Implementation Issues in the Duclos–Jalbert–Araar Decomposition of Redistributive Effect

Abstract

Models decomposing the redistributive effect of fiscal systems into vertical and horizontal effects are extensively used by practitioners. Duclos, Jalbert, and Araar’s model, despite its advantages, has not yet been widely employed in empirical research, possibly due to a relatively challenging implementation procedure that involves the estimation of expected postfiscal incomes. To override these difficulties, the designers of the software DAD have incorporated a module for implementation of the model. However, the application of this module on Croatian tax-benefit system data revealed certain inaccuracies in the results. Carefully unfolding the calculation and estimation procedures needed for implementation of the model, this article instructs practitioners on how to correctly apply the model and helps DAD designers improve their module.

Keywords

redistributive effect vertical equity horizontal inequity prefiscal equals

Duclos, Jalbert, and Araar (2003; henceforth DJA) have designed a comprehensive model to decompose the redistributive effect (RE) of a fiscal system into vertical, classical horizontal inequity (henceforth CHI), and reranking effects. The model is built into the framework of the Atkinson-Gini social welfare function (henceforth AGF), which first converts incomes into utilities employing the Atkinson’s (1970) utility function, and then aggregates utilities using rank-dependent weights, which underlie the S-Gini and S-concentration coefficients proposed by Donaldson and Weymark (1980) and Yitzhaki (1983).¹

The DJA model has certain advantages over its competitors, the widely acknowledged Kakwani’s (1984; henceforth K84) and the Aronson, Johnson, and Lambert’s (1994; henceforth AJL) decompositions of RE. To measure the CHI effect, the researcher must determine the set of counterfactual CHI-free or expected postfiscal incomes (EPIs). While the AJL model relies on the formation of arbitrary groups of close equals in this task, the DJA model employs purposefully designed statistical procedures. Consequently, the implementation of the DJA model requires a certain expertise related to data smoothing and curve-fitting methods. To facilitate the application of the DJA model in empirical analysis, a module for calculation of the DJA indices from the sample data is incorporated into the software DAD (Duclos, Araar, and Fortin 2010; henceforth DAD-DJA).

The use of DAD-DJA in research on the Croatian tax-benefit system revealed certain inaccuracies in the results. Specifically, when the ethical parameter of AGF is set to zero, the CHI effect in the DJA model should be equal to zero by construction. However, the estimated value of the CHI effect was significantly different from zero. Analysis has shown that DAD-DJA produces upward biased estimates of EPIs in the low prefiscal income region. Furthermore, it was revealed that the fitting procedure in DAD-DJA contains a “bug,” producing unreasonably high estimates of EPIs for the top prefiscal income units in the sample.

In an attempt to obtain fully accurate estimates of DJA indices, independent procedures have been developed. They are thoroughly explained in this article to assist practitioners in implementing the DJA model and to help DAD designers improve the working of DAD-DJA. A brief overview of data smoothing methods is provided, accompanied by advice on how to accurately obtain EPIs estimates. Relationships with other measurement models are explained.

The rest of the article is organized as follows: the DJA Model section briefly exposes the elements of the DJA model and its connections with other decompositions. The Calculation of Indices section extensively describes the procedures of data preparation, estimation, and calculation of various elements of the DJA model, and employs them on a simple hypothetical population of four income units. In Application: Croatian Tax-Benefit System section, the procedures are applied to data on the Croatian tax-benefit system, and the results are compared with those obtained by DAD-DJA. The conclusions are presented in the final section.

The DJA Model

Postfiscal income is equal to prefiscal income minus taxes plus benefits. RE is the change of income inequality induced by a fiscal system consisting of taxes and benefits. In measurement terms, we have that $Δ = I (X) - I (N)$ , where $Δ$ represents RE, and $I (X)$ and $I (N)$ are indices of pre- and postfiscal income inequality.

In the DJA model, inequality indices $I (\cdot)$ are derived using the AGF proposed by Araar and Duclos (2003, 2006). For prefiscal income, we have that

W (X, ϵ, ν) = \int_{0}^{1} U (X (p), ϵ) ω (p, ν) d p,

where

ϵ

is the ethical parameter configuring the Atkinson’s (1970) utility function,

U (X (p), ϵ) = {[X (p)]}^{1 - ϵ} / (1 - ϵ)

for

ϵ \neq 1

, and

U (X (p), ϵ) = l n [X (p)]

for

ϵ = 1

, with p denoting the quantiles of the prefiscal income distribution, and

X (p)

the income at quantile p. The term

ν

is another ethical parameter, characterizing the Donaldson and Weymark’s (1980) and Yitzhaki’s (1983) S-Gini rank-dependent weighting scheme,

ω (p, ν) = ν {(1 - p)}^{ν - 1}

. The equally distributed equivalent income is an inverse function of

W (\cdot)

and is obtained as

ξ (X, ϵ, ν) = {[(1 - ϵ) W (X, ϵ, ν)]}^{1 / (1 - ϵ)}

for

ϵ \neq 1

, and

ξ (X, ϵ, ν) = e x p [W (X, ϵ, ν)]

for

ϵ = 1

. Finally, the Atkinson-Gini inequality index is calculated as follows:

I (X) = 1 - ξ (X, ϵ, ν) / μ^{X},

where

μ^{X}

is the mean prefiscal income.

I (N)

is obtained analogously, using the quantiles of the postfiscal income distribution.

The DJA model decomposes RE as follows:

Δ = V - C - R = [I (X) - I (N^{E})] - [I (U^{P}) - I (N^{E})] - [I (N) - I (U^{P})] .

The vertical effect, $V = I (X) - I (N^{E})$ , represents the potential RE or the reduction of inequality that would be achieved by the counterfactual CHI-free system. The discrepancy between potential and actual RE is divided into a CHI effect, $C = I (U^{P}) - I (N^{E})$ , and a reranking effect, $R = I (N) - I (U^{P})$ , which measure two different manifestations of horizontal inequity (HI). The former effect (C) measures HI emerging from violation of the “classical horizontal equity principle,” which says that equals should be treated equally. The latter effect (R) evaluates HI arising from infringement of the “no-reranking principle,” which requires the fiscal process to not change the ranks of income units in transition from pre- to postfiscal income. Take, for example, four households of equal size. A and B have prefiscal incomes of 10 each, whereas C and D have prefiscal incomes of 20 each. Suppose that A, B, C, and D end up with postfiscal incomes of 8, 16, 12, and 24, respectively. Between prefiscal equals (A and B; C and D) CHI has occurred, whereas between prefiscal unequals (B and C) reranking has taken place.

In equation (3), $I (N^{E})$ is the inequality index obtained for EPIs, equal to $N^{E} (p) = \int_{0}^{1} N (q | p) d q$ , where $N (q | p)$ denotes a postfiscal income at the qth quantile among all those income units belonging to the pth quantile of the prefiscal income distribution. $I (U^{P})$ represents the inequality index obtained for expected postfiscal utilities (EPUs) at quantile p, $U^{P} (p, ϵ) = \int_{0}^{1} U (N (q | p), ϵ) d q$ . For $N^{E} (p)$ and $U^{P} (p, ϵ)$ , we obtain the respective social welfare functions $W (N^{E}, ϵ, ν) = \int_{0}^{1} U (N^{E} (p), ϵ) ω (p, ν) d p$ and $W (U^{P}, ϵ, ν) = \int_{0}^{1} U^{P} (p, ϵ) ω (p, ν) d p$ , while the corresponding inequality indices are $I (N^{E}) = 1 - ξ (N^{E}, ϵ, ν) / μ^{N}$ and $I (U^{P}) = 1 - ξ (U^{P}, ϵ, ν) / μ^{N}$ , where $μ^{N}$ is the mean postfiscal income.

