- Email: [email protected]
A critique of latent variable tests of asset pricing models
Journal of Financial Economics 23 (1989) 325-338. North-
*
Simon Universip of Washington, Se&e,
land
WA 98195, USA
Received December 1987, final version received March 1989 Latent variable tests of asset pricing models make assumptions about the joint distribution of observable returns and unobservable benchmark returns. These tests can falsely accept models when a mean-variance eficient portfolio other than the benchmark satisfies the distributional assumptions imposed on the benchmark portfolio. Also, because the assumptions are untestable, there is no way to discover whether a model is being rejected because the assumptions are false. Without these assumptions, however, latent variable tests can be viewed only as tests of distributional hypotheses about mean-variance efficient portfolios of unknown composition.
.
Asset pricing models often predict that some benchmark portfolio will be mean-variance efficient. The returns on these portfolios, however, are usually difficult to measure. To avoid this measurement problem, latent variable tests make assumptions about the joint distribution of the unobservable benchmark returns and other, observable returns. With these assumptions, latent variable tests of asset pricing restrictions can be conducted without a series of benchmark returns. As Roll (1877) emphasizes, however, mean-variance efficient portfolios always exist. Consequently, latent variable tests can falsely act some asset pricing model, because an efficient portfolio other than the ben mark satisfies the distributional assumptions imposed on the benchmark. the other hand, the distributional assumptions ar models do not guide their choice. termine whether this i model, one can never assumptions are false. tests can be viewed only as tests of mean-variance efficient portfolios of unknown *I would like to thank Robert Cumby, Wayne Ferson, Alan Vance Roley, Andrew F. Siegel, Rene Stub, and especially editor, G. William Schwert, for their comments. ‘That is, theoretical modeis that m their choice.
0304-405X/89/$3.50
Q 1989, Elsevier Scitmcc Publishers R.V. (North-
ice. the
*
S. M. Wheutlq:
326
Luterlt cwriuhle tests of met
pricitlg
models
The intuition behind the result can best be explained using Gibbons and erson’s (1985) tests of the mean-variance capital asset pricing mo o will be mean-varia predicts that the market port on all available information.2 ut as Roll points out, portfolio is unobservable. Consequently he states: ‘There is practically no possibility that.. . a test [of this model] can be accomplished.’ Gibbons and Ferson, however, develop latent variable tests of the market portfolio’s mean-variance efficiency that do not require a series of market returns.3 As a result, they conclude that ‘ the[ir] suggested methodology.. . [is] free from Roll’s criticism’. Gibbons and Ferson make an assumption about the conditional joint distribution of observable asset returns and the market return, which they regard as unobservable. With ihis assumption they are able to test restrictions the CAPM imposes on conditional expected returns without a series of market returns. As Roll shows, however, CAPM-like restrictions will hold for any mean-variance efficient portfolio.4 Thus, the restrictions Gibbons and Ferson test will hold whenever an efficient portfolio exists whose return satisfies the distributional assumption they impose on the market return. Their tests, therefore, really work in the following way. First, they make an assumption about the conditional joint distribution of aiset returns and the market return. Next, they test the hypothesis that a mea+variance efficient portfolio exists whose return satisfies this assumption. Finally, if they reject this hypothesis, they also reject the hypothesis that the market portfolio is mean -variance efficient. he problem with this procedure is that if the market portfolio is unobservable, one cannot test assumptions about its distribution. Also, no theoretical rationale exists for choosing one distributional assumption over another. Thus, erson’s tests do not avoid Roll’s critique. Without the assumption they make about the behavior of the market return, their tests can be interpreted only as tests of the hypothesis that an eficient portfolio of unknown composition satisfies the same assumption.
’ Hansen and Richard (1987) investigate relations between cllnditional and unconditional mean-variance efficiency. They show that conditionally mean-variance effi&nt portfolios can be unconditionally mean-variance inefficient. Throughout the paper, mean-variance efftciencv refers to conditional mean-variance efficiency unless otherwise stated. Also, portfolios are defined to be mean-variance efficient if they have t-r;’.,imun, variance of return for a given expected return. ’ Kandel and Stambaugh (1987) shov.3 that tests of the CAPM can be conducted usino an observable proxy if one makes an assumption about the correlation of its return withe the unobservable market rtttim. Similarly, Shanken (1987) shows that tests can be conducted using a vector of returns on obscJt\able proxies with an assumptiun about the multiple correlation bctwccn these returns and the market return. Unfortunately, these correlation cocflicicnts are unobservable andel and Stambaugh and Shankcn do not entirely circumvent Roll’s critique. n i it.
minimum variance portfolio, which has no zero-beta portfolio cr ignores this cxccption. g so doe not altc’r the
S. M. Wheutlq, Latent rwiuhle tests of usse: pricing modeis
(1972) version of the
,
327
because Gibbons and
discusses latent variable tests of other asset pricing models and offers conclusions.
2. P. Description The Black version of the CAP implies that the market portfolio lies on the positively sloped portion of the mean-variance efficient frontier, where this frontier is defined as the locus of mean-variance efficient portfolios in expected return-standard deviation of return space. Gibbons and Ferson develop latent variable tests of the market portfolio’s mean-variance efficiency that can be conducted without observing the market portfolio. Let the market portfolio be mean--%rariance efficient conditional on $_ 1, all information available at time t - 1, with respect to all assets at each time t = 1,2, . . . , T. Then, for any subset of these assets,
j= ?,2,...,
portfolio whose return is uncorrelate
N,
t=1,2 ,..., T,
S. M. Whmtley, Lutent ouriuhle tests of asset pricing models
328
return must exceed the conditional expected return on the zero-beta portfolio.5 From eq. (l), the conditional expected returns on (N - 2) of the left-hand-side assets can be expressed as a linear function of the conditional expected returns on the two remaining left-hand-side assets. It follows that
(2)
E(r,*-%Id-1) =ajtE(r*rrla+-l)~ j= 3,4,...,
N,
t=1,2 ,..., T,
where ail = (pjl- P1,J/<&tflit)- Gibbons and Person’s tests are based on the observation that if restrictions are placed on the behavior of the vector ( a3p aq,, - 9 aNt), then (2) can be tested without a series of returns on the market portfolio. They assume: l
(Al)
l
The vector of conditional betas (&, &,
. . . , &)
is constant over time,
so that the vector (a3r, ad*,. . . , aNI) is also constant over time, and (2) can be rewritten6
j= 3,4,...,
N,
t=1,2 ,..*, T.
