A Multivariate Test of an Equilibrium APT With Time Varying Risk Premia in the Australian Equity

4A Multivariate Test of an EquilibriumAPT with Time Varying Risk Premiain the Australian Equity Market

byRobert W. Faff †

Abstract:

This paper applies an asymptotic principal components technique, developed byConnor and Korajczyk (1988), to test an equilibrium version of the ArbitragePricing Theory (APT), which permits time varying risk premia, using Australianequity data. Cross-equation restrictions imposed by the APT on a multivariateregression of excess returns on derived factors are tested. Both one-step anditerative versions of the technique are used and results are compared to the capitalasset pricing model (CAPM). While the APT appears to perform better than theCAPM, neither model can adequately explain monthly seasonal mispricing inAustralian equities.

Keywords:ARBITRAGE PRICING THEORY; TIME VARYING RISK PREMIA; ASYMPTOTICPRINCIPAL COMPONENTS.

† Department of Accounting and Finance, Monash University, Clayton VIC 3168.

The author wishes to thank the seminar participants at Monash University and the AustralianGraduate School of Management, University of NSW. Particular appreciation is due to TimBrailsford, Justin Wood and two anonymous referees. Support provided by the Centre forResearch in Accounting and Finance and the research assistance of Shih Thin Wong are alsogratefully acknowledged. The author is grateful to Tim Brailsford for providing a series ofThirteen Week Treasury Note rates.

Australian Journal of Management, 17, 2, December 1992, The University of New South Wales

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AUSTRALIAN JOURNAL OF MANAGEMENT December 1992

1. Introduction

The Arbitrage Pricing Theory developed by Ross (1976) relies on two basicassumptions: (a) that returns are generated by a k-factor process; and, (b) that

any investment involving zero risk and zero net wealth will produce zero returns.The consequent pricing theory states that k-factor sensitivities are linearly relatedto expected returns. Empirical tests of the APT have supported anything up to fivepriced factors. Initial work by Roll and Ross (1980) suggested the possible pricingof three or four factors in the United States, while Australian evidence produced byFaff (1988) indicated a three-factor model.

The work of Faff (1988) provides an initial attempt at testing the APT usingAustralian data. Several limitations of that study can be identified, which involvethe issues of nonstationarity, errors-in-variables (EIV), and seasonality in monthlyreturn. First, tests that involve averaging over long time-series assume that theunderlying economic parameters being estimated remain constant over the periodexamined. A period of twelve years was examined by Faff (1988), which isexcessive relative to the standard five-year analysis commonly employed in mostasset pricing tests.1 Second, the two-stage testing procedure employed suffersfrom the well known errors-in-variables problem.2 A final drawback of the cross-sectional testing framework used by Faff (1988) is an inability to incorporate themonthly seasonality in Australian equity returns as documented by Brown, Keim,Kleidon and Marsh (1983) and Wood (1990a).

Connor and Korajczyk (1986, 1988) developed an asymptotic principalcomponents technique, which presents a framework able to overcome thesedifficulties. The two most notable features which distinguish it from the techniqueused by Faff (1988) are, first, that it permits factor risk premiums to vary overtime, and, second, that it involves a principal components analysis of the time-series cross-product matrix (rather than the cross-sectional variance-covariancematrix) of returns. An important consequence of these features is that the commonempirical concern of nonstationarity is greatly diminished. In standard frameworksfor testing asset pricing models [for example, in Faff (1988)] the risk measure(s)and the factor risk premium(s) are all assumed to be constant over a given periodof investigation. In the framework developed by Connor and Korajczyk (1988),however, only the factor risks are assumed to be constant. Moreover, theirframework makes it feasible to assume that factor risks are constant over muchshorter periods of analysis, for example, five years.

______________1. A period of twelve years was necessary in order to maintain sufficient degrees of freedom, that

is, the number of companies included had to exceed the number of time-series observations.See Faff (1988, p.30).

2. In the first stage, a principal components analysis of the cross-sectional variance-covariancematrix of returns provides the estimated (APT) factor loadings. These are then employed asthe independent variables (measured with error) in a cross-sectional regression, with thesample mean return across assets as the dependent variable.

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Vol.17, No.2 Faff: ARBITRAGE PRICING THEORY

Connor and Korajczyk (1988) conducted several multivariate tests of cross-equation restrictions imposed by the APT using U.S. monthly data. They examinedfour non-overlapping five-year subperiods from 1964 to 1983. Generally, theyfound that a five-factor version of the APT was able to explain the January/sizeeffect that is well documented in previous U.S. research. They attributed thissuccess to “seasonality in the estimated risk premiums of the multi-factor modelthat is not captured by the single-factor CAPM” (Connor and Korajczyk 1988,p.288). But it was found that the APT cannot explain non-January specificmispricing. Consequently, they concluded that while neither the CAPM nor theAPT are perfect models, their evidence suggests the APT is a reasonable empiricalalternative.

Connor and Korajczyk (1988) recognised that in some cases the cross-sectional size available may not be sufficiently large to ensure that the bestpossible estimate is obtained from the initial one-step procedure. In thesecircumstances they suggest an iterative variant that is more efficient. This iterativeprocedure involves scaling excess returns according to the estimated standarddeviation of idiosyncratic returns.

The cross-sectional sample size available to Connor and Korajczyk (1988)was very large and consequently they found, in their application, that the iterativeprocedure did not provide much improvement over the one-step procedure. Butthey recognised that “applications with smaller cross-sectional samples may findgreater improvement” (Connor and Korajczyk 1988, p.260). Given the relativelysmall number of firms available in an Australian application, it is quite possiblethat the iterative estimates will produce a nontrivial improvement. Consequently,this paper explores the sensitivity of results using the one-step versus the iterativeversion of the asymptotic principal components procedure.

The aim of this paper is to extend and improve on the empirical examinationof the APT in Faff (1988) by applying a methodology developed by Connor andKorajczyk (1988) which: (a) is feasible with relatively small subperiods; (b) usesa multivariate approach; (c) permits time varying factor risk premiums; and, (d)incorporates monthly seasonality effects.

