Principles of Econometrics, 4th Edition
Page 1
Chapter 3:
Interval Estimation and Hypothesis Testing
Chapter 3
Interval Estimation and Hypothesis
Testing
Principles of Econometrics, 4th Edition
Page 2
Chapter 3:
Interval Estimation and Hypothesis Testing
3.1
Interval Estimation
3.2
Hypothesis Tests
3.3
Rejection Regions for Specific
Alternatives
3.4
Examples of Hypothesis Tests
3.5
The
p
-value
Chapter Contents
Principles of Econometrics, 4th Edition
Page 3
Chapter 3:
Interval Estimation and Hypothesis Testing
3.1
Interval Estimation
Principles of Econometrics, 4th Edition
Page 4
Chapter 3:
Interval Estimation and Hypothesis Testing
•
There are two types of estimates
–
Point estimates
•
The estimate
b
2 is a point estimate of the unknown
population parameter in the regression model.
–
Interval estimates
•
Interval estimation proposes a range of values in
which the true parameter is likely to fall
•
Providing a range of values gives a sense of what
the parameter value might be, and the precision with
which we have estimated it
•
Such intervals are called
confidence intervals
or
interval estimates
3.1
Interval Estimation
Principles of Econometrics, 4th Edition
Page 5
Chapter 3:
Interval Estimation and Hypothesis Testing
•
Let us assume that assumptions SR1-SR6
hold. The normal distribution of b2, the
least squares estimator of
β2, is
•
A standardized normal random variable is
obtained from b2 by subtracting its mean
and dividing by its standard deviation:
3.1.1
The
t
-Distribution
Eq. 3.1
3.1
Interval Estimation
(
29
-
∑
2
2
2
2
,
~
x
x
N
b
i
σ
β
(
29
(
29
1
,
0
~
2
2
2
2
N
x
x
b
Z
i
∑
-
-
=
σ
β
Principles of Econometrics, 4th Edition
Page 6
Chapter 3:
Interval Estimation and Hypothesis Testing
•
We know that:
•
Substituting:
•
Rearranging:
3.1
Interval Estimation
3.1.1
The
t
-Distribution
(
29
95
.
0
96
.
1
2
96
.
1
2
2
2
=
≤
-
-
≤
-
∑
x
x
b
P
i
σ
β
(
29
95
.
0
96
.
1
96
.
1
=
≤
≤
-
Z
P
(
29
(
29
95
.
0
96
.
1
96
.
1
2
2
2
2
2
2
2
=
-
+
≤
≤
-
-
∑
∑
x
x
b
x
x
b
P
i
i
σ
β
σ
Principles of Econometrics, 4th Edition
Page 7
Chapter 3:
Interval Estimation and Hypothesis Testing
•
The two end-points
provide an interval estimator.
•
In repeated sampling 95% of the intervals
constructed this way will contain the true
value of the parameter β2.
•
This easy derivation of an interval estimator
is based on the assumption SR6
and
that we
know the variance of the error term σ2.
3.1
Interval Estimation
3.1.1
The
t
-Distribution
(
29
∑
-
±
2
2
2
96
.
1
x
x
b
i
σ
Principles of Econometrics, 4th Edition
Page 8
Chapter 3:
Interval Estimation and Hypothesis Testing
•
Replacing σ2 with
creates a random
variable
t:
•
The ratio
has a
t
-distribution
with (
N –
2) degrees of freedom, which we
denote as:
Eq. 3.2
3.1
Interval Estimation
3.1.1
The
t
-Distribution
2
σ
ˆ
(
29
(
29
(
29
(
29
2
2
2
2
2
2
2
2
2
2
2
-
-
=
-
=
-
-
=
∑
N
i
t
~
b
se
b
b
r
a
ˆ
v
b
x
x
ˆ
b
t
β
β
σ
β
(
29
2
2
2
b
se
)
b
(
t
β
-
=
(
29
2
~
-
N
t
t
Principles of Econometrics, 4th Edition
Page 9
Chapter 3:
Interval Estimation and Hypothesis Testing
•
In general we can say, if assumptions SR1-SR6 hold in the
simple linear regression model, then
–
The
t
