Group Title: arbitrage model of security returns
Title: The arbitrage model of security returns
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Title: The arbitrage model of security returns an empirical evaluation
Physical Description: v, 145 leaves : ; 28 cm.
Language: English
Creator: Jordan, Bradford Dunson
Publication Date: 1984
Copyright Date: 1984
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Subject: Securities -- Mathematical models   ( lcsh )
Finance, Insurance, and Real Estate thesis Ph. D
Dissertations, Academic -- Finance, Insurance, and Real Estate -- UF
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Thesis: Thesis (Ph. D.)--University of Florida, 1984.
Bibliography: Bibliography: leaves 139-144.
Statement of Responsibility: by Bradford Dunson Jordan.
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General Note: Vita.
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THE ARBITRAGE MODEL OF SECURITY RETURNS:
AN EMPIRICAL EVALUATION







By

BRADFORD DUNSON JORDAN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1984
















TABLE OF CONTENTS


PAGE

ABSTRACT . . . . . . . . . . . . iv

CHAPTER

I. ESSENTIALS OF THE ARBITRAGE MODEL . . . ... 1

Introduction ..................... 1
An Alternative to the CAPM . . .... .. . . .. 2
Testing the APT ......... 4
The Arbitrage Model as a Tool in Financial Research 7
Summary and Overview . . . . . . .... 9

II. PREVIOUS RESEARCH IN MULTI-FACTOR MODELS.. .. . . 11

Introduction . . . ..... .. . .. 11
Applications of Multivariate Statistical Techniques 11
Multiple Regression Models of Security Returns 18
Tests of the Arbitrage Theory . . . . . .. 19
Summary . . . . . . . . . . 23

III. THE ARBITRAGE MODEL: THEORY AND ESTIMATION . .. 25

Introduction ..... . . . . . . . . 25
The Arbitrage Pricing. Theory ......... .. 25
Estimating the Arbitrage Model . . . . . 31
Measuring the Risk Premia .. . . .. ... 35
Summary . . . . . . . . . . . 38

IV. TESTING THE ARBITRAGE THEORY . . . ... .. .40

Introduction . . . . . . . . . ... 40
Factor Analysis of Daily Security Returns ...... 42
Preliminary Analyses of the Arbitrage Model . .. .55
Univariate Results from the Arbitrage Model ..... 72
A Multivariate Test of the APT . . . . . 84
Summary .. ... . .. . . . . . 86

V. AN EVENT STUDY COMPARISON OF THE MARKET MODEL
AND THE ARBITRAGE MODEL .......... 107

Introduction . . . . . . . . . . 107
Data for the Study . . . . . . . . 109
An Event Study Methodology . .. . . . 110
ii








CHAPTER


V. Impact of the Oil Embargo on the Petroleum Refining
and Oil Field Services Groups . . . ... 114
Impact of the Con Ed Dividend Omission on the
Electric Utility Group . . . . . 116
The Financial Services Group in the Period 8/73 -
9/74 . . . . . . . . . . 118
Some Results on the January Effect . . . . 123
Summary . . .. .. . . . . . . . 126

VI. RETURN, RISK AND ARBITRAGE: CONCLUSIONS . . .. 128

Introduction . . . . . . . . . 128
Testing the Arbitrage Theory . . . . . ... 129
Empirical Findings for the Arbitrage Model ... . 133
Implementing the Arbitrage Model . . . . .. 136
Conclusion . . . . . . . . . . . 138

REFERENCES . . . . . . . . . . . . . 139

BIOGRAPHICAL SKETCH . . .. . . .. .. . . . 145













Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


THE ARBITRAGE MODEL OF SECURITY RETURNS:
AN EMPIRICAL EVALUATION

By

BRADFORD DUNSON JORDAN

April, 1984

Chairman: R. H. Pettway
Major Department: Department of Finance, Insurance, and Real Estate

Over the last two decades, the Capital Asset Pricing Model (CAPM)

has emerged as the dominant theoretical basis for much of the research

in financial economics. Because direct observation of the market port-

folio is a pre-requisite for any valid application of the CAPM, it can-

not serve as a theoretical basis for empirical research in securities

markets. The Arbitrage Pricing Theory (APT) is a theoretical alternative

to the CAPM in which the market portfolio plays no particular role.

The purpose of this research is to develop and test a model of the

security return generating process based on the APT.

Particular emphasis is placed on two facets of the proposed arbi-

trage model. First, the central prediction of the APT is an absence of

arbitrage opportunities, the empirical identification of which would

lead to a rejection of the theory. Thus, the first use to which the

model is put is the examination of abnormal performance for the securities

individually and jointly. The second application involves an event

study comparison of the arbitrage model and a popular variant of the

iv








market model. The objective of this comparison is to establish the

stability and usefulness of the arbitrage model against a known bench-

mark. In light of the growing list of empirical anomalies associated

with the market model and the difficulties in application of the CAPM,

an empirically tractable and theoretically sound model of security

returns would be a significant step forward in financial research.

The data used in the study are daily returns for individual

securities from the CRSP file and cover the period 1962 through 1979.

The results indicate substantial support for the APT and the arbitrage

model. Significant arbitrage opportunities are found to occur in less

than 1% of the individual cases, and the hypothesis of jointly zero

abnormal performance cannot be rejected in any case. In the event

study comparison, the arbitrage model was found to work at least as

well as the market model in all cases and was markedly superior in

accounting for the January effect.













CHAPTER I
ESSENTIALS OF THE ARBITRAGE MODEL

Introduction

In the broadest sense, the primary concern of research in financial

economics is the relationship between risk and return in well-organized

markets. While security returns can generally be measured with relative

ease, the determination of an appropriate measure of risk is a far more

difficult question. Over the last two decades, the Capital Asset Pricing

Model (CAPM) has emerged as the dominant theoretical basis for much of

the research in this area. The fundamental result of the CAPM is

straight-forward: the relevant riskiness for any asset is determined

by the standardized covariance of its return with the return on the

market portfolio, i.e., the portfolio consisting of all risky assets

held in proportion to their value.

As a theory, the CAPM is extremely powerful and broadly applicable;

however, no valid test of its empirical content has appeared in the

literature. For reasons discussed in Roll (1977), such a test requires

that the return on the market portfolio be observed directly. Because

it is not technologically possible to obtain the necessary data, it is

unlikely that a valid test will be forthcoming. For the same reason,

any attempt to estimate the parameters of the model introduces bias of

unknown magnitude and direction.

The impetus for this research stems from the need for a model free

of these deficiencies. The purpose of this thesis is to develop and test

an empirically tractable model of security returns which retains the

1






2
intuitive appeal of CAPM-based models without the need for the market

portfolio in estimation. In the next section, the theoretical basis for

such a model is outlined.

An Alternative to the CAPM

The CAPM is a general equilibrium model of perfect markets with

homogeneous investor expectations. In such markets, the CAPM will hold

if investors have quadratic preferences or asset returns possess multi-

variate normal distributions. When these conditions are imposed, several

important results follow. In particular:

1. An asset's expected return is independent of its own
volatility; only that portion of its riskiness which
cannot be diversified away is relevant.

2. All assets with the same non-diversifiable risk have
the same expected return.

3. Asset returns contain two elements, one which is
related to changes in the macro-economy and one
which is unique to the particular asset. It is the
unique portion which is eliminated by diversification.

These propositions collectively form the basis for much of the modern

theory of finance. Curiously, these propositions are often used in

informal derivations to justify the CAPM (see, for example, Brigham

(1983), pp. 158-169). However, if the validity of these results is

assumed a priori, the CAPM is needlessly restrictive. If securities

markets are characterized by risk-averse investors who make decisions

based only on expected returns and risk, then the assets will be priced

as substitutes and the first two results are no more than simple economic

propositions. Any asset which offered compensation for diversifiable

risk would have its price bid up until the premium was eliminated. If

two assets possessing the same non-diversifiable risk had different

expected returns, then investors would sell (or supply) the one with the

lower return and demand the one with the higher return. The relative








prices would adjust until the expected returns were equal. Moreover,

these conditions would hold across any subset of securities. Finally,

that the unique portion of security returns can be eliminated by diversi-

fication is a property of any collection of imperfectly correlated

variables. A certain portion will generally not be diversifiable simply

because, to a greater or lesser extent, all asset returns depend on

general economic conditions.

Ross (1976, 1977) has formalized the kind of reasoning outlined

above in his Arbitrage Pricing Theory (APT). The principal assumption

of the APT is that investors homogeneously view the random return, r.,

on the particular set of assets under consideration as being generated

by a k-factor model of the following form:

r = E + + bil + + bikk + e i 1= ., n (1.1)
where

E. = the expected return on the ith asset

S= the change in the pure interest rate. E[6 ] = 0.
th
6j= The random value of the jth common factor.

E[6.] = 0, j = 1, . ., k.
bij = the sensitivity of the return on asset i to factor j.

e. = the random (unsystematic) portion of r.. E[e.] = 0.

also

E[ei.1K1-0
E[e j 0 0 i j

COV (ei, e ) = 2
1 2 .< "i =j
el
Intuitively, the APT models security returns as a linear function of

Ross's formulation omits this term, implicitly assuming a constant
risk-free rate. Including it allows for the absence of a risk-free asset,
and is similar to Black's (1972) concept of a "zero-beta" portfolio.
This issue is discussed in detail in Chapter III.








some unspecified state variables plus a random component. By appeal to

the law of large numbers, any well diversified portfolio will have

virtually no unsystematic risk. It is interesting to note that any

linear model (including the CAPM) is a special case of the APT. In this

sense, the APT is, as Brennan (1981) has remarked, . a minimalist

model since it predicts no more than the absence of arbitrage opportuni-

ties . [and] is logically prior to our other utility-based models"

(p. 393).

Testing the APT

Because the APT only predicts an absence of arbitrage opportunities,

the identification of such opportunities would lead to a rejection of

the theory. An arbitrage opportunity amounts to a constant non-zero

portion of return not explained by the factors. In efficient markets,

there are two fundamental no-arbitrage properties. First, portfolios

with no net investment and no systematic risk must, on average, have no

return. Second, portfolios with net positive investment and no systematic

risk must have expected returns equal to the pure time value of foregone

consumption. The return on such portfolios should equal the risk-free

rate if such an asset exists; however, the existence of a risk-free

asset is not a requirement of the APT.

A test of the APT requires the estimation of the parameters of

eq. (1.1). Referring to eq. (1.1), if it is assumed that the random

portion of return is completely eliminated, then the no-arbitrage pro-

positions imply the existence of k + 1 weights such that

Ei = o + Ibil + + kbik' (1.2)
where .j is the risk premium on the jth factor and o is the expected

return on all portfolios with no systematic risk (this result is formally

demonstrated in Chapter III). While the APT provides no insight as to








the interpretation of the factor risk premia, it is possible to re-write

(1.2) in a more useful form. Consider a portfolio formed such that

bpl = bP2 = . = bPk = 0. If the portfolio has positive investment,

its expected return is

E = x o

Next, a portfolio is formed with the property that its return is equal

to the risk premium on the first factor; i.e., it is constructed such

that bp1 = 1 and bp2 = . = bPk = 0. If it has no net investment, its

expected return is

E' = .

Repeating this process for every factor, equation (1.2) can be written

E = E Ebil + . + Ekbik. (1.3)

Substituting (3) into (1) and defining Ei as Ei + '., then

ri = E + E'bil + . + E bik + ei. (1.4)
Equation (1.4) is an empirically useful representation of the APT: here

the ex post return on the ith security is expressed as a linear combina-

tion of the "zero beta" return (E") and the returns on the k arbitrage

portfolios. Again, if k is taken to be one and E is interpreted as the

market risk premium, then (1.4) is the ex post two-parameter CAPM (Black

1972)).

Assuming that the returns on the k + 1 arbitrage portfolios can be

determined (discussed in detail in Chapters III and IV), it is possible to

test the APT. To accomplish this, the returns on n assets and the

arbitrage portfolios are collected for some time period. Then for each

security, a time-series regression is estimated of the form
% o ^ 1 ^
r = bo + . kiE + ei (1.5)
Because E measures the "zero beta" return, boi should equal unity.








The intercept term, a, can be interpreted as a measure of abnormal

performance and should not be significantly different from zero. A

simple test of the predictions of the APT would consist of estimating

the parameters of (1.5) subject to the constraints a. = 0 and boi = 1.

The restricted estimate can then be compared to the unrestricted results

using a standard F-test. If the constraint is binding in a substantial

number of cases, then the APT may be rejected in that its predictions

would be inconsistent with the data. Such a procedure, while intuitively

appealing, suffers from at least two drawbacks. This approach has no

objective decision rule. If the hypothesis were rejected in, say, 40%

of the trials, would it then follow that the APT is invalid? Secondly,

this approach requires that the contemporaneous residual covariances

between returns be equal to zero. While this is formally an assumption

of the theory, eq. (1.2) can be expected to hold as an approximation so

long as the residuals are sufficiently independent for the law of large

numbers to be operative. Hence, small, though significant correlations

are not precluded. A number of large correlations would be indicative

of an omitted factorss.

For the reasons outlined above, a valid test of the APT requires

that the cross-sectional dependence among the parameters be considered.

Whether or not eq. (1.2) holds exactly is largely irrelevant (and probably

untestable). With this theory, as with any theory, it is the extent to

which its predictions are consistent with observed phenomena that is of

interest. Brennan (1981) has remarked "[For an adequate test] .

whatt is required is a test of the hypothesis that the intercept terms

for all securities are equal to zero, though such a test may be difficult

to construct" (p. 393). A consistent pattern of non-zero intercepts

would be indicative of arbitrage opportunities, a result at odds with the








arbitrage theory (and, for that matter, most of modern portfolio theory).

Thus, a test of the APT amounts to testing whether the intercept terms

are jointly different from zero, i.e., a pooled time-series and cross-

sectional approach. Such a test is particularly appealing because, as

shown in Chapter IV, it is formally equivalent to testing the following:

H : there exists no well-diversified portfolio with zero
systematic risk and zero net investment which earns
a significantly non-zero return.

vs.

HA: such a portfolio exists.

