1. Calculating polarization indices for population subgroups using Stata
Jan Zwierzchowski

2. “middle class” is a commonly used term, but vague;
Polarisation
Concept of economic polarization has been used since the 1980s to describe the disappearance of the middle income class in the USA;
“middle class” is a commonly used term, but vague;
The notion of economic polarization is frequently used incorrectly to describe the increase of income inequality;
Economic polarisation describes a process in which income focuses on two separate poles or groups (one rich, and the other poor), which causes the disappearance of the middle income class.
Calculating polarization indices for population subgroups using Stata

3. Measures of Polarisation
The median relative polarization index (MRP) was proposed by Morris, Bernhardt and Handcock (1994);
The MRP is based on a comparison of income distributions in two periods:
Where:
Calculating polarization indices for population subgroups using Stata

4. MRP for real data
For individual data (discrete distribution), the MPR index takes the following form:
The above index measures the average absolute deviation from the median of the effect function, normalized to vary between -1 and 1.
Calculating polarization indices for population subgroups using Stata

5. Generalisation: the MRP index for subgroups
The MRP index is normalised between -1 and 1 if and only if it is implemented for the whole population;
There is no reason not to generalise the formula, so that it is capable of assessing of polarisation in population subgroups;
The simplest formula for an index for a subgroup would be:
where G denotes a certain subgroup of the whole population
Calculating polarization indices for population subgroups using Stata

6. Generalisation: the MRP index for subgroups
The proposed formula in case of real data takes form:
Where the normalisation coefficient is equall to:
Calculating polarization indices for population subgroups using Stata

7. Case of panel data
The generalised formula does not inform on the polarisation within a subgroup of individuals, it rather informs on the polarisation in a certain region of the distribution;
What if we have panel data and observe change in the belongingness of units to certain subgroups over time?
Should this additional information be somehow implemented to the analysis?
We believe that it should be implemented, as this kind of information allows for the full assessment of the polarisation within subgroups.
Calculating polarization indices for population subgroups using Stata

8. Further generalisation: MRP index for panel data
The proposed formula for panel data incorporates a history element:
The first element accounts for the change in the distribution shape over time, while the second element measures the individual changes in relative incomes of units belonging to the analysed subgroup.
Calculating polarization indices for population subgroups using Stata

9. Data source
The empirical analysis was based on the polish Social Diagnosis Panel Database;
The Database contains a panel of household which covers the period (over 700 households);
The income category used is the net monthly equivalent income of households
The middle class was defined by two absolute income limits – the lower equal to social minimum, the upper equal to second income tax threshold.
Calculating polarization indices for population subgroups using Stata

10. Calculating generalised MRP indices with Stata
A do file was created co calculate indices;
The do file was written in a way that it can be quickly transformed into ado file;
Standard errors were calculated using bootstraping;
The inference on the significance of estimates is certainly not valid, as their distribution is asymmetrical;
As of today this is still work in progress.
Calculating polarization indices for population subgroups using Stata

11. Empirical results – Poland 2005-2015
Fractions in the distinguished groups:
Year
Class limits (in PLN)
Household fractions y'
y"
wLC
wMC
wHC
2005
1,030
5,610
0.508
0.486
0.006
2015
0.176
0.802
0.021
The fraction of household groups
Subgroup
wEP
wMC
wMC
wNEP
wMC
wLC,HC
Fraction
0.302
0.135
0.167
0.698
0.606
0.093
Calculating polarization indices for population subgroups using Stata

12. Empirical results – Poland 2005-2015
Index type
MRP indices
Standard error
z statistic
p-value
95% confidence interval
0.036
-0.017
0.986
[-0.069; 0,072]
0.037
0.220
0.826
[-0.067; 0,085]
0.121
-0.634
0.526
[-0.335; 0,198]
0.1814
0.341
0.532
0.595
[-0.492; 0,256]
0.048
-0.604
0.546
[-0.120; 0,064]
0.2569*
0.139
1.850
0.032
[-0.007; 0,541] *
0.051
-1.348
0.089
[-0.166; 0,033]
0.0222
0.038
0.586
0.558
[-0.049; 0.099]
0.044
-0.422
0.673
[-0.099; 0.072]
0.0931
0.115
0.808
0.419
[-0.147; 0.300]
Calculating polarization indices for population subgroups using Stata

13. Empirical results – Poland 2005-2015
A fraction of households belonging to the middle class (defined by absolute limits) increased from 48,6% in 2005 to 80,2% in 2015 as a result of growth in real incomes;
The polarisation index was virtually equall to 0 – there was no polarisation of incomes;
The analysis of indices for groups reveal a more intricate picture – there was a strong polarisation in the upper class – the rich grew even richer, while the poor and the middle class converged toward median income.
Calculating polarization indices for population subgroups using Stata

14. A new set of polarisation indices has been proposed;
Conclusions
A new set of polarisation indices has been proposed;
It allows for a more thorough analysis of economic polarisation, in particular, for the analysis in certain distinguished subgroups of the population;
A Stata command is being written, which allows for calculating the new indices using survey panel data;
It is not clear how inference on their significance should be conducted.
Calculating polarization indices for population subgroups using Stata

15. References
Duclos J-Y., Esteban J., Ray D.,(2004). Polarization: Concepts, Measurement, Estimation. Econometrica, Vol. 72, No. 6, pp. 1737–1772;
Handcock M. S., Morris M., (1999). Relative Distribution Methods in the Social Sciences, New York: Springer;
Horrigan H., Haugen S., (1988). The Declining Middle Class: a Sensitivity Analysis, Monthly Labour Rev., No. 111, pp. 3-13
Panek T., Zwierzchowski J. Median Relative Partial Income Polarization Indices: Investigating Economic Polarization in Poland during the year
Calculating polarization indices for population subgroups using Stata