Do Assumptions About Factor Structure Matter in Empirical Tests of the APT?

DOASSUMPTIONS ABOUT FACTOR STRUCTUREMATTER IN EMPIRICAL TESTS OF THE APT?

Ian Garrett and Richard Priestley*

INTRODUCTION

Multifactor models of security returns such as the Arbitrage Pricing Theory(APT) of Ross (1976) offer an alternative to the Sharpe (1964) and Lintner(1965) equilibrium-based CAPM in terms of measuring the expected returnson risky assets. Themultifactor APTmodel of security returns derived byRoss(1976) begins by specifying the following relationship:

Rt = E(R) + BkFkt + ut, (1a)

E(R) = �0 iN + Bkkk, (1b)

where Rt is an N vector of security returns, Fkt is a zero mean, k vector ofobservations on the k pervasive risk factors at time t generated using eitherfactor analysis or observed macroeconomic variables, Bk is an Nxk matrix ofsensitivities of security returns to the factors, ut is an N vector of zero mean,idiosyncratic returns, E(R) is anN vector of expected returns, �0 is the returnon the risk free asset, iN is anN vector of ones and kk is a k vector of (constant)prices of risk associated with the k systematic risk factors, Fk. One of theassumptions made in this model is that the correlation between the factors, F,and idiosyncratic returns, u, is zero. The significance of this is that sinceidiosyncratic returns are uncorrelated with the pervasive factors thecovariance matrix of returns can be decomposed into:

RR = RF + RU, (2)

where RF is the covariance matrix of the factors (representing systematic orpervasive risk) and RU is the covariance matrix of idiosyncratic returns(representing idiosyncratic risk). It is the assumption about the form of RU

that determines whether the factor structure of returns is strict orapproximate. Ross (1976), for example, assumes that RU is a diagonal matrixwhichmeans that idiosyncratic returns are uncorrelated across assets since theoff-diagonal elements of the matrix are zero. In this case, returns are said to

Journal of Business Finance&Accounting, 24(2), March 1997, 0306-686X

ß Blackwell Publishers Ltd. 1997, 108 Cowley Road, Oxford OX4 1JF, UKand 350Main Street, Malden, MA 02148, USA. 249

* The authors are from the Department of Accounting and Finance, Manchester University andthe Department of Economics and Finance, Centre for Empirical Research in Finance, BrunelUniversity. They would like to thank the anonymous referee for helpful comments. The usualdisclaimer applies. (Paper received February 1996, revised and accepted April 1996)

Address for correspondence: Richard Priestley, Centre for Empirical Research in Finance,Department of Economics and Finance, Brunel University, Uxbridge, Middlesex UB8 3PH, UK.

have a strict factor structure. This assumption about returns having a strictfactor structure is implicitly made in many empirical studies of the APT. Forexample, factor analysis extracts factors from RF under the assumption thatRU is diagonal (see, for example, Roll and Ross, 1980) whilst studies whichuse observed macroeconomic variables as the factors usually obtain estimatesof the parameters from OLS regressions which again assumes RU to bediagonal.

As an alternative to returns following a strict factor structure, Chamberlainand Rothschild (1983) show that the APT still holds under the much weakerassumption that RU is nondiagonal, such that idiosyncratic returns arecorrelated across assets, as long as the first k eigenvalues of RR are unboundedas the number of assets approaches infinity whilst the k+1th eigenvalue isbounded, that is, the first k eigenvalues increase as the number of assetsincreases whilst the k+1th eigenvalue of RR is less than the largest eigenvalueof RU. In this case returns have an approximate k factor structure. Theinteresting question here is the extent to which the assumed structure of RU

matters empirically. The potential importance of such an assumption ishighlighted by Connor and Korajczyk (1993) who point out that under astrict factor structure, industry-specific factors which lead to significantcorrelations across idiosyncratic returns of firms within an industry mayactually be treated as pervasive under a strict factor structure. To date,however, this question has received little attention in the literature althoughclearly it is a question that may be of relevance to those studies that use factoranalysis or observed macroeconomic variables as factors with the two stepmethodology of Fama and MacBeth (1973) (see, for example, Chen, Rolland Ross, 1986) since the use of ordinary least squares or weighted leastsquares in the second step regression implicitly imposes a strict factor structureby constraining off-diagonal elements of RU to be zero.The objectives of this paper are to investigate whether returns have a strict