The structure of this paper is as follows. In Section 2 the asymptoticprincipal components technique of Connor and Korajczyk (1988), as applied here,is reviewed. Section 3 presents the multivariate methodology upon which the testsare performed. In Section 4 the data set is described and related issues arediscussed. The penultimate section details the results obtained, while the finalsection provides a summary and conclusion.

2. Arbitrage Pricing and Asymptotic Principal Components

2.1 An Empirical Specification of the APT

Connor and Korajczyk (1988) present an empirical specification of the APTwhich integrates the assumed factor model generating returns with an

equilibrium version of the APT asset pricing equation. Specifically, the underlying

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k-factor model generating returns is assumed to be:

Rit = E (Rit) + B 1i f 1t + B 2i f 2t + . . . + Bki fkt + ε it . (1)

i = 1, . . . , N assets, andt = 1, . . . , T time periods,

the return on asset i in period t ;where: Rit = the expected return on asset i in period t ;E (Rit) = the jth factor sensitivity for asset i, j = 1, . . . , k ;Bji = the realisation of the jth factor in period t, j = 1, . . . , k ;and,

fjt =

the idiosyncratic return for asset i in period t.ε it =

Further, if a risk-free asset exists then the equilibrium version of the APT is givenby:

E (Rit) = RFt + B 1i γ 1t + B 2i γ 2t + . . . + Bki γ kt , (2)

the return on the risk-free asset in period t ; and,where: RFt = the realised risk premium for factor j in period t.γ jt =

Upon substitution of Equation 2 into Equation 1, and rearranging, the empiricalspecification of the APT in excess returns form becomes:

rit = B 1i F 1t + B 2i F 2t + . . . + Bki Fkt + ε it , (3)

(Rit − RFt), that is, the excess return for asset i in period t ;and,

where: rit =

(γ jt + fjt)j = 1, . . . , k, that is, the realised risk premiumplus the factor realisation for factor j in period t.

Fjt =

The focus of the empirical tests which follow is primarily on the empiricalspecification of the APT given in Equation 3. This formulation incorporates factorrisk premium information (γ jt) through the Fjt variables and these are notrestricted in their time-series properties. It is in this sense that Connor andKorajczyk (1988) state that their framework is valid for models with time variationin factor risk premiums. The reader should note, however, that this frameworkdoes not permit the separate identification of the factor risk premiums (γ jt) fromthe associated factor realisations (fjt).

2.2 Asymptotic Principal Components: One Step and Iterative Versions

The asymptotic principal components technique was initially developed by Connorand Korajczyk (1986). It is similar to standard principal components3 except that

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it analyses the time-series cross-product matrix of excess returns as opposed to thecross-sectional variance-covariance matrix of returns. Its asymptotic nature relatesto its reliance on “statistical approximations which are valid as the number ofcross-sectional observations grow large” (Connor and Korajczyk 1986, p.381).Connor and Korajczyk (1986) showed that the first k eigenvectors of the cross-product matrix are approximately (asymptotically) non-singular lineartransformations of the true Fjt variables in Equation 3.4

A drawback of this one-step technique is that there is no objective means ofassessing whether a given cross-sectional sample is sufficiently large so as tojustify the asymptotic nature of the procedure. But Connor and Korajczyk (1988,pp.259–261) provided an “iterative” refinement of the technique which promisedimproved estimation efficiency in smaller samples. The iterative process is asfollows.

a. Form the cross-product matrix (Ω) of excess returns.

b. Calculate the eigenvectors for the cross-product matrix. The first keigenvectors represent proxies for the independent variables in Equation 3.

c. For each individual asset in the sample run a regression of excess returns onthe first k eigenvectors obtained in (b) and calculate the standard deviation ofresiduals.

d. Scale the excess returns of each asset by its associated residual standarddeviation obtained in (c) and form a new scaled cross-product matrix ofexcess returns.

e. Repeat steps (b), (c) and (d) until convergence is achieved.5

3. The Multivariate Framework

3.1 Testing the APT for Mispricing

The tests in this paper are focused on a multivariate regression model that is anaugmented version of Equation 3. Two basic variants are explored which

relate to the existence of unconditional and conditional mispricing. First, a test forthe absence of unconditional mispricing involves Equation 3 augmented by anintercept term (ai) which captures general mispricing in the data, relative to thepricing model specified. The null hypothesis to be tested is the restriction that the

______________3. Refer to Trzcinka (1986) and Faff (1988) for examples of the application of the standard

principal components technique to empirical tests of the APT.4. See Connor and Korajczyk (1986, pp.382–384) for details. Also refer to Connor and

Korajczyk (1988, pp.258–259) for an intuitive discussion in a simple one factor case.5. Connor and Korajczyk (1988) had very large subsample sizes (ranging from 1,487 to 1,745)

and so, not surprisingly, found no benefit from using the iterated variant. Consequently, theydid not indicate any specific criteria for determining convergence.

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intercept is zero across all (p) asset equations, that is, zero unconditionalmispricing:6

H 0 : a 1 = a 2 = a 3 = . . . = ap = 0. (4)

The second variant incorporates monthly seasonal behaviour in stock returns aspotential measures of conditional mispricing. It is desirable to include the monthlyseasonal behaviour in Australia stock returns most relevant for our data period(1974 to 1987). The work of Brown, Keim, Kleidon and Marsh (1983) and Wood(1990a) suggests that seasonals for January, July and August need to be includedin a risk-adjusted analysis.7 Hence, Equation 3 will be augmented by an interceptand three seasonal dummies, that is, Equation 3 plus:

ai(NSEAS) + ai(JAN) DJAN + ai(JUL) DJUL + ai(AUG) DAUG, (5)

the coefficient measuring non-seasonal-specificmispricing:

where: ai(NSEAS) =

the coefficient measuring January-specific mispricing;ai(JAN) = the coefficient measuring July-specific mispricing;ai(JUL) = the coefficient measuring August-specific mispricing;ai(AUG) = a dummy variable taking the value of unity in January,zero otherwise;

DJAN =

a dummy variable taking the value of unity in July, zerootherwise; and,

DJUL =

a dummy variable taking the value of unity in August,zero otherwise.

