Simulation Studies Comparing Dagum and Singh–Maddala Income Distributions

Abstract

Dagum and Singh–Maddala distributions have been widely assumed as models for income distribution in empirical analyses. The properties of these distributions are well known and several estimation methods for these distributions from grouped data have been discussed widely. Moreover, previous studies argue that the Dagum distribution gives a better fit than the Singh–Maddala distribution in the empirical analyses. This study explores the reason why Dagum distribution is preferred to the Singh–Maddala distribution in terms of the akaike information criterion through Monte Carlo experiments. In addition, the properties of the Gini coefficients and the top income shares from these distributions are examined by means of root mean square errors. From the experiments, we confirm that the fit of the distributions depends on the relationships and magnitudes of the parameters. Furthermore, we confirm that the root mean square errors of the Gini coefficients and top income shares depend on the relationships of the parameters when the data-generating processes are a generalized beta distribution of the second kind.

1 Introduction

Kleiber (1996) shows that Dagum and Singh–Maddala income distributions are closely related, although he states that the Dagum distribution should provide a better fit than the Singh–Maddala distribution. Moreover, Tadikamalla (1980) shows that the shape of the Dagum distribution is considerably more flexible than that of the Singh–Maddala distribution. Therefore, the Dagum distribution more often is assumed as the income distribution in empirical analyses. Accordingly, Kleiber (2008) reviews studies that examine income distributions of the Dagum type.

In empirical analyses, grouped data have been utilized widely and many works have examined several distributions from grouped data (e.g., McDonald and Ransom 1979a). Although McDonald and Mantrala (1995) show that the estimates are different if the number of groups is different, even if the same data source is used,¹ the properties of the estimators have been examined rarely. As far as we know, the properties of the estimators have been examined only by McDonald and Ransom (1979b), who assume a gamma distribution. However, their focus is on comparing several estimation methods. In addition, the properties of the model selection criteria have not been examined, in spite of the fact that several model selection criteria, including the akaike information criterion (AIC), have been utilized widely in empirical analyses. Therefore, we consider that the properties of the model selection criteria, which take into account the effect of group numbers, become important when we use these distributions in empirical analyses.

Moreover, the parameters are used not only for their own inferences, but also for the inference of the Gini coefficients (e.g., Atoda et al. 1988; Chotikapanich and Griffiths 2000; Nishino and Kakamu 2011). However, it is well known that the Gini coefficients are highly nonlinear functions of the parameters. As shown by Gelfand et al. (1990), the MCMC method can yield valid inferences on nonlinear functions of the parameters, such as the Gini coefficients. Therefore, it would be worthwhile to examine the properties of the Gini coefficients in a Bayesian framework.²

Recently, researchers of inequality have become concerned with not only the Gini coefficient but also the top income share. For example, Atkinson et al. (2011) examine the history of the top income shares and Brzezinski (2013) examines the properties of the top income share. While the Gini coefficient measures inequality among whole households or individuals, top income share is constructed as ratios (portions) of upper tails to total income of households or individuals. Therefore, top income share might be sensitive to the choice of the underlying hypothetical distributions. Thus, we analyze top income share as an alternative inequality measure to the Gini coefficient.

This study has two objectives. One is to explore the reason why Dagum distribution is preferred to Singh–Maddala distribution in terms of AICs through Monte Carlo experiments, in which the data-generating processes (DGPs) are the generalized beta distribution of the second kind (GB2 distribution). This involves considering the Dagum and Singh–Maddala distributions as special cases. The other objective is to examine the properties of the Gini coefficients and top income shares for these two distributions by means of root mean square errors (RMSEs). From the experiments, we confirm that the fit of the distributions depends on the relationships and magnitudes of the parameters and that the RMSEs of the Gini coefficients and top income shares also depend on the relationships of the parameters.

The rest of this paper is organized as follows. In Sect. 2, we briefly discuss the relationship between the Dagum and Singh–Maddala distributions and set forth the framework for Bayesian inference. Section 3 presents the results obtained from the Monte Carlo simulations. Finally, brief conclusions and remaining issues are presented in Sect. 4.