This is a powerful test; if, for any collection of assets, a single

arbitrage portfolio (out of an arbitrarily large number) can be

identified with a non-zero return, the APT will be rejected. This is a

strongly positivist test as well. The null hypothesis is literally the

central prediction of the theory; thus, it is strictly the content of

theory which is examined, not the assumptions. On the other hand, the

theory is tested against an unspecified alternative; moreover, the test is

conditional on the measurement of systematic risk. As a result, rejecting

the theory does not necessarily invalidate the model. If the view is

adopted that "it takes a model to beat a model," then the return generating

function of eq. (1-4) is interesting in its own right. In the next section,

the use of the model as an alternative to current practice is discussed.

The Arbitrage Model as a Tool in Financial Research

In financial research it is often desirable to specify a model of

security returns which controls for the differential riskiness of the

assets. Once this is accomplished, it is possible to analyze the effect

of other variables (e.g., dividend yields) or events (e.g., unanticipated

information) on security returns. To this end, the so-called "market

model" has been widely employed (see the June, 1983 Journal of Financial








Economics for some recent examples). The return generating process

specified by this model may be written

ri = Ei + (rm E)bi + ei, (1-6)
where (r Em) is the deviation of some broad-based market index from

its expectation.

The popularity of the market model can probably be traced to its

simplicity, intuitive appeal, and similarity to the theoretical CAPM.

However, as pointed out by Ross (1976) and more fully developed by Roll

(1977, 1978), this similarity is more apparent than real. The model is

in many ways closer to the APT than the CAPM; nonetheless, numerous

shortcomings have been identified in the market model's ability to

explain returns (e.g., Ball (1978), Banz (1981), Basu (1977), Reinganum

(1981a)).

The arbitrage model of eq. (1-5) is an empirical alternative to the

market model. Unlike the market model, the arbitrage model has a solid

theoretical basis while retaining a certain simplicity and intuitive

appeal. Thus, a comparison of the usefulness of the arbitrage model

with that of the market model is a logical step.

One of the more popular uses of the market model has been the

residual analysis methodology pioneered by Fama et al. (1969). Mandelker's

(1974) study of the gains from mergers and Jaffee's (1974) research into

the value of inside information are prime examples. The ability of this

methodology to detect abnormal performance (systematic price changes

unexplained by overall market movements) has been studied by Brown and

Warner (1980). Their simulation results indicate that the procedure

works quite well when the event date is known.

Studies of stock price behavior around various types of events are

based on market efficiency. In an efficient market, prices should








adjust rapidly and fully to new information. In this study, the residual

behavior of the two models is compared around several known events.

This comparison addresses two issues. First, because the market model

is known to perform well in this type of study, the substantive results

from the arbitrage model should be similar. Second, the consistency of

the two models with the concept of efficient markets is of interest.

The more consistent model would show greater pre-event adjustment, more

rapid adjustment about the event date, and less drift subsequent to the

event. This comparison also addresses the issue of stability of the

estimated parameters. To the extent that the arbitrage model provides

better resolution of the information in the residuals, it may judged to

be a superior model of the return generating process.

Summary and Overview

The objective of this dissertation is twofold. First, Shanken

(1982) has argued that no truly valid test of any theory of asset returns

has appeared in the literature. The methodology employed in Chapter IV

to test the APT is free of the problems identified in previous work and

is actually quite general. Similar approaches could have broad

applicability. Second, the market model suffers from both theoretical

and empirical deficiencies. An alternative model with a stronger

theoretical foundation and better empirical properties would be a

significant step forward in financial research.

The present study is organized in six chapters. This chapter, the

first, constitutes a brief outline of the need for research in this area

and procedures by which it can be accomplished. Chapter II is a review

of the relevant prior research in multi-factor models. Chapter III

develops both the APT and the arbitrage model, as well as outlines the

methodologies to be employed. In Chapter IV, the results of the tests


a0






10
of the APT are presented. In Chapter V, the empirical properties of

the model as an alternative to the market model are evaluated and

reported. Chapter VI summarizes the major findings, suggests topics

for future research, and concludes this study.











CHAPTER II
PREVIOUS RESEARCH IN MULTI-FACTOR MODELS

Introduction

The Arbitrage Pricing Theory outlined in Chapter I provides a theoreti-

cal foundation for asset pricing without the stringent general equilibrium

restrictions of the CAPM. Despite the theoretical justification and

intuitive reasonableness of multi-factor models, empirical research has

been dominated by the single-index "market" models. An extensive literature

exists on the statistical properties of the model itself, and a number of

authors have employed the model as a means of controlling for differential

asset riskiness or general market conditions. Despite the popularity of

this approach, research has been undertaken in three areas directly

related to the arbitrage model. These areas are (1) purely empirical

applications of multivariate statistical techniques (principally cluster

and factor analysis), (2) multivariate regression models based on a priori

assumptions as to the number and identity of the relevant factors, and

(3) tests of the APT. Much of this research preceded the development of

the APT and it is interesting to re-examine the empirical results obtained

in an arbitrage model context. The next three sections examine this

research and its implications for the arbitrage model.

Applications of Multivariate Statistical Techniques
When a group of variables exhibits a high degree a linear correlation

or "redundancy," several dimension-reducing techniques are available to

summarize the data in a more parsimonious fashion.1 Because security


What follows is intended as a very brief, intuitive description.
A good introduction to cluster analysis may be found in Elton and Gruber
(1970). Factor analysis is taken up in detail in the next chapter.








returns are often highly correlated, cluster and factor analysis have

been applied in efforts to establish the existence of an underlying

structure in the data. With either technique, it is hypothesized that

the variables are elements of a k-dimensional subspace, where k is

"small" relative to the numbers of variables. In either case, k is

unknown a priori.

With cluster analysis, the objective is to assign each variable to

one of k homogeneous groups. In its simplest form, a cluster analysis of

security returns begins with a full rank correlation matrix of returns.

The two securities with the highest correlation are combined into a

single variable, thereby reducing the rank of the correlation matrix by

one. The correlation matrix is then recomputed with the new variable and

the reamining n-2 securities. The two variables with the highest

correlation in the new matrix are combined and so on. The process is

continued in an iterative fashion until no significant correlations remain

between some number of "clusters." However, no completely objective rule

exists for determining the appropriate number of clusters.

In the general factor analysis model, security returns are assumed

to be characterized by a set of hypothetical or latent variables. The

returns are expressed as a linear combination of these variables plus a

random (or unique) portion.2 Like cluster analysis, factor analysis

usually begins with the estimated correlation matrix. Using one of

several techniques, an estimate of the percentage of total variance which

is unique is obtained for each asset. The main diagonal of the correlation



principal component analysis differs from factor analysis. In
component analysis, no distinction is made between random and non-random
portions. This point is discussed in the context of research which has
used this approach.








matrix (consisting of ones) is adjusted by subtracting this estimated

"uniqueness." The result for a particular asset is an estimate of its

"communality," i.e., that portion of its total return which is

systematic. If the errors are assumed to be uncorrelated across

securities, then the resulting adjusted correlation matrix can be

interpreted as an estimate of the common intercorrelation. The next

step is to construct an artificial variable which accounts for a

maximum of the common variance. Next, a second variable (generally

constrained to be orthogonal to the first) is constructed which accounts

for a maximum of the remaining variance. This procedure is continued,

yielding k variables which account for all the estimated common variance.3

Several objective criteria are available for determining k. A discussion

of these is deferred to Chapter III.

Both cluster and factor analysis are generally employed as explana-

tory techniques and results obtained thereby are purely empirical.

However, if the elements of a particular cluster have similar character-

istics, it may be possible to formulate hypotheses for further testing.

Similarly, if a given factor is particularly related to some group of

securities, it may be possible to infer the identity of the factor.

Regardless of the validity of such heuristics, the research examined

below relates to the existence of multiple factors in security returns

as well as the number of relevant dimensions.

One of the earliest studies to employ dimension reducing techniques

was that of Farrar (1962). Farrar applied the principal component

approach to 47 industry groups in an effort to create a relatively small


If the factor model fit perfectly, the reduced correlation matrix
would be rank k. In practice, k is regarded as the approximate rank,
allowing for measurement error and non-linearities.








number of asset groups with low first-order correlations. He found that

the first five components accounted for about 97% of the total joint

variation among the industry groups, with the first component capturing

77% of the total. Examination of his results (p. 41) indicates the

presence of a single, dominant factor with at least two additional

significant factors.

The principal component approach was also applied by Feeney and

Hester (1967). The purpose of their research was to objectively develop

weights for a stock market index. Using the 30 securities in the Dow

Jones Index, they found that the first two components (of the covariance

matrix) accounted for 90% of the total variance, with the first component

accounting for 76%. Interestingly, the correlation between the Dow Jones

Index and the first component was found to be in excess of .99. The

results from the component analysis are nearly identical to those found

by Farrar, despite the different samples and time periods employed.

In 1966, King investigated the nature of the latent structure of

security returns. His work is of particular importance because he

recognizes both the presence of a market factor and the existence of

unsystematic (or unique) effects. The explicit purpose of King's study

was to determine whether inter-relationships among security returns

could be attributed to a market factor and an industry factor correspond-

ing to a two-digit SIC classification. Using a sample of 64 stocks in

six industry groups, King performed both a mixed factor/cluster analysis

and a multi-factor analysis. In the mixed analysis, he extracted the

first factor (the market factor) and clustered the remaining variation.

When the maximum correlation between groups dropped below .20, the

group corresponded exactly to the SIC two-digit classifications. When








a seven factor solution was obtained, the same pattern emerged; all

securities were sensitive to the general market effect and an industry

factor. Also, King found that the first factor accounted for 74% of the

estimated total systematic variation; however, his results differ from

those of Farrar and Feeney and Hester in that the subsequent factors

(particularly the second and third) were not as pronounced. Also, the

relative importance of the market factor in explaining the systematic

or common variation was found to decline over time, from a high of 63%

in the sub-period June 1927 to September 1935 to a low of 37% for the

period August 1952 to September 1960.

At the time of King's study (1966), the Sharpe (1963) single-index

model was gaining popularity as a simplification to the general Markowitz

(1959) portfolio problem. The validity of this model hinges on the

absence of contemporaneous residual correlations among the assets. King's

findings are at odds with this requirement. In a 1973 study, Meyers

extended King's methodology to include less homogeneous industry groups,

as well as the time period 1961-1967. After extracting the first princi-

pal component, Meyers clustered the residual correlation matrix and

found results generally supportive of King's; however, he does identify

a weakening of the industry effects. Meyers then extracted six components

from the residual correlation matrix and reported evidence of industry

effects similar to King's, though with significantly less clarity.

Meyers concludes that King's results overstate the importance of

industry effects, but he concurs in the finding of residual covariance

unexplained by a general market effect.

The relative strength of industry effects was examined in 1977 by

Livingston. In this study, a number of important issues are identified;








in particular, Livingston documents that the principal components

approach is inappropriate in that it tends to extract more common

variance than actually exists.

To determine the magnitude of industry effects, Livingston proceeded

to regress returns from 734 securities (in over 100 industries) on the

S & P Composite Index return. Next, the residual correlation matrix was

examined for significantly non-zero correlations. Within industries,

20% of the correlations were found to be significantly different from

zero, with very few negative elements. Across industries, 6% were

significantly positive and 2% significantly negative. However, some of

the industries examined showed little residual correlation. Livingston

concludes that a single-index model ignores a significant portion of

the co-movement in security returns and that the use of industry indices

should improve the results. Such models have been constructed and are

reviewed below.

The most general conclusion which can be drawn from the King, Meyers,

and Livingston research is that extra-market covariation does exist, but

it is not clear whether the effect is related to industry classification

per se. An alternative explanation could be offered to the effect that

certain types of businesses are particularly sensitive to different

macro-economic factors. If this proposition is correct for the members

of a homogeneous industry group, then an "industry effect" will appear

to exist. Because factors such as interest rates, foreign exchange

rates, inflation, input prices (raw materials and wages), and so on do

not move in lockstep, firms with particular dependencies on any one








factor will exhibit "extra-market" influences. This is simply due to

the averaging implicit in the construction of a market index.4

Studies by Farrell (1974, 1975) and Arnott (1980) have used cluster

analysis to define groups of securities in terms of their return

characteristics as opposed to industry classification. Farrell used

a stratified (across industries) sample of 100 securities. He computed

the residual correlation matrix from a single-index model. These

residuals were clustered until no correlation above .15 remained. The

results of this procedure were four clusters which Farrell labels as

growth, stable, cyclical, and oil. Arnott used 600 securities and a

somewhat less stringent rule to halt the clustering process. His results

indicate five clusters which he labels quality growth, utilities, oil

and related, basic industries, and consumer cyclicals. The results of

the two studies are actually quite similar; the primary difference is

that the Farrell study combines the utility, basic industry, and

consumer cyclical into two clusters, the stable and cyclical. Both of

these studies are generally supportive of a multi-factor model, where

the factors are some set of macro-economic variables rather than simple

industry effects.

The multivariate studies reviewed in this section have, in varying

degrees, a similar result: a single index model ignores potentially

useful information about the co-movement of security returns. The

techniques used in these analyses are all forms of correlation analysis;

no model or theory is employed. In the next section, several models

which attempt to incorporate extra-market information are examined.


In the case of a value-weighted index, the averaging is in terms
of the characteristics of the largest firms versus the most numerous in
the case of an equal-weighted index.








Multiple Regression Models of Security Returns

Several authors have sought to improve the single-index model by

including additional variables. In an early effort, Kalman and Pogue

(1967) compared the ability of single and multiple index models to

recreate the Markowitz efficient frontier and to predict correlation

matrices. Their results indicate little, if any, benefit from a multi-

index approach. Farrell (1974) criticizes the method used by Kalman

and Pogue in constructing the multiple indices, attributing the lack

of success to the high degree of collinearity among the industry indices.

Using the relatively uncorrelated clusters (described in the previous

section) in addition to a general market effect, he reports superior

results when compared to a single index formulation.5

In another study examining the ability of various models to predict

correlation matrices, Elton and Gruber (1973) test ten different models

of security returns. They find that three models outperform all other

techniques, including the single index and several multiple index models.

The three models differ in their assumptions concerning the pattern of

correlation coefficients. The overall mean model sets all coefficients

equal to the average. The traditional industry mean sets all correlations

within an industry equal to the industry average, and all inter-industry

correlations are set equal to their average. The third model is the

same as the traditional industry with the exception that industries are

defined by a principal component solution ("pseudo-industries"). Elton


Farrell extracts the market effect by regressing the cluster
returns on a market index and using the residuals as "explanatory"
variables. This procedure creates orthogonal indices by construction;
however, such an approach is suspect on econometric grounds. It is
difficult to justify the use of random noise (i.e., the residuals)
from one estimation as "explanatory" variables in another.








and Gruber's results indicate that superior forecasting is possible

using information not produced by index models. Unfortunately, their

multi-index models are based on principal component solutions and the

assumption of zero residual correlations is inappropriate.

