An empirical examination of the implications of arbitrage pricing theory

Journal of Banking and Finance 9 (1985) 73-99. North-Holland

AN EMPIRICAL EXAMINATION OF THE IMPLICATIONS OF ARBITRAGE P R I C I N G T H E O R Y Phoebus J. D H R Y M E S Columbia University, New York, NY10027, USA

Irwin FRIEND and N. Bulent GULTEKIN The Wharton School, Philadelphia, PA 19104, USA

Mustafa N. G U L T E K I N New York University, New York, N Y 10003, USA Received November 1983, final version received February 1984 This paper presents a comprehensive set of tests of the implications of the Arbitrage Pricing Theory. We find, unlike previously reported results, a very limited relationship between the expected returns and the covariance (factor loadings) measures of risk. Furthermore, unique variance measures of risk, while generally making only small contributions to the explanation of asset returns, turn out to be significant about as frequently as the coveriance measures of risk - which is inconsistent with the Arbitrage Pricing Theory model. The intercept tests are more mixed but provide only limited support to the model.

1. Introduction

The two key implications of arbitrage pricing theory (APT), as developed by Ross (1976, 1977) and subsequently tested by Roll and Ross (1980) and many others, are first that only covariance measures of risk (beta coefficients on different factors) are relevant to the relative pricing of risky assets, and second that the constant term or intercept in the linear relationship of expected returns on these risky assets to their covariance measures (or factor loadings) is either the risk-free rate of return or zero-beta rate. 1 These two implications characterize all modern capital asset pricing theory, including the well-known capital asset pricing model (CAPM). This paper will present a comprehensive set of tests of both of these implications of the APT, which lead to substantially different conclusions from those drawn by RR. We find a very limited relationship between the 1Under certain conditions, Ingersoll (1981) argues that the intercept in the APT could be a 'zero beta' asset even though a risk-free asset exists. However, this would seem to imply that the market does price risk other than common or factor risk and that arbitrage pricing theory, unlike the CAPM, cannot explain the basic premium between risky and risk-free assets: 0378-4266/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)

74

P.J. Dhrymes et al., Implications of arbitrage pricing theory

expected returns and the covariance measures of risk (factor loadings). Furthermore, unique variance measures of risk, while generally making only small contributions to the explanation of asset returns, turn out to be significant about as frequently a s the covariance measures of risk - - which is inconsistent with the APT model. The intercept tests are more mixed but provide only limited support to the APT. 2 Before discussing in detail our tests of the covariance and intercept implications of the APT, we should note that the procedures followed in these tests, like those used in virtually all previous tests of the APT, are subject to several basic limitations in the application of factor analysis which we have described in an earlier paper [see Dhrymes, Friend and Gultekin (1983)]. As demonstrated in that paper, there is a general non-equivalence of factor analyzing small groups of securities (customarily limited by computer software to 30 or so) and factor analyzing a group of securities sufficiently large for the APT model to hold. As a result, it is found that as one increases the number of securities in the groups to which the APT/factor analytic procedures are applied, the number of 'factors' determined increases. Moreover, this increase in the number of 'factors' with larger security groups cannot readily be explained away by a distinction between 'priced' and 'nonpriced' risk factors. As our earlier paper shows, it is generally impermissible to carry out tests on whether a given 'risk factor is priced' though such tests are invariably found in the standard factor analytic models used to test the APT. It is interesting to note that just as our earlier paper found that the number of 'factors' determined increases with the number of securities in the groups to which the APT/factor analytic procedures are applied, the analysis in this paper indicates that the number of 'factors' increases with the number of time-series observations used to estimate factor coefficients. Our paper is organized in five sections. In section 2, we briefly explain the empirical procedures. Section 3 presents the significance tests for the risk premia and for unique variance as a measure of risk. Section 4 is devoted to intercept tests, and section 5 provides summary and conclusions of our results. 2. The A P T model and empirical tests 3

The APT model, orginated by Ross (1976) and extended by Huberman 2The unique variance and intercept results in our APT tests are consistent with a number of previous tests of the CAPM [e.g., see Friend and Westerfield (1981), and Blume and Friend (1973)]. However, while virtually all previous tests of the CAPM find an intercept which corresponds to a zero-beta rather than risk-free rate, they differ in their implications for the relative importance of covariance and unique variance measures of risk. 3The APT model and its empirical implications are explained in detail in our previous paper, Dhrymes-Friend-Gultekin (1984). The APT model and empirical methodology is summarized here in order to establish our notation and to make the paper as self-contained as possible.

P.J. Dhrymes et al., Implications of arbitrage pricing theory

75

(1982), starts by postulating the return-generating process for securities

r,. = Et. +ft.B+ ut.,

(1)

where r,. is an m-element row vector containing the observed rates of returns at time t for the m securities under consideration; E t is similarly an melement row vector containing the expected (mean) returns at time t. FUrthermore, yr. =ft.B + ut.

(2)

represents the error process at time t, and the APT model assumes that the error process has two components: the idiosyncratic component

ut.,

t = 1,2,...

and the common component

f,.B. Finally, the two components are assumed to have the following properties,

{G.:t=l,2,...} is a sequence of independent identically distributed (i.i.d.) random vectors with E(u,.)=0,

cov(u't.,f',.)=O

and

cov(u',.)=fL

(3)

f2 being a diagonal matrix. Regarding the common component, we specify that 4 {f't.:t= 1,2,...} is a sequence of k-element i.i.d, random vectors with

E(f't.)=O,

cov(f',.) = I.

(4)

It is a consequence of the assertions above that

{(rt.-E,)':t=l,2,...} 4Note that neither ft. nor B are directly observable. The specification in (4) is to eliminate this problem. B, however, is identified only up to left multiplication by an orthogonal matrix.

P.J. Dhrymes et al., Implications of arbitrage pricing theory

76

is a sequence of i.i.d, random vectors with

E[(r,.--Et.)']=O,

c o v [ ( r t . - E t . ) ' ] = B ' B + O =7/,

(5)

where B is a k x m matrix and 7j is a diagonal matrix of size m. Ross shows that, if the number of securities (m) is sufficiently large, there exists a (k + 1)element row vector, %, such that Et. --- c t . B * ,

where B* =

t = 1, 2,..., T,

[e']

(6)

(7)

B'

e being an m-element column of ones. 5 The result in (6) characterizes the noarbitrage condition of the APT model, and we can rewrite (1) with any desired degree of approximation as

rt. = c t . B * +ft.B + u,.,

t = 1, 2,..., T.

(8)

The empirical tests of the APT initiated first by Roll and Ross (1980) are based upon a two-step factor analytic approach. Factor analytic methods, in effect, are suggested by the formulation in (1) and the composition of the covariance matrix (5). In the first step, one determines the number of factors (k) and estimates the elements of B. If T, the number of trading days, is sufficiently large, using factor analysis we estimate B, say by B, and f2 by t). We thus implicitly estimate ~P= B'/3+ ~.

(9)

In the second stage, using B as 'independent variables', one estimates the vector c,., whose elements have the interpretation that c,i is the 'risk premium' attached to the ith factor, i = 1, 2,..., k, while c,0 is the risk-free rate or possibly the return on a zero-beta asset. Thus in the second step for each 5Ross's original formulation of the APT relies on 'diversified or efficient' portfolios. Huberman (1982) provides a more precise definition of the arbitrage condition and shows that the relation in (6) is an approximation. In both approaches, one relies on the strong law of large numbers. Recently, however, a number of authors introduced models under the generic name of 'Arbitrage Pricing Model or Theory'. These authors, see for example Connor (1981) and Dybvig (1983), attempt to derive Ross's APT model in a finite economy. In doing so, however, these authors rely on much more restrictive assumptions than Ross's model. An attractive feature of Ross's model is the minimal assumptions required as in developing demand theory through revealed preferences.

P.J. Dhrymes et al., Implications of arbitrage pricing theory

77

t we may estimate 6 ~',. -- ( n * ~ -

1 n * ' ) - 1 / ~ * ~ - lr't. ,

t=

1,2,..., T.

(10)

If the underlying process is normal and if T is sufficiently large, it can be shown that (approximately)

(4.--ct.)'~ N[O,(B*T- 1B.,)- 1]. Thus, we may view the {U,:t= 1, 2,..., T}, approximately, as drawings from a multivariate normal distribution with mean

c't.:

t = 1,2,..., T

and covariance matrix ( B * T - 1B,,) - 1

There are two crucial testable hypotheses implied by the APT model. First, the intercept term Cto in (10) is the risk-free rate. 7 Second, there is a linear relation between the risk measures embodied in B and the expected returns. As we indicated in our paper, one cannot test unambigously the 'significance' of individual risk premia but one can test unambigously the null hypothesis c*=0

c,.=(Cto, C*).

where

In addition to the two testable hypotheses implied by the APT model, general tests also involve the introduction of other explanatory variables and a test of the hypothesis that the corresponding coefficients are zero. The respecified model in this context is (11)

rt. -= ct.B* + dt.P + Vt.,

where P is a matrix of 'extraneous' variables. Recall that the restriction on empirical evidence implied by (8) is that no other (relevant) economicfinancial variables have any effect on the determination of expected rates of return. This implies that dr. should not be significantly different from zero. 6In the remainder of the paper, we shall group the securities into 42 groups of 30 securities as RR did. In this context, equations should have a superscript indicating the group number. For example, (10) should be rewritten as ~i)' = (/3" ~ 1/3")- 1R.*~ - l~")' i= 1,2, ,42. t.

--t

--t

t

-t.

~

"

We omit the group superscript for simplicity whenever there is no ambiguity. 7The implications of the 'zero-beta' version are discussed further in section 3.

