Specification Errors

1
Specification Errors

• Omission of relevant variables Exclusion of
  relevant variables

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Example

• Suppose Y is the average monthly rent of two
  bedroom apartments. X is the number of closets.
  Z is square footage of the apartment.
• We estimate a regression model with Y as the
  dependent variable and X as the explanatory
  variable. What do you think will happen?

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What Happens?
Y
Z
X
With Z, Pink Area Used to Estimate bx. If Z is
omitted, Green Pink Area Used to Estimate bx.
4
Omitted Variables

• True Model is
• Yi b0b1X1ib2X2i ei
• We estimate
• Yi b0b1X1i mi
• Observe that
• mib2X2iei

5
What Are Consequences?

• Recall the classical assumptions. Which one is
  violated?
• What are the consequences if this assumption is
  violated?

6
Classical Assumptions

• The regression model is linear in the
  coefficients and the error term.
• The error term has zero population mean
• All explanatory variables are uncorrelated with
  the error term
• Observations of the error term are uncorrelated
  with each other (no serial correlation)

• The error term has constant variance (no
  heteroskedasticity)
• No explanatory variable is a perfect linear
  function of the other explanatory variables (no
  perfect multicollinearity)
• The error term is normally distributed.
  (optional).

7
Omitted Variable Bias

• Consistency and unbiasedness depended on these
  two assumptions
• If X1 and X2 are correlated, the error term will
  be correlated with X1.And, the estimated
  coefficient on X1 will be biased.
• The expected sign of the bias will depend on the
  sign on b2 and on whether X1 and X2 are
  positively or negatively correlated.

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Example Yi102Xi-2Ziei
9
Without Z
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Sampling Distribution of b Estimator When
Relevant Z Omitted
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Expected Bias
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(No Transcript)
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Conditions Where b1 is Unbiased

• X2 is irrelevant. The true b20.
• The simple correlation coefficient between X1 and
  X2 is zero.

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What Kind of Bias?

• In equation for the demand for peanut butter, the
  impact on the coefficient of disposable income of
  omitting the price of peanut butter
• In earnings equation for workers, the impact of
  the coefficient of experience of omitting the
  variable for age
• In a production function for airplanes, the
  impact on the coefficient of labor of omitting
  the capital variable
• In an equation for daily attendance at outdoor
  concerts, the impact on the coefficient of the
  weekend dummy variable (1weekend) of omitting a
  variable that measures the probability of
  precipitation at concert time.

15
Another Example

• Consider the following annual model of the death
  rate (per million population) due to coronary
  heart disease in the United States (Yt)
• Yhatt14010.0Ct4.0Et-1.0Mt
• (2.5) (1.0) (0.5)
• N31 (1950-1980) Adj R Squared.678

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Most Likely Cause of Unexpected Sign on M is
Omission of

• Per capita consumption of hard liquor (gallons)
  in year t
• The average fat content () of the meat that was
  consumed in year t
• Percapita consumption of wine and beer (gallons)
  in year t
• Per capita number of miles run in year t
• Per capita open heart surgeries in year 5
• Per capita amount of oat bran eaten in year t

17
Inclusion of Irrelevant Variable

• True Model is
• Yi b0b1X1i ei
• We estimate
• Yi b0b1X1ib2X2i mi
• Observe that
• mi --b2X2iei

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No Bias

• If X2 is irrelevant, b2 0.
• The estimated coefficient on X1 will be
  unbiased
• The variance of estimated coefficient will
  increase.

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Regression Analysis In Action

• Trandel The Bias Due to Omitting Quality When
  Estimating Automobile Demand
• Graddy Do Fast-Food Chains Price Discriminate?