National open university of nigeria introduction to econometrics I eco 355

 
 
Which shows that 
̅
is biased downward (i.e., it underestimates the true 
) in small 
samples. But notice that as n, the sample size, increase indefinitely, the second term in 
(15), the bias factor, tends to be zero. Therefore, asymptotically (i.e., in a very large 
sample),
̅
is unbiased too, that is, lim E(
̅
= 
as 
. It can further be proved 
that 
̅
is also a consistent estimator.
4
; that is, as n increase indefinitely 
̅
converges to 
its true value 
. 
 
4.0 
CONCLUSION 
An alternative to the least-squares method is the method of maximum likelihood (ML).
To use this method, however, one must make an assumption about the probability 

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distribution of the disturbance term 
. In the regression context, the assumption most
popularly made is that 
 
follows the normal distribution. However, under the normality
assumption, the ML and OLS estimators of the intercept and slope parameters of the
regression model are identical. However, the OLS and ML estimators of the variance of
, 
are different. In large samples, however, these two estimators converge. Thus the ML
method is generally called a 
large-sample method. 
The ML method is of broader
application in that it can also be applied to regression models that are nonlinear in the
parameters.
5.0 
SUMMARY 
 
The unit has vividly discussed the maximum likelihood estimation of two variable 
regression model and the method of maximum likelihood corresponds to many well-
known estimation methods in statistics. For example, one may be interested in the heights 
of adult female penguins, but be unable to measure the height of every single penguin in 
a population due to cost or time constraints. Assuming that the heights are normally 
distributed with some unknown mean and variance, the mean and variance can be 
estimated with MLE while only knowing the heights of some sample of the overall 
population. MLE would accomplish this by taking the mean and variance as parameters 
and finding particular parametric values that make the observed results the most probable 
given the model.