P.J. Dhrymes et al., Implications of arbitrage pricing theory

78

In the next section, we investigate whether the risk premia are priced, i.e., whether d,. is a null vector and whether higher-order moments of returns when introduced as 'extraneous' variables as in (11) are priced. The significance tests on the intercept term go are presented in section 4. 3. Empirical results: Contribution of covariance measures vs. unique variance to asset returns

The time series of security returns used in this paper (like RR) consists of daily returns for the July 3, 1962-December 31, 1972 period for each of 1260 New York and American Stock Exchange stocks, providing a sample size (number of observations) ranging from 2509 to 2619 daily returns per security. The first step factor analysis and the second step cross-section analysis are based on 42 groups of 30 stocks each, with the stock selection in these groups determined from an alphabetical ordering. The results of the second step of the analysis, testing whether (any of) the risk premia are priced for the standard five-factor model used by RR and others, are shown in table 1.8 Columns (1)-(6) show the means of the intercept term and of the vector of risk premia obtained from the 'crosssectional' GLS regressions for each t for 42 groups separately. Columns (7) and (8) show the significance tests for the intercept term and the vector of risk premia respectively. Row numbers correspond to the forty-two 30security groups arranged alphabetically. The test statistic for the significance test for the risk premia for each group is given by (for simplicity, we omit the group superscript below) T~*W-

1~*,

Z

~Z5

where

~.__ 1 r - ~ - ~ ~*

(12)

t=l

and

w=!T t =Zl The test statistic is asympototically chi-square with 5 degrees of freedom. Column (8) of table 1 shows that, for the standard five-factor model used by RR and others, in only six (30-security) groups out of 42 (about 14~o) is the SThe first stage of the analysis, i.e., the determination of number of factors and estimation of factor loadings, and the problems associated with dividing the universe of asset are discussed in great detail in our previous paper. While we carried out most of the analysis presented in this paper using one to five factor models separately, we only present the results for the five factor model for the sake of brevity and to be comparable to those results reported by RR and others for the five factor model. Most conclusions regarding a five factor model in this paper are, however, also true for one to four factor models.

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Group

(2)

gl

0.00031 0.02906 0.00033 0.01406 0.00015 0.02496 0.00033 0.04338 0.00022 0.05741 0.00046** 0.01674 0.00023 0.06399 0.00043* 0.03022 0.00043** 0.02555 0.00017 0.07112 0.00030* 0.04828 0.00022 0.01358 0.00035 --0.01068 0.00015 0.09109 0.00017 0.00330 0.00060** -0.00024 0.00041 0.01352 --0.00011 0.01109 0.00033 0.04910 0.00024 0.03398 0.00041** 0.03279 0.00067** -0.01277 0.00004 0.01680 0.00028 0.02564 0.00018 0.04813 0.00056** 0.03205 0.00014 0.06678 0.00046** 0.03424

(1)

C0a

Statistic

--0.00777 0.02373 0.02278 0.00014 0.05473 0.02822 0.04183 --0.04610 --0.01606 0.03367 -0.01387 0.07201 0.04991 --0.04646 0.06522 0.09049 0.00119 0.15558 0.01900 0.04584 -0.00245 -0.07062 0.10468 0.01924 0.01171 -0.07018 -0.00318 -0.00961

(3)

gZ (5)

C4

--0.01885 0.01955 0.04012 0.01230 0.03118 0.07034 0.02896 0.08462 0.04039 0.01538 --0.01043 --0.00628 --0.03783 --0.04399 --0.03291 0.01554 0.02111 0.04431 0.04216 -0.05987 0.01289 -0.03399 --0.04364 --0.00894 --0.03986 . 0.02182 --0.01407 0.02251 -0.03195 0.00244 -0.00043 -0.02093 0.03840 0.00874 0.01691 0.05566 0.01450 -0.00472 0.05198 0.03330 0.03490 0.00535 0.10889 -0.08044 0.04674 -0.06551 0.02261 0.01928 -0.4854 0.02135 0.01329 0.06546 -0.01892 0.01963 -0.00822 0.00370

(4)

C3 0.03965 --0.01387 0.04462 --0.07534 --0.00236 --0.07789 -0.01042 -0.02364 0.04152 --0.00789 0.00652 0.06428 --0.03180 0.02635 0.00603 -0.00129 0.04897 -0.08433 -0.02273 -0.01970 -0.04920 0.06094 -0.08662 -0.06094 -0.01703 0.02552 -0.03929 -0.07811

(6)

C5 1.56 1.44 0.86 1.30 0.76 2.79 0.91 1.77 2.21 0.70 1.67 0.88 1.46 0.91 0.58 2.06 1. 60 -0.29 1.53 1.15 2.20 2. 41 1.15 1.61 0.86 2.48 0.72 2.51

(7) 1.987 2.551 4.594 9.869t 6.122 4.169 8.109 4.432 3.935 8.859 4.095 11:959t? 4.471 8.929 4.329 5.970 2.259 10.737t 2.517 4.963 3.588 9.928t 14.250~t 3.426 3.202 7.448 5.830 6.057

(8)

t(C0)b ~2c 2468 2571 2568 2575 2554 2405 2540 2553 2559 2540 2512 2575 2536 2547 2560 2565 2547 2416 2562 2527 2506 2559 2504 2575 2563 2561 2510 2561

(9)

Number of observations

Table 1 Tests of significance for intercepts and risk premia for 5-factor model using alphabetically ranked groups, 7/12/196212/31/1972.

q

t~

h

-0.00014 0.00008 0.00037* 0.00015 0.00031 0.00064** 0.00028 0.00037* 0.00027 0.00012 0.00011 0.00044** 0.00032 0.00027*

0.12947 0.06412 0.03452 0.08805 0.03286 0.00090 0.06050 0.01042 0.04540 0.07880 0.07281 0.02028 0.02727 0.01926

Cl

-0.03723 -0.00678 0.02932 -0.02079 -0.00259 -0.01326 0.00472 0.03715 0.05001 0.03941 -0.00797 -0.01039 0.03385 0.04634

C2

0.06784 0.02018 -0.02035 0.03852 0.02204 0.05227 -0.03068 -0.02369 --0.02108 -0.01249 -0.02980 0.01125 0.01141 -0.00777

1~3

0.00819 0.00510 -0.04517 -0.03136 0.01301 0.00233 0.02146 -0.04576 -0.06003 0.01153 -0.07269 0.00507 0.06001 0.05096

C4-

0.03151 0.02009 -0.07711 -0.01862 0.00448 -0.01953 -0.09301 -0.04488 0.04478 -0.02106 0.06763 0.06808 0.02950 -0.04919

C5

-0.50 0.42 1.67 0.85 1.53 3.04 1.38 1.85 1.01 0.68 0.38 2.81 1.61 1.50

t(Co) b

12.022tt 3.846 6.574 7.339 0.855 1.976 7.608 3.265 2.959 6.797 7.304 3.444 3.476 5.401

)C2c

2544 2545 2557 2492 2574 2511 2546 2555 2076 2563 1917 2508 2542 2546

Number of observations

ac 0

through t?5 are the arithmetic means of the daily cross sectional regression estimates using the GLS model ?',.=(/~*~-I~*')-IB*'P-lr',.. B*' is [e,B], which is the augmented matrix of factor loadings with unit vector. bt(Co) is the 't-ratio' for the intercept term. 't-ratio' is given by x/~o/So), where ~o=(1/T)~r=~?,o and So= [(l/T)~f=l(~to--?o)2] ½. * and ** indicate intercept terms which are significantly different from zero at 10 and 5 percent levels, respectively. In case of a 't-test', this refers to a bilateral test; if a unilateral test is desired the critical points are 1.65 and 1.31, respectively, for the 5 and 10 percent level of significance tests. cX2 is the test statistic to test the null hypothesis that none of the risk premia are priced. The test statistic is distributed as chi-square with 5 degrees of freedom. The critical values for chi-square distribution with 5 degrees of freedom at 10 and 5 percent significance levels are 9.24 and 11.10, respectively. ~" and I"'["indicate groups for which the null hypothesis is rejected at 10 and 5 percent levels of significance respectively.

29 30 31 32 33 34 35 36 37 38 39 40 41 42

COa

Statistic

Table 1 (continued)

0~ g.

t~

g.