In the special case where $ϵ = 0$ , utilities are identical to incomes: $U (y, 0) = y$ . Therefore, we have that $U (N (q | p), ϵ) = N (q | p)$ across all p and $N (q | p)$ , and it follows that $W (N^{E}, 0, ν) = W (U^{P}, 0, ν)$ and $I (N^{E}, 0, ν) = I (U^{P}, 0, ν)$ . The consequence for the DJA model is that the CHI effect collapses to zero, and the decomposition (3) can be rewritten as

\begin{aligned} Δ (0, ν) = V (0, ν) - R (0, ν) = [I (X, 0, ν) - I (N^{E}, 0, ν)] \\ - [I (N, 0, ν) - I (N^{E}, 0, ν)] \end{aligned} .

It can be shown that $I (X, 0, ν)$ , $I (N, 0, ν)$ , and $I (N^{E}, 0, ν)$ are the S-Gini coefficient of prefiscal income, $G (X, ν)$ , the S-Gini coefficient of post-fiscal income, $G (N, ν)$ , and the S-concentration coefficient of postfiscal income, $D (N, ν)$ , respectively.²

Consequently, $V (0, ν)$ is equal to the S-Gini Kakwani’s (1984) index of vertical effect, $V^{K} (ν) = G (X, ν) - D (N, ν)$ , and $R (0, ν)$ is the S-Gini Atkinson (1980), Plotnick (1981), and Kakwani’s (1984) index of reranking, $R^{A P K} (ν) = G (N, ν) - D (N, ν)$ . The Kakwani’s (1984) decomposition of RE into vertical and horizontal components can be rewritten in S-Gini terms as

Δ (ν) = V^{K} (ν) - R^{A P K} (ν) = [G (X, ν) - D (N, ν)] - [G (N, ν) - D (N, ν)] .

In another special case, where $v = 1$ , the weights $ω (p, ν)$ are equal for all p and the reranking effect disappears. For $ϵ > 0$ , the vertical and CHI effect, $V (ϵ, 1)$ and $C (ϵ, 1)$ , become the indices consistent with the Duclos and Lambert’s (2000) model of HI measurement.³

Calculation of Indices

Data Preparation

A typical research uses the following data for a household or family i: (a) unequivalized pre- and postfiscal incomes, ${\dot{X}}_{i}$ and ${\dot{N}}_{i}$ ; (b) survey frequency (or sampling) weights, f_i ; and (c) equivalence factor β _i . The equivalized pre- and postfiscal incomes are $X_{i} = {\dot{X}}_{i} / β_{i}$ and $N_{i} = {\dot{N}}_{i} / β_{i}$ (hereafter, we deal only with equivalized incomes, calling them plainly pre- and postfiscal incomes). The frequency weights are defined as $ϕ_{i} = f_{i} β_{i}$ . Thus, the equivalence factor β _i is employed both for deriving the equivalized income and for weighting households of different types.⁴

We form the $3 S$ matrix $M^{0}$ , where $S$ is the number of households in the sample:

M^{0} = [\begin{matrix} X_{1} & \dots & X_{i} & \dots & X_{S} \\ N_{1} & \dots & N_{i} & \dots & N_{S} \\ ϕ_{1} & \dots & ϕ_{i} & \dots & ϕ_{S} \end{matrix}] .

To obtain the matrix $M^{x}$ ( $M^{n}$ ), the columns in $M^{0}$ are sorted in increasing order of the values from the first (second) row:

M^{x} = [\begin{matrix} X_{1}^{x} & \dots & X_{i}^{x} & \dots & X_{S}^{x} \\ N_{1}^{x} & \dots & N_{i}^{x} & \dots & N_{S}^{x} \\ ϕ_{1}^{x} & \dots & ϕ_{i}^{x} & \dots & ϕ_{S}^{x} \end{matrix}]; M^{n} = [\begin{matrix} X_{1}^{n} & \dots & X_{i}^{n} & \dots & X_{S}^{n} \\ N_{1}^{n} & \dots & N_{i}^{n} & \dots & N_{S}^{n} \\ ϕ_{1}^{n} & \dots & ϕ_{i}^{n} & \dots & ϕ_{S}^{n} \end{matrix}] .

From $M^{x}$ we take out the values $X_{i}^{x}$ , $N_{i}^{x}$ , and $ϕ_{i}^{x}$ , while from $M^{n}$ the values $N_{i}^{n}$ and φ $_{i}^{n}$ are extracted. Notice that the superscript x (n) denotes that income units are sorted in increasing order of prefiscal (postfiscal) income.

The sample estimates of quantiles p and the weights $ω (p, ν)$ are obtained in the following manner:

\begin{aligned} {\tilde{p}}_{i}^{x} = {(2 Σ)}^{- 1} \sum_{j = 1}^{i} (ϕ_{j}^{x} + ϕ_{j - 1}^{x}) \\ {\tilde{ω}}_{i}^{x, ν} = {(Σ)}^{- 1} {(1 - {\tilde{p}}_{i}^{x})}^{ν - 1}, \end{aligned}

where $Σ = \iSigma_{j = 1}^{S} ϕ_{j}^{x} a n d ϕ_{0}^{x} = 0$ .

When a large group of prefiscal exact equals exists in the sample, one of the inequality indices would be biased if based on the weights ${\tilde{ω}}_{i}^{x, ν}$ , namely, $\tilde{I} (N_{i}^{x}, ϵ, ν, {\tilde{ω}}_{i}^{x, ν})$ from equation (A2) (see later discussion). Therefore, we derive a new set of weights, ${\hat{ω}}_{i}^{x, ν}$ . Assume that income units $i = 1, \dots, Q$ have zero prefiscal incomes, that is, $X_{1}^{x} = X_{2}^{x} = \dots = X_{Q}^{x} = 0$ , and corresponding postfiscal incomes $N_{1}^{x}, N_{2}^{x}, \dots, N_{Q}^{x}$ . The procedure described by equation (8) automatically ascribes to these units the weights that are strictly decreasing in i and ${\tilde{p}}_{i}^{x}$ ; that is, ${\tilde{ω}}_{1}^{x, ν} > \dots > {\tilde{ω}}_{Q}^{x, ν}$ , although all these units have equal prefiscal income and rank. Therefore, we replace them with the new set of weights obtained as follows:

\begin{aligned} {\hat{ω}}_{i}^{x, ν} = {(\sum_{b = 1}^{Q} ϕ_{b}^{x})}^{- 1} \sum_{c = 1}^{Q} ϕ_{c}^{x} \cdot {\tilde{ω}}_{c}^{x, ν}, f o r i = 1, \dots, Q \\ {\hat{ω}}_{i}^{x, ν} = {\tilde{ω}}_{i}^{x, ν}, f o r i > Q \end{aligned} .

Thus, the original weights ${\tilde{ω}}_{i}^{x, ν}$ of prefiscal equals are transformed into their group average. An analogous procedure should be applied to other large groups of prefiscal equals, if they exist in the sample.

