justify as the extent of the bias increases.

For these reasons, our preferred estimates utilize a monotonic transformation called the “ordered quantile normalization” to transform welfare. This method transforms the scaled rank of the welfare variable to conform with a normal distribution. Variants of this method date back more than 70 years (Bartlett, 1947, Van der Waerden, 1952 ) . Among several methods proposed in the literature, the ordered quantile normalization most consistently transforms an underlying variable to follow a normal distribution (Peterson and Cavanough, 2019). We utilize the Ordernorm function in the Bestnormalize R package to implement this transformation. The method produces an exactly normal distribution of the transformed variable when the original welfare measure contains no ties, irrespective of its underlying distribution. While a normally distributed outcome variable does not guarantee normality of the residuals, it helps make the distribution of residuals closer to normal, and the ordered quantile normalization leads to smaller discrepancies between the official national poverty rate estimated from the survey and the weighted mean of the small area estimates.

To calculate small area estimates of the poverty gap, poverty severity, or inequality, it is critical to retransform the estimates back to the original welfare metric, as is done in the ELL method by exponentiating log welfare in each simulation. However, this retransformation is not necessary when estimating headcount poverty rates. This is because households’ poverty status is preserved under any monotonic transformation applied to both the welfare measure and the poverty line threshold. We therefore set the poverty line to the percentile of the transformed welfare measure that corresponds to the assumed national poverty rate, which is 4 percent in Sri Lanka and 20 percent in Tanzania. Because the ordered quantile normalization transformation is based on household ranks within the distribution, applying it entails information loss regarding the shape of the distribution between household ranks, and alters the estimated parameters of the model. To verify that this loss of information is minor compared with the benefits of normalizing the welfare measure, we also consider in the robustness section below a more traditional Box-Cox normalization procedure designed to minimize skewness.

> 18 When the underlying distribution is left-skewed, the skew-minimizing transformation becomes , which is not monotonic. log(−yy+ kk)

16