3

OO

P.J. Dhrymes et al., Implications of arbitrage pricing theory

81

risk premium vector 'significantly' different from zero (3 groups at the 5~ level and 3 groups at the 10~ level). The evidence of table 1 suggests a very substantial failure for one of the crucial implications of the APT model.Thus, just how much explanation for asset returns is afforded by the APT model in the RR context is questionable and certainly at variance with the results they present. 9 Turning now to the question as to whether the use of the (five) factor model exhausts the 'explanation' of the factor return process, the relevant results are in table 2. Previous writers have considered the ability of unique variance to add to the explanation of asset returns provided by common factors to be a critical test of the APT [e.g., RR (1980)1. To determine whether unique variance is priced (or any other variable which is extraneous from the viewpoint of the APT), the obvious procedure which we (and others) have followed is to add that variable to the factor coefficients (loadings) to determine whether it adds in the cross-section explanation of asset returns. The rationale for pursuing this analysis, in spite of the dependence of the results of factor analysis on the size of the groups selected for analysis, is that while it is not possible to determine the true number of factors in the universe of assets from relatively small samples it may still be possible to determine the relative importance of common factors vs. unique variance or other 'extraneous' risk variables in explaining asset returns. There is at least no clear bias in favor of either the factor or other risk variables, and there is no obvious way of avoiding this complication. In column (12) in table 2, we give the relevant statistics for testing the null hypothesis of zero-risk premia, when the factor premia are estimated in conjunction with other extraneous variables' coefficients - - in this instance standard deviation and skewness of own returns. The skewness of returns has been added to these regressions as another 'extraneous' variable because of the well-known complication in estimating the relation between mean returns and standard deviation which may be caused by skewness in the returns' 9In the RR paper there is no counterpart to table 1, so a direct comparison is not possible. RR estimated the risk premia by 6= (/~* ~-1/~.,)-x/~, ~ - i f , , where f is the mean of rates of returns, i.e., f=(1/T)~r=l~ and use the statistic Tr*~-lc*'~X~, where C-V=Bi~ff1[~'ii-6iee']~i71~'i and 6i=1/(e'~i71e). In the RR approach, one relies heavily on the 'truth' of one's assertions relating to the distributional aspects of the cross-sectional GLS estimated coefficients. The approach we employ is more robust to departures from underlying RR procedures. We did, however, repeat the analysis presented in table 1 using the mean returns as shown above and using the test statistics employed by RR. Our results show again that risk premia are priced only in six groups. These groups, using the same group numbers in table 1, are 4, 12, 22, 23, 30 and 39. In table 1, on the other hand, groups with priced risk premia are 4, 12, 18, 23 and 29 with an overlap of three groups between the two methods. This result is again in sharp contrast to those by RR who reported that in 88.1% of the groups at least one factor is priced. Also note that the above test statistic is appropriate when there are no extraneous variables; when there are, as in eq. (11), simply replace in the bracketed equation ~ i i - ~ i e e ' by ~'.-P*'(P*~ffxP*')-XP*, where P*=(e',p'), and p is the matrix of observations on the extraneous variables shown in eq. (11).

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Group

6x

(2)

0.00001 -0.03783 -0.00017 0.00235 -0.00013 0.03656 0.00016 -0.03912 0.00009 0.00147 0.00046** 0.01532 0.00007 0.03959 0.00035 0.02473 0.00049** 0.05007 0.00016 0.03880 0.00011 0.00515 0.00032 0.02547 0.00006 -0.01529 0.00008 -0.07285 0.00017 -0.00678 0.00028 0.02331 0.00047 0.03402 -0.00073* -0.03615 0.00022 0.05679 0.00004 0.02563 0.00028 0.01006 0.00065* -0.01383 -0.00005 0.00615

(1)

Co

Statistic

c2

-0.07908 -0.00961 0.00776 -0.02668 0.02431 0.02302 0.01366 -0.03681 -0.00188 0.01806 -0.03769 0.10158 0.04487 -0.05287 0.03384 0.05987 0.03268 0.05279 0.02509 0.03358 -0.02361 -0.07108 0.07042

(3)

63

-0.01371 0.00857 -0.01452 -0.02482 -0.00108 -0.01061 -0.04117 -0.03594 0.02049 0.03115 0.00571 -0.02315 0.00331 -0.04776 -0.04442 0.01510 0.05450 -0.01240 0.02881 0.03710 0.01270 0.10832 0.03423

(4)

~,,

-0.00420 -0.05965 0.06596 0.04715 0.00853 -0.00669 -0.04679 0.03054 0.04386 -0.06601 -0.03735 -0.00638 -0.00123 -0.03221 -0.00291 -0.01345 0.01245 -0.05942 -0.01251 0.01617 0.01619 -0.07522 -0.06484

(5)

g5

0.01427 -0.03645 0.01809 -0.05703 -0.06110 -0.07785 0.01282 -0.02378 0.04296 -0.00243 -0.01791 0.07869 0.00330 0.03541 -0.00194 -0.01167 0.05608 -0.12489 -0.03451 -0.04285 -0.05447 0.05933 -0.08953

(6)

c6

g7 (8)

0.03485 0.00010 0.03865* 0.00009 0.02348 0.00015 0.02790* 0.00024 0.03392 -0.00027 0.00296 -0.00003 0.01653 -0.00005 0.00187 0.00019 -0.01380 0.00009 0.01694 -0.00019 0.02288 0.00004 -0.01650 0.00000 0.00935 0.00023 0.02288 -0.00045 0.01229 -0.00007 -0.00049 0.00028 -0.01680 0.00022 0.07972**-0.00036 -0.00182 0.00019 0.01242 0.00014 0.02188 -0.00017 0.00049 0.00004 0.01824 -0.00011

(7)

t(Co)b

0.04 -0.49 -0.56 0.58 0.27 2.10 0.19 1.12 2.19 0.57 0.53 1.07 0.21 0.36 0.58 0.67 1.53 -1.65 0.93 0.15 1.29 1.88 -0.21

(9)

t(CG)

2.456 2.178 4.332 2.738 2.245 3.513 3.954 3.429 4.573 3.969 1.936 9.546t 1.005 6.761 2.023 1.745 2.578 8.416 2.803 3.753 2.393 6.974 8.751

t(cv) ~2¢ (11) (12)

1.41 0.40 1.89 0.41 1.57 1 . 0 2 1.75 1 . 3 2 1.38 -1.21 0.13 -0. 23 0.81 - 0 . 2 9 0.08 0.78 -0.83 0.88 0.87 -1.11 1.47 0.31 -0.77 0.01 0.51 1.50 0.86 - 1 . 4 6 0.61 - 0 . 2 5 -0.02 1.12 -0.79 1.33 2.51 - 1 . 3 9 -0.09 1.21 0.78 0.73 1.40 -0. 65 0.02 0.16 1.08 - 0 . 6 4

(10)

4.192 5.932? 3.931 5.349? 2.663 0.054 0.686 0.783 1.041 1.412 3.427 0.594 4.807? 2.251 0.380 1.390 2.213 6.311?? 1.578 1.350 2.006 0.029 1.272

X,.k2d (13)

2468 2571 2568 2575 2554 2405 2540 2553 2559 2540 2512 2575 2536 2547 2560 2565 2547 2446 2562 2527 2506 2559 2504

Number of observations (14)

Table 2 Significance tests of standard deviation and skewness of daily security returns as an alternative hypothesis to 5-factor APT model, 7/12/196212/31/1972.

¢b

¢b

3

.<

OO

0 . 0 0 0 1 5 0 . 0 0 9 6 1 0 . 0 1 3 3 3 0.02624 -0.00006 0.02460 0.00649 -0.07010 0 . 0 0 0 5 1 0.09075 -0.06659 -0.00845 0 . 0 0 0 0 7 -0.02385 -0.06570 -0.03161 0 . 0 0 0 2 4 0.00765 -0.07675 -0.03080 -0.00011 0.01278 -0.00711 -0.01440 -0.00016 0.03900 -0.03632 0.03584 0.00005* -0.01467 0.00712 -0.07649 0 . 0 0 0 0 7 -0.05440 0.01843 -0.00050 0 . 0 0 0 0 4 -0.05363 -0.07487 -0.00650 0 . 0 0 0 3 9 -0.04330 0.01769 0.00395 0 . 0 0 0 1 6 0.00121 -0.03797 -0.04815 -0.00001 0 . 0 0 2 2 5 0.02570 -0.01805 0.00010 0.02402 0.04291 -0.04103 -0.00002 0.05644 0.02344 0.00550 0 . 0 0 0 1 1 0 . 0 7 3 2 8 0.00209 -0.02705 0 . 0 0 0 2 4 -0.00121 -0.01832 0.00615 0 . 0 0 0 2 0 0.00574 0.03139 0.00415 0 . 0 0 0 2 3 0.01782 0.03529 -0.02173

0.00522 -0.01240 0.10648 0.01895 0.00530 0.05113 -0.00207 0.02892 -0.03638 0.01200 0.01076 -0.01024 -0.02191 -0.06405 0.02246 -0.08171 0.00497 0.05180 0.04517

-0.04448 0.00899 0.00010 0.66 -0.02746 0.01814 0.00006 -0.19 -0.00803 -0.02824 0.00051** 1.53 -0.02430 0.03354* -0.00001 0.32 -0.08602 0.03259 0.00009 1.07 -0.01208 0.03532* 0.00001 -0.39 -0.01693 0.01691 0.00017 -0.57 -0.12455 0.02882* 0.00021 0.20 -0.01186 0.00961 0.00021 0.35 0.07720 0.03483 0.00030 0.14 -0.02522 0.02099 0.00022 1.51 -0.12319 0.01615 0.00026 0.55 -0.07575 0.02822* 0.00004 -0.05 0.05175 0.01228 0.00010 0.35 -0.01185 0.01913 -0.00014 -0.09 0.00608 -0.00272 0.00010 0.28 0.03024 0.01898 -0.00001 0.98 0.00824 0.01689 -0.00010 0.78 -0.04707 0.00405 0.00004 1.21

0.56 0 . 3 9 1 . 3 2 3 0.751 0.70 0 . 2 8 2 . 6 7 5 0.922 -1.02 2 . 1 0 11.349~t 4.519 1.87 -0.08 2 . 6 0 5 3.868 1.53 0.66 5 . 9 2 7 5.926t 1.69 0 . 0 5 1 . 8 9 2 3.722 0.72 0.46 1.091 1.366 1.66 1 . 3 4 8 . 5 5 5 7.076tt 0.48 1 . 1 5 2.980 2.013 1.47 1 . 4 0 3 . 1 6 0 6.073t~ 1.01 0 . 9 8 1 . 0 8 9 3.995 0.62 1 . 0 7 8 . 4 2 7 3.541 1.86 0 . 6 1 3 . 0 4 2 4.697t 0.46 1 . 1 5 3 . 4 3 8 3.420 1.10 -0.76 2 . 3 1 4 1.418 -0.09 0 . 3 9 7 . 6 7 4 0.219 1.06 -0.02 0.650 1.267 0.98 -0.55 1 . 4 7 7 1.166 0.18 0 . 1 9 3 . 7 3 3 0.087