DAUG =

In this situation two types of hypotheses can be tested—one relates to the absenceof non-seasonal-specific mispricing and the other relates to the absence ofseasonal-specific mispricing. In the former case, the null hypothesis to be tested

______________6. This assumes that the chosen risk free proxy is appropriate. If this is not so then rejection of

H 0 may reflect that: (a) the APT is inappropriate; and/or, (b) the risk free rate is misspecified.7. Brown et al. (1983) examined the period from March 1958 to June 1981 (of which the

subperiod 1974 to 1981 is relevant to the current study). They found that risk-adjusted returnsindicated seasonalities in January, July and August. Wood (1990a) examined the period 1974to 1988. While this period fully encompasses our data period, the seasonality analysis focusedon raw returns only and, hence, in our risk-adjusted framework, his results are suggestiveonly. He found a significant January and July seasonal across industrial firms and asignificant July and August seasonal for small resource stocks. These results reinforce theJanuary, July and August seasonals found by Brown et al. (1983) and justify their inclusion inour analysis. Wood also found a positive seasonal in April for small resource sector stocks,but since this was a raw return seasonal only and because it was not confirmed by Brown et al.(1983), the April seasonal was not included here.

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is:

H 0 : a 1(NSEAS) = a 2(NSEAS) = a 3(NSEAS) = . . . = ap(NSEAS) = 0. (6)

In the latter case, however, we can test for the absence of individual or jointseasonal mispricing. The null hypotheses for the case of zero individualmispricing are:

H 0 : ai(JAN) = 0; i = 1, . . . , p. (7a)

H 0 : ai(JUL) = 0; i = 1, . . . , p. (7b)

H 0 : ai(AUG) = 0; i = 1, . . . , p. (7c)

The null hypothesis for the case of zero joint mispricing is:

H 0 : ai(JAN) = ai(JUL) = ai(AUG) = 0; i = 1 . . . , p. (8)

The standard procedure for testing these hypotheses in a multivariate settingrequires the estimation of the appropriately specified system of restricted andunrestricted models for each case. Note that the restricted systems estimated inorder to test Hypotheses 4 and 6 are Equation 3 and Equation 3 augmented bythree seasonal dummy variables, respectively. Further, Equation 3 augmented by 5is the common unrestricted system estimated in order to test Hypotheses 7a, 7b, 7cand 8.

The multivariate regression framework requires that returns are multivariatenormally distributed. In this paper the statistical tests are based on a modifiedlikelihood ratio test (MLRT) statistic. The MLRT is given by:8

MLRT = T *

BACAD

HAAI det (Σ u)

det (Σ r)________JAAK

− 1

EAFAG

, (9)

the determinant of the maximum likelihood estimate of theerror covariance matrix from the restricted system;

where: det (Σˆ r) =

the determinant of the maximum likelihood estimate of thecovariance matrix from the unrestricted system;

det (Σˆ u) =

(T − k − p) L p; andT * = the number of equations in the multivariate regressionsystem.

p =

______________8. See Connor and Korajczyk (1988, p.271).

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Given the assumption of normality, the MLRT has an exact small-sample Fdistribution with (p, T − k − p) degrees of freedom.

3.2 Testing the APT Versus the Capital Asset Pricing Model (CAPM)

In order to have a benchmark for comparison, an analogous range of tests as justdescribed are also conducted for the standard CAPM. But a naıve comparison maybe misleading as the APT and CAPM are non-nested models. Connor andKorajczyk (1988, p.288) suggested that “procedures designed to compare non-nested models [similar to those used in Chen (1983)] will improve ourunderstanding of the relative merits of the models”. Following this suggestion, amultivariate extension of the “C test” used by Chen (1983) is performed in thecurrent paper.

Specifically, consider the artificial regression model

rit = α i • rit, APT + (1 − α i) • rit, CAPM + error , (10)

where the independent variables are defined as:

the excess return for cross-section i in the period t, predicted bythe (restricted) empirical specification of the APT in Equation 3;and,

rit, APT =

the excess return for cross-section i in the period t, predicted byan excess returns market model with implied CAPM restrictionsimposed.

rit, CAPM =

In the multivariate regression model framework, the null hypothesis which favoursthe APT over the CAPM is:

H 0 : α1 = α2 = α3 = . . . = αp = 1. (11)

Alternatively, the null hypothesis which favours the CAPM over the APT is:

H 0 : α1 = α2 = α3 = . . . = αp = 0. (12)

These hypotheses can be tested using the MLRT as previously defined.

4. Data

4.1 General Description

The data used in the current work are returns on Australian equity securities,calculated from the Price Relatives File of the Centre for Research in Finance

(CRIF) at the Australian Graduate School of Management. The data set covers the165 month period from January 1974 to September 1987 and is divided into three

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nonoverlapping 55 month subperiods for analysis.9 Securities are included in asubperiod sample if they have a complete price relative history in that period. Thisproduces sample sizes of 303, 158 and 340 for the three subperiods, respectively.The market index employed is the value-weighted index, as supplied by CRIF,while the risk-free rate of return is estimated from a series of monthly observationson Thirteen Week Treasury Notes.

To reduce the dimension of the equation system to feasible proportions,securities in each subperiod are allocated to ten10 equally weighted portfolios11

according to their beginning market capitalisation in each subperiod.12, 13

4.2 Use of Portfolios

It is important to note that portfolios (rather than individual assets) are used for thereason of making the analysis statistically feasible. That is, the use of portfoliosreduces the number of equations in the multivariate analysis to a number which canbe estimated. [This is in contrast with the reason for using portfolios in traditional

______________9. This fourteen-year period is chosen for analysis because January 1974 is the earliest month in

which market capitalisation data is available. The empirical method involves individual firmsbeing grouped into portfolios according to market capitalisation. The subperiod size waschosen so as to provide a balanced and independent coverage of the overall period, whilemaintaining acceptable sample sizes.

10. While the choice of ten portfolios is arbitrary, this number has been chosen in the vastmajority of Australian based (and overseas) studies which have used market capitalisation as aportfolio formation variable. [For example, see Beedles, Dodd and Officer (1988).]

11. It is recognised that a potential computational bias occurs when using equally weightedportfolios based on a size ranking [see Blume and Stambaugh (1983)]. Wood (1990) analysedthis bias based on size deciles of Australian companies. Using all companies in the AGSMdatabase, he found the bias was most severe in the decile of the smallest firms, with relativelylittle bias for the other deciles. Given the data restrictions used here, very few of thecompanies in our sample will correspond to the companies included in Wood’s smallestdecile. Hence, it is argued that, in the current context, the computational bias is minimal andcan safely be ignored.

