Corridor analysis for detection of significant breakout movements
by Mahadevan Krishnamoorthi
A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in
Mechanical & Industrial Engineering
Montana State University
© Copyright by Mahadevan Krishnamoorthi (1996)
Abstract:
Time series form patterns during their movements and have been continuously analyzed for relevance
to forecasting. A neutral corridor is one of many patterns formed by time series. Neutral corridors,
when broken, result in significant upward or downward movements of time series. Therefore, it is
imperative that a forecaster be able to detect such significant movements. Research was undertaken to
identify a quantitative technique for the detection of such significant breakout movements.
Three techniques were analyzed for possible use. The techniques are cusum control charts,
nonparametric tests for location and trends, and a new empirical method developed by the author.
Neutral corridors were identified from historical data corresponding to market securities. Rules were
formed to define significant breakout movements and each test was performed on all sets of data. A
signal of positive or negative trend (breakout movements) was verified by comparison to the actual plot
of the data. Accuracy of a signal was determined based on the actual direction of movement and the
time of movement.
The cusum control charts and the nonparametric tests gave erratic signals of positive and negative
trends and were not confirmed by the actual plot of the time series. Hence these tests cannot be
recommended for the purpose of detecting significant breakout movements. The new empirical method
illustrated a high accuracy of above 90 percent correct breakout calls, and therefore, is highly
recommended for detection of significant movements away from neutral corridors. CORRIDOR ANALYSIS FOR DETECTION OF
SIGNIFICANT BREAKOUT MOVEMENTS
by
Mahadevan Krishnamoorthi
A thesis submitted in partial fulfillment
o f the requirements for the degree
of
Master o f Science
in
Mechanical & Industrial Engineering
MONTANA STATE UNIVERSITY-BOZEMAN
Bozeman, Montana
May 1996
H
APPROVAL
o f a thesis submitted by
Mahadevan Krishnamoorthi
This thesis has been read by each member o f the thesis committee and has been found to be
satisfactory regarding content, English usage, format, citations, bibliographic style, and
consistency, and is ready for submission to the College o f Graduate Studies.
Dr. Paul Schillings
(Signature)
Date
Approved for the Department o f Mechanical ancf Industrial Engineering
Dr. Victor Cundy
(Signature)
Date
Approved for the College o f Graduate Studies
Dr. Robert Brown
(Signature)
Date
Ill
STATEMENT OF PERMISSION TO USE
In presenting this thesis in partial fulfillment o f the requirements for a master’s
degree at Montana State University - Bozeman, I agree that the Library shall make it
available to borrowers under rules o f the Library. I further agree that copying o f this thesis
is allowable only for scholarly purposes, consistent with “fair-use” as prescribed in the U S.
Copyright Law. Requests for permission for extended quotation from or reproduction o f this
thesis in whole or in parts may be granted only by the copyright holder.
Signature
Date
AYOLtj 2, p-j I 9
iv
TABLE OF CONTENTS
Page
1. IN T R O D U C TIO N ...................................................................................................................I
Corridors ......................................................................................................................... 2
2. LITERATURE R E V IE W ........................................................................................................6
3. CONCEPTS OF STATISTICAL T E S T IN G ....................................................................... 8
Approach to Hypothesis Testing ..................................................................................9
Errors Associated with Hypothesis T e stin g ...............................................................10
4. INVESTIGATION OF CUSUM CONTROL C H A R T S ..................................................12
Process Behavior and Control Charts ....................................................................... 12
Adoption to Forecasting............................................................................................... 13
Cusum C h a rts.................................................................................................................14
Advantages o f Cusum C harts................................................................- . . . . 1 4
Traditional Cusum Charts ..............................................................................15
Standardized Cusum Chart ........................................................................... 20
Construction o f the Standardized Cusum C h art...........................................22
Tests with Cusutn Control Charts on Neutral C o rrid o rs........................... 23
R e s u lts ........................................................................................................................... 23
Conclusions .................................................................................................................. 24
5. INVESTIGATION OF NONPARAMETRIC T E S T S ..................................................... 25
Advantages o f Nonparametric Statistics .............................................
25
Disadvantages o f Nonparametric S ta tistic s.............................................................. 26
Tests for Location ....................................................................................................... 26
Ordinary Sign T e s t.......................................................................................... 27
W ilcoxon Signed Rank Test ......................................................................... 29
Tests Against T r e n d ..................................................................................................... 32
Cox-Stuart Test ...............................................................................................34
Modified Cox-Stuart Test ............................................................................. 35
V
TABLE OF CONTENTS - Continued
Page
Kendall's test for tr e n d ...........................................
37
Results ........................................................
40
C o n clu sio n .....................................................................................................................41
6. THE NEW EMPIRICAL M E T H O D ...................
42
Confirmation using Volume o f Transaction ............................................................ 45
R e s u lts ........................................................................................................................... 46
C o n clu sio n s....................................................
47
7. RESULTS AND DISCUSSION ........................................................................................ 48
Discussion on False s ig n a ls ........................................................................................ 48
Traditional Cusum Charts ............................................................................. 48
Standardized Cusum charts ........................................................................... 49
Ordinary Sign T e s t...........................................................................................50
Wilcoxon Signed Rank Test ......................................................................... 50
Cox-Stuart Test ...............................................................................................50
Modified Cox-Stuart Test ............................................................................. 51
Kendall's T e s t......................................................
51
Results o f the New Empirical M eth o d ................................................................. .... 51
C o n clu sio n s.................................................................................................................. 52
A P P E N D IC E S ........................................................................................................................... 53
Appendix
Appendix
Appendix
Appendix
Appendix
Appendix
Appendix
Appendix
Appendix
Appendix
A - Plots o f Neutral C orridors................................................................. 54
B - Results o f Normality Tests ................................................................ 61
C - Traditional Cusum C h a rts ...................................................................63
D - Standardized Cusum Charts .............................................................. 70
E - Results o f Ordinary Sign T e s t s .......................................................... 77
F - Results o f Wilcoxon Signed Rank T e s ts ......................................... 87
G - Results of Cox-Stuart Tests .............................................................. 97
H - Results o f Modified Cox-Stuart Tests ........................................... 107
I - Results o f Kendall’s Tests .................................................................117
J - Results o f New Empirical Method ..................................................127
TABLE OF CONTENTS - Continued
REFERENCES CITED
vii
LIST OF TABLES
Table
Page
1.
Summary o f Type I and Type II Errors .....................................................................11
2.
Results o f the Ordinary Sign Test ..............................................................................30
3.
Results o f W ilcoxon Signed Rank T e s t....................................
33
4.
Results o f the Cox-Stuart T e s t.............................................
36
5.
Results o f the Modified Cox-Stuart T e s t..................................................
38
6.
Results o f the Kendall's T e s t ........................................................
39
7.
Results o f the New Empirical M eth o d .......................................................................47
8.
Results o f Normality T e sts.......................................................................................... 62
9.
Results o f Ordinary Sign Test on Market Security 2
10.
Results o f Ordinary Sign Test on Market Security 3 ............................................... 79
11.
Results o f Ordinary Sign Test on Market Security 4 .................................................81
12.
Results o f Ordinary Sign Test on Market Security 5 ............................................... 82
13.
Results o f Ordinary Sign Test on Market Security 6
14.
Results o f Ordinary Sign Test on Market Security 7 ............................................... 86
15.
Results o f W ilcoxon Signed Rank Test on Market Security 2 ...............................88
16.
Results o f Wilcoxon Signed Rank Test on Market Security3 ...............................89
............................................. 78
............................................. 83
17. Results o f Wilcoxon Signed Rank Test on Market Security 4 ....................................91
18.
Results o f Wilcoxon Signed Rank Test on Market Security 5 ............................... 92
Vlll
LIST OF TABLES - Continued
Table
Page
19.
Results o f Wilcoxon Signed Rank Test on Market Security 6 .............................. 93
20.
Results o f Wilcoxon Signed Rank Test on Market Security 7 .............................. 96
21.
Results o f Cox-stuart Test on Market Security 2 ..................................................
22.
Results o f Cox-stuart Test on Market Security 3 ......................................................99
23.
Results o f Cox-stuart Test on Market Security 4 ....................................................101
.24.
Results o f Cox-stuart Test on Market Security 5 .................................................... 102
25.
Results o f Cox-stuart Test on Market Security 6 ....................................................103
26.
Results o f Cox-stuart Test on Market Security 7 ......................................... - . . . . 106
27.
Results o f Modified Cox-stuart Test on Market Security 2 ................................... 108
28.
Results o f Modified Cox-stuart Test on Market Security 3 ................................... 109
29.
Results o f Modified Cox-stuart Test on Market Security 4 ................................... I l l
30.
Results o f Modified Cox-stuart Test on MarketSecurity 5 ................................... 112
3 1.
Results o f Modified Cox-stuart Test on Market Security 6 ..............
32.
Results o f Modified Cox-stuart Test on Market Security I ................................... 116
33.
Results o f Kendall's Test on Market Security 2 ......................................................118
34.
Results o f Kendall's Test on Market Security 3 ......................................................119
35.
Results of Kendall's Test on Market Security 4 ......................................................121
98
113
ix
LIST OF TABLES - Continued
Table
Page
36.
Results o f Kendall's Test on Market Security 5 ...................................................... 122
37.
Results o f Kendall's Test on Market Security 6 ...................
38.
Results o f Kendall's Test on Market Security 7 ...................................................... 126
39.
Result o f Test with New Empirical Method on Market Security 2
40.
Result o f Test with New Empirical Method on Market Security 3 ..................... 129
41.
Result o f Test with New Empirical Method on Market Security 4 ..................... 130
42.
Result o f Test with New Empirical Method on Market Security 5 ............. ..
43.
Result o f Test with New Empirical Method on Market Security 6 ..................... 132
44.
Result o f Test with New Empirical Method on Market Security 7 ..................... 133
123
................... 128
13 1
X
LIST OF FIGURES
Figure
Page
1.
Neutral Corridor formed by M arket Security 8 ..................................................
3
2.
Neutral Corridor formed by Market Security I ................................................
18
3.
Traditional Cusum Chart for Market Security I ................................................
19
4.
Standardized Cusum Chart for Market Security I
21
5.
Neutral Corridor formed by M arket Security 2 .........................
55
6.
Neutral Corridor formed by M arket Security 3 ...............................................
56
7.
Neutral Corridor formed by M arket Security 4 ...............................................
57
8.
Neutral Corridor formed by Market Security 5 ...............................................
58
9.
Neutral Corridor formed by M arket Security 6 ...............................................
59
10.
Neutral Corridor formed by M arket Security 7 ...............................................
60
11.
Traditional Cusum Chart for Market Security 2 ...............................................
64
12.
Traditional Cusum Chart for M arket Security 3 ...............................................
65
13.
Traditional Cusum Chart for M arket Security 4 ...............................................
66
14.
Traditional Cusum Chart for M arket Security 5 ...............................................
67
15.
Traditional Cusum Chart for M arketSecurity 6 ...............................................
68
16.
Traditional Cusum Chart for MarketSecurity 7 ...............................................
69
17.
Standardized Cusum Chart for Market Security 2
71
18.
Standardized Cusum Chart for Market Security 3
.........................................
.........................................
72
xi
LIST OF FIGURES - Continued
Figure
Page
19.
Standardized Cusum Chart for Market Security 4
..........................................
73
20.
Standardized Cusum Chart for Market Security 5
..........................................
74
21.
Standardized Cusum Chart for Market Security 6
..........................................
75
22.
Standardized Cusum Chart for M arket Security 7
..........................................
76
x ii
ABSTRACT
Time series form patterns during their movements and have been continuously
analyzed for relevance to forecasting. A neutral corridor is one of many patterns formed
by time series. Neutral corridors, when broken, result in significant upward or downward
movements of time series. Therefore, it is imperative that a forecaster be able to detect
such significant movements. Research was undertaken to identify a quantitative technique
for the detection o f such significant breakout movements.
Three techniques were analyzed for possible use. The techniques are cusum control
charts, nonparametric tests for location and trends, and a new empirical method developed
by the author. Neutral corridors were identified from historical data corresponding to
market securities. Rules were formed to define significant breakout movements and each
test was perform ed on all sets of data. A signal of positive or negative trend (breakout
movements) was verified by comparison to the actual plot of the data. Accuracy of a signal
was determined based on the actual direction of movement and the time of movement.
The cusum control charts and the nonparametric tests gave erratic signals of
positive and negative trends and were not confirmed by the actual plot of the time series.
Hence these tests cannot be recommended for the purpose of detecting significant breakout
movements. The new empirical method illustrated a high accuracy o f above 90 percent
correct breakout calls, and therefore, is highly recommended for detection of significant
movements away from neutral corridors.
I
CHAPTER I
INTRODUCTION
Time series form patterns during their movements. These patterns have been
continuously analyzed for relevance to forecasting. By studying the nature o f previous
turning points, it is possible to develop some characteristics that can help identify
movements o f time series. Studies o f patterns for purposes o f forecasting are based on the
assumption that history repeats itself. The art o f forecasting - fo r it is an art - is to identify
trend changes a t an early stage and to maintain a posture until the weight o f evidence
indicates that the trend has reversed [Pring, 11], It is worthwhile to note that time series
never duplicate their performance exactly, but the recurrence o f similar characteristics is
sufficient to enable forecast analysts to identify important junctures.
Here, data obtained from market securities have been used for analysis. Reasons for
using data from market securities are fourfold:
1.
Ease o f availability
2.
Accurate and timely information
3.
Availability o f real time data, for verification.
4.
Reluctance o f business people to release privy information.
Movements o f market securities can be classified as long (primary), intermediate,
and short term. Primary movements typically work out in a period o f I to 3 years.
Intermediate movements usually develop over a period o f 3 weeks to as many months.
2
sometimes longer. Short term movements last less than 3 weeks. Studies presented in this
paper are primarily confined to identification o f intermediate term movements in the market
securities.
As in a court o f law or when testing a hypothesis, a trend in presumed innocent
(continuing) until proven otherwise. The "evidence" is the objective element in forecasting.