or an approximate factor structure and to analyze the empirical importance ofthe assumption about the factor structure that returns are assumed to follow.To address these issues we examine the behaviour of the eigenvalues of RR andRU in conjunction with tests of the significance of the cross correlationsbetween the off-diagonal elements of RU for a sample of UK security returnsin order to determine whether returns have a strict or an approximate factorstructure. We then allow for alternative specifications of RU to examinewhether this affects the number of statistically significant observed factors.As a robustness check on our empirical results, we also undertake the testdeveloped by Connor and Korajczyk (1993) for the appropriate number offactors in an approximate factor model. To anticipate the results, we find thatreturns are best described by an approximate factor structure and when this isallowed for, six factors carry significant prices of risk, a result that is consistentwith the findings of the Connor and Korajczyk (1993) test. In contrast, whenreturns are constrained to have a strict factor structure, no factors carry

250 GARRETT AND PRIESTLEY

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significant prices of risk. Thus, the assumption about the behaviour ofidiosyncratic returns is of crucial importance in terms of identifying thenumber of observed systematic risk factors that are priced.

DATA, ESTIMATION ANDRESULTS

In order to analyze the nature of the factor structure of returns and to examinewhether the specification of RU matters, we use the nonlinear time seriesmethodology discussed in Burmeister and McElroy (1988) and McElroy andBurmeister (1988). This approach treats the APT as a system of nonlinear,seemingly unrelated regression (NLSUR) equations and enjoys advantagesover the often-used Fama and MacBeth (1973) two step procedure. Inparticular, the two step procedure typically proceeds by first estimating thesensitivities (B) using time series regressions of returns on a constant and thefactors and then estimating the prices of risk (�) in a second step cross sectionalregression and as such is prone to the errors in variables (EIV) problemwhich,when corrected for, can lead to very different conclusions than those drawnfrom uncorrelated estimates (see Shanken, 1992). In contrast to this, thenonlinear time series approach estimates both the prices of risk (k) and thesensitivities (B) jointly, thereby avoiding the errors in variables problem. Inaddition to mitigating the EIV problem, the nonlinear time seriesmethodology allows for alternative specifications of RU in a straightforwardmanner, as will be seen below, and thus facilitates straightforward comparisonof the effects of assumptions about different factor structures empirically. Thenonlinear time series methodology proceeds by substituting (1b) into (1a) andstacking the equations for theN securities, to give:

Rÿ �0 = {IN[(k0iT) + F]}B + u, (3)

whereR is anNTx1 vector of security returns, � is a kx1 vector of prices of risk,F is a Txk matrix of observations on the k factors, B is an Nkx1 vector ofsensitivities, IN is an NxN identity matrix and is the Kronecker productoperator. If the system is not supplemented by an equation for the marketportfolio, from which the equity market risk premium can be calculated, andthe market portfolio is not included as a factor, then the NLSUR estimatorsfor the APT model are those values of B and k that solve the followingproblem:

min�;B

u0�R̂ ÿ1U IT�u; �4�

where u is derived from (3), R̂ ÿ1U is the estimated residual (idiosyncratic

return) covariance matrix from estimating (3) with (k0 iT)B replaced by aconstant and IT is a TxT identity matrix. The system can be extended in astraightforward fashion to include an equation for the market portfolio.