12. Given that the empirical set up involves size portfolios and regression models that includemonthly seasonal dummy variables, it seems natural to use the current analysis to shed somelight on how well the APT can explain size related anomalies. But it should be noted that suchanomalies have invariably been found most pronounced in the very small firms, whereas thesample used here is clearly biased toward larger firms. This large firm bias is induced by theneed to have complete price histories of all firms analysed. Unfortunately, with Australiandata there is a relatively high incidence of missing data which tends to be more commonamongst small firms. While this is less of a problem for U.S. data as used by Connor andKorajczyk (1988), it is interesting to note that even they lost 30% of the available firms forthis reason (p.261, footnote 4).

13. The average beginning of subperiod size of companies in the smallest firm portfolio is $2.18million compared to an average size of $450.64 million for the largest firm portfolio. Thisrepresents a two hundred fold difference. Notwithstanding the comments made in theprevious footnote regarding the large firm sample bias, one would still expect to observe arelative firm size effect.

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(univariate) CAPM tests of the 1960s and 1970s. In these tests, portfolios wereformed to attenuate the problem of errors-in-variables (EIV), introduced by thewell known two-stage testing approach.]

But the use of portfolios does come at a cost. The cost is that it will biastests in favour of the null model to the extent that individual pricing errors offseteach other in the selected portfolios. It is this concern which justifies selection ofsize-based portfolios, since there is ample prior evidence suggesting a strong andsystematic CAPM pricing error for assets of similar size. Moreover, size-basedportfolios have been found to provide a wide range of return and risk acrossportfolios. This is desirable for testing purposes because the variation of returnand risk determines the degree of potential asset pricing information contained inthe data. That is, in the extreme, if return and risk did not vary at all acrossportfolios, then the data would contain no information on how asset prices areformed. In this case, tests would be pointless.

One could argue from the above that the tests should ideally be conducted onindividual assets. While this is not impossible, it requires that a strong assumptionbe made regarding the covariance structure of returns. Indeed, Connor andKorajczyk (1988) conducted such tests (assuming that the covariance matrix onindividual asset returns is “block diagonal”). But they concluded that their testslacked power because of the large number of assets used relative to the smallnumber of (monthly) observations used in the time-series dimension. It is for thisreason that tests using individual assets were not conducted in the current paper.

5. Empirical Results

5.1 The Error of Approximation in the Factor Estimates: Simulation Evidence

As indicated earlier in Section 2.2, the asymptotic principal componentsapproach involves using eigenvectors from the cross-product matrix of excess

returns as proxies for the true factor matrix. But the validity of this applicationrelies on the number of assets being large enough such that the error ofapproximation is immaterial. Connor and Korajczyk (1988) investigated thepotential significance of this error by simulating “true” factors in their data andrunning regressions of the eigenvectors on the “true” factor matrix. On the basis oftheir simulation results, Connor and Korajczyk (1988, p.269) concluded “that theasymptotic principal components technique provides accurate estimates of thepervasive economic factors”.

The number of assets available and used in the current paper is a far smallernumber than that used by Connor and Korajczyk (1988). Consequently, the errorof approximation in the factor estimates would seem to be a greater problem here.To assess the significance of the error, an identical simulation exercise is repeatedusing one subperiod of our data.

The simulation is performed using the 55 month subperiod from August1978 to February 1983, in which the sample size is 158. The “true” factors aretaken to be the first five eigenvectors extracted from the original data for this

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subperiod. Based on these “true” factors, a series of (simulated) returns is formedcomposed of a factor component and an idiosyncratic component. Thecorrelational structure of returns is constructed to have an approximate factorstructure, that is, non-zero idiosyncratic cross-sectional correlation (ρ) is permitted.The cross–sectional correlation values considered are: ρ = 0.0, 0.1, 0.3, 0.5, 0.7,0.8 and 0.9.14

The degree of error in the factor estimates is gauged by the R 2 valueobtained from regressing the estimated factor on five “true” factors. A perfect R 2

value of unity indicates a factor estimate with zero error. These regressions wererun across five iterations for each estimated factor. The average, maximum andminimum R 2 values across the different ρ values for each of the first fiveestimated factors are reported in Table 1.

The R 2 values are very large for low to intermediate values of ρ (ρ = 0.0,0.1, 0.3 and 0.5). For ρ = 0.7 only the first three estimated factors show high R 2

values, whereas for ρ = 0.8 and 0.9 only the first estimated factor reveals high R 2

values. These results are very similar to those found by Connor and Korajczyk(1988), at least in that the significance of the error becomes increasingly importantas the assumed level of idiosyncratic correlation increases. Not surprisingly,however, due to the much smaller sample size used here, the importance of theerror of approximation is more significant than in the U.S. study. Connor andKorajczyk (1988) found that the error appeared to be a problem only in the case ofρ = 0.9 for the estimated factors four and five. But they argued that given the firstfive factors have been extracted, this high level of correlation is “implausiblylarge”. The results in Table 1 indicate that for our data the error of approximationis of concern at least for ρ = 0.8 and perhaps also for ρ = 0.7. It is argued here thateven these values for ρ are unrealistically high and that for more reasonable valuesof ρ (< 0.7) the error of approximation in factor estimates is acceptable. Thisclaim is likely to be even stronger in the other two subperiods since the sample sizein both cases is double that available in the subperiod used in the simulationexercise. In any case, the results that are reported in the following sections shouldbe read with due caution.

5.2 One-Step Versus Iterated Factor Estimates

The first empirical issue to be resolved is whether it is necessary to go beyond theinitial one-step factor estimates and perform an iterated analysis. It was found thatconsiderable instability is observed between the initial and iterated estimates.While this indicates that the iterated optimal estimates may be desirable [incontrast to the results of Connor and Korajczyk (1988) with U.S. data], somecriteria for determining convergence of these factor estimates must be specified.

______________14. Refer to Connor and Korajczyk (1988, pp.267–268) for a detailed description of the

simulation framework employed.

