Re: st: Quantile Regression Coefficient Investigation for Individual Observations

[David Hoaglin <dchoaglin@gmail.com>]

Hi, George.

The additional information is helpful.  Thank you.

Though you probably do not intend it that way, I could read the
description of the EKC as longitudinal.  Your data are
cross-sectional.  If the EKC was formulated from a longitudinal point
of view, cross sectional data may show a different pattern of
behavior.

I am not an expert on quantile regression, but I don't see why it
would be desirable to use that approach with your data.  What happens
when you use ordinary regression?

A U-shaped pattern may not be quadratic.  If the distribution of per
capita income is reasonable, you could explore the pattern by setting
up categories of per capita income (perhaps as many as 10 of them) and
using the corresponding dummy variables (for the categories other than
the reference category) as the predictors (instead of per capita
income and per capita income squared).  Plotting the coefficients of
those dummy variables against the midpoints of the categories should
give you an idea of the pattern, after adjusting for the contributions
of the other variables.

So far, I don't see a reason for fitting quantile regressions and
identifying the counties that happen to have a zero residual for a
particular quantile regression.  I also don't see the logic of making
statements about whether the coefficients of per capita income and per
capita income squared are correct.

David Hoaglin

On Fri, Jan 18, 2013 at 8:57 AM, George Bentley
<george.bentley@uconn.edu> wrote:
> 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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