I am currently studying the procedure to calculate the GDI, especially for regions within the country, such as municipalities, provinces, etc. The problem I am dealing with right now is the possibility of having an IDG value higher than the HDI value. According to theory, the maximum value GDI can reach is the HDI value, when there is no difference between male and female achievements because GDI is precisely a measure of average achievement to reflect the inequalities between men and women. On the other hand, whenever female and male achievements in any of the four components measured by the indexes (life expectancy, literacy rates, school enrollment rate or income per capita) are different, GDI value has to be lower than HDI value.

In our office in Honduras, we have developed a methodology to calculate both indexes (HDI and GDI) at municipal level, estimating values of income and life expectancy through modeling (using as source of data household surveys and the national census), but we are facing odd results regarding the GDI, which for several municipalities is higher than HDI. For this reason, I have been investigating whether other countries have faced this same issue and I found out that many countries have published their NHDR showing data tables where for several provinces or regions GDI is higher than HDI, for instance, Kenya (2006), Philippines (2005), Panama (2004) and El Salvador (2003).

I have been wondering whether these results are correct and why is this happening. One of the reasons I have considered, although very rare and theoretically impossible, is that mathematically there is a possibility that GDI is higher than HDI. This is due to the fact that in the equations to calculate the equally distributed indexes the proportions used correspond to the total population in opposition to the specific population from which the indicator is extracted (for example, the indicator of literacy rate is really computed from population above 15 years old, not from the total population; enrollment rates are computed from population in school age only, etc.). Is not uncommon that the proportions of specific population differ from the proportion of total population, thus, there could be some combinations that would lead to an HDI lower than the GDI. Additionally, using different achievement equations in the case of life expectancy (maximum and minimum values change when we disaggregate by gender) and using a logarithmic function in the achievement equation for income contribute to break the mathematical identity between HDI and GDI calculation formulae.

In order to visualize this problem, I have performed some calculation tests for HDI and GDI, going from a case where all component values and proportions are the same to a case where all these differ (please see attached file). Although, values chosen as input are arbitrary, this exercise has been useful to figure some trends and you will be able to see the last result (test 8) yields a value of GDI higher that the value of HDI. In the same attached table, you will be able to find, preliminary data for some cases I have been working on for my country. Here the calculation problem derives from the life expectancy component only, mainly due to the differences in the achievement equations.
 
From the Honduran cases in the example shown in the table, another issue rises up: If in the computation we only take into account life expectancy, education level and income, we cannot generalize that whenever GDI value is lower than HDI, it means the women are the disadvantaged group. In the data shown for Honduras for example, it can bee seen that if women perform better in life expectancy and education there is a possibility that even though they have lower incomes than men, their total HDI is higher than HDI for males (see these calculations on the same table). The sole value of GDI does not indicate clearly which is the disadvantaged group.

Finally, the question I would like to post here is how can we treat these specific cases or what would be an advisable way to overcome this inconsistency.

Thank you very much for your help!

Regards,