Principles of Econometrics, 4t
h
Edition
Page 1
Chapter 3:
Interval Estimation and Hypothesis Testing
Chapter 3
Interval Estimation and Hypothesis
Testing

Principles of Econometrics, 4t
h
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
3.6 Linear Combinations of Parameters
Chapter Contents

Principles of Econometrics, 4t
h
Edition
Page 3
Chapter 3:
Interval Estimation and Hypothesis Testing
3.1
Interval Estimation

Principles of Econometrics, 4t
h
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 often called
confidence intervals
.
–
We prefer to call them
interval estimates
because the term ‘‘confidence’’ is widely
misunderstood and misused
3.1
Interval Estimation

Principles of Econometrics, 4t
h
Edition
Page 5
Chapter 3:
Interval Estimation and Hypothesis Testing
The normal distribution of b
2
, the least squares
estimator of
β
2
, is
A standardized normal random variable is
obtained from b
2
by subtracting its mean and
dividing by its standard deviation:
3.1.1
The
t
-Distribution
±
²
¸
¸
¹
·
¨
¨
©
§
³
¦
2
2
2
2
,
~
x
x
N
b
i
V
E
±
²
±
²
1
,
0
~
2
2
2
2
N
x
x
b
Z
i
¦
³
³
V
E
Eq. 3.1
3.1
Interval Estimation

Principles of Econometrics, 4t
h
Edition
Page 6
Chapter 3:
Interval Estimation and Hypothesis Testing

Principles of Econometrics, 4t
h
Edition
Page 7
Chapter 3:
Interval Estimation and Hypothesis Testing
We know that:
Substituting:
Rearranging:
±
²
95
.
0
96
.
1
2
96
.
1
2
2
2
¸
¸
¸
¹
·
¨
¨
¨
©
§
d
³
³
d
³
¦
x
x
b
P
i
V
E
±
²
95
.
0
96
.
1
96
.
1
d
d
³
Z
P
±
²
±
²
95
.
0
96
.
1
96
.
1
2
2
2
2
2
2
2
¸
¹
·
¨
©
§
³
´
d
d
³
³
¦
¦
x
x
b
x
x
b
P
i
i
V
E
V
3.1
Interval Estimation
3.1.1
The
t
-Distribution

Principles of Econometrics, 4t
h
Edition
Page 8
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
β
2
.
This easy derivation of an interval estimator is
based on the assumption SR6
and
that we know
the variance of the error term
σ
2
.
±
²
¦
³
r
2
2
2
96
.
1
x
x
b
i
V
3.1
Interval Estimation
3.1.1
The
t
-Distribution