# Consistency

```Research Method
Lecture 4 (Ch5)
OLS Asymptotics
1
OLS Asymptoticc
OLS asymptotics are the analyses of OLS
properties when the sample size (n)
increases to infinity. We will talk about
the concept of (i) consistency and (ii)
asymptotic normality.
2
Consistency
Consistency is a similar concept as the
unbiasedness.
Unbiasedness: Given the sample size n, the
expected value of the estimator ˆ j is equal
to the true value βj.
Consistency: The estimator ˆ j approaches to the
true value βj as the sample size n increases to
infinity.
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Why we need the concept of
consistency?
Often, unbiasedness is difficult to achieve.
But consistency is easier to achieve under
less strict conditions.
Econometrician consider that consistency
is the minimum requirement for any
estimators.
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Theorem 5.1: Consistency of OLS
Under assumptions MLR.1 through MLR.4,
OLS estimators ˆ j is consistent for βj for
j=0,1,…,k. That is:
p lim(ˆ j )   j for j  1,2,...,k
Proof: See front board
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Consistency can be achieved under less
strict assumptions, given below.
Assumptions MLR.4’
E(u)=0 and cov(xj,u)=0 for j=0,1,2,…,k
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Asymptotic normality
In the previous handout, we assumed that
the error term is normal (MLR.6) in order
to do the hypothesis testing.
But in many cases, normality assumption
is not appropriate.
We want to conduct hypothesis testings
while making no assumption about the
distribution of the error term.
Asymptotic normality result (in the next
slide) will shows that using t-test is just
fine for any type of distribution.
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Theorem 5.2 Asymptotic normality
Under Gauss-Markov Assumptions (MLR.1
through MLR.5), the distribution of the
following will approach to N(0,1) as
sample size increases to infinity. That is:
ˆ j   j
se( ˆ j )
a
~
N (0,1)
Or an equivalent notation is:
ˆ j   j
se( ˆ j )
d


N (0,1) as n 
 
Proof: See front board
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Theorem 5.2 tells us that, even if we do
not know the distribution of the error
term u, we can use the usual t-test in a
usual way to conduct hypothesis testing.
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Lagrange Multiplier Statistic
(or nR2-statistic)
Remember that F test relies on the
normality assumption about u.
There is a test of the exclusion restrictions
that does not need the normality
assumption.
This uses LM-statistic (or often called n-Rsquared statistic)
This is a test of exclusion restrictions.
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I explain the procedure by using the
following example
Y= β0+β1x1+β2x2+β3x3+β4x4+u --------------(1)
H0: β2=0, β4=0
H1: H0 is not true
Next slide shows the procedure
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The procedure
(i)Regress the restricted model. That is, Y=
β0+β1x1+β2x2+u. Then, get the residual, u~ .
(ii)Regress u~ on all the independent variables.
That is u~  0  1x1  2 x2  3 x3  4 x4  e . Then compute
R-squared. Call this Ru2.
(iii)Compute LM=n Ru2. The asymptotic
distribution of LM-stat is chi-squared
distribution with df equal to number of
equations in H0. That is
a
LM ~
p
# equations in H0. In
this example q=2.
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(iv) Set the significance level  . This is
usually set at 0.05.
(v) Find the cutoff point such that
P(χ2q>c)=  .
(vi) Reject if LM is greater than the cutoff
number. This is illustrated in the next
slide.
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The density of χ2q
1- 

c
Rejection region
The cutoff points can be found in
the table in the next slide.
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SouthWestern/Cengage
Learning
15
Example
Using crime1.dta, consider the following
model.
Narr86=β0+β1pcnv+β2avgsen+β3tottime+β4ptime86+
β5qemp86+u
Narr86: the number of time a man was arrested until 1986
Pcnv: proportion of prior arrests leading to conviction
Avgsen: average sentence served from past conviction
Tottime: total time the man has spent in prison prior to 1986
Ptime86: month spent in prison in 1986
Qemp86:number of quarters in 1986 during which the man was legally
employed.
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Test if avgsen and tottime have no effect
on narr86 once the other factors have been
controlled for.
That is test the following hypothesis. (Use
LM statistic instead of F-test)
H0: β2=0,β3=0
H1: H0 is not true.
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