Other studies have used information beyond a general market effect

in estimation. Rosenberg (1974) assumed the general validity of the

single index approach, but he used a number of firm-specific descriptorr"

variables to obtain forecasts of the parameters. Lloyd and Schick (1977)

have tested a two index model proposed by Stone (1974), where the

additional index is composed of debt instruments. Langetieg (1978)

adopted an approach similar to Farrell's, using orthogonalized industry

indices to measure gains from mergers. All of these studies find benefits

in the use of extra-market information, but they lack a theoretical

underpinning. The arbitrage theory provides this missing element, and

studies incorporating it directly are reviewed in the next section.

Tests of the Arbitrage Theory

The first published study of the APT is credited to Gehr (1975). Gehr

constructed two samples of 360 monthly returns, one consisting of 24

industry indices and the other of 41 individual companies. He next

obtained a three component solution for the 41 companies. The industry

returns were then regressed on the components to estimate the sensitivity

coefficients. A second-pass regression of the mean industry index returns

against the coefficients was performed as the final step. Of the estimated

risk premia, only one is found to be generally significant.

An empirical anomaly associated with the market model has been

investigated by Reinganum (1981b) and Banz (1981). When portfolios are

formed based on firm size, small firms earn significantly greater rates








of return, even after accounting for difference in estimated betas.

Reinganum (1981a) has examined the same question using an arbitrage

model. Essentially, Reinganum forms a set of control portfolios based

on ranked factor loadings. Then, the returns on the control portfolios

are subtracted from corresponding individual security returns. The

resulting excess returns are ordered into deciles based on market equity

values, and the average excess return is computed for each decile.

Reinganum's results are similar to those found using the market model:

portfolios of small firms offer a risk-adjusted return significantly

greater than the portfolios of large firms. Thus, Reinganum rejects the

arbitrage model as an empirical alternative to the simpler market model.

Oldfield and Rogalski (1981) have examined the influence of factors

estimated from Treasury bill returns on common stock returns. As a first

step, they gather Treasury bill returns for 1 to 26 week maturities.

The one week return is then subtracted from the subsequent maturities to

calculate excess weekly returns. The one week rate is reserved for the

risk-free rate. Next, the excess T-bill returns are factored and factor

scores are computed.6 Next, individual common stock returns are regressed

on the factor scores, yielding a set of sensitivity coefficients. The

stocks are then randomly assigned to intermediate portfolios, and the

covariance matrix of the returns among the portfolios is calculated.

Using this covariance matrix, a minimum variance portfolio is calculated

for each factor with the property that a particular portfolio is sensitive

to that factor, with a zero loading on the others. Additionally, a

minimum variance portfolio is formed with no sensitivity to any factor


6Factor scores are estimates of the population factors; hence they
constitute a time-series of measurements of the factors.








(a "zero-beta" portfolio). The weekly returns on these factor port-

folios is computed, and these are used in time-series regressions to

re-estimate sensitivity coefficients.

Their first result from the procedure is that significant correlation

exists between common stock returns and the factor portfolio returns.

Next, the authors run a cross-sectional regression of the weekly inter-

mediate portfolio returns and their factor loadings in each of 639 weeks.

They then compare the mean regression coefficient for a particular factor

with the mean return on the factor portfolio. They argue that the two

should be equal, and find no statistical difference. By including an equal

weighted market portfolio, the authors find that the significance of

the factor portfolios is greatly diminished, a result which they attri-

bute partially to the collinearity between the variables. The authors

report that the estimated risk-free rate is significantly less than the

corresponding T-bill rate, while the cross-sectional intercepts are not

different from zero.

Fogler, John, and Tipton (1981) have also attempted to relate the

returns on debt and equity instruments in the context of the arbitrage

theory. The basic data for this study were excess monthly returns on

100 securities divided into seven groups. The first four groups were

selected on the basis of Farrell's cluster analysis, consisting of

stocks classified as growth, stable, cyclical, and oil. The other

three groups correspond to the pseudo-industries developed by Elton and

Gruber (both studies are reviewed in a previous section). The authors

next calculate excess monthly returns on a value-weighted market index,

a three month Treasury bond index, and a long-term Aa utility bond index.

The excess returns were calculated by subtracting the return on a one








month Treasury bond. Next, the excess returns on the securities were

regressed on the three indices; of the three, only the market index had

generally significant coefficients. The authors report that some of the

groups display consistent signs on other indices; however, no non-parametric

results were included.

In a second part of their study, Fogler, John and Tipton extract a

principal component solution from the 100 securities, retaining the first

three. They then examined the canonical correlation between the components

and the three indices. From this analysis, one important result emerges:

the correlation between the three components and the market index is near-

ly perfect. Also, in some sub-periods there is a statistically signifi-

cant relationship between the components and the three month Treasury bond

yield. Whether or not the authors have achieved their goal of "imparting

economic meaning to the stock returns factors" (p. 327) is difficult to

say; yet they implicitly establish an important empirical result; namely,

the return on the overall market can be decomposed without loss of

information about the market while potentially including other relevant

information. Thus, while their study is not actually a test of the APT,

it nonetheless suggests a certain empirical rationale for the theory.

A final study deserving of particular attention is that of Roll and

Ross (1980). This study is a straightforward extension of Gehr's

methodology. The authors form 42 portfolios of 30 securities each, using

ten years of daily security returns. A factor solution is then obtained

for each group. For each group, a cross-sectional GLS regression of mean

returns on the factor loadings is estimated. The authors report that at

least three factors of the five used are "priced" in the results. Next,

an additional variable, the standard deviation of return, is included in








the cross-sectional regressions. After correcting for the positive

dependence between sample mean and sample standard deviation arising

from the positive skewness in daily returns (by using non-overlapping

samples), little support is found for the hypothesis that returns are

related to total volatility. As a final test, Roll and Ross test for

cross-sectional differences in the intercepts from the cross-sectional

regressions. To do so, they employed Hotelling's T2 statistic to account

for cross-sectional dependencies in the estimates. Their results indicate

no significant difference, lending support to the APT.

Summary

The preceding three sections have reviewed research in three areas--

purely empirical analysis of stock market groups, multiple regression

models based on a priori knowledge of the relevant variables, and studies

testing the APT, either directly or indirectly. Of the multivariate

studies, the results obtained from a variety of different approaches are

consistent in that they generally indicate that a single index model

ignores significant facets of security returns. This conclusion is

reinforced by the multiple regression studies in that the additional

variables specified add significant explanatory or predictive power despite

their ad hoc nature.

The APT offers, in principle, an empirical alternative. The studies

published to date using it all suffer from serious methodological flaws;

in addition, no tractable multi-index model based on the APT has been

forthcoming. Because many of the methodological problems in the literature

stem from a misapplication of factor analysis, a discussion of them is

deferred to the next two chapters where the application of factor analysis

to security returns is addressed. Problems also arise in the development






24

of testable hypotheses in an arbitrage pricing framework and with the

nature of the appropriate return generating function. These three

issues--factor analysis of security returns, testable hypotheses of the

APT, and the structuring of a return generating function--are inter-related

to the extent that the validity of any one of the three depends on the

other two. In other words, the theoretical justification for a multi-

factor return generating function obtained from a factor analysis is

found in the APT. However, a test of the APT requires a return function

obtained from a factor analysis procedure. Finally, a number of factor

analysis procedures are available; the choice of a particular one

depends on both the APT and the desired form of the empirical model

derived therefrom. The next chapter considers each of the subjects

independently before combining them into the arbitrage model.














CHAPTER III
THE ARBITRAGE MODEL: THEORY AND ESTIMATION

Introduction

In the previous two chapters, the need for an alternative model of

security returns was established and evidence for the validity of a

multi-factor representation was examined. In the first section of this

chapter, the theoretical basis for such a model is illustrated. In the

second section, the relationship between the APT and the general factor

analysis model is developed. The results of these sections are used to

derive an empirical model of returns and to establish the testable

hypotheses of the APT.

The Arbitrage Pricing Theory

The APT was originally proposed by Ross (1976, 1977). A simplified

approach was derived by Huberman (1982). The theory has been generalized

and extended by Ingersoll (1982). The exposition in this section draws

heavily from these three sources.

The principal assumption of the APT is that investors homogeneously

view the random returns, r, on the particular set of assets under consid-

eration as being generated by a k-factor linear model of the following

form:

r = E + B6 + e, (3-1)

where


Strictly speaking, complete homogeneity of investor expectations
is not required. Ross (1976) has established that the existence of non-
negligible agents with upward bounded relative risk aversion and homogeneous
opinions about expected returns are sufficient. As Ross notes, however,
translating ex post occurences into ex ante anticipations will require
homogeneity.








En x 1 = = the expected returns .on the n assets

B = = the sensitivity of the return on asset i to
nx k bJ changes in common factor j (factor loadings)

k x 1 = the random values of the k common factors

en x = = the random (unsystematic) portion of r.

It is also assumed that

E[e] = 0

E[6] = 0

E[es'] = 0
_2 -2
E[ee'] = = ij < i=
0 i j.

In other words, the deviation of the return on asset i from its expecta-

tion is a linear combination of the random values of the k factors and a

unique, residual component. The residuals are assumed to be independent

of the factors and mutually uncorrelated. In the absense of a riskless

asset with a constant certain return, eq. (3-1) may be written

r = E + A + e, (3-2)

where
(k + 1) x = < .> = the random values of the k common factors
S x with as the change in the "zero beta"
return

An (k+l) = = The augmented factor loading matrix

i = <1> = the sum vector (a column vector of ones).

Heuristically, the arguments underlying the APT begin with the considera-

tion of a portfolio vector, x, chosen such that x'a = o. The components

of x are the dollar amounts invested in each asset. Since the total invest-

ment is zero by construction, all purchases (long positions) are financed






27

by sales (short positions).2 If x is a well-diversified portfolio with

each xi of order 1/ in absolute magnitude, then by the law of large numbers,

the dollar return on x is

x'r = x'E + x'AS + x'e

-x'E + (x'A)6. (3-3)

If x is chosen to have no systematic risk as well, then the return is

x'r x'E. (3-4)

Taking a to be any non-zero scalar, then ax is an arbitrage portfolio.

If it is assumed that the random portion of (3-3) can be completely

eliminated by diversification, then (3-4) holds with equality and it must

be the case that

x'r = x'E = 0, (3-5)

or unbounded certain profits are possible by increasing the scale (a) of

the arbitrage operation. If this condition holds for all portfolios

constructed in the manner described above, then there exist constants

= x' such that
E = AX, (3-6)

where A is the augmented factor loading matrix. Algebraically, (3-6) is

simply the statement that all vectors orthogonal (perpendicular) to A are

orthogonal to E if and only if E is in the span of the columns of A. This

result and several others can be illustrated by introducing the following

notation:


2In the absense of restrictions on short selling, such portfolios
can always be constructed. Even with short selling restrictions,
investors with positive net holdings can, in effect, engage in such
activities by buying and selling. Letting w be the dollar amounts
invested in the n assets (with w'e = W, the investor's net wealth),
then, assuming no transactions costs, the difference between w and
any other portfolio, w, is an arbitrage portfolio: w + x = w.
Thus, an investor who changes his relative investments is implicitly
purchasing an arbitrage portfolio.








S = span {A}, where A is assumed to have full column rank

S = set of all vectors orthogonal to S with orthogonal basis

x = < .., x n-k-l>.

By construction, then

S IS = TR

SnS = {0}

x'A = 0 i = 1, ..., n-k-I

S0 i = j
j = 0 i j.

Equation (3-6) follows from the no-arbitrage assumption; either EFS or

arbitrage is possible. To see this, note that E can always be written

E = AX + z, (3-7)

where zeS But z is itself an arbitrage portfolio with return

z'E = (z'A)X + z'z = z'z f 0. (3-8)

So (3-6) must hold to prevent arbitrage.

Following Huberman (1982), the results obtained above can be extended

to the case where the residual portion of return is not completely elimi-

nated. The objective is to establish an upper bound on the sum of the

squared deviations from the pricing relationship (3-6). The APT considers

a sequence of economies with increasing numbers of risky assets. The n

economy has n risky assets whose returns are generated by a k-factor

model, where k is a fixed number. Arbitrage is defined as the existence

of a subsequence, z n, of arbitrage portfolios with the properties

lim zE = (3-9)
n-'

lim var(z'E) = 0. (3-10)
n-fm

Intuitively, arbitrage possibilities exist whenever increasing profits at

diminishing risk are obtainable as the number of assets grows. Put another






29

way, the reward to volatility ratio increases without limit. To preclude

such occurrences, there must be an upper bound to

(En AXnn)' (En An ), (3-11)

the sum of squared deviations in the nth economy. Referring to (3-4) and

assuming that z is scaled such that z'z increases to infinity with n,

(the subscripts are understood)

z'r = z'E + z'e, (3-12)

substituting (3-7) for z'E

z'r = z'z + z'e. (3-13)

Letting a be a scalar between 1/2 and 1 and defining y = (z'z)- then

Yz is an arbitrage portfolio with expected return and variance

Yz'E = Yz'z = (z'z) 1-c = E[z] (3-14)

and
2 -2 2 -2 1-2a
z'E[ee']z = < a y (z'z) < (z'z) (3-15)

By construction

lim E[z] = = (3-16)
n*=
2
lim a = 0. (3-17)
n+~m

For example, if a is taken to be 3/4, the expected return increases with

the fourth root of z'z and the variance decreases with the square root.

The reward to volatility ratio is

E[z] > (z'z)1- (3-18)
1/2- = zl a
a (z'z)1/2-a

which does not have a lower bound unless z'z is bounded. From (3-7),

z = E Ax; hence the foregoing suffices to show that unlimited deviations

from the pricing relationship (3-6) give rise to arbitrarily large profits.

If such profits are precluded by assumption then it follows that

(E Ax)' (E Ax) < M. (3-19)







Inequality (3-19) indicates that the permissible sum of squared

deviations is less than some finite number for any number of assets. As

a consequence, as the number of assets increases, the approximation

improves. The reverse is, of course, for any finite set of assets, the

approximation can be quite poor.