2575 2563 2561 2510 2561 2544 2545 2557 2492 2574 2511 2546 2555 2076 2563 1917 2508 2542 2546

loadings, and c6 and c7 are the coefficients for the standard deviation and skewness. bt(Co) is the 't-ratio' for the intercept term and t(c6) and t(cT) are the 't-ratios' for the coefficients of the standard deviation and skewness, respectively; the 't-ratio' for the ith coefficient is given by x/~(~/s~,), where ~ = ( 1 / T ) ~ = ~ Y t , and s~=[(1/T)~L~(5,i-~/)2]~. , and ** indicate regression coefficients which are significantly different from zero at 10 and 5 percent significance levels, respectively. In the case of 't-test' this refers to a bilateral test. If a unilateral test is desired, the critical points are 1.65 and 1.31, respectively, for the 5 and 10 percent levels of significance tests. ~Zz is the test statistic to test the null hypothesis that none of the risk premia is priced. The test statistic is distributed as chi-square with 5 degrees of freedom. The critical values for chi-square distribution with 5 degrees of freedom at 10 and 5 percent significance levels are 9.24 and 11.10, respectively. d X,.k 2 is the test statistic to test the null hypothesis that c6 and c7 are not different from zero. It is distributed as chi-square with 2 degrees of freedom. The critical values for chi-square distribution with 2 degrees of freedom at 10 and 5 percent significance levels are 4.61 and 5.99, respectively, t and t t indicate the groups for which the null hypothesis is rejected at 10 and 5 percent significance levels, respectively.

[e:B':tr:K] which is the augmented matrix of factor loadings with unit vector, (e), standard deviation (a), and skewness (k) of the securities over the T sample period (i.e., for the ith security, at =[1/T~,=~(r,--?~)z]~), Co is the intercept term, Cl through c s are the regression coefficients for the factor

"g0 through c7 are the arithmetic means of the daily cross sectional regression estimates using the GLS model 5~ =(/~**~-~/~**')- 1/~**~-lr,t B**' is

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

,,7

t~

t~

'X:I

,l

84

P.J. Dhrymes et al,, Implications of arbitrage pricing theory

distribution. In only two (out of 42) groups is the null hypothesis of zero risk premia rejected at conventional significance levels. 1° Interestingly enough, column (13) of table 2, which gives the relevant statistics for the test that the coefficients of standard deviation and skewness are zero, shows that the null hypothesis is rejected at the 5~ level in three cases (groups) and at the 10~ level in five additional cases (groups). Thus, in 34 out of 42 groups standard deviation and skewness cannot be said to have any perceptible influence on the return generating process, while a similar statement can be made in forty out of forty-two groups for factor risk premia. Overall, the implications of the test results reviewed in this section are not very favorable to the APT model. Our results, also, do not fully accord with those of RR; in testing for extraneous variables, however, they did not use a sample of contiguous days. This is not likely to explain the difference in our results but further investigation of this issue may be warranted. To minimize the problem of 'spurious' correlation, RR order their daily returns chronologically and estimate mean returns from data for days 1, 7, 13, 19,..., factor loadings from days 3, 9, 15, 21,..., and standard deviations from days 5, 11, 17, 23, .... Following their procedure, we obtain the results presented in table 3, which indicate that the null hypothesis of zero-risk prernia is rejected at the ten percent level in only nine of the 42 groups (five at the 0.05 level), while the hypothesis of no effect of own standard deviation on return is rejected in only six groups (four at the 0.05 level), 1~ The moderate difference in results from table 2 may represent the use of noncontemporaneous rather than contemporaneous measures of risk and returns, the cut in the number of observations by a factor of about 85~ is a reflection of the skipped dates caused by the use of non-contemporaneous measures, and the omission of a skewness variable. There are, however, some peculiarities in the results that should be noted. For example, when we use factor loadings (without the standard deviation) as independent variables using the non-contiguous observations, the risk premia are 'priced' in only six groups. 12 Similarly, when we use the standard deviation as a single inde1°Incidentally, when we introduce standard deviation as the only extraneous variable, the null hypothesis that the coefficient of standard deviation is zero in the GLS regressions is rejected for 13 groups while the hypothesis that the risk prernia vector is null is rejected in seven groups. It is not clear whether the changes in these results are due to the dependency among higher moments because of the deviation of the daily stock returns from normality or due to the (high) correlation among standard deviation, skewness and factor loadings when they are all used together as 'independent' variables. 11RR use mean returns as dependent variables instead of running the GLS cross-sectional regressions for the trading day t-- 1, 7, 13,.... If we follow this procedure and employ the test statistics shown in footnote 6, we obtain the following result: Risk premia are priced only in two groups (groups 13 and 29) while standard deviations are 'priced' in eight groups (groups 4, 14, 20, 23, 24, 29, 34 and 39). 12These groups generally do not correspond to the same groups with significant risk premia when we use contiguous data shown in table 1. When we use non-contiguous data, groups 13, 14, 18, 22, 34 and 36 have significant risk premia. In table 1, using contiguous data, however, groups with significant risk premia are 4, 12, 18, 22, 23, and 29.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Group

0.00099* 0.00060 0.00045 -0.00054 0.00015 0.00007 0.00089 0.00017 0.00053 0.00034 0.00020 0.00018 0.00003 -0.00011 -0.00017 0.00052 0.00058 0.00044 -0.00021 -0.00129 0.00057 0.00073 -0.00050 -0.00069 -0.00015 0.00189** 0.00013 0.00075

COa (1)

Statistic

-0.02411 -0.00454 -0.17124 -0.02213 -0.09791 0.00437 --0.04200 -0.02944 -0.07900 0.03434 -0.03672 -0.03591 0.12780 -0.11654 -0.09831 -0.07376 --0.02186 0.08343 -0.05289 --0.03907 0.06336 -0.07700 -0.04523 -0.07787 0.04040 -0.05467 0.05224 -0.03774

Cl (2) 0.00670 0.01327 0.02814 0.09559 0.09584 0.04252 --0.04707 0.05188 -0.08816 -0.02655 0.07978 0.05075 0.12752 0.07841 -0.01429 0.03307 0.04008 0.40931 -0.02693 0.03057 -0.02380 -0.09576 -0.06315 0.11208 -0.12453 0.00439 0.02934 0.00948

C2 (3) -0.05770 0.01530 -0.12179 0.07184 -0.11564 0.02140 --0.04845 -0.05772 0.01476 0.05775 0.06514 0.05685 0.05378 0.09985 -0.03814 -0.03441 -0.09331 0.18353 -0.00581 -0.11749 -0.02260 0.11271 0.00428 -0.02357 0.01534 -0.02094 0.07143 0.01466

C3 (4) 0.15431 -0.03833 0.03573 -0.11728 0.07680 0.02987 --0.01184 0.02201 0.08874 0.06443 0.02431 0.03812 0.04074 -0.12782 0.02788 -0.15271 -0.02177 0.15576 0.00910 -0.11754 -0.01822 0.03082 -0.02717 0.02810 -0.05517 -0.02889 0.11874 0.09488

C4 (5) -0.08796 0.01432 -0.02036 -0.04389 -0.04102 0.09399 --0.10839 -0.00618 0.04990 --0.05139 0.02649 --0.08505 0.01661 -0.05442 -0.10313 0.01455 -0.07517 0.03324 -0.05837 -0.12559 -0.01598 -0.17165 -0.08207 0.02198 -0.01148 0.06853 -0.03272 0.00169

C5 (6) 0.00439 -0.00991 0.04864 0.02279 0.04145 0.01407 -0.00322 0.01863 0.00939 --0.00121 0.00802 0.01022 0.00283 0.08869 0.03393 0.01728 -0.00294 -0.10599 0.04438 0.07009 -0.00826 0.02698 0.07216 0.05678 0.06603 -0.05180 -0.00395 -0.00753

C6 (7) 1.77 0.84 0.96 -0.86 0.26 0.14 1.14 0.30 1.26 0.81 0.42 0.41 0.07 -0. 21 -0. 31 0.90 0.94 0.47 -0.37 -0.22 1.09 1.38 -0.71 -1.37 -0.23 2.49 0.30 1.39

t(Co)b (8)

~2¢ (10)

5.306 0.13 0.398 -0.30 6.907 1.61 5.704 0.53 7.836 1.32 1.386 & 0.35 5.212 --0.09 1.967 0.42 4.987 0.28 3.276 --0.04 2.818 0.26 4.273 0.29 9.6527 0.10 2.10"* 10.767tt 0.85 5.145 0.49 4.779 --0.09 4.325 -2.26** 17.404tt 1.10 2.139 2.10"* 9.117 2.436 --0.24 11.874tt 0.68 1.928 1.66" 5.660 1.92" 6.495 1.34 1.592 --0.99 3.743 0.11 3.403 --0.19

t(C6) (9)

415 432 431 432 429 404 428 429 428 425 422 432 425 428 426 433 433 432 433 428 424 427 419 432 433 430 420 428

Number of observations (11)

Table 3 Significance tests of standard deviation of daily security returns as an alternative hypothesis to 5-factor APT models; noncontemporaneous data: 7/12/62-12/31/72.

