3.3 Omitted Variable Bias

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3.3 Omitted Variable Bias
-When a valid variable is excluded, we
UNDERSPECIFY THE MODEL and OLS estimates are
biased -Consider the true population model
-Assume this satisfies all 4 assumptions and that
we are concerned with x1 -if we exclude x2, our
estimation becomes
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3.3 Omitted Variable Bias
-From (3.23) we know that
-where Bhats come from regressing y on ALL xs
and deltatilde comes from regressing x2 on
x1 -since deltatilde depends on independent
variables, it is considered fixed -we also know
from Theorem 3.1 that Bhats are unbiased
estimators, therefore
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3.3 Omitted Variable Bias
-From this we can calculate Btildes bias

• -this bias is often called OMITTED VARIABLE BIAS
• -From this equation, B1tilde is unbiased in two
  cases
• B20 x2 has no impact on y in the true model
• deltatilde0

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3.3 Deltatilde0
-deltatilde is equal to the covariance of x1 and
x2 over the variance of x1, all in the
sample -deltatilde is equal to zero only if x1
and x2 are uncorrelated -therefore if they are
uncorrelated, B1hat is unbiased -it is also
unbiased if we can show that
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3.3 Omitted Variable Bias
-As B1hats bias depends on B2 and deltatilde,
the following table summarizes the possible
biases
Corr(x1,x2)gt0 Corr(x1,x2)lt0
B2hatgt0 Positive Bias Negative Bias
B2hatlt0 Negative Bias Positive Bias
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3.3 Omitted Variable Bias
-the SIZE of the bias is also important, as a
small bias may not be cause for
concern -therefore the SIZE of B2 and deltatilde
are important -although B2 is unknown, theory
can give us a good idea about its
sign -likewise, the direction of correlation
between x1 and x2 can be guessed through
theory -a positive (negative) bias indicates that
given random sampling, on average your estimates
will be too large (small)
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3.3 Example

• Take the true regression

Where pasta taste depends on experience making
pasta and love -While we can measure years of
experience, we cant measure love, so we find
that
What is the bias?
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3.3 Example
We know that the true B2 should be positive love
improves cooking We can also support a positive
correlation between experience and love, if you
love someone you spend time cooking for
them Therefore B1hat will have a positive
bias However, since the correlation between
experience and love is small, the bias will
likewise be small
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3.3 Bias Notes
-It is important to realize that the direction of
bias is ON AVERAGE -a positive bias on average
may underestimate in a given sample If
There is an UPWARD BIAS If
There is a DOWNWARD BIAS And B1tilde is BIASED
TOWARDS ZERO if it is closer to zero than B1
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3.3 General Omitted Bias

• Deriving the direction of omitted variable bias
  with more independent variables is more difficult
• -Note that correlation between any explanatory
  variable and the error causes ALL OLS estimates
  to be biased.
• -Consider the true and estimated models

x3 is omitted and correlated with x1 but not
x2 Both B1tilde and B2tilde will always be biased
unless x1 and x2 are uncorrelated
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3.3 General Omitted Bias

• Since our x values can be pairwise correlated, it
  is hard to derive the bias for our OLS estimates
• -If we assume that x1 and x2 are uncorrelated, we
  can analyze B1tildes bias without x2 having an
  effect, similar to our 2 variable regression

With this formula similar to (3.45), the previous
table can be used to determine bias -Note that
much uncorrelation is needed to determine bias
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3.4 The Variance of OLS Estimators

• -We now know the expected value, or central
  tendency, of the OLS estimators
• -Next we need information on how much spread OLS
  has in its sampling distribution
• -To calculate variance, we impose a
  HOMOSKEDASTICITY (constant error variance)
  assumption in order to
• Simplify variance formulas
• Give OLS an important efficiency property

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Assumption MLR.5(Homoskedasticity)
The error u has the same variance given any
values of the explanatory variables. In other
words,
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Assumption MLR.5 Notes

• -MLR. 5 assumes that the variance of the error
  term, u, is the SAME for ANY combination of
  explanatory variables
• -If ANY explanatory variable affects the errors
  variance, HETEROSKEDASTICITY is present
• -The above five assumptions are called the
  GAUSS-MARKOV ASSUMPTIONS
• -As listed above, they apply only to
  cross-sectional data with random sampling
• -time series and panel data analysis require
  more complicated, related assumptions

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Assumption MLR.5 Notes

• If we let X represent all x variables, combining
  assumptions 1 through 4 give us

Or as an example
MLR. 5 can be simplified to
Or for example
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3.4 MLR.4 vs. MLR.5
Assumption MRL. 4 says that the expected value
of y, given X, is linear in the parameters but
it certainly depends on x1, x2,.,xk. Assumptio
n MLR. 5 says that the variance of y, given X,
does not depend on the values of the independent
variables. (bold added)
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Theorem 3.2(Sampling Variances of the OLS Slope
Estimators)
Under assumptions MLR. 1 through MRL. 5,
conditional on the sample values of the
independent variables,
For j 1, 2,,k, where Rj2 is the R-squared from
regressing xj on all other independent variables
(and including an intercept) and SST is the total
sample variation in xj
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Theorem 3.2 Notes
Note that all FIVE Gauss-Markov assumptions were
needed for this theorem Homoskedasticity (MLR. 5)
wasnt needed to prove OLS bias The size of
Var(Bjhat) is very important -a large variance
leads to larger confidence intervals and less
accurate hypothesis tests