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 MULTIFACTOR 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 prerequisite 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 wellorganized
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
straightforward: 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 CAPMbased 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 nondiversifiable risk have
the same expected return.
3. Asset returns contain two elements, one which is
related to changes in the macroeconomy 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. 158169). However, if the validity of these results is
assumed a priori, the CAPM is needlessly restrictive. If securities
markets are characterized by riskaverse 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 nondiversifiable 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 kfactor 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.1K10
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
riskfree rate. Including it allows for the absence of a riskfree asset,
and is similar to Black's (1972) concept of a "zerobeta" 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 utilitybased 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 nonzero
portion of return not explained by the factors. In efficient markets,
there are two fundamental noarbitrage 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 riskfree
rate if such an asset exists; however, the existence of a riskfree
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 noarbitrage 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 rewrite
(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 twoparameter 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 timeseries 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 Ftest. 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 crosssectional 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 nonzero 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 timeseries 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 welldiversified portfolio with zero
systematic risk and zero net investment which earns
a significantly nonzero 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 nonzero 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. (14) 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 socalled "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, (16)
where (r Em) is the deviation of some broadbased 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. (15) 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 preevent 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 multifactor 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 MULTIFACTOR 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 multifactor models, empirical research has
been dominated by the singleindex "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 reexamine 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 dimensionreducing 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 kdimensional 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 n2 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 nonrandom
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 nonlinearities.
number of asset groups with low firstorder 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 interrelationships among security returns
could be attributed to a market factor and an industry factor correspond
ing to a twodigit SIC classification. Using a sample of 64 stocks in
six industry groups, King performed both a mixed factor/cluster analysis
and a multifactor 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 twodigit 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 subperiod 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) singleindex
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 19611967. 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 nonzero 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 singleindex model ignores a significant portion of
the comovement 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 extramarket 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
macroeconomic 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 "extramarket" 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 singleindex 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 multifactor model, where
the factors are some set of macroeconomic 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 comovement 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 extramarket information are examined.
In the case of a valueweighted index, the averaging is in terms
of the characteristics of the largest firms versus the most numerous in
the case of an equalweighted index.
Multiple Regression Models of Security Returns
Several authors have sought to improve the singleindex 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 interindustry
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 ("pseudoindustries"). 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
multiindex 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 firmspecific 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 extramarket 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 secondpass 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 riskadjusted 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
riskfree rate. Next, the excess Tbill 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 timeseries of measurements of the factors.
(a "zerobeta" portfolio). The weekly returns on these factor port
folios is computed, and these are used in timeseries regressions to
reestimate 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 crosssectional 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 riskfree rate is significantly less than the
corresponding Tbill rate, while the crosssectional 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 pseudoindustries developed by Elton and
Gruber (both studies are reviewed in a previous section). The authors
next calculate excess monthly returns on a valueweighted market index,
a three month Treasury bond index, and a longterm 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 nonparametric
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 subperiods 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 crosssectional 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 crosssectional regressions. After correcting for the positive
dependence between sample mean and sample standard deviation arising
from the positive skewness in daily returns (by using nonoverlapping
samples), little support is found for the hypothesis that returns are
related to total volatility. As a final test, Roll and Ross test for
crosssectional differences in the intercepts from the crosssectional
regressions. To do so, they employed Hotelling's T2 statistic to account
for crosssectional 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 multiindex 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
issuesfactor analysis of security returns, testable hypotheses of the
APT, and the structuring of a return generating functionare interrelated
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
multifactor 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 kfactor linear model of the following
form:
r = E + B6 + e, (31)
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. (31) may be written
r = E + A + e, (32)
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 welldiversified 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. (33)
If x is chosen to have no systematic risk as well, then the return is
x'r x'E. (34)
Taking a to be any nonzero scalar, then ax is an arbitrage portfolio.
If it is assumed that the random portion of (33) can be completely
eliminated by diversification, then (34) holds with equality and it must
be the case that
x'r = x'E = 0, (35)
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, (36)
where A is the augmented factor loading matrix. Algebraically, (36) 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 nkl>.
By construction, then
S IS = TR
SnS = {0}
x'A = 0 i = 1, ..., nkI
S0 i = j
j = 0 i j.
Equation (36) follows from the noarbitrage assumption; either EFS or
arbitrage is possible. To see this, note that E can always be written
E = AX + z, (37)
where zeS But z is itself an arbitrage portfolio with return
z'E = (z'A)X + z'z = z'z f 0. (38)
So (36) 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 (36). 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 kfactor
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 = (39)
n'
lim var(z'E) = 0. (310)
nfm
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 ), (311)
the sum of squared deviations in the nth economy. Referring to (34) 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, (312)
substituting (37) for z'E
z'r = z'z + z'e. (313)
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) 1c = E[z] (314)
and
2 2 2 2 12a
z'E[ee']z = < a y (z'z) < (z'z) (315)
By construction
lim E[z] = = (316)
n*=
2
lim a = 0. (317)
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 (318)
1/2 = zl a
a (z'z)1/2a
which does not have a lower bound unless z'z is bounded. From (37),
z = E Ax; hence the foregoing suffices to show that unlimited deviations
from the pricing relationship (36) give rise to arbitrarily large profits.