Alternatively, we could use the original weights and randomize the order of income units within each group of exact prefiscal equals. This procedure would reduce the bias to an insignificant level, but each possible ordering of income units would still result in different values of $\tilde{I} (N_{i}^{x}, ϵ, ν, {\tilde{ω}}_{i}^{x, ν}) .$ Therefore, for purposes of consistency and transparency, the use of weights ${\hat{ω}}_{i}^{x, ν}$ is recommended. For all the other indices, it is irrelevant whether the weights ${\tilde{ω}}_{i}^{x, ν}$ or ${\hat{ω}}_{i}^{x, ν}$ are employed, because the income vectors they are based upon (namely, $X_{i}^{x}$ , $U_{i, ϵ}^{P}$ , and $N_{i}^{E}$ ) have identical values within a group of exact prefiscal equals.

Finally, analogous to the preceding procedures, the estimates ${\hat{ω}}_{i}^{n, ν}$ are obtained from $ϕ_{i}^{n}$ (for calculation of the inequality index based on $N_{i}^{n}$ ).

Indices of Inequality

The following equations show how to obtain utilities, the Gini-Atkinson welfare index, and the inequality index for prefiscal incomes $X_{i}^{x}$ , when $ϵ \neq 1$ :

\begin{aligned} U (X_{i}^{x}, ϵ) = {(X_{i}^{x})}^{1 - ϵ} / (1 - ϵ) \\ \hat{W} (X_{i}^{x}, ϵ, ν; {\hat{ω}}_{i}^{x, ν}) = \sum_{j = 1}^{S} U (X_{j}^{x}, ϵ) \cdot ϕ_{j}^{x} \cdot {\hat{ω}}_{j}^{x, ν} \\ \hat{I} (X_{i}^{x}, ϵ, ν; {\hat{ω}}_{i}^{x, ν}) = 1 - {[(1 - ϵ) \hat{W} (X_{i}^{x}, ϵ, ν; {\hat{ω}}_{i}^{x, ν})]}^{\frac{1}{1 - ϵ}} / \hat{μ} (X_{i}^{x}) . \end{aligned}

where $\hat{μ} (X_{i}^{x}) = {(Σ)}^{- 1} \sum_{j = 1}^{S} ϕ_{j}^{x} \cdot X_{j}^{x}$ is the mean prefiscal income. To shorten the presentation, the formulas referring to the case where $ϵ = 1$ are omitted. Analogously, the utilities and indices for postfiscal incomes $N_{i}^{n}$ are obtained, as shown by equation (A1) in the appendix.

$\hat{I} (X_{i}^{x}, ϵ, ν, {\hat{ω}}_{i}^{x, ν})$ and $\hat{I} (N_{i}^{n}, ϵ, ν, {\hat{ω}}_{i}^{n, ν})$ from (10) and (A1) are the sample estimates of the indices of pre- and postfiscal income inequality, $I (X)$ and $I (N)$ . As equation (3) indicates, the application of the DJA model requires the estimates of two other indices, $I (N^{E})$ and $I (U^{P})$ , derived from EPIs, $N^{E} (p)$ , and EPUs, $U^{P} (p, ϵ)$ .

To obtain the sample estimates of EPUs, $U_{i, ϵ}^{P}$ , we should smooth a data set ${\{X_{i}^{x}, U (N_{i}^{x}, ϵ); ϕ_{i}^{x}\}}_{i = 1}^{S}$ . For each value of $ϵ$ , we estimate the regression relationship $U (N_{i}^{x}, ϵ) = m^{P, ϵ} (X_{i}^{x}) + σ_{i}$ , to obtain the approximation ${\overset{⌣}{m}}^{P, ε} (X_{i}^{x})$ . Subsequently, the vector of fitted values is calculated as $U_{i, ε}^{P} = {\overset{⌣}{m}}^{P, ε} (X_{i}^{x})$ .

However, the following identity says that the whole procedure of estimating EPUs can be circumvented, saving the practitioner’s time and energy in sensitivity analysis using multiple scenarios for $v$ and $ϵ$ . Genuinely, the sample estimate of $I (U^{P})$ is equal to $\hat{I} (U_{i, ϵ}^{P}, ϵ, ν, {\hat{ω}}_{i}^{x, ν})$ from equation (A5), but it can be derived more simply by the inequality index $\hat{I} (N_{i}^{x}, ϵ, ν, {\hat{ω}}_{i}^{x, ν})$ from equation (A3), because of the following equality:

\hat{I} (U_{i, ϵ}^{P}, ϵ, ν; {\hat{ω}}_{i}^{x, ν}) = \hat{I} (N_{i}^{x}, ϵ, ν; {\hat{ω}}_{i}^{x, ν}) .

To understand why (11) holds, recall that $U^{P} (p, ϵ) = \int_{0}^{1} U (N (q | p), ϵ) d q$ and notice that the theoretical values $N (q | p)$ are represented by the sample values $N_{i}^{x}$ . Imagine the population consisting of two groups of exact prefiscal equals: B income units have prefiscal income $X_{(1)}^{x}$ and postfiscal incomes $N_{1}^{x}, \dots, N_{B}^{x}$ , while F income units have prefiscal income $X_{(2)}^{x} > X_{(1)}^{x}$ and postfiscal incomes $N_{B + 1}^{x}, \dots, N_{B + F}^{x}$ . Their rank-dependent weights are ${\hat{ω}}_{1}^{x, ν} = \dots = {\hat{ω}}_{B}^{x, ν} = {\hat{ω}}_{(1)}^{x, ν}$ and ${\hat{ω}}_{B + 1}^{x, ν} = \dots = {\hat{ω}}_{B + F}^{x, ν} = {\hat{ω}}_{(2)}^{x, ν}$ . For given $ϵ$ , the values of expected postfiscal utilities, $U_{i, ϵ}^{P}$ , are obtained as averages of utilities $U_{(1), ϵ}^{P} = B^{- 1} \sum_{j = 1}^{B} U (N_{j}^{x}, ϵ)$ and $U_{(2), ϵ}^{P} = F^{- 1} <$ . The welfare is then obtained by (A5) as

\begin{aligned} \hat{W} (U_{i, ϵ}^{P}, ϵ, ν; {\hat{ω}}_{i}^{x, ν}) & = B \cdot {\hat{ω}}_{(1)}^{x, ν} \cdot B^{- 1} \sum_{j = 1}^{B} U (N_{j}^{x}, ϵ) \\ + F \cdot {\hat{ω}}_{(2)}^{x, ν} \cdot F^{- 1} \sum_{j = B + 1}^{B + F} U (N_{j}^{x}, ϵ) \\ = {\hat{ω}}_{(1)}^{x,} \cdot \sum_{j = 1}^{B} U (N_{j}^{x}, ϵ) + {\hat{ω}}_{(2)}^{x,} \cdot \sum_{j = B + 1}^{B + F} U (N_{j}^{x}, ϵ) . \end{aligned}

Observe that according to (A3) we would obtain the identical result for $\hat{W} (N_{i}^{n}, ϵ, ν; {\hat{ω}}_{i}^{n, ν})$ . Consequently, $\hat{I} (U_{i, ϵ}^{P}, ϵ, ν; {\hat{ω}}_{i}^{x, ν})$ and $\hat{I} (N_{i}^{x}, ϵ, ν; {\hat{ω}}_{i}^{x, ν})$ are also identical.

