Re: st: Quantile Regression Coefficient Investigation for Individual Observations

[George Bentley <george.bentley@uconn.edu>]

Hi David,

Thank you for your response.  I would be happy to supply more
information to better detail what I am hoping to achieve.

I am investigating an empirical model called the Environmental Kuznets
Curve (EKC), it is used to model the relationship between a measure of
environmental quality/degradation and a measure of income.  When
investigating the relationship between environmental quality and
income if an EKC exists it appears as a U-shaped curve (quality on y
axis, income on x axis).  The general idea is that at low levels of
income prior to economic development very little damage occurs to the
environment, then as economic development and consequent increases in
income take place there is greater damage to environment/decrease in
environmental quality, finally it is believed an inflection point is
reached where further increases in income correspond to improved
levels of environmental quality (better technology, greater demand for
cleaner environment, etc.).


The functional form of the EKC is captured in the EKC income
polynomial consisting the per capita income term and per capita income
term squared.  In this scenario I am treating agriculture as an
environmental quality measure so if an EKC exists there should be a
negative sign on the per capita income term and positive sign on the
per capita income squared term.  The county area, county population,
and county topography are acting as controls.

You are correct, each county of the 153 counties has a value for per
capita income, per capita income squared, county area, county
population, and county topography.  My rationale and train of thought
for using quantile regression, please correct me if this is wrong is
as follows.

After running the code in my original email and constructing the
quantile plot I could see that there were cases where the coefficient
on the per capita income term is negative and the coefficient on the
per capita income squared term is positive.  At this point I went into
Stata and began running single quantiles individually finding that the
quantile range where the values match what I am looking for is
.52-.70.  As I ran each quantile from .52 to .70 individually I
generated the residuals for each observation.  After reading that in
quantile regression there are as many observations with residuals of
zero as indepedent variables (5 indepedent variables + 1 intercept
term in my case) I found the six observations in each quantile with a
residual of zero.  I believed that I could then say for these six
counties (in a single quantile) the coefficients on the per capita
income term and per capita income squared term were correct.  Is this
the wrong way to look at the outputs?

Again, I appreciate any help and insight you can offer.

Thank you,
George Bentley

On Thu, Jan 17, 2013 at 4:32 PM, David Hoaglin <dchoaglin@gmail.com> wrote:
> Hi, George.
>
> It would help if you gave some more information about your analysis
> and its goals.
>
> Am I correct that your data consist of one observation for each of the
> 153 counties?
>
> What is the motivation for the quantile regressions?
>
> Each of the quantile regressions has an overall coefficient for per
> capita income and an overall coefficient for per capita income
> squared, but not separate coefficients for the individual counties.
>
> How confident are you that per capita income and per capita income
> squared capture the functional relation between a given quantile and
> per capita income?  Does the way in which the coefficients of those
> two predictors vary across the quantiles make substantive sense?  Many
> departures from a linear relation are not quadratic.  You may be able
> to get the data to guide the choice of functional form.
>
> I hope this discussion is helpful.
>
> David Hoaglin
>
> On Wed, Jan 16, 2013 at 6:24 PM, George Bentley
> <george.bentley@uconn.edu> wrote:
>> Good evening,
>>
>> I am new to Stata software and Statalist.  So far I have been using
>> Stata solely for quantile regression.  I am able to run quantile
>> regression without issue but would like to delve deeper and am unsure
>> of how to proceed.  In a project, I am working with 153 counties where
>> agricultural cover is the dependent variable and the independent
>> variables are per capita income, per capita income squared, county
>> area, county population, and county topography.
>>
>> Here is an excerpt of my Stata code:
>>
>> quietly qreg  laagrpc  larea  lapop  topo  lainc  lainc2, quantile(.1)
>> estimates store QR_10
>> quietly qreg  laagrpc  larea  lapop  topo  lainc  lainc2, quantile(.2)
>> estimates store QR_20
>> quietly qreg   laagrpc  larea  lapop  topo  lainc  lainc2, quantile(.3)
>> estimates store QR_30
>> quietly qreg   laagrpc  larea  lapop  topo  lainc  lainc2, quantile(.4)
>> estimates store QR_40
>> quietly qreg   laagrpc  larea  lapop  topo  lainc  lainc2, quantile(.5)
>> estimates store QR_50
>> quietly qreg   laagrpc  larea  lapop  topo  lainc  lainc2, quantile(.6)
>> estimates store QR_60
>> quietly qreg   laagrpc  larea  lapop  topo  lainc  lainc2, quantile(.7)
>> estimates store QR_70
>> quietly qreg   laagrpc  larea  lapop  topo  lainc  lainc2, quantile(.8)
>> estimates store QR_80
>> quietly qreg   laagrpc  larea  lapop  topo  lainc  lainc2, quantile(.9)
>> estimates store QR_90
>> estimates table QR_10 QR_20 QR_30 QR_40 QR_50 QR_60 QR_70 QR_80 QR_90,
>> b(%7.3f) se
>>
>>
>> Excerpt of Estimates Table (Standard errors removed):
>>
>>                                  Q = 0.1    Q = 0.2     Q = 0.3     Q
>> = 0.4     Q = 0.5     Q = 0.6     Q = 0.7     Q = 0.8     Q = 0.9
>> Per capita Income        61.581     76.360       65.790      50.775
>>    3.250       -18.945      -6.001      30.653      17.331
>>
>> Per Capita Income Sq. -3.039      -3.764        -3.260       -2.553
>>   -0.209         0.840        0.196       -1.579      -0.885
>>
>>
>>  From my estimates table I can see that the per capita income term has
>> a negative coefficient at quantile = 0.6 and quantile = 0.7.  Also, I
>> can see that the per capita income squared term has a positive
>> coefficient at quantile = 0.6 and quantile = 0.7.  My main goal is to
>> determine the counties having a negative coefficient on the per capita
>> income term and a positive coefficient on the per capita income
>> squared term simultaneously.  Is this possible and is there any advice
>> as to how I should proceed?  Thank you for any assistance in advance.
>>
>> Thank you,
>> George Bentley
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