The constant-beta assumption (Al) is an assumption about the joint distribution of a vector of retutrns and the return on the market, conditional on the information ++ r. Latent variable tests treat the market return as unobservable, so one cannot test this assumption. Tests of the CAPM that require series of returns on the market portfolio typically make the similar assumption that unconditional betas are constant over time [see, for example, Gibbons (1982), Stambaugh (1982), and Shanken (1985a)]. These tests, however, treat the market return as observable, so one can test that assumption. Although assumption (Al) cannot be tested, one can construct examples where it will hold for a subset of assets within the framework of Cox, Ingersoll, and Ross (1985). One can also construct examples within this framework, however, where it will uot hold. or example, if conditional expected returns depend on state variables, the conditional variance-covariance matrix of returns is constant over time, and returns are conditionally uncorrelated with the state variables, then the market portfolio will be mean-variance effi&nt, its composition will change ovec time, and the const;pnt-beta assumption will not hold. ‘Gibbons and Ferson do not test this restriction. ‘The constant-beta assumption (Al) is a sufficient, but not necessary, condition for the vector CY,,,~,) to be constant over time. For simplicity, the remaindk of the paper ignores this !~J,.q$,r.**. distinction.
S. Ad. Wheatlqc: Luteut uariubfe tests ofasset pricing models
329
Gibbons and Ferson also assume: 2)
Expectations of returns are linear in the information x,_ 1 Cx,_ 1 is an L x 1 vector, that is,
rjr= Six,_1 + where
Ejl)
j= 1,2,...,
N,
&-
1,
wher
r=1,2 ,..., T,
is an L X 1 vector of constants and E(&jr1x,_
1) = 0.
As they note, this linearity assumption can be tested. The constant-beta and linearity assumptions (Al) and (A2) together allow the tested without a series of returns on the market portfoli under the constant-beta assumption implies eq. (3), which in turn imposes the following restriction on the matrix (6,, 8,, . . . , a,!: Hl:
Sj-Sl=~j(S*-Sl),
j= 3,4 ,..., N.
Hypothesis H, implies that the matrix (8, - S,, 6, - 6,, . . . ,6, - 6,) has ran one. If L 2 2 and N 2 3, this rank condition imposes (L - l)( N - 2) testable restrictions on the parameters of the multivariate regression (4). These restrictions can be tested without observing the market portfolio. 2.2. Interpretation As Roll shows, CAPM-like restrictions will hold for any mean-varianse efficient portfolio. For this reason, the rank condition N, will hold whenever a mean-variance efficient portfolio exists that satisfies the constant-beta assumption, and expected returns are linear in the information x,+’ Consequently, tests of the rank condition are really tests for the existence of an efficient portfolio whose return satisfies the distributional assumption imposed on the market return.* If one assumes that conditional market betas are constant, one can interpret rejections of the rank condition as rejections of the C4PM. This is because, if no mean-variance efficient portfolio satisfies the constant-beta assumption, but market betas are constant, the market portfolio cannot be mean-variance efficient. Thus, interpretations of latent variable tests as tests of the CAP rely on the assumption t at conditional mar 7This portfc “Io ~11d lie on an eficient frontier constructed from the N assets or from the N assets togethe, with a further M assets. These additional assets could even be hypothetical. ‘Tests of the rank condition H, as tests of the hypothesis that an eflicient portfolio satisfies the constant-beta assumption can be without power. To see this, consider a portfolio of N assets. demonstrates that if this portfolio satisfie> a CAPM-like relation, it will be mea efficient relative to those assets. Roll’s demonstration, however. depends on betas relation to the conditional joint distribution of the pcrtfolio’s return and the N asset return>. Without this relation, CAPMike restrictions do not imply meanvariable tests treat conditio:r,aI Greta meee cons?ants. For this reason cient portfolio satisfies the const hold even when no mean-variance
SM. Wheutky, Lutent mk.dde tests of usset pricirtg models
330
f the market portfolio is unobserv
en it is true, becaus
: : Le
tests whether such efficient portfolios exist.
3.I.
escription
le example will illustrate how latent variable tests of the can be misleading. ‘These tests can mislead either when the distribussumption imposed on the market return is false, or when the return on an efficient portfolio other than the market satisfies the assumption. As Gibbons and Ferson do not reject the CAEM, I use their data to test the othesis that mean-variance efficient portfolios other than the market fy the distributional assumption they impose on the market portfolic. erson use daily returns on the individual firms in the Dow he rank condition I-I,. ey include in the information set search in Security Prices (C RSI?) return on the Center for als one if day t is a Monday and ones 30, betas relative to every ient frontier constructed from those assets are io on this frontier 3611satisfy the distributional assumption imposed on the market return (Al). None of these mean-variance portfolios will be the arket nortfolio, however, because the Dow are only a subset of t universe of assets. For any set of l&rassets, tive to every e cient portfolio constructed from those assets will be e following assumptions ho1
a
eters,
is an IV X N matrix of
s.
Es
33
constant ratio assu These assumptions
i
efkient portfolio constructed from Lose assets are consta
Thus in this example, latent variable tests of the power; that is, the ower and size of the tests wi
will be without
3.2. Evidence 3.2. I. Test p_yoces-wes This section te its whether Gibbons and erson’s data are consistent with the example, by -rsrng their data to test the constant-ratio assum 4). GibboSts aad Ferson assume that expectations of retur in the informaticn .x~_~,that is, the lagged return on the value-weighted index rmt _ 1, and a du nmy 0, that equals one if day t is a ay an zero otherwise. l[t fo% +wst”ilat
332
ielaxe
Wheutley, Luterlt oariuhle tests of met pricing mode/s
t is assumed instead that.
82, are 30 X 30 matrices of parameters itional variance-covariance matrix of return on the value-weighted i e distribution of the return vector rr conditional only on information x,+ it is simil in spirit to volatility models estimated by (1987). Although not reported here, tests ench, Schwert, and Stambau uncovered no significant linear relation between the conditional variancematrix of returns and other tra sformations of x,__~,for example, The constant-ratio assumptions (A3) and (A4) impose restrictions on the 6,), a,, and 9,. Assumption (A3), that conditional exmatrices (S,&.., pected return ratios are constant, imposes the restriction $ = Q,, j = n the other hand, assumptron (A4), that , . . . ,30, where yj is a scalar. 233 conditional return volatility ratios re constant, imposes the restriction AZ,= I/&?,, where 4 is also a scalar. lo These assumptions are tested using Lagrange lier statistics and dily return data for the Dow Jones 30 taken from the istics resemble similar statistics (1979) derive for tests of hets are used because they do not of unconstraineo estimates, which is costly. istributions of Lagrange multiplier statistics often do not mptotic distributions. Consequently, bootstrap simulaproperties of the test statistics and maximum fron (1982)]. The results, described in the apultiplier statistics for tests of (A3) and (A4) n when critical values are based on their
ecausc: it is unliikely 01:)be true, this model wzb
33?