Estimation of EPIs and Utilities

Unlike the estimation of EPUs, the evaluation of EPIs cannot be avoided. To obtain the sample estimates of $N^{E} (p)$ , we must smooth a data set ${\{X_{i}^{x}, N_{i}^{x}; ϕ_{i}^{x}\}}_{i = 1}^{S}$ , that is, approximate the mean response curve $m^{E}$ in the regression relationship $N_{i}^{x} = m^{E} (X_{i}^{x}) + δ_{i}$ . The estimates of EPIs are then obtained as $N_{i}^{E} = {\overset{⌣}{m}}^{E} (X_{i}^{x})$ , where ${\overset{⌣}{m}}^{E} (X_{i}^{x})$ is the approximation of $m^{E}$ .

The estimation of EPIs represents the greatest challenge in the implementation of the DJA model. Although parametric models (such as polynomial regression) can be appropriate for some data sets, it is better to rely on nonparametric approaches, assuming no a priori functional relationship between post- and prefiscal incomes. One such approach is the “kernel-weighted local polynomial regression” (KWLPR). A description of the method can be found in Fan and Gijbels (1996), Wand and Jones (1995), Keele (2008), and Härdle (1990), while the software applications include Stata 12 (function lcpoly), R (function loess, package lokern, etc.), and XploRe (function lpregxest).

The choice of the degree of polynomial (p), the type of the kernel function, and the size of the kernel half-bandwidth rests on the analyst. For $p = 0$ , KWLPR becomes the “Nadaraya-Watson estimator” (NWE), while for $p = 1$ we obtain the “local linear estimator” (LLE). Fan and Gijbels (1996) explain that the odd degree polynomials achieve the best balance between bias and variability and automatically correct the boundary problem.

Another interesting smoothing technique came to light during the research: the “Fourier series in trigonometric form” (FSTF), which is a sum of the sine and cosine functions describing a periodic signal (Faunt and Johnson 1992). The estimation procedure is programmed in Matlab R2011b’s Curve Fitting Toolbox 3.2, which contains several other fitting methods, such as smoothing splines.

DAD-DJA and supporting documentation⁵ do not inform us which fitting method is used to obtain EPIs for estimation of DJA indices. However, DAD incorporates a separate module, “Non Parametric Regression,” enabling us to estimate EPIs independently of DAD-DJA. Two basic methods are offered: NWE and LLE (henceforth, DAD-NWE and DAD-LLE). Experimentation with different options and choices offered by the module revealed that in estimating EPIs DAD-DJA in fact employs DAD-LLE, using the default set of parameters.

Before moving further, we offer the following advice to help judge whether the estimates $N_{i}^{E}$ are appropriate for use in the DJA model implementation.

Although the fitting methods and their software implementations ensure optimality in the statistical sense, the analyst still has the freedom and the responsibility to change some of the parameters or the whole estimation method if the results contradict her or his knowledge of the appropriate shape of the EPIs curve. An example is a too “wiggly” curve, in which case we have to “stretch” it, perhaps by raising the kernel half-bandwidth. Another example may be the existence of certain kinks or local minimums (maximums) we are aware of, which are not reflected by the EPIs estimate.

For certain data points, the programmed fitting procedures may produce irregular results. Some software tools are “smart” in such cases, leaving a blank space instead of the estimate, while others are not. Anyway, if this happens, we should fill in the corresponding estimate manually, using the best-guess approach.

A simple preliminary test of the correctness of the approximation ${\overset{⌣}{m}}^{E} (X_{i}^{x})$ is to check whether the mean value of the estimated values $N_{i}^{E}$ is approximately equal to the mean of the sample values $N_{i}^{x},$ that is, if

\hat{μ} (N_{i}^{E}) = \hat{μ} (N_{i}^{x}) .

The discussion in The DJA Model section indicated that when $ϵ = 0$ , we have that $U (N_{i}^{x}, 0) = N_{i}^{x}$ and $U_{i, ϵ}^{P} = N_{i}^{E}$ . Therefore, equation (11) becomes

\hat{I} (N_{i}^{E}, 0, ν; {\hat{ω}}_{i}^{x, ν}) = \hat{I} (U_{i, ϵ}^{P}, 0, ν; {\hat{ω}}_{i}^{x, ν}) = \hat{I} (N_{i}^{x}, 0, ν; {\hat{ω}}_{i}^{x, ν}) .

From equation (13) follows another test: the inequality indices $\hat{I} (N_{i}^{E}, 0, ν, {\hat{ω}}_{i}^{x, ν})$ , obtained by equation (A4), for different values of parameter $ν$ should be (approximately) equal to the inequality indices obtained for $\hat{I} (N_{i}^{x}, 0, ν; {\hat{ω}}_{i}^{x, ν})$ . Otherwise, the estimates of the DJA indices would be biased, and we should try to obtain an alternative configuration of ${\overset{⌣}{m}}^{E} (X_{i}^{x})$ .

Decompositions

Having defined all the indices needed, we can present RE and its decompositions in terms of sample estimate formulas. RE is obtained as $\hat{Δ} = \hat{I} (X_{i}^{x}) - \hat{I} (N_{i}^{n})$ . According to the DJA model from (3), RE is decomposed as follows:

\begin{aligned} \hat{Δ} & = \hat{V} - \hat{C} - \hat{R} \\ = [\hat{I} (X_{i}^{x}) - \hat{I} (N_{i}^{E})] - [\hat{I} (U_{i, ϵ}^{P}) - \hat{I} (N_{i}^{E})] - [\hat{I} (N_{i}^{n}) - \hat{I} (U_{i, ϵ}^{P})] \\ = [\hat{I} (X_{i}^{x}) - \hat{I} (N_{i}^{E})] - [\hat{I} (N_{i}^{x}) - \hat{I} (N_{i}^{E})] - [\hat{I} (N_{i}^{n}) - \hat{I} (N_{i}^{x})] . \end{aligned}

where the last row in equation (14) arrives from the property (11), by which $\hat{I} (U_{i, ϵ}^{P}) = \hat{I} (N_{i}^{x})$ . The differences in the brackets, that is, $\hat{V} = \hat{I} (X_{i}^{x}) - \hat{I} (N_{i}^{E})$ , $\hat{C} = \hat{I} (U_{i, ϵ}^{P}) - \hat{I} (N_{i}^{E}) = \hat{I} (N_{i}^{x}) - \hat{I} (N_{i}^{E})$ , and $\hat{R} = \hat{I} (N_{i}^{n}) - \hat{I} (U_{i, ϵ}^{P}) = \hat{I} (N_{i}^{n}) - \hat{I} (N_{i}^{x})$ , are respectively the sample estimates of the vertical, CHI, and reranking effects of the DJA model. Note that, if $\tilde{I} (N_{i}^{x})$ is used instead of $\hat{I} (N_{i}^{x})$ , the CHI and reranking effects would be biased. Furthermore, if the estimates $N_{i}^{E}$ are inappropriate, the vertical and CHI effect would be biased.

