Stochastic dominance in stock market 2

7
Chapter 2
Size and Power of Some Stochastic Dominance Tests

2.1 Introduction
Consumers choose from among uncertain alternatives to maximize their utility. One
of the necessary conditions for optimizing behavior in economic analysis is that an
optimal portfolio cannot be inferior to another feasible portfolio for any increasing
utility function. This can be characterized in terms of stochastic dominance (SD)
between distributions of returns. The theory of SD provides a systematic framework
for analyzing economic behavior under uncertainty.

Although the SD methodology which focuses on economic decisions theory
has been developed for more than three decades, while powerful SD tests have been
only available recently. In the literature, Levy and Sarnat (1970, 1972), Joy and Porter
(1974), Wingender and Groff (1989), Seyhun (1993) implement the SD rules
empirically, but they have not discussed the testing procedure for SD and no statistical
tests have been done. There are two broad classes of SD tests. One is
minimum/maximum statistic while the other is based on distribution values computed
on a set of grid points. McFadden (1989) first develops a SD test using the
minimum/maximum statistic. Later, Klecan, McFadden and McFadden (1991, KMM),
Kaur, Rao and Singh (1994, KRS) also propose some tests using the
minimum/maximum statistic. On the other hand, Anderson (1996), and Davidson and
Duclos (2000, DD) are the commonly used SD tests that compare the underlying
distributions at a finite number of grid points. Although there have been differences in

8
SD test methods in prior studies, the literature is rather silent on the performance of
SD tests. Recently, Tse and Zhang (2004) present Monte Carlo studies to examine the
size and power of some SD tests when the underlying distributions are independent.

However, little is known about the size- and power-performance of SD tests when the
underlying distributions are correlated.

Zheng and Cushing (2001) point out that the conventional inference procedure
usually requires samples to be independently drawn. Nevertheless, most of the
economics or finance issues involve studies of dependent data, for example, the
momentum puzzle discussed by Jegadeesh and Titman (1993, 2001) and Fong, Lean
and Wong (2004) where the portfolio of winners is highly correlated with the
portfolio of losers. Without confirmation of the size and power of the SD tests in the
dependent situation, it will be difficult for academicians and practitioners to apply the
SD tests to most of the empirical finance and economics studies. As the previous
literature that study the performance of different SD tests overlook the dependent
situation, this study attempts to fill the gap to examine the size- and power-
performance of several commonly used SD tests when the underlying distributions are
correlated. Findings of this study will enable the academicians and practitioners to
apply the SD tests efficiently in the dependent situations as required in most of the
empirical issues. Monte Carlo simulations are used to compare the size- and power-
performance of three commonly used SD tests, including DD, Anderson and KRS.
Specifically, this study attempts to determine which SD test is the most appropriate to
be used for empirical research when the underlying distributions are correlated.


9
In addition, the influence of heteroskedasticity on the size and power of SD
tests as well as the issue of choosing the number of grids used for SD tests are
included in this study. Tse and Zhang (2004) show that the DD test has superior size-
and power-performance compared to both Anderson and KRS tests when the
underlying distributions are independent. The simulation results in this study show
that the DD test also has good performance in both the power and size when the
underlying distributions are correlated. In addition, the results show that

heteroskedasticity significantly reduces the power of all SD tests. But, the DD test
still has a good power-performance for large sample.

The choice of the optimal number of grid points is an important issue for SD
tests that compute the test statistics using different grid points. Too few grid points
may not able to reveal more information from the distributions while too many grid
points will violate the independence assumption of the grid statistics (Stoline and Ury
1979). Tse and Zhang (2004) propose to use 10-15 equal-distance major grids to
satisfy the independence assumption for different grids. However, this may result in
missing the detection of SD behavior for the distributions between any two
consecutive major grids as the distance of these grids could be quite big due to the
small number of grid points being used. This study follows the suggestion by Tse and
Zhang (2004) using 10-15 equal-distance major grids. Moreover, it is suggested to use
at least 10 equal-distance minor grids between any two consecutive major grids to
ensure non-omission of important information between the major grids. The critical
values for 100-150 grids which will violate the independence assumption of different
grids cannot be used in this situation. Thus, it is suggested to use the critical values
based on 10-15 major grids for all the major and minor grids to satisfy the

10
independence assumption of grids. The simulation results support the suggestion of
adding minor grid points. Furthermore, the results also show that the selection of the
critical values based on 10-15 grids is appropriate for both major and minor grids.

