Bauer, Larry L.

Epperson, James E.

Kaiser, Eddie H.

Wells, Gary J.

English

Research Papers in Economics

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~~R1QJLTURAL  ECONOMICS AND RURAL SOCIOLOGY..t  --- -z-...-or :p!'M'SON' 
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working paper 
GIANNINI  FOUNDAnON  Olt" 
AGRlCUL~~ 
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S.~~6  1984  - '-' 
WP082783 
A  SUGGESTED  CRITERION  FOR  SELECTING 
BETWEEN  E-V  AND  STOCHASTIC 
DOMINANCE  ANALYSIS  TOOLS 
by 
Gary  J.  Wells,  Larry  L.  Bauer, 
Eddie Kaiser,  and James  Epperson* 
clemson university 
south carolina agncultural experiment station WP082783  August  1982 
A  SUGGESTED  CRITERION  FOR  SELECTING 
BETWEEN  E-V  AND  STOCHASTIC 
DOMINANCE  ANALYSIS  TOOLS 
by 
Gary  J.  Wells,  Larry L.  Bauer, 
Eddie Kaiser,  and James  Epperson* 
*Associate Professor,  Professor,  Assistant Professor,  Clemson 
University  and Associate Professor,  University of  Georgia,  respec-
tively. 
Working Papers are not  subject  to  formal  review within  the Depart-
ment  of Agricultural Economics  and  Rural  Sociology. A SUGGESTED  CRITERION  FOR  SELECTING  BETWEEN  E-V 
AND  STOCHASTIC  DOMINANCE  ANALYSIS  TOOLS 
Recent  works  by Kramer  and  Pope,  and  Musser et al.,  compared  and 
contrasted traditional mean-variance  (E-V)  analysis  and  stochastic  domi-
nance  as  techniques  of assessing  risk preferences.  Both techniques  have 
strengths  and  weaknesses.  It seems  appropriate  to  establish a  criterion 
of choosing between  the  techniques  so  as  to  capitalize  on  the  strengths. 
The  objective of this note  is to  suggest  such  a  criterion. 
The  characteristics of E-V  analysis  and  stochastic  dominance  are 
well  documented  and  detailed  coverage will not be presented here.  Both 
techniques  are  aimed at eliminating inefficient choices  from  the  set of 
all possible  choices  available to  the  decision maker  (Hanoch  and  Levy, 
p.  335).  Assumptions  about the  risk averseness  of the decision maker 
provide  the  framework  for  these  techniques  to  decide  between efficient 
and  inefficient choices.  Additionally,  E-V  analysis  assumes  that only 
the first two  moments  of  a  probability distribution are  important in the 
decision of  choice.  This  assumption  implies  a  normal probability dis-
tribution unless  the decision maker's utility function exhibits  increas-
ing risk aversion as  wealth increases  (Feldstein,  p.  6;  Tobin,  p.  13). 
Therefore,  if the alternatives  are normally distributed,  E-V  analysis 
provides  a  powerful tool,  especially if the  analysis  involves  a  continu-
ous  choice set. 
Stochastic dominance  is best suited for  discrete  choice  efficiency 
analysis,  but if a  continuous  choice efficiency is being made,  it must 
consider  an infinite number  of choice alternatives  (Kramer  and  Pope). Stochastic  dominance  has  the ability to  consider all moments  of the 
distribution,  therefore,  it is able  to  cope  with  skewness  in particular. 
The  inability of E-V  analysis  to  do  this  has  left it open to  criticism. 
Until  a  technique  is developed  as  flexible  as  E-V  analysis  however,  the 
researcher needs  to  be  cautious  in discarding it for  risk analysis. 
The  key  weakness  of E-V  analysis  hinges  on its use  of only the 
first two  moments  of the distribution of alternatives.  It,  therefore, 
would  seem  appropriate  to test the  distribution for  evidence  of non-nor-
mality.1  An  appropriate  small  sample test is  the  Shapiro-Wilk W-statis-
tic which  has  been  shown  to provide  an effective test for normality in 
sample  sizes  as  small  as  twenty  (Shapiro  and  Wilk).  This  is particu-
larly relevant  since  sample  sizes  are  often relatively small  in situ-
ations  where  E-V  or stochastic  dominance  may  be  used.  If larger samples 