The evidence is derived from the use o f one or more forecasting techniques. Not all o f them
work for all situations. The "art" consists o f combining forecasting techniques into an overall
picture and recognizing the resemblance o f that picture to market patterns.
Corridors
Corridors, are one o f the most important and often analyzed patterns. Corridors could
be classified as neutral, positive or negative corridors depending on the direction of
movement o f the time series within a corridor. A neutral corridor (or sideways trend) is
essentially horizontal or transitional, which usually separates two major market movements.
To the forecaster, the neutral corridor has great significance because it marks the turning
point between major market movements. The phenomenon o f neutral corridors and their
formation is described below. Figure I illustrates a typical neutral corridor formed by a
market security.
Suppose a time series moving in the upward direction reaches the top at point A
(Figure I) and then starts moving down. The point A corresponds to the highest value, the
market security reached within that trading period (usually a day). Now, if the time series
MARKET SEC. 8
11/06/95
LINE 1-1
LINE 2-2
30 N
Figure I.
Neutral Corridor formed bg Market Securitg 8
06
4
reaches point B and reverses its direction o f movement, to go up, a neutral corridor is likely
to be formed. The point B corresponds to the lowest value the market security reached in that
trading period. The formation o f the neutral corridor is confirmed only some time after this
picture, including points A and B, is formed. Now, if the time series, while moving up, does
not go above point A, but changes direction and moves past the center o f the vertical distance
between points A and B, point A is called a "valid top." A line (1-1 in Figure I), is drawn
parallel to the X axis from p o in ts and termed the "upper boundary" [Schillings, 12]. When
the time series once again changes direction from a downward to an upward movement
moving past the center line, we are permitted to draw a line from point B. The second line,
(2-2 in Figure I) is also drawn parallel to the X axis, and is termed the "lower boundary."
The point B is now called a "valid bottom."
Appendix A illustrates more examples o f
neutral corridors formed using the above method.
Neutral corridors sometimes include micro corridors within their boundaries. These
micro corridors may exhibit positive, negative or sideways trends. However, for purposes of
analysis one can assume randomness o f the oscillations within the boundaries o f a neutral
corridor.
Neutral corridors are broken only by significant movements o f the time series. A
significant movement is defined as "a movement outside the boundary in either direction,
culminating at a distance o f more than the width o f the corridor." The culmination o f a
movement is observed when the time series retracts by more than half o f the total distance
moved in the primary direction o f movement. Understanding the concept o f significant
movements is imperative because time series often cross the boundaries o f corridors
5
momentarily and then move back into the corridors. Such moves are quite misleading and
may result in incorrect decisions. Figure 9 shown in Appendix A illustrates temporary
movements out o f a corridor.
Research was undertaken to eliminate subjectivity in detecting significant movements
away from neutral corridors. Here, an effort has been made to formulate a quantitative
technique to consistently identify significant movements. This research may be classified as
part o f the continuing search for the "holy grail" or perfect technique.
Three techniques have been analyzed for possible use to detect significant
movements. The techniques are cusum control charts, nonparametric tests for location and
trends, and an empirical method developed by the author.
6
CHAPTER 2
LITERATURE REVIEW
A neutral corridor is one o f many time series patterns that are often analyzed by
forecasters. Although patterns resembling neutral corridors have been observed and
considered for many years, the author and Schillings [Schillings, 12] found no published
material that analyzed quantitative aspects for the detection o f significant breakout
movements. It is worth mentioning that the patterns studied in this paper were referred to
for the first time, as neutral corridors, by Schillings. Bring [Bring, 11] calls these patterns
transition zones or rectangles.
Currently, there is one technique used for analyzing patterns resembling neutral
corridors which concentrates on the position o f the time series values and corridor lengths.
If a neutral corridor is formed at a market top, then the time series is usually expected to
undergo a reversal and revert to a negative trend [Bring, 11]. Ifth e corridor is formed at the
bottom, then the time series is expected to revert to a positive trend. Analysis based on the
length o f the corridor leads to an estimate o f how far the time series will go up or down after
the breakout.
Bring finds that some practitioners (references not cited) define a movement away
from a rectangle by a three percent penetration o f either boundary as “significant.” This
filters out only some o f the misleading moves, but does not ensure consistency in detection
o f statistically significant breakout movements. A movement o f half the width o f the corridor
7
away from either boundary is also termed “significant" Both rules are only heuristics
followed by some forecasters.
The review of published literature related to corridor analysis, as defined in this paper,
does not indicate the availability of a quantitative technique. Hence, research was undertaken
to formulate a quantitative technique for the detection of statistically significant breakout
movements from time series neutral corridors.
8
CHAPTER 3
CONCEPTS OF STATISTICAL TESTING
Tests, usually called hypothesis tests or significance tests, often play a major part in
statistical investigations. The basic idea o f most statistical techniques is to increase our
knowledge about populations using information in samples taken from them. In statistical
testing, we are concerned with examining the truth, or otherwise, o f hypotheses about some
feature(s) o f one or more populations. A statistical hypothesis is a statement about a
population; for example its form or shape, or some aspect thereof; the numerical value o f one
or more parameters; and so forth [Gibbons, 4]. A hypothesis set consists o f two statements,
called the null hypothesis, H0 and the alternative hypothesis, H 1. These statements must be
mutually exclusive.
Whether or not particular hypotheses are eventually assessed to be reasonable, will
depend on the weight o f evidence contained in the sample data taken from population(s). In
hypotheses testing, we never claim to prove anything completely (beyond doubt) by means
o f statistical test: we simply pronounce a judgement based on the available evidence, and
give an assessment o f the strength o f that evidence [Neave, 9], Sometimes, the evidence may
be so overwhelming that a hypothesis may be regarded as proved (or disproved) “for all
practical purposes.”
9
Approach to Hypothesis Testing
For example, if the null hypothesis is stated as “The populations under consideration
do not differ in persistence”, then an alternative could be stated as “A motivated population
exhibits more persistence for a task than a population with no motivation.” As a
mathematical statement, if p, and P2 are parameters representing average persistence for the
populations with motivation and without motivation, respectively, then the null hypothesis
is P1= P2 and the alternative is p, > P2. The alternative hypothesis here is called one-sided
(or one-tailed or right-tailed), because it states a particular direction o f inequality. When the
alternative is stated as p, < P2, it is called left-tailed and when it is stated as p, not equal to
P2, then it is called two sided.
A decision to accept or reject any hypothesis is made on sample evidence according
to some statistical test procedure and test statistic. The test procedure may also be called
one-sided or two-sided, according to whether the alternative is one-sided or two-sided. One­
sided tests are either left-tailed or right-tailed, depending on the direction o f the alternative
hypothesis, H 1. The test statistic should be consistent with, and appropriate for, the type o f
alternative, the data available for analysis, and the assumptions the investigator is willing to
make about the population. By central limit theorem, the sampling distribution o f the sample
mean (test statistic in this case) is a normal distribution.
Once the test statistic is chosen, its value is calculated for the data obtained. Then the
investigator can use this value and the sampling distribution o f the test statistic to determine
a quantity called the P-value, or the associated probability. The P-value is the probability,
10
when H0 is true, o f obtaining a value o f the test statistic which is equal to or “more extreme”
in the appropriate direction than its critical value [Gibbons, 4],
W hen the investigator wishes to make a statistical decision, whether to reject or
accept H0, the decision can be based on the magnitude o f the P-value in the following
method. Because the P-value is found from the probability distribution o f the test statistic
under the null hypothesis, a very small P-value implies that a sample result this extreme,
when H0 is true, occurs only very rarely, by chance phenomena. Sampling error is primarily
due to two causes: (I) sample does not encompass the entire population, and (2) each
member o f the population is not equally accessible to the sample. When the P-value is
critically small, then the investigator can state that the data do not support the null
hypothesis, that rejection is “statistically significant.” When the P-value is less than 0.05
the result is probably significant and if it is less than 0.01 then the result is highly significant.
In both cases the statistical decision is to “Reject the null hypothesis, H0."
Suppose that the P-value is large, then the sample result offers no convincing
evidence that the statement made in H0 about the population is false. In other words, the data
do not deny the null hypothesis. The statistical decision is “Fail to reject the null hypothesis,
H0 ."
Errors Associated with Hypothesis Testing
It is imperative that we properly address the question “How rare is the rare event?”,
while discussing the decision process. The probability value taken as the cutoff between a
rare event and a likely occurrence is frequently called the level o f significance o f the test
11
[Gibbons, 4], The value for the cutoff is primarily a matter o f personal choice, but it should
reflect the investigator’s feeling about the cost and the consequence o f error.
Two types o f error may occur. The first error occurs when the null hypothesis is
rejected, while it is actually true and is called, Type I error, a . The second error, called Type
II error, P occurs, when the null hypothesis is accepted while it is actually false. These
definitions are summarized below.
T able I.
Sum m ary o f Type I and Type II E rro rs
Actual Situation
Decision Taken
H 0 TRU E (Probability)
H 0 False (Probability)
Accept H 0
Correct Acceptance ( I -a)
Type II Error (P)
R eject H 0
Type I Error (a)
Correct Rejection (I-P)
If rejecting a true hypothesis would be considered a serious mistake, a small value
for a would be appropriate. However, a very small value o f a may not be good either,
because Type I error and Type II errors interact. When the probability o f one type of error
decreases, the probability o f the other increases, but disproportionately because o f the non­
linearity o f the distribution function. This is to be distinguished from the trivial case at p0
where a + P = 1.0. Thus if a Type II error has serious consequences, a larger a value might
be advisable. Once the value for a is chosen and the test result gives a P-value, the null
hypothesis H0 is rejected if P <= a , but not otherwise. It is important to note that in statistical
analysis, P-values can be an aid to making a research conclusion, but only in concert with the
judgement and intelligence o f the investigator.
12
CHAPTER 4
INVESTIGATION OF CUSUM CONTROL CHARTS
Process Behavior and Control Charts
A process is set o f causes and conditions that repeatedly come together to transform
inputs into outcomes [Moen, 8]. The inputs may include people, methods, material,
equipment, environment, and information. The outcome is some product or service. Process
behaviours are often studied with the use o f control charts. Control charts are graphical
displays o f statistics plotted according to the order o f their observation [Devor, 2]. A process
is said to be in statistical control if there are only random patterns on a control chart. A
succession o f data emanating from a process, which is under statistical control, will exhibit
variability due to a constant set of causes that are inherent in the process. These causes,
usually called common causes, can also be thought o f as causes leading to variability in the
process.
Data observed from processes, often fall into a predictable pattern o f variation, such
as a normal distribution, easily described by simple statistical measures, namely, a mean and
a standard deviation. These measures serve as a model to predict process behavior, when the
process is subject only to a set o f common causes. When a process is subject only to a set
o f common causes, data observed from the process will fall within the control limits o f the
13
process. Control limits are boundaries constructed on either side o f the mean o f the process,
usually at a distance o f three standard deviations.
For process control, data collected over time may be used to develop a statistical
model as long as the data are collected while the process is subject to a set o f common
causes. Hence, if we can develop a model for the process measurements, then, when a major
disturbance or an abnormal situation affects the process, the ensuing data will not conform
to that model. The data from the abnormal situation will stand out clearly from the commoncause variability pattern. To be able to distinguish abnormal data from data generated during
normal operating conditions, we should consider the process as it evolves over time.
Adoption to Forecasting
Data within a neutral corridor can be thought o f as data generated by a process. Chisquared tests for normality, were performed on sets o f data and the results show that data
within neutral corridors exhibit normality. All chi-square tests were performed at five percent
level o f significance and the results of some o f them are listed in Appendix B. A mean and
a standard deviation can now be computed and used to construct control charts for visual
analysis o f ■existence o f special causes. If nonrandom patterns or the existence o f data
generated by abnormal situations are observed, then it is concluded that all the data points
in the given set no longer fall within a neutral corridor.
A particular control chart that can be effectively used for purposes o f predicting
movements out o f the boundaries o f the neutral corridors is the cusum chart, where “cusum”
stands for “cumulative sum .”
Cusum Charts
Cusum charts are becoming widely used in the industry because they are powerful,
versatile, and have the ability to quickly detect small changes in a process mean [Lucas, 7],
The use o f cumulative sums o f sample data for on-line process control was developed by E.
S. Page [Page, 10] and G. A. Barnard. The principle behind these charts is based on a method
using the sequential likelihood ratio test [Wald, 13], where, as each new sample becomes
available, a test is conducted to determine whether the process mean deviates by at least a
specified amount from the target value.
Advantages of Cusum Charts
The most fundamental advantages o f cusum charts are threefold:
i.
Ease with which changes in mean level can be detected, either by observing points
that lie outside the control limits or by a change in the slope o f the chart.
ii.
Ability to locate a point o f change on a chart as that point at which the change in
slope occurs.
iii.
Efficiency over the standard control chart (Shewhart control chart) for changes of
about 0.5a and 2.0a, which means that in this region changes can be detected
approximately twice as quickly, when compared with Shewhart control charts
[Ewan, 3]. These advantages make the cusum charts suited to analyze processes
expected to have small and sustained deviations in the mean.
15
Traditional Cusum Charts
Common to all cusum control charts that are developed according to the sequential
likelihood ratio is the idea of hypothesis testing between two alternative quality levels, one
acceptable and the other rejectable. These cusum charts are usually referred to as
traditional cusum charts [Devor, 2], To construct traditional cusum charts both acceptable
and rejectable quality levels must be specified.
The necessity to specify the shifts
(acceptable and rejectable quality levels) in advance, presents a serious concern in the
application of these charts. However, the success of traditional cusum control charts
depends mainly on the accurate estimation of both the target process parameter value and
the size o f the shift the control chart is designed to detect.
Therefore, the use of
traditional cusum charts is suited only when accurate estimation of target process
parameter and shift are possible. Instead a modified version of the traditional cusum chart
called the standardized cusum chart can be used. The method of preparation and analysis
o f the standardized cusum chart will be discussed later.
The traditional cusum control chart is developed according to the concept of
hypothesis testing, where the hypotheses, for a one-sided cusum chart, are stated as
follows:
H 0: ft = Ho
H 1: H = Hi
(Hi > Ho) ,
where h is the mean o f X and both Type I and Type II errors are specified beforehand. H 1is
accepted only if there is a significant rejection o f H0 If rejection is not significant then H0
stands unrejected.