This chapter is organized as follows. The next section discusses expected
utility theory. Section 3 introduces SD theoretical framework and section 4 describes
some commonly used SD tests to be examined in this study. Section 5 explains the
procedure of Monte Carlo simulations while section 6 exhibits and discusses the
findings. The conclusion is in section 7.

2.2 Expected Utility Theory
Since SD theory is developed on the foundation of expected utility paradigm, it will
be beneficial for us to have some knowledge about expected utility theory beforehand.
Investor's utility function determines combination of the optimum portfolio held by
the investor and how much he will invest. The utility value of a prospect is a real
number that represents the relative desirability of the prospect. If L is a set of
prospects each defined over a fixed set of consequences and R is the set of real
numbers, then u is a function mapping L into R.

Let L
i
be the prospect associated with act A
i
in a decision problem, then the
relative desirability of the act is measured by the utility value of its prospect, u(L
i
).
Following the definition of utility, the best act is the one for which u(L
i
) is a
maximum. This is the most general criterion of choice in decision theory and is called
utility maximization. Understanding of the properties of utility function can enlighten
the process of rational choice and reduce the chances of making bad decisions.

11

2.2.1 Properties of Utility Function
The first property of utility function is non-satiation or more is preferred to less. It
means that the utility of x + 1 dollar is always higher than the utility of x dollars. If
utility increases as wealth increases, then the first derivative of utility, with respect to

wealth, is positive. Thus, the first restriction placed on the utility function is a positive
first derivative denoted as
(
)
0' >WU where
(
)
WU is the utility function.


The second property of utility function is an assumption about investor's taste
for risk. Three assumptions of risk are risk aversion, risk neutrality, and risk seeking.
Risk aversion means that an investor will reject a fair gamble because the disutility of
the loss is greater than the utility of an equivalent gain. Risk aversion implies that the
second derivative of utility with respect to wealth is negative denoted as
()
0"
<
WU .
A function where an additional unit increase is less valuable than the last unit increase
is a function with a negative second derivative. Risk neutrality means that an investor
is indifferent to accept or reject a fair gamble. For the investor to be indifferent
between investing and not investing, the expected utility of investing is same as the
expected utility of not investing. Risk neutrality implies a zero second derivative
denoted as
()
0" =WU
. Risk seeking means that an investor will select a fair gamble.
Risk-seeking investors have utility functions with positive second derivatives denoted
as

()
0" >WU . The expected utility of investment must be higher than the expected
utility of not investing.


12
The third property of utility function is an assumption about how the investor's
preferences change with a change in wealth. If the investor increases the amount
invested in risky assets as wealth increases, then the investor is said to exhibit
decreasing absolute risk aversion. If the investor's investment in risky assets is
unchanged as wealth changes, then the investor is said to exhibit constant absolute
risk aversion. Finally, if the investor invests fewer dollars in risky assets as wealth
increases, then the investor is said to exhibit increasing absolute risk aversion.
()
()
()
WU
WU
WA
'
"−
=
can be used to measure an investor’s absolute risk aversion. The
derivative of A(W) with respect to wealth, A'(W), is an appropriate measure of how
absolute risk aversion behaves with respect to changes in wealth. Relative risk
aversion, R(W), can be used as well to measure the changes in percentage instead of
absolute amount.

2.3 Stochastic Dominance Theoretical Framework
SD theorem has become an important tool in the analysis of choice under uncertainty.

Dominance principles have important role in understanding and solving the portfolio
problem in economics and finance. The SD selection rules for risk averters have been
well studied in the literature, see for example, Markowitz (1952), Tobin (1958),
Fishburn (1964), Hadar and Russell (1969), Hanoch and Levy (1969), Whitmore
(1970), Tesfatsion (1976), Meyer (1977), Wong and Li (1999) and Li and Wong
(1999). There are three basic SD rules as stated in the following definitions.




