The model is developed assuming the relationship between household income or expenditure and poverty correlates is linear and that there is an error in projection.<sup>13</sup> Equation (a.1) shows this relationship: ln �� ���′���� (a.1) where ln �� refers to a natural logarithm of household income or expenditure of household _h_ , �� is a ���1� vector of poverty correlates of household _h_ , � is a ���1� vector of coefficients of poverty correlates, � is the number of variables, and �� is a projection error. In principle, SWIFT estimates the linear formula by regressing the natural logarithm of household income or expenditure on a set of poverty correlates in household survey data that include both household income/expenditure and poverty correlates. The regression model becomes a formula, with which household expenditure or income will be projected into a data set that has only poverty correlates. The latter data set will be collected by a SWIFT survey. A SWIFT survey collects the poverty correlates. To improve accuracy of projections, SWIFT adopts approaches used in machine learning, poverty mapping, and multiple imputation. More details are available in the annex of the guidelines for SWIFT (Yoshida, et al., 2015).<sup>14</sup> The SWIFT modeling process includes multiple steps to improve the ability of the formula to project household income or expenditures by adjusting the coefficients (�� and estimating the distributions of both the coefficients and the projection errors. No formula is perfect, so inclusion of the projection error is essential. Indeed, estimating the distribution of the projection error is key for estimating poverty rates and their standard errors.

# _Cross Validation_

Since consumption patterns can differ significantly across areas and population groups, the SWIFT team makes efforts to create a model that is specific to the areas and population groups of interest. Such an adjustment is good to create the model tailored to the needs, but can cause potentially large bias in poverty estimates because the sample used for creating a model declines by focusing on the specific group of population. “Over-fitting” is one of such problems. The over-fitting problem means that while a model can perform well within the sample developed for the model, it can perform badly outside the data set. In a sense, the model over-fits the data set used to develop it. To detect this problem, the SWIFT team conducts a cross-validation analysis. The cross-validation approach separates data used for developing the model from those used for evaluating the model fitness.

More specifically, a household survey data set is split randomly into 10 subsamples. Each of these subsamples is called a “fold.” A consumption model is estimated from nine folds by running a stepwise Ordinary Least Square (OLS) regression.<sup>15</sup> The stepwise OLS regression means that a statistical package searches for an OLS regression model where all variables are statistically significant, at a given p-value level. We use STATA and its stepwise selection model. The nine folds used for developing a model are known as “Training Data”.

> 13 This does not mean SWIFT does not use a non-linear model, but it means that SWIFT’s formula is linear in variables created in the data set. Since some variables can be squares of other variables, SWIFT’s formula can be nonlinear. One of the typical examples is that SWIFT uses household size and household size squared in a formula. 14 Yoshida, N., R. Munoz, A. Skinner, C. Kyung-eun Lee, M. Brataj, W. Durbin, D. Sharma, and C. Wieser. (2015). SWIFT Data Collection Guidelines version 2. The World Bank. 15 Or weighted least squares.

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