The likelihood ratio can be expressed in terms o f cc and P as,
—£—<
likelihood ratio o f the data
I P
I-G
a
The above equation can be used to construct a one-sided cusum control chart for detecting
upward shifts in the mean. If we have a time ordered sequence o f independent sample
observations, X „ X2,
, Xt, from a normal population with variance <tx2 and an uncertain
mean p, it can be shown that the ratio will take the form
-^ ln -
I -a
-<S X — I n
I-P
a
where A1 - p, - p, is the difference between the two hypotheses means or the shift in the
process mean from p0, and
A1'
S f = E [Xi - ( P ^ ) J
is the cumulative sum o f the sample deviations o f the data from the average o f the two
means. However, when it is desirable to detect a process mean shift in either direction from
the target, as with predictions o f significant movements away from neutral corridors, one
could use a pair of one-sided cusum charts to monitor the process for upward and downward
shifts, separately. Let the two off-target mean values o f interest be denoted by p, > p0 and
P2 < Po with a Type II error P and a Type I error 2a. Note, that the Type II errors for the
upward and downward shifts can be different, but with the neutral corridors, the selection of
two different p values is not meaningful.
By combining two one-sided decision criteria, it can be shown that the upper and
lower control limits for the centered cumulative sum, H(Xj - p), are
Q2x
U C L = -In
(
Al
LCL-
a
O2*
I n (a
A2
2
2
where A1= p, - p0 and A2= p 2- P0. Because the equations for the control limits are functions
o f the sample size t, the upper and lower control limits are linear trend lines for the cusum
chart. An example o f the resulting cusum chart is shown in Figure 3 and the corresponding
neutral corridor is shown in Figure 2.
The general model for the cusum chart above is that of a sequential sampling scheme
shown in Figure 3. There are two alternative hypotheses for rejection o f the null hypothesis:
H0: P = P0
H 1: P = Po + A1,
and
H2: p = p0 + A2.
Only the upper and lower trend lines are used for the detection o f shifts in the mean.
In every case cusums are cumulative sums of sample deviations from the target process mean
Po, where p, for a neutral corridor as defined earlier, is the value o f the center line o f the
neutral corridor.
MARKET SEC.
0 1 /1 3 /9 5
I
49.5
49.0
48.5
48.0
47.5
47.0
46.5
46.0
BUY (51)
45.5
45.0
44.5
44.0
43.5
43.0
42.5
42.0
41.5
41.0
40.5
40.0
39.5
39.0
0 05
Figure 2.
Neutral C orrid o r formed bq Market S ecuritg I
PERIODS
Figure 3.
Traditional Cusum Chart for Market Security I
20
Standardized Cusum Chart
Cusum charts can be used for individual measurements, when these measurements
exhibit a normal distribution with mean p. and standard deviation a . Given the individual
measurements, X i,
i = I, 2, 3,
. . . , t, we can standardize them by applying the
transformation
X i-H
Zr
o
The sum o f Z ’s will have a normal distribution with mean = 0 and variance = t {since the
variance o f each Zi is 1.0, the variance o f t independent ZirS will be (t)(l .0) = t [Devor, 2]}.
We can write the cusum as follows:
2Z,
Vt
where, St* also follows a normal distribution. This allows us to plot the St* on a control chart
with constant control limits (usually ± 3) and a centerline o f zero. Thus a control chart
simpler than the traditional cusum control chart can be constructed for the cusum defined
above. Figure 4 shows a standardized cusum chart which has been plotted on the same
neutral corridor data (Figure 2) used for plotting the traditional cusum chart.
(f)
o
PERIODS
Figure 4.
Standardized Cusum Chart for Market Security I
22
Construction of the Standardized Cusum Chart
i.
Compute X-bar and sx from the data. These are used as point estimators o f the
true mean and true standard deviation o f the X ’s respectively.
EX.
X b a r = -----:
k
ii.
Standardize all the X ’s into Z ’s for i = 1 ,2, 3, . . . , k:
_ {X.-Xbar)
iii.
Sum the Z ’s cumulatively for each t, where t = 1 ,2, 3, . . . , k:
sumt = DZi
iv.
Obtain the standardized cusum for each t, where t = 1, 2, 3, . . . , k:
,,Suml
'
23
v.
vi.
Plot the St* on the standardized cusum chart, where
Centerline
= 0
UCL
= 3
LCL
= -3
Interpret the chart, looking especially for possible trends in the sums.
Tests with Cusum Control Charts on Neutral Corridors
More than 15 neutral corridors have been tested for possible indications o f significant
movements away from one o f the boundaries (upper or lower boundary) o f the neutral
corridor (abnormal situations), using the traditional and standardized cusum charts. One set
o f data, which was tested using the traditional cusum chart is shown in Figure 2. These data
represent a neutral corridor formed by open, high, low and close prices o f a market security.
The data used to plot the cusum charts, the
X i1S,
correspond to the closing prices o f that
particular security.
The value o f the center line o f the neutral corridor was taken as p0, and the values of
upper and lower boundaries o f the neutral corridor were set equal to P1and P2, respectively.
The standard deviation o f the closing prices within the neutral corridor was used for all
calculations. Type I error values, a , are set to 0.05 and Type II error values, (I, are set to 0.10.
Results
The traditional cusum chart shown in Figure I has a number o f points that lie outside
the upper control limit, indicating a significant movement away from the upper boundary of
24
the neutral corridor starting from the forty-third time period. The corresponding significant
movement can be seen in Figure 2, the actual neutral corridor.
Signal, for significant movement away from the upper boundary, was given by the
standardized cusum chart at time period 32. Figure 2 shows a temporary movement away
from the upper boundary during time periods 31 and 32. However, movement in time
periods 3 1 and 32 can be considered valid because a significant movement occurs in the
same direction within a short period of time, say, within 12 periods. Therefore, points falling
outside the upper control limit, on the standardized cusum chart gave a valid signal.
In the standardized cusum chart points between the eighth and thirteenth time period
fall outside the upper control limit indicating a significant movement away from the neutral
corridor. Comparing this indication o f significant movement to the neutral corridor (Figure
2), it is obvious, that the signal is false. Appendix D shows cusum charts which are
examples o f both successful and failed signals o f movements away from neutral corridors.
Conclusion
Signals, o f movements away from neutral corridors, were given by traditional and
standardized cusum charts, when there was actually no significant movement. The author
considers these erroneous movements contradictory, which leads to the failure o f the concept
that cusum control charts could be used to detect significant movements away from neutral
corridors.
CHAPTER 5
IN V ESTIG A TIO N O F N O N PA RAM ETRIC TESTS
Usually statistical inferential procedures contain parametric procedures which
depend upon certain distributional assumptions about population(s). Some widely used
parametric tests are t-test, ANOVA5and regression analysis. Because the populations) does
not always meet the underlying assumptions for parametric tests, inferential procedures are
needed whose validity does not depend on rigid assumptions. In many instances,
nonparametric statistical procedures fill this need.
Statistical inferences not concerned with parameter values are called nonparametric.
Inferences whose validity does not depend on a specific probability model, such as a
binomial or normal, is called distribution-free. While nonparametric and distribution-free are
not synonymous, and in fact refer to two different aspects o f a statistical inference,
procedures o f either type are frequently known as nonparametric methods.
Advantages of Nonparametric Statistics
o
Often nonparametric procedures are computationally simple because they exist for
the measurement scales o f nominal and ordinal only,
o
Most nonparametric procedures depend on very few assumptions,
o
Assumptions are often “weaker” than those o f parametric procedures.
26
Disadvantages of Nonparametric Statistics
o
W hen a parametric is appropriate, using a nonparametric test is less efficient,
o
Some procedures are computationally demanding.
Two types o f nonparametric tests have been considered for possible use to detect
significant movements away from neutral corridors. The two types are tests for location and
tests against trend.
Tests for Location
A measure of central tendency in the population is a location parameter. The best
known location parameters are the mean and the median. Both represent a "typical" or
"average" value of a sample variable, or a central value of a distribution, but typify
different concepts of centrality.
For a finite population, the mean is determined by
summing all the values and dividing by the number of elements, while the median is the
middle case o f the ordered values. Therefore, by definition, the probability that a variable
exceeds its median is equal to the probability that the variable is exceeded by the median.
The mean is influenced by extreme values, since each possible value directly affects
the mean, while only ordering affects the median. The statistics that correspond to these
location parameters are the sample mean and the sample median.
In nonparametric
statistics, the median is often the sample measure of central tendency that has desirable
distribution properties.
27
The value o f the center line of the neutral corridor is selected as the sample
measure of central tendency. For a symmetric distribution, inferences concerning the value
of the center line can be considered equivalent to inferences concerning the median and
mean. However, in the case of neutral corridors, due to non-symmetry, the value of the
center line is more meaningful than the mean or the median. The study of inferences
concerning the chosen measure of central tendency was conducted using two tests for
location. They are the ordinary sign test and the Wilcoxon signed rank test.
Ordinary Sign Test
The sign test is a quick and simple test for the location parameter. Here, the
numerical data are dispensed with and only limited information, summarized b y ' + ' and
'-' signs, is retained. Hence the sign test involves very little computation and is found
suitable to make quick inferences concerning location parameters. The assumptions for the
sign test are:
i.
The measurement scale is at least nominal.
ii.
The observations can be classified into non-overlapping sets while set
union exhausts all possibilities. The sets are labeled ' + ' and '-'.
Suppose there are n independent observations, which are later classified into two
categories according to their relative magnitudes. After classification the data will consist
o f n dichotomous observations. These two categories are termed as "success", denoted
by ' + ' and "failure", denoted by '- '. The data to be analyzed are the frequencies in the
two categories.
28
The occurrences of success ( + ) or failure (-) are assumed to follo,w a Bernoulli
process [Gibbons, 4],
The only population parameters relevant are the probability of
success, denoted by P +, and the probability of failure, P . The ordinary sign test can be
used for the following types of inference concerning these parameters.
Let C0 denote the
initial value o f the center line o f the neutral corridor and C the value o f the center line
calculated after the observation of additional data. The hypothesis set to be tested is one
of the following:
H0:
C = C0
H 1:
C >
C0 , to check whether or not the time series has moved away from
the upper boundary of the neutral corridor.
or
H 0:
C
= C0
H 1:
C < C0 , to check whether or not the time series has moved away from
the lower boundary of the neutral corridor.
Suppose we have n observations in the period of analysis, each point is classified
as a ' + ' o r ' - ' depending on the location o f the observation. If an observation is above the
center line then it is classified as ' + ' and if the observation is below to center line it is
classified as
Points falling on the center line (give zero differences) are omitted from
all calculations. These derived data can be regarded as random samples drawn from a
dichotomous population of plus and minus signs. The test statistics are defined as follows:
S+ = Number of plus signs among the (Xi - C0)
S„ = Number of minus signs among the (Xi - C0).
29
Because the sampling distributions of S+ and S. are each binomial with parameter 0 - 0.5
if H0 is true, the P-Values could be found from the appropriate binomial distribution. The
P-values are found using the binomial distribution when n < 20 and the normal
approximation when n > 20. Table 2 lists the results of the sign test performed on the
data shown in Figure 2.
Wilcoxon Signed Rank Test
Because the sign test for location utilizes only the signs of the differences between
each observation and the measure of central tendency (center line o f neutral corridors), the
magnitudes of the differences are not considered. If this information is available from the
data measured on a ratio scale, a test procedure that takes into account the size of relative
magnitudes might be expected to give better performance [Gibbons, 4], The Wilcoxon
signed rank test statistic uses both the signs and the magnitudes of differences to influence
the inference. The only additional assumption required is that of symmetry about the true
measure of central tendency. Because data from a neutral corridor belong to the ratio
scale, the W ilcoxon signed rank test can be applied.
For C0, the center line value of a neutral corridor, the hypotheses set to be tested
is one of the following:
H0:
C = C0
H 1:
C >
C0 , to check whether or not the time series has moved away from
the upper boundary of the neutral corridor, or
30
Table 2.
Results of the Ordinary Sign Test
Data pertains to ==> MARKET SECURITY I
Points in neutral corridor ==>
Upper boundary =>
16
44.625
TOTAL
POINTS
NON-ZERO
POINTS
POINTS
ABOVE
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
13
14
15
16
17
17
17
17
18
19
20
21
22
23
24
25
26
27
27
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
Lower boundary
POINTS
BELOW
4
4
4
4
4
5
6
7
7
7
7
7
7
7
7
7
7
7
8
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
PROB./
ZVAL
0.9936
0.9962
0.9978
0.9987
2.8587
2.5797
2.3145
2.0617
2.2200
2.3730
2.5211
2.6646
2.8040
2.9394
3.0713
3.1997
3.3249
3.4471
3.2285
3.0167
3.1400
3.2607
3.3787
3.4943
3.6076
3.7187
3.8277
3.9347
4.0398
4.1431
4.2447
4.3446
4.4429
4.5396
4.6349
4.7288
4.8214
4.9126
=>
42 .5
RECOMMENDATION
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
31
H0:
C
H1:
C < C0 , to check whether or not the time series has moved away from
= C0
the lower boundary o f the neutral corridor.
Suppose we have n observations in the period o f analysis, each point is classified
as a ' + ' o r ' - ' depending on the location of the observation. If an observation is above the
center line then it is classified as ' + ', and if it is below the center line it is classified as
Points which lie on the center line (zero differences) are omitted from all calculations.
To compare the absolute magnitude values o f the differences D, = Xi - C0, we
temporarily ignore the signs and rank the absolute values ID1IJ D2I, . . . J DnIaccording
to relative magnitude. In other words, rank I is given to the smallest absolute difference
IDiI, rank 2 to the second smallest, . . . and rank n to the largest. The sum o f the ranks
assigned to the absolute values of those differences whose original sign is plus, called
positive ranks, should be approximately equal to the sum of the ranks o f those absolute
differences that are originally minus, called negative ranks [Gibbons, 4].
