with Nq1 and Nq2 denoting the points corresponding to the Q1 and Q2 quantiles, respectively.
This statistic has been proposed as measure of income inequality. For example, if Q1 = 0.2 and Q2 = 0.8, then this ratio compares the income of the bottom 20% relative to the income of the upper 20% percent.
The economist Palma has specifically suggested the Palma index which is based on Q1 = 0.4 and Q2 = 0.9. That is, the income of the bottom 40% is compared to the income of the top 10%. This is based on empirical studies which indicate the stability of incomes of the middle 50% (from roughly the 40th to the 90th decile) across countries. This measure has been proposed as an alternative to the Gini index. The Gini index indicates difference in income inequality, but is not particularly informative about where those differences occur. The Palma index is sensitive to the differences in the tails (i.e., the bottom 40% and the top 10%). The Cobham and Sumner article gives a number of arguments for preferring the Palma index to the Gini index.
The desired quantiles can be set with the commands
LET XQDENOM = <value>
Note that Dataplot will take the larger value for the numerator and the smaller value for the denominator. If these values are not given, Dataplot will use 0.9 for the numerator and 0.4 for the denominator (i.e., the Palma index).
Currently this command is restricted to non-negative data. If any negative numbers are encountered in the data, an error will be reported and the statistic will not be computed.
where <par> is a parameter where the computed interdecile ratio is stored;
<var> is a response variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET XQNUM = 0.8
In the context of this command, we are actually interested in the NI and the NI1P. For the denominator, we sum the points from 1 to NI and add REM*Y*N(NI1P). For the numerator, we sum the points from NI to N and subtract REM*Y(NI1P).
If a different algorithm is used to computed the desired quantiles, a slightly different result could be returned for the interdecile ratio statistic. Note:
Palma (2011), "Homogeneous Middles vs. Hererogeneous Tails, and the End of the 'Inverted-U': It's All About the Share of the Rich", Development and Change, 42(1), pp. 87-153.
. Step 1: Read the Data . . The first data set follows a normal distribution . while the second follows a lognormal distribution . skip 25 read zarr13.dat y1 read lgn.dat y2 . . Step 2: Compute the statistic for both data sets . let xqnum = 0.9 let xqdenom = 0.4 let decrat1 = interdecile ratio y1 let decrat2 = interdecile ratio y2 . let xqnum = 0.8 let xqdenom = 0.2 let decrat3 = interdecile ratio y1 let decrat4 = interdecile ratio y2 . . Step 3: Print the results . let decrat1 = round(decrat1,2) let decrat2 = round(decrat2,2) let decrat3 = round(decrat3,2) let decrat4 = round(decrat4,2) print "num: 0.9, den: 0.4" print "decrat1: ^decrat1" print "decrat2: ^decrat2" print "num: 0.8, den: 0.2" print "decrat3: ^decrat3" print "decrat4: ^decrat4"The following output is generated
num: 0.9, den: 0.4 decrat1: 0.25 decrat2: 0.31 num: 0.8, den: 0.2 decrat3: 1.01 decrat4: 1.26