Principles of Econometrics, 4t
h
Edition
Page 1
Chapter 3:
Interval Estimation and Hypothesis Testing
Chapter 3
Interval Estimation and
Hypothesis Testing
Walter R. Paczkowski
Rutgers University
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
Eq. 3.1
3.1
Interval
Estimation
2
2
2
2
,
~
x
x
N
b
i
1
,
0
~
2
2
2
2
N
x
x
b
Z
i
Principles of Econometrics, 4t
h
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
95
.
0
96
.
1
2
96
.
1
2
2
2
x
x
b
P
i
95
.
0
96
.
1
96
.
1
Z
P
95
.
0
96
.
1
96
.
1
2
2
2
2
2
2
2
x
x
b
x
x
b
P
i
i
Principles of Econometrics, 4t
h
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 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
.
3.1
Interval
Estimation
3.1.1
The
t
-
Distribution
2
2
2
96
.
1
x
x
b
i
Principles of Econometrics, 4t
h
Edition
Page 8
Chapter 3:
Interval Estimation and Hypothesis Testing
Replacing σ
2
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
ˆ
2
2
2
2
2
2
2
2
2
2
2
~
r
a
ˆ
v
N
i
t
b
se
b
b
b
x
x
b
t
2
2
2
b
se
b
t
2
~
N
t
t
Principles of Econometrics, 4t
h
Edition
Page 9
Chapter 3:
