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:
y   0  1 x1   2 x2  u (3.40)
-Assume this satisfies all 4 assumptions and that
we are concerned with x1
-if we exclude x2, our estimation becomes:
~
~
y  0  1 x1
3.3 Omitted Variable Bias
-From (3.23) we know that:
~
~
ˆ
ˆ
1  1   2 (3.43)
-where Bhats come from regressing y on ALL x’s
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:
~
~
E ( 1 )  1   2 (3.45)
3.3 Omitted Variable Bias
-From this we can calculate Btilde’s bias:
~
~
~
ˆ
Bias ( 1 )  E ( 1 )  1   2 (3.46)
-this bias is often called OMITTED VARIABLE
BIAS
-From this equation, B1tilde is unbiased in two
cases:
1) B2=0; x2 has no impact on y in the true model
2) deltatilde=0
3.3 Deltatilde=0
-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:
E( x2 | x1 )  E( x2 )
3.3 Omitted Variable Bias
-As B1hat’s bias depends on B2 and deltatilde, the
following table summarizes the possible biases:
Corr(x1,x2)>0
Corr(x1,x2)<0
B2hat>0
Positive Bias
Negative Bias
B2hat<0
Negative Bias
Positive Bias
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)
3.3 Example
Take the true regression:
Pasta   0  1Experience   2 Love  u
(ie)
Where pasta taste depends on experience
making pasta and love
-While we can measure years of experience, we
can’t measure love, so we find that:
Paˆsta  5.3  0.4Experience
What is the bias?
(ie)
3.3 Example
Paˆsta  5.3  0.4Experience
(ie)
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
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 E ( ~ )  
1
1
There is an UPWARD BIAS
~
If
E ( 1 )  1
There is a DOWNWARD BIAS
And B1tilde is BIASED TOWARDS ZERO if it is
closer to zero than B1
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:
y   0  1x1   2 x 2  3 x 3  u
~ ~
~
~
y   x  x
0
1 1
2
2
(3.49)
(3.50)
x3 is omitted and correlated with x1 but not x2
Both B1tilde and B2tilde will always be biased
unless x and x are uncorrelated
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 B1tilde’s bias without x2 having an
effect, similar to our 2 variable regression:
~
E ( 1 )  1   3
 (x  x )x
 (x  x )
i1
i1
1
i3
1
2
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
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
1) Simplify variance formulas
2) Give OLS an important efficiency property
Assumption MLR.5
(Homoskedasticity)
The error u has the same variance
given any values of the explanatory
variables. In other words,
Var (u | x1 , x2 ,..., xk )  
2
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 error’s
variance, HETEROSKEDASTICITY is present
-The above five assumptions are called the
GAUSS-MARKOV ASSUMPTIONS
-As listed above, they apply only to crosssectional data with random sampling
-time series and panel data analysis require
more complicated, related assumptions
Assumption MLR.5 Notes
If we let X represent all x variables, combining
assumptions 1 through 4 give us:
E (y | X)   0  1x1   2 x 2  ...   k x k
Or as an example:
E (U | X)   0  1I   2Time  3Love
MLR. 5 can be simplified to:
Var ( y | X )  
2
Var (U | X )  
2
Or for example:
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.”
“Assumption MLR. 5 says that the variance of y,
given X, does not depend on the values of
the independent variables.”
(bold added)
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,
2

ˆ
Var (  j ) 
(3.51)
2
SST j (1  R j )
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
2
sample variation in xj: SST 
(x  x j )
j

ij
Theorem 3.2 Notes
Note that all FIVE Gauss-Markov assumptions
were needed for this theorem
Homoskedasticity (MLR. 5) wasn’t 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