The existence of a finite bound on the arbitrage pricing relationship

suggests a natural test of the APT. For A and E given, it is possible to

determine X such that the sum of the squared deviations from the pricing

relationship is minimized, i.e., minimize

(E AA)'(E AX). (3-20)

Equation (3-20) is simply the OLS estimate of E on B. The resulting sum of

squared errors could be compared to the bound in (3-19) and the APT

rejected if the bound is exceeded. Such a test, however, requires that

an a prior bound be established.

In the absence of such a specification, several authors (notably Roll

and Ross (1980)) have attempted to verify the APT by establishing the

linearity of (3-6). As pointed out by Shanken (1982), the linearity of

(3-6) is not literally an implication of the APT for any finite collection

of assets. For any such collection, (3-6) is an approximation and will

have a finite bound on the sum of the squared errors. To take a polar

case, in the absence of any linear relationship for a particular set of

assets, the bound would be equal to the sum of the squared expected returns.

Such a result would not necessarily invalidate the APT because the permis-

sible upper bound on the pricing relationship is not known a priori;

hence, an examination of the degree of linearity in the pricing relationship

is without power to reject the APT.

The central prediction of the APT is the absence of arbitrage oppor-

tunities where arbitrage is defined as a non-zero return on a well





31

diversified portfolio with no net investment and no factor risk. The

empirical identification of such opportunities would lead to a rejection

of the theory. In the next section, the estimation of a k-factor model

to be used in testing the APT is discussed.

Estimating the Arbitrage Model

The APT is a theory of the structure of asset returns. In the

theoretical development of the previous section, it was assumed that

the matrix of factor loadings (and the number of factors) was known.

Because the APT provides no insight into the nature or number of the

factors, it will be necessary to infer both from observed security returns.

Techniques for accomplishing this fall under the general heading of factor

analysis. In this section, a particular type of factor analysis due to

Lawley (1940) is outlined, following Joreskog (1967) and Lawley and

Maxwell (1971).

It is easiest to conceive of factor analysis as a form of linear

regression in which the number and identity of the regressors is unknown.

Factor analysis is then a technique by which (for k regressors) the

coefficients and variables of the linear model are simultaneously deter-

mined. Based on the assumptions of the previous section, the covariance

matrix of returns is

E[rr'] = E[B5 + e)(B6 + e)']

= BE(66')B' + E[ee']

= B B; + 4, (3-21)

where i is the covariance matrix of the factors. Let Q be a matrix

satisfying Q'Q = I and Q Q' = I.3 Equation (3-21) can be written


3Such a matrix exists for any symmetric matrix of full rank. The
columns of Q are the eigenvectors of 4 scaled by the square root of
their respective eigenvalues. See Frieberg et al. (1979).







E[rr'] = B(Q'Q) 6 (Q'Q)'B' +

= (BQ')(Q'B)' +

= B*B*' + p. (3-22)

Thus, without loss of generality, it can be assumed that the factors are

mutually uncorrelated with unit variances. When the factors have this

relationship, the insertion of any full rank orthonormal matrix leads to

a mathematically equivalent solution. Thus there is an infinity of

mathematically equivalent factor loading matrices. A linear transformation

of this type is termed a rotation and geometrically amounts to a rigid

motion of the factor axes to a new set of coordinates. This indetermi-

nateness is not a problem with the APT since no interpretation of the

factors is necessary; researchers are free to choose any convenient

orientation. Naturally, this lack of uniqueness would make the task of

"identifying" the factors difficult and any interpretations of them suspect.

If it is assumed that the vectors 6 and e follow multivariate normal

distributions, then the elements of the sample covariance matrix S have

a Wishart distribution with t 1 degrees of freedom (t is the number of

observations). In this case it is possible to obtain maximum likelihood

estimates for B and t (again, for a given k). Following the usual technique

of maximizing the log-likelihood function, it can be shown (Lawley and

Maxwell (1971), p. 26) that an equivalent procedure is to minimize the

function

F(B,P) = Inl Z + tr(S-I1) In s n, (3-23)

where Z is the hypothetical covariance matrix and n is the number of

variables. No direct solution exists, so numerical techniques are used

to find the minimum value and the resulting estimates B and .

The principal attraction of the maximum likelihood approach is that

it allows a test for the number of common factors. Denote by L(n) the





33

maximum of the likelihood function for k unrestricted and let L(; ) be

the maximum under the null hypothesis of exactly k factors. If X is the

ratio of the restricted maximum to the unrestricted, it is well known

(e.g., Mendenhall and Schaeffer (1973)) that -21nx converges in distribu-

tion to x with degrees of freedom equal to the number of parameters or

functions of parameters assigned specific values under the null hypothesis.

In the case of a factor analysis, the number of parameters estimated in the

unrestricted model is the sum of n variances and the 1/2(n2 n) unique

covariances, for a total of 1/2n(n + 1). There are nk unknowns in B and

n unknowns in P. Without further restrictions, the matrix B is not

uniquely defined. For computational reasons, it is convenient to require

that B'PB be a diagonal matrix. This has the effect of imposing 1/2k(k-1)

restrictions; hence the total number of free parameters is nk + n 1/2k(k-1)

and the degrees of freedom are

1/2n(n + 1) nk n + 1/2k(k 1) =

1/2[(n-k)2 (n + k)]. (3-24)
2
In fact, the value of the computed X statistic is simply t times the

minimand of eq. (3.23), explaining its use in the estimation procedure.

The maximum likelihood procedure provides for a test of k = k ,

where ko is a prespecified number of factors. In essence, the hypothetical

covariance matrix constructed using only k0 factors is compared to the

saturated (sample) covariance matrix and if the discrepancy is found to

be sufficiently small, the hypothesis is not rejected. In the usual

exploratory case, it is not possible to specify a pre-determined value of

k; instead, what is desired is an estimate of the dimension of the model.

The procedure adopted is to begin with a small hypothesized value of k.

If the hypothesis is rejected, k is increased by one and the test repeated.

The dimension of the model is taken to be the smallest value of k which






2
yields a non-significant x at a predetermined significance level.

Because the ultimately determined value of k depends on a sequence

of prior tests, the assumptions of the classical Neyman-Pearson theory

are violated. Thus, the test can only be interpreted as a test of

sufficiency, and practice has shown that the value of k arrived at by

this procedure using conventional significance levels is greater than

the number of relevant factors; hence, the test should be regarded as

conservative in the sense that it is unlikely to lead to an underestimate

of the true number of factors (Lawley and Maxwell (1971), Harman (1976),

Horn and Engstrom (1979)). With this approach, it is important to note
2
that significant x values lead to the fitting of more factors; hence,

greater significance levels will lead to fewer factors being retained.

Horn and Engstrom's (1979) results from a related criterion indicate

that, for large samples, significance levels in excess of .999 are

warranted.

Because of the tendency for the maximum likelihood approach to

over-estimate the number of substantially important dimensions, two other

criteria will be examined. Akaike (1973, 1974) has proposed an information

theoretic loss function as an extension to the likelihood approach and

Schwarz (1978) has developed a large-sample Bayesian criterion. All three

criteria are related. Schwarz (p. 461) indicates that his and Akaike's

approach amount to adjusting the maximum likelihood estimator. If M(kj)

is the value of the likelihood function for k factors, then Akaike's

procedure results in the selection of k such that InM(k-) Inkj is

largest and Schwarz's criterion results in the selection of k such that

InM(k.) 1/2kjlnt is maximized. In large samples, the three criteria

can lead to very different estimates of k. Schwarz's criterion will lead

to smallest estimate and the maximum likelihood procedure the largest.







The results from applying each of these standards to daily security

returns are reported in the next chapter.

Several other points about the maximum likelihood approach are of

interest. A consequence of the weighting scheme implicit in the procedure

is that the resulting estimates are scale-free; hence, correlations may

be used instead of covariances. In regard to the distributional assump-

tions underlying the approach, Howe (1955) has demonstrated that the

same loadings and residual variances result from maximizing the determinant

of the estimated common correlation matrix; thus the approach is valid

as a descriptive measure regardless of the underlying distribution. The

principal drawback relates to the computational resources required to

obtain the solution. The CPU time required varies exponentially and

appears to be proportional to the fourth or fifth power of the number of

variables. Sample sizes are thus limited. Also, convergence of the

numerical algorithms employed need only be local; thus the solution obtained

may not be the global maximum. In any event, once the number of factors

and their associated loadings are determined, it is still necessary to

measure the factor risk premia. This topic is the subject of the next

section.

Measuring the Risk Premia

The final step in constructing a multi-factor model of security

returns is to use the information from the factor analysis to estimate

the time-series behavior of the factor risk premia. In this section,

the estimates of the factor loading matrix, B, and the diagonal residual

covariance matrix, i, are taken to be fixed; hence the carats are dropped.

Also, it is assumed that the estimates of B and P are obtained from the

covariance matrix, so the typical element of B is the covariance of the

return on security i with common factor j. The typical element of P is





36

the uniqueness for asset i multiplied by its variance, i.e., the portion

of its variance not associated with the common factors.

The approach adopted here is similar to methods used by Oldfield and

Rogaliski (1981), though with some important differences. The general

technique employed is to partition the observations into two groups: an

estimation (base) period and a test period. The data from the base period

are used to obtain a factor solution. Next, for each factor, a portfolio

is constructed with unit sensitivity to that factor subject to the

constraints that it use no wealth and that it have no correlation with the

other factors. Also, a zero beta portfolio is formed possessing positive

investment and no correlation with the factors. Without further constraints,

the weight vectors for each of the portfolios are not unique; for any

particular one there would n-k-l linearly independent choices plus any

number of linear combinations. For a particular factor, then, the weight

vector is chosen such that its unsystematic portion is minimized. For

example, the weights for the zero beta portfolio, x are the solution to

minimizing x'px subject to x' = 1 and x'b = x'b = . x'b = 0,
0 0 0 0 0 k ok
where the b. are the loadings on the ith factor. This program is then

repeated for each factor. More generally letting ci be the standard unit

vector, the weights for the k + 1 portfolios, x, are the solutions to

Min x'.Wx. i = 0, . ., k

x.

s.t. xA = ci.

The solution is found by introducing k + 1 Lagrange multipliers and

minimizing

L(xi, xi) = xi.xi 2\i (x.A ci), (3-25)







where Xi is the row vector of the multipliers. Thus

L(xi,i) xi = 0 = > xi A = 0. (3-26)

L(xi Xi)/ri = 0 = > xIA ci = 0. (3-27)

Defining x as , Aas and I as the identity matrix , the

solutions for all k + 1 weight vectors can be written as

' A A= 0 (3-26a)

x'A I = 0. (3-27a)

Eliminating the multipliers and solving for x yields

X = -1A(A'-1A)-1. (3-28)

Once the portfolio weights are obtained as in (3-28), the data from the

base period are discarded. This is done to avoid the circularity inherent

in using the same data to fit and test the model. In particular, it is

desirable for testing purposes to have estimates with known distributions.

For example, one of the tests performed by Roll and Ross (1980) consists

of regressing mean returns cross-sectionally on the factor loadings. A

standard t-test is used to evaluate the significance of the estimated

risk premia. Clearly, any results obtained by such a procedure depend on

the previously determined factor solution and especially on the number of

factors determined to be significant. Such a procedure seems little dif-

ferent from determining a one factor solution and running such a regression,

then obtaining a two-factor solution and repeating the regression, and so

on until the kth factor is insignificant. A discussion of the problems with

such pre-test estimators may be found in Judge et al. (1980, pp. 54-94).

Once the weights are obtained, the returns on the arbitrage port-

folios are calculated using the test period data. Letting R be the n x t

matrix of individual security returns, the arbitrage portfolio returns are

R'X = R'-1A'(A'~-1A)-1. (3-29)

Estimates obtained in this fashion have some interesting features. First,





38

they are equivalent to the estimates obtained from running cross-sectional

GLS regressions of the security returns on the factor loadings for each

day, and the mean returns on the arbitrage portfolios are equivalent to

regression coefficients of mean security returns on the factor loadings.

There is no question of "statistical" difference as considered by Oldfield

and Rogalski (1981). Second, the estimated weights are actually just

factor scoring coefficients estimated with Bartlett's approach, and are

unbiased estimates of the true factors (Lawley and Maxwell (1971), p. 109).

Bartlett's method also produces factor estimates that are univocal, i.e.,

uncorrelated with the other factors (Harman (1976), p. 385). Finally,

the use of the inverse of the residual variances in (3-29) has the effect

of correcting for the heteroscedasticity of the residuals and, all other

things being equal, places greater weight on those securities with

greater common variances.

The final step in estimating the arbitrage model is to use the factor

risk premia as independent variables in time-series regressions of the

form

rt = + bo + bji1 + . + bjkbk + ejt, (3-30)

for each security. Equation (3-30) is the basis for the tests of the next

chapter and the event study comparison with the market model in Chapter V.

Summary

In this chapter, the theory and estimating procedure underlying the

arbitrage model were discussed. The result is a model in which security

returns are linearly related to a set of unspecified, though measurable,

latent variables. The next step is to compare the empirical results

obtained from its use with the predictions of the underlying theory. To

the extent that they are not in accord, the model loses one of its prime

justifications. The last step in developing the arbitrage model is of a





39

more practical nature. Theoretical consideration aside, the arbitrage

model is somewhat more involved than other models (and more expensive

to use), so its performance relative to simpler models is a subject worth

investigating.












CHAPTER IV
TESTING THE ARBITRAGE THEORY

Introduction

In this chapter, attention turns to the empirical issues of the

arbitrage model. Previous chapters have addressed the need for a multi-

factor model and examined evidence suggesting its appropriateness. In

the last chapter, an arbitrage model was specified as an empirical analog

to the APT. Because the content of the model stems directly from the

predictions of the arbitrage theory, it becomes a natural vehicle for

establishing the general validity of the APT.

For all of its simplicity and intuitive appeal, the arbitrage theory

is rather limited; all that is indicated is that the return generating

process has an approximate linear dimension less than the number of risky

assets in the economy. It is important to realize, however, that the

dimension reduction is the content of the theory, not the factors them-

selves. If the dimension of the structure of security returns is known,

the factors are implicit in that structure. In this sense, they are

best viewed as continuous versions of Arrow-Debreu pure or "primitive"

securities. In the continuously distributed case, the number of states

is equal to the number of securities; hence, n linearly independent

securities are necessary to exactly span the state space. The arbitrage

theory amounts to the assertion that the state space is approximately

spanned by k + 1 linearly independent vectors, and that the degree

of approximation improves as the number of securities increases. Interest

in the theory, then, should not stem from the possibility of interpreting

40







the factors as some collection of macro-variates. In fact, there is

nothing in the theory which suggests that such an interpretation exists,

certainly the theory does not require it.