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Table 1

A Simulation Comparison of the Asymptotic Principal Components

Factor Estimates Versus “True” Factors For a Five Factor Model_________________________________________________________________

ρ Est. Factor Average R 2 Maximum R 2 Minimum R 2

_________________________________________________________________

0.0 1 0.981 0.985 0.9722 0.968 0.988 0.9533 0.951 0.976 0.8984 0.895 0.910 0.8505 0.871 0.882 0.835

0.1 1 0.978 0.982 0.9692 0.955 0.972 0.9283 0.930 0.956 0.8694 0.899 0.918 0.8915 0.860 0.881 0.852

0.3 1 0.972 0.973 0.9722 0.963 0.980 0.9223 0.900 0.960 0.8004 0.900 0.921 0.8895 0.846 0.911 0.829

0.5 1 0.959 0.967 0.9312 0.958 0.964 0.9373 0.809 0.827 0.8014 0.751 0.781 0.7225 0.652 0.822 0.344

0.7 1 0.876 0.919 0.8392 0.847 0.887 0.7853 0.544 0.569 0.4944 0.323 0.745 0.1625 0.293 0.739 0.147

0.8 1 0.757 0.824 0.6872 0.141 0.436 0.0313 0.197 0.245 0.1534 0.271 0.314 0.2055 0.371 0.438 0.342

0.9 1 0.479 0.531 0.4442 0.103 0.283 0.0383 0.092 0.122 0.0544 0.179 0.221 0.0905 0.096 0.189 0.049

_________________________________________________________________

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The overriding criterion used in this study is that a “high” correlation must beobserved between factor estimates from consecutive iterations.15 Following thisrequirement, the necessary number of iterations for each subperiod were: ten forthe 1974/78 subperiod, seven for the 1978/83 subperiod, and five for the 1983/87subperiod.16

5.3 Seasonality in Factor Mean Returns

As discussed earlier, a major advantage of the Connor and Korajczyk framework isthat it allows for time variation in factor risk premiums. One specific case of suchvariation is monthly seasonality. This is of particular interest due to theprominence that monthly seasonality has achieved in the vast literature on capitalmarket anomalies. The existence of monthly seasonals in factor risk premiums canpotentially help explain the puzzle of observed seasonal patterns in share returns.Connor and Korajczyk (1988) tested for January seasonality in U.S. factor meanreturns and found very strong evidence in favour of seasonality in the first fourfactors. Moreover, they attributed this seasonality as an important reason for theAPT’s ability, in subsequent tests, to explain away January seasonal mispricing.

A similar test of seasonality in the mean factor returns is conducted in thecurrent study by regressing each statistically estimated factor (Fjt) on a constantand three dummy variables representing the months of January, July and August,viz:

Fjt = C 0 + C 1 DJAN + C 2 DJUL + C 3 DAUG + error, (13)

the realised risk premium plus the factorrealisation for factor j in period t, statisticallyestimated using asymptotic principalcomponents:

where: Fjt =

C 0,C 1,C 2,C 3 = unspecified regression coefficients; and,DJAN , DJUL , DAUG = dummy variables as defined for Equation 5,

previously.

______________15. In the first subperiod, for example, perfect positive correlation between the ninth and tenth

iteration factor estimates was obtained for factors one through eight. More generally,correlations in all subperiods between the final two iterations were mostly around 0.98 andabove.

16. It is important to recognise that while the iterative procedure offers potential gains inestimation efficiency, it may come at the cost of “estimation risk”. The Connor andKorajczyk (1988) analysis assumes that the true idiosyncratic variances are known, whereas inpractice we use sample estimates. While these estimates are asymptotically valid, the fixed Tfacing the empirical researcher creates the estimation risk problem. [See Connor andKorajczyk (1988, p.260).] Consequently, this may place a limitation on the validity of theiterative estimates and so they should be treated with some caution.

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The results from estimating this regression are reported in Table 2 and indicateonly very weak evidence of any monthly seasonality in the factor mean returns. InTable 2, for each subperiod, Panel A presents the results based on the initialestimates of Fjt , while Panel B presents the results based on the iterated estimates.It can be seen that there is no consistent or strong seasonal pattern acrosssubperiods either for the initial or for the iterated factor estimates. There is someevidence of a January seasonal in the first subperiod, with three (two) out of fiveiterated (initial) factors proving significant at the 5% level. But January does notappear important in either of the later subperiods. July seasonality is apparent inonly one factor in each of the first and last subperiods using initial estimates and inonly one factor in the last subperiod using iterated estimates. August seasonality isevident in at least one factor in every subperiod except the first (for the iteratedestimates).

The strength of this evidence of seasonality in mean factor risk premiums isdisappointing. But the Connor and Korajczyk (1988) APT framework does allowfor unspecified time variation in factor risk premium and the results of tests in thismore general environment are still of great interest. It is to these we now turn ourattention.

5.4 Tests of the APT: Unconditional Mispricing

Presented in Table 3 are the modified likelihood ratio test (MLRT) statistics for theabsence of unconditional mispricing (the null hypothesis in Equation 4) in a fivefactor APT model [APT (5)] and a ten factor APT model [APT (10)].17 Forcomparative purposes, the results for similar Sharpe-Lintner CAPM tests (using avalue-weighted index return) are also provided in Table 3.

It can be seen that for the CAPM and also for the initial APT (5) andAPT (10) specifications, despite some minor subperiod support, in the overallperiod the aggregate test statistics indicate strong rejection of the null hypothesis ofno unconditional mispricing. But in the case of both iterated APT models there isstrong evidence that the null cannot be rejected in either the early subperiod or thelater subperiod. This non-rejection is likewise evident in the overall period at the10% level and in particular seems to favour the APT (5) model more.18

______________17. The five [APT (5)] and ten [APT (10)] factor versions of the APT were conservatively chosen

as the focus of analysis. While prior evidence supports three or four factors at most, it is notnecessarily the case that factors will be economically important in the order that they areextracted by the principal components technique. For example, Faff (1988) found that thefirst, third and eighth factors (eigenvectors) seemed to be important. Moreover, theirimportance may vary over time in an unpredictable manner. Connor and Korajczyk (1988)also examined APT (5) and APT (10).

18. It is prudent to employ the 0.10 critical level for aggregate multivariate tests following theanalysis of Gibbons and Shanken (1987, p.393).