2 Dagum and Singh–Maddala Income Distributions

Dagum (1977) and Singh and Maddala (1976) distributions are special cases of the GB2 distribution proposed by McDonald (1984). The probability density function (PDF) of the GB2 distribution with parameters $a > 0$, $b > 0$, $p > 0$, and $q > 0$ is defined as

$$\begin{aligned} f_{\textit{GB2}}(x) = \frac{a x^{ap - 1}}{\displaystyle b^{ap}B(p,q)\left[ 1 + \left( \frac{x}{b} \right) ^{a} \right] ^{p+q}},\ x > 0, \end{aligned}$$

where $B(\cdot ,\cdot )$ is a beta function, b is a scale, and a, p, and q are shape parameters. In particular, parameter p makes the shape of the upper tail change and parameter q makes the shape of the lower tail change. The distribution proposed by Majumder and Chakravarty (1990) is simply a reparameterization of the GB2 distribution.

This four-parameter family nests most of the previously considered income distributions as special or limiting cases. Our notations are as follows: $\textit{GB2}(a,b,p,1) = \mathcal {DA}(a,b,p)$ and $\textit{GB2}(a,b,1,q) = \mathcal {SM}(a,b,q)$, where $\textit{GB2}(a,b,p,q)$, $\mathcal {DA}(a,b,p)$, and $\mathcal {SM}(a,b,q)$ are GB2 distribution with parameters a, b, p, and q; the Dagum distribution has parameters a, b, and p; and the Singh–Maddala distribution has parameters a, b, and q. These relationships and related distributions are summarized in Kleiber and Kotz (2003) in detail. In addition, Kleiber (1996) shows that $\displaystyle X \sim \mathcal {SM}(a,b,r) \Leftrightarrow \frac{1}{X} \sim \mathcal {DA}\left( a, \frac{1}{b}, r \right)$. Therefore, we can confirm that the Dagum and Singh–Maddala distributions are related closely.

The PDF and cumulative density function (CDF) of the Dagum distribution are expressed as

$$\begin{aligned} f_{DA}(x)= & {} \frac{a p x^{a p - 1}}{\displaystyle b^{a p} \left[ 1 + \left( \frac{x}{b} \right) ^{a} \right] ^{p + 1}},\ x > 0, \end{aligned}$$

(1)

$$\begin{aligned} F_{DA}(x)= & {} \left[ 1 + \left( \frac{x}{b} \right) ^{-a} \right] ^{-p},\ x > 0 , \end{aligned}$$

(2)

respectively. On the other hand, the PDF and CDF of the Singh–Maddala distribution are given as

$$\begin{aligned} f_{SM}(x)= & {} \frac{a q x^{a-1}}{\displaystyle b^{a} \left[ 1 + \left( \frac{x}{b} \right) ^{a} \right] ^{1+q}},\ x > 0, \end{aligned}$$

(3)

$$\begin{aligned} F_{SM}(x)= & {} 1 - \left[ 1 + \left( \frac{x}{b} \right) ^{a} \right] ^{-q},\ x > 0 , \end{aligned}$$

(4)

respectively. As shown above, these two distributions have closed-form expressions in their CDFs, unlike the GB2 distribution; furthermore, their CDFs are invertible. Therefore, these two distributions are tractable and it is possible that the statistical inference is implemented using these PDFs and CDFs.³

To estimate the parameters of the income distribution from grouped data, Kakamu and Nishino (2014) show that the Bayesian approach is preferred to the maximum likelihood method. Therefore, we take the Bayesian approach and introduce the likelihood function, following Nishino and Kakamu (2011). Let ${\varvec{\theta }}$ be the vector of parameters of the relevant income distribution and let $\mathbf {x}= (x_{1},\ x_{2},\ldots ,\ x_{k})$ be the vector of observed income in the form of grouped data. Then, $x_{i}$s are strictly known. In the general form, the likelihood function given by the PDF and CDF is expressed as follows

$$\begin{aligned}&L(\mathbf {x}| {\varvec{\theta }})\nonumber \\&\quad = n! \frac{F(x_{1})^{n_{1} - 1}}{(n_{1} - 1)!} f(x_{1}) \left\{ \prod _{i = 2}^{k} \frac{\left( F(x_{i}) - F(x_{i - 1}) \right) ^{n_{i} - n_{i - 1} - 1}}{(n_{i} - n_{i - 1} - 1)!} f(x_{i}) \right\} \nonumber \\&\quad \times \frac{\left( 1 - F(x_{k}) \right) ^{n - n_{k}}}{(n - n _{k})!}, \end{aligned}$$

(5)

where $n_{i}$ is the number of observations when income is less than $x_{i}$ and n is the number of total observations. If we substitute (1) and (2) into (5), then we could derive an inference for the Dagum distribution. Similarly, if we use (3) and (4), we could draw an inference for the Singh–Maddala distribution.

