National open university of nigeria introduction to econometrics I eco 355

 
 
 
 
 
(iv) 
CONSISTENCY 
Consistency is the idea that, as the sample becomes infinitely large, the parameter 
estimate given by a procedure such as OLs converges on the true parameter value. This is 
obviously true when estimator is unbiased, as shown in our previous discussion above, as 
consistency is really just a weaker form of unbiasedness. However, the proof above rests 
on our assumption 3, that the X variables are fixed. If we relax this assumption it is no 
longer possible to prove the unbiasedness of OLs but we can still establish that it is a 
consistent estimator. That is, when we relax assumption 3, OLS is no longer a BLU 
estimator but it is still consistent. We showed in equation (29) in this module that 
̂
Let us divide the top and the bottom of the last term by n, we 
have:
̂
Finally, using the law of large numbers, we know that 
coverages to its 
expectation, which is 
. Similarly 
converges to 
. So as n 
tend to infinity (i.e. n 
 
̂
, which is equal to the true 
population parameter 
if 
(that is if 
uncorrelated). Thus 
̂
is a consistent estimator of the true population parameter 
. 
 
4.0 
 CONCLUSION 
The unit has critically examines assumption 5 and 6 of the CLRM and we can concludes 
that efficiency and consistency of CLRM as shown that is the best property of the model 
the value of 
̂ ̂
respectively. 
5.0 
SUMMARY 
It should be noted by all the students that this unit make a continuation of the classical 
linear regression model (CLRM) and it is also called the GAUSS-MARKOV 

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THEOREM. However, the unit try to proof the efficiency and consistency of classical 
linear regression model (CLRM) in one case while the unit also make a difference 
between the efficiency and BLUEness.