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Table 2

Test Statistics and p-values for the Hypotheses That

the Conditional Mean Factor Risk Premium in January, July

and August, is Equal to the Unconditional Mean Factor Risk Premium_________________________________________________________________________

FactorTime Period 1 2 3 4 5

_________________________________________________________________________

A. Initial Estimates

1974/1–1978/7 Jan 2.43(0.019)

Jan1 2.382 ns3 ns ns Jul 2.36(0.021)4 (0.022)

Aug 2.10(0.040)

1978/8–1983/2 ns ns ns Aug 1.74 ns(0.089)

1983/3–1987/9 Jul 3.84 ns Aug 3.08 ns ns(0.000) (0.003)

B. Iterated Estimates

1974/1–1978/7 Jan 2.49 Jan 2.03 ns Jan 2.45 ns(0.016) (0.048) (0.018)

1978/8–1983/2 ns ns ns Aug 2.72 ns(0.009)

1983/3–1987/9 ns Jul 3.30 ns Aug 1.84 Aug 2.60(0.002) (0.071) (0.012)

_________________________________________________________________________

Notes: 1. Month having significant difference from unconditional mean factor riskpremium.

2. t-statistic from Equation 13 for month with significantly different (at a 10%level) mean factor risk premium.

3. No significant differences at the 10% critical level are found for the meanfactor risk premiums.

4. The p-value is given in parentheses.

Connor and Korajczyk (1988, p.271) caution that comparing test results betweenthe non-nested CAPM and APT models “can be misleading . . . since a model thatactually fits better (smaller values of | a |) may be rejected if the deviations are

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Table 3

Modified Likelihood Ratio Test (MLRT) Statistics for the Absence

of Unconditional Mispricing in the CAPM, APT (5) and APT (10)1

_______________________________________________________

APT (5) APT (10)Time Period CAPM Initial Iterated Initial Iterated

_______________________________________________________

1974/1–1978/7 1.972 2.09 0.52 2.48 0.81(0.061)3 (0.049) (0.866) (0.023) (0.621)

1978/8–1983/2 2.71 1.19 2.07 1.75 2.06(0.011) (0.326) (0.051) (0.108) (0.056)

1983/3–1987/9 1.91 2.93 0.96 2.64 1.20(0.069) (0.008) (0.492) (0.016) (0.324)

Overall period 3.074 2.62 0.32 3.10 1.00aggregate test (0.001) (0.004) (0.375) (0.001) (0.158)_______________________________________________________

Notes: 1. Each subperiod uses ten portfolios sorted on marketvalue.

2. Modified likelihood ratio test (MLRT) statistic [see Rao(1973, p.555)] which has an F distribution with(numerator, denominator) degrees of freedom of:(10, 44) for CAPM, (10, 40) for APT (5), and (10, 35) forAPT (10).


4. This value is the standard normal variate equivalent forthe aggregated subperiod F-statistics.

measured more precisely (that is, the test has more power)”. They suggested thatsome evidence of this problem would be revealed by plotting the average estimatesof the mispricing parameter (average a ’s) for each portfolio, across the differentmodels. Given the similarity of our test results for the CAPM versus the initial(one-step) APT (5) and APT (10) models, comparative plots of these three shouldhighlight the extent of the problem in this study.19 Hence, in Figure 1, a plot of the

______________19. It should be noted that there is a bias in the mispricing estimates which is induced by errors-

in-variables. It can be shown that this bias is a function of the covariance between theapproximation errors and the factor sensitivities. [See Connor and Korajczyk (1988, p.270).]The bias may be a problem here due to the small samples involved. Hence, the plotspresented in the Figures and the results reported later should be read with this in mind.

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average unconditional mispricing for the CAPM and for the one-step versions ofAPT (5) and APT (10) are given.

Mispricing(% p.a.)

Size Portfolio

-10

0

10

20

30

40

1 2 3 4 5 6 7 8 9 10

CAPMAPT (5)APT (10)

..........................................................

.........................

Figure 1

Unconditional mispricing, 1974–1987

Generally, the plots reveal that CAPM has a higher average mispricing, particularlyfor the smallest company portfolio. But for all three models a discernible “smallfirm effect” is evident, although it is clearly strongest for the CAPM. This wouldsuggest that the APT is being rejected with smaller pricing errors because they arebeing estimated more precisely.

5.5 Tests of the APT: Conditional Mispricing

As suggested in Section 3.1, we can also test for the absence of conditionalmispricing in terms of the January, July and August seasonal behaviour observedin Australian data. The results of these tests for the individual seasonal effects,involving hypotheses (7a), (7b) and (7c), are reported in Table 4. It is apparentfrom Table 4 that none of the models have much difficulty explaining individuallyJanuary, July or August related conditional mispricing. The subperiod and overallperiod test statistics are very small indeed, relative to conventionally appliedcritical values. But this analysis can be extended in two important ways. First, ina similar fashion to the results of Table 3, it may be instructive to examine therelative plots of average seasonal-specific mispricing for evidence of differentialtesting power across models. The second approach is to conduct a test of the(stronger) joint hypothesis of zero January, July and August conditional

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Table 4

MLRT Statistics for the Absence of Conditional Mispricing (Individual

January, July and August Seasonality) in the CAPM, APT (5) and APT (10)1

______________________________________________________________________

Time Period Null CAPM APT (5) APT (10)Hypothesis2 Initial Iterated Initial Iterated

______________________________________________________________________

1974/1–1978/7 ai(JAN) = 0 1.663 1.41 1.86 1.21 1.94(0.124)4 (0.216) (0.084) (0.324) (0.076)

ai(JUL) = 0 1.04 0.70 1.42 0.78 2.14(0.426) (0.715) (0.208) (0.650) (0.050)

ai(AUG) = 0 1.32 1.61 1.62 1.20 1.97(0.251) (0.142) (0.140) (0.329) (0.071)

1978/8–1983/2 ai(JAN) = 0 0.94 0.70 0.77 0.74 1.07(0.507) (0.715) (0.658) (0.682) (0.413)

ai(JUL) = 0 0.35 0.17 0.19 0.39 0.66(0.959) (0.998) (0.996) (0.940) (0.753)

ai(AUG) = 0 0.47 0.46 0.51 0.77 0.44(0.902) (0.903) (0.873) (0.658) (0.916)