The APT is not a causal theory; security returns are merely associa-

ted with some collection of measurements. These measurements are best

interpreted as indices, the level of which have no intrinsic meaning. The

best that can be said is that the indices are aggregate measures of the

information sets and expectations of market participants. Changes in

the levels of the factors are consequences of the arrival and assimila-

tion of information and its role in the formulation of market expecta-

tions, a process about which little is known.

While the theory is mute in regard to the nature and number of the

factors, estimates derived therefrom must have certain properties if the

theory is to be empirically testable or practically usuable. Formally,

all that is required from the theory is that

lim k = 0. (4-1)

n-*o- n

Thus, for any finite collection of assets, the number of factors can be

quite large, and that number can change as assets are added. In the

non-stable case, where the number of factors and the factor loadings

change as assets are added, the theory is probably of limited practical

interest. What is hoped instead is that the economy has a fixed, finite

number of sources of risk and that k<
theory in this light is inversely related to the ratio in (4-1), at least

for a particular market.

The arguments above suggest that the practical interest in the APT

is related to the degree of empirical parsimony possible through its use.





42

This subject is of interest in its own right and is taken up in the next

section. Once the dimension of the model is fixed, subsequent sections

report various univariate results obtained from the model, and, finally,

a multivariate test of the APT. While a fixed dimension and relative

stability of the factor loadings across securities are sufficient for

research in this area to be interesting, one additional requirement is

that the estimates contain a degree of intertemporal stability sufficient

to justify their use vis-a-vis other, simpler models. Evidence relating

to this stability is a by-product of the results of this chapter; however,

a discussion of the implications is deferred to Chapter V.

Factor Analysis of Daily Security Returns

The purpose of this section is to arrive at an estimate of the number

of relevant factors. The basic data consist of daily holding period returns

including dividends on nearly 5000 New York Stock Exchange and American

Stock Exchange listed securities extracted from the Center for Research

in Security Prices (CRSP) (1983) daily returns file. The computations

are performed on the University of Georgia IBM 370 (MVS/OS) using the

maximum likelihood factor routine in the Statistical Analysis System (1982),

1982b version.

The first analysis performed is similar to that of Roll and Ross (1980).

Returns for the first 1250 trading days in the CRSP file (7/3/62 6/19/67)

were assembled for the first 30 securities (alphabetically) with a complete

return series over the period. A maximum likelihood solution was obtained

for one factor, two factors, and so on up to eight factors. The results

are summarized in Table 4-1.












TABLE 4-1

SUMMARY INFORMATION


FACTOR ANALYSIS RESULTS


Number of Securities: 30

Number of Observations: 1250

Sample Period: 7/3/62 6/19/67


Number Comouted Degrees Probability Schwarz's Akaike's
of x2 of K Factors Bayesian Information
Factors Value Freedom Sufficient Criterion Criterion


1 526.78 405 <.0001 479.98 652.10

2 447.97 376 .0063 543.69 630.74

3* 371.70 348 .1830 605.09 609.86

4* 321.38 321 .4836 675.99 613.15
5* 282.11 295 .6953 748.92 625.58

6* 243.94 270 .8709 818.80 637.07

7*+ 214.85 246 .9248 889.70 655.73

8*+ 187.45 223 .9600 957.87 674.06














TABLE 4-1 (continued)

COMMUNALITY ESTIMATES


Number Total Percentage of Total Estimated Communality Attributaole to
of Estimated Each Factor
Factors Communality
(Percent) 1 2 3 4 6 7 3


1 17.60 100
2 19.57 91.47 8.53
3* 19.34 88.24 11.76
4- 21.11 81.34 10.70 7.96
5* 22.44 76.81 9.94 7.40 5.84
6' 21.92 70.97 12.21 9.31 7.51
7* 20.73 64.27 15.46 10.53 9.75
8* 83.39 56.66 19.47 13.37 5.89 1.87





SQUARED CANONICAL CORRELATIONS


Number Squared Canonical Correlations for Each Factor with the Variables
of I (Percent)
Factors 1 2 3 4 5 6 7 8

1 84.08
2 84.30 33.37
3* 100.00 78.96 33.33
4" 100.00 79.40 33.65 27.38
5* 100.00 79.69 33.67 27.44 22.98
6* 100.00 100.00 73.78 32.62 25.95 22.94
7* 100.00 100.00 100.00 65.70 31.54 23.88 22.52
8- 100.00 92.79 81.56 75.24 57.25 29.81 26.34 22.93


Indicates a Heywood Case
+ Indicates a lack of convergence after 15 iterations, convergence is approximate.





45

Several interesting results emerge from this analysis. First, at

the 5 percent level, the hypothesis that three factors are sufficient

cannot be rejected (Table 4-1). Akaike's criterion reaches its minimum

at three as well, but Schwarz's criterion only picks up the dominant

first factor. The three factor result agrees with that ultimately

obtained by Roll and Ross (1980). A problem that can arise with the

algorithm employed is the potential for Heywood cases. A Heywood case

occurs when the factor model is a perfect fit for one or more of the

variables (i.e., a communality of one). The likelihood function is dis-

continuous at such points. The solution adopted here is to delete the

offending variable(s) and fit k 1 (or k minus the number of eliminated

variables) factors. Heywood cases arise from the numerical algorithm;

at each iteration, the securities are weighted by the reciprocal of their

uniqueness. Variables with greater communalities are thus given greater

weight. If there are too many factors relative to the number of variables,

the uniqueness can approach zero, assigning an extremely large weight to

a particular security. For example, in the three factor solution, the

communality for one of the securities moves from .49 to 1.00 in five

iterations. It is removed from the sample and convergence is established

in two additional iterations, where covergence requires that no changes

in the communality estimates exceed .001 in absolute magnitude. Once

the variable is deleted, two factors are fit to the remaining 29 variables.

The communality estimates in Table 4-1 refer to the variance explained

by the factors collectively and individually. They are obtained in the

following way: the hypothetical population correlation matrix is repro-

duced using k factors after weighting each variable by the reciprocal of

its uniqueness. The percentage of variance explained by the factors is

just the trace of reproduced correlation matrix divided by the trace of





46

the sample correlation matrix (which is simply the number of variables).

So, in the two factor solution, an estimated 19.57 percent of the total

variance is systematic. Of that systematic portion, the first factor

accounts for 91.47 percent with the second accounting for the remaining

8.53 percent. Information regarding the adequacy of the sample size is

contained in Table 4-1. The squared canonical correlations for each

factor with the variables are measures of the extent to which the factors

can be predicted from the variables and can be interpreted as squared

multiple correlation coefficients. Inspecting Table 4-1, the first
2
factor can be predicted with reasonable accuracy (R = 88.24%), but

there are insufficient variables to accurately measure the others. The

perfect correlations stem from the Heywood cases. The Heywood cases and

low multiple correlations for factors beyond the first indicate a need

for a larger sample.

In an attempt to improve the results, the sample size was doubled to

60 securities, selected sequentially from the CRSP file beginning with

the thirty-first security with complete returns data. The results are

reported in Table 4-2.

With 60 securities, the results are not drastically different from

those obtained with the smaller sample. Again, Schwarz's criterion

points to the single dominant factor. Based on Akaike's criterion, a

five factor solution is optimal, though only slightly better than a four

factor representation, and, depending on the significance level chosen,

the maximum likelihood criterion would also indicate five factors at the

most. The communality estimates obtained with the five factor solution

are quite similar to those obtained with three factors in the smaller

sample, indicating again that about 20 percent of the total variance is

systematic.












TABLE 4-2

SUMMARY INFORMATION


FACTOR ANALYSIS RESULTS


Number Computed Degrees Probability Schwarz's Akaike's
of x2 of K Factors Bayesian Information
Factors Value Freedom Sufficient Criterion Criterion


1 2235.56 1710 <.0001 1566.12 2516.54
2 1920.02 1651 <.0001 1616.36 2314.28
3 1779.84 1593 .0007 1752.23 2288.44
4 1638.73 1536 .0339 1883.99 2259.50
5 1522.11 1480 .2180 2024.60 2253.39
6 1432.15+ 1425 .4419 2175.20 2272.38
7 1344.32+ 1371 .6915 2323.26 2291.43
8 1264.53 1318 .8515 2471.82 i 2316.62
i


Number of Securities: 60

Number of Observations: 1250

Sample Period: 7/3/62 6/19/67















TABLE 4-2 (continued)

COMMUNALITY ESTIMATES


.lumoer Total Percentage of Total Estimated Communality Attributable to Eacn
of Estimated Factor
Factors Communality
(Percent) 1 2 3 4 5 6 7 8


S14.25 100.00
2 16.27 90.17 9.83
3 17.47 85.12 9.35 5.53
4 18.66 80.56 8.91 5.35 5.18
5 19.86 76.81 8.58 5.22 4.96 4.43
6+ 21.66 72.25 8.41 6.79 4.58 4.43 3.56
7+ 22.20 70.47 8.02 5.62 4.56 4.41 3.53 3.40
8 22.99 68.37 7.83 5.C6 1.60 4.27 3.41 3.38 3.08


SQUARED CANONICAL CORRELATIONS


Number Squared Canonical Correlations for Each Factor Aith the Variables
or (Percent
Factors 1 2 3 4 5 6 7 8


1 89.53
2 89.80 48.98
3 89.92 49.50 36.69
4 90.02 49.96 37.45 36.70
5 90.15 50.56 38.35 37.13 34.55
6+ 90.37 52.21 46.86 37.31 36.54 31.62
7+ 90.37 51.64 42.80 37.77 37.01 31.96 31.15
8 90.41 51.93 41.10 38.81 37.37 32.01 31.80 29.79


+Indicates a lack of convergence after 15 iterations, convergence is approximate.





49

As indicated in Table 4-2, there is a significant increase in the

squared canonical correlation for the second factor, rising from about

33 percent to 50 percent. The increased sample size also eliminates the

troublesome Heywood cases, with no communalities exceeding .50. The

algorithm converges rapidly for up to five factors, generally requiring

no more than five iterations. Although convergence was achieved for the

eight factor solution (fifteen iterations), the six and seven factor

attempts did not converge. The conclusion from this analysis is that no

more than five factors are needed to account for the systematic inter-

correlation for sixty securities. These analyses were repeated for

several different samples with the same general results: not more than

five factors are needed.

As a final check, the two samples were combined with ten other

securities for a total of one hundred. The results in Table 4-3 indicate

that the five factor solution originally indicated is insufficient.

As indicated in Table 4-3, factor solutions for up to six factors

converge rapidly (as few as three iterations); however, the maximum likeli-

hood criterion indicates that even a seven factor solution is inadequate.

With eight or nine factors, the algorithm was unable to find even an

approximate solution. Since Horn and Engstrom's (1979) results indicate

the use of a larger significance level with the larger sample, an eleven

or even ten factor solution is indicated. Akaike's criterion indicates

six factors with Schwarz's criterion again identifying only the dominant

factor. The use of the larger sample results in general improvement in

the predictability of the first several factors; however, the communality

totals are relatively unchanged.

The results from the different sized samples contain some similari-

ties. First, the Bayesian criterion always indicates a single factor.





































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Based on the description in Chapter III, this is not surprising; all

other things equal, large numbers of observations (1250 in this case) will

tend to greatly reduce the number of significant factors in this approach.

The classical maximum likelihood estimate behaves in a relatively predict-

able fashion as well with the number of factors determined to be significant

varying with the sample size. In fact, the number of factors appears to

be roughly proportional to the number of securities at 8 to 10 percent.

Akaike's criterion is relatively stable, indicating five factors with

sixty securities and five to six with one hundred.

An important similarity between the results is the behavior of the

estimated total communality; one conclusion which can be drawn is that

the portion of daily security returns which is systematic is approximately

20 percent. Of this portion, the market factor accounts for about 75

percent or 15 percent of the total variance. If the one factor solution

is viewed as the best possible single index, then the inclusion of multiple

indices can result in a 33 percent (.05/.15) improvement in systematic

risk estimates. Moreover, inspection of the factor loadings indicates

that this potential gain is not uniformly spread among the securities; in

some cases, the variance explained by factors beyond the first is greater

than the single factor explanatory power--evidence of significant "extra-

market" sources of risk.

As a final attempt at objectively determining the approximate

dimension of the model, the analysis of the one hundred securities was

extended to include an additional five hundred observations per security.

The results were virtually identical with those obtained in the previous

trial and are not reported here. The very similar results stem from the

fact that there is little difference in the sample correlation matrices

despite the different lengths of time.








When the sample size is increased to 150 securities, the computational

requirements become prohibitive, with a six factor solution requiring more

than forty minutes of CPU time. In comparison, the six factor solution

with 100 securities required only twelve minutes.

This section began in an attempt to arrive at an objective determina-

tion of the number of factors. Based on analyses of different sample sizes,

only one criterion is completely consistent regarding the number of factors.

Schwarz's Bayesian criterion indicates that the basic intuition of the

market model is correct; namely, there is a single dominant factor present

in security returns. On the other hand, the communality estimates indicate

significant gains to be had from additional information, particularly for

certain securities. The fact that the variance explained by factors beyond

the first is concentrated in a subset of the securities improves the case

for a multi-index approach. If the securities in the subset are relatively

homogeneous, then single-index models may systematically mis-price their

riskiness. This potential bias could account for anomalies associated

with the market model. Some evidence for the validity of this speculation

is offered in Chapter V.

For the reasons discussed above, the dimension of the arbitrage model

must be specified subjectively. The decision was made to continue this

research using six factors, the number indicated by Akaike's criterion.

This choice also agrees with the five extra-market clusters identified by

Arnott (1980). If anything, six factors would appear to be more than

enough. Referring to Table 4-3, factors beyond the fourth individually

account for less than one percent of the total variation and less than

five percent of the estimated communality.