Tests of distributional assumptions from August 17.1962 t r,r=~~,+6,,D,+~~,~~,r-l
y ret
ow Jones 30 using
c P-k +p
j= 1,2,...,
’
(Elr.+....eJ&
N(O,Q,+
30,
t= 1,2 . . . . . T.
Q,r,z,,_,).
where .r/, is the jth asset’s return at time t, 0, equals one if day I 199 Xo~tiay and zero o r,,, _ 1 1s the return on the CRSP value-weighted index at time t - 1, 6, = ( vector of parameters, Q0 and kl are 30 x 30 matrices of parameters, an parameters. Panel A : Lagrange
Assumption Conditional expected return ratios constant, 8, = y, 8,. j = 2.3,. . . ,30
(A3) (A4)
Conditional return volatility ratios constant, s2, = \c1&
multiplier
statistics
Subperiod -8/17/623/14/6711/16/716/9/763/13/67 11/H/71 6/8/76 12/31/80 (ir= 1149) (T= 1151) (T= 1150) (T= 115 68.933 (0.172)
58.756 (0.492)
68.253 (0.180)
2020.037 (0.000)
883.357 (0.220)
YOL.449 (0.800)
Panel B: Maximum
Parameter
Estimate
0.012
0.092 (0.013)
0.037
s 21
0.195 (0.007)
0.072
rcI’
0.356 (0.000)
0.047
s 11h
-
1056.358 4862.201 (0.660) (0.011)
11/16/716/8/76 (T= 1150)
-6/9/7612/32/80 (T= 1151)
Standard Standard Overall test Standard Standard statistic” error Estimate error Estimate error Estimate error
0.020 (0.096)
601 b
3/14/6711/15/71 (T= 1151)
265.083 (0.109)
likelihood estimates
Subperiod 8/17/623/13/67 (T= 1149)
69.141 (0.184)
Sum of subperiod
0.094
0.028
0.067 (0.056)
0 035
0.038 (0.046)
0.019
4.469 (G.OOQ)
0.071
- 0.326 (0.000)
0.085
- 0.097 (0.021)
(I.042
6.590 (O.Qm
(0.001) - 0.323 (0.000)
0.473 (0.000) 0.165 (0.000)
0.076
0.500 (0.000)
0.061
0.328 (0.000)
0.064
11.127 (0.000)
0.023
0.150 (O.oO)
0.018
0.173 (0.000)
0.029
14.524 (0.000) --
“The overall test-statistic for SoI is given by (l/@)Xy=, $IL/s($,I, ). where M is the 9umber of subperiods (four), Polk is the maximum likelihood estimate of S,,, in subperio k, and ~(a,,,, I is its standard error. Overall test statistics for a,,, 62,, and 4 are computed in the sa ‘Estimates of U& and a,, and their standard errors have been ‘Estimates of JI and their standard errors have been multiplied by 1G 4.
334
S. M. Wheutlqr; L.utent rwicrhle tests of usset prichg
models
the four subperiods, but is overwhelmingly rejected in the first subperiod, leading to a rejection over all subperiods. Panel B of table 1 indicates that the restrictions that conditional expected returns are constant and that the conditional variance-covariance matrix of retu ns is constant are rejected in all subperiods and overall at conventional significance levels. These results show that the example described in this section is consistent with Gibbons and Ferson’s data in three of the four subperiods. That is, the hypothesis that every efficient portfolio constructed from the Dow Jones 30 ion imposed on the market is not r satisfies the distributional assu hus while Gib and Ferson do not reject the CA those periods. least three of the four subperiods the data are consistent with the hypothesis that their tests are without power.
Latent variable tests of other asset pricing models, for example, the intertemporai CAPM (ICAPM) of Eucas (1978) and Breeden (I979), are subject to the same problems of interpretation as those discussed in this paper. For any set of assets, the ICAPM implies that ;he portfolio of those assets maximally correlated with a representative individual’s marginal utility must be mean-variance efficient relative to that set [see Breeden (1979) and Hansen, chard, and Singleton (1581)]. Consequently, as Gibbons and Ferson note, latent variable tests of the ICAPM do not differ from latent variable tests of the mean-variance CAPPL The tests, however, rely on different assumptions. The assumption necessary to interpret latent variable tests of the ICAPM ncerns the conditional joint distribution of asset returns and the return on axi correlation portfolio. Without observing this portfolio, one cannot test distributional assumptions for its return.13 atent variable tests of multifactor asset pricing models are also subject to lems of interpretation. Shanken (1985b, 1987) and Campbell t tests of a multifactor model can also be viewed as tests of the mean-variance efficiency of a portfolio spanned by those factors. It follows at relaxing constraints on the number of factors corresponds to relaxing the distributional assum posed on the spanned portfolio. Without observe cannot test distributional assumptions for the that latent variable tests rely on assumptions turns. These assumptions 1sof individual behavior. eedcn. Gibbons, an
itzenbcrgcr ( 1989) and
SM. Wheutlqv, Lutertt rwiuhle tests of met pricing model5
335
us, if latent variable t s is because the assum
oints, I use Gibbons and mean-variance CAP hich are supposed to circumv assumption that is the basis for Gibbons and the same variable, the market return, that , but unobservable. Thus, their tests a s a simple, but restri ztive, example in which latent variable are without power. Tests show that Gibbons and Ferson’s data are consistent with this examplt,: in three of the four periods they examine. Although it is difficult to view latent variable tests as tests of asset models, they can be given an alternative interpretation. If it is assum some asset pricing model holds, latent variable tests can be viewe hypotheses about the behavior of risk over time. Even when viewed this way, however, they can be without power. For example, within the context of the CAPM, section 3’s illustration shows that when risk measured relative to the market is time-varying, risk measured relative to other mean-variance efficient portfolios can be constant.