To proceed with the Bayesian analysis, we need to assume the prior distribution as $\pi ({\varvec{\theta }})$. Given the likelihood function (5) and the prior distribution $\pi ({\varvec{\theta }})$, the posterior distribution is expressed as

$$\begin{aligned} \pi ({\varvec{\theta }}|\mathbf {x}) \propto \pi ({\varvec{\theta }}) L(\mathbf {x}| {\varvec{\theta }}). \end{aligned}$$

Using this posterior distribution, posterior inference via the MCMC method proposed by Chotikapanich and Griffiths (2000) is implemented.⁴

Finally, we specify the prior distribution as $\pi ({\varvec{\theta }}_{SM}) \propto b^{-1}$, which is the same assumption as in Chotikapanich and Griffiths (2000). In addition, we assume that the prior distribution for the Dagum distribution is $\pi ({\varvec{\theta }}_{DA}) \propto b^{-1}$.

3 Monte Carlo Experiments

We now explain the setup for the Monte Carlo simulations. First, we set the number of observations as n = 10,000 and 100,000 to evaluate the effect of the number of observations. In addition, we assume the number of groups as quintile ($k=4$) and decile ($k=9$) to evaluate the effects of the number of groups.

Given n and k, we assume the true DGP to be a GB2 distribution, and L samples of $x_{i}$ for $i=1,2,\ldots ,k$ are generated. That is, we perform L simulation runs for the Dagum and Singh–Maddala distributions; in this section, $L = 500$.

The simulation procedure in this section is as follows:

• (i) Given $a = 3.0$; $b = 6.0$; $p = 0.5,\ 0.6,\ 0.7,\ldots ,\ 2.9,\ 3.0$; and $q = 0.5,\ 0.6,\ 0.7,\ldots ,\ 2.9,\ 3.0$, we generate random numbers $x_{j}$s, $j = 1, 2, \ldots , n$ from the GB2 distribution.⁵

• (ii) Sort the random numbers in ascending order and pick up $x_{i}$, which corresponds to the $n_{i}$-th observation, $i = 1, 2, \ldots , k$.

• (iii) Given $\mathbf {x}= (x_{1}, x_{2}, \ldots , x_{k})$, we obtain the estimates using the Dagum and Singh–Maddala distributions. In the MCMC procedure, we run the MCMC algorithm, with 7000 iterations excluding the first 1, 000 iterations.

• (iv) We repeat (i)–(iii) L times, where $L = 500$, as mentioned above.

• (v) From L estimates, we compute the AIC and RMSE for Gini coefficients and top income shares, as follows

  $$\begin{aligned} \text {AIC}= & {} -2 L(\mathbf {x}| {\varvec{\theta }}^{(l)}) + 2 p,\\ \text {RMSE}_{\text {G}}= & {} \left( \frac{1}{L} \sum _{l=1}^{L} (G^{(l)} - G)^{2} \right) ^{\frac{1}{2}},\\ \text {RMSE}_{\text {T}}= & {} \left( \frac{1}{L} \sum _{l=1}^{L} (T^{(l)} - T)^{2} \right) ^{\frac{1}{2}}, \end{aligned}$$

  where ${\varvec{\theta }}^{(l)}$, $G^{(l)}$, and $T^{(l)}$ represent the estimators of ${\varvec{\theta }}$, the Gini coefficient and the 10 % top income share in the l-th simulation run, respectively, and p denotes the number of parameters.

We compare the fit of the distributions by the AICs and the accuracy of the estimators of the Gini coefficient and 10 % top income share from the Dagum and Singh–Maddala distributions by RMSEs through the Monte Carlo studies. All the results reported here are generated using Ox version 7.00 (OS_X_64/U) (see Doornik 2009).