Clearly, if H0 is true, we would expect that sums of the ranks o f the negative and
positive differences to be roughly equal. If the sum of positive ranks is much larger than
the sum o f negative ranks, most o f the ranks belong to positive differences, and the data
support the alternative H1: C > Q . A larger sum o f positive ranks indicate that more
observations lie above the center line and have higher magnitudes o f differences. This
situation is analogous to a higher momentum of the time series towards or away from the
upper boundary o f the neutral corridor.
32
Similarly, a large sum of negative ranks reflects the situation in which large ranks
are associated primarily with negative differences, and the data support H 1: C < C0. A
larger sum of negative ranks indicates that more observations lie below the center line and
have higher magnitudes o f differences. This situation is analogous to a higher momentum
towards or away from the lower boundary of the neutral corridor.
The W ilcoxon test statistic is defined as either
T + = sum of positive ranks
T
= Sum o f negative ranks.
Note, that T + and T are both defined as nonnegative integers, and
T++ T_-n ( n + 1 ) / 2 .
Tables that provide the critical values [Harter, 5] are available and have been used for
determining the P-values.
The P-values are found using the null distribution for a
W ilcoxon signed rank test when n < 50 and the normal approximation when n > 50.
Table 3 lists results o f Wilcoxon signed rank test performed on data shown in Figure 2.
Tests Against Trend
There are two types o f trends: upward trend and downward trend. A series of
observations is said to exhibit an upward trend if the magnitudes of the later observations
tend to be greater than those o f the earlier observations. The data exhibit a downward
trend if the earlier observations tend to be larger than the later observations. This simple
33
Table 3.
Results of Wilcoxon Signed Rank Test
Data pertains to ==> MARKET SECURITY I
Number of points in neutral corridor =
Value of Center Line =
Periods
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
Nonzero Diff.
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
16
43.5625
Sum 1+' Ranks
134
152
171
187.5
205
212
222.5
238
259
283
308
334
362.5
386.5
415.5
445.5
478.5
512
533.5
558.5
591
629
668
708
749
791
834
878
923
969
1016
1064
1113
1163
1214
1266
1319
1373
Sum '-' Ranks
19
19
19
22.5
26
41
53.5
62
66
68
70
72
72.5
78.5
80.5
82.5
82.5
83
96.5
107.5
112
112
112
112
112
112
112
112
112
112
112
112
112
112
112
112
112
112
Recommendation
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
34
idea has been used to construct some tests against trend. Here, three tests have been
studied to determine their possible use for detection of significant movements away
from neutral corridors.
Cox-Stuart Test
The Cox-Smart test for trend, is a modification of the sign test. To use it, we panone of the earlier observations with one of the later observations. When the later
observations exceed the earlier observation, we replace the pair by a minus sign. When
the earlier observation is greater than the later observation, we replace the pair by a plus
sign. A preponderance of plus signs suggests a downward trend, and a preponderance of
minus signs suggests an upward trend [Daniel, I]. If positive and negative signs occur in
equal number, no trend is present. The hypothesis set to be tested is one of the following:
H0:
There is no upward trend
H 1:
There is an upward trend
or
H0:
There is no downward trend
H 1:
There is a downward trend
Suppose the time series has a set of observations denoted by X 1, X2, . . . , % .
Pairs are first formed as
(X1, X 1+c), (X2, X2+c), . . . , (Xn.c, Xn) where,
C = n/2 when n is an even number, and C = (n + l) /2 when n is an odd number. For
example, if n = 6, X 1 = 2, X2 = 3, X3 = 5, X4 = 8, X5 = 9, and X6 = 10, then C =
35
6/2 = 3. The pairs in this case are (2,8), (4,10), (6,12). If n = 7, and we add the
observation X7 = 12 to the other six observations, then C = (7 + l)/2 = 4. The pairs in
this case are (2, 9), (4,10), (6,11). Note that the middle term, 8 is not used. This is always
the case when n is an odd number. A plus sign replaces each pair (Xi, Xi+C) for which Xi
is greater than for which X i+C is greater than X i [Daniel, I]. Pairs leading to zero
differences are omitted from the analysis. The number of pairs yielding to nonzero
differences is equal to m.
The test statistic depends on the hypotheses being tested. The test statistic for the
first hypotheses set is the number of plus signs, and the test statistic for the second set of
hypotheses is the number of minus signs. The probabilities of observing the number of
plus or minus signs is obtained from the binomial distribution. Table 4 lists the result of
the Cox-Stuart test performed on the data shown in Figure 2.
Modified Cox-Stuart Test
A slight modification was made to the Cox-Stuart test to increase the sensitivity of
the test, permitting the magnitudes of the differences to influence the test statistic. This
method combines the principles of the Cox-Stuart test and the Wilcoxon signed rank test.
The hypotheses sets used here, are the same as the hypotheses sets used for the Cox-Stuart
test.
Similar to the W ilcoxon signed rank test, the absolute magnitude values of the
differences between the elements in each pair are ranked according to relative magnitude.
While ranking, signs of the differences are temporarily ignored and all differences are
36
Table 4 .
Results of the Cox-Stuart Test
Data pertains to ==> MARKET SECURITY I
Initial, Sample size =
54
Number of points in neutral corridor =
POINTS
+ SIGNS
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
I
I
2
2
2
2
4
4
5
5
7
7
7
7
8
8
11
11
11
11
12
12
10
10
7
7
10
10
10
10
5
5
3
3
3
3
3
3
4
- SIGNS
7
7
6
6
8
8
7
7
7
7
6
6
5
5
5
5
4
4
5
5
3
3
7
7
10
10
10
10
■12
12
16
16
16
16
19
19
21
21
23
PROB./ZVAL
0.035
0.035
0.145
0.145
0.055
0.055
0.274
0.274
0.387
0.387
0.500
0.500
0.387
0.387
0.291
0.291
0.059
0.059
0.105
0.105
0.018
0.018
0.315
0.315
0.315
0.315
0.588
0.588
-0.405
-0.405
-2.379
-2.379
0.002
0.002
-3.390
-3.390
-3.654
-3.654
-3.637
16
RECOMMENDATION
Positive Trend !
Positive Trend !
Negative Trend !
Negative Trend !
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
!
!
!
!
!
!
!
!
!
37
treated as a single array of numbers. The sum of the ranks assigned to the absolute values
o f those differences whose original sign is plus, called positive ranks, should be
approximately equal to the sum of the ranks o f those absolute differences that are
originally minus, called negative ranks. The distribution of the test statistic is the same
as the W ilcoxon signed rank test. The author suggested this modification and tested it.
Table 5 lists the results of the modified Cox-Stuart test performed on the data shown in
Figure 2.
Kendall's test for trend
Here we arrange the data in the ascending order of their magnitude and rank them
as T 1, T2, . . . , T n . After ranking all observations, each rank is compared to the time
order of that observation. Suppose it is believed that the observed data gradually increase
over the period in consideration, the null hypothesis will be stated as "There is no trend
in the data over the period of observation". The alternative hypothesis will be stated as
"There is an upward trend in the data over the period of observation". The existence of
such a trend would increase the probability of arrangements of the ranks T 1, T2, . . . , Tn,
where large values of Ti, would occur later.
Under the alternatives large values of Ti will occur for large values of z" and small
values o f T i for small values o f i, so that the differences (T , - i)2 will tend to be small.
This means that the differences between the rank of the observations and the time of
observation will be reduced. The distribution for the test statistic has been tabled for n <
38
Table 5.
Results of the Modified Cox-Stuart Test
DATA PERTAINS TO ==> MARKET SECURITY I
INITIAL SAMPLE SIZE =
54
NUMBER OF POINTS IN NEUTRAL CORRIDOR =
PERIODS
NONZERO DIFF..
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
8
8
8
10
10
11
11
12
12
13
13
12
12
13,
13
15
15
16
16
15
15
17
17
17
17
20
20
22
22
21
21
19
19
22
22
24
24
27
SUM '+' RANKS
1.5
4.5
4.5
8.5
8.5
22
22
38
38
50
50
45.5
45.5
64
64
95.5
95.5
87.5
87.5
88.5
88.5
97.5
97.5
72
72
87
87
77.5
77.5
40.5
40.5
20.5
20.5
25.5
25.5
20
20
27
16
SUM '-' RANKS
34.5
31.5
31.5
46.5
46.5
44
44
40
40
41
41
32.5
32.5
27
27
24.5
24.5
48.5
48.5
31.5
31.5
55.5
55.5
81
81
123
123
175.5
175.5
190.5
190.5
169.5
169.5
227.5
227.5
280
280
351
RECOMMENDATION
Positive
Positive
Positive
Positive
Positive
Trend
Trend
Trend
Trend
Trend
Negative Trend
Negative Trend
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
trend
trend
trend
trend
trend
trend
trend
trend
trend
39
Table 6.
Results of the Kendall's Test
DATA PERTAINS TO ==> MARKET SECURITY I
Initial Sample size =
PERIODS
TEST STAT.
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
.392.5
442.5
472
737.5
1032
1494
1979.5
2410
2765.5
3043.5
3341.5
3660
3860
4462
4853.5
5268.5
5346.5
5621
.6647.5
7639
8429
8429
8429
8435
8445
8459
8489
8529.5
8615
8829.5
8879
8922.5
8922.5
8922.5
8922.5
8922.5
8928.5
8940.5
54
RECOMMENDATION
Positive
Positive
Positive
Positive
trend
trend
trend
trend
!
!
!
!
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
!
!
!
!
!
!
!
!
!
!
!
!
!
!
40
11 [Harter, 5], For sufficiently large n, the distribution is approximately normal [Lehman,
6]. Table 6 lists the results o f the test performed on the data shown in Figure 2.
Results
From Table 2, it can be seen that the ordinary sign test gave a signal indicating a
positive trend from the seventeenth period. The data illustrated in Figure 2, lies within the
neutral corridor until the fiftieth period. Comparing the signal and the plot o f the data it was
concluded that the signal for positive trend is false. Table 3, results o f Wilcoxon signed rank
test, also gave a signal for positive trend from seventeenth period. This is also a false
signal because the plot shows that the data lie within the neutral corridor until period 50.
Hence we know that both tests for location gave false signals for the data set, corresponding
to market security I .
Results o f Cox-Stuart test (Table 4) gave signals for positive trends at the sixteenth
and seventeenth time periods and also from the forty-sixth time period. There were also
signals for negative trends at the thirty-sixth and thirty-seventh time periods. Data shown in
Figure I lie within the neutral corridor until time period 50. This proves that the first signal
for positive trend and the signal for negative trend are false. Table 5, results o f the modified
Cox-Stuart test, also gave two signals for positive trends and a signal for negative trend
between these positive signals. Comparing with the plot o f the data and using the same
logic, it can be seen that the signals for the first positive trend and the negative trend are
false.
41
Results o f Kendall's test, (Table 5), gave signals o f positive trends from the
seventeenth to the twentieth periods. A signal for positive trend was also given starting from
the forty-first period. Comparing these signals with the plot o f the data (Figure 2) it was
concluded that the signal for the first positive trend was false.
All nonparametirc tests discussed above were tested on more than 15 sets o f data
(neutral corridors), where erratic signals were received on 12 sets o f data. Appendices E, F,
G, H and I, show the results o f the nonparametric tests performed on selected sets o f data.
Conclusion
All nonparametric tests gave erratic signals o f positive and negative trends when there
were none. This ratio o f false signals to correct signals was very high (12 to 3). The mixture
o f positive and negative signals and improper timing o f the correct signals (positive trend for
data in Figure 2) led to the failure o f these tests. Therefore, the author concludes that these
nonparametric tests should not be used to detect significant movements away from neutral
corridors.
42
CHAPTER 6
THE NEW EMPIRICAL METHOD
Failure o f the cusum charts and all nonparametric tests led to a change in thinking.
Traditional methods were not found suitable, but the concept o f hypotheses testing was
considered important and promising. The concept o f hypotheses testing has been used to
venture into a pristine zone and an empirical method has been developed to identify a
quantitative approach to detect significant breakout movements away from neutral corridors.
Hypotheses used in this test are similar to the hypotheses stated for the cusum control
chart procedure. The hypotheses are
H q: >
=
Ho
H 1: n = Ho + A1,
and
H2: h = H o '
where,
Ho represents the center line of the neutral corridor; H o + ^ Represents the upper boundary
o f the corridor; Ho ~ A2 represents the lower boundary o f the corridor.
The magnitudes o f distances between each data point and the hypothetical center
line of the neutral corridor were monitored. These magnitudes were calculated separately
for points above and points below the center line. The sum o f the magnitudes o f distances
from the center line indicates the cumulative strength of the time series in each direction.
43
The cumulative sum might also be considered as an index o f momentum in that particular
direction. Steps to calculate the momentum index and a corresponding threshold value,
similar to a critical value, are listed below:
o
Identify neutral corridor and values o f upper and lower boundaries using the
method described in Chapter I .
o
Count the number o f periods that elapsed before the upper and lower boundaries
w ere formed. This count will henceforth be called the “number o f points in the
neutral corridor. ”
For example, points A and B form the upper and lower
boundaries in Figure 2, and the number of points in the neutral corridor is sixteen,
o
Compute the half width o f the neutral corridor. The half width o f the corridor in
Figure I is calculated as
HW = (difference between point A and point B) / 2
o
Classify the data, closing prices o f the market security, into two groups.
The first group consists o f points above the center line and the other consists of
points below. If an observation lies on the center line that observation is ignored.
Classification o f the data in Figure 2 until the seventeenth time period gave 13
observations above the center line (first group) and 4 observations below center line
(second group).
o
Calculate the differences between each observation and the value o f the center line.
44
Compute the ratio between the differences and the half width o f the corridor. The
ratio indicates the measure o f how far away an observation is from the center
line.These differences are termed a, 's. Let 0.1875 be the difference between the
first observation and the center line. Note that the first observation is below the
center line and hence belongs to the second group. The ratio for the first
observation is calculated as P 1 = a, / H W .
Calculate the momentum index for each group of points using the formula
'£a.*Pl-b
The value of 1.5 for the exponential power was determined empirically after
tests on several neutral corridors.
