CHAPTER 8: HYPOTHESIS TESTING
References:
Textbook, Chapter 10, Sections 10.1, 10.2, 10.3, 10.4, 10.5, 10.6, 10.7, 10.8.
8.1 Hypothesis Testing
Example [Finance]:
There are two Stocks
X
and
Y:
Before making a decision on investment,
we would like to know (a) which has a higher expected return; (b) which is more risky?
Question (a) can be formulated as the following hypotheses:
The null hypothesis
H
0
:
°
X
=
°
Y
:
The alternative hypothesis
H
A
:
°
X
6
=
°
Y
:
Based on a data, if we can design a test and reject the null hypothesis
H
0
and accept
H
A
;
then
we can claim that the expected return of
X
is not equal to the expected return of
Y:
Question (b) can be formulated as the following hypotheses:
The null hypothesis
H
0
:
±
2
X
=
±
2
Y
:
The alternative hypothesis
H
A
:
±
2
X
6
=
±
2
Y
:
Again, if we reject
H
0
and accept
H
A
;
then we can conclude that
X
and
Y
are not equally risky.
Suppose after doing the above two hypotheses testing, we conclude that
X
has a higher
expected return and a smaller risk, then we will invest on
X
rather than on
Y:
Example [Wealth E/ect]
: A consumption function is given by
Y
=
²
+
³X
+
´W
+
µ;
where
Y
is the consumption,
X
is the labor income,
W
is the wealth, and
µ
a random disturbance
representing other omitted variables (e.g.
family structures).
The parameters
²; ³
and
´
are
constants.
We are interested in whether the wealth has e/ect on the consumption.
For this
purpose, we can formulate the hypotheses as
H
0
:
´
= 0
:
H
A
:
´
6
= 0
:
If the null hypothesis
H
0
is true, then the wealth has no e/ect on the consumption. If
H
A
is
true, then there exists a wealth e/ect.
1
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Example [Constant Return to Scale]:
The CobDouglas production function is speci°ed
as
F
(
L; K
) =
cL
°
K
±
;
where
Y
is output,
L
is labor and
K
is capital. Suppose
²
+
³
= 1
;
then
F
(
¶L; ¶K
)
=
c
(
¶L
)
°
(
¶K
)
±
=
¶
a
+
±
cL
°
K
±
=
¶F
(
¶; K
)
;
i.e. increasing both
L
and
K
by
¶
units will also increase output by
¶
units. This is called the
constant return to scale (CRS).
The null hypothesis
H
0
:
²
+
³
= 1
:
The alternative hypothesis is
H
A
:
²
+
³
6
= 1
:
The null is the CRS, the alternative hypothesis
H
A
says that the production technology is not
CRS.
Remarks:
(i) The null hypothesis
H
0
usually formulates some simple economic relationships among
population parameters
·
. These relationships may be predicted by the economic theory.
(ii) The alternative hypothesis
H
A
is a di/erent hypothesis against the null hypothesis
H
0
:
In other words, it is the hypothesis that we hope to accept if the null hypothesis
H
0
is rejected.
8.2 Tests of the mean of a Normal Distribution when
±
2
is known
Assumptions:
(i) The random sample
X
n
=
f
X
1
; X
2
; :::; X
n
g
is from a normal population
N
(
°; ±
2
)
;
where
°
is unknown.
(ii)
±
2
is known.
Hypotheses:
H
0
:
°
=
°
0
:
H
A
:
°
6
=
°
0
:
where
°
0
is a given real number (e.g.
°
0
= 5)
:
Question:
How to design a test for these hypotheses?
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 Fall '10
 Null hypothesis, Hypothesis testing, Statistical hypothesis testing, Type I and type II errors, Xn

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