8/10/2019 Chapter 1 Interpretation of a Regression Equation %28EC220%29 (1)
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Christopher Dougherty
EC220 - Introduction to econometrics(chapter 1)
Slideshow: interpretation of a regression equation
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 1). [Teaching Resource]
2012 The Author
This version available at: http://learningresources.lse.ac.uk/127/
Available in LSE Learning Resources Online: May 2012
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INTERPRETATION OF A REGRESSION EQUATION
The scatter diagram shows hourly earnings in 2002 plotted against years of schooling,defined as highest grade completed, for a sample of 540 respondents from the National
Longitudinal Survey of Youth.
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Highest grade completed means just that for elementary and high school. Grades 13, 14,and 15 mean completion of one, two and three years of college.
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Grade 16 means completion of four-year college. Higher grades indicate years ofpostgraduate education.
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. reg EARNINGS S
Source | SS df MS Number of obs = 540
-------------+------------------------------ F( 1, 538) = 112.15
Model | 19321.5589 1 19321.5589 Prob > F = 0.0000
Residual | 92688.6722 538 172.283777 R-squared = 0.1725
-------------+------------------------------ Adj R-squared = 0.1710
Total | 112010.231 539 207.811189 Root MSE = 13.126
------------------------------------------------------------------------------EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
S | 2.455321 .2318512 10.59 0.000 1.999876 2.910765
_cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444
------------------------------------------------------------------------------
INTERPRETATION OF A REGRESSION EQUATION
This is the output from a regression of earnings on years of schooling, using Stata.
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. reg EARNINGS S
Source | SS df MS Number of obs = 540
-------------+------------------------------ F( 1, 538) = 112.15
Model | 19321.5589 1 19321.5589 Prob > F = 0.0000
Residual | 92688.6722 538 172.283777 R-squared = 0.1725
-------------+------------------------------ Adj R-squared = 0.1710
Total | 112010.231 539 207.811189 Root MSE = 13.126
------------------------------------------------------------------------------EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
S | 2.455321 .2318512 10.59 0.000 1.999876 2.910765
_cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444
------------------------------------------------------------------------------
INTERPRETATION OF A REGRESSION EQUATION
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For the time being, we will be concerned only with the estimates of the parameters. Thevariables in the regression are listed in the first column and the second column gives the
estimates of their coefficients.
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. reg EARNINGS S
Source | SS df MS Number of obs = 540
-------------+------------------------------ F( 1, 538) = 112.15
Model | 19321.5589 1 19321.5589 Prob > F = 0.0000
Residual | 92688.6722 538 172.283777 R-squared = 0.1725
-------------+------------------------------ Adj R-squared = 0.1710
Total | 112010.231 539 207.811189 Root MSE = 13.126
------------------------------------------------------------------------------EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
S | 2.455321 .2318512 10.59 0.000 1.999876 2.910765
_cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444
------------------------------------------------------------------------------
INTERPRETATION OF A REGRESSION EQUATION
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In this case there is only one variable,S, and its coefficient is 2.46. _cons, in Stata, refers tothe constant. The estimate of the intercept is -13.93.
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Here is the scatter diagram again, with the regression line shown.
INTERPRETATION OF A REGRESSION EQUATION
SEARNINGS 46.293.13 ^
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SEARNINGS 46.293.13 ^
What do the coefficients actually mean?
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SEARNINGS 46.293.13 ^
To answer this question, you must refer to the units in which the variables are measured.
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SEARNINGS 46.293.13 ^
Sis measured in years (strictly speaking, grades completed),EARNINGSin dollars perhour. So the slope coefficient implies that hourly earnings increase by $2.46 for each extra
year of schooling.
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SEARNINGS 46.293.13 ^
We will look at a geometrical representation of this interpretation. To do this, we willenlarge the marked section of the scatter diagram.
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You should ask yourself whether this is a plausible figure. If it is implausible, this could bea sign that your model is misspecified in some way.
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SEARNINGS 46.293.13 ^
For low levels of education it might be plausible. But for high levels it would seem to be anunderestimate.
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SEARNINGS 46.293.13 ^
What about the constant term? (Try to answer this question yourself before continuing withthis sequence.)
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SEARNINGS 46.293.13 ^
Literally, the constant indicates that an individual with no years of education would have topay $13.93 per hour to be allowed to work.
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This does not make any sense at all. In former times craftsmen might require an initialpayment when taking on an apprentice, and might pay the apprentice little or nothing for
quite a while, but an interpretation of negative payment is impossible to sustain.
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A safe solution to the problem is to limit the interpretation to the range of the sample data,and to refuse to extrapolate on the ground that we have no evidence outside the data range.
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With this explanation, the only function of the constant term is to enable you to draw theregression line at the correct height on the scatter diagram. It has no meaning of its own.
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Another solution is to explore the possibility that the true relationship is nonlinear and thatwe are approximating it with a linear regression. We will soon extend the regression
technique to fit nonlinear models.
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Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 1.4 of C. Dougherty,
In troduct ion to Econ ometr ics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspxor the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
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