1983/3–1987/9 ai(JAN) = 0 0.93 1.16 1.07 1.48 1.25(0.515) (0.350) (0.407) (0.192) (0.297)

ai(JUL) = 0 1.25 0.65 0.72 0.49 0.88(0.291) (0.764) (0.699) (0.881) (0.563)

ai(AUG) = 0 1.03 1.03 0.89 1.07 1.02(0.440) (0.441) (0.553) (0.413) (0.446)

Overall period ai(JAN) = 0 0.635 0.35 0.70 0.49 1.26aggregate test (0.263) (0.364) (0.243) (0.310) (0.104)

ai(JUL) = 0 –0.58 –2.38 –1.34 –1.80 0.46(0.719) (0.991) (0.910) (0.964) (0.322)

ai(AUG) = 0 –0.27 –0.05 –0.11 0.15 0.13(0.607) (0.519) (0.544) (0.441) (0.447)

______________________________________________________________________

Notes: 1. Each subperiod uses ten portfolios sorted on market value.

2. i = 1, 2, . . . , 10.

3. Modified likelihood ratio test (MLRT) statistic [see Rao (1973, p.555)]which has an F distribution with (numerator, denominator) degrees offreedom of: (10, 41) for CAPM, (10, 37) for APT (5), and (10, 32) forAPT (10).


5. This value is the standard normal variate equivalent for the aggregatedsubperiod F-statistics.

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mispricing. These two aspects are now considered in turn.In Figures 2 to 5, average mispricing relating to the conditional seasonal

model specifications are plotted for the CAPM versus the initial five and ten factorspecifications. Figure 2 shows the average non-seasonal-specific mispricing forthe three models. This is similar to Figure 1, showing relatively larger mispricingevident for the CAPM, particularly for the smallest firm portfolio.

Figure 3 reveals average January-specific conditional mispricing. In contrastto the results of Connor and Korajczyk (1988), the APT does not provide anadequate explanation of this seasonality. Significant mispricing is observedparticularly in the mid-sized portfolios (20% to 30% per annum) for both theCAPM and APT models.

In Figure 4, average conditional mispricing that is specific to July is plottedfor the ten size portfolios. Here, very large average mispricing of around 90% and60% per annum is observed for the CAPM for the smallest two firm size portfolios,respectively. This mispricing is much larger than the maximum mispricing (10%to 20% per annum) found for the APT models. While a small firm effect is stillevident for the APT, it is clearly much less pronounced than for the CAPM.

Finally, in Figure 5 the average August-specific-conditional mispricing isdisplayed. A small firm effect is evident for all models, but again it is lessprominent for the APT. In general all these seasonal specific plots reinforce thesuggestion that the APT tests have stronger power because of the more preciseestimation of mispricing coefficients.

As discussed earlier, a further extension of the seasonality mispricing issueis achieved by testing the hypothesis (given by Equation 8 previously) of joint zeromispricing in January, July and August. The results of this analysis are found inTable 5.

In stark contrast to the results of Table 4, in virtually all subperiods, andindeed in the overall period, test statistics for all models provided a resoundingrejection of the null hypothesis. That is, the data will not support the conclusionthat January, July and August mispricing is jointly zero, whereas we could rejecttheir individual mispricing effects.20 Also contained in Table 5 are the results oftesting the null hypothesis of zero non-seasonal-specific mispricing (given byEquation 6). It finds support only in the iterated version of APT (5).

5.6 Tests of the APT versus the CAPM

Finally, in Table 6 results for the non-nested tests described in Section 3.2 areprovided. In Panel A, the univariate results for each individual portfolio are

______________20. The very strong rejection of the CAPM and APT models due to the existence of (joint) January,

July and August mispricing provides solid support for the initial choice of this monthlyseasonal structure in returns. This set of seasonals has provided sufficient power for the teststo reject the null models.

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Mispricing(% p.a.)

Size Portfolio

-5

0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10

CAPMAPT (5)APT (10)

.......................................................................

................................................

Figure 2

Conditional mispricing: non-seasonal-specific, 1974–1987

Mispricing(% p.a.)

Size Portfolio

-20

-10

0

10

20

30

40

1 2 3 4 5 6 7 8 9 10

CAPMAPT (5)

APT (10)

......

.................................

.............................................................................

...........................................

Figure 3

Conditional mispricing: January-specific, 1974–1987

- 252 -


Mispricing(% p.a.)

Size Portfolio

-20

0

20

40

60

80

100

1 2 3 4 5 6 7 8 9 10

CAPM

APT (5)APT (10)

.................................................

...........................................

.....................

Figure 4

Conditional mispricing: July-specific, 1974–1987

Mispricing(% p.a.)

Size Portfolio

-20

-10

0

10

20

30

40

1 2 3 4 5 6 7 8 9 10

CAPM

APT (5)

APT (10)

...............................................................

.................

......

.

Figure 5

Conditional mispricing: August-specific, 1974–1987

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Table 5

MLRT Statistics for the Absence of Conditional Mispricing (joint

January, July and August Seasonality) in the CAPM, APT (5) and APT (10)1

_________________________________________________________________________

Time Period Null CAPM APT (5) APT (10)Hypothesis2 Initial Iterated Initial Iterated

_________________________________________________________________________

1974/1–1978/7 ai(NSEAS) = 0 1.573 1.66 0.79 1.97 1.34(0.152)4 (0.129) (0.642) (0.071) (0.250)

ai(JAN) = ai(JUL) 5.18 5.24 7.27 4.52 11.07= ai(AUG) = 0 (0.000) (0.000) (0.000) (0.000) (0.000)

1978/8–1983/2 ai(NSEAS) = 0 2.28 1.33 1.98 1.69 1.66(0.031) (0.250) (0.065) (0.126) (0.133)


1983/3–1987/9 ai(NSEAS) = 0 1.67 2.79 1.24 2.42 1.44(0.122) (0.011) (0.299) (0.028) (0.206)


Overall period ai(NSEAS) = 0 2.345 2.36 0.97 2.61 1.50aggregate test (0.010) (0.009) (0.166) (0.004) (0.066)


_________________________________________________________________________

Notes: 1. Each subperiod uses ten portfolios sorted on market value.