FACTOR STRUCTURES ANDTHE APT 251


However, if this is the case, themarket portfoliomust be treated as endogenousand the system estimated by nonlinear three stage least squares (NL3SLS). Inthis case, the minimisation problem becomes:

min�;B

u0�R̂ ÿ1U Z�Z0Z�ÿ1Z0�u; �5�

whereZ is a matrix of instrumental variables. An important point to note hereis that R can be non-diagonal, thereby allowing for idiosyncratic returns to becontemporaneously cross correlated or it can be restricted to be diagonal,imposing a strict factor structure. The parameter standard errors are givenbyXÿ1 where:

X � G0�R̂ ÿ1U Z�Z0Z�ÿ1Z0�G; �6�

where G is the matrix of partial derivatives of u with respect to theparameters. From (5) and (6), it can be seen that the specification of RU maymake a significant difference to reported results, both in terms of estimatedprices of risk and their significance.1

We begin our empirical analysis by determining whether returns are bestdescribed by a strict or an approximate factor structure. A natural way toinvestigate this is to examine the behaviour of the eigenvalues of the returncovariance matrix RR relative to the largest eigenvalue of RU since, fromChamberlain and Rothschild (1983), if returns have an approximate k factorstructure the first k eigenvalues of RR should increase without bound as thenumber of assetsN increases whilst the k+1th eigenvalue of RR is less than thelargest eigenvalue of RU, and to examine the significance of contemporaneouscross correlations amongst idiosyncratic returns.

In order to examine the behaviour of the eigenvalues of RR and RU, we firstneed an estimate of the idiosyncratic return covariance matrix which weobtain by estimating an unrestricted version of the linear factor model, thatis, RU is the variance-covariance matrix of u in the modelRtÿ�0 = A + BktFkt+ ut. The excess returns vectorRÿ�0 used in the estimation of the linear factormodel consists of excess returns on sixty nine individual securities traded onthe London Stock Exchange together with excess returns on a broad stockmarket index (the FTAll Share Index) to proxy for the market portfolio, givinga system of seventy equations to be estimated. In the spirit of the pioneeringwork by Chen, Roll and Ross (1986), the factors we use are generated frommacroeconomic variables. The factors generated from the underlyingeconomic and financial variables are listed in Table 1 and reflectunanticipated shocks to the relevant variable such that they are stationary,zero mean, white noise processes.2 Full details about the macroeconomic dataand themodels used to generate the factors can be found in the Appendix. Thedata is monthly and covers the period January 1980 to August 1993. Since themarket portfolio enters the model as an explanatory factor that may have asignificant price of risk and the system is also supplemented by an equation



for the market portfolio to facilitate calculation of the equity market riskpremium, the proxy for the market portfolio to facilitate calculation of theequity market risk premium, the proxy for the market portfolio is treated asendogenous and the system is estimated using NL3SLS where, following thesuggestions in Amemiya (1977), the instruments used are the exogenousfactors and their squares, returns and squared returns on the S&P500 Index,returns and squared returns on the UKUnlisted Securities Market Index and thefitted values and squared fitted values from a regression of the excess return onthe market portfolio on the macroeconomic factors.

Table 2 summarises the behaviour of the first ten eigenvalues of RR as thenumber of securities, N, is increased and reports the first two eigenvalues ofRR and the first (largest) eigenvalue of RU for N = 69. The general behaviourof the eigenvalues as the number of securities is increased appears to conformto that when returns have an approximate factor structure, that is, theeigenvalues increase without bound as the number of assets is increased. Ifreturns do have an approximate factor structure then we should also expectto see significant contemporaneous correlation of the residuals from the linearfactor model since, under an approximate factor structure, the idiosyncraticreturn covariance matrix is nondiagonal.

Table 1

Variables that Define the Factors

Factor Macroeconomic/Financial Dickey-Fuller �(1)Variable Unit Root Test

f1(�1) Unanticipated Shocks to Default Risk ÿ11.799 0.103 [0.173]f2 (�2) Unanticipated Shocks to Real Industrial

Production ÿ13.588 ÿ0.068 [0.385]f3 (�3) Unanticipated Shocks to the Exchange

Rate ÿ12.748 ÿ0.004 [0.957]fe (�4) Unanticipated Shocks to the Money

Supply ÿ15.308 ÿ0.184 [0.018]f5 (�5) Unanticipated Shocks to the Money

Supply ÿ15.234 ÿ0.176 [0.024]f6 (�6) Unexpected Inflation ÿ14.414 ÿ0.125 [0.109]f7 (�7) Change in Expected Inflation ÿ14.603 ÿ0.116 [0.130]f8 (�8 Unanticipated Shocks to the Term