13
Definition 1:
Suppose that
an asset X has a cumulative distribution function (CDF) F(x), and an
asset Y has a CDF G(y); both with the support [a,b] with a < b. Let
() ()
∫
=
+
x
a
ii
dttFxF
1
, and
() ()
∫
=
+
y

a
ii
dttGyG
1
with a ≤ x, y ≤ b, for i = 1, 2. (2.1)
1)
The asset X stochastically dominates the asset Y at first order, denoted by YX
1
f ,
or
GF
1
f if and only if
(
)
(
)
xGxF
11
≤ for all possible returns, x;
2)
the asset X stochastically dominates the asset Y at second order, denoted by
YX
2
f , or GF
2
f if and only if
(
)
(

)
xGxF
22
≤
for all possible returns, x; and
3) the asset X stochastically dominates the asset Y at third order, denoted by YX
3
f ,
or
GF
3
f
if and only if
(
)
F
µ
≥
(
)
G
µ
and
(
)
(
)
xGxF
33
≤

for all possible returns x
where
∫
==
b
a
xxdFXEF )()()(
µ
and
∫
==
b
a
xxdGYEG )()()(
µ
.

Definition 2:
Let {
s
U : s = 1, 2, 3} denotes a set of utility functions, where

()
()
{
}
siuuU
i
i
s

, ,1,01:
1
=≥−=
+
and
()
()
xu
i
is the i
th
derivative of u(x). The extended sets of utility functions are:
{}
{}
{}
.convex is ':
,concave and increasing is :
,increasing is :
23
2
1
uUuU
uuU
uuU
EE
E
E
∈=
=
=

The following theorem is the main result describes the basic relation between
utility functions and distribution functions.

14
Theorem 1 (Li and Wong 1999):
Let X and Y be the random variables with CDF, F
and G respectively. Suppose u is a utility function. For s = 1, 2 and 3,
YX
s
f , if and
only if
() ()
GuFu ≥
for any u in U such that
E
ss
UUU ⊆⊆
where

( ) () ()
., , GFHxdHxuHu
b
a
==
∫


One may refer to Hanoch and Levy (1969), Rothschild and Stiglitz (1970),

Hadar and Russell (1969, 1971), Bawa (1975), Tesfatsion (1976), Meyer (1977),
Fishburn and Vickson (1978) and Li and Wong (1999) for the proof and the detail
explanation of the theorem.

SD test enables the application of SD within the theoretical framework
circumscribed by the definitions and theorem laid down above. Application of SD in
the comparison of different assets is important, because the above theorem shows that
SD is equivalent to the choice of assets by utility maximization. SD also has a wide
range of application in economics and finance, such as the study of market efficiency
discussed in Liao and Chou (1995), Post (2003), and Post and Vliet (2004), the event
study discussed in Larsen and Resnick (1999), the study of mutual fund performance
by Kjetsaa and Kieff (2003), and the study of momentum strategy performance by
Post and Levy (2004) and Fong, Lean and Wong (2004).

2.4 Various Stochastic Dominance Tests
Varian (1983) and Green and Srivastava (1986) are the earliest work to establish
statistical tests for the hypothesis of utility maximization when the distributions of
returns are known. McFadden (1989) introduces a statistical test on empirical

15
distributions using SD statistical methodology by assuming that the paired
observations are drawn from two independent distributions with equal sample sizes,
and that the observations are independent over time. The hypothesis is set up as X
dominates Y against the alternative hypothesis that X does not dominate Y. McFadden
(1989) points out that even if the variables are not statistically independent, SD is a
well-defined relationship between the marginal distributions. However, he indicates
that the distribution of the testing statistic for SD depends on the joint distribution of
paired observations, and that the use of this test would be inappropriate in application
when the underlying distributions are dependent.

Thereafter, different SD tests are introduced in the literature and they will be
discussed in the following sub-sections with the assumption that the SD is studied
between two dependent random variables, X and Y, where
F
and G are their
corresponding CDF.

KMM extend McFadden test by relaxing the assumption of independent
underlying distributions. This extension is important to studies in the area of finance
because most financial variables are dependent. The first- and second-order KMM test
statistics are, respectively,
()
(
)
[]
(
)
(
)
[
]
(
)
,max,maxmin
1111
*
xFxGxGxFd
xx
−−= and
()

(
)
[]
(
)
(
)
[
]
(
)
.max,maxmin
2222
*
xFxGxGxFs
xx
−−=

Under the null hypothesis that X stochastically dominates Y at first (second) order,
(
)
**
sd
should not be greater than 0. KMM show that the test statistics converge in
distribution to a maximum of a Gaussian process with a covariance function of
ρ,

16
where the maximum is taken over a set of domain which makes the two distributions
equivalent.

