Download Econ 399 Chapter5

5. Consistency
We cannot always achieve unbiasedness of
estimators.
-For example, σhat is not an unbiased
estimator of σ
-It is only consistent
-Where unbiasedness cannot be achieved,
consistency is the minimum requirement for an
estimator
-Consistency requires MLR. 1 through MLR.4, as
well as no correlation between x’s
5. Intuitive Consistency
While the actual proof of consistency is
complicated, it can be intuitively explained
-Each sample of n observations produces a Bjhat
with a given distribution
-MLR. 1 through MLR. 4 cause this Bjhat to be
unbiased with mean Bj
-If the estimator is consistent, as n increases the
distribution becomes more tightly distributed
around Bj
-as n tends to infinity, Bjhat’s distribution
collapses to Bj
5. Empirical Consistency
In general,
If obtaining more data DOES NOT get us closer
to our parameter of interest…
We are using a poor (inconsistent) estimator.
-Fortunately, the same assumptions imply
unbiasedness and consistency:
Theorem 5.1
(Consistency of OLS)
Under assumptions MLR. 1 through
MLR. 4, the OLS estimator Bjhat is
consistent for Bj for all j=0, 1,…,k.
Theorem 5.1 Notes
While a general proof of this theorem requires
matrix algebra, the single independent variable
case can be proved from our B1hat estimator:
ˆ1
(x


 (x
ˆ1  1 
i1
 x1 ) y i
i1
 x1 )
2
(x

n
1  (x
n
1
i1
i1
(5.2)
 x1 )u i
 x1 )
2
Which uses the fact that yi=B0+B1xi1+u1 and
previously seen algebraic properties
Theorem 5.1 Notes
Using the law of large numbers, the numerator
and denominator converge in probability to the
population quantities Cov(x1,u) and Var(x1)
-Since Var(x1)≠0 (MLR.3), we can use probability
limits (Appendix C) to conclude:
plim ˆ1  1  Cov(x 1 , u) Var ( x1 )
(5.3)
plim ˆ   (since Cov(x , u )  0)
1
1
1
Note that MLR.4, which assumes x1 and u aren’t
correlated, is essential to the above
-Technically, Var(x1) and Var(u) should also be
less than infinity
5. Correlation and Inconsistency
-If MLR. 4 fails, consistency fails
-that is, correlation between u and ANY x
generally causes all OLS estimators to be
inconsistent
-”if the error is correlated with any of the
independent variables, then OLS is biased and
inconsistent”
-in the simple regression case, the
INCONSISTENCY in B1hat (or ASYMPTOTIC BIAS)
is: plim ˆ    Cov(x , u) Var ( x )
(5.4)
1
1
1
1
5. Correlation and Inconsistency
-Since variance is always positive, the sign of
inconsistency depends on the sign of covariance
-If the covariance is small compared to the
variance, the inconsistency is negligible
-However we can’t estimate this covariance
as u is unobserved
5. Correlation and Inconsistency
Consider the following true model:
y   0  1 x1   2 x2  v
Where we satisfy MLR.1 through MLR.4 (v has a
zero mean and is uncorrelated with x1 and x2)
-By Theorem 5.1 our OLS estimators (Bjhat) are
consistent
-If we omit x2 and run an OLS regression, then
~
u=B2x2+v and
plim 1  1   21
(5.5)
Cov
(
x
,
x
)
1
2
1 
(5.6)
Var ( x1 )
5. Correlation and Inconsistency
Practically, inconsistency can be viewed the same
as bias
-Inconsistency deals with population covariance
and variance
-Bias deals with sample covariance and variance
-If x1 and x2 are uncorrelated, the delta1=0 and
B1tilde is consistent (but not necessarily
unbiased)
5. Inconsistency
-The direction of inconsistency can be calculated
using the same table as bias:
Corr(x1,x2)>0
Corr(x1,x2)<0
B2hat>0
Positive Bias
Negative Bias
B2hat<0
Negative Bias
Positive Bias
5. Inconsistency Notes
If OLS is inconsistent, adding observations does
not fix it
-in fact, increasing sample size makes the
problem worse
-In the k regressor case, correlation between one
x variable and u generally makes ALL coefficient
estimators inconsistent
-The one exception is when xj is correlated
with u but ALL other variables are uncorrelated
with both xj and u
-Here only Bjhat is inconsistent