Calculate threshold value using the formula
HPiZtn1-5
where, n = number of points in the group (above or below center line). The
concept of threshold value was derived based on the assumption that the momentum
index attains a maximum, when all points in a group lie on the boundary, given
that no point has crossed the boundary. Suppose the momentum index is being
calculated for the group o f points above the center line. H ere, the maximum
possible value o f the momentum index will be attained when all points lie on
the upper boundary.
Compute the ratio of the momentum index and the threshold value for that group
of observations.
It can be concluded that the group o f points have gained enough momentum to
cross the corresponding boundary, when the ratio is greater than I .
Tnnfirmatinn using Volume o f Transaction
The above method considers only the momentum attained by the closing prices.
Studies show a definite relation between the price movements and the volume of
m o r i o n Volume shows the intensity of changes in human attitudes and fears. Volume
moves up when prices move up or down indicating interest from the major buyers and
sellers. For example, the level of enthusiasm implied by a price rise on low volume is not
nearly as strong as that implied by a similar price advance accompanied by very high
volume [Bring, 11]. The concept of volume of price relation is used here to obtain a
confirmation.
Information on the volume of transactions is available only for securities of
individual companies. Therefore, the confirmation of high volume transactions can be
obtained only for such securities. For example in Figure 2 it was observed drat sixteen
observations were needed to form the boundaries of the corridor (represented by points A
and B). The median volume is identified from the set of the first sixteen volumes of
transactions. A value of 150 percent of this median volume has been considered a
•‘substantial increase in volume* and the value of “150 percent of the median volume*
46
was also empirically determined after analyzing several market securities. The median
volume for the data shown in Figure 2 is 671,950 units.
Another aspect of price-volume relationship is also helpful in confirming significant
breakout movements. Volume often leads the price movement and serves as a warning for
imminent price movements. If there is a substantial increase in volume (150 % of median
volume), one or two periods before a signal is received from the price momentum, such
increases are also considered valid confirmations o f significant breakout movements.
Both price momentum and volume increase were used to detect significant breakout
movements away from neutral corridors. The price momentum was used to detect
significant movements when information on volumes of transaction was unavailable for
securities like market indices and mutual funds.
Results
Table 7 shows the results of the test performed on the corridor in Figure 2. The
data shown in Figure 2 correspond to an individual company. Therefore, information on
volume was available and used to obtain a confirmation. A signal for positive trend was
given starting from time period 51. Observation of the plot of data (Figure 2) shows that
the time series started the significant breakout movement from the forty-eighth period of
observation. From the comparison it was concluded that the empirical method gave a
correct and timely signal.
47
Twenty one corridors obtained from historical market data were studied. Nineteen
tests gave signals in the correct direction and at the right time. The accuracy of these tests
was verified by comparing the signals to actual movements of the time series.
Table 7:
Results of New Empirical Method on Market Security I
Points in neutral corridor ==>
Upper boundary =>
NUMBER OF
POINTS
16
44.625
Lower boundary =>
42.5
TEST STATISTIC
RATIO
SIGNAL
51
1.476549
Positive trend
52
2.271212
Positive trend
53
2.647483
Positive trend
54
2.970027
Positive trend
Conclusions
The new empirical method gave nineteen correct signals out of the 21 tests, placing
the accuracy of the method at 90 percent. Although, a 90 percent accuracy is considered
high, any new testing method can be accepted for use, only if it performs consistently over
a wide range o f conditions. To verify the consistency in performance, tests are being
conducted with real-time data from the market. Neutral corridors that are currently being
analyzed include market securities that do not contain information on volume of
transaction.
48
CHAPTER?
RESULTS AND DISCUSSION
Three techniques were investigated for possible use in the detection of significant
breakout movements away from neutral corridors. They were cusum control charts,
nonparametric tests for location and trend, and a new empirical method developed by the
author. A n accurate result was defined as the availability of a signal indicting a positive
or negative trend at the right time. Comparison and a brief discussion of the results are
presented below.
Discussion on False signals
Incorrect signals were of two types. The first type of false signal was the indication
of a positive or negative trend when there was none. The other false signal was the
indication of a positive or negative trend much before or after the actual movement
occurred.
Traditional Cusum Charts
Traditional cusum charts also gave similar false signals. Appendix C shows
selected plots of traditional cusum charts constructed on data corresponding to market
securities.
Charts plotted on data from market securities 3 and 5 gave signals of positive
49
trends very early compared to the actual movement of the time series. Thus the two signals
were considered false. A plot of the data from market security 6 gave no signals.
However, it can be seen from the plot of the actual data (Figure 7, Appendix A) that the
times series attained a significant movement starting from period 97. Although traditional
cusum charts gave correct and timely signals for market securities 1 , 2 , 4 and 7, the ratio
o f false signals to correct signals was quite high. Ten out of the 15 traditional cusum
charts gave false signals. Hence, traditional cusum charts cannot be used for detection of
significant breakout movements.
Standardized Cusum charts
Standardized cusum charts constructed for market securities 2 and 7 (shown in
Appendix D) gave accurate and timely signals. However most of the standardized cusum
charts shown in Appendix D and the chart shown in Chapter 3, gave out of control signals
indicating positive or negative trends. These false signals were often received much before
the time series actually moved in that direction. However, the direction of movement
indicated by these standardized cusum charts was correct, the time of movement indicated
did not match with the actual time of movement. Plots of neutral corridors, not shown
here, twelve out of fifteen, also gave erratic signals. Therefore, the author concluded that
standardized cusum charts cannot be used to detect significant breakout movements away
from neutral corridors.
50
Ordinary Sign Test
Appendix E lists selected results of sign tests conducted on market securities. The
sign test often gave signals of significant movements much earlier than actual movements.
Such false signals were encountered in data corresponding to market securities I, 2, 3, .4,
5, and 6. The test performed on data from market security 7 gave the only correct and
timely signal, leading to a very low accuracy. More than twelve out of the fifteen tests
gave false signals. Hence, the author concluded that the ordinary sign test can not be used
for detection of significant movements away from neutral corridors.
Wilcoxon Signed Rank Test
Wilcoxon signed rank tests also gave false signals in many instances. Appendix F
lists selected results of tests performed on market securities. From the results we can see
that signals of significant movements were given before the actual movements of the time
series. These results are similar to the results of ordinary sign tests. Therefore, the
possibility o f using W ilcoxon signed rank test was also ruled out.
Cox-Stuart Test
Appendix G shows selected results of the Cox-Stuart tests performed on market
securities. Positive and negative signals were received within the period of analysis for
data corresponding to market securities I and 2. Results on market securities 3, 4, 5, and
6 gave signals of significant movement before the actual movement o f times series. Thus,
the accuracy of the Cox-Stuart test also fell below expectations which led to rejection.
51
Modified Cox-Stuart Test
Appendix H lists selected results o f the modified Cox-Stuart tests. Results of the
tests were very similar to the results obtained during Cox-Smart tests. The signal from
tests on market security 7 was the only correct signal. Hence, this test was also rejected.
Kendall's Test
Appendix I lists selected results of the Kendall's tests. Results on market security
7 was a correct and timely signal. The movement signals on all other market securities
were either very early or were a mixture of positive and negative trends leading to a very
low accuracy. Therefore, the Kendall's test cannot be used to detect significant breakout
movements away from neutral corridors.
Results of the New Empirical Method
Twenty-one neutral corridors were analyzed using the new empirical method.
Nineteen of them gave accurate and timely signals. The neutral corridors were formed
from historical data corresponding to market securities. Data that were tested by cusum
charts and nonparametric tests were included in the twenty-one tests to facilitate
comparison o f the techniques.
Appendix J shows selected results of the new empirical method. Results of tests
on market security 6 gave signals of positive trend before a positive trend actually started.
Tests on all other market securities shown in Appendix A gave accurate and timely
52
signals.
The accuracy o f the new empirical method can be computed as 90 percent
(19/21), and therefore was concluded to be the best method.
Conclusions
Acceptance of any method depends on the robustness of the method. Robustness
is defined as the ability to consistently perform over a variety of situations. Here, the
consistency has to be tested over different types o f time series. Tests are currently being
conducted with real-time data obtained from market indices and mutual funds. Note, that
these data sets do not contain information on volume of transactions.
In addition to these
sets of data, more neutral corridors formed for market securities o f individual companies
are also being analyzed. Results from the ongoing tests will help to confirm the accuracy
o f results derived from historical data.
Although the accuracy has been determined as 90 percent, the results depend
largely on proper identification of neutral corridors. Improper identification of the upper
and lower boundaries could result in inflated or deflated boundary values. The inflation
or deflation o f the boundary values will affect the sensitivity o f the empirical method.
Thus proper identification of neutral corridors is imperative for accuracy.
As a conclusion to this research, the author wants the reader to understand and
assimilate the following quote from Pring, a leading market analyst [Pring, 11].
"The continual search for the "holy grail" or a perfect indicator, will undoubtedly
continue, but it is unlikely that such an indicator will ever be developed. Even if it were,
news of its discovery would soon be disseminated and the technique will gradually be
discounted."
53
APPENDICES
54
APPENDIX A
PLOTS OF NEUTRAL CORRIDORS
MARKET SEC. 2
0 4 /0 5 /9 5
Ln
cn
BUY (3 8 )
30 F
Figure 5.
06
27 M
06
Neutral Corridor formed bg Market Security 2
MARKET SEC. 3
BUY (33?
30
Figure 6 .
F
Neutral Corridor formed bg Market Securitg 3
0 3 /2 2 /9 5
MARKET SEC. 4
BUY (2 2 )
30
Figure 7 .
N
Neutral Corridor formed by Market Security 4
12/18/95
MARKET SEC. 5
30 F
Figure 8.
06
Neutral Corridor formed bg Market Securitg 5
0 3 /2 9 /9 5
MARKET SEC. 6
11/21/95
BUY (9 4 )
Figure 9.
Neutral Corridor formed by Market Security 6
MARKET SEC. 7
SELL (1 6 )
30 N
Figure 10.
Neutral Corridor formed bg Market Security 7
12/29/95
61
APPENDIX B
RESULTS OF NORMALITY TESTS
62
Table 8.
Results of Normality Tests
No. of Observations
I
37
0.118677
2
37
0.199360
3
32
0.512036
4
20
0.943950 **
5
41
0.261792
6
90
0.647873
7
20
0.802360 **
Note:
**
P-value
Market Security
Indicates insufficient degrees of freedom for Chi-Square test.
K-S test was performed.
63
APPENDIX C
TRADITIONAL CUSUM CHARTS
30
-30 ----------------------------------------------------------------------------------------------------0
10
20
30
40
PERIODS
Figure 11.
Traditional Cusum Chart for Market Security 2
50
PERIODS
Figure 12.
Traditional Cusum Chart for Market Security 3
Figure 13. Traditional Cusum Chart for Market Security 4
I j 20
PERIODS
Figure 14. Traditional Cusum Chart for Market Security 5
PERIODS
Figure 15.
Traditional Cusum Chart for Market Security 6
30
-40 ---------------------------------------------------------------------------------------------------------0
5
Figure 16.
10
15
PERIODS
20
Traditional Cusum Chart for Market Security 7
25
30
70
APPENDIX D
STANDARDIZED CUSUM CHARTS
PERIODS
Figure 17.
Standardized Cusum Chart for Market Security 2
PERIODS
Figure 18.
Standardized Cusum Chart for Market Security 3
PERIODS
Figure 19.
Standardized Cusum Chart for Market Security 4
12
O
10
20
30
PERIODS
Figure 20.
Standardized Cusum Chart for Market Security 5
40
O
20
40
60
80
' PERIODS
Figure 21.
Standardized Cusum Chart for Market Security 6
100
PERIODS
Figure 22.
Standardized Cusum Chart for Market Security 7
APPENDIX E
RESULTS OF ORDINARY SIGN TESTS
78
Table 9 . RESULTS OF ORDINARY SIGN TEST ON MARKET SECURITY 2
Points in neutral corridor ==>
Upper boundary =>
5
45.5
Lower boundary
=>
43.75
TABULATION OF RESULTS
TOTAL
POINTS
NON-ZERO
POINTS
POINTS
ABOVE
POINTS
BELOW
PROB./
ZVAL
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
6
7
8
9
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
24
25
26
27
28
29
30
31
32
33
34
35
36
37
4
4
5
6
6
7
8
9
10
11
11
11
11
11
11
11
11
11
11
11
11
12
12
13
14
15
16
17
18
19
20
21
22
23
2
3
3
3
3
3
3
3
3
3
4
5
6
7
8
9
10
11
12
13
13
13
14
14
14
14
14
14
14
14
14
14
14
14
0.8906
0.7734
0.8555
0.9102
0.9102
0.9453
0.9673
0.9807
0.9888
0.9935
0.9824
0.9616
0.9283
0.8811
0.8204
0.7483
0.2400
0.0213
0.2294
0.4287
0.4287
0.2200
0.4118
0.2117
0.0189
0.2043
0.3834
0.5568
0.7248
0.8878
1.0461
1.2001
1.3500
1.4960
RECOMMENDATION
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
TREND
TREND
TREND
TREND
TREND
TREND
!
!
!
!
!
!
79
Table 10. RESULTS OF ORDINARY SIGN TEST ON MARKET SECURITY 3
Points in neutral corridor ==>
Upper boundary =>
5
58.5
Lower boundary
=>
56.125
TABULATION OF RESULTS
TOTAL
POINTS
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
, 33
34
35
36
37
38
39
40
NON-ZERO
POINTS
POINTS
ABOVE
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
5
5
5
5
5
5
6
6
7
8
9
10
11
11
12
12
12
13
14
15
16
16
16
17
18
19
20
21
22
23
24
25
26
27
28
POINTS
BELOW
I
2
3
4
5
6
6
’7
7
7
7
7
7
8
8
9
10
10
10
10
10
11
12
12
12
12
12
12
12
12
12
12
12
12
12
PROB./
ZVAL
0.9844 .