This appendix presents simulation evidence of the properties of the Lagrange multiplier statistics and maximum likelihood estimators. For each subperiod a separate simulation of 250 replications is performed with parameter values chosen to match estimates computed over the subperiod. The T x 30 matrix of residuals from the multivariate regression (5) estimated under (A3) and (A4), is used to produce disturbances. For each t&met, the t th row of this matrix is divided by (1 + $w:,_,), while each column’s mean is subtracted from the to enstixe that the expected value of a randomly selec column’s eleme element is zero. isturbances are generated by sa rmed residual matrix with replacement, an at time t by (1 + Jlr,$_,), with 4 equal t subpeiiuu. --2 Y!ith these &stnrbSnces, return the restriction (A3) im
336
S. M. Wheutlqrl, Luterlt curiuhle tests of usset pricing models
Table 2 Bootstrap results using daily returns on the Dow Jones 30. The bootstrap simulations investigate the properties of Lagrange multip?ier statistics for tests of (83) 6, = y, 6, and (A4) & = #s2,, and maximum likelihood estimators for 6, and \cI.The resdts are based on 250 replications, and parameter values that match estimates computed using daily returns on the Dow Jones 30 from August 17,1962 to March 13.1967. j=1,2, . . . . 30, t=1,2 ,..., 1149, r,,=~~,+S1,D,+~~,r~,,_l+~,r.
where r,, is the jth asset’s return at time t, D, equals one if day t is a Monday and zero otherwise, rnrf- 1 is the return on the CRSP value-weighted index at time t - 1, SJ = ( ao,, a,,. $, )’ is a vector of parameters, &, and s2, are 3(! x 30 matrices of parameters, and y, and 4/ are scalar parameters. Lagrange multiplier statistic for (83) (A4) Lagrange mu1t ipiier statistic for tA3) (A4) Maximum likelihood estimator of
Fraction rejected at 10% significance level
Fraction rejected at 5% significance level
Fraction rejected at 1% significance level
0.552 I.000
0.124 1.000
0.080 1.000
0.024 1.000
Simulated mean
Theoretical asymptotic standard error
Simulated standard error
Studentized range
59.010 864.422
10.770 30.463
12.048 148.581
5.664 8.519’
Theoretical asymptotic mean
Simulated mean
Theoretical asympiotic standard error
Simulated standard error
Studentized range
0.020 0.091’ 0.195 0.356
0.020 - 0.093 O.&O 0.343
0.012 0.037 0.072 0.047
6.692h 5.574 5.689 5.982
Theoretical asymptotic mean 58.000 464.000
801
s 11 821 J/
Fraction rejected at 50% significance level”
-
0.011 0.037 0.070 0.024
.- ~
“Lagrange multiplier statistics for tests of (A3) and (A4) are compared distributions. ‘Exceeds the 0.95 fractiie of the distribution of the studentized range. ‘Exceeds the 0.99 fractiie of the distribution of the studentized range.
to x& and xih4
its simulated standard error is d residuals from the multivariate regtes-
s.
the maximum likeli
Wheatlqr:
estimator for 4 is biased downw
distribution, the smaller the number of assets N.
Black. Fischer, 1972, Capital market equilibrium with restricted borrowing, Journal of Business 45.444-454. Breeden, Douglas T., 1979, An intertemporal asset pricing model with stochastic consumption and investment opportunities, Journal of Financial Economics 7.265-296. Breeden, Douglas T., Michael R. Gibbons, and Robert H. Litzenberger, 1989, Empirical tests of the consumption-oriented CAPM, Journal of Finance, forthcoming. Breusch, Trevor S. and Adrian R. Pagan, 1979, A simple test for heteroskedasticity and random coefficient variation, Econometrica 47,1287-1294. Campbell, John Y., 1987, Stock returns and the term structure, Journal of Financial Economics 18,373-400. Campbell, John Y. and Richard I-L Clarida, 1987, The term structure of Euromarket interest rates: An empirical investigation, Journal of Monetary Economics 19, 25-44. Cox, John C., Jonathan Ingersoll, Jr., and Stephen A. Ross, 1985, An intertemporal general equilibrium model of asset prices, Econometrica 53, 363-384. Cumby, Robert E., 1987, Consumption risk and international asset returns: Some empirical evidence, National Bureau of Economic Research working paper no. 2383. Cumby, Robert E., 1988, Is it risk? Explaining deviations from uncovered interest parity. Journal of Monetary Economics 22,279-299. Efron. Bradley, 1982, The jackknife, the bootstrap and other resampling plans (Society for Industrial and Applied Mathematics, Philadelphia, PA). Ferson. Wayne E., Stephen R. Foerster, and Donald B. Keim. 1987, Tests of asse with changing expectations, Working paper (University of Pennsylvania, Phil French, Kenneth R., G. William Schwert. and Robert F. Stambaugh, 1987, Expected sto and volatility, Journal of Financial Economics 19, 3-29. Gibbons, Michael R., 1982, Multivariate tests of financial models: A Financial Economics 10, 3-27. Gibbons, Michael X. and expectations and an 71 217-236. Godfrey, Leslie G., 1978, Testing for multiplicative hcteroskcdastic
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S. M. Wheutlq,; Luterrtwiahle tests of usset pricirtg models
Hansen, I_,ars P. and Scott F. Richard, 1987, The role of conditioning information in deducing testable restrictions implied by dynamic asset pricing models, Econometrica 55, 587-613. Mansen, ears P., Scott F. Richard, and Kenneth J. Singleton, 1981, Econometric implications of the intertemporal capital asset pricing model, IJnpublished paper (Carnegie-Mellon University, Pittsburgh. PA). Hodriclk, Robert J. and Sanjay Srivastava, 1984, An investigation of risk and return in forward foreign exchange, Journal of International Money and Finance 3, S-29. Jorion, Philippe and Albert0 Giovannini, 1987, Interest rates and risk premia in the stock market and in the foreign exchange market, Journal of International Money and Finance 6, 107-123. Kandel, Shmuel and Robert F. Stambaugh, 1987, On correlations and inferences about mean-variance efficiency, Journal of Financial Economics 18, 61-90. kucas, Robert E., Jr., 1978, Asset prices in an exchange economy, Econometrica 46, 1429-1446. Merton, Robert C.. 1973, An intertemporal capital asset pricing model, Econometrica 41. 867-887. Roll, Richard W., 1977, A critique of the asset pricing theory’s tests - Part 1: On past and potential testability of the theory, Journal of Financial Economics 4, 129-176. Shanken, Jay, 1985a. Multivariate tests of the zero-beta CAPM, Journal of Financial Economics 14, 327-348. Shanken, Jay, 1985b. Multi-beta CAPM or equilibrium APT? A reply, Journal of Finance 40, 1189-1196. Shanken. Jay. 1987, Multivariate proxies and asset pricing relations: Living with the Roll critique, Journal of Financial Economics 18, 91-110. Siariibaugh, K0bert F., i%t, an tiae cxc;ti&ri ,:’ assets from tests of the two-parameter modei: A sensitivity analysis, Journal of Financtai Economics 10, 237-268. Stambaugh, Robert F., 1988, The information in forward rates: Implications for models of the term structure, Jourirll of Financial Economic, 21, 41-70. Wheatley, Simon M., 198P3, Some tests of the consumption-based asset pricing model, Journal of Monetary Economics 2.1 193-215. Wheatley, Simon M.. 1988b, !Lme tests of international equity market integration, Journal of Financial Economics 21. 177-212.