However, Shin (1994) points out that the KMM test is not robust to tails. He
claims that the KMM test may produce result accepted in the first-order null
hypothesis but rejected in the second-order null hypothesis, especially when the min-
max of CDF difference (or integrated CDF difference) is attained in the left tail of
distribution. This is because a few negative extreme observations may cause the min-
max of the CDF difference small and the min-max of the integrated CDF difference
large. Recently, Linton, Maasoumi and Whang (2003) find that sub-sampling
bootstrap technique is better than a traditional bootstrap method when computing the
critical values of the KMM test.

On the other hand, Barrett and Donald (2003) extend McFadden test by
developing a Kolmogorov-Smirnov (KS) test applied to two independent samples
with possibly unequal sample sizes. They state the hypotheses as
() ()
[
]
() ()
[]
.,0 somefor :
,,0 allfor :
1
0
zzzGzFH
zzzGzFH
ss
s
ss
s
∈>

∈≤

This hypothesis is formed similar as in McFadden (1989). The KS test statistic is
() ()
[]
zGzF
N
N
K
ss
z
s
ˆ
ˆ
sup
2
ˆ
21
2
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
,

where
()
()
() ()
()
()
∑∑
=
−
+
=
−
+
−
−
=−
−
=
N
i
s
is
N
i
s
is
yz
sN
zGxz
sN

zF
1
1
1
1
!1
1
ˆ
and,
!1
1
ˆ
;
N is the sample size, s = 1, 2, 3 and
(
)
zz ,0max
=
+
.


17
They point out that tests based on comparisons at a fixed number of grid
points are potentially inconsistent, since only a subset of the restrictions implied by
SD are considered. Nevertheless, Barrett and Donald (2003) claim that their test is
consistent as it compares the objects at all points. However, due to the computation
complexity of these methods, it will not be included in the Monte Carlo experiment
here.

In the rest of this chapter, the size and power of the null hypothesis,
,:
0 ss
GFH =

will be investigated, against any of the following alternatives,
, :
,:
,,but , :
2
1
FGH
GFH
FGGFGFH
sA
sA
ssssA
f
f
ff
//
≠

where
s = 1, 2 or 3.
1
As some SD tests focus on
1A
H or
2A

H only, their corresponding
nulls are set as
. :
and , :
0
2
0
1
FGH
GFH
sA
sA
f
f
/
/


2.4.1 Kaur, Rao and Singh (1994)
The KRS test is a two-sample test comparing
(
)
xF
2
and
(
)
xG
2
to test for second-

order SD under the hypotheses below:
.::
,::
222
0
2
212
0
1
FGHversusFGH
GFHversusGFH
AA
AA
ff
ff
/
/




1
s can be set as any integer but it only be discussed up to 3 in this study.

18
Let
X
1
, X
2

, …, X
n
be a random sample from the distribution and
()
xF is the
empirical CDF based on this random sample. Let
X
(i)
, i = 1, 2, …, n, denote the i
th
-
order statistic of the sample. Let
Y
1
, Y
2
, …, Y
n
be a random sample from another
distribution and
()
xG is the empirical CDF based on this random sample. Let Y
(j)
, j =
1, 2, …, n,
denote the j
th
-order statistic of the sample. It is assuming that
()
xF and

()
xG are continuous and have finite second moments.

The test statistics are
{
}
)(inf
0
2
xZT
n
bx
M
≤≤
=
and
{
}
)(inf
0
2
*
xZT
n
bx
M
−=
≤≤
,
where

()
()
(
)
() ()
xS
n
xS
n
xGxF
xZ
GnFn
n
2
,
2
,
22
11
ˆ
ˆ
+
−
= , and
() () ()
[]
.,;,,
1
2
1

2
2
,
GFHyzhxHhx
n
xS
n
i
iHn
==−−=
∑
=
+

KRS define
()
0=xZ
n
when x is less than the minimum observation of the combined
sample. The null hypothesis,
0
1A
H will not be rejected if
2
*M
T <
α
Z where
α
Z is (1-α)

th

percentile of the standard normal distribution.

The KRS test is consistent and has an upper bound
α on the asymptotic size. It
also allows unequal sample sizes from two populations. It is evaluated over the full
support of the population rather than at certain grid points. However, the weakness of
KRS test is that it can only be used to test second-order SD, because the extension of
KRS algorithm to
s = 3 is intractable.