0.9375
0.8555
0.7461
0.6230
0.7256
0.6128
0.7095
0.6047
0.6964
0.7728
0.8338
0.8811
0.8204
0.8684
0.6765
0.4477
0.6464
0.8369
1.0200
1.1963
0.9815
0.7748
0.9470
1.1137
1.2752
1.4319
1.5841
1.7321
1.8762
2.0167
2.1536
2.2873
2.4179
2.5456
RECOMMENDATION
POSITIVE TREND !
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
TREND
TREND
TREND
TREND
TREND
TREND
TREND
!
!
!
!
!
!
I
80
Table 10. RESULTS OF ORDINARY SIGN TEST ON MARKET SECURITY 3
-Continued
Points in neutral corridor ==>
Upper boundary =>
5
58.5
Lower boundary
=>
56.125
TABULATION OF RESULTS
TOTAL
POINTS
NON-ZERO
POINTS
POINTS
ABOVE
POINTS
BELOW
PROB./
ZVAL
41
42
43
44
45
4.6
47
48
49
50
51
52
53
54
55
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
12
12
12
12
12
12
12
12
12
12
12
12
12
12
12
2.6706
2.7929
2.9127
3.0302
3.1454
3.2585
3.3695
3.4785
3.5857
3.6911
3.7948
3.8968
3.9972
4.0961
4.1935
.
RECOMMENDATION
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
81
Table 11. RESULTS OF ORDINARY SIGN TEST ON MARKET SECURITY 4
Points in neutral corridor ==>
3
Upper boundary = > 8 6
Lower boundary
=>
83.5
TABULATION OF RESULTS
TOTAL
POINTS
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
NON-ZERO
POINTS
3
3
4
5
6
7
8
9
10
11
12
13 '
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
POINTS
ABOVE
3
3
4
5
6
6
7
8
8
8
8
8
8
8
8
9
10
11
12
13
14
15
16
.17
18
19
20
POINTS
BELOW
0
0
0
0
0
I
I
I
2
3
4
5
6
7
8
8
8
8
8
8
8
8
8
8
8
8
8
PROB./
ZVAL
1.0000
1.0000
1.0000
1.0000
1.0000
0.9922
0.9961
0.9980
0.9893
0.9673
0.9270
0.8666
0.7880
0.6964
0.5982
0.6855
0.7597
0.8204
0.8684
1.1129
1.3005
1.4805
1.6534
1.8200
1.9808
2.1362
2.2867
RECOMMENDATION
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
!
!
!
!
!
!
!
!
!
!
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
TREND
TREND
TREND
TREND
TREND
!
!
!
!
!
82
Table 12. RESULTS OF ORDINARY SIGN TEST ON MARKET SECURITY 5
Points in neutral corridor ==>
Upper boundary =>
7
47.75
Lower boundary
=>
43.75
TABULATION OF RESULTS
TOTAL
POINTS
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
NON-ZERO
POINTS
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
POINTS
ABOVE
POINTS
BELOW
PROB./
ZVAL
4
5
6
7
8
9
10
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
0.6367
0.7461
0.8281
0.8867
0.9270
0.9539
0.9713
0.9408
0.9616
0.9755
0.9846
0.9904
0.9941
2.4222
2.5797
2.7315
2.8782
3.0200
3.1575
3.2909
3.4206
3.5468
3.6697
3.7897
3.9068
4.0212
4.1331
4.2427
4.3500
4.4552
4.5584
4.6597
4.7592
RECOMMENDATION
POSITIVE TREND !
POSITIVE TREND !
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
83
Table 13. RESULTS OF ORDINARY SIGN TEST ON MARKET SECURITY 6
Points in neutral corridor ==>
Upper boundary =>
3
71.875
Lower boundary
=>
69.375
TABULATION OF RESULTS
TOTAL
POINTS
NON-ZERO
POINTS
POINTS
ABOVE
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
4
5
6
7
8
8
9
10
11
12
13
14
15
16
17
17
18
19
20
21
21
22
23
23
24
25
26
27
28
29
30
31
31
32
33
I
I
I
2
3
3
3
3
4
5
6
6
6
7
8
8
9
10
11
12
12
12
12
12
13
14
15
15
15
15
15
15
15
16
17
POINTS
BELOW
3
4
5
5
5
5
6
7 •
7
7
7
8
9
9
9
9
9
9
9
9
9
10
11
11
11
11
11
12
13
14
15
16
16
16
16
PROS./
ZVAL
0.9375
0.9688
0.9844
0.9375
0.8555
0.8555
0.9102
0.9453
0.8867
0.8062
0.7095
0.7880
0.8491
0.7728
0.6855
0.6855
0.5927
0.6762
0.7483
0.6765
0.6765
0.4477
0.2294
0.2294
0.4287
0.6200
0.8041
0.5966
0.3969
0.2043
0.0183
0.1976
0.1976
0.0177
0.1915
RECOMMENDATION
NEGATIVE TREND !
NEGATIVE TREND !
84
Table 13. RESULTS OF ORDINARY SIGN TEST ON MARKET SECURITY 6 Continued
Points in neutral corridor ==>
Upper boundary =>
3
71.875
Lower boundary
=>
69.375
TABULATION OF RESULTS
TOTAL
POINTS
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
NON-ZERO
POINTS
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
56
57
58
59
60
61
62
63
64
65
POINTS
ABOVE
17
18
19
20
21
22
23
24
25
26
27
28
2.9
30
31
32
33
34
35
36
37
38
39
40
40
40
40
41
41
41
41
41
41
41
POINTS
BELOW
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
16
17
18
18
19
20
21
22
23
24
PROB./ '
ZVAL
0.1915
0.3601
0.5240
0.6833
0.8384
0.9896
1.1369
1.2807
1.4212
1.5585
1.6927
1.8241
1.9528
2.0789
2.2026
2.3238
2.4429
2.5597
2.6745
2.7874
2.8983
3.0074
3.1148
3.2205
3.2205
3.0597
2.9019
3.0074
2.8531
2.7016
2.5527
2.4064
2.2625
2.1210
RECOMMENDATION
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
POSITIVE
-TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
TREND
85
Table 13. RESULTS OF ORDINARY SIGN TEST ON MARKET SECURITY 6 Continued
Points in neutral corridor ==>
Upper boundary =>
71.875
3
,
Lower boundary
=>
69.375
TABULATION OF RESULTS
TOTAL
POINTS
NON-ZERO
POINTS
POINTS
ABOVE
POINTS
BELOW
PROB./
ZVAL
RECOMMENDATION
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
41
41
41
41
41
41
41
41
41
41
41
41
41
42
43
44
45
46
47
48
49
50
51
52
53
25
26
27
28
29
30
31
32
33
34
35
36
37
37
37
37
37
37
37
37
37
37
37
37
37
1.9818
1.8448
1.7099
1.5771
1.4462
1.3173
1.1903
1.0651
0.9416
0.8198
0.6997
0.5812
0.4642
0.5738
0.6820
0.7889
0.8945
0.9989
1.1020
1.2040
1.3048
1.4045
1.5031
1.6006
1.6971
POSITIVE TREND !
POSITIVE TREND !
POSITIVE TREND !
POSITIVE TREND !
86
Table 14. RESULTS OF ORDINARY SIGN TEST ON MARKET SECURITY 7
Points in neutral corridor ==>
Upper boundary =>
5
63.75
Lower boundary
=>
60.75
TABULATION OF RESULTS
TOTAL
POINTS
NON-ZERO
POINTS
POINTS
ABOVE
POINTS
BELOW
PROB./
ZVAL
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
I
2
3
3
3
3
3
4
4
4
4
5
6
7
8
8
8
8
8
8
8
3
3
3
4
5
6
7
7
8
9
10
10
10
10
10
11
12
13
14
15
16
0.9375
0.8125
0.6563
0.7734
0.8555
0.9102
0.9453
0.8867
0.9270
0.9539
0.9713
0.9408
0.8949
0.8338
0.7597
0.8204
0.8684
1.1129
1.3005
1.4805
1.6534
'
RECOMMENDATION
NEGATIVE TREND !
NEGATIVE TREND I
NEGATIVE TREND !
87
APPENDIX F
RESULTS OF WILCOXON SIGNED RANK TESTS
88
Table 15. RESULTS OF WILCOXON SIGNED RANK TEST ON MARKET
SECURITY 2
Initial Sample size =
39
Number of points in neutral corridor =
Value of Center Line =
Periods
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
Nonzero Diffs.
6
7
8
9
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
24
25
26
27
28
29
30
31
32
33
34
35
36
37
5
44.625
Sum 1+ 1 Ranks
10
10.5
16
24.5
24.5
34
43
53
66
77.5
87
96.5
103
107
109
111
112
113
114
120.5
120.5
136
146.5
161
181.5
210.5
240.5
271.5
297
328
355.5
390.5
426.5
463.5
Sum '-' Ranks
11
17.5
20
20.5
20.5
21
23
25
25
27.5
33
39.5
50
64
81
99
119
140
162
179.5
179.5
189
204.5
217
224.5
224.5
224.5
224.5
231
233
239.5
239.5
239.5
239.5
Recommendation
Positive Trend
89
Table 16. RESULTS OF WILCOXON SIGNED RANK TEST ON MARKET
SECURITY 3
Initial Sample size =
55
Number of points in neutral corridor =
Value of Center Line =
Periods
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
Nonzero Diffs.
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
5
57.3125
Sum '+' Ranks
16.5
21.5
26.5
27.5
30.5
34
43.5
49.5
58
69
85
96.5
109
120
132.5
137
139
158
178
197
217
220.5
230.5
259.5
288.5
319.5
351.5
384.5
418.5
453.5
489.5
526.5
564.5
603.5
643.5
Sum '-' Ranks
4.5
6.5
9.5
17.5
24.5
32
34.5
41.5
47
51
51
56.5
62
70
77.5
94
114
118
122
128
134
157.5
175.5
175.5
176.5
176.5
176.5
176.5
176.5
176.5
176.5
176.5
176.5
176.5
176.5
Recommendation
-
-
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
90
Table 16. RESULTS OF WILCOXON SIGNED RANK TEST ON MARKET
SECURITY 3 - Continued
Initial Sample size =
55
Number of points in neutral corridor =
Center :
Line =
Value of i
Periodsi
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
Nonzero Diffs
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
5
57.3125
Sum 1+ 1 Ranks
684.5
726.5
769.5
813.5
858.5
904.5
951.5
999.5
1043.5
1092.5
1143.5
1195.5
1248.5
1302.5
1357.5
Sum 1- 1 Ranks
176.5
176.5
176.5
176.5
176.5
176.5
176.5
176.5
181.5
182.5
182.5
182.5
182.5
182.5
182.5
Recommendation
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
91
Table 17. RESULTS OF WILCOXON SIGNED RANK TEST ON MARKET
SECURITY 4
Initial Sample size =
30
Number of points in neutral corridor =
3
Value of Center Line = 8 4 . 7 5
Periods
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Nonzero Diffs.
3
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Sum 1+ 1 Ranks
6
6
10
15
21
26
34
43
50.5
57.5
65
65
71
71
74.5
84.5
100.5
119.5
139.5
160.5
182.5
205.5
229.5
254.5
280.5
307.5
335.5
Sum 1- 1 Ranks
0
0
0
0
0
2
2
2
4.5
8.5
13
26
34
49
61.5
68.5
70.5
70.5
70.5
70.5
70.5
70.5
70.5
70.5
70.5
70.5
70.5
Recommendation
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Trend
Trend
Trend
Trend
Trend
Trend
Trend
92
Table 18. RESULTS OF WILCOXON SIGNED RANK TEST ON MARKET
SECURITY 5
Initial Sample size =
40
Number of points in neutral corridor =
Value of Center Line =
Periods
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
Nonzero Diffs.
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
7
45.75
Sum '+ 1 Ranks
25.5
33
43
53.5
65
77.5
89.5
99.5
115
131.5
149.5
168.5
188.5
209.5
231.5
254.5
278.5
303.5
329.5
356.5
384.5
413.5
443.5
474.5
506.5
539.5
573.5
608.5
644.5
681.5
719.5
758.5
798.5
Sum 1- 1 Ranks
10.5
12
12
12.5
13
13.5
15.5
20.5
21
21.5
21.5
21.5
21.5
21.5
21.5
21.5
21.5
21.5
21.5
21.5
21.5
21.5
21.5
21.5
21.5
21.5
21.5
21.5
21.5
21.5
21.5
21.5
21.5
Recommendation
Positive Trend
Positive Trend
Positive Trend
Positive Trend
Positive Trend
Positive Trend
Positive Trend
Positive Trend
Positive..Trend
Positive Trend
Positive Trend
Positive Trend
Positive Trend
Positive Trend
Positive Trend
.Positive Trend
Positive Trend
Positive Trend
Positive Trend
Positive Trend
Positive Trend
Positive Trend
Positive Trend
Positive Trend
Positive Trend
Positive Trend
Positive Trend
Positive Trend
Positive Trend
Positive Trend
93
Table 19. RESULTS OF WILCOXON SIGNED RANK TEST ON MARKET
SECURITY 6
Initial Sample size =
97
Number of points in neutral corridor =
Value of Center Line =
Periods
4
5
6
7
8
9
10
11
12
13
•14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
Nonzero Diffs
4
5
6
7
8
8
9
10
11
12
13
14
15
16
17
17
18
19
20
21
. 21
22
23
23
24
25
26
27
28
29
30
. 31
31
32
33
3
70.625
Sum 1+ 1 Ranks
2
2
3
7
12
12
13
13
18.5
30.5
41.5
42
47.5
60.5
72
72
83
93.5
106.5
121.5
121.5
132.5
144
144
165
190
216
226
227
227
229.5
239.5
239.5
270.5
302
Sum 1-' Ranks
8
13
18
21
24
24
32
42
47.5
47.5
49.5
63
72.5
75.5
81
81
88
96.5
103.5
109.5
109.5
120.5
132
132
135
135
135
152
179
208
235.5
256.5
256.5
257.5
259
Recommendation
94
Table 19. RESULTS OF WILCOXON SIGNED RANK ON MARKET
SECURITY 6 - Continued
Initial Sample size =
97
Number of points in neutral corridor =
Value of Center Line =
Periods
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
Nonzero Diffs.