*
Simon Universip of Washington, Se&e,
land
WA 98195, USA
Received December 1987, final version received March 1989 Latent variable tests of asset pricing models make assumptions about the joint distribution of observable returns and unobservable benchmark returns. These tests can falsely accept models when a mean-variance eficient portfolio other than the benchmark satisfies the distributional assumptions imposed on the benchmark portfolio. Also, because the assumptions are untestable, there is no way to discover whether a model is being rejected because the assumptions are false. Without these assumptions, however, latent variable tests can be viewed only as tests of distributional hypotheses about mean-variance efficient portfolios of unknown composition.
.
Asset pricing models often predict that some benchmark portfolio will be mean-variance efficient. The returns on these portfolios, however, are usually difficult to measure. To avoid this measurement problem, latent variable tests make assumptions about the joint distribution of the unobservable benchmark returns and other, observable returns. With these assumptions, latent variable tests of asset pricing restrictions can be conducted without a series of benchmark returns. As Roll (1877) emphasizes, however, mean-variance efficient portfolios always exist. Consequently, latent variable tests can falsely act some asset pricing model, because an efficient portfolio other than the ben mark satisfies the distributional assumptions imposed on the benchmark. the other hand, the distributional assumptions ar models do not guide their choice. termine whether this i model, one can never assumptions are false. tests can be viewed only as tests of mean-variance efficient portfolios of unknown *I would like to thank Robert Cumby, Wayne Ferson, Alan Vance Roley, Andrew F. Siegel, Rene Stub, and especially editor, G. William Schwert, for their comments. ‘That is, theoretical modeis that m their choice.
0304-405X/89/$3.50
Q 1989, Elsevier Scitmcc Publishers R.V. (North-
ice. the
*
S. M. Wheutlq:
326
Luterlt cwriuhle tests of met
pricitlg
models
The intuition behind the result can best be explained using Gibbons and erson’s (1985) tests of the mean-variance capital asset pricing mo o will be mean-varia predicts that the market port on all available information.2 ut as Roll points out, portfolio is unobservable. Consequently he states: ‘There is practically no possibility that.. . a test [of this model] can be accomplished.’ Gibbons and Ferson, however, develop latent variable tests of the market portfolio’s mean-variance efficiency that do not require a series of market returns.3 As a result, they conclude that ‘ the[ir] suggested methodology.. . [is] free from Roll’s criticism’. Gibbons and Ferson make an assumption about the conditional joint distribution of observable asset returns and the market return, which they regard as unobservable. With ihis assumption they are able to test restrictions the CAPM imposes on conditional expected returns without a series of market returns. As Roll shows, however, CAPM-like restrictions will hold for any mean-variance efficient portfolio.4 Thus, the restrictions Gibbons and Ferson test will hold whenever an efficient portfolio exists whose return satisfies the distributional assumption they impose on the market return. Their tests, therefore, really work in the following way. First, they make an assumption about the conditional joint distribution of aiset returns and the market return. Next, they test the hypothesis that a mea+variance efficient portfolio exists whose return satisfies this assumption. Finally, if they reject this hypothesis, they also reject the hypothesis that the market portfolio is mean -variance efficient. he problem with this procedure is that if the market portfolio is unobservable, one cannot test assumptions about its distribution. Also, no theoretical rationale exists for choosing one distributional assumption over another. Thus, erson’s tests do not avoid Roll’s critique. Without the assumption they make about the behavior of the market return, their tests can be interpreted only as tests of the hypothesis that an eficient portfolio of unknown composition satisfies the same assumption.
’ Hansen and Richard (1987) investigate relations between cllnditional and unconditional mean-variance efficiency. They show that conditionally mean-variance effi&nt portfolios can be unconditionally mean-variance inefficient. Throughout the paper, mean-variance efftciencv refers to conditional mean-variance efficiency unless otherwise stated. Also, portfolios are defined to be mean-variance efficient if they have t-r;’.,imun, variance of return for a given expected return. ’ Kandel and Stambaugh (1987) shov.3 that tests of the CAPM can be conducted usino an observable proxy if one makes an assumption about the correlation of its return withe the unobservable market rtttim. Similarly, Shanken (1987) shows that tests can be conducted using a vector of returns on obscJt\able proxies with an assumptiun about the multiple correlation bctwccn these returns and the market return. Unfortunately, these correlation cocflicicnts are unobservable andel and Stambaugh and Shankcn do not entirely circumvent Roll’s critique. n i it.
minimum variance portfolio, which has no zero-beta portfolio cr ignores this cxccption. g so doe not altc’r the
S. M. Wheutlq, Latent rwiuhle tests of usse: pricing modeis
(1972) version of the
,
327
because Gibbons and
discusses latent variable tests of other asset pricing models and offers conclusions.
2. P. Description The Black version of the CAP implies that the market portfolio lies on the positively sloped portion of the mean-variance efficient frontier, where this frontier is defined as the locus of mean-variance efficient portfolios in expected return-standard deviation of return space. Gibbons and Ferson develop latent variable tests of the market portfolio’s mean-variance efficiency that can be conducted without observing the market portfolio. Let the market portfolio be mean--%rariance efficient conditional on $_ 1, all information available at time t - 1, with respect to all assets at each time t = 1,2, . . . , T. Then, for any subset of these assets,
j= ?,2,...,
portfolio whose return is uncorrelate
N,
t=1,2 ,..., T,
S. M. Whmtley, Lutent ouriuhle tests of asset pricing models
328
return must exceed the conditional expected return on the zero-beta portfolio.5 From eq. (l), the conditional expected returns on (N - 2) of the left-hand-side assets can be expressed as a linear function of the conditional expected returns on the two remaining left-hand-side assets. It follows that
(2)
E(r,*-%Id-1) =ajtE(r*rrla+-l)~ j= 3,4,...,
N,
t=1,2 ,..., T,
where ail = (pjl- P1,J/<&tflit)- Gibbons and Person’s tests are based on the observation that if restrictions are placed on the behavior of the vector ( a3p aq,, - 9 aNt), then (2) can be tested without a series of returns on the market portfolio. They assume: l
(Al)
l
The vector of conditional betas (&, &,
. . . , &)
is constant over time,
so that the vector (a3r, ad*,. . . , aNI) is also constant over time, and (2) can be rewritten6
j= 3,4,...,
N,
t=1,2 ,..*, T.