Other techniques are available for arriving at a decision in regard

to the number of factors. The two most popular are Cattell's (1966) scree


_~








chart and Kaiser's (1960) eigenvalue criterion. Both approaches are

based on the relative magnitudes of the eigenvalues of the sample correla-

tion matrix. Cattell's method amounts to inspecting a chart of the

eigenvalues plotted in order of decreasing magnitude and looking for

breaks in the pattern. Because the average eigenvalue of a correlation

matrix is unity, Kaiser's rule is to retain all eigenvalues with values

greater than one. Neither of these approaches is especially enlightening

for the securities data; the only clear break in the ordered eigenvalues

occurs at the second factor and the number of eigenvalues exceeding one

is quite large (23 out of 100 in one case).

In summary, there is sufficient noise in the ex post realized returns

that it is impossible to objectively determine the number of factors

beyond the first. Grouping techniques would no doubt reduce the noise

content, but the results of this section indicate that randomly formed

portfolios would probably swamp the extra-market components. It appears

likely that the number of factors present ex ante is less than six, and

evidence for this will be forthcoming in the tests of the APT. In the

next section, the time periods and sampling techniques used in this study

are described along with some properties of the six factor solutions in

different samples.

Preliminary Analyses of the Arbitrage Model

The basic data used to test the APT once again consist of daily

security returns from the CRSP file. Three non-overlapping base periods

were chosen for the factor analyses, each covering 1250 trading days

(about five years). Within each of three base periods, three samples of

100 securities each were created by taking every tenth security with a

complete returns series from the CRSP file beginning with the first,

second and third listed securities. Interval sampling was used to avoid






56
undue concentration in certain industries; for example, alphabetic selection

would result in one group being dominated by financial institutions--over

thirty have names beginning with the letters of "Fi." To avoid confusing

the samples, the notational convention of designating the time periods

as 1, 2, and 3 and the samples as a, b, and c is adopted; thus, sample

2b is the second sample in the second time period. The exact dates for

the base periods are

Base Period 1: 7/3/62 6/19/67

Base Period 2: 7/20/67 7/11/72

Base Period 3: 7/12/72 6/23/77.

Once the data were obtained, a six factor maximum likelihood solution

was obtained for each of the nine groups. Table 4-4 summarizes the results.

The total communality estimates in the first two base periods are all

between 20 and 25 percent; however, the third period totals are greater;

between 26 and 32 percent of the total variance is systematic. The market

factor accounts for 71 to 80 percent of the total communality, with

increased importance in the third period.

As discussed in Chapter III, any orthogonal linear transformation of

the initial solution generates a mathematically equivalent result.

Because the APT is designed to explain cross-sectional differences in

security returns, a weighted Varimax rotation (Cureton and Mulaik, 1975)

was used to increase the cross-sectional variation in the factor loadings.

With this rotation, the factor loadings for the individual securities are

first weighted by reciprocal of their uniqueness estimates; then, an

orthogonal transformation matrix is determined such that the variance of

the loadings on a particular factor is maximized. By maximizing the

variance of the column loadings, the larger estimates are increased and

the smaller estimates decreased. The weighted Varimax rotation also





















TABLE 4-4
SUMMARY INFORMATION


Base Period 1: 7/3/62 6/19/67
Base Period 2: 6/20/67 7/11/72
Base Period 3: 7/12/72 6/23/77
Trading Days per Period: 1250
Samples per period: 3
Number of Securities per Sample: 100


INITIAL SOLUTION


BASE PERIOD 1


BASE PERIOD 2


BASE PERIOD 3


SAMPLE a b c a b c a b c


TOTAL
COMMUNALITY 22.05 24.10 21.41 21.13 22.45 21.90 26.10 31.75 23.07


PERCENTAGE
OF TOTAL a b1 c a b c a b
FOR EACH FACTOR

1 70.97 76.65 75.41 76.44 77.15 75.58 78.37 78.48 79.48
2 10.00 7.77 7.94 8.05 6.75 9.16 7.23 7.43 6.87
3 7.44 5.18 5.67 4.59 5.47 4.69 4.36 4.87 4.16
4 4.27 3.79 4.53 4.05 3.95 4.01 4.07 3.36 3.58
5 3.86 3.32 i 3.44 3.51 3.44 3.41 3.11 3.19 3.19
6 3.46 3.09 1 3.02 3.36 3.25 3.15 2.86 2.67 2.72







increases the "gain" on factors two through six at the expense of the

first factor. As indicated in Table 4-5, the total communality is spread

fairly evenly over the first four factors as a result. This effect

should make it possible to measure the factors beyond the first with

greater accuracy.

The three test periods in this study are, for each sample, the five

hundred trading days subsequent to the base periods. The exact dates are

Test Period 1: 6/20/67 7/25/69

Test Period 2: 7/12/72 7/08/74

Test Period 3: 6/24/77 6/18/79.

To calculate the returns on the arbitrage portfolios, the weights are

obtained using eq. (3-28)

X = 4-A(A-1A)1 (3-28)

where A is the augmented factor loading matrix and is the diagonal

matrix of residual variances. The arbitrage portfolio returns over the

test periods are calculated using eq. (3-29)

R'X = R'V-1A(A -A)-1, (3-29)

where R is the 100 by 500 matrix of daily returns. As noted in Chapter

III, the arbitrage portfolio returns calculated in this fashion are

identical to a time-series of coefficients obtained from 500 GLS cross-

sectional regressions, with the zero beta return as the time series of

estimated intercept terms. Summary univariate statistics are reported

for the nine samples in Tables 4-6 through 4-14.

In the first test period, covering 7/67 7/69, the zero beta return

is insignificantly different from zero in all three samples. Point esti-

mates of the average daily return range from a negative 2.6 basis points

to a positive 3 basis points, with standard deviations of 60 to 70 basis

points. Averaging the three point estimates and annualizing the results,































TABLE 4-5
VARIMAX ROTATED SOLUTION


BASE PERIOD 1


BASE PERIOD 2


BASE PERIOD 3


SAMPLE a b c a b c a b c

TOTAL
COMMUNALITY 22.05 24.10 21.41 21.13 22.45 21.90 26.10 31.75 28.07


PERCENTAGE
OF TOTAL a b c a b c a b c
FOR EACH FACTOR

1 17.62 32.99 25.94 26.11 22.94 33.65 23.31 29.64 27.53
2 17.20 27.19 19.23 25.03 20.23 29.14 21.03 24.88 23.16
3 21.37 15.00 20.06 19.73 16.42 19.73 19.91 18.79 15.02
4 19.42 13.64 14.21 16.95 19.55 9.35 17.78 14,24 15.34
5 14.57 6.12 15.04 7.40 12.77 4.80 12.07 8.50 10.50
6 9.82 5.07 4.50 4.78 5.06 3.34 5.90 3.95 8.46
i ----------_____ ____ ____ ____ ___ \ ____ ____ ____ ____I








a return in the neighborhood of 1 percent is indicated, at a time when

money market rates varied from four to six percent (Federal Reserve

Bulletin, 1970). In the second test period (7/72 6/74), the zero beta

returns are larger and vary from 2 to 7 basis points per day. An

approximate annual yield of 18 percent is obtained by averaging the three.

Once again, the standard deviations of the estimates are large relative

to the point values. Also, the standard deviations are similar to each

other, ranging from 89 to 106 basis points. A low estimate of .4 basis

points per day was obtained in the third period (7/77 6/79) with a high

estimate of 5 points. The average of the three is about 2 points per day

or a 7.2 percent annual yield which compares favorably with the 7.19

percent yield on 90-day Treasury Bills for the year 1978 (Federal Reserve

Bulletin, 1979). The standard deviations range from 49 to 63 basis points.

Because of the variability in the zero beta return, the estimates are not

especially reliable; however, the average estimates obtained in the first

and third periods are fairly close to the then prevailing interest rates.

One problem with the GLS estimates as opposed to OLS estimates is that

unbiasedness is achieved at the expense of greater variance in the estimated

factor scores. When the zero beta returns are estimated using OLS, the

averages in basis points per day for the three periods are 2.2, 3.53, and

1.47 respectively. The estimate in the first period is quite close to the

1968 daily return on 90-day Treasury Bills which averaged 2.18 points

per day (Federal Reserve Bulletin, 1970). The estimate in the second

period is still high, though less so than with the GLS estimate, and the

third period estimate appears low. It is difficult to generalize from

these results; however, there appears to be no tendency for the estimates

to be consistently higher than some measure of the risk-free rate, unlike






















TABLE 4-6
SUMMARY INFORMATION

Test Period: la
Number of Trading Days: 500
Number of Securities: 100




UNIVARIATE STATISTICS FOR TEST PERIOD


CORRELATIONS AMONG ARBITRAGE PORTFOLIO RETURNS


Average Standard Computed Significance
Portfolio Daily Deviation T Level
Return Statistic


:ero Beta .00006 .0067 .199 .842
Arbitrage *1 .1128 2.0702 1.219 .224
Arbitrage 2 .0317 1.2539 .566 .572
Arbitrage 43 -.1503 1.0789 -3.115 .002
Arbitrage A4 .0223 1.1348 .440 .660
Arbitrage C5 -.0120 1.3648 .197 .844
Arbitrage '6 .0577 1.3576 .950 .343


Portfolio Zero Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage
Beta #1 #2 #3 A4 05 '6


:ero Beta 1.000
Arbitrage 1i .750 1.000
Arbitrage ': .190 .109 1.000
Arbitrage -3 .074 .036 .280 1.000
Arbitrage -1 .084 .006 .109 .044 1.000
Arbitrage 5 .192 .002 .166 .201 .041 1.000
Arbitrage 6 .160 .220 .152 .141 .111 .175 1.000





















TABLE 4-7
SUMMARY INFORMATION

Test Period: Ib
Number of Trading Days: 500
Number of Securities: 100




UNIVARIATE STATISTICS FOR TEST PERIOD


Average Standard Computed Significance
Portfolio Daily Deviation T Level
Return Statistic


Zero Beta -.00026 .0061 .944 .345
Arbitrage 41 -.1180 .9780 -2.698 .007
Arbitrage 42 .1947 2.0155 2.160 .031
Arbitrage '3 .0143 1.3830 .232 .818
Arbitrage 44 .0232 1.0805 .480 .631
Arbitrage 45 .0186 1.3815 .300 .764
Arbitrage 6 .0933 1.6222 1.286 .199


CORRELATIONS AMONG ARBITRAGE PORTFOLIO RETURNS


Portfolio Zero Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage
Beta #1 #2 #3 4 45 =b


Zero Beta 1.000
Aroitrage #1 .265 1.000
Arbitrage 2 .765 .301 1.000
Arbitrage #3 .347 .058 .233 1.000
Arbitrage '4 .171 .255 .090 .059 1.000
Arbitrage -5 .116 .075 .191 .007 .041 1.000
Arbitrage #6 .308 .113 .236 .223 .209 .112 1.000




















TABLE 4-8
SUMMARY INFORMATION

Test Period; Ic
Number of Trading Days: 500
Number of Securities: 100




UNIVARIATE STATISTICS FOR TEST PERIOD



Average Standard Computed Significance
Portfolio Daily Deviation T Level
Return Statistic


Zero Beta .0003 .0064 1.054 .293
Arbitrage l -.0787 1.0224 -1.721 .086
Arbitrage :2 .0566 1.7255 .734 .463
4rbitrage :3 .1169 1.5585 1.677 .094
Arbitrage '4 .00C6 1.4533 .010 .992
Arbitrage cS -.0873 1.3578 -1.440 .151
Arbitrage 46 -.0363 1.4333 .566 .571


CORRELATIONS AMONG ARBITRAGE PORTFOLIO RETURNS



Portfolio Zero Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage
Beta #1 #2 03 #4 45 si


:ero Beta 1.000
Arbitrage Cl .351 1.CO0
Arbitrage 2 .623 .310 1.000
Arbitrage e3 .039 .003 .013 1.C00
Arbitrage e4 .312 .221 .103 .001 1.0CO
Arbitrage :S .324 .383 .161 .406 .1C1 1.000
Arbitrage 76 .138 .070 .160 .168 .108 .098 1.000





















TABLE 4-9
SUMMARY INFORMATION

Test Period: 2a
Vumber of Trading Days: 500
Number of Securities: 100




UNIVARIATE STATISTICS FOR TEST PERIOD


CORRELATIONS AMONG ARBITRAGE PORTFOLIO RETURNS


Average Standard Computed Significance
Portfolio Daily Deviation T Level
Return Statistic


:ero Beta .0002 .0106 .465 .642
Arbitrage 1 .1659 1.9190 1.933 .054
Arbitrage -2 -.0016 1.5070 .024 .981
Arbitrage ;3 .0110 1.4200 .173 .863
Arbitrage *4 .0175 1.2895 .303 .762
Arbitrage 5 .0271 1.5663 .386 .699
Arbitrage e6 .0345 1.5421 .501 .617


Portfolio Zero Arbitrage Arbitrage Arbitrage Aritrage Aroitrrage Arbitrage
Beta #1 #2 O3 4 *5 =C


:ero Beta 1.000
Arbitrage CI .695 1.000
Arbitrage C2 .327 .270 1.000
Arbitrage 43 .370 .301 .136 1.000
Arbitrage *4 .082 .101 .151 .233 1.000
Arbitrage #5 .168 .014 .050 .002 .086 1.000
Arbitrage 6 .066 .089 .047 .088 .362 .040 1.000





















TABLE 4-10
SUMMARY INFORMATION

Test Period: 2b
Number of Trading Days: 500
Number of Securities: 100




UNIVARIATE STATISTICS FOR TEST PERIOD


CORRELATIONS AMONG ARBITRAGE PORTFOLIO RETURNS


Portfolio Zero Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage
Beta #1 #2 #3 '4 '5 'b


Zero Beta 1.000
Arbitrage -1 .438 1.000
Arbitrage 2 .417 .113 1.000
Arbitrage 43 .144 .262 .154 1.000
Arbitrage 44 .088 .002 .053 .319 1.000
Arbitrage i5 .329 .042 .074 1.75 .039 1.000
Arbitrage 46 .192 .250 .077 1.82 .075 .119 1.000


Average Standard Computed Significance
PortDolio Dally Deviation T Level
Return Statistic


:ero Beta .0006 .0095 1.426 .154
Aroitrage I .0079 1.3987 .126 .900
Arbitrage 2 .1534 1.5485 2.215 .027
Arbitrage #3 .0191 1.3076 .327 .744
Arbitrage '4 -.0111 1.3230 .188 .851
Aroitrage 45 -.0421 1.5497 .607 .544
Arbitrage i6 .0136 1.2597 .241 .810