19
2.4.2 Anderson (1996)
The Anderson test applies trapezoidal rule to approximate the required integrals to
estimate
()
xH
i
in (2.1) with
ii
FH
=
or
i
G . He proposes a nonparametric test
2
for the
first three orders of SD with omnibus test for the differences in two distributions. It
provides simple tests of all three forms of SD of interest. Suppose the range of a

random variable
Y is partitioned into k mutually exclusive categories. Let
ij
p be the
probability of finding an observation in the
i
th
category for the j
th
population, j = X, Y.
Assume that x and y are the empirical frequency vectors with size
X
n
and
Y
n
drawn
from populations
X and Y respectively. Thus,
()
),0(~ Ω−= mN
n
y
n
x
iv
a
YX
,
where

YX
YX
nn
nnn
m
/)( +
= ,
).1(~)('and
21
−Ω
−
kvmv
a
χ


One may refer to the Monte Carlo study in Appendix 1 of Anderson (1994) for
the degree of this statistics behaves according to its asymptotic distribution in small
sample. Essentially the results suggest that little of the size and power characteristics
are lost when the underlying parameters remain unspecified.
)1(~)('
21
−Ω
−
kvmv
a
χ
is
easily established as the likelihood ratio test for the difference in two distributions are
based upon the multinomial distribution of x (Anderson 1995).

2
Note that the test assumes independent sampling.

20
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
+++
++
+

=
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
=
k
Ff
ddddddd
ddddd
ddd
d
II

0 0
0 00
5.0and
1 111


11.
0 011
0 001
Define
433221
33221
221
1
,
where d
j
is the length of the j
th
interval. The Anderson test is used to examine the
following hypotheses:
()
. :
,:
,0:
, 0)(:
:

12
11
0
XYH
YXH
ppIH
ppIH
SDorderFirst
A
A
YXfA
YXf
f
f
≠−
=−
−

.:
,:
, 0)(:
,0)(:
:
22
21
0
XYH
YXH
ppIIH
ppIIH

SDorderSecond
A
A
YXfFA
YXfF
f
f
≠−
=−
−


.:
,:
,0)(:
,0)(:
:
32
31
0
XYH
YXH
ppIIIH
ppIIIH
SDorderThird
A
A
YXfFFA
YXfFF
f

f
≠−
=−
−

The test statistic is
()
1,0~
),(
)(
'*
N
iiHmH
ivH
A
a
ss
s
s
i
Ω
=
,
fFfs
IIIH ,where =

and
fFF
III which represent first-, second- and third-order SD tests respectively. As
⎟

⎟
⎠
⎞
⎜
⎜
⎝
⎛
Ω
XX
n
pN
n
x
,~ and
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
Ω
YY
n
pN
n
y
,~,
*

Ω
is the estimate of Ω such that

21
⎟
⎟
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎜
⎜
⎝
⎛
−−−
−−−
−−−
=Ω
)1(


)1(
)1(
21

22212
12111
kkkk
k
k
pppppp
pppppp
pppppp
n
,
with
p is estimated by
YX
nn
yx
p
+
+
=
*
. Proof of consistent estimation is available in
the literature (see, for example, Rao 1973 and White 1984). However, Barrett and
Donald (2003) claim that the Anderson test may produce biased and inconsistent
estimator as it could easily lead one not to reject a false
H
0
.

2.4.3 Davidson and Duclos (2000)
DD derive the asymptotic sampling distribution of various estimators - frequently

used to order distributions with application in poverty and inequality - to test any
order of SD for any two random variables,
X and Y with CDF
F
and G respectively
based on conditional moments of distributions, in which the size is well controlled by
studentized maximum modulus (SMM).
3
Difference hypotheses are defined with
respect to the finite set of value
i
z :
.:
,:
,,but , somefor , )()(:
, allfor , )()(:
2
1
0
XYH
YXH
XYYXzzDzDH
zzDzDH
sA
sA
ssii
s
Yi
s
XA

ii
s
Yi
s
X
f
f
ff
//
≠
=


Let ),()(
1
1
zFzD
X
= and
(
)
(
)
zGzD
Y 1
1
= , then
,2,,,)()(
1
≥==

∫
∞−
−
sYXHduuDzD
z
s
H
s
H
or


3
The theorem and proof of DD test are in the Appendix A.

22
.,,)()(
)!1(
1
)(
1
YXHudHuz
s
zD
z
i
ss
H
=−
−

=
∫
∞−
−

Y is stochastically dominates X at order s if )()( zDzD
s
Y
s
X
≥ for all z ≥ 0. Suppose
there are
N independent paired observations,
i
x and
i
y from the same population. The
DD test statistic is:
)(
ˆ
)(
ˆ
)(
ˆ
)(
zV
zDzD
zT
s
s