If such profits are precluded by assumption then it follows that
(E Ax)' (E Ax) < M. (319)
Inequality (319) 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). (320)
Equation (320) is simply the OLS estimate of E on B. The resulting sum of
squared errors could be compared to the bound in (319) 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 (36). As pointed out by Shanken (1982), the linearity of
(36) is not literally an implication of the APT for any finite collection
of assets. For any such collection, (36) 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 nonzero 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 kfactor 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, (321)
where i is the covariance matrix of the factors. Let Q be a matrix
satisfying Q'Q = I and Q Q' = I.3 Equation (321) 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. (322)
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 loglikelihood 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(SI1) In s n, (323)
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(k1)
restrictions; hence the total number of free parameters is nk + n 1/2k(k1)
and the degrees of freedom are
1/2n(n + 1) nk n + 1/2k(k 1) =
1/2[(nk)2 (n + k)]. (324)
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 predetermined 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 nonsignificant 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 NeymanPearson 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
overestimate 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 largesample 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 scalefree; 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 multifactor model of security
returns is to use the information from the factor analysis to estimate
the timeseries 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 nkl 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), (325)
where Xi is the row vector of the multipliers. Thus
L(xi,i) xi = 0 = > xi A = 0. (326)
L(xi Xi)/ri = 0 = > xIA ci = 0. (327)
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 (326a)
x'A I = 0. (327a)
Eliminating the multipliers and solving for x yields
X = 1A(A'1A)1. (328)
Once the portfolio weights are obtained as in (328), 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 crosssectionally on the factor loadings. A
standard ttest 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 twofactor solution and repeating the regression, and so
on until the kth factor is insignificant. A discussion of the problems with
such pretest estimators may be found in Judge et al. (1980, pp. 5494).
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. (329)
Estimates obtained in this fashion have some interesting features. First,
38
they are equivalent to the estimates obtained from running crosssectional
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 (329) 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 timeseries regressions of the
form
rt = + bo + bji1 + . + bjkbk + ejt, (330)
for each security. Equation (330) 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 ArrowDebreu 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 macrovariates. 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. (41)
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
nonstable 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 (41), 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 visavis other, simpler models. Evidence relating
to this stability is a byproduct 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 41.
TABLE 41
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 41 (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 41). 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 41 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 41. 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 41, 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 thirtyfirst security with complete returns data. The results are
reported in Table 42.
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 42
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 42 (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 42, 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 43 indicate
that the five factor solution originally indicated is insufficient.
As indicated in Table 43, 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 powerevidence 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 multiindex approach. If the securities in the subset are relatively
homogeneous, then singleindex models may systematically misprice 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 extramarket clusters identified by
Arnott (1980). If anything, six factors would appear to be more than
enough. Referring to Table 43, 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 extramarket 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 nonoverlapping 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 institutionsover
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 44 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 crosssectional differences in
security returns, a weighted Varimax rotation (Cureton and Mulaik, 1975)
was used to increase the crosssectional 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 44
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 45, 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. (328)
X = 4A(A1A)1 (328)
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. (329)
R'X = R'V1A(A A)1, (329)
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 timeseries 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 46 through 414.
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 45
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 90day 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 90day 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 riskfree rate, unlike
TABLE 46
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 47
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 48
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 49
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 410
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 411
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 412
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 413
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 414
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 46 through 414 are the estimated returns
on the arbitrage portfolios and their standard deviations. In the first
period, one portfolio in sample a has a significantly nonzero 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
nonzero 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 overestimated. 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 nonsingular transformation. In practical terms, the significance
of the risk premia are a function of the rotation chosen, using an oblique
(nonorthogonal) 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 intercorrelation 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 crosssectional 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 meanvariance 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 timeseries regressions of
the form specified in (330)
rjt = _j + bjooot + bjlt + . + bj666t + ejt, (330)
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 meanvariance 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 nonarbitrage 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 subperiods, 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 415 to
423.
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 415
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 twotailed Ttest.
First 250
Trading Days
Second 250
Trading Days
TABLE 416
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 twotailed Ttest.
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 417
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 twotailed Ttest.
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 418
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 twotailed Ttest.
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 419
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 twotailed Ttest.
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 twotailed Ttest.
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 421
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 twotailed Ttest.
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 422
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 twotailed Ttest.
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 423
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 twotailed Ttest.
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 subperiods. 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 subperiod 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 crosssectional 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
timeseries estimates of eq. (330) can be brought together in matrix
notation as
R = AB + E, (42)
where
R =