National open university of nigeria introduction to econometrics I eco 355

 
 OBJECTIVES 
At the end of this unit, you should be able to: 

understand confidence interval for 
and 

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88 

interpret the confidence interval 
3.0.
 
 MAIN CONTENT 
3.1. Confidence Interval for 
 
With the normality assumption
 
for 
u„ 
the OLS estimators 
and 
are themselves 
normally distributed with means and variances given therein. Therefore, for example, the 
variable. 
̂
( ̂
)
 
( ̂
)√∑
̂
as noted in (4.3.6), is a standardized normal variable. It therefore seems that we can use 
the normal distribution to make probabilistic statements about 
provided the true 
population variance 
is known. If 
is known, an important property of a normally 
distributed variable with mean 
 
and variance 
is that the area under the normal curve 
between 
it ± 
 
is about 68 percent, that between the limits 
 
± 2
is about 95 percent, and 
that between 
 ± 
 
is about 99.7 percent.

But 
is rarely known, and in practice it is determined by the unbiased
 
estimator 
̂
. If 
we replace 
 
by 
̂
, (5.3.1) may be written as 
̂
( ̂
)
( ̂
)√∑
̂
̂
where the se (
̂
) now refers to the estimated standard error. It can be shown that the 
t 
variable thus defined follows the 
t 
distribution with n – 2 df. [Note the difference between 
(4.3.1) and (4.3.2).] Therefore, instead of using the normal distribution, we can use the 
t 
distribution to establish a confidence interval for 
̂
as follows: 
Pr 
(—
t 
) = 1 —
(4.3.3)
where the 
t 
value in the middle of this double inequality is the 
t 
value given by (4.3.2) 
and where 
is the value of the 
t 
variable obtained from the 
t 
distribution for 
/2 level 
of significance and n — 2 df; it is often called the 
critical 
t 
value at 
/2 level of 
significance. Substitution of (4.3.2) into (4.3.3) yields 
*
̂
( ̂
)
+