Chapter 1 Interpretation of a Regression Equation %28EC220%29 (1)

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]

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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

------------------------------------------------------------------------------


This is the output from a regression of earnings on years of schooling, using Stata.

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. reg EARNINGS S


-------------+------------------------------ F( 1, 538) = 112.15

Model | 19321.5589 1 19321.5589 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.1710

Total | 112010.231 539 207.811189 Root MSE = 13.126


-------------+----------------------------------------------------------------

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

------------------------------------------------------------------------------


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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


-------------+------------------------------ F( 1, 538) = 112.15

Model | 19321.5589 1 19321.5589 Prob > F = 0.0000


-------------+------------------------------ Adj R-squared = 0.1710

Total | 112010.231 539 207.811189 Root MSE = 13.126


-------------+----------------------------------------------------------------

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

------------------------------------------------------------------------------


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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.


SEARNINGS 46.293.13 ^


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What do the coefficients actually mean?


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To answer this question, you must refer to the units in which the variables are measured.


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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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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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For low levels of education it might be plausible. But for high levels it would seem to be anunderestimate.



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What about the constant term? (Try to answer this question yourself before continuing withthis sequence.)



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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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Chapter 1 Interpretation of a Regression Equation %28EC220%29 (1)