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
56
57
58
59
60
61
62
63
64
65
66
3
70.625
Sum 1+ 1 Ranks
302
322.5
342.5
378
413
437.5
470.5
508.5
547.5
587.5
630
672
717
763
810
858
907
957
1008
1058
1111
1159
1205.5
1255.5
1255.5
1294.5
1331.5
1376.5
1405.5
1420
1444
1462.5
1475
1480
1487
Sum '-' Ranks
259
272.5
287.5
288
290
303.5
309.5
311.5
313.5
315.5
316
318
318
318
318
318
318
318
318
320
320
326
334.5
340.5
340.5
358.5
379.5
393.5
424.5
471
509
553.5
605
665
724
Recommendation
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
95
Table 19. RESULTS OF WILCOXON SIGNED RANK ON MARKET
SECURITY 6 - Continued
Initial Sample size =
97
Number of points in neutral corridor =
Value of Center Line =
Periodsi
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
Nonzero Diffs
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
3
70.625
Sum '+' Ranks
1492.5
1497.5
1500
1505
1521.5
1528.5
1534
1539
1544.5
1565
1602.5
1637
1694.5
1749.5
1813
1892
1975
2059
2144
2227
2314
2402
2488
2569
Sum 1- 1 Ranks
785.5
848.5
915
980
1034.5
1099.5
1167
1236
1305.5
1361
1400.5
1444
1465.5
1490.5
1508
1511
1511
1511
1511
1514
1514
1514
1517
1526
Recommendation
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
96
Table 20. RESULTS OF WILCOXON SIGNED RANK TEST ON MARKET
SECURITY 7
Initial Sample size =
26
Number of points in neutral corridor =
Value of Center :
Line =
Periods
6
7
8
9
10
11
12
13
14
15
•16
17
18
19
20
21
22
23
24
25
26
Nonzero Diffs .
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
5
62.25
Sum 1+ 1 Ranks
3.5
7.5
10.5
12.5
15
17
20
29
30
30
30
40
54
69
77.5
77.5
77.5
77.5
77.5
77.5
77.5
Sum '-' Ranks
6.5
7.5
10.5
15.5
21
28
35
37
48
61
75
80
82
84
93.5
112.5
132.5
153.5
175.5
198.5
222.5
Recommendation
Negative trend
Negative trend
97
I
APPENDIX G
RESULTS OF COX-STUART TESTS
98
Table 21. RESULTS OF COX-STUART TEST ON MARKET SECURITY 2
Initial Sample size =
39
Number of points in neutral corridor =
POINTS
+ SIGNS
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
2
I
I
I
I
2
2
I
I
0
0
2
2
2
2
4
4
6
6
9
9
10
10
12
12
12
12
11
11
9
9
8
8
8
8
- SIGNS
0
2
2
3
3
3
3
4
4
7
7
6
6
7
7
6
6
4
4
3
3
2
2
2
2
3
3
4
4
6
6
7
7
10
10
PROB./ZVAL
0.250
0.500
0.500
0.313
0.313
0.500
0.500
0.188
0.188
0.008
0.008
0.145
0.145
0.090
0.090
0.377
0.377
0.377
0.377
0.073
0.073
0.019
0.019
0.006
0.006
0.018
0.018
0.059
0.059
0.304
.0.304
0.500
0.500
0.407
0.407
5
RECOMMENDATION
Positive Trend !
Positive Trend !
Negative
Negative
Negative
Negative
Negative
Negative
Trend
Trend
Trend
Trend
Trend
Trend
!
!
!
!
!
!
99
Table 22. RESULTS OF COX-STUART TEST ON MARKET SECURITY 3
Initial Sample size =
55
Number of points in neutral corridor =
POINTS
+ SIGNS
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
0
I
I
3
3
4
4
4
4
5
5
6
6
4
4
3
3.
5
5
5
5
6
6
9
9
8
8
7
7
5
5
4
4
5
5
6
6
6
6
- SIGNS
2
2
2
I
I
I
I
2
2
2
2
2
2
4
4
7
7
5
5
6
6
6
6
5
5
7
7
9
9
12
12
14
14
13
13
13
13
15
15
PROB./ZVAL
5
RECOMMENDATION
0.250
0.500
0.500
0.313
0.313
0.188
0.188
0.344
0.344
0.227
0.227
0.145
0.145
0.637
0.637
0.172
0.172
0.623
0.623
0.500
0.500
0.613
0.613
0.212
0.212
0.500
0.500
0.402
0.402
0.072
0.072
0.015
0.015
0.048
0.048
0.084
0.084
-1.942
-1.942
Positive
Positive
Positive
Positive
Trend
Trend
Trend
Trend
!
!
!
!
Positive Trend !
Positive Trend !
100
Table 22. RESULTS OF COX-STUART TEST ON MARKET SECURITY 3 ’
Continued
Initial Sample size =
55
Number of points in neutral corridor =
POINTS
+ SIGNS
44
45
46
47
48
49
50
51
52
53
54
55
4
4
4
4
3
3
3
3
2
2
0
0
- SIGNS
18
18
19
19
20
20
21
21
23
23
27
27
PROB./ZVAL
-2.963
-2.963
-3.107
-3.107
-3.524
-3.524
-3.654
-3.654
-4.180
-4.180
-5.177
-5.177
5
RECOMMENDATION
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
1 01
Table 23. RESULTS OF COX-STUART TEST ON MARKET SECURITY 4
Initial Sample size =
30
Number of points in neutral corridor =
POINTS
+ SIGNS
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
I
I
I
2
2
I
I
2
2
3
3
5
5
6
6
8
8
7
7
8
8
7
7
6
6
4
4
4
- SIGNS
0
I
I
I
I
2
2
3
3
3
3
2
2
2
2
I
I
3
3
3
3
5
5
7
7
10
10
11
PROB./ZVAL
0.500
0.750
0.750
0.500
0.500
0.500
0.500
0.500
0.500
0.656
0.656
0.227
0.227
0.145
0.145
0.020
0.020
0.172
0.172
0.113
0.113
0.387
0.387
0.500
0.500
0.090
0.090
0.059
3
RECOMMENDATION
Negative Trend !
Negative Trend !
•
102
Table 24. RESULTS OF COX-STUART TEST ON MARKET SECURITY 5
Initial Sample size =
40
Number of points in neutral corridor =
POINTS
+ SIGNS
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
0
I
I
2
2
2
2
3
3
3
3
3
3
4
4
3
3
I
I
I
I
0
0
0
0
0
0
0
0
0
0
0
0
0
- SIGNS
3
3
3
3
3
3
3
3
3
4
4
5
5
6
6
8
8
10
10
11
11
13
13
14
14
16
16
17
17
18
18
19
19
20
PROB./ZVAL
0.125
0.313
0.313
0.500
0.500
0.500
0.500
0.656
0.656
0.500
0.500
0.363
0.363
0.377
0.377
0.113
0.113
0.006
0.006
0.003
0.003
0 . 000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0 . 000
7 '
RECOMMENDATION
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
103
Table 25. RESULTS OF COX-STUART TEST ON MARKET SECURITY 6
Initial Sample size =
97
Number of points in neutral corridor =
POINTS
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
+ SIGNS
I
I
I
I
I
I
I
I
I
0
0
I
’ I
4
4
3
3
3
3
3
3
4
4
5
5
5
5
5
5
4
4
7
7
9
9
9
9
6
- SIGNS
0
0
0
I
I
3
3
4
4
5
5
5
5
4
4
5
5
5
5
7
7
8
8
7
7
8
8
9
9
12
12
10
10
8
8
9
9
12
PROB./ZVAL
0.500
0.500
0.500
0.750
0.750
0.313
0.313
0.188
0.188
0.031
0.031
0.109
0.109
0.637
0.637
0.363
0.363
0.363
0.363
0.172
0.172
0.194
0.194
0.387
0.387
0.291
0.291
0.212
0.212
0.038
0.038
0.315
0.315
0.500
0.500
0.593
0.593
0.119
3
RECOMMENDATION
Positive Trend !
Positive Trend I
Positive Trend !
Positive Trend !
104
Table 25. RESULTS OF COX-STUART TEST ON MARKET SECURITY 6 Continued
Initial Sample size =
97
Number of points in neutral corridor =
POINTS
41
42
43
44
45
46
47 ■
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71 '■
72
73
74
75
76
77
+ SIGNS
6
7
7
9
9
9
9
8
8
6
6
8
8
8 •
8
8
8
7
7
6
6
7
■ 7
9
9
6
6
5
5
5
5
8■
8
10
10
11
11
- SIGNS
12
14
14
12
12
14
14
16
16
19
19
18
18
18
18
19
19
22
22
23
23
23
23
22
22
27
27
28
28
30
30
27
27
27
27
26
26
PROS./ZVAL
0.119
-1.506
-1.506
-0.633
-0.633
-1.022
-1.022
-1.613
-1.613
-2.580
-2.580
-1.942
-1.942
-1.942
-1.942
-2.098
-2.098
-2.767
-2.767
-3.138
-3.138
-2.903
-2.903
-2.317
-2.317
-3.638
-3.638
-3.986
-3.986
-4.209
-4.209
-3.195
-3.195
-2.778
-2.778
-2.450
-2.450
3
RECOMMENDATION
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
105
Table 25. RESULTS OF COX-STUART TEST ON MARKET SECURITY 6
Continued
Initial. Sample size =
97
Number of points in neutral corridor =
POINTS
+ SIGNS
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
13
13
17
17
19
19
21
21
25
25
25
25
23
23
23
23
23
23
23
23
- SIGNS
24
24
22
22
22
22
21
21
16
16
18
18
21
21
22
22
24
24
24
24
PROB./ZVAL
-1.792
-1.792
-0.785
-0.785
-0.453
-0.453
0.015
0.015
-1.390
-1.390
-1.052
-1.052
-0.286
-0.286
-0.134
-0.134
-0.131
-0.131
-0.131
-0.131
3
RECOMMENDATION
Positive Trend !
Positive Trend !
'I
106
Table 26. RESULTS OF COX-STUART TEST ON MARKET SECURITY 7
Initial Sample size =
26
Number of points in neutral corridor =
POINTS
+ SIGNS
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
2
2
2
I
I
2
2
3
3
4
4
5
5
5
5
6
6
5
5
9
9
10
- SIGNS
0
I
I
2
2
3
3
3
3
3
3
3
3
4
4
4
4
6
6
3
3
3
PROB./ZVAL
0.250
0.500
0.500
0.500
0.500
0.500
0.500
0.656
0.656
0.500
0.500
0.363
0.363
0.500
0.500
0.377
0.377
0.500
0.500
0.073
0.073
0.046
5
RECOMMENDATION
Negative Trend !
107
APPENDIX H
RESULTS OF MODIFIED COX-STUART TESTS
108
Table 27. RESULTS OF MODIFIED COX-STUART TEST ON MARKET
SECURITY 2
INITIAL SAMPLE SIZE =
39
NUMBER OF POINTS IN NEUTRAL CORRIDOR =
PERIODS
NONZERO DIFFS.
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
3
3
4
4
5
5
5
5
7
7
8
8
9
9
10
10
10
10
12
12
12
12
14
14
15
15
15
15
15
15
15
15
18
18
SUM 1+' RANKS
I
I
1.5
1.5
3.5
3.5
I
I
0
0
6
6
12
12
23.5
23.5
35
35
58
58
61
61
89.5
89.5
108.5
108.5
86
86
68.5
68.5
60.5
60.5
68.5
68.5
5
SUM 1- ' RANKS
5
5
8.5
8.5
11.5
11.5
14
14
28
28
30
30
33
33
31.5
31.5
20
20
20
20
17
17
15.5
15.5
11.5
11.5
34
34
51.5
51.5
59.5
59.5
102.5
102.5
RECOMMENDATION
Positive Trend
Positive Trend
Negative
Negative
Negative
Negative
Negative
Negative
trend
trend
trend
trend
trend
trend
109
Table 28. RESULTS OF MODIFIED COX-STUART TEST ON MARKET
SECURITY 3
INITIAL SAMPLE SIZE =
55
NUMBER OF POINTS IN NEUTRAL CORRIDOR =
PERIODS
NONZERO DIFFS.
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
3
3
4
4
5
5
6
6
7
7
8
8
8
8
10
10
10
10
11
11
12
12
14
14
15
15
16
16
17
17
18
18
18
18
19
19
21
21
SUM '+' RANKS
I
I
6
6
12
12
17.5
17.5
24
24
24
24
16.5
16.5
22
22
27.5
27.5
28
28
31
31
49.5
49.5
55
55
57
57
41.5
41.5
39
39
30
30
26
26
25.5
25.5
5
SUM '-' RANKS
5
5
4
4
3
3
3.5
3.5
4
4
12
12
19.5
19.5
33
33
27.5
27.5
38
38
47
47
55.5
55.5
65
65
79
79
111.5
111.5
132
132
141
141
164
164
205.5
205.5
RECOMMENDATION
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
trend
trend
trend
trend
trend
trend
trend
trend
HO
Table 28. RESULTS OF MODIFIED COX-STUART TEST ON MARKET
SECURITY 3 - Continued
INITIAL SAMPLE SIZE =
55
NUMBER OF POINTS IN NEUTRAL CORRIDOR =
PERIODS
NONZERO DIFFS.
44
45
46
47
48
49
50
51
52
53
54
55
22
22
23
23
23
23
24
24
25
25
27
27
SUM '+ ' RANKS
19.5
19.5
18
18
10.5
10.5
12.5
12.5
9
9
0
0
5
SUM '- ' RANKS
233.5
233.5
258
258
265.5
265.5
287.5
287.5
316
316
378
378
RECOMMENDATION
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
Ill
Table 29. RESULTS OF MODIFIED COX-STUART TEST ON MARKET
SECURITY 4
INITIAL SAMPLE SIZE =
30
NUMBER OF POINTS IN NEUTRAL CORRIDOR =
PERIODS
NONZERO DIFFS.
