55
Lecture 22  Statistical inference
1) We have learned how to calculate
estimators
, and we have discovered what their
PDF
is.
Now we must
return to the original purpose for finding estimators.
After all, what we really want to know are the true
values of the
parameters
.
What we want to learn how to do now is to use these estimators to make
inferences
concerning the true parameter values
.
This is known as
statistical inference
.
There are two kinds of operations one can perform in statistical inference
.
The first is to create what is
known as a
confidence interva
l
.
A confidence interval is simply a range of values within which we have a
certain level of confidence that the true parameter value lies.
Obviously, we will have to use our estimator
to form the confidence interval.
The second type of operation one can perform is known as
hypothesis
testing
.
In hypothesis testing, we begin with a hypothesized value for the true parameter, and then use the
estimator to ascertain whether the hypothesis can be accepted or rejected.
We will look at both of these
types of operations, beginning with the idea of the confidence interval.
We saw last time that every estimator has a normal distribution, is unbiased, and has a variance
.
We may
write this as:
$
~
,
$
B
N B
i
i
B
i
σ
2
e
j
In picture terms, we have the following:
.95
.68
B
B
−
196
.
$
σ
E B
B
$
e j
=
B
B
+
σ
$
B
B
−
σ
$
B
B
+
196
.
$
σ
The probabilities in the above picture are gotten from the table in the back of the text, check them out for
yourself!
Using the above picture, we can say the following:
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P B
B
B
B
B
1
1
1
1
1
68
−
≤
≤
+
F
H
I
K
=
σ
σ
$
$
.
$
P B
B
B
B
B
1
1
1
196
196
95
1
1
−
≤
≤
+
F
H
I
K
=
.
$
.
.
$
$
σ
σ
Now with a little algebra, this last equation may be rewritten as:
P B
B
B
B
B
$
.
$
.
.
$
$
1
1
1
196
196
95
1
1
−
≤
≤
+
F
H
I
K
=
σ
σ
Look at the last formula closely.
What it says in words is that the true parameter will lie within 1.96
standard deviations of the estimated coefficient with probability =.95.
We would call this a
95 percent
confidence interval
.
It is an interval within which the true parameter value lies with a certain level of
confidence, in this case with a probability =.95.
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 Spring '11
 YongJinPark
 Econometrics, Normal Distribution, 2$, 2 degrees, 20 degrees, 28 degrees, 2.048 $

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