Structure of Interest Rates ÿ12.404 0.021 [0.782]f9 (�9) Unanticipated Shocks to Commodity

Prices ÿ14.384 ÿ0.125 [0.110]f10 (�10) Returns on the Market Portfolio

Notes:Figures in the Dickey-Fuller column are t-tests testing the hypothesis that �= 0 in the regression yt= � + �ytÿ1 + ut. Reject the null hypothesis of a unit root if jtj>2.879. �(1) is the first orderautocorrelation coefficient and the figures in square parentheses are probability (p) values.Reject H0: �(1) = 0 at the 1% level if p<0.01.



Table 3 reports contemporaneous cross-correlation coefficients for theidiosyncratic returns of the first fifteen companies in the sample. The tableclearly reveals that there are a number of correlations that are significant atthe 10% level or less which suggests that an approximate factor structuremay be more appropriate. However, whilst the behaviour of the eigenvaluesand correlations across idiosyncratic returns suggest that returns have anapproximate factor structure, it is interesting to note that the contradictionidentified by Trzcinka (1986) and noted by Connor and Korajczyk (1993) interms of determining the number of factors k that are priced occurs here. Forexample, as can be seen in Table 2, the first two eigenvalues from the returncovariancematrix for the sixty nine securities are 35.066 and 10.181 whilst thelargest eigenvalue for the idiosyncratic return covariance matrix is 10.632.Using the bounded condition, this suggests that only one factor is pervasive.However, analyzing the percentage changes in the first ten eigenvalues of RR

as N is increased from 40 to 69 shows that the first ten eigenvalues increasequite substantially which, on the unboundedness condition, suggests that allten factors may be pervasive.

The evidence presented thus far is very suggestive of an approximate factorstructure for returns but it does not help us in determining the number offactors that are priced. Therefore, Table 4 reports the estimated prices of riskfor the APT specification of the linear factor model allowing for idiosyncraticreturns to be contemporaneously correlated across assets. As can be seen,when we allow idiosyncratic returns to be correlated across assets six factorscarry prices of risk that are significant at the 10% level or less, these beingunexpected inflation, industrial production, the money supply, default risk,the exchange rate and excess returns on the market portfolio. The

Table 2

Percentage Change in the First Ten Eigenvalues (�) of the Return CovarianceMatrixRR as the Number of Securities is Increased from 40 to 69 and the Two

Largest Eigenvalues of RRRelative to the Largest Eigenvalue of RU

�1 +63%�2 +13%�3 +19%�4 +18%�5 +32%�6 +34%�7 +46%�8 +39%�9 +32%�10 +39%

Largest eigenvalue of RR: 35.066 Largest eigenvalue of RU: 10.632Second largest eigenvalue of RR: 10.181



Table 3

Correlations Amongst the Idiosyncratic Returns of the First 15 Companies

Comp. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15No.

1 1.002 ÿ0.20b 1.003 0.07 ÿ0.18b 1.004 0.12 ÿ0.04 0.05 1.005 0.05 ÿ0.17b 0.19 ÿ0.03 1.006 ÿ0.01 ÿ0.28c ÿ0.04 ÿ0.14a 0.07 1.007 ÿ0.22c ÿ0.01 ÿ0.15a 0.04 ÿ0.14a ÿ0.07 1.008 0.15b ÿ0.24c 0.11 0.13a ÿ0.01 ÿ0.02 0.06 1.009 0.15b ÿ0.12 ÿ0.02 0.01 0.11 ÿ0.04 ÿ0.14b 0.09 1.0010 0.01 ÿ0.07 0.03 0.07 0.00 ÿ0.26c 0.16b 0.17b ÿ0.09 1.0011 0.11 ÿ0.03 0.08 ÿ0.04 0.13a 0.04 ÿ0.21c ÿ0.01 0.06 ÿ0.09 1.0012 0.13a ÿ0.08 ÿ0.13a ÿ0.03 0.00 0.01 0.01 0.15a 0.02 0.11 ÿ0.08 1.0013 0.12 ÿ0.06 0.01 0.14a 0.19b 0.05 ÿ0.13a ÿ0.09 ÿ0.05 0.32c ÿ0.34c ÿ0.14a 1.0014 0.03 0.07 ÿ0.00 0.11 0.08 ÿ0.01 ÿ0.12a 0.05 ÿ0.04 0.09 0.18b ÿ0.03 0.11 1.0015 0.07 ÿ0.05 0.46c 0.04 0.09 ÿ0.05 ÿ0.29c ÿ0.01 0.13a ÿ0.15a 0.27c ÿ0.06 ÿ0.09 ÿ0.05 1.00