O~

t~

e~ r~

0.00018 0.00019 0.00038 0.00003 0.00111* -0.00044 0.00071 0.00098* 0.00074 0.00051 0.00100 -0.00004 0.00022 0.00080*

29 30 31 32 33 34 35 36 37 38 39 40 41 42

--0.13478 -0.04442 0.00842 -0.03827 -0.19078 -0.00491 -0.06688 -0.01521 0.07372 -0.09809 -0.01016 0.02641 -0.14077 -0.12312

(2)

Cl

-0.13379 -0.02895 0.00563 0.02999 0.01458 -0.03774 0.04559 -0.01726 -0.03011 -0.03131 -0.14150 0.01499 -0.14295 -0.00619

(3)

(~2

-0.00940 -0.03361 -0.01885 --0.05944 -0.02095 -0.07499 0.04169 0.11693 0.04755 -0.06356 0.03102 0.02034 -0.05267 0.04561

(4)

t~3

0.03946 -0.02834 0.10888 0.02771 0.04010 -0.02103 0.02055 0.03626 -0.01725 -0.00175 -0.14756 0.03701 -0.03433 0.01546

(5)

C4

0.08805 -0.01248 0.08047 0.00270 -0.05740 0.21029 -0.00387 -0.01267 0.00506 -0.14972 0.04998 -0.01879 0.03332 -0.05611

(6)

C5

0.06431 0.03596 0.00268 0.03946 0.03513 0.05335 0.02911 0.00765 -0.03559 0.02985 -0.01082 0.00973 0.05317 0.02817

(7)

~'6

0.45 0.04 0.73 0.05 1.99 -0.83 1.36 1.90 0.95 1.07 1.07 -0.08 0.38 1.71

(8)

t(c0) b

2.04** 1.01 0.07 1.25 0.93 1.53 0.89 0.22 -0.87 0.83 -0.18 0.32 1.51 0.63

(9)

t(c6)

Number of

10.367#t 0.930 3.464 1.684 6.305 10.649t 2.303 10.730t 1.214 9.464 t 4.887 0.601 10.130tt 3.041

(10)

Z 2e

429 434 427 422 431 422 430 431 433 435 433 418 422 431

(11)

observations

~go through c6 are the arithmetic means of the daily cross sectional regression estimates using the GLS model ?'=,. (~*~-1/~*')-1/~*~-1r~.. /~*' is [e:B':tr] which is the augmented matrix of factor ioadings with unit vector, (e), and standard deviation (tr), of the securities over the sample period (i.e., for the ith security, ai= [(I/T)~It=I {r,-p])2]'), Co is the intercept term, cl through c5 are the regression coefficients for the factor loadings, and c6 is the coefficient for the standard deviation. bt(c0) is the 't-ratio' for the intercept term and t(£6) is the 't-ratio' for the coefficient of the standard deviation; the 't-ratio' for the ith coefficient is given by x/~(~o/So), where ~o=(1/T)~f=l ~,0 and So= [ ( 1 / T ) ~ L 1 (tTt0-~o)2] ~. * and ** indicate intercept terms which are significantly different from zero at 10 and 5 percent levels, respectively. In the case of the 't-test', this refers to a bilateral test; if a unilateral test is desired the critical points are 1.65 and 1.31, respectively, for the 5 and 10 percent levels of significance tests. ~;t2 is the test statistic to test the hypothesis that none of the risk premia are priced. The test statistic is distributed as chi-square with 5 degrees of freedom. The critical values for chi-square distribution with 5 degrees of freedom at 10 and 5 percent significance levels are 9.24 and 11.I0, respectively, t and t t indicate the groups for which the null hypothesis is rejected at 10 and 5 percent significance levels, respectively, t and t t indicate the groups for which the null hypothesis is rejected at 10 and 5 percent significance levels, respectively.

(1)

Group

~'0 a

Statistic

Significance tests of standard deviation of daily security returns as an alternative hypothesis to 5-factor APT models; noncontemporaneous data: 7/12/62-12/31/72.

Table 3 (continued)

t~

g~

¢-a t~

oo O~

P.J. Dhrymes et al., Implications of arbitrage pricing theory

87

pendent group, again six groups have significant coefficients. ~3 On the other hand, when both factor loadings and standard deviation are included as explanatory variables, the risk premia are priced in nine groups while standard deviation is significant in six groups. The findings in table 3, therefore, do not provide much evidence to support the argument that factor loadings rather than standard deviation represent the appropriate measure of risk. To make a more direct comparison of the results of the analysis of contemporaneous and non-contemporaneous measures of returns and risk, the regressions in table 4 include skewness as an additional explanatory variable, thus duplicating the form of the table 2 regressions. However, as in table 3, non-contemporaneous dates are used, with mean return estimated from days 1, 9, 17, 25,..., factor coefficients from days 3, 11, 19, 17,..., standard deviation from days 5, 11, 21, 29,..., and skewness from days 7, 15, 23, 31, .... Now, with eight days skipped between successive measures of return and risk, the number of observations is again reduced by about 25~ from that covered by table 3 and by close to 90~ from table 2. In this analysis, the hypothesis of zero-risk premia is rejected at the 10~ level in only four of the 42 groups (with one group just below the 10~ level) while the hypothesis of both zero standard deviation and skewness effects is not to be rejected in any group (with five groups just below the 10~ level). As we pointed out earlier, however, some care should be exercised in interpreting this set of results. For example, when we use the factor loadings as 'independent' variables in a similar fashion to the GLS regression presented in table 1 while using non-contemporaneous measures of returns as in table 4 (i.e., mean returns are estimated from days 1, 9, 25 etc. and factor loadings from 3, 11, 19, 27 by skipping eight days between successive measures of return and risk measures), we find only three groups where risk premia are 'priced': groups 4, 8 and 18. Interestingly enough, when standard deviations (estimated using the days 5, 13, 21, etc.) are used, the null hypothesis of zero-risk premia is rejected in three groups (groups 8, 17 and 18) while the hypothesis of no effect of standard deviation is rejected in two groups (groups 4 and 14). Combining these results, there is some but not very strong evidence that both common risk factors and residual or unique risk do affect asset returns, without much basis for differentiating between the relative importance of the common and unique risk factors. There is some difference in the results based on the longer time-series of observations and contemporaneous returns and risk measures, and those based on the much smaller number of observations associated with non-contemporaneous measures of returns and risk, but it is not clear which set of results is the more reliable. 13These groups are 4, 13, 14, 26, 34 and 39.

1 2 3 4 5 6 7 8 9 10 I1 12 13 14 15 16 17 18 19 20 21 22 23 24

Group

--0.00008 -0.00048 --0.00013 -0.00165** -0.00061 0.00041 --0.00012 0.01538** 0.00061 0.00061 0.00040 -0.00013 0.00011 -0.00086 -0.00004 0.00008 --0.00025 0.00070 0.00000 0.00034 0.00051 0.00039 -0.00044 -0.00059

(1)

Coa

Statistic

--0.04619 --0.04117 --0.01304 0.11777 0.06664 0.03194 0.13684 0.10100 --0.06532 0.06904 0.13101 0.01798 --0.01842 --0.02420 0.09089 --0.07677 0.15655 --0.12471 --0.01752 --0.00331 --0.01681 0.06499 -0.10119 --0.06228

(2)

C1

0.00425 -0.07897 0.01808 0.03226 0.00465 --0.03190 0.03990 --0.04631 0.06192 --0.10191 --0.01571 0.06065 0.03414 0.00600 0.06216 --0.10831 --0.04481 --0.16435 0.05483 -0.00904 0.01832 -0.01569 0.04108 -0.01757

(3)

C2

--0.00876 0.09634 --0.03945 0.13202 --0.03694 --0.06846 0.02068 --0.15065 0.02783 0.03406 0.01356 --0.05358 0.04750 --0.20458 0.01531 0.00788 0.00365 0.02881 0.05131 0.00191 -0.00501 0.00563 --0.01166 --0.05986

(4)

C3

0.05141 0.01139 --0.04506 0.21100 0.10578 --0.13760 0.11361 --0.09078 0.07496 0.03583 0.10147 --0.15187 --0.02283 --0.02669 0.02127 --0.02945 0.10033 0.01429 0.09073 --0.08533 0.03339 0.07549 0.03031 --0.04411

(5)

C4

--0.03852 -0.02757 0.04618 -0.12113 -0.06371 --0.07362 -0.00602 -0.25432 -0.04868 0.06961 --0.11766 0.03213 0.02149 --0.04640 0.12839 -0.16867 --0.15934 0.07976 0.05009 0.00699 0.07314 -0.01108 0.00147 --0.04452

(6)

~'5

0.02977 0.02444 0.02409 0.05347 0.05383 0.01978 --0.01237 -0.04594 0.00750 --0.03086 --0.04237 0.02995 --0.00276 0.06839 --0.03221 0.04318 0.06459 0.02621 --0.01578 -0.00273 --0.00394 --0.04399 0.03960 0.03906

(7)

C6

--0.00023 0.00010 0.00012 0.00030 -0.00044 0.00005 --0.00039 0.00046 --0.00033 0.00017 0.00043 --0.00030 --0.00028 --0.00025 --0.00019 0.00030 0.00069 --0.00089 -0.00026 0.00007 --0.00012 0.00019 --0.00004 0.00024

(8)

C7

--0.13 --0.63 --0.21 --2.51 -0. 68 0.74 --0.17 2.17 1.16 0.87 0.61 -0.22 0.22 --1.63 --0.06 0.11 -0.34 0.52 0.00 0.45 0.88 0.52 --0.65 -0.94

(9)

t(co)b (11)

t(c7 )

0.77 --0.81 0.59 0.30 0.68 0.71 1.60 0.96 1.48 - 1 . 1 6 0.44 0.15 --0.35 --1.28 --0.86 1.25 0.24 --1.20 --0.58 0.47 --0.90 1.44 0.64 --0.96 --0.10 --1.02 1.64"--0.65 --0.58 --0.53 1.14 1.04 --1.56 1.77 0.46 - 1.92 --0.37 - 0 . 8 2 -0.07 0.30 - 0 . 1 0 --0.59 -0.82 0.96 1.06 - 0 . 0 7 1.01 0.45

(10)

t(c6 )

0.852 4.902 1.413 9.687* 3,220 5.824 4.740 13.312"* 4,125 2.441 6.715 5.908 1.014 7.512 4.263 5.045 10.343" 10.166" 2.877 1.036 1.347 1.461 3.658 1.703

(12)

gc2e

1.446 0.449 0.937 4.225 3.055 0.294 2.010 1.945 1.444 0.387 2,219 1.086 1.037 3.060 1.064 2.153 4.564 4.354 1.553 0.102 0.365 1.140 1.128 1.179

2k Z~, (13)

312 328 326 324 324 305 318 324 325 319 319 326 324 324 324 325 321 186 325 320 318 323 315 327

Number of observations (14)

Table 4 Significance tests of standard deviation and skewness of daily security returns as an alternative hypothesis to 5-factor APT models; noncontemporaneous data: 7/12/62-12/31/72.