The constant-beta assumption (Al) is an assumption about the joint distribution of a vector of retutrns and the return on the market, conditional on the information ++ r. Latent variable tests treat the market return as unobservable, so one cannot test this assumption. Tests of the CAPM that require series of returns on the market portfolio typically make the similar assumption that unconditional betas are constant over time [see, for example, Gibbons (1982), Stambaugh (1982), and Shanken (1985a)]. These tests, however, treat the market return as observable, so one can test that assumption. Although assumption (Al) cannot be tested, one can construct examples where it will hold for a subset of assets within the framework of Cox, Ingersoll, and Ross (1985). One can also construct examples within this framework, however, where it will uot hold. or example, if conditional expected returns depend on state variables, the conditional variance-covariance matrix of returns is constant over time, and returns are conditionally uncorrelated with the state variables, then the market portfolio will be mean-variance effi&nt, its composition will change ovec time, and the const;pnt-beta assumption will not hold. ‘Gibbons and Ferson do not test this restriction. ‘The constant-beta assumption (Al) is a sufficient, but not necessary, condition for the vector CY,,,~,) to be constant over time. For simplicity, the remaindk of the paper ignores this !~J,.q$,r.**. distinction.
S. Ad. Wheatlqc: Luteut uariubfe tests ofasset pricing models
329
Gibbons and Ferson also assume: 2)
Expectations of returns are linear in the information x,_ 1 Cx,_ 1 is an L x 1 vector, that is,
rjr= Six,_1 + where
Ejl)
j= 1,2,...,
N,
&-
1,
wher
r=1,2 ,..., T,
is an L X 1 vector of constants and E(&jr1x,_
1) = 0.
As they note, this linearity assumption can be tested. The constant-beta and linearity assumptions (Al) and (A2) together allow the tested without a series of returns on the market portfoli under the constant-beta assumption implies eq. (3), which in turn imposes the following restriction on the matrix (6,, 8,, . . . , a,!: Hl:
Sj-Sl=~j(S*-Sl),
j= 3,4 ,..., N.
Hypothesis H, implies that the matrix (8, - S,, 6, - 6,, . . . ,6, - 6,) has ran one. If L 2 2 and N 2 3, this rank condition imposes (L - l)( N - 2) testable restrictions on the parameters of the multivariate regression (4). These restrictions can be tested without observing the market portfolio. 2.2. Interpretation As Roll shows, CAPM-like restrictions will hold for any mean-varianse efficient portfolio. For this reason, the rank condition N, will hold whenever a mean-variance efficient portfolio exists that satisfies the constant-beta assumption, and expected returns are linear in the information x,+’ Consequently, tests of the rank condition are really tests for the existence of an efficient portfolio whose return satisfies the distributional assumption imposed on the market return.* If one assumes that conditional market betas are constant, one can interpret rejections of the rank condition as rejections of the C4PM. This is because, if no mean-variance efficient portfolio satisfies the constant-beta assumption, but market betas are constant, the market portfolio cannot be mean-variance efficient. Thus, interpretations of latent variable tests as tests of the CAP rely on the assumption t at conditional mar 7This portfc “Io ~11d lie on an eficient frontier constructed from the N assets or from the N assets togethe, with a further M assets. These additional assets could even be hypothetical. ‘Tests of the rank condition H, as tests of the hypothesis that an eflicient portfolio satisfies the constant-beta assumption can be without power. To see this, consider a portfolio of N assets. demonstrates that if this portfolio satisfie> a CAPM-like relation, it will be mea efficient relative to those assets. Roll’s demonstration, however. depends on betas relation to the conditional joint distribution of the pcrtfolio’s return and the N asset return>. Without this relation, CAPMike restrictions do not imply meanvariable tests treat conditio:r,aI Greta meee cons?ants. For this reason cient portfolio satisfies the const hold even when no mean-variance
SM. Wheutky, Lutent mk.dde tests of usset pricirtg models
330
f the market portfolio is unobserv
en it is true, becaus
: : Le
tests whether such efficient portfolios exist.
3.I.
escription
le example will illustrate how latent variable tests of the can be misleading. ‘These tests can mislead either when the distribussumption imposed on the market return is false, or when the return on an efficient portfolio other than the market satisfies the assumption. As Gibbons and Ferson do not reject the CAEM, I use their data to test the othesis that mean-variance efficient portfolios other than the market fy the distributional assumption they impose on the market portfolic. erson use daily returns on the individual firms in the Dow he rank condition I-I,. ey include in the information set search in Security Prices (C RSI?) return on the Center for als one if day t is a Monday and ones 30, betas relative to every ient frontier constructed from those assets are io on this frontier 3611satisfy the distributional assumption imposed on the market return (Al). None of these mean-variance portfolios will be the arket nortfolio, however, because the Dow are only a subset of t universe of assets. For any set of l&rassets, tive to every e cient portfolio constructed from those assets will be e following assumptions ho1
a
eters,
is an IV X N matrix of
s.
Es
33
constant ratio assu These assumptions
i
efkient portfolio constructed from Lose assets are consta
Thus in this example, latent variable tests of the power; that is, the ower and size of the tests wi
will be without
3.2. Evidence 3.2. I. Test p_yoces-wes This section te its whether Gibbons and erson’s data are consistent with the example, by -rsrng their data to test the constant-ratio assum 4). GibboSts aad Ferson assume that expectations of retur in the informaticn .x~_~,that is, the lagged return on the value-weighted index rmt _ 1, and a du nmy 0, that equals one if day t is a ay an zero otherwise. l[t fo% +wst”ilat
332
ielaxe
Wheutley, Luterlt oariuhle tests of met pricing mode/s
t is assumed instead that.
82, are 30 X 30 matrices of parameters itional variance-covariance matrix of return on the value-weighted i e distribution of the return vector rr conditional only on information x,+ it is simil in spirit to volatility models estimated by (1987). Although not reported here, tests ench, Schwert, and Stambau uncovered no significant linear relation between the conditional variancematrix of returns and other tra sformations of x,__~,for example, The constant-ratio assumptions (A3) and (A4) impose restrictions on the 6,), a,, and 9,. Assumption (A3), that conditional exmatrices (S,&.., pected return ratios are constant, imposes the restriction $ = Q,, j = n the other hand, assumptron (A4), that , . . . ,30, where yj is a scalar. 233 conditional return volatility ratios re constant, imposes the restriction AZ,= I/&?,, where 4 is also a scalar. lo These assumptions are tested using Lagrange lier statistics and dily return data for the Dow Jones 30 taken from the istics resemble similar statistics (1979) derive for tests of hets are used because they do not of unconstraineo estimates, which is costly. istributions of Lagrange multiplier statistics often do not mptotic distributions. Consequently, bootstrap simulaproperties of the test statistics and maximum fron (1982)]. The results, described in the apultiplier statistics for tests of (A3) and (A4) n when critical values are based on their
ecausc: it is unliikely 01:)be true, this model wzb
33?
