The True Regression

1
The True Regression

• In the population regression model,
  Yi?0?1Xiui, ?0 and ?1 are the true and unknown
  values of the constant term and slope
  coefficient.
• ui is an unknown random variable that represents
  all variables (other than X) that have an effect
  on Y.
• We collect data on Y and X for the purpose of
  getting estimates of ?0 and ?1

2
How Observations Arise

• Given the population or true model,
  Yi?0?1Xiui, we think of
• Drawing a set of observations on X
• Making a behind the scenes calculation of the
  corresponding values of Y using the parameters of
  the model and the random error
• After getting the values of Y, we can use both
  the Ys and the Xs to run a regression to
  ESTIMATE ?0 and ?1.

3
Observations (Continued)

• What would happen if we could somehow run the
  experiment over again?
• It is easiest to think about having the same
  values of X (Xs fixed in repeated samples),
  but this is not necessary
• Would make a second behind the scenes
  calculation of the values of Y
• This time, however, the values of Y obtained
  would be different because the errors (ui) would
  be different (Why?)
• A second regression would produce different
  estimates of ?0 and ?1

4
Observations (Continued)

• We could at least imagine running the experiment
  over indefinitely many times.
• This idea leads to treating estimates of the
  constant term and slope coefficient as random
  variables governed by a probability or sampling
  distribution
• The idea that estimates of ?0 and ?1 are random
  variables is perhaps the most important concept
  in this course and provides the link back to ECO
  3411

5
Least Squares Estimates

• Under what circumstances do the least squares
  estimates accurately and precisely estimate ?0
  and ?1?
• Assumptions
• The conditional mean of ui given Xi is equal to
  zero E(ui?Xi)0, i1,2,..,n
• Xi and Yi are identically and independently
  distributed (i.i.d.)
• Xi and Yi have four moments (technical
  requirement)

6
Why Have Assumptions?

• Identify properties of the least squares
  estimates in a pure case.
• Econometrics is really about what steps to take
  to avoid violating assumptions than about the
  mechanics of computing estimates.
• In this regard, there are important parallels
  with microeconomic theory
• Perfect competition

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E(ui?Xi)0, i1,2,..,n
Population or True Regression
Y
E(Yi?Xi)?0?1Xi
X
X1
X2
X3
8
E(ui?Xi)0, i1,2,..,n

• This assumption says two important things
• For any given value of X (for example, Xi), the
  mean of the random variable ui is equal to zero
• The mean of ui is equal to zero no matter what
  value of X happens to be drawn. Thus, u and X
  are uncorrelated X is uncorrelated with all
  other causes of Y.

9
(Xi,Yi) are i.i.d., i1,2,,n

• This assumption is really a statement about how
  the observed sample is drawn.
• A useful way to think of it is to say that each
  observation that we have in a regression (an
  Xi,Yi pair) is an i.i.d. draw from their joint
  distribution
• It is as if the sample was drawn by simple random
  sampling. The values for each observation drawn
  is unaffected by the values obtained on previous
  or subsequent draws.

10
Implications of Assumptions

• OLS gives unbiased and consistent estimates of
  the parameters, ?0 and ?1
• OLS estimates are efficient in the sense that no
  other linear estimator of ?0 and ?1 has a smaller
  variance.
• By the central limit theorem, OLS estimators are
  normally distributed provided the sample is
  large.

11
Central Limit Theorem

• Let the distribution of ui be arbitrary.
  Alternatively, suppose that we have no idea what
  the distribution of ui might look like
• Regardless of the distribution of ui, the
  distribution of OLS estimators of ß0 and ß1 will
  become approximately normal as n becomes large

12
Example 1

• Problem 4.2, p. 133 reports results of
  Wageiß0ß1Maleiui
• What are the some determinants of wage other than
  gender? Are these determinants uncorrelated with
  gender?
• Schooling
• Years of work experience
• Public sector (government) employment
• Disabilities

13
Example 2

• Another example in the text regresses test scores
  of children on class size Scoresiß0ß1Class
  sizeiui
• What factors other than class size might
  determine test scores? Are these uncorrelated
  with class size?
• Socioeconomic background
• Teacher experience/quality
• Student access to computers

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Example 3

• Is it more expensive because of environmental
  regulations to drill an oil/natural gas well on
  federal or private property?
• Factors to control
• Remoteness
• Characteristics of reserves
• Quantity of environmental resources
• Regional differences in attitudes toward resource
  development
• Policy differences between federal land
  management agencies

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Regulatory Framework

• National Environmental Policy Act
• Toxic Substances Control Act
• Resource Conservation and Recovery Act
• Comprehensive Environmental Response,
  Compensation, and Liability Act
• Threatened and Endangered Species Act
• Etc., etc,

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Experimental Design

• Wyoming Checkerboard
• 40 mile wide strip of land20 miles on either
  side of the Union Pacific Railroad right-of-way
• Pacific Railway Acts of 1862 and 1864
• Granted odd-numbered section of land (including
  mineral rights) to UPR
• Retained even-numbered sections of land as
  federal property

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Randomization of Land Ownership

• Federal and private sections
• Equally remote
• Similar geological characteristics
• Similar quantities of environmental assets
• BLM is federal land manager
• Small, sparsely populated arealittle geographic
  variation in attitudes toward development

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Drilling Cost Data

• I.H.S. Energy Group, Inc. uses APS well numbers
  to merge cost data with supplementary data on
  well characteristics
• Data on about 325,000 onshore U.S. wells,
  1987-1999
• Data available for 1404 wells in the Checkerboard
  (excluding 59 wells on State property)
• Available commercially (about 1.42 per record)

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Results

• Regress drilling cost of wells (in thousands of
  dollars) on a dummy variable indicating whether
  well is on federal or private property. Federal
  equals 1 if well is on federal property zero
  otherwise
• Cost885.0201.0xFederal ûi
• Interpretation?