The McGraw-Hill Series Economics essentials of economics brue, McConnell, and Flynn Essentials of Economics

Part One
Single-Equation Regression Models
where 
µ
1
=
mean of 
Y
1
µ
2
=
mean of 
Y
2
σ
1
=
standard deviation of 
Y
1
σ
2
=
standard deviation of 
Y
2
ρ
=
coefficient of correlation between 
Y
1
and 
Y
2
4.2. By applying the second-order conditions for optimization (i.e., second-derivative test), show that
the ML estimators of
β
1
,
β
2
, and
σ
2
obtained by solving Eqs. (9), (10), and (11) do in fact
maximize the likelihood function in Eq. (4).
4.3. A random variable 
X
follows the 
exponential distribution
if it has the following probability
density function (PDF):
f
(
X
)
=
(1
/θ
)
e
−
X
/θ
for
X
>
0
=
0
elsewhere
where 
θ >
0 is the parameter of the distribution. Using the ML method, show that the ML
estimator of 
θ
is 
ˆ
θ
=
X
i
/
n
, where 
n
is the sample size. That is, show that the ML estimator
of 
θ
is the sample mean 
¯
X
.
4.4. Suppose that the outcome of an experiment is classified as either a success or a failure. Letting
X
=
1 when the outcome is a success and 
X
=
0 when it is a failure, the probability density, or
mass, function of 
X
is given by
p
(
X
=
0)
=
1
−
p
p
(
X
=
1)
=
p
, 0
≤
p
≤
1
What is the maximum likelihood estimator of 
p
, the probability of success?
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Beware of testing too many hypotheses; the more you torture the data, the more likely they are
to confess, but confession obtained under duress may not be admissible in the court of scientific
opinion.
1
As pointed out in Chapter 4, estimation and hypothesis testing constitute the two major
branches of classical statistics. The theory of estimation consists of two parts: point
estimation and interval estimation. We have discussed point estimation thoroughly in the
previous two chapters where we introduced the OLS and ML methods of point estimation.
In this chapter we first consider interval estimation and then take up the topic of hypothesis
testing, a topic intimately related to interval estimation.