Setting $ϵ = 0$ and following (4) and (5), we can calculate the S-Gini K84 decomposition as

\begin{aligned} \hat{Δ} = \hat{V^{K}} (ν) - \hat{R^{A P K}} (ν) = [\hat{I} (X_{i}^{x}, 0, ν; {\hat{ω}}_{i}^{x, ν}) - \hat{I} (N_{i}^{x}, 0, ν; {\hat{ω}}_{i}^{x, ν})] \\ - [\hat{I} (N_{i}^{n}, 0, ν; {\hat{ω}}_{i}^{n, ν}) - \hat{I} (N_{i}^{x}, 0, ν; {\hat{ω}}_{i}^{x, ν})] . \end{aligned}

Simple Hypothetical Example

We return to the example of four hypothetical households from The DJA Model section to illustrate how the DJA model implementation procedures work. There are two groups of prefiscal equals in the sample: A and B with prefiscal income of 10 each belong to the lower quantile, whereas C and D with prefiscal income of 20 each belong to the upper quantile of prefiscal income distribution.

The first column in table 1 shows the “original” weights ${\tilde{ω}}_{i}^{x, ν}$ obtained by (8) for $ν = 2$ ; observe that A (C) obtains larger weight than B (D), although they belong to same prefiscal quantile. Therefore, analogous to the procedure from equation (9), we obtain the new set of weights, ${\tilde{ω}}_{i}^{x, ν}$ : for A and B (C and D), we have ${\hat{ω}}_{1}^{x} = {\hat{ω}}_{2}^{x} = ({\tilde{ω}}_{1}^{x} + {\tilde{ω}}_{2}^{x}) / <$ .

Table 1.

Hypothetical Population: Weight, Incomes, and Utilities

#	${\tilde{ω}}_{i}^{x}$	${\hat{ω}}_{i}^{x}$	$X_{i}^{x}$	$N_{i}^{x}$	$N_{i}^{n}$	$N_{i}^{E}$	$U (X_{i}^{x})$	$U (N_{i}^{x})$	$U (N_{i}^{n})$	$U (N_{i}^{E})$	$U_{i, ϵ}^{P}$
1 (A)	0.438	0.375	10	8	8	12	6.32	5.66	5.66	6.93	6.83
2 (B)	0.313	0.375	10	16	12	12	6.32	8.00	6.93	6.93	6.83
3 (C)	0.188	0.125	20	12	16	18	8.94	6.93	8.00	8.49	8.36
4 (D)	0.063	0.125	20	24	24	18	8.94	9.80	9.80	8.49	8.36
SUM	1	1	60	60	60	60	30.54	30.38	30.38	30.83	30.38

Note. Weights are obtained for $ν = 2$ ; utilities are obtained for $ϵ = 0.5$ .

As equation (11) explains, the estimate of $I (U^{P})$ can be obtained in two ways: by $\hat{I} (U_{i, ϵ}^{P})$ or by $\hat{I} (N_{i}^{x})$ . In the former case, we first obtain the values $U_{i, ϵ}^{P}$ , which are the sample estimates of $U^{P} (p, ϵ) = \int_{0}^{1} U (N (q | p), ϵ) d q$ . The values $N (q | p)$ for the lower (upper) quantile are represented by $N_{1}^{x}$ and $N_{2}^{x}$ ( $N_{3}^{x}$ and $N_{4}^{x}$ ), and the corresponding utilities are $U (N_{1}^{x})$ and $U (N_{2}^{x})$ [ $U (N_{3}^{x})$ and $U (N_{4}^{x})$ ]. The expected postfiscal utility at each quantile is simply the average utility of units belonging to the corresponding quantile: $U_{1}^{P} = U_{2}^{P} = [U (N_{1}^{x}) + U (N_{2}^{x})] / 2$ and $U_{3}^{P} = U_{4}^{P} = [U (N_{3}^{x}) + U (N_{4}^{x})] / 2$ , for the lower and the upper quantiles, respectively.

According to (A5), for $ν = 2$ and $ϵ = 0.5$ we have that

\hat{W} (U_{i, ϵ}^{P}, ϵ, ν; {\hat{ω}}_{i}^{x, ν}) = 2 \cdot U_{1}^{P} \cdot {\hat{ω}}_{1}^{x} + 2 \cdot U_{3}^{P} \cdot {\hat{ω}}_{3}^{x} = 7.21,

and $\hat{I} (U_{i, ϵ}^{P}, ϵ, ν; {\hat{ω}}_{i}^{x, ν}) = 0.133$ . Substituting back previously obtained utility terms into the expression for welfare, we obtain

\begin{aligned} \hat{W} (U_{i, ϵ}^{P}, ϵ, ν; {\hat{ω}}_{i}^{x, ν}) = [U (N_{1}^{x}) + U (N_{2}^{x})] \cdot {\hat{ω}}_{1}^{x} \\ + [U (N_{3}^{x}) + U (N_{4}^{x})] \cdot {\hat{ω}}_{3}^{x} \end{aligned} = 7.21

which is identical to the result that would be obtained by (A3):

\begin{aligned} \hat{W} (N_{i}^{x}, ϵ, ν; {\hat{ω}}_{i}^{x, ν}) = U (N_{1}^{x}) \cdot {\hat{ω}}_{1}^{x} + U (N_{2}^{x}) \cdot {\hat{ω}}_{2}^{x} + U (N_{3}^{x}) \cdot {\hat{ω}}_{3}^{x} \\ + U (N_{4}^{x}) \cdot {\hat{ω}}_{4}^{x} \end{aligned} = 7.21.

Thus, our hypothetical example confirms the identity (11). On the other hand, indices based on the ‘wrong’ weights, ${\tilde{ω}}_{i}^{x, ν}$ , would produce quite a different picture. By (A2), we have

\begin{aligned} \tilde{W} (N_{i}^{x}, ϵ, ν; {\tilde{ω}}_{i}^{x, ν}) = U (N_{1}^{x}) \cdot {\tilde{ω}}_{1}^{x} + U (N_{2}^{x}) \cdot {\tilde{ω}}_{2}^{x} \\ + U (N_{3}^{x}) \cdot {\tilde{ω}}_{3}^{x} + U (N_{4}^{x}) \cdot {\tilde{ω}}_{4}^{x} \end{aligned} = 6.89,

and $\tilde{I} (N_{i}^{x}, ϵ, ν; {\tilde{ω}}_{i}^{x, ν}) = 0.210$ .

The estimate of $I (N^{E})$ is obtained as $\hat{I} (N_{i}^{E}, ϵ, ν; {\hat{ω}}_{i}^{x, ν})$ from (A4), which is in turn based on sample estimates $N_{i}^{E}$ of $N^{E} (p) = \int_{0}^{1} N (q | p) d q$ . In this example, we can simply calculate $N_{1}^{E} = N_{2}^{E} = (N_{1}^{x} + N_{2}^{x}) / 2$ and $N_{3}^{E} = N_{4}^{E} = (N_{3}^{x} + N_{4}^{x}) / 2$ . According to (A4),

\hat{W} (N_{i}^{E}, ϵ, ν; {\hat{ω}}_{i}^{x, ν}) = 2 \cdot N_{1}^{E} \cdot {\hat{ω}}_{1}^{x} + 2 \cdot N_{3}^{E} \cdot {\hat{ω}}_{3}^{x} = 7.32,

and $\hat{I} (N_{i}^{E}, ϵ, ν; {\hat{ω}}_{i}^{x,}) = 0.108$ .

All inequality indices for $ν = 2$ and $ϵ = 0.5$ are presented in the first column of table 2, together with the DJA decomposition results. Although the vertical effect of the hypothetical system is positive ( $\hat{V} = 0.081$ ), the RE is negative ( $\hat{Δ} = - 0.052$ ), because the CHI effect ( $\hat{C} = 0.026$ ) and especially the reranking effect ( $\hat{R} = 0.107$ ) are very large.

Table 2.

Indices Obtained for Hypothetical Population

			DAD-DJA
	$ν = 2$ , $ϵ = 0.5$	$ν = 2$ , $ϵ = 0$	$ν = 2$ , $ϵ = 0$
$\hat{I} (X_{i}^{x})$	0.188	0.167	0.167
$\hat{I} (N_{i}^{n})$	0.240	0.217	0.217
$\tilde{I} (N_{i}^{x})$	0.210	0.183	0.183
$\hat{I} (N_{i}^{x})$	0.133	0.100
$\hat{I} (U_{i, ϵ}^{P})$	0.133	0.100
$\hat{I} (N_{i}^{E})$	0.108	0.100	0.500
$\hat{Δ} = \hat{I} (X_{i}^{x}) - \hat{I} (N_{i}^{n})$	−0.052	−0.050	−0.050
$\hat{V} = \hat{I} (X_{i}^{x}) - \hat{I} (N_{i}^{E})$	0.081	0.067	−0.333
$\hat{C} = \hat{I} (N_{i}^{x}) - \hat{I} (N_{i}^{E})$	0.026	0.000
$\hat{R} = \hat{I} (N_{i}^{n}) - \hat{I} (N_{i}^{x})$	0.107	0.117
$\tilde{C} = \tilde{I} (N_{i}^{x}) - \hat{I} (N_{i}^{E})$	0.102	0.083	−0.317
$\tilde{R} = \hat{I} (N_{i}^{n}) - \tilde{I} (N_{i}^{x})$	0.030	0.033	0.033

Another set of CHI and reranking effects is derived using $\tilde{I} (N_{i}^{x}),$ which is based on the weights ${\tilde{ω}}_{i}^{x, ν}$ . They show a completely different picture of the relative contributions of CHI ( $\tilde{C} = 0.102$ vs. $\hat{C} = 0.026$ ) and reranking ( $\tilde{R} = 0.030$ vs. $\hat{R} = 0.107$ ) to the overall HI. We have indicated that the weights ${\hat{ω}}_{i}^{x, ν}$ are the “right ones,” but to demonstrate this in our example, we have to obtain the indices for $ν = 2$ and $ϵ = 0$ , shown in the second column of table 2.

Recall that equation (13) says that the inequality indices based on $N_{i}^{x}$ , $U_{i, ϵ}^{P}$ , and $N_{i}^{E}$ must be equal when $ϵ = 0$ ; this is true for $\hat{I} (N_{i}^{x}) = 0.100$ , but not for $\tilde{I} (N_{i}^{x}) = 0.183$ . The difference $\tilde{I} (N_{i}^{x}) - \hat{I} (N_{i}^{x}) = 0.083$ presents by how much the CHI effect, which should be zero when $ϵ = 0$ , is overestimated if the weights ${\tilde{ω}}_{i}^{x, ν}$ are used in computation of the inequality index for postfiscal incomes $N_{i}^{x}$ .

Finally, we look at how DAD-DJA deals with this small hypothetical case. The DAD supporting documentation tells us that the estimate of $I (U^{P})$ is obtained by the index based on $N_{i}^{x}$ ; its value for $ν = 2$ and $ϵ = 0$ (0.183) is identical to $\tilde{I} (N_{i}^{x})$ . This indicates that DAD-DJA does not envisage the presence of exact equals in the sample. The third column in table 2 shows the other results. The index $\hat{I} (N_{i}^{E})$ diverges highly from our estimate (0.500 vs. 0.100), but this may be due to the small sample size. We deal with the DAD-DJA estimates of EPIs in the next section.

Application: Croatian Tax-Benefit System

Data

We analyze the fiscal system consisting of social security contributions (SSC) for the pension, health, and unemployment insurance funds, personal income tax and surtax (PITS), public pensions, and cash social benefits.⁶ The data on incomes come from the Croatian household budget survey (Anketa o potrošnji kućanstava; APK) for 2008, whose sample contains 3,108 households. Since APK registers only net incomes of household members, the amounts of prefiscal income, PITS, and SSC are obtained by a microsimulation model.

Postfiscal income of household i is obtained as ${\dot{N}}_{i} = {\dot{X}}_{i} - {\dot{T}}_{i} + {\dot{B}}_{i}$ , where ${\dot{X}}_{i}$ , ${\dot{T}}_{i}$ , and ${\dot{B}}_{i}$ are prefiscal income, the sum of all taxes paid, and the sum of all benefits received. To obtain $X_{i}$ and $N_{i}$ , ${\dot{X}}_{i}$ and ${\dot{N}}_{i}$ are deflated by the equivalence factor $β_{i}$ according to the “modified OECD scale,” $β_{i} = 1 + 0.5 (a_{i} - 1) + 0.3 c_{i}$ , where $a_{i}$ and $c_{i}$ are number of adults and children in household i.

Before analyzing the results of the DJA decomposition, we observe the features of the data set. The dots in the scattergram (figure 1) are the post-fiscal and prefiscal incomes of sample income units, expressed in terms of the mean prefiscal income (mpfi). The full line shows EPIs obtained by KWLPR (see the next section for details on estimation). The dotted line represents the cumulative density, which tells us, for each prefiscal income X, the proportion of all income units having prefiscal income below X (on the right axis). We can observe that quite a large proportion of units, about 7 percent, have zero prefiscal income (group A), while the next 13 percent of units have prefiscal income below 10 percent of mpfi (group B).

The mean postfiscal incomes of groups A and B are 64 percent and 54 percent of mpfi, respectively. Observe that the EPIs curve is decreasing on the interval [0, 0.1]. The following three facts taken together can explain the curious feature that the mean postfiscal income is decreasing. First, for the majority of pensioners’ households, a public pension is the only source of income; since public pensions are benefits in the current scenario, the pre-fiscal income of most pensioners’ households is zero. Second, majority of households with zero prefiscal income (group A) are pensioners’ households. Third, pensions are on average higher than other social benefits.

Estimation of EPIs and the Decomposition

The indices of the DJA decomposition are estimated by three models, using three different fitting methods described in Estimation of EPIs and Utilities subsection.

In model A, EPIs are estimated by KWLPR programmed in Stata 12. Following Bilger (2008), we use the third-degree local polynomials, employing the Epanechnikov kernel. The optimal half-bandwidth of the kernel obtained by the program was equal to 6.7 percent of mpfi, and it was increased by one half. In model B, EPIs are obtained using FSTF programmed in Matlab R2011b’s Curve Fitting Toolbox 3.2. The number of harmonics is set to 7; the “Trust-Region” algorithm is employed with the robust fitting option turned off. In both models, the top five prefiscal income units are excluded from the fitting process, and their values of $N_{i}^{E}$ are set to the values of $N_{i}^{x}$ . The size of the half-bandwidth in model A and the number of harmonics in model B are chosen to minimize the bias $\hat{I} (N_{i}^{x}, 0, ν; {\hat{ω}}_{i}^{x, ν}) - \hat{I} (N_{i}^{E}, 0, ν; {\hat{ω}}_{i}^{x, ν})$ . For the estimates of $I (U^{P})$ , we used $\hat{I} (N_{i}^{x}, ϵ, ν; {\hat{ω}}_{i}^{x, ν})$ with weights ${\hat{ω}}_{i}^{x, ν}$ obtained by equation (9).