,I

OO OO

3 b

m’

- 0.07330 0.14328 -0.14939 -0.00125 0.11317 0.0042 1 - 0.07273 -0.13213 0.00044 0.06780

-0.06722 -0.01413 0.09599 -0.20141 0.03192 - 0.02393 0.11003

0.0009 1 0.0005 1 0.00002 0.00005 0.00021 - 0.00024 0.00032 0.00086 0.00104 0.00054

- 0.00009 0.00026 0.00048 - 0.00002 0.00054 0.0008 1 - 0.00003

-0.00949 -0.14027 - 0.07723 0.06519 0.02794 - 0.06348 -0.08287

- 0.002 13 0.06086 0.01273 -0.09576 0.14355 0.02324 0.00911 -0.02647 -0.00911 -0.00275 0. I 1166 - 0.04898 - 0.04000 0.07361 0.02990 0.11212 0.02261 -0.09082 -0.09675 -0.05941 0.12718 -0.07014 -0.10060 0.04146 0.20776 -0.07133 0.02542 -0.10280 0.04404 0.07365 - 0.04567 -0.02951 0.00247 0.03567 -0.09850 0.01887 0.04268 0.11000 0.01987 0.15964 It is not possible -0.11132 0.03269 -0.10251 0.12121 -0.14495 -0.01567 0.03458 - 0.04648 0.10490 -0.12187 0.08570 - 0.00921 -0.01567 -0.11384 0.03250 - 0.05060 - 0.10806 0.10338 -0.07542 0.0319.6 0.10678

- 0.04826 -0.03829 0.07225 0.01191 -0.04586 0.03537 0.02882 0.00776 0.03215 0.01819 to estimate 0.04246 0.01260 - 0.0415 1 0.05975 0.01600 0.00334 0.04398

0.00059 -0.00012 -0.00047 -0.00009 -0.00011 -0.00079 - 0.00025 - 0.00026 -0.00009 -0.00023 live factors 0.00013 -0.00065 0.00008 -0.00008 -0.00017 - 0.00030 -0.00033 3.603 7.648 5.154 6.145 2.763 3.622 6:717

0.90 0.45 0.40 0.30 - 1.24 0.27 0.83 - 1.10 - 0.03 1.24 - 0.46 0.98 0.51 -0.57 1.11 0.07 -0.82 - 0.06 1.22 - 1.20 -0.15

4.822 4.840 4.576 6.560 8.926 3.054 2.752 2.347 2.371 5.619

- 1.03 -0.79 1.71 0.32 - 1.23 0.91 -0.11 0.18 -0.84 -0.41

1.97 - 0.48 - 1.56 -0.23 -0.39 - 1.82 - 1.36 - 0.48 -0.33 - 1.03

1.34 0.67 0.03 0.09 0.45 -0.38 0.55 1.46 1.49 0.80

325 322 316 324 321 323 324 313 326 316 321 261 324 243 320 321 322

4.016 1.005 3.532 0.166 1.531 4.001 1.952 0.228 1.026 1.332 1.371 1.591 1.242 1.703 0.532 0.687 2.490

is [e:B’:c:

a& through C-, are the arithmetic means of the daily cross sectional regression estimates using the GLS model ~;.=(B**Y-‘~**‘)-‘ACHY-‘r;.. B**’ k] which is the augmented matrix of factor loadings with unit vector, (e), standard deviation (c), and skewness (k) of the securities over the sample period, i.e., for the ith security, CJ~ = [(l/T)C,T_i (rli -I=~)~]!. co is the intercept term, c1 through cg are the regression coefficients for the factor loadings, and cg and c, are the coefficients for the standard deviation and skewness. bt(~,) is the ‘t-ratio’ for the intercept term and t(c,J and t(c,) are the ‘t-ratios’ for the coefficients of the standard deviation and skewness, respectively; the ‘t-ratio’ for the ith coefficient is given by fi(ci/sii), where ci =(l/T)~~~= 1 zri and sii = [(l/T)xir, 1(& -Ci)2]‘s * and ** indicate regression coefficients which are significantly different from zero at 10 and 5 percent significance levels, respectively. In the case of ‘t-test’ this refers to a bilateral test. If a unilateral test is desired, the critical points are 1.65 and 1.31, respectively, for the 5 and 10 percent levels of significance tests. “x,” is the test statistic to test the null hypothesis that none of the risk premia are priced. The test statistic is distributed as chi-square with 5 degrees of freedom. The critical values for chi-square distribution with 5 degrees of freedom at 10 and 5 percent significance levels are 9.24 and 11.10, respectively. * and ** indicate the groups for which the null hypothesis is rejected at 10 and 5 percent significance levels, respectively. d~z,k is the test statistic to test the null hypothesis that cg and c, are not different from zero. It is distributed as chi-square with 2 degrees of freedom. The critical values for chi-square distribution with 2 degrees of freedom at 10 and 5 percent significance levels are 4.61 and 5.99, respectively. t and tt indicate the groups for which the null hypothesis is rejected at 10 and 5 percent significance levels, respectively. eProgram does not converge for a 5-factor model, therefore it was not possible to estimate the matrix of factor loadings.

25 26 27 28 29 30 31 32 33 34 35’ 36 37 38 39 40 41 42

90

P.J.

Dhrymes et al., Implications of arbitrage pricing theory

An even more interesting difference in results occurs when we carry out the usual kind of factor analysis, assuming a maximum of five factors as RR and many others have done, and obtain the results presented in table 5 based on the same 404--434 observations per group of stocks covered in table 3. Table 5 presents chi-square tests on one to five factors for each of the 42 groups covered. These results differ substantially from those obtained in table Table 5 Chi-square test of the hypothesis that k-factors generate daily security returns# Group 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

k= 1

k= 2

k= 3

k= 4

k= 5

453.70 (0.0473) 457.293 (0.0370) 441.319 (0.1033) 419.850 (0.2950) 423.092 (0.2579) 498.343 (0.0010) 519.254 (0.0001) 490.315 (0.0023) 462.003 (0.0262) 470.477 (0.0135) 513.172 (0.0002) 583.007 (0.0001) 469.107 (0.0151) 422.381 (0.2658) 483.943 (0.0042) 413.160 (0.3790) 466.258 (0.0190) 483.016 (0.0046) 437.856 (0.1255) 790.796 (0.0001)

402.043 (0.1703) 392.832 (0.2647) 387.114 (0.3352) 363.311 (0.6713) 374.163 (0.5171) 413.097 (0.0910) 426.983 (0.0355) 436.133 (0.0174) 406.805 (0.1317) 414.257 (0.0846) 458.534 (0.0023) 427.877 (0.0332) 415.323 (0.0791) 354.756 (0.7780) 402.347 (0.1677) 368.261 (0.6025) 402.390 (0.1673) 424.365 (0.0430) 375.290 (0.5006) 413.251 (0.0901)

353.742"~t (0.4045) 333.713 (0.6998) 336.933 (0.6549) 312.867 (0.9121) 331.430 (0.7300) 351.131 (0.4429) 376.392 (0.1416) 391.226 (0.0548) 352.560 (0.4218) 366.918 (0.2328) 408.658 (0.0138) 371.846 (0.1816) 364.262 (0.2636) 300.538 (0.9687) 249.639 (0.4652) 323.222 (0.8256) 347.936 (0.4909) 378.034 (0.1287) 331.702 (0.7265) 330.028 (0.7479)

312.763 (0.6185) 293.982 (0.8580) 292.767 (0.8691) 266.314 (0.9883) 290.551 (0.8879) 303.295 (0.7535) 325.004 (0.4271) 347.804 (0.1455) 312.926 (0.6160) 321.703 (0.4784) 363.286 (0.0519) 324.494 (0.4350) 324.473 (0.4353) 264.465 (0.9906) 309.479 (0.6678) 278.680 (0.9576) 308.706 (0.6791) 337.613 (0.2511) 296.328 (0.8348) 282.079 (0.9425)

227.363 (0.7622) 257.117 (0.9457) 255.158 (0.9548) 231.355 (0.9975) 259.333 (0.9338) 267.543 (0.8728) 287.528 (0.6113) 306.496 (0.3104) 277.505 (0.7603) 281.716 (0.7012) 315.512 (0.1968) 278.030 (0.7533) 290.120 (0.5693) 228.555 (0.9984) 269.182 (0.8572) 242.120 (0.9891) 270.165 (0.8473) 296.657 (0.4619) 255.297 (0.0541) 249.456 (0.9746)