--





















TABLE 4-11
SUMMARY INFORMATION

Test Period- 2c
Number of Trading Days: 500
Number of Securities: 100




UNIVARIATE STATISTICS FOR TEST PERIOD



Average Stanaard Computed Significance
Portfolio Daily Deviation T Level
Return Statistic


:ero Beta .0007 .0089 1.786 .075
Arbitrage -1 .0953 1.4644 1.455 .146
Arbitrage p: .0626 1.3501 1.037 .300
Arbitrage #3 .0138 1.0822 .284 .776
Arbitrage 4 -.0893 1.5654 -1.275 .203
Arbitrage 05 .0647 1.5594 .928 .354
Arbitrage #6 -.0674 1.6286 .925 .355


CORRELATIONS AMONG ARBITRAGE PORTFOLIO RETURNS



Portfolio 2ero Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage
Beta FI #2 S3 a4 '5 .1


Zero Beta 1.000
Arbitrage I .219 1.000
Arbitrage ': .123 .255 1.000
Arbitrage 03 .108 .107 .399 1.000
Arbitrage '4 .387 .047 .090 .112 1.000
Arbitrage t5 .219 .033 .095 .192 .085 1.000
Arbitrage -6 .048 .025 .336 .149 .170 .1029 1.000




















TABLE 4-12
SUMMARY INFORMATION

Test Period: 3a
Number of Trading Days: 500
Number of Securlties: 100




UNIVARIATE STATISTICS FOR TEST PERIOD


Average Standard Computed Significance
Portfolio Daily Deviation T Level
Return Statistic


:ero Beta .00007 .0063 .253 .800
Arbitrage l .0474 .9139 1.160 .247
Arbitrage #2 .1099 1.0036 2.450 .016
Arbitrage 03 -.0104 1.0073 .217 .829
Arbitrage '4 -.0287 .9802 .655 .513
Arbitrage 45 -.0053 .9601 .123 .902
Arbitrage #6 -.0649 .9340 -1.550 .122


CORRELATIONS AMONG ARBITRAGE PORTFOLIO RETURNS


?ortfolio Zero Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage
Beta 1 #2 53 i '0 "b


:ero Beta 1.000
Arbitrage 1 .128 1.000
Arbitrage '= .251 .094 1.000
Arbitrage '3 .737 .020 .018 1.000
Arbitrage 4 .302 .053 .125 .263 1.000
Arbitrage S .291 .110 .114 .160 .264 1.300
Arbitrage '6 .163 .005 .102 .155 .158 .132 1.000




















TABLE 4-13
SUMMARY INFORMATION

Test Period: 3b
Number of Trading Days: 500
Number of Securities: 100




UNIVARIATE STATISTICS FOR TEST PERIOD


CORRELATIONS AMONG ARBITRAGE PORTFOLIO RETURNS


Average Standard Computed Significance
Portfolio Daily Deviation T Level
Return Statistic


:ero Beta .0005 .0068 1.535 .125
Arbitrage al .0843 1.0709 1.760 .079
Arbitrage #2 -.0087 .9648 .201 .841
Arbitrage #3 -.0623 .9539 -1.460 .145
Arbitrage #4 -.0228 .9241 .551 .582
Arbitrage #5 -.0104 .9786 .239 .811
Arbitrage #6 -.0050 .9335 .120 .905


Portfolio Zero Arbitrage Arbitrage Arbira bitrae Arbitrage Arbitrage
Beta #1 #2 #3 4 s5 -6


Zero Beta 1.000
Arbitrage '1 .189 1.000
Arbitrage '2 .185 .102 1.000
Arbitrage #3 .742 .053 .278 1.000
Arbitrage .210 .123 .035 .043 1.000
Arbitrage c5 .554 .035 .084 .401 .044 1.000
Arbitrage 6 .100 .014 .003 .014 .018 .130 1.000




















TABLE 4-14
SUMMARY INFORMATION

Test Period: 3c
Number of Trading Days: 500
Number of Securities: 100




UNIVARIATE STATISTICS FOR TEST PERIOD


Average Standard Computed Significance
Portfolio Daily Deviation T Level
Return Statistic


:ero Beta .0004 .0049 .181 .856
Arbitrage 41 .0812 .9404 1.930 .054
Arbitrage =: .0132 .9073 .325 .745
Arbitrage 03 -.0329 1.0015 .734 .453
Arbitrage 4 -.0059 .8027 .164 .870
Arbitrage 5 .0296 .9363 .706 .480
Arbitrage 06 .0194 .9542 .454 .650


CORRELATIONS AMONG ARBITRAGE PORTFOLIO RETURNS


Portfolio Zero Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage
Beta 01 42 '3 -4 45 *6


:ero Beta 1.000
Arbitrage i .365 1.000
Arbitrage 1: .109 .177 1.000
Arbitrage =3 .224 .208 .020 1.000
Arbitrage 41 .029 .004 .122 .201 1.000
Arbtrage 05 .440 .120 .085 .015 .022 1.000
Arbitrage =6 .165 .098 .032 .157 1.,6 .074 1.000





70

the same estimates reported in studies of the market model (e.g., Fama

and MacBeth, T973).

Also reported in Tables 4-6 through 4-14 are the estimated returns

on the arbitrage portfolios and their standard deviations. In the first

period, one portfolio in sample a has a significantly non-zero return

while b and c each have two such portfolios (at a 10% level). Samples

2a and 2b each contain one significant return, and 2c contains none. In

the third period, each sample has one portfolio with a statistically

non-zero return.

In the base periods, the hypothetical factors were constructed such

that they were mutually uncorrelated with unit variances. Estimates of

the factor scores will in general possess neither property exactly. In

the first two periods, the variances generally exceed one, while in the

third period they are relatively close.

More troublesome are the correlations of the arbitrage portfolio

returns with the zero beta return; in every case, there is at least one

arbitrage portfolio with a substantial negative correlation. This unanti-

cipated result is difficult to explain. However, if the risk premia are

defined as excess returns above the zero beta return, then an inverse

relationship implies that the nominal premia are not constant and decrease

when the zero beta return increases. Alternatively, noise in the ex post

data may give rise to differential measured sensitivities to the zero beta

return. In this case, an extra factor may appear to exist and the appro-

priate dimension of the model would be over-estimated. The extra factor

would not be priced, however. A purely empirical explanation is readily

apparent when the structure of the arbitrage portfolios is examined. The

portfolios with large negative correlations correspond to factors which

tend to be dominated by public utilities, a group traditionally considered







to be interest rate sensitive. This should not necessarily be interpreted

as evidence for an interest rate factor. The utilities are regulated,

they have substantial dividend yields, and they tend to be large. Thus,

the phenomenon could relate to regulatory risk or lag, taxation of dividends,

a size effect, or some other common characteristic. Whatever the explana-

tion, the phenomenon is persistent and the collinearity between the zero

beta return and the arbitrage returns may lead to econometric difficulty.

The results of this section indicate considerable volatility in the

estimated zero beta returns and the risk premia, most of which are not

statistically different from zero. It is important to emphasize here that

no conclusions concerning the APT can be drawn at this point. The statis-

tical significance (or lack thereof) of the risk premia cannot be inter-

preted as evidence for or against the APT for two reasons. First, on a

theoretical level, such evidence relates to the linearity of the pricing

relationship which for reasons discussed in Chapter III is not the relevant

issue. Second, the risk premia are not uniquely determined empirically.

The factor analysis uniquely determines the space into which returns are

projected; the orientation of the factors within that space is only unique

up to non-singular transformation. In practical terms, the significance

of the risk premia are a function of the rotation chosen, using an oblique

(non-orthogonal) rotation will generally result in greater numbers of

"significant" factors at the expense of greater correlations among the

risk premia. In fact, with the Promax rotation (Lawley and Maxwell, 1971),

the degree of factor inter-correlation is to some extent controllable, and

it can simply be increased until most of the premia are significant. This

inherent indeterminateness renders attempts at identifying the number of

relevant factors by cross-sectional regression meaningless. In sum, the







requisite degree of linearity in the pricing relationship is unknown

a priori and empirically indeterminate.

The result of a factor analysis is a unique estimate of a common

factor space and the residual variances for the securities. Testing the

APT requires examination of this unique information. Because the zero

beta portfolio is constructed to be a member of the space perpendicular

to the common factor space, it is unaffected by rotations of the factors

within that space. When the additional requirement of minimum residual

variance is imposed, the zero beta portfolio is uniquely defined. In

fact, in the hypothetical population, the zero beta portfolio is the

global minimum variance portfolio, and is thus mean-variance efficient.1

The implications of this fact are pursued in the next section.

Univariate Results from the Arbitrage Model

In this section, the portfolio returns created in the previous

section are used as independent variables in time-series regressions of

the form specified in (3-30)

rjt = _j + bjooot + bjlt + . + bj666t + ejt, (3-30)

j = 1, . ., 100

t = 1 ...., 250

251, . ., 500.

The intercept term is a measure of abnormal performance and will not

differ from zero unless arbitrage opportunities exist. The coefficient

of the zero beta return should be one, and the other coefficients should

be generally significant. Mathematically, the coefficient of the zero

beta return would be exactly equal to one if the testing were done in


1This fact is pointed out by Ingersoll (1982).







the base period using the actual residual covariance matrix to obtain

the weights. In this case, the portfolio is mean-variance efficient in

the sample and has an exactly zero correlation with any arbitrage port-

folio. Because of this the estimated coefficient, bo is equal to the

covariance of the return on security j with the zero beta return, divided

by the variance of the zero beta return. It is easily verified (e.g.,

Roll, 1976) that the global minimum variance portfolio has the property

that its covariance with any non-arbitrage portfolio is equal to its own

variance; hence, an "estimated" coefficient would be unity with a zero

error. As it is employed here, the zero beta portfolio return is an

estimate of the unobservable minimum variance portfolio for all risky

assets of the type under consideration.2

In the actual estimation, each of the three test periods is divided

into two sub-periods, each of which covers 250 trading days. This was

done in order to examine the stability of the estimates and the extent

to which they deteriorate over time. The total number of estimated

equations is 1800, consisting of nine samples of 100 securities each and

two subperiods per sample. The results are reported in Tables 4-15 to

4-23.

In each of the tables, several items are tabulated. The first is

the number of securities for which significant intercepts were found.

The second and third are the number of estimates of b which differ from

zero and one respectively. In the first 250 trading days of sample la,

three of the 100 securities had significant intercepts, and 85 had significant

coefficients on the zero beta portfolio, eight of which were significantly


This argument is extendable to all risky assets in the economy if
it is possible to hedge any type of systematic risk using the subset of
assets from which the factors were obtained.









TABLE 4-15

SUMMARY INFORMATION


Test Period: la

Number of Trading Days: 500

Number of Securities: 100


UNIVARIATE TESTS OF APT


Hypothesis Rejections


ccO
S=0
0
b = 0
bc=1


3
85

8


Hypothesis Rejections


= 0 1
b = 0 88
1 16
: 1 16
0


NUMBER OF EQUATIONS WITH k SIGNIFICANT FACTORS

(not including zero beta portfolio)

First 250 Second 250
Trading Days Trading Days



k # of eq. k # of eq.

0 1 0 2
1 9 1 13
2 12 2 20
3 18 3 14
4 21 4 22
5 23 5 17
6 16 6 12


Note: Significance level is .05 for two-tailed T-test.


First 250
Trading Days


Second 250
Trading Days








TABLE 4-16

SUMMARY INFORMATION


UNIVARIATE TESTS OF APT


Second 250
Trading Days


Hypothesis Rejections


S0 0
b= 0 90

b = 1 18
o1


NUMBER OF EQUATIONS WITH k SIGNIFICANT FACTORS

(not including zero beta portfolio)

First 250 Second 250
Trading Days Trading Days


k # of eq.

0 0
1 8
2 17
3 24
4 22
5 20
6 9


k # of eq.

0 1
1 11
2 20
3 25
4 17
5 19
6 7


Note: Significance level is .05 for two-tailed T-test.


Test Period: lb

Number of Trading Days: 500

Number of Securities: 100


First 250
Trading Days


Hypothesis Rejections


a= 0 2
b =0 87

b =1 11
o








TABLE 4-17

SUMMARY INFORMATION


Test Period: Ic

Number of Trading Days: 500

Number of Securities: 100


UNIVARIATE TESTS OF APT


Second 250
Trading Days


Hypothesis Rejections


a= 2
b = 0 95

b =1 18
0


NUMBER OF EQUATIONS WITH k SIGNIFICANT FACTORS

(not including zero beta portfolio)


Second 250
Trading Days


k # of eq.

0 0
1 4
2 23
3 26
4 23
5 18
6 6


Note: Significance level is .05 for two-tailed T-test.


First 250
Trading Days


First 250
Trading Days


k # of eq.

0 2
1 11
2 22
3 19
4 21
5 21
6 4








TABLE 4-18

SUMMARY INFORMATION


UNIVARIATE TESTS OF APT


Second 250
Trading Days


NUMBER OF EQUATIONS WITH k SIGNIFICANT FACTORS

(not including zero beta portfolio)


Second 250
Trading Days


k # of eq. k #

0 1 0
1 4 1
2 15 2
3 20 3
4 25 4
5 23 5
6 12 6


Note: Significance level is .05 for two-tailed T-test.


Test Period: 2a

Number of Trading Days: 500

Number of Securities: 100


First 250
Trading Days


Hypothesis Rejections


= 0 0
S= 0 94

b 1 24
0 ___________


Hypothesis Rejections


a= 0 0
= 0 89

S b 1 13
L 0


First 250
Trading Days








TABLE 4-19

SUMMARY INFORMATION


Test Period: 2b

Number of Trading Days: 500

Number of Securities: 100


UNIVARIATE TESTS OF APT


Second 250
Trading Days


Hypothesis Rejections


a: 0 0
b = 0 88

b = 1 19
0__ _


NUMBER OF EQUATIONS WITH k SIGNIFICANT FACTORS

(not including zero beta portfolio)


First 250
Trading Days


Second 250
Trading Days


k # of eq. k

0 1 0
1 8 1
2 10 2
3 24 3
4 21 4
5 19 5
6 17 6


Note: Significance level is .05 for two-tailed T-test.