Y
s
X
s
−
=
,
where
,)(
)!1(
1
)(
ˆ
,)(
)!1(
1
)(
ˆ
1
1
1
1
−
+
=
=
−
+
∑
∑

−
−
=
−
−
=
s
N
i
i
s
Y
N
i
s
i
s
X
yz
sN
zD
xz
sN
zD

()
).(
ˆ
2)(
ˆ

)(
ˆ
)(
ˆ
,)(
ˆ
)(
ˆ
)(
))!1((
11
)(
ˆ
,)(
ˆ
)(
))!1((
11
)(
ˆ
,)(
ˆ
)(
))!1((
11
)(
ˆ
,
1
1

1
2
,
1
2)1(2
2
2
1
)1(2
2
zVzVzVzV
zDzDyzxz
sN
N
zV
zDyz
sN
N
zV
zDxz
sN
N
zV
s
YX
s
Y
s
X
s

N
i
s
Y
s
X
s
i
s
i
s
YX
N
i
s
Y
s
i
s
Y
s
X
N
i
s
i
s
X
−+=
⎥

⎦
⎤
⎢
⎣
⎡
−−−
−
=
⎥
⎦
⎤
⎢
⎣
⎡
−−
−
=
⎥
⎦
⎤
⎢
⎣
⎡
−−
−
=
∑
∑
∑
=

−
+
−
+
=
−
+
=
−
+


To test for SD, the distributions should be examined for the full support but
this is empirically impossible. Thus, H
0
is tested on a selected finite number of values
of z in the distribution. Bishop, Formby and Thistle (1992) suggest union-intersection
test for multiple comparisons. One advantage of the multiple-comparison approach is
the ease in dealing with dependent samples (Barrett and Donald 2003). Significance is
determined asymptotically by the critical values of the SMM distribution with k and ∞
degree of freedom. The following decision rules are adopted:

23
.accept somefor )( and somefor )(
.accept somefor )( and allfor )(
.accept somefor )( and allfor )(
.accept , ,1for)(
,,
2,,
1,,

0,
A
k
i
sk
i
s
A
k
i
sk
i
s
A
k
i
sk
i
s
k
i
s
HiMzTiMzTIf
HiMzTiMzTIf
HiMzTiMzTIf
HkiMzTIf
αα
αα
αα
α

∞∞
∞∞
∞∞
∞
>−>
><−
>−<
=<

These decision rules show that the null hypothesis is accepted at α% level if the
absolute value of DD statistics at every grid point is less than the critical value of
SMM. If the DD statistics at every grid point is less than the critical value of SMM
and the negative value of DD statistics at some grid points is larger than the critical
value of SMM, then we will accept the alternative hypothesis A1. The same approach
applies to the other two alternative hypotheses. For the simplicity of simulation,
s
i
A
can take the role of
s
T
and the decision rules above can be applied to
s
i
A as well.

2.5 Monte Carlo Simulation
Monte Carlo simulation is employed to study the size and power of the DD, Anderson
and KRS tests. The experiment is carried out according to a linear return-generating
process

4
which is expressed as
),1,0(~,ln
1,
,
Nrer
p
p
r
iid
ptjtptjj
tj
tj
jt
++=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
−
βα
(2.2)
where
tj
p

,
is the price of stock portfolio j at time t,
pt
r
is a market return from an
index at time t, for example, the Dow Jones Industrial Index and
jt
e is the error term.
Without loss of generality, it is assumed that
)1,0(~ Nr
iid
pt
, and
(
)
.0, =
ptjt
recorr This
is a simple but well-known single-factor market model. Two series of returns,
i
r and
j
r
which are generated from the return generating process can be written as:


4
Chow (2001) uses similar process for his simulation.

24

).1,0(~,
),1,0(~,
Nrerr
Nrerr
iid
pjpjjj
iid
pipiii
++=
++=
βα
βα


This study examines the size and power of SD tests under both
homoskedasticity and heteroskedasticity situations, in which:
)1,0(~ Ne
iid
jt
, or
γ
σ
tjt
e
=
, (2.3)
where
,ln
2
1

ln),1,0(~
2
1
2
τσθσγ
+=
−tt
N and
)1,0(~ N
τ
. Specifically, the following
situations for the linear return-generating process are considered:
a)

i
e and
j
e are independent, and
b)

i
e and
j
e are dependent,
such that
,1
2
ρρ
−+= aee
ij

),1,0(~ Na
and
(
)
ji
eecorr ,
=
ρ
, where both
i
e and
j
e

can be either homoskedastic or heteroskedastic. In both situations, the returns,
i
r and
j
r are generated from the return generating process (2.2). They are dependent when
both
i
β
and
j
β
are not zero and/or
i
e and
j

e are dependent. They are independent if
both
i
β
and
j
β
are zero and
i
e and
j
e are independent. Since the independent case
with homoskedasticity process has been studied in Tse and Zhang (2004), this study
only report the situations in which the returns,
i
r
and
j
r
are dependent.
5


It is well-known that (i) if
i
α
>
j
α
and

i
β
=
j
β
, then
ji
rr
1
f ; and (ii) if
i
α
=
j
α
and
i
β
<
j
β
, then
ji
rr
2
f
(Li and Wong 1999). In this study, SD between the
series with different values of α and β are tested to cover the first two orders of SD.

5
As the simulation results for the independent situation with heteroskedasticity processes are similar
with the dependent situation, they are not reported in this chapter.

25
Figures 2.1 and 2.2 depict the CDFs for different values of α and β and their
corresponding second- and third-order integrals as defined in (2.1).



26





27







28
Figure 2.1a to Figure 2.1c depict the CDFs and their corresponding second-
and third-order integrals with
i
α
= 0.2,

j
α
= -0.2 and
i
β
=
j
β

= 1.0. These figures
show that
i
r stochastically dominates
j
r
at all the first three orders. On the other hand,
when
i
α
=
j
α
= 0.0,
i
β
= 0.0 and
j
β

= 2.0, Figure 2.2a shows no first-order SD

between the two series. But, Figure 2.2b and Figure 2.2c show that
i
r stochastically
dominates
j
r at second and third orders respectively. The greater the difference
between
i
α
and
j
α
(
i
β
and
j
β
) is, the clearer the domination between two series. It
is noted that the values of α and β are chosen from the range of the estimated means
and the estimated standard deviations of all the industrial indices from the US market,
so that all
i
r
and
j
r
being studied in this chapter represent the returns of some
empirical portfolios. In addition, this study covers both homoskedasticity and
heteroskedasticity processes. Hence, it can be said that these simulations cover most,

if not all, of the empirical situations.

Since the results of independent and dependent situations are similar, this
chapter only reports the results obtained when
i
e and
j
e are dependent. The value of
ρ used in this simulation is 0.8, which is the empirical correlation coefficient between
portfolios of winners and losers in the US stock market during the period from 1965
to 2000. Other values of ρ also be tested and it is found that the results of simulation
are not sensitive to varying values of ρ.

There is no theoretical definitive solution for determining the optimal number
of grid points to maximize the power of a SD test. The number of grid points used in
empirical studies is often decided by the rule of thumb. Zheng, Formby, Smith and

29
Chow (2000) recommend 10 to 15 ordinates in empirical applications for their
normalized SD test. More information on the distributions would be revealed by the
use of more ordinates. But, as pointed out by Sidak (1967) and Hochberg (1974), the
SMM test is conservative if these statistics are not independent. Besides, the
independence assumption of the grid statistics would be violated by the presence of
excessive grid points (Stoline and Ury 1979). On the other hand, too sparse a number
in grid points could miss out the detection of SD behavior for the distributions
between any two consecutive grids as the distance of any two consecutive grids could
be quite big due to the small number of grids being used.

Ideally, we should choose sufficient large number of grids to reveal more
information without compromising the grid point’s independence property. In view of

this, Tse and Zhang’s (2004) recommendation of making k major intervals where k =
6, 10 and 15 is used in this study. In addition, it is suggested that each major interval
is then be partitioned into 10 (or more) equal-distance minor intervals to ensure non-
omission of important information between any consecutive major grid points. In the
simulations, the two generated series are combined and sorted in ascending order.
This combined sample is used to determine the grid points. The generated series are
divided into k major intervals and each major interval is then partitioned to ten minor
intervals. Nevertheless, the critical values for 10k grids cannot be used as this will
violate the independence assumption of different grids. Thus, it is suggested to use the
critical values based on only k major grids for all the major and minor grids to satisfy
the independence assumption of grids. The critical values of the SMM statistics can
be found in Stoline and Ury (1979).