Table 5 (continued) Group

k= 1

k= 2

k= 3

k= 4

k= 5

21

563.209 (0.0001) 412.645 (0.3858) 454.095 (0.0462) 467.248 (0.0175) 490.444 (0.0023) 542.929 (0.0001) 521.078 (0.0001) 481.667 (0.0052) 476.338 (0.0083) 463.798 (0.0229) 507.081 (0.0004) 521.824 (0.0001) 486.613 (0.0033) 436.848 (0.1326) 420.554 (0.2867) 403.948 (0.5054) 473.172 (0.0108) 495.461 (0.0014) 467.352 (0.0174) 577.196 (0.0001) 489.261 (0.0026) 440.180 (0.1102)

485.505 (0.0001) 353.812 (0.7885) 399.254 (0.1962) 407.925 (0.1237) 421.278 (0.0534) 479.078 (0.0002) 431.653 (0.0249) 423.573 (0.0455) 420.528 (0.0562) 408.864 (0.1172) 438.487 (0.0144) 443.379 (0.0094) 413.938 (0.0863) 377.064 (0.4748) 339.520 (0.9117) 351.977 (0.8081) 407.247 (0.1285) 438.274 (0.0146) 412.639 (0.0936) 488.062 (0.0001) 421.790 (0.0515) 370.691 (0.5676)

416.897 (0.0065) 307.047 (0.9443) 354.161 (0.3984) 349.453 (0.4680) 361.323 (0.3002) 422.721 (0.0037) 360.941 (0.3051) 369.595 (0.2040) 370.799 (0.1918) 360.331 (0.3131) 380.113 (0.1138) 381.277 (0.1060) 357.909 (0.3456) 327.255 (0.7815) 291.750 (0.9872) 309.968 (0.9295) 365.407 (0.2501) 385.492 (0.0810) 369.170 (0.2084) 421.972 (0.0040) 373.552 (0.1658) 321.550 (0.8422)

367.379 (0.0379) 268.916 (0.9843) 318.036 (0.5362) 305.851 (0.7194) 315.734 (0.5724) 367.148 (0.0386) 307.578 (0.6953) 317.919 (0.5381) 322.841 (0.4606) 315.609 (0.5744) 341.525 (0.2062) 338.363 (0.2421) 309.374 (0.6693) 291.020 (0.8841) 251.937 (0.9983) 273.635 (0.9740) 318.935 (0.5221) 337.585 (0.2515) 329.619 (0.3581) 369.508 (0.0320) 327.893 (0.3835) 275.815 (0.9677)

315.921 (0.1924) 236.094 (0.9951) 281.903 (0.6984) 272.172 (0.8258) 279.839 (0.7283) 315.920 (0.1924) 266.674 (0.8806) 277.105 (0.7656) 279.254 (0.7365) 274.218 (0.8020) 304.734 (0.3359) 296.901 (0.4580) 269.358 (0.8555) 248.496 (0.9771) 223.479 (0.9993) 240.091 (0.9916) 285.446 (0.6443) 291.758 (0.5424) 298.123 (0.4382) 275.786 (0.7827) 283.900 (0.6683) 242.491 (0.9886)

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

Number of times the null hypothesis is accepted at 5~ significance level 11 20 30 38

42

aNumber of degrees of freedom for the chi-square test for each group is ½[(30-k) 2 - 30 - k ] . Number of securities in each group is 30. Number of observations for each security is 434. No security has more than 110 missing observations. Values in parentheses indicate the p-value associated with the statistic, i.e., the probability that the test statistic (under the null hypothesis) will assume a value at least as large as the statistic obtained in this particular test.

92

P.J. Dhrymes et al., Implications of arbitrage pricing theory

1 of our earlier paper based on 2618 observations. It should be pointed out, a's noted in our earlier paper, that with 2618 observations we find a larger number of groups for which the five-factor decomposition is inadequate than do RR, which may be attributable either to the very large number of missing observations for some securities in the RR sample or to the greater precision of our computer software (SAS) or both. However, in any case, in a recent independent analysis, Cho, Elton and Gruber (1982) found results similar to ours.

The sensitivity of the factor results to the number of time-series observations used to estimate mean returns and the different risk measures for each stock included in the analysis is indicated by the tabulation below which shows the number of times (out of 42) that the null hypothesis of at most k (1-5) factors is accepted at the 5~ significance level based on 404-434 non-contiguous observations contrasted with the number of times for 2618 contiguous observations. (The non-contiguous observations cover the same time period as the contiguous observations but include only data for days 1, 7, 13, 19... or roughly one-sixth of the total number of daily observations.) Number of times null hypothesis is accepted at 5 ~ significance level.

404-434 non-contiguous observations 2618 contiguous observations

k=l

k=2

k=3

k=4

k=5

11

20

30

38

42

0

1

11

29

36

Obviously, more factors are indicated by the analysis based on the larger number of observations. The reason for this disparity in results is not clear but one possible explanation is that sampling error is so large that only with a very large number of observations can we adequately cope with it. TM Another possible explanation is the type of stationarity assumption implicit in the estimations of factors from two different sets of days. The stationarity assumption is likely to be a less important basis for explaining the difference in results than the size of the sample of observations because both sets of days cover the same overall period. A more definitive determination of the 14It might also be noted that the number of Heywood cases (i.e., degenerate cases where one of the 'factors' is perfectly correlated with a security) increases to 22 groups when factor loadings are estimated from non-contiguous data, resulting in the major reduction in the number of observations. When we use all 2618 contiguous observations, we encounter 11 Heywood cases. In order to avoid Heywood cases, we substituted new securities whenever a Heywood case appeared. Results presented so far are materially the same whether one includes the Heywood cases in the regressions or not. Also note that there are no Heywood cases when one extracts only one factor and the number of Heywood cases are far fewer with a smaller number of factors extracted.

P.J. Dhrymes et al., Implications of arbitrage pricing theory

93

relative importance of these two different explanations is provided by the summary data in table 6, which presents an analysis of the number of factors obtained from daily observations covering (1) every other day for the entire period, (2) every day in the first half of the period, (3) every day in the second half of the period, (4) the first 400 or so days in the period, and (5) the last 400 or so days in the period. Clearly, as the number of observations increases from 400 to 1309 to 2618, the number of 'factors' estimated increases markedly. Even with a sample of observations as large as 1309, the number of 'factors' is appreciably lower than with 2618 observations. On the other hand, with the number of observations held constant, the time period covered seems to have less influence on the number of factors estimated. The results are as sensitive to the number of observations used in estimating mean returns and risk as they are to the number of securities used in each group (the latter having been shown in our earlier paper [DFG (1984)]. Table 6 Effect of number of observations on the factor structure. Number of Times null hypothesis is accepted at 5~ significance leveP

Contiguous observationsb All 2618 daily observations for entire period 1309 observations for first half of period 1309 observations for second half of period First 436 observations Last 436 observations Non-contiguous observations Every other observation for entire period (1309 observations) Every ninth observation for entire period (404-434) observations)

k=l

k=2

k=3

k=4

k=5

0

1

11

29

36

1

13

26

34

39

0 7 16

9 22 26

20 36 36

32 38 41

38 41 41

0

9

23

33

38

11

20

30

38

42

~Hypothesis is that k-factors generate daily security returns. See table 5 for further explanation of this table. bMissing observations do not exceed 90 for each security for the entire period July 3, 1963December 3l, 1972.

3. Testing the intercept in the cross-section APT pricing relationship Another important implication of the APT, which it shares with other capital asset pricing models, is that the 'constant' term in the relation r,. = ct.B* + f t . B + ut.,

P.J. Dhrymes et al., Implications of arbitrage pricing theory

94

i.e., the term Cto, corresponds to the risk-free rate, or at least a zero beta asset. As we pointed out in our earlier paper, the operational procedure for obtaining ct.B* is time invariant, which would argue strongly that ct. is time invariant. In turn this could imply that the 'risk-free' rate is also time invariant - - which is rather far-fetched and questionable. Despite this and the other reservations expressed earlier regarding the testability of the APT, we proceeded to carry out a test of this particular set of implications. We felt it particularly appropriate since RR and others carry out a test on the equality of intercept terms for adjacent groups only, instead of a test on equality of intercept terms for all groups. Recall from the last section that once the matrix B~ of factor loadings for the ith group has been estimated on the basis of a five-factor model, we obtain GLS estimators by ~i)' = (/~* ~ t/~*')- 1/~*~ ar~i)',

i= 1,2,...,42,

(13)

where / ~ , = F e'-] '

t = 1,2,..., T.

~lii = BiBi + ~i,

L J'

If daily returns are normal and if the sample size T on the basis of which/3i and O~ are estimated is large - - which it is in the present context - - then we would expect that, approximately,

~i )''~ N[c't., (B*~i 15*')- 1],

(14)

The important thing to realize here is that the covariance matrix is time invariant. Hence, we can treat the 41 ) as 'observations' from a population with mean c~. and a constant covariance matrix. Hence, defining z ~ ! ) - ~ i ) - ~ 1) --

t.

t.

,

i=2, 3,

"'"

,42

(15)

we have that under the APT model, approximately, z~!)' ~ N(0, K,,),

(16)

where Kii is an appropriate time invariant covariance matrix. Extracting the first element therefore, we find

z~)~N(O, Koo.,),

i = 2 , 3 .... ,42.