are available  (e.g.,  larger than 50),  then a  test such  as  the  Kolmogo~ 
rov-Smirnov D-statistic can be  used  to test for normality.  (Both sta-
tis  tics are  available in the Statistical Analysis  System.) 
Use  of a  test for  normality would  provide  a  decision criterion when 
economic  considerations  failed  to  suggest characteristics of the distri-
bution of alternatives.  Kramer  and  Pope  reason that,  "returns both in 
and  out of the  farms  program are not normally distributed  (or  symmetric) 
because  the program  serves  to  reduce  the probability of  low  income 
events  (p.  120)."  The  implication is that  farm  commodity programs 
result in a  distribution skewed  toward  higher  incomes  since the  lower 
incomes  are effectively "chopped off."  Such  economic  reasoning would, ' 
of  course,  have  to  take precedence  over the statistical test for  normal-
ity.2 Musser et al.,  on  the other hand,  did not have  any  ~ priori eco-
nomic  reasoning  to  expect particular  characteristir~  ~n the 
" 
2 distributions  they were  considering.  They  were  investigating an 
integrated pest management  production  system.  Because  of  the  lack of  a 
priori guidance  they  chose  to  contrast results  from  E-V  and  stochastic 
dominance  analysis.  This  is  a  situation where the! priori use  of a 
test of normality would  have  provided useful  in selecting a  particular 
analytical tool. 
Table  1  contains  the  Shapiro-Wilk test results  for  the Musser et 
al.,  IPM  study.  Of  the  four  management  levels  included,  the Shapiro-
Wilk test indicates that number  three is not  normally distributed and 
is,  in fact,  positively skewed.  This  conclusion lends  support to  the 
concern  expressed by Musser et al.  relative to the  skewness  of manage-
ment  level three  (p.  122-123).  They note  that "a  favorable  skewness  is 
not  sufficient for  stochastic  dominance  to  conflict with E-V  analysis--
-the desirable positive  skewness  of level three  was  not sufficient to 
overcome  its unfavorable mean  and variance in reference  to  four  (pp. 
122-123)." 
Table  1.  Shapiro-Wilk  "w"  Statistic Test for Normality of 
Georgia  IPM  Study. 
Management 
Level 
I 
II 
III 
IV 
"w"  Statistic for 
Detrended Data 
0.960 
0.940 
0.972 
Net Profit 
Probability of 
Smaller Value 
0.41 
0.16 
0.01 
0.63 
a.  Indicates  a  possible  departure  from  normality at the  99  percent 
level of confidence. 
3 Another  aspect of stochastic dominance  versus  E-V  analysis  is also 
graphically presented  in the Musser  et al.  study.  They  state: 
In an E-V  framework,  level  two,  which  is the most  conven-
tional pesticide treatment method,  and  level  four,  which  has  the 
highest  level of  IPM,  are both efficient.  More  risk-averse pro-
ducers  could be  hypothesized to  likely use  level  two,  while  less 
risk-averse producers  would  be  hypothesized to adopt  level four 
(p.  123). 
Stochastic  dominance,  on  the  other hand,  concluded  that only man-
agement  level  four  was  efficient.  The  reason Musser  et al.  concluded 
that management  level  two  would  be  selected by  risk-averse people  is 
because it has  a  smaller variance  and,  of  course,  a  smaller mean  than 
level  four.  Porter and  Gaumnitz  empirically found  that low  variance, 
low  return choices  are more  often included in the efficient set with E-V 
analysis  than with stochastic  dominance  analysis.3  Stochastic dominance 
is  a  more  powerful  analytical  tool  in that it is based  on  both  a  neces-
sary and  sufficient condition and  thereby not functional  form  dependent, 
and  does  not  incorrectly include  low  variance,  low  return options  in the 
efficient set as  E-V  sometimes  does.  Thus,  with E-V  analysis  the more 
risk-averse producer would  be  led astray.  In such cases,  the Shapiro-
Wilk test would  indicate to  the  scientist E-V  analysis  is probably not 