Tests of distributional assumptions from August 17.1962 t r,r=~~,+6,,D,+~~,~~,r-l
y ret
ow Jones 30 using
c P-k +p
j= 1,2,...,
’
(Elr.+....eJ&
N(O,Q,+
30,
t= 1,2 . . . . . T.
Q,r,z,,_,).
where .r/, is the jth asset’s return at time t, 0, equals one if day I 199 Xo~tiay and zero o r,,, _ 1 1s the return on the CRSP value-weighted index at time t - 1, 6, = ( vector of parameters, Q0 and kl are 30 x 30 matrices of parameters, an parameters. Panel A : Lagrange
Assumption Conditional expected return ratios constant, 8, = y, 8,. j = 2.3,. . . ,30
(A3) (A4)
Conditional return volatility ratios constant, s2, = \c1&
multiplier
statistics
Subperiod -8/17/623/14/6711/16/716/9/763/13/67 11/H/71 6/8/76 12/31/80 (ir= 1149) (T= 1151) (T= 1150) (T= 115 68.933 (0.172)
58.756 (0.492)
68.253 (0.180)
2020.037 (0.000)
883.357 (0.220)
YOL.449 (0.800)
Panel B: Maximum
Parameter
Estimate
0.012
0.092 (0.013)
0.037
s 21
0.195 (0.007)
0.072
rcI’
0.356 (0.000)
0.047
s 11h
-
1056.358 4862.201 (0.660) (0.011)
11/16/716/8/76 (T= 1150)
-6/9/7612/32/80 (T= 1151)
Standard Standard Overall test Standard Standard statistic” error Estimate error Estimate error Estimate error
0.020 (0.096)
601 b
3/14/6711/15/71 (T= 1151)
265.083 (0.109)
likelihood estimates
Subperiod 8/17/623/13/67 (T= 1149)
69.141 (0.184)
Sum of subperiod
0.094
0.028
0.067 (0.056)
0 035
0.038 (0.046)
0.019
4.469 (G.OOQ)
0.071
- 0.326 (0.000)
0.085
- 0.097 (0.021)
(I.042
6.590 (O.Qm
(0.001) - 0.323 (0.000)
0.473 (0.000) 0.165 (0.000)
0.076
0.500 (0.000)
0.061
0.328 (0.000)
0.064
11.127 (0.000)
0.023
0.150 (O.oO)
0.018
0.173 (0.000)
0.029
14.524 (0.000) --
“The overall test-statistic for SoI is given by (l/@)Xy=, $IL/s($,I, ). where M is the 9umber of subperiods (four), Polk is the maximum likelihood estimate of S,,, in subperio k, and ~(a,,,, I is its standard error. Overall test statistics for a,,, 62,, and 4 are computed in the sa ‘Estimates of U& and a,, and their standard errors have been ‘Estimates of JI and their standard errors have been multiplied by 1G 4.
334
S. M. Wheutlqr; L.utent rwicrhle tests of usset prichg
models
the four subperiods, but is overwhelmingly rejected in the first subperiod, leading to a rejection over all subperiods. Panel B of table 1 indicates that the restrictions that conditional expected returns are constant and that the conditional variance-covariance matrix of retu ns is constant are rejected in all subperiods and overall at conventional significance levels. These results show that the example described in this section is consistent with Gibbons and Ferson’s data in three of the four subperiods. That is, the hypothesis that every efficient portfolio constructed from the Dow Jones 30 ion imposed on the market is not r satisfies the distributional assu hus while Gib and Ferson do not reject the CA those periods. least three of the four subperiods the data are consistent with the hypothesis that their tests are without power.
Latent variable tests of other asset pricing models, for example, the intertemporai CAPM (ICAPM) of Eucas (1978) and Breeden (I979), are subject to the same problems of interpretation as those discussed in this paper. For any set of assets, the ICAPM implies that ;he portfolio of those assets maximally correlated with a representative individual’s marginal utility must be mean-variance efficient relative to that set [see Breeden (1979) and Hansen, chard, and Singleton (1581)]. Consequently, as Gibbons and Ferson note, latent variable tests of the ICAPM do not differ from latent variable tests of the mean-variance CAPPL The tests, however, rely on different assumptions. The assumption necessary to interpret latent variable tests of the ICAPM ncerns the conditional joint distribution of asset returns and the return on axi correlation portfolio. Without observing this portfolio, one cannot test distributional assumptions for its return.13 atent variable tests of multifactor asset pricing models are also subject to lems of interpretation. Shanken (1985b, 1987) and Campbell t tests of a multifactor model can also be viewed as tests of the mean-variance efficiency of a portfolio spanned by those factors. It follows at relaxing constraints on the number of factors corresponds to relaxing the distributional assum posed on the spanned portfolio. Without observe cannot test distributional assumptions for the that latent variable tests rely on assumptions turns. These assumptions 1sof individual behavior. eedcn. Gibbons, an
itzenbcrgcr ( 1989) and
SM. Wheutlqv, Lutertt rwiuhle tests of met pricing model5
335
us, if latent variable t s is because the assum
oints, I use Gibbons and mean-variance CAP hich are supposed to circumv assumption that is the basis for Gibbons and the same variable, the market return, that , but unobservable. Thus, their tests a s a simple, but restri ztive, example in which latent variable are without power. Tests show that Gibbons and Ferson’s data are consistent with this examplt,: in three of the four periods they examine. Although it is difficult to view latent variable tests as tests of asset models, they can be given an alternative interpretation. If it is assum some asset pricing model holds, latent variable tests can be viewe hypotheses about the behavior of risk over time. Even when viewed this way, however, they can be without power. For example, within the context of the CAPM, section 3’s illustration shows that when risk measured relative to the market is time-varying, risk measured relative to other mean-variance efficient portfolios can be constant.