Notes:a Denotes significant at the 10% level.b Denotes significant at the 5% level.c Denotes significant at the 1% level.

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approximate factor model estimates the equity market risk premium3 to be inthe region of approximately 0.3% per month which translates to an annualequity market risk premium of approximately 3.5% in excess of the risk freerate. Compare this with the results in Table 5, where the residual covariancematrix is constrained to be diagonal. In this case, none of the factors carrysignificant prices of risk, implying an equity market risk premium of 0%.Moreover, if we assume that the six factors that are priced in the approximatefactor model are priced in the strict factor model the corresponding equitymarket risk premium is approximately 0.4% per month, some 25% higherthan that estimated from the model with a nondiagonal idiosyncratic returncovariance matrix. Therefore, clearly the assumption about whether returnshave a strict or an approximate factor structure matters empirically, both interms of estimated prices of risk and their significance.

As a final robustness check, we calculated the test statistic proposed byConnor and Korajczyk (1993) which is designed to determine the number offactors that should be significant in an approximate factormodel. Assume that

Table 4

Estimates of the Prices of Risk for the Approximate Factor Modelö Estima-tion is by Nonlinear Three Stage Least Squares

�1 (Unexpected Inflation) ÿ0.00446c(0.00125)

�2 (Expected Inflation) 0.00098(0.00064)

�3 (Industrial Production) ÿ0.00107a(0.00057)

�4 (Retail Sales) 0.00008(0.00039)

�5 (Money Supply) 0.00200b

(0.00098)�6 (Commody Prices) 0.00352

(0.00297)�7 (Term Structure) 0.00111

(0.00076)�8 (Default Risk) ÿ0.00052c

(0.00019)�9 (Exchange Rate) ÿ0.01644c

(0.00521)�10 (Market Portfolio) 0.00180a

(0.00105)Estimated Equity Market Risk Premium (% per month) 0.281

Notes:Figures in parentheses below the estimated prices of risk are standard errors.a Denotes significant at the 10% level.b Denotes significant at the 5% level.c Denotes significant at the 1% level.



there are k systematic risk factors F and another factor, say F*, which is notpervasive. The test proceeds by estimating the following factor models:

Rt = A + BkFkt+ et, (7a)

Rt = A*+ BkFkt + B*F*t + e*t, (7b)

and then calculating the squared residuals (adjusted for degrees of freedom)from both models for each asset, denote these by �̂it and �̂�it, and calculatingthe cross sectional means of �̂it in odd months and subtracting from this thecross sectional means of �̂�it in even months to generate a vector of differencesin means which under the assumption of an approximate k factor structureshould be equal to zero. The test for the number of factors is then simply a testto see whether the difference is zero upon addition of an extra factor. The

Table 5

Estimates of the Prices of Risk for the Strict Factor Modelö Estimation is byNonlinear Three Stage Least Squares

�1 (Unexpected Inflation) ÿ0.00400(0.02018)

�2 (Expected Inflation) 0.00236(0.12630)

�3 (Industrial Production) ÿ0.00022(0.00835)

�4 (Retail Sales) ÿ0.00026(0.00738)

�5 (Money Supply) ÿ0.00041(0.01421)

�6 (Commodity Prices) ÿ0.01014(0.06008)

�7 (Term Structure) 0.00148(0.01609)

�8 (Default Risk) ÿ0.00062(0.00342)

�9 (Exchange Rate) ÿ0.02295(0.10164)

�10 (Market Portfolio) 0.00218(0.02377)