4
5
6
7
8
9
10
11
12
13
14
•15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
2
2
3
3
3
3
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
13
14
14
15
SUM '+' RANKS
2
2
3
3
3
3
6
6
11
11
19
19
28
28
39
39
46.5
46.5
39.5
39.5
38
38
28.5
28.5
18
18
14
3
SUM '-' RANKS
I
I
3
3
3
3
9
9
10
10
9
9
8
8
6
6
8.5
8.5
26.5
26.5
40
40
62.5
62.5
87
87
106
RECOMMENDATION
Negative
Negative
Negative
Negative
Trend
Trend
Trend
Trend
Positive trend
Positive trend
Positive trend
112
Table 30. RESULTS OF MODIFIED COX-STUART TEST ON MARKET
SECURITY 5
INITIAL SAMPLE SIZE =
40
NUMBER OF POINTS IN NEUTRAL CORRIDOR =
PERIODS
NONZERO DIFFS.
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
4
4
5
5
5
5
6
6
7
7
8
8
10
10
11
11
11
11
12
12
13
13
14
14
16
16
17
17
18
18
19
19
20
.
SUM '+' RANKS
I
I
3
3
5
5
7.5
7.5
9
9
10
10
16
16
14
14
3
3
I
I
0
0
0
0
0
0
0
0
0
0
0
0
0
7
SUM '- ' RANKS
9
9
12
12
10
10
13.5
13.5
19
19
26
26
39
39
52
52
63
63
77
77
91
91
105
105
136
. 136
153
153
171
171
190
190
210
RECOMMENDATION
-
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
1 13
Table 31. RESULTS OF MODIFIED COX-STUART TEST ON MARKET
SECURITY 6
INITIAL SAMPLE SIZE =
97
NUMBER OF POINTS IN NEUTRAL CORRIDOR =
PERIODS
NONZERO DIFFS.
4
5
6
7
8
I
I
2
2
4
4
5
5
5
5
6
6
8
8
8
8
8
8
10
10
12
12
12
12
13
13
14
14
16
16
17
17
17
17
18
18
18
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
SUM '+' RANKS
I
I
2
2
3
3
2
2
0
0
3
3
17
17
14.5
14.5
11
11
8
8
19
19
27
27
33.5
33.5
22
22
28.5
28.5
68.5
68.5
78.5
78.5
70
70
65
3
SUM '-' RANKS
0
0
I
I
7
7
13
13
15
15
18
18
19
19
21.5
21.5
25
25
47
47
59
59
51
51
57.5
57.5
83
83
107.5
107.5
84.5
84.5
74.5
74.5
101
101
106
RECOMMENDATION
Positive Trend
Positive Trend
Positive
Positive
Positive
Positive
Trend
Trend
Trend
Trend
114
Table 31. RESULTS OF MODIFIED COX-STUART TEST ON MARKET
SECURITY 6 - Continued
INITIAL SAMPLE SIZE =
97
NUMBER OF POINTS IN NEUTRAL CORRIDOR =
PERIODS
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
NONZERO DIFFS.
18
21
21
21
21
23
23
24
24
25
25
26
26
26
26
27
27
29
29
29
29
30
30
31
31
33
33
33
33
35
35
35
35
37
37
37
37
SUM '+' RANKS
65
85.5
85.5
93.5
93.5
112
112
93.5
93.5
85.5
85.5
88.5
88.5
78.5
78.5
59.5
59.5
54
54
62
62
70.5
70.5
74.5
74.5
64
64
64
64
72
72
108
108
188.5
188.5
233.5
233.5
3
SUM '- ' RANKS
106
145.5
145.5
137.5
137.5
164
164
206.5
206.5
239.5
239.5
262.5
262.5
272.5
272.5
318.5
318.5
381
381
373
373
394.5
394.5
421.5
421.5
497
497
497
497
558
558
522
522
514.5
514.5
469.5
469.5
RECOMMENDATION
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
Trend
115
Table 31. RESULTS OF MODIFIED COX-STUART TEST ON MARKET
SECURITY 6 - Continued
INITIAL SAMPLE SIZE =
97
NUMBER OF POINTS IN NEUTRAL CORRIDOR =
PERIODS
78
79
80
81
82
83
84
85
86
87
88
89
' 90
91
92
93
94
95
96
97
NONZERO DIFFS.
37
37
39
39
' 41
41
42
42
41
41
43
43
44
44
45
45
47
47
47
47
SUM
' + '
RANKS
255
255
342.5
342.5
413.5
413.5
464.5
464.5
447.5
447.5
494.5
494.5
483
483
472
472
511
511
486.5
486.5
3
SUM '-' RANKS
448
448
437.5
437.5
447.5
447.5
438.5
438.5
413.5
413.5
451.5
451.5
507
507
563
563
617
617
641.5
641.5
RECOMMENDATION
Positive Trend
Positive Trend
116
Table 32. RESULTS OF MODIFIED COX-STUART TEST ON MARKET
SECURITY 7
INITIAL. SAMPLE SIZE =
26
NUMBER OF POINTS IN NEUTRAL CORRIDOR =
PERIODS
6
7
8
9
10
11
12
13
14
15
16
17
'18
19
20
21
22
23
24
25
26
NONZERO DIFFS.
3
3 •
3
3
5
5
6
6
7
7
8
8
9
9
10
10
11
11
12
12
13
SUM '+' RANKS
4
4
2
2
5
5
8
8
17
17
26
26
28
28
32.5
32.5
45
45
63.5
63.5
77
5
SUM '-' RANKS
RECOMMENDATION
2
2
4
4
10
10
13
13
11
11
10
10
17
17
22.5
22.5
21
21
14.5
14.5
.14
Negative Trend
Negative Trend
Negative Trend
117
APPENDIX I
RESULTS OF KENDALL'S TESTS
118
Table 33. RESULTS OF KENDALL'S TEST ON MARKET SECURITY 2
Initial Sample size =
PERIODS
6
I
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
TEST STAT.
26
48
53.5
53.5
105.5
107
115
127.5
127.5
178
344
534.5
765
1039
1401.5
1801
2263
2767
3313.5
3753
4142.5
4347.5
4829
5198
5343.5
5343.5
5345.5
5351.5
5499.5
5559.5
5735
5787.5
5787.5
5787.5
39
RECOMMENDATION
Positive
Positive
Positive
Positive
trend
trend
trend
trend
!
!
!
!
Negative
Negative
Negative
Negative
Negative
Negative
trend
trend
trend
trend
trend
trend
!
!
!
!
!
!
Positive trend !
Positive trend !
Positive trend !
119
Table 34. RESULTS OF KENDALL'S TEST ON MARKET SECURITY 3
Initial Sample size =
PERIODS
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
TEST STAT.
20
50
90.5
162.5
232.5
316.5
364.5
466
570
676.5
676.5
825
990.5
1184.5
1400.5
1745
2193.5
2332
2479.5
2729
2996
3600
4166
4215.5
4295.5
4381.5
4381.5
4381.5
4383.5
4388
4394
4402
4402
4402
4402
4402.5
4431
4444.5
4450
4450
4498
55
RECOMMENDATION
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
!
!
!
!
!
!
!
!
!
!
!
!
I
120
Table 34. RESULTS OF KENDALL'S TEST ON MARKET SECURITY 3 Continued
Initial Sample size =
PERIODS
47
48
49
50
51
52
53
54
55
TEST STAT.
4502
4675.5
5516.5
6114
6450
6605.5
6873
7063.5
7253.5
55
RECOMMENDATION
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
trend
trend
trend
trend
trend
trend
trend
trend
trend
!
!
!
!
!
!
!
!
!
121
Table 35. RESULTS OF KENDALL'S TEST ON MARKET SECURITY 4
Initial Sample size =
PERIODS
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
TEST STAT.
14
31.5
41.5
54
77.5
149.5
173
173
299
455
622.5
832.5
1070.5
1342.5
1640.5
1829
1865.5
1865.5
1867.5
1873.5
1873.5
1873.5
1897.5
1913
1927
1927
1927
30
RECOMMENDATION
Negative
Negative
Negative
Negative
Negative
Negative
trend
trend
trend
trend
trend
trend
!
!
!
!
!
!
Positive
Positive
Positive
Positive
Positive
trend
trend
trend
trend
trend
!
!
!
!
!
122
Table 36. RESULTS OF KENDALL'S TEST ON MARKET SECURITY 5
Initial Sample size =
PERIODS
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
TEST STAT.
33.5
61.5
85.5
124
171.5
228.5
326.5
484.5
575.5
677
677
677
696.5
721
735
735
735
746.5
746.5
754.5
766.5
788.5
792
796.5
822
891.5
1053.5
1288
1502
1502
1504
'1508
1510.5
40
RECOMMENDATION
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
123
Table 37. RESULTS OF KENDALL'S TEST ON MARKET SECURITY 6
Initial Sample size =
PERIODS
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
TEST STAT.
15.5
30
40
40
40.5
62.5
114.5
199.5
239.5
239.5
241.5
451.5
580
590.5
630
763.5
895
1053
1215.5
1359
1601.5
1933.5
2277
2599
2652.5
2652.5
2654.5
3306.5
4298.5
5354.5
6451
7298.5
7912.5
7920
7940
8662.5
9308.5
10008
10008
10072.5
10856
97
RECOMMENDATION
Positive trend !
Positive trend !
Positive trend !
Positive trend !
124
Table 37. RESULTS OF KENDALL'S TEST ON MARKET SECURITY 6 Continued
Initial Sample size =
PERIODS
.45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
TEST STAT.
11095
11171
11255.5
11375
11381.5
11505.5
11505.5
11505.5
11505.5
11511.5
11527
11541 •
11563.5
11831
11877
12594
13651
14459.5
16691
19345.5
22237
24305.5
27870
32284
36391
41067
45921.5
51033.5
56287.5
61685.5
67232
72932
78773.5
84521.5
90635
96908.5
103356.5
109929
116202.5
97
RECOMMENDATION
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
I
!
!
!
Negative trend !
Negative trend !
125
Table 37. RESULTS OF KENDALL'S TEST ON MARKET SECURITY 6 Continued
Initial Sample size =
PERIODS
84
85
86
87
88
89
90
91
92
93
94
95
96
97
TEST STAT.
121067
126268.5
128226
130789
132475.5
132821.5
132821.5
132821.5
132823.5
133225
133283.5
133479.5
133940
134600.5
97
RECOMMENDATION
Negative
Negative
Negative
Negative
trend
trend
trend
trend
!
!
!
!
126
Table 38. RESULTS OF KENDALL S TEST ON MARKET SECURITY 7
Initial Sample size =
PERIODS
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
TEST STAT.
46
58
74
134
186
277.5
349.5
385.5
. 545.5
755.5
993.5
1049
1049
1051
1165
1573
2019
2525
3077
3677
432.7
26
RECOMMENDATION
Negative trend !
Negative trend !
Negative trend !
127
APPENDIX J
RESULTS OF NEW EMPIRICAL METHOD
128
Table 39. RESULT OF TEST WITH NEW EMPIRICAL METHOD ON
MARKET SECURITY 2
Points in neutral corridor ==>
Upper boundary =>
5
45.5
Lower boundary =>
NUMBER OF
POINTS
TEST STATISTIC
RATIO
38
39
1.208636
2.426475
SIGNAL
Positive trend !
Positive trend I
43.75'
129
Table 40. RESULT OF TEST WITH NEW EMPIRICAL METHOD ON
MARKET SECURITY 3
Points in neutral corridor ==>
Upper boundary =>
NUMBER OF
POINTS
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
51
52
53
54
55
5
58.5
Lower boundary =>
TEST STATISTIC
RATIO
I .099998
1.32438
1.52925
1.866536
2.264048
2.725309
3.365859
4.018759
4.626633
4.905608
5.393173
5.901108
6.6604
6.964272
7.536541
7.594529
■ 7.146586
7.324341
7.433609
7.596066
7.781936
SIGNAL
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
56.125
130
Table 41. RESULT OF TEST WITH NEW EMPIRICAL METHOD ON
MARKET SECURITY 4
Points in neutral corridor ==>
Upper boundary =>
NUMBER OF
POINTS
22
23
24
25
26
27
28
29
30
3
86
Lower boundary =>
TEST STATISTIC
RATIO
1.254642
1.306354
2.150102
2.961543
3.048129
3.340749
3.704159
5.090116
7.359515
SIGNAL
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
trend
trend
trend
trend
trend
trend
trend
trend
trend !
83.5
131
Table 42. RESULT OF TEST WITH NEW EMPIRICAL METHOD ON
MARKET SECURITY 5
Data correspond to ===> MARKET SECURITY 5
Points in neutral corridor ==>
Upper boundary =>
NUMBER OF
POINTS
28
29
30
31
32
33
34
35
36
37
38
39
40
7
47.75
Lower boundary =>
TEST STATISTIC
RATIO
1.136184
1.192074
1.292359
1.384931
1.458309
1.49471
1.503407
1.48166
1.477168
1.702656
1.82248
1.948561
2.141253
SIGNAL
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
!
!
!
!
!
!
!
!
!
!
!
!
!
43.75
132
Table 43. RESULT OF TEST WITH NEW EMPIRICAL METHOD ON
MARKET SECURITY 6
Data correspond to ===> MARKET SECURITY 6
Points in neutral corridor ==>
Upper boundary =>
NUMBER OF
POINTS
54
55
56
57
58
59
92
93
94
95
96
97
3
71.875
Lower boundary =>
TEST STATISTIC
RATIO
1.042072
1.08077
1.176833
1.250869
1.243901
1.28476
I.764774E+10
I.728758E+10
I .694183E+10
I .660963E+10
I .629022E+10
I .598286E+10
SIGNAL
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
Positive
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
trend
!
!
I
!
!
!
!
!
!
!
!
!
69.375
1 33
Table 4 4 . RESULT OF TEST WITH NEW EMPIRICAL METHOD ON
MARKET SECURITY 7
Data correspond to ===> MARKET SECURITY 7
Points in neutral corridor ==>
Upper boundary =>
NUMBER OF
POINTS
16
22
23
24
25
26
5
63.75
Lower boundary =>
TEST STATISTIC
RATIO
1.102831
1.722097
3.073235
4.58988
6.201894
7.920076
SIGNAL
Negative
Negative
Negative
Negative
Negative
Negative
trend
trend
trend
trend
trend
trend
!
!
!
!
!
!
60.75
134
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