Fig. 1

Selected frequency from the AIC. Note The ratios with which the Dagum distributions are selected by AIC are displayed. The top left side shows the results from quintile data and n = 10,000, the top right side shows those from decile data and n = 10,000, the bottom left side shows those from quintile data and n = 100,000, and the bottom right side shows those from decile data and n = 100,000, respectively

Figure 1 shows the selected frequencies of the Dagum distribution compared with the Singh–Maddala distribution, where the AICs are smaller.⁶ From the figure, the standard results can be confirmed: the selected frequency increases as the number of observations or the number of groups increase, and when the DGPs are true. Moreover, we can establish that the selected frequencies are almost the same if $p = q$. On the other hand, the Singh–Maddala distributions are selected if $p < q$ and $p > 1.0$, while the Dagum distributions are selected if $p > q$ and $q > 1.0$. However, if $p < 1.0$ or $q < 1.0$, the opposite results can be confirmed in some regions, that is, the Singh–Maddala distributions are likely to be selected if $p > q$, while the Dagum distributions are likely to be selected if $p < q$ and the frequencies depend on the relationship between the parameters p and q. Therefore, we can conclude that the fit of the distributions seems to depend on the relationships and magnitudes of the parameters p and q if we compare the models by AIC. In fact, for example, McDonald and Mantrala (1995) estimate the parameters of Dagum, Singh–Maddala, and GB2 distributions and show that the Dagum distribution is preferred to the Singh–Maddala distribution when the parameters of GB2 distribution are $p = 0.5186$ and $q = 1.358$. From the figure, we can confirm that the Dagum distribution is preferred to the Singh–Maddala distribution at this point.

Fig. 2

Differences of RMSEs of Gini coefficients. Note The differences of the RMSEs from Singh–Maddala and Dagum distributions are displayed. Therefore, if the differences are larger than zero, the RMSEs from Dagum distribion are smaller than those from Singh–Maddala distribution and vice versa. The top left side shows the results from quintile data and n = 10,000, the top right side shows those from decile data and n = 10,000, the bottom left side shows those from quintile data and n = 100,000, and the bottom right side shows those from decile data and n = 100,000, respectively

Fig. 3

Differences of RMSEs of top income shares. Note The differences of the RMSEs from Singh–Maddala and Dagum distributions are displayed. Therefore, if the differences are larger than zero, it means that the RMSEs from Dagum distribution are smaller than those from Singh–Maddala distribution and vice versa. The top left side shows the results from quintile data and n = 10,000, the top right side shows those from decile data and n = 10,000, the bottom left side shows those from quintile data and n = 100,000, and the bottom right side shows those from decile data and n = 100,000

Figure 2 shows the differences of the RMSEs of the Gini coefficients. From the figure, similar tendencies can be found, even if the number of observations n and/or the number of groups are changed. First, we find that the differences of the RSMEs are almost zero if $p = q$, $p > 1$, and $q > 1$. Second, the RMSEs from the Singh–Maddala distributions are smaller than those from the Dagum distributions if $p < q$ and $p > 1$, while the RMSEs from the Dagum distributions are smaller than those from the Singh–Maddala distributions if $p < q$ and $q > 1$. Moreover, the difference increases when parameters p and q move far away from $p = q$ and the differences become largest when p or q approaches 1. In other words, the RMSEs from the true distribution are smaller than those from the competing distribution. However, exceptions appear in the case of $p < 1$ or $q < 1$. In the case of $p < 1$, we find the parameter region where $q < 1.5$, while in the case of $q < 1$, the differences become smaller and cross zero, that is, the RMSEs from the Singh–Maddala distributions are smaller than those from the Dagum distributions. Therefore, we can conclude that the accuracy of the distribution seems to depend on the RMSE relationship of the parameters p and q, which is the same as the results from the AIC.

Figure 3 shows the differences of the RMSEs of the top income shares. From the figure, similar tendencies can be found, even if the number of observations n and/or the number of groups are changed and the tendencies are almost the same as the results from the Gini coefficients. First, we find that the differences of the RSMEs are almost zero if $p = q$, $p > 1$, and $q > 1$. Second, the RMSEs from the Singh–Maddala distributions are smaller than those from the Dagum distributions if $p < q$ and $p > 1$, while the RMSEs from the Dagum distributions are smaller than those from the Singh–Maddala distributions if $p < q$ and $q > 1$. Moreover, the differences increase when parameters p and q move far away from $p = q$ and the differences becomes largest when p or q approaches 1. In other words, the RMSEs from the true distribution are smaller than those from the competing distribution. However, exceptions appear in the case of $p < 1$ or $q < 1$. In the case of $p < 1$, we find the parameter region where $q < 1.5$, while in the case of $q < 1$, the differences become smaller and cross zero, that is, the RMSEs from the Singh–Maddala distributions are smaller than those from the Dagum distributions. Therefore, we can conclude that the accuracy of the distribution seems to depend on the RMSE relationship of the parameters p and q, which is the same as the results from the AIC and Gini coefficients.