Estimated Equity Market Risk Premium 0.000(% per month, no significant factors)Estimated Equity Market Risk Premium 0.378(% per month assuming that the six significant factors in theapproximate factor model are significant in the strict factor model)

Notes:Figures in parentheses below the estimated prices of risk are standard errors.a Denotes significant at the 10% level.b Denotes significant at the 5% level.c Denotes significant at the 1% level.



results from this test suggest that for our sample of returns and factors, thenumber of priced factors is at most seven, the p-value based on the Newey-West robust covariance matrix being 0.056 for the addition of the seventhfactor and 0.114 for the addition of the eighth. This evidence, coupled withour earlier findings, serves to reinforce the empirical significance ofassumptions about factor structure.

CONCLUSION

In this paper we have investigated the role of assumptions about the form offactor structures in empirical testing of multifactor models such as the APTthat use observed macroeconomic variables as the systematic risk factors. Byanalyzing the behaviour of the eigenvalues from the return and idiosyncraticreturn covariance matrices we have examined whether returns are bestdescribed by an approximate factor structure and whether this actually hasany effect empirically on the estimated prices of risk and their significance.The results based on an analysis of the eigenvalues of both the return andidiosyncratic return covariance matrices and tests of the significance ofcontemporaneous cross correlations amongst idiosyncratic returns suggestthat the returns on securities traded on the London Stock Exchange are bestdescribed by an approximate factor structure where idiosyncratic returns areallowed to be contemporaneously correlated across assets. In terms of theempirical significance of this, we find that when idiosyncratic returns areallowed to be correlated across assets, as is consistent with an approximatefactor structure, six factors carry significant prices of risk, a finding that isreinforced by the Connor and Korajczyk (1993) test for the number of factorsin an approximate factor model. However, when we assume that returns havea strict factor structure, none of the factors carry significant prices of risk.Clearly, in terms of the significance of the estimated prices of risk, theassumption about whether or not idiosyncratic returns are correlated acrossassets matters. This finding, moreover, carries over to the estimated equitymarket risk premium, with quite different estimates being generated from thetwo models.

These findings suggest that the specification of the covariance matrix ofidiosyncratic returns matters in empirical tests of the APT. This result hasimportant implications, for Shanken (1992) has shown that once a correctionfor the Errors in Variables (EIV) problem inherent in the use of the two stepmethodology is made to the t-statistics testing the significance of the factors inChen, Roll, and Ross (1986), the factors become insignificant. This, at firstglance, questions the ability of multifactor models to explain the movementsof risky asset returns. However, using a technique free from the EIV problemwe have also reported results similar in spirit to Shanken's, that is, when theempirical version of the APT is estimated using a strict factor structure there is



little evidence of statistically significant factors. However, far from this beingevidence against multifactor models such as the APT per se, the evidencepresented here would suggest that it is rather symptomatic of the assumedform of the idiosyncratic return covariance matrix.

APPENDIX

All of the factors used in this paper are generated frommodels that have time varying parameters.Following Priestley (1996), the models are either simple unobserved components models or timevarying parameter autoregressive models to reflect the fact that agents learn and update theirexpectations over time. The model chosen is that which makes the residuals white noise. In eachcase, the Kalman Filter is used to estimate the models. The unobserved component models takethe following form:

yt = y*t + ut,

y*t = y*tÿ1 + et,while the autoregressive models with time varying parameters take the following general form:

yt = xt�t + ut,

�t = �tÿ1 + et,

where xt is aTÿpxKmatrix of observations on the lagged dependent variable and, for bothmodels,ut is the factor. The data for the macroeconomic variables and the models used to generate thefactors are listed below:

Unexpected Inflation �tÿEtÿ1(�t), where � is the change in the log of theUnited KingdomRetail Price Index (All Items). The model used to generate theexpectation is an AR(1) model.

Expected Inflation Et(�t+1)ÿEtÿ1(�t), where � and the model used to generate theexpectations are as defined above.

Real Industrial Production Log of United Kingdom industrial production deflated by theUnitedKingdom Producer Price Index; unobserved components model.