The aim of model C is to replicate the results obtained by DAD-DJA. We employ DAD-LLE to estimate EPIs, with all observations included in the fitting process. To estimate $I (U^{P})$ , DAD-DJA also uses the index based on $N_{i}^{x}$ , but does not envisage the possibility of pre-fiscal exact equals. To play down the bias in the calculation of reranking and CHI effects, we randomize the order of income units within the group of zero prefiscal equals; these data are then put into DAD-DJA to obtain the original estimates, while in the replication we calculate $\tilde{I} (N_{i}^{x}, ϵ, ν; {\tilde{ω}}_{i}^{x, ν})$ .

Before moving on to the results, let us look at the shapes of the different EPIs curves, shown in figure 2, concentrating first on the bottom part of the income distribution. While both A and B reflect the initial fall in EPI, discussed earlier, C does not, that is, its EPIs curve is rather flat on the whole interval. For prefiscal income of zero all estimates are roughly the same, but in the prefiscal income interval [0.025, 0.42] of mpfi, C’s EPIs lie above those estimated by A and B. On the prefiscal income interval [0, 0.5] of mpfi, the mean of EPIs obtained by A (B) is 0.5764 (0.5775) of mpfi, which is very close to the mean postfiscal income for actual values, equal to 0.5762. On the other hand, the mean of EPIs obtained by C is 0.5881, or 2 percent above the actual mean. This suggests that C overestimates EPIs for the lowest incomes. For prefiscal incomes above 0.5 of mpfi, the EPIs of B and C are almost identical, while the EPIs curve of A is “more flexible” and intertwining the other two curves.

Models A and B convincingly pass the test from equation (12), as the ratios $\hat{μ} (N_{i}^{E}) / \hat{μ} (N_{i}^{x})$ in table 3 are 0.999732 and 1.0, respectively. On the other hand, for method C, $\hat{μ} (N_{i}^{E})$ is 1.75 percent higher than $\hat{μ} (N_{i}^{x})$ . This is partly the consequence of the earlier noticed overestimation on the interval [0, 0.5] of mpfi. However, there is another feature that is particularly odd: the estimate $N_{i}^{E}$ for the two income units with top prefiscal incomes are 2.4 and 6.4 times larger than their respective actual postfiscal incomes $N_{i}^{x}$ ! In this case, we can talk about a “bug” in DJA-LLE, which seriously damages the estimate of $\hat{μ} (N_{i}^{E})$ , which will lead to biased estimates of vertical and CHI effects, as we will soon observe. Table 3 shows the results of the DJA decomposition for $ν = 2$ and $ϵ = 0$ . All three models obtain equal values of $\hat{I} (X_{i}^{x})$ and $\hat{I} (N_{i}^{n})$ . Models A and B obtain the value of $\hat{I} (N_{i}^{x})$ equal to 0.244337, which is almost insignificantly different from the value of $\tilde{I} (N_{i}^{x})$ obtained by model C, thanks to randomizing the order of income units within the group of zero prefiscal equals. If the units within the group of zero prefiscal equals were, by chance, sorted in increasing (or decreasing) order of postfiscal income, the difference $|\tilde{I} (N_{i}^{x}) - \hat{I} (N_{i}^{x})|$ for the given data set could be as high as 0.001183. The reranking and CHI effects in model C could be seriously biased.

Table 3.

Decomposition of Redistributive Effect for $ν = 2$ and $ϵ = 0$

	A	B	C	A	B	C
				As percentage of $\hat{I} (X_{i}^{x})$
$\hat{I} (X_{i}^{x})$	0.506665	0.506665	0.506665	100.00	100.00	100.00
$\hat{I} (N_{i}^{n})$	0.291511	0.291511	0.291511	57.54	57.54	57.54
$\tilde{I} (N_{i}^{x})$			0.244328			48.22
$\hat{I} (N_{i}^{x})$	0.244337	0.244337		48.22	48.22
$\hat{I} (N_{i}^{E})$	0.244540	0.244517	0.251432	48.26	48.26	49.62
						As percentage of $\hat{Δ}$
$\hat{Δ}$	0.215154	0.215154	0.215154	100.00	100.00	100.00
$\hat{V}$	0.262124	0.262148	0.255233	121.83	121.84	118.63
$\hat{C}$	−0.000203	−0.000180	−0.007104	−0.09	−0.08	−3.30
$\hat{R}$	0.047174	0.047174	0.047183	21.93	21.93	21.93
$\hat{μ} (N_{i}^{E}) / \hat{μ} (N_{i}^{x})$	0.999732	1.000000	1.017489

The estimates $\hat{I} (N_{i}^{E})$ obtained by A and B are close to the value of $\hat{I} (N_{i}^{x})$ , as expected from equation (13); the differences $\hat{I} (N_{i}^{x}) - \hat{I} (N_{i}^{E})$ are −0.000203 and −0.000180, or between −0.09 percent and 0.08 percent of the corresponding RE ( $\hat{Δ}$ ). For the model C, the difference $\hat{I} (N_{i}^{x}) - \hat{I} (N_{i}^{E})$ is no less than −0.007104, or −3.3 percent of RE, meaning that the bias produced by C is about 35 times larger than the bias of A and B.

Model C thus underestimates the vertical effect by about 3 percent of RE for $ν = 2$ and $ϵ = 0$ . This underestimation is even larger (somewhat smaller) for $ν = 1.5$ and $ϵ = 0$ ( $ν = 3$ and $ϵ = 0$ ) and amounts to 5.9 (2.4) percent of RE, as table 4 indicates. Among the scenarios shown in table 4, the differences in the estimates of vertical effect obtained by models C and A (B) are lowest when $ν = 2$ and $ϵ = 0.9$ , equaling 0.8 (0.9) percent of RE.

Table 4.