appropriate.  Porter and  Gaumnitz  warn  that E-V  analysis  may  not yield 
accurate  decisions  for highly risk-averse decision makers.  They  also 
conclude  that the  less  risk-avers~ decision makers  will be  more  indiffe-
rent between  the use  of E-V  and  stochastic dominance  analysis because  in 
higher variance,  higher  return ranges  these  techniques  do  not systemati-
cally vary in their choice  of an efficient set. 
With  the availability of these  research tools  and  the powerful 
small  sample  tests  for normality,  it would  seem  that scientists do, 
indeed,  have  the  capability to  choose  the  technique best able  to  analyze 
4 a  distribution of possible  outcomes.  E-V  analysis  has  an  advantage  over 
stochastic dominance  when  considering  a  continuous  choice  set.  There-
fore,  if the  Shapiro  Wilk  test does  not  indicate  a  departure  from  nor-
mality,  choice  of E-V  analysis  may  be  advantageous,  especially since it 
is more  widely understood  and  research results  may  be  expected to 
receive wider application. 
5 Footnotes 
1.  There are,  of  course,  distributions  other than the  normal  that 
are  fully described by the first two  moments.  The  normal,  however,  has 
the  advantage  of  combining  two  normal  distributions produces  a  normal 
distribution. 
2.  Kramer  and  Pope  later conclude  that "large  and  small  outcomes 
are  truncated by the program  (p.  125)."  This  would  indicate that even 
though the  distributions  of interest may  not be  normal,  evidence  is not 
available  to  suggest they are  not  symmetric.  In fact,  truncation of the 
tails  could be  of little consequence.  A Shapiro-Wilk test for  normality 
of the probability distributions  used  in the Kramer  and  Pope  work  failed 
to  indicate a  departure  from  normality in any  of the  alternative  commod-
ity program participation levels  they considered. 
3.  Porter and  Gaumnitz  were  comparing  E-V  analysis with  second 
degree  stochastic  dominance  (SSD)  for  stock portfolios.  SSD  includes 
beyond  the  assumption more  is preferred to less that decision makers  are 
risk-averse.  See  Hadar  and Russell  for  a  discussion of SSD. 
6 7 
References 
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Hanoch,  G.  and  H.  Levy.  "The  Efficiency Analysis  of Choices  Involving 
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Musser,  Wesley  W.  et al.  "An  Economic  Examination of  an  Integrated Pest 
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36(1969): 13-14 SOUTH  CAROLINA  AGRICULTURAL  EXPERIMENT  STATION, 
CLEMSON  UNIVERSITY, CLEMSON,  SOUTH  CAROLINA 
W.  C.  GODLEY, ASSOCIATE DEAN AND  DIRECTOR 
S.  C. EXPERIMENT STATION 
lUTHER P.  ANDERSON, DEAN, COllEGE OF AGRICULTURAL SCIENCES 
The  South  Carolina  Agricultural  Experiment  Station  is  a  cooperative  program  financed  from  federal 
and  state  funds.  It is  the  policy  of  the  Experiment  Station  to  comply  fully  with  the  regulations  of  Title 
VI,  the  Civil  Rights  Act  of  1964.  Complaints  may  be  filed  with  the  Director,  S.  C.  Agricultural  Experi-
ment  Station,  Clemson  University,  Clemson,  S.  C.  29631. 

An Economic Examination of an Integrated Pest Management Production System with a Contrast Between E-V and Stochastic Dominance Analysis.&quot;

Mean-Variance Analysis in the Theory of Liquidity Preference and Portfolio Selection.&quot;

Participation in Farm Commodity Programs: A Stochastic Dominance Analysis.&quot;

The Efficiency Analysis of Choices Involving Risk.&quot;

Wilko &quot;An Analysis of Variance Test for Normality.&quot;

A Suggested Criterion for Selecting between E-V and Stochastic Dominance Analysis Tools

A Suggested Criterion for Selecting between E-V and Stochastic Dominance Analysis Tools

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