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# Re: st: Quantile Regression Coefficient Investigation for Individual Observations

 From George Bentley To statalist@hsphsun2.harvard.edu Subject Re: st: Quantile Regression Coefficient Investigation for Individual Observations Date Wed, 30 Jan 2013 10:56:21 -0500

```On Mon, Jan 28, 2013 at 10:22 AM, George Bentley
<george.bentley@uconn.edu> wrote:
>
> Hi David,
>
> My apologies for the delayed response but I used your advice and thought
> about my end goal.  I have decided that identifying EKC conformity on a
> county by county basis is not necessary.  Due to the presence of spatial
> autocorrelation in the indicators I am going to use an spatial error
> regression model.  Thank you again for your assistance, it is appreciated.
>
> Best,
> George
>
>
> On Fri, Jan 18, 2013 at 10:35 PM, David Hoaglin <dchoaglin@gmail.com>
> wrote:
>>
>> Hi, George.
>>
>>
>> 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.
>> >>
>> >> 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
>> >>> as to how I should proceed?  Thank you for any assistance in advance.
>> >>>
>> >>> Thank you,
>> >>> George Bentley
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>
>
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```