Summarizing the results, if the DGPs are GB2 distribution, the case in which $p = 1$ and $q = 1$ plays an important role and the selected frequencies or RMSEs from the Gini coefficients or the top income shares appear as a type of saddle point; moreover, around the point, the performance of the AIC, Gini coefficients, and top income shares change. Therefore, it is important to check whether parameters p and/or q in the GB2 distribution are one, even when we choose the Dagum and Singh–Maddala distributions as income distribution.

4 Conclusion

This study compared the fit of the distributions in terms of AIC and the accuracy of the estimators of the Gini coefficients and top income shares from the Dagum and Singh–Maddala distributions by RMSEs through Monte Carlo experiments. From the experiments, we confirmed that the fit of the distributions depends on the relationships and magnitudes of the parameters p and q and that the RMSEs of the Gini coefficients and top income shares depend on the relationships of the parameters when the DGPs are GB2 distribution. Therefore, a possible reason why the Dagum distribution is preferred to the Singh–Maddala distribution is that the estimated parameters lie in the desirable regions where the Dagum distribution is preferred when the GB2 distribution is estimated.

Finally, we mention issues that remain to be analyzed. We examined the fit of the distributions only by AIC. However, there are several other model selection procedures or tests. Therefore, it is worthwhile examining other methods, including, for example, the marginal likelihood by Chib and Jeliazkov (2001) and the Bayesian goodness-of-fit test by Johnson (2004). Moreover, we assume GB2 distribution as the DGP. However, the true income data might not be generated from the GB2 distribution but from the double Pareto-lognormal distribution by Reed and Jorgensen (2013) or some mixture of distributions. Therefore, it would be worthwhile to examine the performance of the Dagum and Singh–Maddala distribution when the true GDP is not GB2. However, we believe that our findings from the Monte Carlo experiments represent an interesting first step.

Appendix: Markov chain Monte Carlo Methods

In this appendix, we introduce a Markov chain Monte Carlo (MCMC) method to estimate the parameters of the distributions, which is proposed by Chotikapanich and Griffiths (2000) for the Singh–Maddala distribution. As the estimation procedure of the Dagum distribution is the same as that of the Singh–Maddala distribution, we continue the explanation focusing on the Singh–Maddala distribution.

Chotikapanich and Griffiths (2000) first proposed the MCMC method to estimate the parameters of the distribution from grouped data using a random walk Metropolis–Hastings (RWMH) algorithm. However, it is sometimes difficult to estimate the parameters using their algorithm. Therefore, some modifications are required. Their algorithm with some modifications is as follows.

• 1. Generate a candidate value ${\varvec{\theta }}_{SM}^{new}$ from $\mathcal {N}({\varvec{\theta }}_{SM}^{(m-1)}, c^{2} \varvec{\Sigma })$, where c is a tuning parameter and $\varvec{\Sigma }$ is the maximum likelihood covariance estimate.⁷

• 2. Compute

  $$\begin{aligned} \alpha \left( {\varvec{\theta }}_{SM}^{(m-1)}, {\varvec{\theta }}_{SM}^{new} \right) = \min \left\{ 1, \frac{ \pi ( {\varvec{\theta }}_{SM}^{new} | \mathbf {x}) }{\pi ( {\varvec{\theta }}_{SM}^{(m - 1)} | \mathbf {x}) } \right\} , \end{aligned}$$

  and if any of the elements of ${\varvec{\theta }}_{SM}^{new}$ fall outside the feasible parameter region, then $\alpha \left( {\varvec{\theta }}_{SM}^{(m-1)}, {\varvec{\theta }}_{SM}^{new} \right) = 0$.

• 3. Generate a value u from $\mathcal {U}(0,1)$, where $\mathcal {U}(a,b)$ is a uniform distribution on the interval (a, b).

• 4. If $u \le \alpha \left( {\varvec{\theta }}_{SM}^{(m-1)}, {\varvec{\theta }}_{SM}^{new} \right)$, set ${\varvec{\theta }}_{SM}^{(m)} = {\varvec{\theta }}_{SM}^{new}$, otherwise ${\varvec{\theta }}_{SM}^{(m)} = {\varvec{\theta }}_{SM}^{(m-1)}$.

• 5. Return to step 1, with m set to $m+1$.