Real Retail Sales Log of United Kingdom Retail Sales deflated by the RPI; AR(1)model.

RealMoney Supply Log of United Kingdom currency in circulation deflated by the RPI;AR(2) model.

Commodity Prices Log of the IMFAll Commodity Price Index; unobserved componentsmodel.

Term Structure Itÿitÿ1 where I is the yield on United Kingdom Government longterm bonds and i is the yield on United Kingdom Government shortterm bonds: AR(1) model.

Default Risk Defined as the difference between the FTA Debenture and LoanStocks Redemption Yield and the yield on United KingdomGovernment long term bonds; unobserved components model.

Exchange Rate Log of the Sterling effective exchange rate; unobserved componentsmodel.

Market Portfolio Returns on the FTAll Share Indexminus the one month Treasury BillRate.

All data was obtained from Datastream.

NOTES

1 There is also a possibility that the results may be affected by small sample problems. McElroyand Burmeister (1988) point out that the estimators of � andBwill exist as long asNT>NK+Kwhere N is the number of equations, T is the number of observations and K is the number of



prices of risk to be estimated. For the system considered here,N= 70,T= 164 andK= 10. This,coupled with the estimation ofN variances andN(Nÿ1)/2 covariances leaves 8,285 degrees offreedom for the system which gives 118 degrees of freedom per equation if they are allocatedequally across the system.On the basis of this it seems that the small sample problem is unlikelyto be an important one.

2 This is confirmed by the unit root tests and tests of the significance of the first orderautocorrelation coefficient for the factors also reported in Table 1. The unit root nullhypothesis is strongly rejected in all cases whilst at the 1% level of significance, none of thefactors display significant autocorrelation.

3 The equity market risk premium is calculated as �jbMj�j where j = factors, 1, 3 5, 8 and 9 (seeTable 4) and bMj represents the sensitivity of excess returns on the market to the jth factor. Thesensitivities and prices of risk used in the calculation come from re-estimating the model withonly the six significant factors included.

REFERENCES

Burmeister, E. and M.B. McElroy (1988), `Joint Estimation of Factor Sensitivities and RiskPremia for the Arbitrage Pricing Theory', Journal of Finance, Vol. 43, pp. 721^35.

Chamberlain, G. and M. Rothschild (1983), Àrbitrage and Mean Variance Analysis on LargeAsset Markets', Econometrica, Vol. 51, pp. 1281^304.

Chen, N-F., R. Roll and S.A. Ross (1986), Èconomic Forces and the Stock Market', Journal ofBusiness, Vol. 59, pp. 383^403.

Connor, G. and R.A. Korajczyk (1993), À Test for the Number of Factors in an ApproximateFactor Model', Journal of Finance, Vol. 48, pp. 1263^91.

Fama, E.F. and J.D.MacBeth (1973), `Risk, Return and Equilibrium: Empirical Tests', Journal ofPolitical Economy, Vol. 71, pp. 607^36.

Lintner, J. (1965), `The Valuation of Risk Assets and the Selection of Risky Investments in StockPortfolios and Capital Budgets', Review of Economics and Statistics, Vol. 47, pp. 13^37.

McElroy, M.B. and E. Burmeister (1988), Àrbitrage Pricing Theory as a Restricted NonlinearMultivariate Regression Model: ITNLSUR Estimates', Journal of Business and EconomicStatistics, Vol. 6, pp. 29^42.

Priestley, R. (1996), `The APT, Macroeconomic and Financial Factors and ExpectationsGenerating Processes', Journal of Banking and Finance, Vol. 20, pp. 869^90.

Ross, S.A. (1976), `The Arbitrage Theory of Capital Asset Pricing', Journal of Economic Theory,Vol. 13, pp. 341^60.

Shanken, J. (1992), Òn the Estimation of Beta-pricingModels', Review of Financial Studies, Vol. 5,pp. 1^33.

Sharpe,W.F. (1964), `Capital Asset Prices: ATheory ofMarket EquilibriumUnder Conditions ofRisk', Journal of Finance, Vol. 19, pp. 425^42.

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Do Assumptions About Factor Structure Matter in Empirical Tests of the APT?