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.

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 $m+1$.