This appendix presents simulation evidence of the properties of the Lagrange multiplier statistics and maximum likelihood estimators. For each subperiod a separate simulation of 250 replications is performed with parameter values chosen to match estimates computed over the subperiod. The T x 30 matrix of residuals from the multivariate regression (5) estimated under (A3) and (A4), is used to produce disturbances. For each t&met, the t th row of this matrix is divided by (1 + $w:,_,), while each column’s mean is subtracted from the to enstixe that the expected value of a randomly selec column’s eleme element is zero. isturbances are generated by sa rmed residual matrix with replacement, an at time t by (1 + Jlr,$_,), with 4 equal t subpeiiuu. --2 Y!ith these &stnrbSnces, return the restriction (A3) im
336
S. M. Wheutlqrl, Luterlt curiuhle tests of usset pricing models
Table 2 Bootstrap results using daily returns on the Dow Jones 30. The bootstrap simulations investigate the properties of Lagrange multip?ier statistics for tests of (83) 6, = y, 6, and (A4) & = #s2,, and maximum likelihood estimators for 6, and \cI.The resdts are based on 250 replications, and parameter values that match estimates computed using daily returns on the Dow Jones 30 from August 17,1962 to March 13.1967. j=1,2, . . . . 30, t=1,2 ,..., 1149, r,,=~~,+S1,D,+~~,r~,,_l+~,r.
where r,, is the jth asset’s return at time t, D, equals one if day t is a Monday and zero otherwise, rnrf- 1 is the return on the CRSP value-weighted index at time t - 1, SJ = ( ao,, a,,. $, )’ is a vector of parameters, &, and s2, are 3(! x 30 matrices of parameters, and y, and 4/ are scalar parameters. Lagrange multiplier statistic for (83) (A4) Lagrange mu1t ipiier statistic for tA3) (A4) Maximum likelihood estimator of
Fraction rejected at 10% significance level
Fraction rejected at 5% significance level
Fraction rejected at 1% significance level
0.552 I.000
0.124 1.000
0.080 1.000
0.024 1.000
Simulated mean
Theoretical asymptotic standard error
Simulated standard error
Studentized range
59.010 864.422
10.770 30.463
12.048 148.581
5.664 8.519’
Theoretical asymptotic mean
Simulated mean
Theoretical asympiotic standard error
Simulated standard error
Studentized range
0.020 0.091’ 0.195 0.356
0.020 - 0.093 O.&O 0.343
0.012 0.037 0.072 0.047
6.692h 5.574 5.689 5.982
Theoretical asymptotic mean 58.000 464.000
801
s 11 821 J/
Fraction rejected at 50% significance level”
-
0.011 0.037 0.070 0.024
.- ~
“Lagrange multiplier statistics for tests of (A3) and (A4) are compared distributions. ‘Exceeds the 0.95 fractiie of the distribution of the studentized range. ‘Exceeds the 0.99 fractiie of the distribution of the studentized range.
to x& and xih4
its simulated standard error is d residuals from the multivariate regtes-
s.
the maximum likeli
Wheatlqr:
estimator for 4 is biased downw
distribution, the smaller the number of assets N.
Black. Fischer, 1972, Capital market equilibrium with restricted borrowing, Journal of Business 45.444-454. Breeden, Douglas T., 1979, An intertemporal asset pricing model with stochastic consumption and investment opportunities, Journal of Financial Economics 7.265-296. Breeden, Douglas T., Michael R. Gibbons, and Robert H. Litzenberger, 1989, Empirical tests of the consumption-oriented CAPM, Journal of Finance, forthcoming. Breusch, Trevor S. and Adrian R. Pagan, 1979, A simple test for heteroskedasticity and random coefficient variation, Econometrica 47,1287-1294. Campbell, John Y., 1987, Stock returns and the term structure, Journal of Financial Economics 18,373-400. Campbell, John Y. and Richard I-L Clarida, 1987, The term structure of Euromarket interest rates: An empirical investigation, Journal of Monetary Economics 19, 25-44. Cox, John C., Jonathan Ingersoll, Jr., and Stephen A. Ross, 1985, An intertemporal general equilibrium model of asset prices, Econometrica 53, 363-384. Cumby, Robert E., 1987, Consumption risk and international asset returns: Some empirical evidence, National Bureau of Economic Research working paper no. 2383. Cumby, Robert E., 1988, Is it risk? Explaining deviations from uncovered interest parity. Journal of Monetary Economics 22,279-299. Efron. Bradley, 1982, The jackknife, the bootstrap and other resampling plans (Society for Industrial and Applied Mathematics, Philadelphia, PA). Ferson. Wayne E., Stephen R. Foerster, and Donald B. Keim. 1987, Tests of asse with changing expectations, Working paper (University of Pennsylvania, Phil French, Kenneth R., G. William Schwert. and Robert F. Stambaugh, 1987, Expected sto and volatility, Journal of Financial Economics 19, 3-29. Gibbons, Michael R., 1982, Multivariate tests of financial models: A Financial Economics 10, 3-27. Gibbons, Michael X. and expectations and an 71 217-236. Godfrey, Leslie G., 1978, Testing for multiplicative hcteroskcdastic
338
S. M. Wheutlq,; Luterrtwiahle tests of usset pricirtg models
Hansen, I_,ars P. and Scott F. Richard, 1987, The role of conditioning information in deducing testable restrictions implied by dynamic asset pricing models, Econometrica 55, 587-613. Mansen, ears P., Scott F. Richard, and Kenneth J. Singleton, 1981, Econometric implications of the intertemporal capital asset pricing model, IJnpublished paper (Carnegie-Mellon University, Pittsburgh. PA). Hodriclk, Robert J. and Sanjay Srivastava, 1984, An investigation of risk and return in forward foreign exchange, Journal of International Money and Finance 3, S-29. Jorion, Philippe and Albert0 Giovannini, 1987, Interest rates and risk premia in the stock market and in the foreign exchange market, Journal of International Money and Finance 6, 107-123. Kandel, Shmuel and Robert F. Stambaugh, 1987, On correlations and inferences about mean-variance efficiency, Journal of Financial Economics 18, 61-90. kucas, Robert E., Jr., 1978, Asset prices in an exchange economy, Econometrica 46, 1429-1446. Merton, Robert C.. 1973, An intertemporal capital asset pricing model, Econometrica 41. 867-887. Roll, Richard W., 1977, A critique of the asset pricing theory’s tests - Part 1: On past and potential testability of the theory, Journal of Financial Economics 4, 129-176. Shanken, Jay, 1985a. Multivariate tests of the zero-beta CAPM, Journal of Financial Economics 14, 327-348. Shanken, Jay, 1985b. Multi-beta CAPM or equilibrium APT? A reply, Journal of Finance 40, 1189-1196. Shanken. Jay. 1987, Multivariate proxies and asset pricing relations: Living with the Roll critique, Journal of Financial Economics 18, 91-110. Siariibaugh, K0bert F., i%t, an tiae cxc;ti&ri ,:’ assets from tests of the two-parameter modei: A sensitivity analysis, Journal of Financtai Economics 10, 237-268. Stambaugh, Robert F., 1988, The information in forward rates: Implications for models of the term structure, Jourirll of Financial Economic, 21, 41-70. Wheatley, Simon M., 198P3, Some tests of the consumption-based asset pricing model, Journal of Monetary Economics 2.1 193-215. Wheatley, Simon M.. 1988b, !Lme tests of international equity market integration, Journal of Financial Economics 21. 177-212.