First 250
Trading Days


Hypothesis Rejections


= 0 0
b 0 96

bo 1 13








TABLE 4- 20

SUMMARY INFORMATION


Test Period: 2c

Number of Trading Days: 500

Number of Securities: 100


UNIVARIATE TESTS OF APT


Second 250
Trading Days


NUMBER OF EQUATIONS WITH k SIGNIFICANT FACTORS

(not including zero beta portfolio)


Second 250
Trading Days


k # of eq.

0 0
1 2
2 13
3 13
4 28
5 30
6 14


Note: Significance level is .05 for two-tailed T-test.


First 250
Trading Days


Hypothesis Rejections


= 0 0
b 0 95

b 1 21
0


Hypothesis Rejections


S 0 0
b= 0 86

b =1 15
0


First 250
Trading Days


k # of eq.

0 1
1 7
2 7
3 25
4 21
5 23
6 15








TABLE 4-21

SUMMARY INFORMATION


UNIVARIATE TESTS OF APT


First 250
Trading Days


Second 250
Trading Days


Hypothesis Rejections


S= 0
b 0

b =1
0


0
87

15


NUMBER OF EQUATIONS WITH k SIGNIFICANT FACTORS

(not including zero beta portfolio)

First 250 Second 250
Trading Days Trading Days



k # of eq. k # of eq.

0 1 0 2
1 6 1 5
2 22 2 7
3 19 3 28
4 20 4 19
5 14 5 21
6 18 6 18


Note: Significance level is .05 for two-tailed T-test.


Test Period: 3a

Number of Trading Days: 500

Number of Securities: 100


Hypothesis Rejections


a=0 2
b 0 85

b 1l 20
o









TABLE 4-22

SUMMARY INFORMATION


UNIVARIATE TESTS OF APT


First 250
Trading Days


Second 250
Trading Days


Hypothesis Rejections


S0 0
b = 0 89

b = 1 17
0


NUMBER OF EQUATIONS WITH k SIGNIFICANT FACTORS

(not including zero beta portfolio)


First 250
Trading Days


Second 250
Trading Days


k # of eq. k #

0 2 0
1 11 1
2 17 2 12
3 14 3 2:
4 22 4 2E
5 25 5 2:
6 9 6 1(


Note: Significance level is .05 for two-tailed T-test.


Test Period: 3b

Number of Trading Days: 500

Number of Securities: 100


Hypothesis Rejections


a=0 0
b= 0 79

b = 1 11
ob ________








TABLE 4-23

SUMMARY INFORMATION


Test Period: 3c

Number of Trading Days: 500

Number of Securities: 100


UNIVARIATE TESTS OF APT


Second 250
Trading Days


Hypothesis Rejections


a = 0 0
b 0 87

S=1 22
So


NUMBER OF EQUATIONS WITH k SIGNIFICANT FACTORS

(not including zero beta portfolio)

First 250 Second 250
Trading Days Trading Days


k # of eq.

0 0
1 5
2 10
3 20
4 23
5 23
6 19


Note: Significance level is .05 for two-tailed T-test.


First 250
Trading Days


Hypothesis Rejections


S=0 0
b 0 83

b = 1 12
0


k # of eq.

0 3
1 7
2 13
3 17
4 24
5 23
6 13







different from one. Also reported are the number of equations which

were found to have various numbers of significant coefficients on the arbi-

trage portfolios. The majority of securities have from two to five such

coefficients.

The results are in substantial, though not complete, agreement with

the predictions of the APT. Significant intercepts occur less than 1% of

the time; the zero beta return is significant in about 90% of the trials,

and differs from one in about 16% of the trials. Over 90% of the securities

have significant coefficients on three or more of the arbitrage portfolios.

There does not appear to be any significant deterioration in explanatory

power over the second sub-periods. Moreover, the average R2's agree with

the initial communalities; for example, the communality estimated in

sample la was 22.05% and the average R2 from the first sub-period was

20.04%. These summary measures are to some extent misleading in that

they understate the degree of conformity of the results with the APT.

There is a definite tendency for the model to work quite well for the

majority of the securities and work poorly for a minority. Typically,

the intercept is insignificant at any conventional level, b0 is within

one standard error of its predicted value, and several other coefficients

are significant at any conventional level. The coefficient estimates are

frequently in excess of three standard errors away from zero. The results

in this section are generally in accord with the APT. That the model

occasionally works poorly is not surprising; the theory is itself an

approximation and is expected to have low explanatory power for some subset

of the securities under consideration. While the predictions of the APT

appear to be supported by the data, the results in this section do not

account for cross-sectional dependencies in the estimates, and no general

conclusion can be drawn about the central prediction of the theory, namely,






84

an absence of arbitrage opportunities. This is the subject of the next

section.

A Multivariate Test of the APT

In this section, consideration turns to the question of whether the

intercepts and zero beta coefficients are jointly different from their

predicted values. A good reference for the multivariate techniques

employed in this section may be found in Timm (1975). The individual

time-series estimates of eq. (3-30) can be brought together in matrix

notation as

R = AB + E, (4-2)

where


R = = the 250 x 100 of security returns

6= = the 250 x 8 matrix of augmented portfolio returns

B = = the 8 x 100 matrix of estimated coefficients

E = = the 250 x 100 matrix of residuals.

It is assumed that

E[E] = 0

E[R] = AE

V[R] = It 0 E.

In this case

B = (AA )- 'R,

and an unbiased estimate of c is

= (R AB)1 (R B)
t-k-l


(4-3)


= R'(I A(A' A) A' )R. (4-4)
t-k-l

The columns of B in (4-3) are the usual univariate OLS estimates. Even

though the equations are distrubance related, the regressors are identical


----





85

in each equation and thus OLS on each equation is efficient (Thiel (1971),

p. 309). The hypothesis of interest is

H : b = 0, 1 (4-5)
O -0
vs.

HA: a, b o 0, 1

where a is the first row of B and b is the second. If it is assumed that
-o
the rows of R possess multivariate normal distributions, then this hypothe-

sis can be tested by comparing restricted and unrestricted sum of squares

and cross-products. The hypothesis can be written as

H : CBM = r,

where C is the matrix of restrictions operating within the individual

equations with M operating across equations. From Timm (1975), the error

sums of squares and cross-products are

Q = M'R'[I A(A' )-1 A] RM,

and the sums due to the hypothesis are

Qh = (CBM r)' [C(A'A)1C']-1 (CBM r).
The total sums of squares and cross-products under the null hypothesis is

Qt = Qe + Qh With these definitions, four statistics based on the eigen-
values of Q e-Qh or Qt1 Qh are available. Letting yi be the ordered eigen-

values of Q e1 h' then the following statistics are estimated:

Wilk's Lambda = |Qe1 Qt = 1 +n


Pillai's Trace = tr(QQt-) = E i
1 + Yi

Hotelling-Lawley Trace = tr(Q e-Qh) =Q -i

Roy's Greatest Root = yl.

No general preference for any one of the four can be established; however,

in the case where the null hypothesis is of the form of (4-5), the F







approximation for Wilk's Lambda is exact and Roy's criterion is an upper

bound. When the hypothesis is of the form H : a = 0 or H : b = 1, all
o o -o
four criteria are equivalent.

The results from the securities data for the three hypotheses

S= 0, b = 1, and a, b = 0, 1 are reported in Tables 4-24 to 4-32. For

first two hypotheses only Wilk's Lambda is reported because of the equiva-

lence of the four criteria. Examining the results, the restriction a = 0

is not binding in any of the trials; thus it does not appear to be possible

to form portfolios with no factor risk or net investment with significantly

non-zero returns. On the other hand, the restriction b = 1 is binding in
-o-
every case. This suggests the existence of portfolios with no factor risk

and returns that exceed the zero-beta return. Such an operation requires

a positive investment, however. This finding may be attributable to the

difficulties in estimating the zero beta return described above. When the

two hypotheses are combined, the results indicate some dependency between

the estimates; however, no unambiguous conclusion concerning the validity

of the hypothesis can be reached. Depending on the significance level

and test statistic, the hypothesis appears to be binding in about one

third of the trials, indicating significant though not complete agreement

with the predictions of the APT.

Summary

The purpose of this chapter has been to elevate the APT from a

relatively unstructured theory to a concrete model of security returns.

Also developed were testable hypotheses of the APT with the emphasis

placed on the conformity of the uniquely determined parts of the estimated

structure with the predictions of the theory. The results of this positiv-

ist approach are strongly supportive of the basic content of the theory:

it is not possible to form portfolios with no net investment and no factor


















TABLE 4-24
SUMMARY INFORMATION

Test Period: ia
Number of Trading Days: 500
Number of Securities: 100

FIRST 250 TRADING DAYS

TEST OF Ho: 0 o


TEST OF Ho: 1
0 -0 -


TEST OF H : _, = 0,1


Test Computed Computed Numerator Denominator Significance
Statistic Value F Statistic CF OF Level

dilk's Lambda .361 1.072 186 300 .296
Pillai's Trace .781 1.041 186 302 .377
Hoteliing-Lawley 1.376 1.102 186 298 .226
Roy's Greatest Root 1.575 1.575 93 151 .007



Note: F Statistic for Roy's Greatest Root is an upper bound.
F Statistic for Wilk's Lamba is exact.














TABLE 4-24 (continued)

SECOND 250 TRADING DAYS

TEST OF HO: = o


TEST OF H : b= 1
0 -0 -


TEST OF H: I, = 0,1



--st I Comuted Computed Numerator Denominator Significance
Statistic Value F Statistic OF DF Level


.ilk's Lambda .281 1.429 186 300 .003
Pillai's Trace .879 1.274 186 302 .031
Hotelling-Lawley 1.986 1.591 186 298 .001
Roy's -reatest Root 1.638 2.660 93 151 ..001


















TABLE 4-25
SUMMARY INFORMATION


Test Period: lb
Number of Trading Days: 500
Number of Securities: 100

FIRST 250 TRADING DAYS

TEST OF H : = o


TEST OF HF: I 1
0 -0 -


TEST OF Ho: a, = .,1


Test Computed Computed Numerator Denominator Significance
Statistic Value F Statistic DF DF Level

Wilk's Lambda .308 1.291 186 300 .025
Pillai's Trace .863 1.233 186 302 .054
Hotelling-Lawley 1.686 1.350 186 298 .011
Roy's Greatest Root 1.236 2.006 93 151 .001



Note: F Statistic for Roy's Greatest Root is an upper bound.
F Statistic for Wilk's Lamba is exact.
















TABLE 4-25 (continued)

SECOND 250 TRADING DAYS

TEST OF H: o


TEST OF H: b- 1
o-0 -


TEST OF Ho: a, = 0,1



es I Computed Computed Numerator Cenominator Significance
Statistic Value F Statistic OF DF Level


Wilk s 'Labda .287 1.398 186 300 .005
Pillai's Trace .874 1.260 186 302 .038
Hotelling-Lawley 1.925 1.542 IB6 298 .001
Roy's greatestt 0oot 1.568 2.546 93 151 .001


















TABLE 4-26
SUMMARY INFORMATION


Test Period: Ic
Number of Trading Days: 500
Number of Securities: 100


FIRST 250 TRADING OAYS

TEST OF H : a =
0


TEST OF Ho: b = 1
0 .-, -


TEST OF H : a, b = o,1
0 _0 -


Test Computed Computed Numerator Denominator Significance
Statistic Value F Statistic OF CF Level


Wilk's Lambda .287 1.399 186 300 .005
Pillai's Trace .908 1.350 186 302 .010
Hotelling-Lawley 1.809 1,449 186 298 .002
Roy's Greatest Root 1.278 2.075 93 151 .001



Note: F Statistic for Roy's Greatest Root is an upper bound.
F Statistic for Wilk's Lamba is exact.















TABLE 4-26 (continued)

SECOND 250 TRADING DAYS

TEST OF H : = o


TEST OF Ho: b -


TEST OF H: 3, bO = 0,1
0 -s-


it Computed Computed Numerator Denominator Significance
Statistic V alue F Statistic DF OF Level


Ailk's Lambda .286 1.420 186 300 .005
Pillai s Trace .905 1.341 186 302 .012
Hotellig-Lawley 1.828 1.464 186 298 .002
Roy's Greatest Root 1.324 2.149 93 151 .001


















TABLE 4-27
SUMMARY INFORMATION

Test Period: 2a
Number of Trading Days: 500
Number of Securities: 130

FIRST 250 TRADING JAYS

TEST OF H : a=


TEST OF Ho: b =
0 ~


TEST OF H : I, -* ,1


Test Computed Computed Numerator Denominator Significance
Statistic Value F Statistic OF OF Level

Wilk's Lambda .274 1.469 186 300 .002
Pillai's Trace .855 1.213 186 302 .069
Hotelling-Lawley 2.179 1.745 186 298 .001
Roy's Greatest Root 1.936 3.143 93 151 .001



Note: F Statistic for Roy's Greatest Root is an upper bound.
F Statistic for Wilk's Lamba is exact.














TABLE 4-27 (continued)

SECOND 250 TRADING DAYS

TEST OF H : = o


Test Computed Computed Numerator Denominator Significance
Statistic Value F Statistic OF OF Level

Wilk's Lambda .789 .431 93 150 I.999

TEST OF Ho: = 1


Test Computed Computed Numerator Denominator Signifizance
Statistic Value F Statistic OF OF Level

Wilk's Lambda .466 1.846 93 150 .001


TEST OF Ho: i, = c,1


SComputed Computed Numerator Denominator Significance
Statistic Value F Statistic DF OF Level


Wilk's Lanbda .372 1.033 186 300 .399
Pillai's Trace .737 .947 186 302 .657
Hotel!1ng-Liwley 1.399 1.121 186 298 .190
Roy's greatest Root 1.145 1.859 93 151 .001


















TABLE 4-28
SUMMARY INFORMATION

Test Period: 2b
Number of Trading Days: 500
Number of Securities: 100


FIRST 250 TRADING DAYS

TEST OF H : a = o


TEST OF H : = 1
a -' -


TEST OF He: p, b 0,1



Test Computed Computed Numerator Denominator Significance
Statistic Value F Statistic OF OF Level


Wilk's Lambda .384 .988 186 300 531
Pillai's Trace .706 .886 186 302 .816
Hotelling-Lawley 1.366 1.094 186 298 .245
Roy's Greatest Root 1.163 1.889 93 151 .001



Note: F Statistic for Roy's Greatest Root is an upper bound.
F Statistic for Wilk's Lamba is exact.


-------




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