30
2.6 Findings and Discussion
Each experiment uses 10,000 Monte Carlo runs with four sample sizes of 50, 100, 500
and 1000 respectively. First, the size of each SD test when two distributions are
identical, i.e. when
i
α
=
j
α
= 0.0 and
i
β
=
j
β

= 1.0 has been examined. Then, the
power of these tests with different
i
α
and
j
α
and/or different
i
β
and
j
β
also be tested.
Monte Carlo simulations are conducted for returns generated from either the
homoskedasticity process or heteroskedasticity process in (2.3) and the results are
discussed in the following sub-sections.

2.6.1 Homoskedasticity Process
Table 2.1 reports the empirical sizes of the DD, Anderson and KRS tests under the
null hypothesis,
0
H
, for the homoskedasticity process.

Table 2.1: Empirical Sizes of the DD, Anderson and
KRS Tests for Homoskedasticity Process

Parameters of Distributions:
i

α
= 0.0,
j
α
= 0.0,
i
β
= 1.0,
j
β

= 1.0
s = 1 s = 2 s = 3
N k DD A KRS DD A KRS DD A KRS
6 0.0260 0.0002 0.0155 0.0000 0.0078 0.0000
10 0.0111 0.0021 0.0075 0.0000 0.0034 0.0000
50
15 0.0036 0.0023
0.0000
0.0039 0.0000
0.0000
0.0014 0.0000
0.0000
6 0.0778 0.0001 0.0138 0.0000 0.0071 0.0000
10 0.0221 0.0000 0.0059 0.0000 0.0026 0.0000
100
15 0.0129 0.0000
0.0000
0.0024 0.0000
0.0007

0.0010 0.0000
0.0012
6 0.0883 0.0001 0.0177 0.0000 0.0105 0.0000
10 0.0425 0.0000 0.0072 0.0000 0.0031 0.0000
500
15 0.0216 0.0000
0.0000
0.0020 0.0000
0.0004
0.0010 0.0000
0.0048
6 0.1021 0.0003 0.0206 0.0000 0.0099 0.0000
10 0.0474 0.0000 0.0061 0.0000 0.0049 0.0000
1000
15 0.0255 0.0000
0.0000
0.0037 0.0000
0.0017
0.0009 0.0000
0.0059
Average 0.0401 0.0004 0.0000 0.0089 0.0000 0.0007 0.0045 0.0000 0.0030
Notes: The figures are the empirical sizes of the DD, Anderson (A) and KRS tests.
N is the sample sizes, k is the number of major grid points; s is the order of SD.


31

The empirical size is estimated by calculating the percentage of rejections of
the null hypothesis (i.e. the simulated statistic is greater than the critical value
6

) when
the null hypothesis is true. The simulation results show that all the three SD tests are
conservative as their empirical sizes are less than 0.05 except the case for the DD test
with s = 1, k = 6 and N = 100, 500 and 1000. From the table, it is noted that the
empirical sizes of the DD test are closer to the critical values, while the empirical
sizes of both the KRS and Anderson tests are close to zero. These results are
consistent with the results in Tse and Zhang (2004) for independent distributions.

In addition, the size of the DD test decreases as k increases from 6 to 15, and
the size of DD test decreases from first-order to third-order SD (except for sample
size 50 at k = 15). Overall, the DD test’s biggest size is 0.1021 and the smallest size is
0.0009 in this simulation. The empirical size of the DD test is closest to the nominal
size of 5% with N = 1000, k = 10 and s = 1. On the other hand, the effect of sample
size on the size-performance of DD test is inconsistent. For example, size increases as
samples become larger for the first-order SD. However, size decreases and then
increases as samples become larger for the second-order SD. For the KRS test, its
biggest size is in sample size of 1000 and its smallest size is in sample size of 50. In
contrast to the DD test, the sizes of KRS test increase from first- to third-order SD.
With the exception of case of s = 1 and N = 50 which sees the sizes of the Anderson
test increase as k increases, all other sizes of Anderson test in the simulations are
either zero or very close to zero, and therefore, no further comparison can be made.



6
The DD test follows the SMM distribution while both the Anderson and KRS tests follow the
standardized normal distribution. At the nominal size of 5%, the SMM critical values of k for 6, 10 and
15 are 2.928, 3.254 and 3.487 respectively while the Z
0.025
is 1.96 for KRS tests.