# e-TA 12: Measures of Inequality

Welcome to a new issue of e-Tutorial. This time we focus on measures
of inequality. We will suggest some basic methods to calculate the Hill
estimator, the Lorenz curve, and the Gini coefficient. The data set to
be used is the same from the problem set 4. ^{1}

# Data

The first thing you need is to download the phuzics panel data set, called *phuzics10.txt* from the Econ 508 web site. Save it in your preferred directory.

The next step is loading the Data in Stata. If you saved it in your hard drive you can load it by typing:

`infile id yr phd sex rphd ru y Y s using "phuzics10.txt", clear`

drop if id==.

xtset id yr

# Hill Estimator

Here we compute the Hill estimator mentioned in Prof. Koenker’s Lecture 19 and “Appendix A - Concentration of Productivity in Phuzics Scholarship”.The idea is to calculate the index of concentration “\(\alpha\)” for the years between 1970 and 1990, and check if there is a trend. As mentioned in the note, if there is an unbiased positive trend, you can infer that phuzics productivity is becoming less concentrated, and the field is becoming less scientific (according to Parzen’s definition of the term). An unbiased negative trend would mean the reverse. The function below can calculate the alpha-coefficient of concentration for a given year (say, 1970). You are expected to adjust the code such that you can reproduce the experiment for the other periods:

`drop if yr~=70`

`gsort -Y`

gen Yratio=Y/Y[11]

drop if Y<Y[10]

gen a=log(Yratio)

egen b=sum(a)

scalar alpha=(b/10)^(-1)

scalar list

` alpha = .73199431`

You are asked to repeat this for the remaining years, and then plot the coefficients along time.

# Bootstrap Bias-Correction for the Hill Estimator:

In STATA, you can implement the bootstrap bias correction for the Hill estimator, suggested in Prof. Koenker's "Appendix A - Concentration of Productivity in Phuzics Scholarship", by using the command bsample. I will do that for one year, say, 1970, and then you can replicate the experiment for other years. The strategy is as folows:### Numerator \(\alpha_{N}\):

The first step is to calculate the numerator of the formula on page 2 of the note “Appendix A - Concentration of Productivity in Phuzics Scholarship”, called alpha_N. This is done as follows.

- Open the original data set;

`infile id yr phd sex rphd ru y Y s using "phuzics10.txt", clear`

drop if id==.

- Drop all observations not included in your interval of interest, say not included in 1970-1990;

`drop`

`if yr<70 | yr>90`

- Compute the Hill estimator for this pooled sample.

`gsort -Y`

gen Yratio=Y/Y[11]

drop if Y<Y[10]

gen a=log(Yratio)

egen b=sum(a)

scalar alpha=(b/10)^(-1)

scalar list

` alpha = 9.4789316`

### Denominator \({\hat{\alpha}_{n_{t}}}\):

- Open the original data set;

`infile id yr phd sex rphd ru y Y s using "phuzics10.txt", clear`

drop if id==.

- Drop all observations not included in your interval of interest:

`drop`

`if yr<70 | yr>90`

- Get the sample size of the respective year of interest;

` sort yr `

by yr: sum Y

- Generate a bootstrapped sample (with replacement) of the same size of the year you are working with. For example, because 1970 has 11 observations, you type

`bsample[11]`

this will generate a bootstrapped sample with 11 observations drawn from the pooled sample.

- Calculate the Hill estimator for this bootstrapped sample.

` gsort -Y `

gen Yratio=Y/Y[11]

drop if Y<Y[10]

gen a=log(Yratio)

egen b=sum(a)

scalar alpha=(b/10)^(-1)

scalar list

This routine will give you one bootstrapped alpha for the year 1970.
You need to repeat the experiment "B" times (say, 20 times), and get
"B" (say, 20) different bootstrapped alphas for 1970. After that, take
the average of those "B" (say, 20) alphas and use this number as the
denominator of the formula on page 2 of Prof. Koenker's "Appendix A -
Concentration of Productivity in Phuzics Scholarship". Finally, you need to apply the formula: multiply the original Hill estimator of 1970 by the pooled Hill estimator (here called "numerator") and divide it by the average bootstrapped estimator for 1970 (here called "denominator"), so that you find a bias-corrected Hill estimator for the year 1970.

This procedure is required for every year in the period 1970-1990. Each year will have its respective corrected alpha. The final step is to plot those corrected alphas along time, and check if there is any trend.

# Lorenz Curves and Gini Coefficient

STATA provides add-ins to calculate these statistics. You can type

`help lorenz`

`help`

`glcurve`

This will provide a menu of add-in programs related to the topic.
You need to install the selected package and the help files. You are
free to decide whether to use add-ins or to write your own code. The
most important thing, though, is to understand the theoretical
background of such measures, which we will discuss in class.

Please send comments to bottan2@illinois.edu or srmntbr2@illinois.edu↩