Decomposition of Redistributive Effect for Different Combinations of $ν$ and $ϵ$

	A	B	C	A	B	C
	$ν = 1.5$ and $ϵ = 0$			As percentage of $\hat{Δ}$
$\hat{Δ}$	0.143358	0.143358	0.143358	100.00	100.00	100.00
$\hat{V}$	0.167811	0.167880	0.159522	117.06	117.11	111.28
$\hat{C}$	−0.000139	−0.000070	−0.008432	−0.10	−0.05	−5.88
$\hat{R}$	0.024592	0.024592	0.024596	17.15	17.15	17.16
	$ν$ $= 3$ and $ϵ = 0$			As percentage of $\hat{Δ}$
$\hat{Δ}$	0.287304	0.287304	0.287304	100.00	100.00	100.00
$\hat{V}$	0.374837	0.374568	0.368302	130.47	130.37	128.19
$\hat{C}$	−0.000258	−0.000527	−0.006815	−0.09	−0.18	−2.37
$\hat{R}$	0.087792	0.087792	0.087813	30.56	30.56	30.56
	$ν$ $= 1.5$ and $ϵ = 0.5$			As percentage of $\hat{Δ}$
$\hat{Δ}$	0.303136	0.303136	0.303136	100.00	100.00	100.00
$\hat{V}$	0.350112	0.350366	0.343239	115.50	115.58	113.23
$\hat{C}$	0.023835	0.024088	0.016965	7.86	7.95	5.60
$\hat{R}$	0.023142	0.023142	0.023138	7.63	7.63	7.63
	$ν$ $= 2$ and $ϵ = 0.5$			As percentage of $\hat{Δ}$
$\hat{Δ}$	0.352147	0.352147	0.352147	100.00	100.00	100.00
$\hat{V}$	0.422453	0.422651	0.416784	119.96	120.02	118.36
$\hat{C}$	0.026372	0.026570	0.020713	7.49	7.55	5.88
$\hat{R}$	0.043934	0.043934	0.043924	12.48	12.48	12.47
	$v = 2$ and $ϵ = 0.9$			As percentage of $\hat{Δ}$
$\hat{Δ}$	0.577556	0.577556	0.577556	100.00	100.00	100.00
$\hat{V}$	0.670484	0.670828	0.665771	116.09	116.15	115.27
$\hat{C}$	0.051673	0.052018	0.047002	8.95	9.01	8.14
$\hat{R}$	0.041255	0.041255	0.041213	7.14	7.14	7.14
	$ν$ $= 1$ and $ϵ = 0.9$			As percentage of $\hat{Δ}$
$\hat{Δ}$	0.592622	0.592622	0.592622	100.00	100.00	100.00
$\hat{V}$	0.635048	0.635456	0.627648	107.16	107.23	105.91
$\hat{C}$	0.042426	0.042834	0.035025	7.16	7.23	5.91
$\hat{R}$	0.000000	0.000000	0.000000	0.00	0.00	0.00

Conclusion

Models decomposing the RE of fiscal systems into vertical and horizontal effects are extensively used by practitioners. The DJA (2003) model, despite its advantages over some other models, such as the Kakwani’s (1984) and the Aronson, Johnson and Lambert’s (1994) decompositions of RE, has not yet been broadly employed in empirical research. The reason may be the relatively complex implementation procedure, which involves nonparametric methods in estimation of EPIs.

To override these estimation and calculation difficulties, the designers of the software DAD have incorporated a module for estimation of the DJA model indices, here referred to as DAD-DJA. However, as the application data on the Croatian tax-benefit system indicates, DAD-DJA produces somewhat inaccurate estimates of EPIs, resulting in biased values of DJA model indices. This article carefully explains the estimation procedures needed to obtain the indices of the DJA model, and the problems occurring in DAD-DJA implementation.

The estimates of EPIs are obtained by two fitting methods: kernel-weighted local polynomial regression and Fourier series in trigonometric form. Both achieve reasonable fit of the data at stake, unlike the method built into DAD-DJA, which seems to overestimate EPIs at the bottom region of prefiscal income distribution. Furthermore, we have realized that the fitting procedure in DAD-DJA contains a “bug,” producing unreasonably high estimates of EPIs for the top prefiscal income units in the sample.

We have shown how the estimation of EPUs can be circumvented, saving a practitioner time when doing multiple-scenario analysis. Instead of estimating EPUs for each different value of parameter $ϵ$ , the index of inequality based on EPUs can be obtained simply using postfiscal incomes ordered according to prefiscal incomes. In this procedure, however, caution must be taken in the presence of large groups of exact prefiscal equals: they should all be ascribed identical ranking weights. Otherwise, an estimate of reranking effect would be biased.

Footnotes

Appendix

Acknowledgment

The author would like to thank two anonymous referees, Jean-Yves Duclos and Slavko Bezeredi, for their very useful comments and suggestions.

Declaration of Conflicting Interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

Notes

References

Araar

Abdelkrim

Duclos

Jean-Yves

. 2003. “An Atkinson-Gini Family of Social Evaluation Functions.” Economics Bulletin 3:1–16.

Araar

Abdelkrim

Duclos

Jean-Yves

. 2006. “An Atkinson-Gini Family of Social Evaluation Functions: Theory and Illustration Using Data from the Luxembourg Income Study.” Public Finance 54:191–209.

Aronson

J. Richard

Johnson

Paul

Lambert

Peter J.

. 1994. “Redistributive Effect and Unequal Income Tax Treatment.” The Economic Journal 104:262–70.

Atkinson

Anthony B.

1970. “On the Measurement of Inequality.” Journal of Economic Theory 2:244–63.

Atkinson

Anthony B.

1980. “Horizontal Equity and the Distribution of the Tax Burden.” In The Economics of Taxation, ed. Aaron

Henry J.

Boskins

Michael J.

, 3–18. Washington, DC: Brookings Institution.

Bilger

Marcel

. 2008. “Progressivity, Horizontal Inequality and Reranking Caused by Health System Financing: A Decomposition Analysis for Switzerland.” Journal of Health Economics 27:1582–93.

Donaldson

David

Weymark

John A.

. 1980. “A Single-Parameter Generalization of Gini Indices of Inequality.” Journal of Economic Theory 22:67–86.

Duclos

Jean-Yves

Araar

Abdelkrim

. 2006. Poverty and Equity Measurement, Policy and Estimation with DAD. Kluwer Academic Publishers.

Duclos

Jean-Yves

Araar

Abdelkrim

Fortin

Carl

. 2010. “DAD: A Software for Distributive Analysis/Analyse Distributive.” MIMAP programme, International Development Research Centre, Government of Canada, and CIRPÉE, Université Laval (version: DAD 4.6 of October 2010).

10.

Duclos

Jean-Yves

Jalbert

Vincent

Araar

Abdelkrim

. 2003. “Classical Horizontal Inequity and Reranking: An Integrated Approach.” Research on Economic Inequality 10:65–100.

11.

Duclos

Jean-Yves

Lambert

Peter J.

. 2000. “A Normative and Statistical Approach to Measuring Classical Horizontal Inequity.” Canadian Journal of Economics 33:87–113.

12.

Ebert

Udo

. 1999. “Using Equivalent Income of Equivalent Adults to Rank Income Distributions.” Social Choice and Welfare 16:233–58.

13.

Fan

Jianqing

Gijbels

Irène

. 1996. Local Polynomial Modelling and Its Applications. London: Chapman and Hall.

14.

Faunt

Lindsay M.

Johnson

Michael L.

. 1992. “Analysis of Discrete, Time-sampled Data Using Fourier Series Method.” Methods in Enzymology 210:340–56.

15.

Härdle

Wolfgang

. 1990. Applied Nonparametric Regression. Cambridge: Cambridge University Press.

16.

Kakwani

Nanak C.

1984. “On the Measurement of Tax Progressivity and Redistributive Effect of Taxes with Applications to Horizontal and Vertical Equity.” Advances in Econometrics 3:149–68.

17.

Keele

Luke

. 2008. Semiparametric Regression for the Social Sciences. London: Wiley & Sons.

18.

Plotnick

Robert

. 1981. “A Measure of Horizontal Equity.” The Review of Economics and Statistics 63:283–88.

19.

Wand

Matt P.

Jones

Chris M.

. 1995. Kernel Smoothing. London: Chapman and Hall.

20.

Yitzhaki

Shlomo

. 1983. “On an Extension of the Gini Inequality Index.” International Economic Review 24:617–28.

21.

Yitzhaki

Shlomo

Olkin

Ingram

. 1991. “Concentration Curves and Concentration Indices.” Lecture Notes—Monograph Series 19:380–92.