2. i = 1, 2, . . . , 10.

3. Modified likelihood ratio test (MLRT) statistic [see Rao (1973, p.555)]which has an F distribution with (numerator, denominator) degrees offreedom of: (10, 41) for CAPM, (10, 37) for APT (5), and (10, 32) forAPT (10).


5. This value is the standard normal variate equivalent for the aggregatedsubperiod F-statistics.

reported, based on the estimation of a regression given by Equation 10. Generally,the results indicate support for the APT in all but the largest size portfolios. Forexample, consider the third subperiod. The analysis involving Portfolio 1 (smallestcompanies) through to Portfolio 5 gives support to the one-step APT (5) model(H 0 : α = 1) and rejects the CAPM (H 0 : α = 0). In the case of Portfolio 6 throughto Portfolio 9, neither model is supported. It is only in the case of Portfolio 10

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Table 6

A Non-nested Comparison of APT (5) Versus CAPM1

______________________________________________________________

1974/1–1978/7 1978/8–1983/2 1983/3–1987/9α i Initial Iterated Initial Iterated Initial Iterated

______________________________________________________________

A. Univariate Analysis

α1 0.9901 0.9963 1.1376 0.9592 1.0133 1.0540(smallest) (0.042)2 (0.069) (0.087) (0.122) (0.076) (0.093)

α21.0557 0.9835 1.0914 1.1668 1.0587 1.1596(0.085) (0.088) (0.104) (0.256) (0.060) (0.138)

α31.0978 1.0381 0.9684 1.0068 0.9386 1.0025(0.106) (0.107) (0.143) (0.171) (0.087) (0.133)

α41.0370 1.0571 1.0577 1.2060 1.0675 1.1235(0.107) (0.091) (0.216) (0.216) (0.116) (0.229)

α51.0220 1.0072 1.0313 1.0547 0.9965 1.0410(0.094) (0.073) (0.129) (0.127) (0.068) (0.094)

α60.9823 0.9840 0.9671 1.2173 0.6921 1.0298(0.085) (0.067) (0.166) (0.206) (0.095) (0.116)

α70.8976 0.8893 0.9641 0.9997 0.6587 1.1247(0.093) (0.110) (0.057) (0.050) (0.112) (0.164)

α80.9124 0.9096 0.8073 1.0874 0.3737 0.8940(0.067) (0.080) (0.136) (0.125) (0.112) (0.159)

α90.7491 0.8157 0.6547 1.0078 0.2385 0.8649(0.061) (0.068) (0.141) (0.142) (0.077) (0.086)

α10 0.3129 0.5549 0.1432 0.5818 0.0815 0.5380(largest) (0.067) (0.065) (0.103) (0.108) (0.066) (0.093)

B. Multivariate Analysis

H 0 : α i = 1 3.623 2.93 4.03 0.94 11.75 2.08(0.001)4 (0.007) (0.001) (0.507) (0.000) (0.047)

i = 1, 2, . . . , 10[that is, APT (5)]______________________________________________________________

Notes: 1. The non-nested comparison is based on the C-test framework ofDavidson and MacKinnon (1981).

2. Standard Errors in parentheses.

3. Modified likelihood ratio test (MLRT) statistic [see Rao (1973,p.555)] which has an F distribution with (10, 44) degrees of freedom.

4. The p-values are in parentheses.

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(largest companies) that the CAPM is supported and the APT rejected.21 For theiterated APT (5) model, Portfolio 1 through to Portfolio 9 support the APT andreject the CAPM, while Portfolio 10 supports neither model. These results tend toconfirm earlier analysis suggesting that the APT models are superior at pricing allbut the larger firms, whereas CAPM (at best) only has some ability in pricing thelargest firms.

In Panel B of Table 6, the associated multivariate statistics for the non-nested tests of APT (5) across the ten portfolios are presented. The APT (5) modelcan only be clearly accepted for the middle subperiod and is marginal in the thirdsubperiod (in both cases for the iterated five factor model). In unreported results itis (not surprisingly) found that similar multivariate tests of the converse situation,that is, CAPM as the null hypothesis (that is, H 0 : α1 = α 2 = α 3 = . . . = α 10 = 0 ),show absolutely no support for the null in any subperiod or indeed overall.

6. Conclusions and Summary

The asymptotic principal components technique, developed by Connor andKorajczyk (1988), was used to provide factor estimates necessary for the

testing of an equilibrium version of the arbitrage pricing theory (APT) with timevarying risk premia. Both one-step and iterative versions of the method were used.The tests were based on cross-equation restrictions imposed by the APT on amultivariate regression of excess returns on the estimated factors. This workextends and improves the previous research by Faff (1988) in three major ways.First, the empirical technique has allowed examination of smaller subperiodswhich involves a more realistic assumption regarding stationarity. Second, byusing the multivariate approach, rather than the traditional two-stage method, theerrors-in-variables problem is considerably reduced. Finally, following the workof Brown, Keim, Kleidon and Marsh (1983) and Wood (1990a), the analysisincorporates dummy variables for monthly seasonality effects.

In this paper, tests are performed using Australian monthly equity return datain the period 1974 to 1987. In general, the results provided only weak support forthe APT model. Unconditional mispricing cannot be rejected for the initial one-step versions of a five factor and a ten factor APT nor for the CAPM. In contrast,the iterated versions of the APT models do not reveal significant unconditionalmispricing. Conditional mispricing was tested by augmenting the multivariateregression model with January, July and August seasonal dummy variables. Theabsence of individual seasonal mispricing (that is, January or July or August)cannot be rejected for any model. But joint seasonal mispricing (that is, January

______________21. Note that there are four possible sets of conclusions. We can: (a) reject the APT and accept

the CAPM; (b) accept the APT and reject the CAPM; (c) reject both models; or (d) accept bothmodels.

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and July and August) is significant for all models. Finally, multivariate non-nestedtests of the restrictions imposed by CAPM versus APT favour an (iterated) fivefactor APT. While the APT appears to perform better than the CAPM, neithermodel can adequately explain seasonal mispricing in Australian equities.

(Date of receipt of final typescript: December 1992.)

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A Multivariate Test of an Equilibrium APT With Time Varying Risk Premia in the Australian Equity

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