In general, z~ ) is correlated with .,u) but their covariance is also time invariant. Thus, let z-t0

P.J. Dhrymes et al., Implications of arbitrage pricing theory

z*o = ~"Z t o

95

2')

,~tO , ' ' ' ,

and observe that z*d,~ N(0, Qo), where Q0 is an appropriate time invariant covariance matrix. Clearly we can estimate the mean vector and covariance matrix by

0'

----

1

Tt

T __~ Zto, *'

1

T

Oo=-~t=atZ, Z " *'o--Z,o)(Z*o--5~) -*'"

(17)

and employ the test statistic

T£,Qo 1=*, ~o ~ z ~

(18)

to test the hypothesis that the intercepts of the 42 groups are equal.t 5 In this instance, the test statistic turns out to be T~*Qff t~3,= 34.4 and thus the hypothesis is accepted. This accords with the results of RR who interpret this finding as an endorsement of the APT model. In this connection we should point out that applying the same tests for the equality of constant terms in a one-factor model yields the statistic Z21=43.30

which similarly implies acceptance. The same conlusion is reached when we use a two, three or four factor model. Now acceptance of such a hypothesis while confirming an implication of the APT model does not tell us very much; for example, this hypothesis would be accepted even if all or nearly all intercepts were zero. Such a situation would cast some doubts on the usefulness of the APT. Thus, next we tested the hypothesis that all intercepts are zero. This is done through the statistic Tc~

T.T7

vv

=~t '~" )~22, 0- - 1 CO

(19)

15Note that the mean of the daily regression coefficients c~i) can easily be estimated using mean return data. If one strictly relies on the model and assumes time stationarity, one can also obtain the covariance matrix of the estimators from mean returns. We have chosen to work with the daily regression coefficients, however, since this represents a procedure that is more robust to departures from stationarity.

P.J. Dhrymes et al., Implications of arbitrage pricing theory

96

where 1

c~ = (~ol), C~o2), • • • , C~o4"2)),

T

c~) = ~- ~ '-toZ(i),

(20)

t=l

1

T

f.Vo T,=~ 1 =

~*

-, t ~,

~t,o = (Cto ,~(i);(2) ,"tO , " " , ~o42)).

(21)

The test statistic in this case is T?*I7Vo lc~'= 67.8 and thus, the hypothesis is rejected. 16 However, rejection of such a hypothesis only means that there is at least one coefficient which can be said to be non-zero. To clarify this issue, we examine table 1, which gives the mean intercepts and the corresponding 't-ratios' in the 42 groups. Even a casual perusal of the table shows that at the 0.1 level of significance only 13 of the intercepts can be said to be non-zero with a bilateral test. Using a unilateral test we find 24 'significant' intercepts. Thus, we are not really violating the meaning of the empirical evidence if we state that at best (from the point of view of the APT model) the evidence is ambiguous and at worst that one of the implications of the model is contradicted by the empirical envidence. The (mean) risk free rate computed from the 7th root of weekly Treasury Bill rates is clearly positive and its standard deviation does not support the hypothesis of a zero daily risk free rate. In fact, the mean weekly rate on Treasury Bills over this period is 0.00084166 with standard deviation 0.0002344, which is clearly significantly different from zero. The associated mean daily rate computed as the 7th root of one plus the weekly rate minus one is 0.0001204 and is also significantly different from zero. We also test directly the hypothesis that the intercept is the risk-free rate using the weekly T-bill rate. The test is identical to the one in eq. (15) except we redefine z~!) z~ ) =?I~) - ~ ,

i= 1, 2,..., 42,

where rft is the seventh root of the (one plus) weekly Treasury bill yield observed every Thursday. In this formulation, we assume that the seventh root of the T-bill yield observed on a Thursday is the daily yield on 16Since m o s t readers are most familiar with the normal distribution, one may use the normal a p p r o x i m a t i o n (X~ - r)/x/~"~ N(O, 1) This would yi,~eld,in the present instance, (67.8 - 42)/w/-~ = 2.8, while in the previous case we have (34.4-41)/~/82 = -0.72. Thus, in the first case we reject and in the second case we accept at the 10~ significance level.

P.J. Dhrymes et al., Implications of arbitrage pricing theory

97

Thursday and the next four trading days. 17 This assumption is the least objectionable when one considers the alternatives. There is no clear way of choosing a daily risk-free rate among observable daily rates, such as Federal Funds rates, daily T-bill rates, etc. We decide against using daily T-bill rates because of the serious measurement errors. Testing directly the hypothesis that the intercept is the risk-free rate when the weekly T-bill rate is used for this purpose, we reject this hypothesis. The test statistic, Z22, in this case is 48.87. We also test the same hypothesis by comparing the seventh root of the (one plus) T-bill yield on every Thursday with the intercept term for the same day to avoid the assumption that T-bill rates are constant for a week. This procedure results in a smaller number of observations (one fifth of the previous case). We reject the hypothesis in this case as well (Z22 = 51.62). This result is quite similar to that generally found in tests of the CAPM, where it has generally been concluded that a zero-beta must be substituted for the risk-free rate to explain the empirical results. However, it should be noted that the zero-beta is not a very satisfactory substitute for the risk-free rate since it is simply a statistical construct until economic theory is brought to bear on its meaning. To a substantial extent, this has been done for the CAPM which has been reformulated theoretically to adjust in a statistically testable manner for the effects of inflation and other variables (e.g., human wealth, differences in lending and borrowing rates, etc.), which would be expected to affect the relation between returns and risk, including the intercept. This has not been done for the APT and it is difficult to see how it could be. It is, in any case, not clear how much importance one should attribute to the intercept test results even if they are consistent with a zerobeta version of the APT model since the risk premia vector has empirically been shown in the previous section to yield few significant results. 4. Conclusions

In a previous paper, we pointed out that the factor analytic procedures which have been used to test the APT are seriously flawed for a number of reasons, including notably the dependence of the number of 'factors' found on the number of assets included in the group which is factor-analyzed and the inability to directly test whether a given 'factor' is priced. As a byproduct of the present paper, which attempts to test the two key implications of the APT - - the irrelevance of unique variances (and other variables uniquely affecting a particular asset) to the explanation of asset returns, and 17To be precise, the T-bill yield is measured every Thursday on a T-bill which matures on the next Thursday. This yield is compared with the intercept terms estimated for Thursday and for the four trading days following the Thursday on which we measure the T-bill yield (i.e., Friday, Monday, Tuesday and Wednesday).

98

P.J. Dhrymes et al., Implications of arbitrage pricing theory

the risk-free or zero-beta interpretation of the intercept in the cross-section mean return-risk relationships, we have found that the number of factors determined is a positive function not only for the number of assets factoranalyzed but also of the number of observations used to estimate mean returns and risk. These basic problems create doubt on the testibility of the APT by proper econometric procedures given the present state of the literature. Setting aside these basic objections, we adopted the general methodology used in previous empirical tests of the APT and sought through more extensive statistical analysis to test one of the important implications of the model, viz., that the intercept terms cl~) are, on average, the same in all groups which would be true if the intercepts were either the risk-free or zerobeta rates of return. This implication is not rejected by the empirical evidence; on the other hand, the same evidence suggests that on average Cl~o) is insignificantly different from zero for most groups. This, of course, runs contrary to the interpretation the APT model places on this coefficient. Moreover, these intercepts are significantly different from the risk-free rate interpreted as the appropriate Treasury Bill rate. Second, and more importantly, we find that the risk premia vector is not significant in most groups (at least 36 out of 42), indicating a lack of linear relationship between the expected rates of return and the measures of risk parameters implied by the APT model. Furthermore, when (own) standard deviation or (own) standard deviation and skewness are introduced into the asset-return function, they turn out to be 'significant' as frequently as the factors suggested by RR and other authors, although generally yielding insignificant coefficients. We conclude that the evidence on the usefulness of the APT model is at best mixed. Further work and a different methodology is needed to probe more deeply into its implications.

References Blume, Marshall E. and Irwin Friend, 1973, A new look at the capital asset pricing model, Journal of Finance, March. Cho, C. Chinlyring, Edwin J. Elton and Martin J. Gruber, 1982, On the robustness of the Roll and Ross APT methodology, unpublished manuscript (Graduate School of Business Administration, New York University, NY). Connor, Gregory, 1981, Asset prices in a well-diversified economy, Working paper (School of Organization and Management, Yale University, New Haven, CT). Dhrymes, Phoebus J., Irwin Friend and N. Bulent Gultekin, 1984, A critical reexamination of the empirical evidence on the arbitrage pricing theory, Journal of Finance 39, no. 2, 323-346. Dybvig, Philip, 1983, An explicit bound on individual assets' deviation from APT pricing in a finite economy, forthcoming in Journal of Financial Economics. Friend, Irwin and Randolph Westerfield, 1981, Risk and capital asset prices, Journal of Banking and Finance 5, no. 3, 291-315. Huberman, Gur, 1982, A simple approach to arbitrage pricing theory, Journal of Economic Theory 28, no. 1, 133-151.

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99

Ingersoll, Jonathan E., Jr., 1981, Some results in the theory of arbitrage pricing, unpublished manuscript (Graduate School of Business, University of Chicago, Chicago, IL). Roll, Richard and Stephen A. Ross, 1980, An empirical investigation of the arbitrage pricing theory, Journal of Finance 35, Dec., 1073-1103. Ross, Stephen A., 1976, The arbitrage theory of capital asset pricing, Journal of Economic Theory 13, Dec., 341-360. Ross, Stephen A., 1977, Return, risk and arbitrage, in: Irwin Friend and James L. Brickler, eds., Risk and return in finance (BaUinger, Cambridge, MA).